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2-pro-object | https://ncatlab.org/nlab/source/2-pro-object | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
In the context of [[2-category theory]], the concept of a **2-pro-object** is the [[categorification]] of the concept of a [[pro-object]]. A 2-pro-object in a 2-category, $\mathcal{C}$, is a 2-functor (or diagram) indexed by a 2-cofiltered 2-category. These 2-pro-objects form a 2-category, $2Pro(\mathcal{C})$, which is closed under small 2-cofiltered pseudolimits.
Pre-composition with the inclusion $c:\mathcal{C} \to 2Pro(\mathcal{C})$ is an [[equivalence of 2-categories]]:
$$
c^{\ast}: Hom(2Pro(\mathcal{C}),Cat)_+ \to Hom(\mathcal{C},Cat),
$$
where $Hom(2Pro(\mathcal{C}),Cat)_+$ is the full subcategory whose objects are those 2-functors that preserve small 2-cofiltered pseudolimits ([Descotte & Dubuc, Thrm 2.4.2](#DescotteDubuc)).
## References
* {#DescotteDubuc} [[Maria Emilia Descotte]], [[Eduardo Dubuc]], _A theory of 2-pro-objects (with expanded proofs)_, ([arXiv:1406.5762](https://arxiv.org/abs/1406.5762))
* [[Maria Emilia Descotte]], _A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)_, ([arXiv:2010.10636](https://arxiv.org/abs/2010.10636)) |
2-pullback | https://ncatlab.org/nlab/source/2-pullback | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# $2$-pullbacks
* table of contents
{: toc}
An ordinary [[pullback]] is a [[limit]] over a [[diagram]] of the form $A \to C \leftarrow B$. Accordingly, a __2-pullback__ (or **2-fiber product**) is a [[2-limit]] over such a diagram.
## Definition
Saying that "a 2-pullback is a 2-limit over a [[cospan]]" is in fact a sufficient definition, but we can simplify it and make it more explicit.
A __$2$-pullback__ in a [[2-category]] is a square
$$
\array{
P
& \overset{p}{\to}
&
A
\\
\mathllap{^q}\big\downarrow
& \cong &
\big\downarrow\mathrlap{^f}
\\
B
& \underset{g}{\longrightarrow} & C
}
$$
which commutes up to [[2-isomorphism]], and which is [[universal property|universal]] among such squares in a [[2-category theory|2-category theoretic]] sense. This means that
1. given any other such square
$$
\array{
Z & \overset{v}{\longrightarrow} &A
\\
\mathllap{^w}\big\downarrow
& \cong &
\big\downarrow\mathrlap{^f}
\\
B & \underset{g}{\longrightarrow} &
C
}
$$
which commutes up to [[2-isomorphism]], there exists a morphism $u \colon Z\to P$ and isomorphisms $p u \cong v$ and $q u \cong w$ which are [[coherence|coherent]] with the given ones above, and
1. given any [[pair]] of morphisms $u,t \colon Z\to P$ and [[2-morphisms]] $\alpha \colon p u \to p t$ and $\beta \colon q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-morphism $\gamma \colon u\to t$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.
## Equivalence of definitions
The simplification in the above explicit definition has to do with the omission of an unnecessary structure map. Note that an ordinary pullback of $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ comes equipped with maps $P\overset{p}{\to} A$, $P\overset{q}{\to} B$, and $P\overset{r}{\to} C$, but since $r = f p$ and $r = g q$, the map $r$ is superfluous data and is usually omitted. In the 2-categorical case, where identities are replaced by isomorphisms, it is, strictly speaking, different to give merely $p$ and $q$ with an isomorphism $f p \cong g q$, than to give $p$, $q$, and $r$ with isomorphisms $r \cong f p$ and $r \cong g q$. However, when 2-limits are considered as only defined up to equivalence (as is the default on the nLab), the two resulting notions of "2-pullback" are the same. In much of the 2-categorical literature, the version with $r$ specified would be called a __bipullback__ and the version with $r$ not specified would be called a __bi-iso-comma-object__.
The unsimplified definition would be: a __$2$-pullback__ in a [[2-category]] is a diagram
$$\array{P & \overset{p}{\to} &A \\
^q\downarrow & \searrow & \downarrow^f\\
B& \underset{g}{\to} &C }$$
in which each triangle [[commuting square|commutes]] up to [[2-isomorphism]], and which is [[universal property|universal]] among such squares in a [[2-category theory|2-category theoretic]] sense.
This means that
1. {#AnyOther} given any other such square
$$\array{
Z & \overset{v}{\longrightarrow} &A
\\
\mathllap{^w}\big\downarrow
& \searrow &
\big\downarrow\mathrlap{^f}
\\
B & \underset{g}{\longrightarrow} &C
}$$
in which the triangles commute up to [[2-isomorphism]], there exists a [[1-morphism]] $u\colon Z \to P$ and 2-isomorphisms $p u \cong v$ and $q u \cong w$ which are [[coherence|coherent]] with the given ones above, and
2. given any [[pair]] pf morphisms $u,t\colon Z \to P$ and 2-morphisms $\alpha\colon p u \to p t$ and $\beta\colon q u \to q t$ such that $f \alpha = g \beta$ (modulo the given isomorphism $f p \cong g q$), there exists a unique 2-cell $\gamma\colon u \to t$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.
To see that these definitions are equivalent, we observe that both assert the [[representable functor|representability]] of some [[2-functor]] (where "representability" is understood in the 2-categorical "up-to-equivalence" sense), and that the corresponding 2-functors are equivalent.
* In the simplified case, the functor $F_1\colon K^{op}\to Cat$ sends an object $Z$ to the category whose
* objects are squares commuting up to isomorphism, i.e. maps $v\colon Z\to A$ and $w\colon Z\to B$ equipped with an isomorphism $\mu\colon f v \cong g w$, and whose
* morphisms from $(v,w,\mu)$ to $(v',w',\mu')$ are pairs $\phi\colon v\to v'$ and $\psi\colon w\to w'$ such that $\mu' . (f \phi) = (g \psi) . \mu$.
* In the unsimplified case, the functor $F_2\colon K^{op}\to Cat$ sends an object $Z$ to the category whose
* objects consist of maps $v\colon Z\to A$, $w\colon Z\to B$, and $x\colon Z\to C$ equipped with isomorphisms $\kappa\colon f v \cong x$ and $\lambda\colon x\cong g w$, and whose
* morphisms from $(v,w,x,\kappa,\lambda)$ to $(v',w',x',\kappa',\lambda')$ are triples $\phi\colon v\to v'$, $\psi\colon w\to w'$, and $\chi\colon x\to x'$ such that $\kappa' . (f \phi) = \chi . \kappa$ and $\lambda' . \chi = (g \psi) . \lambda$.
We have a canonical [[pseudonatural transformation]] $F_2\to F_1$ that forgets $x$ and sets $\mu = \lambda . \kappa$. This is easily seen to be an [[equivalence]], so that any representing object for $F_1$ is also a representing object for $F_2$ and conversely. (Note, though, that in order to define an inverse equivalence $F_1\to F_2$ we must choose whether to define $x = f v$ or $x = g w$.)
## Variations
2-pullbacks can also be identified with [[homotopy pullbacks]], when the latter are interpreted in $Cat$-enriched homotopy theory.
### Strict 2-pullbacks
{#StrictPullback}
If we are in a [[strict 2-category]] and all the coherence isomorphisms ($\mu$, $\kappa$, $\lambda$, etc.) are required to be identities, and $u$ in property (1) is required to be unique, then we obtain the notion of a **strict 2-pullback**. This is an example of a [[strict 2-limit]]. Note that since we must have $x = f v = g w$, the two definitions above are still the same. In fact, they are now even isomorphic (and determined up to isomorphism, rather than equivalence).
In literature where "2-limit" means "strict 2-limit," of course "2-pullback" means "strict 2-pullback."
Obviously not every 2-pullback is a strict 2-pullback, but also not every strict 2-pullback is a 2-pullback, although the latter is true if either $f$ or $g$ is an [[isofibration]] (and in particular if either is a [[Grothendieck fibration]]). A strict 2-pullback is, in particular, an ordinary pullback in the underlying 1-category of our strict 2-category, but it has a stronger universal property than this, referring to 2-cells as well (namely, part (2) of the explicit definition).
### Strict weighted limits
If the coherence isomorphisms $\mu$, $\kappa$, $\lambda$ in the squares are retained, but in (1) the isomorphisms $p u \cong r$ and $q u \cong s$ are required to be identities and $u$ is required to be unique, then the simplified definition becomes that of a **[[strict iso-comma object]]**, while the unsimplified definition becomes that of a **strict pseudo-pullback**. (Iso-comma objects are so named because if the isomorphisms in the squares are then replaced by mere morphisms, we obtain the notion of (strict) [[comma object]]).
Every [[strict iso-comma object]], and every strict pseudo-pullback, is also a (non-strict) 2-pullback. In particular, if strict iso-comma objects and strict pseudo-pullbacks both exist, they are equivalent, but they are *not* isomorphic. (Note that their strict universal property determines them up to isomorphism, not just equivalence.) In many strict 2-categories, such as [[Cat]], 2-pullbacks can naturally be constructed as either strict iso-comma objects or strict pseudo-pullbacks.
### Lax versions
Replacing the isomorphism $\mu$ in the simplified definition by a mere transformation results in a [[comma object]], while replacing $\kappa$ and $\lambda$ in the unsimplified definition by mere transformations results in a [[lax pullback]]. In a [[(2,1)-category]], any [[comma object]] or [[lax pullback]] is also a 2-pullback, but this is not true in a general 2-category. Note that comma objects are often misleadingly called lax pullbacks.
## Related concepts
* [[comma object]]
[[!redirects 2-pullback]]
[[!redirects 2-pullbacks]]
[[!redirects 2-fiber product]]
[[!redirects 2-fiber products]]
[[!redirects bipullback]]
[[!redirects bipullbacks]]
[[!redirects bi-pullback]]
[[!redirects bi-pullbacks]]
[[!redirects pseudopullback]]
[[!redirects pseudopullbacks]]
[[!redirects pseudo-pullback]]
[[!redirects pseudo-pullbacks]]
[[!redirects pseudo pullback]]
[[!redirects pseudo pullbacks]]
[[!redirects strict pseudopullback]]
[[!redirects strict pseudopullbacks]]
[[!redirects strict pseudo-pullback]]
[[!redirects strict pseudo-pullbacks]]
[[!redirects strict pseudo pullback]]
[[!redirects strict pseudo pullbacks]]
|
2-representation | https://ncatlab.org/nlab/source/2-representation | [[!redirects 2-representations]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _2-representation_ of a [[2-group]] $G$ is an [[infinity-action]] of $G$ on a [[2-vector space]] $V$. The higher analog of the [[representation]] of a [[group]].
## Related concepts
* [[representation]], [[infinity-representation]]
## References
* [[Zhen Huan]], _2-Representations of Lie 2-groups and 2-Vector Bundles_ ([arXiv:2208.10042](https://arxiv.org/abs/2208.10042))
On [[2-representations]] of the [[string 2-group]] on [[2-vector spaces]] and the construction of the [[stringor bundle]]:
* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]
Reviewed in:
* [[Konrad Waldorf]], *The stringor bundle*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#WaldorfMar2023)]
|
2-rig | https://ncatlab.org/nlab/source/2-rig |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# $2$-rigs
* table of contents
{: toc}
## Idea
The notion of _$2$-rig_ is supposed to be a [[categorification]] of that of a [[rig]]. Several inequivalent formalizations of this idea are in the literature.
Just as a rig is a multiplicative [[monoid]] whose underlying set also has a notion of addition, so a $2$-rig is a [[monoidal category]] whose underlying category also has a notion of addition, and we can describe this notion of addition in a few different ways.
Note that we don\'t expect a $2$-rig to have additive inverses; by the same argument as in the [[Eilenberg swindle]], they are unreasonable to expect. However, in a monoidal [[abelian category]], we have as close to additive inverses as is reasonable and so a categorification of a [[ring]].
Compare also the notion of [[rig category]].
## Definitions
{#Definitions}
Since [[categorification]] involves some arbitrary choices that will be determined by the precise intended application, there is a bit of flexibility of what exactly one may want to call a _2-ring_. We first list some immediate possibilities of classes of monoidal and enriched categories that one may want to think of as 2-rings:
* [Enriched monoidal categories](#EnrichedMonoidalCategories)
But a central aspect of an ordinary ring is the [[distributivity law]] which says that the product in the ring preserves sums. Since sums in a 2-ring are given by [[colimits]], this suggests that a 2-ring should be a cocomplete category which is compatibly [[monoidal category|monoidal]] in that the the tensor product preserves colimits:
* [Compatibly monoidal cocomplete categories](#MonoidalCompleteCateories)
But there are still more properties which one may want to enforce, notably that homomorphisms of 2-rings form a [[2-abelian group]]. This is achieved by demanding the underlying category to be not just cocomplete by [[presentable category|presentable]]:
* [Compatibly monoidal presentable categories](#CompatiblyMonoidalPresentableCategories).
### Enriched monoidal categories
{#EnrichedMonoidalCategories}
1. A __$2$-rig__ might be an [[Ab-enriched category]] which is [[enriched monoidal category|enriched monoidal]].
2. A __$2$-rig__ might be an [[additive category]] which is enriched monoidal.
3. A __$2$-rig__ might be a [[distributive monoidal category]]: a monoidal category with finite [[coproducts]] such that the monoidal product distributes over the coproducts.
4. A __$2$-rig__ might be a [[closed monoidal category]] with finite coproducts.
5. Finally, a __$2$-ring__ is a monoidal [[abelian category]].
Note that (2) is a special case of both (1) and (3), which are independent. (4) is a special case of (3), by the [[adjoint functor theorem]]. (5) is a special case of (2), of course.
### Compatibly monoidal cocomplete categories
{#MonoidalCompleteCateories}
In ([Baez-Dolan](#BaezDolan)) the following is considered:
+-- {: .num_defn #BD2Rig}
###### Definition
A _2-rig_ is a [[monoidal category|monoidal]] [[colimit|cocomplete category]] where the [[tensor product]] respects [[colimits]].
=--
One can define [[braided monoidal category|braided]] and [[symmetric monoidal category|symmetric]] 2-rigs in this sense (and indeed, also in the other senses listed above). In particular, there is a [[2-category]] $\mathbf{Symm2Rig}$ with:
* symmetric monoidal cocomplete categories where the monoidal product distributes over colimits as objects,
* symmetric monoidal cocontinuous functors as 1-morphisms,
* symmetric monoidal natural transformations as 2-morphisms.
### Compatibly monoidal presentable categories
{#CompatiblyMonoidalPresentableCategories}
The following refines the [above](#MonoidalCompleteCateories) by demanding the underlying category of a 2-ring to be not just cocomplete but even a [[presentable category]]. This was motivated in ([CJF, remark 2.1.10](#CJF)).
+-- {: .num_defn }
###### Definition
Write
$$
2 Ab \in 2Cat
$$
for the [[2-category]] of [[presentable categories]] and [[colimit]]-preserving [[functors]] between them.
=--
([CJF, def. 2.1.8](#CJF))
+-- {: .num_remark}
###### Remark
By the [[adjoint functor theorem]] this is equivalently the 2-category of presentable categories and [[left adjoint]] functors between them.
=--
+-- {: .num_example #CategoryOfModulesAs2AbelianGroup}
###### Example
Given an ordinary [[ring]] $R$, its [[category of modules]] $Mod_R$ is presentable, hence may be regarded as a 2-abelian group.
=--
([CJF, example 2.1.5](#CJF))
+-- {: .num_prop}
###### Proposition
The 2-category $2Ab$ is a [[closed monoidal 2-category|closed]] [[symmetric monoidal 2-category]] with respect to the [[tensor product]] $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ such that for $A,B, C \in 2Ab$, $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full [[subcategory]] of [[functor category]] $Hom_{Cat}(A \times B, C)$ on those that are [[bilinear function|bilinear]] in that they preserve [[colimits]] in each argument separately.
=--
See also at [[Pr(∞,1)Cat]] for more on this.
+-- {: .num_example }
###### Example
For $\mathcal{C}$ a [[small category]], the [[category of presheaves]] $Set^{\mathcal{C}}$ is [[presentable category|presentable]] and
$$
Set^{\mathcal{C}_1} \boxtimes
Set^{\mathcal{C}_2}
\simeq
Set^{\mathcal{C}_1 \times \mathcal{C}_2}
\,.
$$
=--
+-- {: .num_example #TensorProductModuleCategoriesAsOf2AbelianGroups}
###### Example
For $R$ a [[ring]] the [[category of modules]] $Mod_R$ is presentable and
$$
Mod_{R_1} \boxtimes Mod_{R_2}
\simeq
Mod_{R_1 \otimes R_2}
\,,
$$
=--
([CJF, example 2.2.7](#CJF))
+-- {: .num_prop #EilenbergWattsTheorem}
###### Proposition
For $R_1, R_2$ two rings,
the category of 2-abelian group homomorphisms between the [[categories of modules]] is [[natural equivalence|naturally equivalent]] to that of $R_1$-$R_2$-[[bimodules]] and their [[intertwiners]]:
$$
(-)\otimes (-)
\;\colon\;
{}_{R_1}Mod_{R_2}
\stackrel{\simeq}{\to}
Hom_{2Ab}(Mod_{R_1}, Mod_{R_2})
\,.
$$
The equivalence sends a bimodule $N$ to the functor given by the [[tensor product]] over $R_1$:
$$
(-) \otimes N \;\colon\;
Mod_{R_1} \to Mod_{R_2}
\,.
$$
=--
This is the [[Eilenberg-Watts theorem]].
+-- {: .num_defn #2RingAsCompatiblyMonoidalPresentableCategory}
###### Definition
Write
$$
2Ring \in 2Cat
$$
for the [[2-category]] of [[monoid objects]] [[internalization|internal]] to $2 Ab$. An [[object]] of this 2-category we call a **2-ring**.
Equivalently, a 2-ring in this sense is a [[presentable category]] equipped with the structure of a [[monoidal category]] where the [[tensor product]] preserves [[colimits]].
=--
([CJF, def. 2.1.8](#CJF))
+-- {: .num_example}
###### Example
The category [[Set]] with its [[cartesian product]] is a 2-ring and it is the [[initial object]] in $2Ring$.
=--
([CJF, example 2.3.4](#CJF))
+-- {: .num_example}
###### Example
The category [[Ab]] of [[abelian groups]] with its standard [[tensor product of abelian groups]] is a 2-ring.
=--
+-- {: .num_example #CommutativeRingGivesCommutative2Ring}
###### Example
For $R$ an ordinary [[commutative ring]], $Mod_R$ equipped with its usual [[tensor product of modules]] is a commutative 2-ring.
=--
+-- {: .num_example}
###### Example
For $R$ an ordinary [[ring]] and $Mod_R$ its ordinary [[category of modules]], regarded as a 2-abelian group by example \ref{CategoryOfModulesAs2AbelianGroup}, the structure of a 2-ring on $Mod_R$ is equivalently the structure of a [[sesquiunital sesquialgebra]] on $R$.
If $R$ is in addition a [[commutative ring]] that $Mod_R$ is a commutative 2-ring and is canonically an $Ab$-[[2-algebra]] in that
$$
Ab \simeq Mod_{\mathbb{Z}} \to Mod_R
\,.
$$
=--
([CJF, example 2.3.7](#CJF))
+-- {: .num_defn}
###### Definition
For $A$ a 2-ring, def. \ref{2RingAsCompatiblyMonoidalPresentableCategory}, write
$$
2Mod_A \in 2Cat
$$
for the [[2-category]] of [[module objects]] over $A$ in $2Ab$.
This means that a 2-module over $A$ is a [[presentable category]] $N$ equipped with a functor
$$
A \boxtimes N \to N
$$
which satisfies the evident action property.
=--
([CJF, def. 2.3.3](#CJF))
+-- {: .num_example}
###### Example
Let $R$ be an ordinary [[commutative ring]] and $A$ an ordinary $R$-[[associative algebra|algebra]]. Then by example \ref{CategoryOfModulesAs2AbelianGroup} $Mod_A$ is a 2-abelian group and by example
\ref{CommutativeRingGivesCommutative2Ring} $Mod_R$ is a commutative ring. By example \ref{TensorProductModuleCategoriesAsOf2AbelianGroups}
$Mod_R$-[[2-module]] structures on $Mod_A$
$$
Mod_R \boxtimes \Mod_A \to Mod_A
$$
correspond to colimit-preserving functors
$$
Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}
$$
that satisfy the action property. Such as presented under the [[Eilenberg-Watts theorem]], prop. \ref{EilenbergWattsTheorem}, by $R \otimes_{\mathbb{Z}} A$-$A$ [[bimodules]]. $A$ itself is canonically such a bimodule and it exhibits a $Mod_R$-[[2-module]] structure on $Mod_A$.
=--
## Properties
### Initial object
{#PropertiesInitialObject}
+-- {: .num_remark}
###### Remark
The analog role in 2-rigs to the role played by the [[natural numbers]] among ordinary [[rigs]] should be played by the standard [[categorification]] of the natural numbers: the [[category of finite sets]]. One is therefore inclined to demand that a reasonable definition of 2-rigs should be such that $FinSet$ is the [[initial object]] (in the suitably higher categorical sense) in the [[2-category]] of 2-rigs.
For the notion in def. \ref{BD2Rig} this was conjectured by [[John Baez]], for the notion in def. \ref{2RingAsCompatiblyMonoidalPresentableCategory} this is asserted in ([Chirvasitu & Johnson-Freyd, example 2.3.4](#CJF)).
=--
### Tannaka duality
{#TannakaDuality}
[[!include structure on algebras and their module categories - table]]
## Related concepts
* [[2-algebraic geometry]]
* A further slight variant of compatibly monoidal cocomplete categories is that of _monoidal [[vectoids]]_.
* [[distributivity for monoidal structures]]
* [[prime spectrum of a monoidal stable (∞,1)-category]]
## References
The proposal that a 2-ring should be a compatibly monoidal cocomplete category is due to
* {#BaezDolan} [[John Baez]], [[James Dolan]], _Higher-dimensional algebra III: $n$-categories and the algebra of opetopes_, _Adv. Math._ **135** (1998), 145-206. ([arXiv](http://arxiv.org/abs/q-alg/9702014))
The proposal that a 2-ring should be a compatibly monoidal presentable category is due to
* {#CJF} [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))
see also
* {#Brandenburg14} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](http://arxiv.org/abs/1410.1716))
This is related to
* [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_.
A similar notion is that of "monoidal [[vectoid]]" due to
* [[Nikolai Durov]], _Classifying vectoids and generalisations of operads_, Proc. of Steklov Inst. of Math. __273__:1, 48-63 (2011) [arxiv/1105.3114](http://arxiv.org/abs/1105.3114)), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273
The role of presentable categories as higher analogs abelian groups in the context of [[(infinity,1)-categories]] have been made by [[Jacob Lurie]], see at _[[Pr(infinity,1)Cat]]_.
Another, more algebraic, notion of a categorical ring is introduced in
* M. Jibladze , T. Pirashvili, _Third Mac Lane cohomology via categorical rings_, J. of homotopy and related structures, __2__(2), 2007, 187–221 [pdf](http://www.emis.de/journals/JHRS/volumes/2007/n2a10/v2n2a10.pdf) [math.KT/0608519](http://arxiv.org/abs/math/0608519)
[[!redirects 2-rig]]
[[!redirects 2-rigs]]
[[!redirects 2-ring]]
[[!redirects 2-rings]]
[[!redirects 2Rig]]
[[!redirects 2Ring]]
|
2-sheaf | https://ncatlab.org/nlab/source/2-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _2-sheaf_ is the generalization of the notion of [[sheaf]] to the [[higher category theory]] of [[2-categories]]/[[bicategories]].
A 2-category of 2-sheaves forms a [[2-topos]].
+-- {: .num_remark }
###### Remark on terminology
A _2-sheaf_ is a higher sheaf of [[categories]]. More restrictive than this is a higher sheaf with values in [[groupoids]], which would be a _[[(2,1)-sheaf]]_. Both these notions are often referred to as **[[stack]]**, or sometimes "stack of groupoids" and "stack of categories" for definiteness. But moreover, traditionally a [[stack]] (in either flavor) is considered only over a [[1-site]], whereas it makes sense to consider [[(2,1)-sheaves]] more generally over [[(2,1)-sites]] and 2-sheaves over [[2-sites]].
Therefore, saying "2-sheaf" serves to indicate the full generality of the notion of higher sheaves in [[2-category theory]], as opposed to various special cases of this general notion which have traditionally been considered.
=--
## Definition
Let $C$ be a [[2-site]] having finite [[2-limit]]s (for convenience). For a covering family $(f_i:U_i\to U)_i$ we have the comma objects
+--{: style="text-align:center"}
<svg xmlns="http://www.w3.org/2000/svg" width="10em" height="10em" viewBox="-30 -20 180 150">
<desc>Comma Square</desc>
<defs>
<marker id="svg295arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="5" orient="auto">
<path d="M 0 0 L 10 5 L 0 10 z"/>
</marker>
</defs>
<g font-size="16" markdown="1">
<foreignObject x="-10" y="0" width="45" height="25">$(f_i/f_j)$</foreignObject>
<foreignObject x="100" y="0" width="20" height="20">$U_i$</foreignObject>
<foreignObject x="0" y="100" width="20" height="20">$U_j$</foreignObject>
<foreignObject x="100" y="100" width="20" height="20">$U$</foreignObject>
<foreignObject x="110" y="50" width="20" height="25">$f_i$</foreignObject>
<foreignObject x="50" y="110" width="20" height="25">$f_j$</foreignObject>
<foreignObject x="-20" y="50" width="20" height="30">$q_{i j}$</foreignObject>
<foreignObject x="50" y="-20" width="20" height="30">$p_{i j}$</foreignObject>
<foreignObject x="60" y="60" width="20" height="25">$\mu_{i j}$</foreignObject>
</g>
<g fill="none" stroke="#000" stroke-width="1.5" marker-end="url(#svg295arrowhead)">
<line x1="40" y1="10" x2="90" y2="10"/>
<line x1="30" y1="110" x2="90" y2="110"/>
<line y1="30" x1="10" y2="90" x2="10"/>
<line y1="30" x1="110" y2="90" x2="110"/>
<line x1="65" y1="55" x2="55" y2="65"/>
</g>
</svg>
=--
We also have the [[nlab:double comma object|double comma objects]] $(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)$ with projections $r_{i j k}:(f_i/f_j/f_k)\to (f_i/f_j)$, $s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)$, and $t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)$.
Now, a functor $X:C^{op} \to Cat$ is called a **2-presheaf**. It is **1-separated** if
* For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$ and $a,b: x\to y$, if $X(f_i)(a) = X(f_i)(b)$ for all $i$, then $a=b$.
It is **2-separated** if it is 1-separated and
* For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$, given $b_i:X(f_i)(x) \to X(f_i)(y)$ such that $\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)$, there exists a (necessarily unique) $b:x\to y$ such that $b_i = X(f_i)(b)$.
It is a **2-sheaf** if it is 2-separated and
* For any covering family $(f_i:U_i\to U)_i$ and any $x_i\in X(U_i)$ together with morphisms $\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)$ such that the following diagram commutes:
$$\array{X(r_{i j k})X(p_{i j})(x_i) &
\overset{X(r_{i j k})(\zeta_{i j})}{\to} &
X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} &
X(s_{i j k})X(p_{j k})(x_j)\\
^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\
X(t_{i j k}) X(p_{i k})(x_i) &
\underset{X(t_{i j k})(\zeta_{i k})}{\to} &
X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} &
X(s_{i j k}) X(q_{j k})(x_k)}$$
there exists an object $x\in X(U)$ and isomorphisms $X(f_i)(x)\cong x_i$ such that for all $i,j$ the following square
commutes:
$$\array{
X(p_{i j})X(f_i)(X) & \overset{\cong}{\to}
& X(p_{i j})(x_i)\\
^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\
X(q_{i j})X(f_j)(x) & \underset{\cong}{\to} &
X(q_{i j})(x_j).}$$
A 2-sheaf, especially on a 1-site, is frequently called a **[[stack]]**. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that $\mu_{i j}$ and $\zeta_{i j}$ _need not be invertible_.
Note, though, they must be invertible as soon as $C$ is (2,1)-site: $\mu_{i j}$ by definition and $\zeta_{i j}$ since an inverse is provided by $\iota_{i j}^*(\zeta_{i j})$, where $\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)$ is the symmetry equivalence.
If $C$ lacks finite limits, then in the definitions of "2-separated" and "2-sheaf" instead of the comma objects $(f_i/f_j)$, we need to use arbitrary objects $V$ equipped with maps $p:V\to U_i$, $q:V\to U_j$, and a 2-cell $f_i p \to f_j q$. We leave the precise definition to the reader.
A 2-site is said to be **subcanonical** if for any $U\in C$, the representable functor $C(-,U)$ is a 2-sheaf. When $C$ has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel [[2-polycongruence]]. In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.
The 2-category $2Sh(C)$ of 2-sheaves on a small 2-site $C$ is, by definition, a [[Grothendieck 2-topos]].
## Properties
### Characterization of over $(n,r)$-sites
If the underlying [[2-site]] happens to be an [[(n,r)-site]] for $n$ and/or $r$ lower than 2, there may be other equivalent ways to think of 2-sheaves.
A [[2-topos]] with a [[2-site]] of definition that happens to be just a 1-site or [[(2,1)-site]] is _1-localic_ or _(2,1)-localic_.
#### Over a 1-site
Over a 1-site, the [[Grothendieck construction]] says that [[2-functors]] on the site are equivalent to [[fibered categories]] over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See _[References -- In terms of fibered categories](#InTermsOfFiberedCategories)_.
Also, over a 1-site a 2-sheaf is essentially a _[[indexed category]]_. Therefore stacks over 1-sites can also be discussed in this language, see notably the work ([Bunge-Pare](#BungePare)).
In particular, if the 1-site $C$ is a [[topos]], then every topos _over_ $C$ as its [[base topos]] (a $C$-topos) induces an [[indexed category]].
+-- {: .num_prop }
###### Proposition
If $C$ is a topos and $E$ is a $C$-topos, then (the [[indexed category]] corresponding to) $E$ is a 2-sheaf on $C$ with respect to the [[canonical topology]].
=--
This appears as ([Bunge-Pare, corollary 2.6](#BungePare)).
Moreover, over a [[1-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the (suitably defined) [[2-category]] of [[internal categories]] in the underlying [[1-topos]]. See _[References -- In terms of internal categories](#ReferencesInTermsOfInternalCategories)_.
#### Over a $(2,1)$-site -- As internal categories
Over a [[(2,1)-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the [[2-category]] of [[internal (infinity,1)-categories]] in the corresponding [[(2,1)-topos]].
This is discussed at _[2-Topos -- In terms of internal categories](2-topos#InTermsOfInternalCategories)_.
## Examples
### Codomain fibrations / sheaves of modules
A classical class of examples for 2-sheaves are [[codomain fibrations]]
over suitable sites, or rather their [[tangent categories]]. As discussed there, this includes the case of sheaves of categories of [[modules]] over sites of [[algebra over an algebraic theory|algebras]].
+-- {: .num_prop }
###### Proposition
For $C$ an [[exact category]] with [[finite limits]], the [[codomain fibration]] $Cod : C^I \to C$ or equivalently (under the [[Grothendieck construction]]), the self-[[indexed category|indexing]] of $C$ is a 2-sheaf with respect to the [[canonical topology]].
=--
This is for instance ([Bunge-Pare, corollary 2.4](#BungePare)).
## Related concepts
* [[presheaf]] / [[sheaf]] / [[cosheaf]]
* **2-sheaf** / [[stack]]
* [[(∞,1)-sheaf]] / [[∞-stack]]
* [[(∞,2)-sheaf]]
* [[(∞,n)-sheaf]]
* [[descent]]
## References
Historically, the original definition of _[[stack]]_
included the case of category-valued functors, hence of 2-sheaves, in:
* [[Jean Giraud]], *Cohomologie non abélienne* Grundlehren **179**, Springer (1971) [[doi:10.1007/978-3-662-62103-5](https://www.springer.com/gp/book/9783540053071)]
* [[Jean Giraud]], _Classifying topos_, in: [[William Lawvere]] (ed.) _Toposes, Algebraic Geometry and Logic_, Lecture Notes in Mathematics **274**, Springer (1972) [[doi:10.1007/BFb0073964](https://doi.org/10.1007/BFb0073964)]
### In terms of categories internal to sheaf toposes
{#ReferencesInTermsOfInternalCategories}
Category-valued stacks as [[internal categories]] in the underlying [[sheaf topos]]:
* {#BungePare} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4 (1979) 373-399 [[numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]
* {#Bunge} [[Marta Bunge]], *Stack completions and Morita equivalence for categories in a topos*, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4, (1979) 401-436 [[numdam](http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0), [MR558106](http://www.ams.org/mathscinet-getitem?mr=558106)]
* {#JoyalTierney} [[André Joyal]], [[Myles Tierney]], section 3 of: *Strong stacks and classifying spaces*, in: *Category Theory* ([[Como]], 1990), Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]
> (establishing the [[canonical model structure on Cat]] in the internal generality)
### In terms of fibered categories
{#InTermsOfFiberedCategories}
A discussion of stacks over [[1-sites]] in terms of their [[Grothendieck construction|associated]] [[fibered categories]] is in
* [[Angelo Vistoli]], _Notes on Grothendieck topologies, fibered categories and descent theory_ ([pdf](http://homepage.sns.it/vistoli/descent.pdf))
### 2-Sites
The above text involves content transferred from
* [[Michael Shulman]], _[[michaelshulman:2-site]]_
2-sites were earlier considered in
* [[Ross Street]], _[[StreetCBS]]_
[[!redirects 2-sheaves]] |
2-site | https://ncatlab.org/nlab/source/2-site |
This entry is about the notion of [[site]] in [[2-category]] theory. For the notion "bisite" of a 1-categorical site equipped with two coverages see instead [[separated presheaf]].
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
The notion of [[2-site]] is the generalization of the notion of [[site]] to the [[higher category theory]] of [[2-categories]] ([[bicategories]]).
Over a 2-site one has a [[2-topos]] of [[2-sheaves]].
## Definition
A **[[coverage]]** on a [[2-category]] $C$ consists of, for each object $U\in C$, a collection of families $(f_i: U_i\to U)_i$ of morphisms with codomain $U$, called _covering families_, such that
* If $(f_i:U_i\to U)_i$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $(h_j:V_j\to V)_j$ such that each composite $g h_j$ factors through some $f_i$, up to isomorphism.
This is the 2-categorical analogue of the 1-categorical notion of [[nlab:coverage|coverage]] introduced in the [[nlab:Elephant|Elephant]].
A 2-category equipped with a coverage is called a **2-site**.
## Examples
* If $C$ is a [[michaelshulman:regular 2-category]], then the collection of all singleton families $(f:V\to U)$, where $f$ is eso, forms a coverage called the _regular coverage_.
* Likewise, if $C$ is a [[michaelshulman:coherent 2-category]], the collection of all finite jointly-eso families forms a coverage called the _coherent coverage_.
* On $Cat$, the _canonical coverage_ consists of all families that are jointly essentially surjective on objects.
## Saturation conditions
A **pre-Grothendieck coverage** on a 2-category is a coverage satisfying the following additional conditions:
* If $f:V\to U$ is an equivalence, then the one-element family $(f:V\to U)$ is a covering family.
* If $(f_i:U_i\to U)_{i\in I}$ is a covering family and for each $i$, so is $(h_{i j}:U_{i j} \to U_i)_{j\in J_i}$, then $(f_i h_{i j}:U_{i j}\to U)_{i\in I, j\in U_i}$ is also a covering family.
This is the 2-categorical version of a [[Grothendieck pretopology]] (minus the common condition of having actual [[pullbacks]]).
Now, a **sieve** on an object $U\in C$ is defined to be a functor $R:C^{op}\to Cat$ with a transformation $R\to C(-,U)$ which is objectwise fully faithful (equivalently, it is a [[fully faithful morphism]] in $[C^{op},Cat]$). Equivalently, it may be defined as a subcategory of the [[slice 2-category]] $C/U$ which is closed under precomposition with all morphisms of $C$.
Every family $(f_i\colon U_i\to U)_i$ generates a sieve by defining $R(V)$ to be the full subcategory of $C(V,U)$ on those $g:V\to U$ such that $g \cong f_i h$ for some $i$ and some $h:V\to U_i$. The following observation is due to [StreetCBS](#StreetCBS).
+--{: .num_lemma}
###### Lemma
A 2-presheaf $X:C^{op}\to Cat$ is a 2-sheaf for a covering family $(f_i:U_i\to U)_i$ if and only if
$$X(U) \simeq[C^{op},Cat](C(-,U),X) \to [C^{op},Cat](R,X)$$
is an equivalence, where $R$ is the sieve on $U$ generated by $(f_i:U_i\to U)_i$.
=--
Therefore, just as in the 1-categorical case, it is natural to restrict attention to covering _sieves_. We define a **Grothendieck coverage** on a 2-category $C$ to consist of, for each object $U$, a collection of sieves on $U$ called covering sieves, such that
* If $R$ is a covering sieve on $U$ and $g:V\to U$ is any morphism, then $g^*(R)$ is a covering sieve on $V$.
* For each $U$ the sieve $M_U$ consisting of _all_ morphisms into $U$ (the sieve generated by the singleton family $(1_U)$) is a covering sieve.
* If $R$ is a covering sieve on $U$ and $S$ is an arbitrary sieve on $U$ such that for each $f:V\to U$ in $R$, $f^*(S)$ is a covering sieve on $V$, then $S$ is also a covering sieve on $U$.
Here if $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, $g^*(R)$ denotes the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $g h$ factors, up to isomorphism, through some morphism in $R$.
As in the 1-categorical case, one can then show that every coverage generates a unique Grothendieck coverage having the same 2-sheaves.
## Properties
* The [[2-category]] of [[2-sheaf|2-sheaves]] on a 2-site is a [[Grothendieck 2-topos]].
* If $C$ is a [[1-category]] regarded as a 2-category with only [[identity morphism|identity]] [[2-morphisms]], then a coverage (pretopology, topology) on $C$ reduces to the usual notion of [[coverage]], [[Grothendieck pretopology]], or [[Grothendieck topology]].
## Related concepts
* [[(n,r)-site]]
* [[1-site]]
* **2-site**, [[(2,1)-site]]
* [[(∞,1)-site]]
* [[model site]], [[simplicial site]]
## References
Strict 2-sites were considered in
* [[Ross Street]], _Two-dimensional sheaf theory_ J. Pure Appl. Algebra __23__ (1982), no. 3, 251-270, [MR83d:18014](http://www.ams.org/mathscinet-getitem?mr=644277), <a href="http://dx.doi.org/10.1016/0022-4049(82)90101-3">doi</a>
Bicategorical 2-sites in
* [[Ross Street]], [[zoranskoda:Characterization of Bicategories of Stacks]], [MR84d:18006](http://www.ams.org/mathscinet-getitem?mr=682967), p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM __962__, 1982
{: #StreetCBS }
See also _[[StreetCBS]]_.
More discussion is in
* [[Michael Shulman]], _[[michaelshulman:2-site]]_
[[!redirects 2-sites]]
[[!redirects (2,2)-site]]
[[!redirects (2,2)-sites]]
[[!redirects bisite]]
[[!redirects bisites]]
[[!redirects Grothendieck 2-topology]]
[[!redirects 2-Grothendieck topology]]
[[!redirects Grothendieck 2-topologies]]
[[!redirects 2-Grothendieck topologies]]
[[!redirects Grothendieck 2-pretopology]]
[[!redirects 2-Grothendieck pretopology]]
[[!redirects Grothendieck 2-pretopologies]]
[[!redirects 2-Grothendieck pretopologies]]
[[!redirects 2-coverage]]
[[!redirects 2-coverages]]
[[!redirects topology on a 2-category]]
[[!redirects topologies on 2-categories]]
[[!redirects Grothendieck topology on a 2-category]]
[[!redirects Grothendieck topologies on 2-categories]]
[[!redirects coverage on a 2-category]]
[[!redirects coverages on 2-categories]]
[[!redirects topology on a bicategory]]
[[!redirects topologies on bicategories]]
[[!redirects Grothendieck topology on a bicategory]]
[[!redirects Grothendieck topologies on bicategories]]
[[!redirects coverage on a bicategory]]
[[!redirects coverages on bicategories]]
|
2-spectral triple | https://ncatlab.org/nlab/source/2-spectral+triple |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
An ordinary [[spectral triple]] is, as discussed there, the abstract algebraic data characterizing [[supersymmetric quantum mechanics]] on a [[worldline]] and thereby spectrally encoding an effective (possibly [[non-commutative geometry|non-commutative]], hence "[[non-geometric background|non-geometric]]") [[target space]] [[geometry]].
Ordinary [[Riemannian geometry]] with [[spin structure]] is the special case of this where the [[Hilbert space]] in the spectral triple is that of [[square integrable function|square integrable]] [[sections]] of the [[spinor bundle]] on a [[spin structure|spin]] [[Riemannian manifold]] and the operator $D$ acting on that is the standard [[Dirac operator]], hence the "supercharge" of the worldline supersymmetry of the [[spinning particle]].
In generalization of this, a "2-spectral triple" should be the analogous algebraic data that encodes the [[worldsheet]] theory of a [[superstring]] propagating on a [[target space]] geometry which is a generalization of [[Riemannian geometry]] with ([[twisted string structure|twisted]]) [[string structure]].
Of course such data is just that of a [[2d superconformal field theory]], realized locally by, for instance, a [[vertex operator algebra]] or by a [[conformal net]] of [[local net of observables|local observables]]. But for emphasis it may be useful to speak of such data as constituting a "2-spectral triple", for emphasizing more the important and intricate relation to the concept of [[spectral triples]], which in much of the literature seems to be unduly ignored.
| [[quantum system]] | [[supercharge]] | formalization | algebra |
|----------------------|---------|------------------------|-----|
| quantum [[spinning particle]] | [[Dirac operator]] | [[spectral triple]] | [[operator algebra]]
| quantum [[spinning string]] | [[Dirac-Ramond operator]] | [[2d SCFT]] | [[vertex operator algebra]] |
That the 0-mode sector of a [[2d SCFT]] -- hence the quantum point [[particle]] limit of a quantum [[superstring]] dynamics -- yields a [[spectral triple]] was maybe first highlighted in ([Fröhlich-Gawędzki 93](#FroehlichGawedzki93)) by way of a series of concrete examples, such as the [[WZW model]].
Here the role of the [[Dirac operator]] of the spectral triple is played by the [[Dirac-Ramond operator]] of the [[superstring]], hence the operator whose [[index]] (in the [[large volume limit]]) is the [[Witten genus]].
That hence the superstring quantum theory should be regarded as a kind of higher spectral triple was maybe first suggested in ([Chamseddine 97](#Chamseddine97)), together with arguments that the associated [[spectral action]] indeed reproduces the [[action functional]] of the string's [[target space]] [[effective quantum field theory|effective]] [[supergravity]] theory. An exposition of this perspective is in ([Fröhlich-Grandjean-Recknagel 97, section 7.2](#FroehlichGrandjeanRecknagel97)).
From [Fröhlich 92, p. 11](non-perturbative+quantum+field+theory#Froehlich92):
> I still have hopes, perhaps romantic ones, that [[string theory]], or something inspired by it, will come back to life again. I believe it is interesting to attempt to formulate string theory in an "invariant" way, quite like it is useful to formulate geometry in a coordinate-independent way. One might, for example, start with a family $\mathcal{F}$, of [[von Neumann algebra factor|hyperfinite type $III_1$ von Neumann algebras]] -- to be a little technical -- [[conformal net|indexed by]] intervals of the circle with non-empty complement (or of the super-circle). It may pay to formulate the starting point using the language of [[sheaves]]. $[...]$ This structure determines a [[braided monoidal category|braided monoidal]] [[C*-category]] with unit, ...; briefly, a quantum theory. From a combination of such [[tensor categories]] (left and right movers) one would attempt to reconstruct (symmetries of) _physical space-time_. String amplitudes would correspond to [[morphisms|arrows]] ([[intertwiners]]) of the [[tensor category]]. $[...]$ it would provide a general way of thinking about string theory that does not presuppose knowing the target space-time of the theory.
Later it was shown more formally ([Roggenkamp-Wendland 03](#RoggenkampWendland03)), reviewed in ([Roggenkamp-Wendland 08](#RoggenkampWendland08)), that there is a precise algebraically formalization of taking the "point particle limit" of a quantum string, by sending its [[vertex operator algebra]] to a spectral triple obtained by suitably retaining only [[worldsheet]] 0-modes.
In ([Soibelman 11](#Soibelman11)) this was used as a means to systematically study the large volume limit of [[effective quantum field theory|effective]] string [[spacetimes]] (and hence aspects of the [[landscape of string theory vacua]]) by studying the spectral geometries (i.e. the Connes-style noncommutative geometries) of the spectral triples arising from the string's point particle limit this way.
Now, since there is information lost in passing from a stringy "2-spectral triple" (a [[2d SCFT]]) to its underlying point particle [[spectral triple]], not all spectral triples are to be expected to have a lift to a 2-spectral triple (possibly corresponding to a [[UV-completion]] of the corresponding target space [[effective field theories]]).
In view of this, it is noteworthy that the spectral triple of the [[Connes-Lott-Chamseddine model]] shares a few key properties with the 2d SCFTs considered in [[string phenomenology]]:
The [[Connes-Lott-Chamseddine model]] is an encoding in a spectral triple of the [[standard model of particle physics]] coupled to [[gravity]] realized as a kind of spectral [[Kaluza-Klein compactification]] on a non-commutative fiber space down to ordinary 4d [[Minkowski spacetime]] (or possibly its [[Wick rotation|Wick rotated]] Euclidean version). In order for this to work out, it turns out that the compactified non-commutative fiber space needs to have [[KO-dimension]] equal to $6$. (Here the fiber space is classically just a ("non-commutative") point, but it appears as the singular collapsing limit of a space of finite dimension. This actual dimension is the [[KO-dimension]].)
Hence the claim of the [[Connes-Lott-Chamseddine model]] is that if the [[standard model of particle physics]] is encoded as a singular limit of a [[Kaluza-Klein compactification]] modeled via a [[spectral triple]] then the [[dimensions]] of the [[KK-compactification]] are
$$
4 + 6 \;\;\; (mod\;8)
$$
with 4-dimensional base space and 6-dimensional fiber space, to a total of a 10-dimensional [[spacetime]] at high energy (after uncompactification of the fiber).
This, of course, is precisely the dimensionality of the target spacetime of [[perturbative string theory vacua]] for the critical [[superstring]].
This point was highlighted in [Connes 06, p. 8](#Connes06):
> {#Connes06OnRelationToStringVacua} When one looks at the table (7.2) of Appendix 7 giving the [[KO-dimension]] of the finite space $[$ i.e. the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$ $]$ one then finds that its [[KO-dimension]] is now equal to 6 [[modulo]] 8 (!). As a result we see that the [[KO-dimension]] of the [[Cartesian product|product space]] $M \times F$ $[$ i.e. of 4d [[spacetime]] $M$ with the [[noncommutative geometry|noncommutative]] [[KK-compactification]]-[[fiber]] $F$$]$ is in fact equal to $10 \sim 2$ [[modulo]] 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an "[[effective field theory|effective]]" [[KK-compactification|compactification]] at least for low energies. But 10 is also 2 [[modulo]] 8 which might be related to the observations of [Lauscher-Reuter 06](#LauscherReuter06) about [[gravity]].
Algebraically, this arises from the fact that the [[BRST complex]] for the [[superstring]] [[worldsheet]] theory is consistent (has BRST differential squaring to 0) precisely if the corresponding [[2d SCFT]] has conformal [[central charge]] 15, and each spacetime dimension contributes $1 \tfrac{1}{2}$ to this central charge (a contribution of 1 from each bosonic direction, and another $\tfrac{1}{2}$ for the corresponding fermionic contribution).
## Examples
### Flop transition
There is at least evidence that there is a continuous path in the space of 2-spectral triples that starts and ends at a point describing the ordinary geometry of a complex 3-dimensional [[Calabi-Yau space]] but passes in between through a 2-spectral triple/2d SCFT (a [[Gepner model]]) which is not the $\sigma$-model of an ordinary geometry, hence which describes "noncommutative 2-geometry" (to borrow that terminology from the situation of ordinary spectral triples). This is called the [[flop transition]] (alluding to the fact that the geometries at the start and end of this path have different [[topology]]). This was further expanded on and used for the mathematical study of the [[large volume limit]] of [[string theory]] [[vacua]] in ([Soibelman 11](#Soibelman11)).
## Related concepts
* [[spectral triple]]
* [[spectral action]]
* [[D-brane geometry]]
* [[automorphism of a 2-spectral triple]]
* [[swampland]]
## References
{#References}
An early observation that the 0-mode sector of a [[2d SCFT]] is a [[spectral triple]], demonstrated in a series of concrete examples, is
* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawędzki]], _Conformal Field Theory and Geometry of Strings_, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))
The suggestion to understand, conversely, the [[string theory|string]]'s [[worldvolume]] [[2d SCFT]] as a higher spectral triple is due to
* {#Chamseddine97} [[Ali Chamseddine]], _An Effective Superstring Spectral Action_, Phys.Rev. D56 (1997) 3555-3567 ([arXiv:hep-th/9705153](http://arxiv.org/abs/hep-th/9705153)),
which claims to show that the corresponding [[spectral action]] reproduces the correct effective background action known in [[string theory]]. A more expository account of this perspective is in
* {#FroehlichGrandjeanRecknagel97} [[Jürg Fröhlich]], Oliver Grandjean, [[Andreas Recknagel]], section 7 of _Supersymmetric quantum theory, non-commutative geometry, and gravitation_, Lecture Notes Les Houches (1995) ([arXiv:hep-th/9706132](http://arxiv.org/abs/hep-th/9706132)).
A more formal derivation of how ordinary [[spectral triples]] arise as point particle limits of [[vertex operator algebra]]s for [[2d SCFTs]] then appears in
* {#RoggenkampWendland03} [[Daniel Roggenkamp]], [[Katrin Wendland]], _Limits and Degenerations of Unitary Conformal Field Theories_, Commun.Math.Phys. 251 (2004) 589-643 ([arXiv:hep-th/0308143](http://arxiv.org/abs/hep-th/0308143))
summarized in
* {#RoggenkampWendland08} [[Daniel Roggenkamp]], [[Katrin Wendland]], _Decoding the geometry of conformal field theories_, Proceedings of the 7th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgaria ([arXiv:0803.0657](http://arxiv.org/abs/0803.0657))
A brief indication of some ideas of [[Yan Soibelman]] and [[Maxim Kontsevich]] on this matter is at
* [[Urs Schreiber]], [_Spectral triples and graph field theory_](http://golem.ph.utexas.edu/category/2007/06/had_the_pleasure_of_talking.html)
Further development of this and application to the study of the [[large volume limit]] of [[superstring]] [[vacua]] is in
* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ ([pdf](http://www.math.ksu.edu/~soibel/nc-riem-3.pdf), [pdf](https://ncatlab.org/schreiber/files/SoibelmanCFTandRicciCurvature.pdf)), in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)
based on
* [[Maxim Kontsevich]], [[Yan Soibelman]], section 2 of _Homological mirror symmetry and torus fibrations_, Proceedings of KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry ([arXiv:math/0011041](http://arxiv.org/abs/math/0011041), [spire](http://inspirehep.net/record/536540/))
(discussing aspects of [[homological mirror symmetry]]).
Analogous detailed discussion based not on the [[vertex operator algebra]] description of local [[CFT]] but on the [[AQFT]] description via [[conformal nets]] is in
* {#CHKL09} [[Sebastiano Carpi]], Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral triples and the super-Virasoro algebra_, Commun.Math.Phys.295:71-97 (2010) ([arXiv:0811.4128](http://arxiv.org/abs/0811.4128))
where [[2d SCFTs]] are related essentially to [[local nets]] of [[spectral triples]].
Exposition of these results is in
* [[Urs Schreiber]], _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_, PhysicsForums--Insights (2016)
See also the references at [[geometric model for elliptic cohomology]].
That the [[Connes-Lott models]] could be [[effective field theory]]-limits of [[perturbative string theory vacua]] is also mentioned in
* {#Connes06} [[Alain Connes]], p. 8 of _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))
[[!redirects 2-spectral triples]] |
2-topos | https://ncatlab.org/nlab/source/2-topos |
+-- {: .rightHandSide}
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###Context###
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of **2-topos** is the generalization of the notion of [[topos]] from [[category theory]] to the [[higher category theory]] of [[2-categories]].
There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of [[Grothendieck topos]] or of [[elementary topos]], and in the latter case what axioms one chooses to take as the basis for generalization.
In contrast, [[(n,1)-topos|(2,1)-toposes]] are much better understood.
A _Grothendieck 2-topos_ is a [[2-category]] of [[2-sheaves]] over a [[2-site]].
A _Grothendieck (2,1)-topos_ is a [[(2,1)-category]] of [[(2,1)-sheaves]] over a [[(2,1)-site]].
See also [[higher topos theory]].
## Properties
### Characterization of 2-sheaf 2-toposes
The [[2-toposes]] of [[2-sheaves]] over a [[2-site]] are special among all 2-toposes, in direct generalization of how [[sheaf toposes]] ("[[Grothendieck toposes]]") are special among all [[toposes]]. In that case, _[[Giraud's theorem]]_ famously characterizes sheaf toposes. This characterization has a 2-categorical analog: the _[[2-Giraud theorem]]_.
### $(n,r)$-Localic 2-toposes
A [[2-sheaf]] [[2-topos]] is "$(n,r)$-localic" or "$(n,r)$-truncated" if it has an [[(n,r)-site]] of definition.
In particular a $(2,1)$-localic 2-topos is the same as a [[(2,1)-topos]].
### In terms of internal categories
{#InTermsOfInternalCategories}
Given a 2-topos $\mathcal{X}$, regard it is a [[2-site]] by equipping it with its [[canonical topology]].
+-- {: .num_defn}
###### Definition
Write $Cat(\mathcal{X})$ for the 2-category of _[[internal categories]]_ in $\mathcal{X}$, precisely: the 2-category of [[2-congruences]] and internal [[anafunctors]] between them (see [here](http://ncatlab.org/nlab/show/2-congruence#2CategoryOf2Congruences)).
=--
+-- {: .num_theorem #2SheavesAsInternalCategories}
###### Theorem
For $\mathcal{X}$ a 2-topos of [[2-sheaves]] on a [[2-site]], there is an [[equivalence of 2-categories]]
$$
\mathcal{X} \simeq Cat(\mathcal{X})
\,.
$$
If $\mathcal{X}$ is $(2,1)$-localic, with a [[(2,1)-site]] of definition $C$, then there is already an equivalence
$$
\mathcal{X} \simeq Cat(Sh_{(2,1)}(C))
$$
with the 2-category of categories internal to the underlying [[(2,1)-topos]].
If $\mathcal{X}$ is $1$-localic, with 1-site of definition, then there is even already an equivalence
$$
\mathcal{X} \simeq Cat(Sh(C))
$$
with the internal categories in the underlying [[sheaf topos]].
=--
+-- {: .proof}
###### Proof
By the [[2-Giraud theorem]], $\mathcal{X}$ is an [[exact 2-category]]. With this, the first statement is [this theorem](2-congruence#nCongExidempotent) at _[[2-congruence]]_.
By the discussion at [[n-localic 2-topos]], a 2-sheaf 2-topos has _[[core in a 2-category|enough groupoids]]_ precisely if it has a [[(2,1)-site]] of definition, and has _[[core in a 2-category|enough discretes]]_ precisely if it has a 1-site of definition.
With this the second and third statement is [this theorem](2-congruence#nCongOnGroupoidsAndDiscretes) at _[[2-congruence]]_.
=--
+-- {: .num_remark}
###### Remark
The noteworthy point about theorem \ref{2SheavesAsInternalCategories} is that for an ambient context which is a $(2,1)$-localic [[(2,1)-topos]], the straightforward morphisms of [[internal categories]], hence the notion of [[internal functors]], needs no further [[localization]]. This is in stark contrast to the situation for an ambient [[1-category]]. The generalization of this phenomenon is discussed at _[[category object in an (∞,1)-category]]_.
=--
## Examples
### The archetypical 2-topos
The archetypical 2-topos is [[Cat]]. This plays the role for 2-toposes as [[Set]] does for [[1-toposes]].
### Internal categories in a $(2,1)$-topos
Given any [[(2,1)-topos]] $\mathcal{X}$, the [[2-category]] $Cat(\mathcal{X})$ of [[internal (infinity,1)-category|internal categories]] in $\mathcal{X}$ ought to be a 2-topos. But it seems that at the moment there is no proof of this in the literature.
For literature on [[internal categories]] in [[1-toposes]] see at _[[2-sheaf]]_.
## Related concepts
* [[elementary theory of the 2-category of categories]] ([[ETCC]])
* [[formal category theory]]
[[!include flavors of higher toposes -- list]]
## References
An introduction is in
* [[Mike Shulman]], _[[michaelshulman:What is a 2-topos]]?_
Early developments include
* [[Ross Street]], _Two dimensional sheaf theory_, J. Pure and Appl. Algebra 24 (1982) 2Opp.
* [[Dominique Bourn]], _Sur les ditopos_, C. R. Acad. Sci. Paris 279, 911–913 (1974).
A detailed discussion from the point of view of [[internal logic]] is at
* [[Mike Shulman]], _[[michaelshulman:2-categorical logic]]_
Discussion of the 2-categorical [[Giraud theorem]] for [[2-sheaf]] 2-toposes is in
* [[Ross Street]], _[[zoranskoda:Characterization of Bicategories of Stacks]]_ Category theory (Gummersbach 1981) LNM 962, 1982, MR0682967 (84d:18006)
* [[Mike Shulman]], _[[michaelshulman:2-Giraud theorem]]_
Discussion of the [[elementary topos]]-analog of 2-toposes is in
* [[Mark Weber]], *Yoneda structures from 2-toposes*, Appl Categor Struct **15** (2007) 259–323 [[doi:10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2), [pdf](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)]
A notion of "flat 2-functor" (cf [[Diaconescu's theorem]]) perhaps relevant to the "points" of 2-toposes is in
* M.E. Descotte, [[Eduardo Dubuc]], M. Szyld, _On the notion of flat 2-functors_, arXiv:[1610.09429](https://arxiv.org/abs/1610.09429)
Discussion of 2-classifiers for 2-toposes is in
* [[Luca Mesiti]], _2-classifiers via dense generators and Hofmann-Streicher universe in stacks_ [[arXiv:2401.16900](https://arxiv.org/abs/2401.16900)]
[[!redirects 2-toposes]]
[[!redirects 2-topoi]]
[[!redirects (2,1)-topos]]
[[!redirects (2,1)-toposes]]
[[!redirects (2,2)-topos]]
[[!redirects (2,2)-toposes]]
[[!redirects Grothendieck 2-topos]]
[[!redirects Grothendieck 2-toposes]]
|
2-topos theory | https://ncatlab.org/nlab/source/2-topos+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The theory of [[2-topos]]es, generalizing [[topos theory]] from [[category theory]] to [[2-category theory]].
## Related concepts
* [[topos theory]]
* **2-topos theory**
* [[(∞,1)-topos theory]]
* [[higher topos theory]]
## References
The analog of the [[Elephant]] for 2-topos theory still needs to be written.
For some speculations and further references, see [[michaelshulman:2-categorical logic|this page]].
|
2-trivial model structure | https://ncatlab.org/nlab/source/2-trivial+model+structure |
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Every [[strict 2-category]] $K$ with finite [[strict 2-limits]] and finite strict 2-colimits becomes a [[model category]] (or, rather, its underlying 1-category does) in a canonical way, where:
* The weak equivalences are the [[equivalences]].
* The fibrations are the morphisms that are representably [[isofibrations]], i.e. the morphisms $e\to b$ such that $K(x,e)\to K(x,b)$ is an isofibration for all $x\in K$.
* The cofibrations are determined.
We call it the **2-trivial model structure**, as it is a 2-categorical analogue of the [[trivial model structure]] on any 1-category. It can be said to regard $C$ as an [[(∞,1)-category]] with only trivial [[k-morphisms]] for $k \geq 3$.
## Properties
* Every object is fibrant and cofibrant.
* By duality, any such category has another model structure, with the same weak equivalences but where the cofibrations are the iso-cofibrations and the fibrations are determined. In $Cat$, the two model structures are the same.
## Examples
* In [[Cat]], this produces the [[canonical model structure]].
* If $T$ is an [[accessible functor|accessible]] [[strict 2-monad]] on a [[locally finitely presentable category|locally finitely presentable]] [[strict 2-category]] $K$. Then the category $T Alg_s$ of strict $T$-[[algebra over a monad|algebras]] admits a [[transferred model structure]] from the 2-trivial model structure on $K$. The cofibrant objects therein are the [[flexible algebra]]s.
## Related pages
* In model categories built from various kinds of [[topological spaces]], there are often analogous [[model structure on topological spaces#Hurewicz (or Strøm) Model Structure|Hurewicz model structures]]. These are not actually examples of a 2-trivial model structure (for instance, the 2-category of spaces, continuous functions and homotopy classes of homotopies does not have finite limits as a 2-category, or even as a 1-category), but they share a common intuition and can sometimes be obtained as two instances of a more general construction.
## References
* [[Steve Lack]], *Homotopy-theoretic aspects of 2-monads*, [arXiv](http://arxiv.org/abs/math.CT/0607646)
[[!redirects trivial model structure on a 2-category]]
[[!redirects trivial model structure for a 2-category]]
|
2-type > history | https://ncatlab.org/nlab/source/2-type+%3E+history | < [[2-type]]
[[!redirects 2-type -- history]] |
2-type theory | https://ncatlab.org/nlab/source/2-type+theory |
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###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
As [[type theory]] has [[categorical semantics]] in 1-[[categories]], _2-type theory_ has semantics in [[2-categories]]. There are, potentially, many different kinds of "2-type theory" with different uses and semantics. 2-type theory is closely related to (and sometimes the same as) *[[directed type theory]]*.
## Applications
* The "mode theories" in some general approaches to [[modal type theory]] and [[adjoint type theory]] are a form of 2-type theory, where the 2-cells represent a general form of "structural rules" acting on modal judgments.
* The 2-cells in 2-type theory can also be used to model rewriting, e.g. the process of $\beta$-reduction.
## Related concepts
* [[type theory]], [[logic]]
* **2-type theory**, [[2-logic]]
[[directed homotopy type theory]]
* [[(∞,1)-type theory]], [[(∞,1)-logic]]
## References
On 2-type theory:
* [[R.A.G. Seely]], *Modeling computations: a 2-categorical framework*, LICS 1987 ([pdf](http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf))
* {#Garner09} [[Richard Garner]], *Two-dimensional models of type theory*, Mathematical structures in computer science 19.4 (2009): 687-736 ([doi:10.1017/S0960129509007646](https://doi.org/10.1017/S0960129509007646), [pdf](https://www.irif.fr/~mellies/mpri/mpri-ens/articles/garner-two-dimensional-models-of-type-theory.pdf))
* [[Tom Hirschowitz]], *Cartesian closed 2-categories and permutation equivalence in higher-order rewriting*. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2013, 9 (3), pp.10. ([pdf](https://hal.archives-ouvertes.fr/hal-00540205v2/document))
* [[Mike Shulman]], _[[michaelshulman:2-categorical logic]]_
* Philip Saville, *Cartesian closed bicategories: type theory and coherence*. PhD thesis, 2020. [pdf](https://arxiv.org/pdf/2007.00624.pdf).
Application to [[adjoint logic]] and [[modal type theory]]:
* [[Daniel Licata]], _Dependently Typed Programming with Domain-Specific Logics_ PhD Thesis (2011) ([pdf](http://www.cs.cmu.edu/~drl/pubs/thesis/thesis.pdf)) - chapter 7
* {#LicataShulman} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_, in _[Logical Foundations of Computer Science 2016](http://lfcs.info/lfcs-2016/)_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf), [slides](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint-lfcs-slides.pdf))
* [[Daniel Licata]], [[Mike Shulman]], [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))
See also:
* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 12 (2022) 1–14, [[doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334)]
[[!redirects type 2-theory]]
[[!redirects 2-dimensional type theory]]
|
2-vector bundle | https://ncatlab.org/nlab/source/2-vector+bundle |
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###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _2-module bundle_ / _2-vector bundle_ is a [[fiber ∞-bundle]] whose typical fiber is a [[2-module]]/[[2-vector space]].
## Definition
Let $R$ be a [[commutative ring]], or more generally an [[E-∞ ring]]. By the discussion at [[2-vector space]] consider the [[2-category]]
$$
2 Vect_R \simeq Alg_R
$$
[[equivalence of 2-categories|equivalent]] to that whose objects are [[associative algebras]] (or generally [[A-∞ algebra|algebras]]) $A$ over $R$, (being placeholders for the 2-vector space $A Mod$ which is the category of [[modules]] over $A$) whose [[1-morphisms]] are [[bimodules]] between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose [[2-morphisms]] are [[homomorphisms]] between those.
Under [[Isbell duality]] and by the discussion at _[Modules -- as generalized vector bundles](module#RelationToVectorBundlesInIntroduction)_ we may think of this 2-category as being that of (generalized) 2-vector bundles over a [[space]] called $Spec R$.
## Examples
* [[line 2-bundle]]
## Related concepts
* [[stringor bundle]]
* [[vector bundle]]
* [[principal 2-bundle]]
* [[BDR 2-vector bundle]]
* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]
## References
### Via completing $n$-tunles of vector bundles
See at *[[BDR 2-vector bundle]]*.
### As algebra bundles with bimodule bundles between them
{#AsAlgebraBundlesWithBimoduleBundlesBetweenThem}
The notion of 2-vector bundles based on regarding 2-vector spaces as algebras with bimodules between them ([here](2-vector+space#ReferencesAsAlgebrasWithBimodules)) is first discussed in
* [[Urs Schreiber]], [[Konrad Waldorf]], §4.4 of: *Connections on non-abelian Gerbes and their Holonomy*, Theory Appl. Categ., **28** 17 (2013) 476-540 [[arXiv:0808.1923](https://arxiv.org/abs/0808.1923), [tac:28-17](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html)]
and much further developed in
* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *The insidious bicategory of algebra bundles* [[arXiv:2204.03900](https://arxiv.org/abs/2204.03900)]
The example of the [[stringor bundle]]:
* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]
Reviewed in:
* [[Konrad Waldorf]], *The stringor bundle*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#WaldorfMar2023)]
[[!redirects 2-vector bundles]]
[[!redirects 2-module bundle]]
[[!redirects 2-module bundles]]
|
2-vector space | https://ncatlab.org/nlab/source/2-vector+space |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Content#
* table of contents
{:toc}
## Idea
The concept of a _$2$-vector space_ is supposed to be a [[vertical categorification|categorification]] of the concept of a [[vector space]]. As usual in the game of 'categorification', this requires us to think deeply about what an ordinary vector space really is, and then attempt to categorify that idea.
### What is a vector space?
There are at least three distinct conceptual roles which vectors and vector spaces play in mathematics:
1. A vector is a _column of numbers_. This is the way vector spaces appear in quantum mechanics, sections of line bundles, elementary linear algebra, etc.
1. A vector is a _direction in space_. Vector spaces of this kind are often the infinitesimal data of some global structure, such as tangent spaces to manifolds, Lie algebras of Lie groups, and so on.
1. A vector is an element of a [[module]] over the base [[ring]]/[[field]].
The first of these may be thought of as motivating the notion of
* [Kapranov-Voevodsky $2$-vector space](#KV2VectorSpace)
the second the notion of
* [Baez-Crans $2$-vector spaces](#BaezCrans2VectorSpaces)
the third the notion of
* [$2$-Modules](#AbstractApproach).
### Kapranov–Voevodsky $2$-vector spaces
{#KV2VectorSpace}
These were introduced in [Kapranov & Voevodsky 1991](#KapranovVoevodsky91).
The idea here is that just as a vector space can be regarded as a [[module]] over the [[ground field]] $k$, a $2$-vector space $W$ should be a [[category]] which is a [[monoidal category module]] with some nice properties (such as being an abelian category) over a suitable [[monoidal category]] $V$ which plays the role of the categorified ground field. There is then an obvious [[bicategory]] of such module categories.
In fact, Kapranov and Voevodsky defined a **Kapranov--Voevodsky $2$-vector space** as an abelian $\Vect$-module category equivalent to $\Vect^n$ for some $n$.
While this definition makes a lot of sense it does not give an abstract characterization of 2-vector spaces. That is, it is hardly different to simply defining a 2-vector space as a category equivalent to $Vect^n$.
Because Kapranov--Voevodsky $2$-vector spaces categorify the idea of a vector space as a 'state-space' of a system, they are the notion of $2$-vector space which feature on the right hand side of extended TQFTs (functors from higher [[cobordism]] categories to higher vector spaces).
An example of a Kapranov--Voevodsky $2$-vector space is $Rep(G)$, the category of [[representations]] of a finite group $G$.
### Baez–Crans $2$-vector spaces
{#BaezCrans2VectorSpaces}
These were explicitly described in [Baez & Crans 2004](#BaezCrans04).
A **Baez--Crans $2$-vector space** is defined as a [[internal category|category internal]] to [[Vect]]. They categorify the idea of a vector as a 'direction in space', and crop up when considering the _infinitesimal directions_ of a structure, such as in higher [[Lie theory]]. In fact, (following for instance from an extension of the [[Dold-Kan correspondence|Dold-Kan theorem]] by Brown and Higgins), [[strict omega-category|strict omega-categories]] internal to $\Vect$ are equivalent to chain complexes in non-negative degree and can be regarded as strict $Disc(k)$-$\infty$-modules. This allows to conceive much of [[homological algebra]] and many of the structures appearing in higher [[Lie theory]] -- for instance the definition of $L_\infty$-[[L-infinity-algebra|algebras]], as being about $\infty$-vector spaces. Regarding a chain complex as an $\infty$-vector space is useful conceptually for understanding the meaning of some constructions on chain complexes, while of course chain complexes themselves are well suited for direct computation with the $\infty$-vector spaces which they are equivalent to. (See also the remark about different notions of 2-vector spaces further below.)
They were also independently introduced and studied [Forrester-Barker (2004)](#Forrester-Barker).
## $2$-modules and $2$-linear maps as algebras and bimodules
{#AbstractApproach}
It is possible to conceive of 2-vector spaces of the Kapranov--Voevodsky and Baez--Crans type from a single unified perspective. Namely, by regarding the [[ground field]] itself as a [[discrete category]] we can think of it as a [[monoidal category]]. A $Disc(k)$-module category is a category whose space of objects and space of morphisms are both $k$-modules – ordinary vector spaces! – such that all structure morphisms (source, target, identity, composition) respect the $k$-action – hence are linear maps. These are categories internal to $\Vect_k$ which are equivalent to chain complexes of vector spaces concentrated in degree 0 and 1.
In other words, a Baez--Crans $2$-vector space can be thought of as a Kapranov--Voevodsky $2$-vector space, if one 'categorifies' the ground field by simply regarding it as a discrete monoidal category.
For $V$ a general symmetric [[closed monoidal category]] the full bicategory of all [[monoidal category modules]] over $V$ is in general hard to get under control, but what is more tractable is the sub-bicategory which may be addressed as the bicategory of $V$-modules _with basis_ namely the category $V-Mod$ in the sense of [[enriched category theory]] with
* objects are categories $C$ [[enriched category|enriched over]] $V$, to be thought of as placeholders for their categories of [[modules]], $Mod_C := [C,V]$
* morphisms $C \to D$ are [[bimodules]] $C^{op}\otimes D \to V$;
* $2$-morphisms are natural transformations.
Notice that all $V$-categories $Mod_C$ of modules over a $V$-category $C$ are naturally [[copower|tensored]] over $V$ and hence are [[monoidal category modules]] over $V$. In analogy to how a vector space $W$ (a $k$-module) is equipped with a basis by finding a set $S$ such that $W \simeq [S,k]$, we can think of a general [[monoidal category module]] $W$ over $V$ to be equipped with a basis by providing an [[equivalence]] $W \simeq [C,V]$, for some $V$-category $C$. In this sense $V-Mod$ is the category of $V$ 2-vector spaces with basis.
All of the examples on this page are special cases of this one.
### $\Vect$-enriched categories
According to the above a $Vect$-[[enriched category]] $C$ can be regarded as a basis for the $Vect$-module $Mod_C = [C,Vect]$. A $Vect$-enriched category is just an [[algebroid]]. If it has a single object it is an [[algebra]] and $Mod_C$ is the familiar category of modules over an algebra.
Notice that, by the very definition of [[Morita equivalence]], two algebras (algebroids) have equivalent module categories, and hence can be regarded as different bases for the same $\Vect$ $2$-vector space, iff they are Morita equivalent.
$Vect$-enriched categories as models for 2-vector spaces appear in
* Jacob Lurie, _On the classification of topological field theories_ ([pdf](http://www.math.harvard.edu/~lurie/papers/cobordism.pdf)) (see example 1.2.4)
* B. Toën, G. Vezzosi, _A note on Chern character, loop spaces and derived algebraic geometry_, ([arXiv](http://arxiv.org/abs/0804.1274), p. 6)
$2$-vector spaces in the sub-bicategory of algebras ($Vect$-enriched categories with a single object), bimodules and intertwiners are discussed in
* U. Schreiber, _AQFT from $n$-functorial QFT_ ([arXiv](http://arxiv.org/abs/0806.1079)) (appendix A)
and
* U. Schreiber and K. Waldorf, _Connections on non-abelian gerbes and their holonomy_ ([arXiv](http://arxiv.org/abs/0808.1923))
Some blog discussion of this point is at [2-Vectors in Trodheim](http://golem.ph.utexas.edu/category/2007/11/2vectors_in_trondheim.html).
### $Ch(Vect)$-enriched categories
More generally one can replace vector spaces by complexes of vector spaces and consider $Ch(Vect)\Mod$ as a model for the $2$-category of $2$-vector spaces (with basis): its objects are [[dg-category|dg-categories]].
It is argued in
* B. Toën, G. Vezzosi, _A note on Chern character, loop spaces and derived algebraic geometry_, ([arXiv](http://arxiv.org/abs/0804.1274), p. 6)
that the generalization from $Vect\Mod$ to $Ch(Vect)\Mod$ is necessary to have a good notion of higher sheaves of sections of 2-vector bundles, i.e. of higher coherent sheaves.
### Revisiting Kapranov–Voevodsky 2-vector spaces
Upon further restriction of $\Vect\Mod$ to 2-vector spaces whose basis is a _discrete category_, namely a set $S$ (or the $Vect$-enriched category over $S$ which has just the [[ground field]] object sitting over each element of $S$) one arrives at $Vect$-modules of the form
$$
[S, Vect] = Mod_{k^n} \simeq (Vect)^n
$$
(where $k^n$ denotes the algebra of diagonal $n\times n$-matrices). These are precisely Kapranov--Voevodsky $2$-vector spaces.
### Elgueta $2$-vector spaces
Another notion of 2-vector space which also includes Kaparanov--Voevodsky as particular instances is given in
* Josep Elgueta, _Generalized 2-vector spaces and general linear 2-groups_ ([arXiv](http://arxiv.org/abs/math/0606472))
The idea is to categorify the construction of a vector space as the space of finite linear combinations of elements in any set $S$. Instead of $S$, we start now with any category $C$, and take first the free $k$-linear category generated by $C$, and next the additive completion of this. Kapranov--Voevodsky $2$-vector spaces are recovered when $C$ is discrete. In some cases this gives nonabelian and even non-[[Karoubian category|Karoubian]] (i.e., nonidempotent complete) categories. This is the case, for instance, when we take as $C$ the one-object category defined by the additive monoid of natural numbers. The 2-vector space this category generates can be identified with the category of free $k[T]$-modules, which is nonKaroubian.
### Infinite-dimensional K-V 2-vector spaces
We can regard the objects of the $n$-dimensional Kapranov--Voevodsky $2$-vector space $Vect^n$ – which are $n$-tuples of vector spaces – as vector bundles over the finite set of $n$ elements. This has an obvious generalization to vector bundles over any topological space – in terms of modules these are the finitely generated projective modules of the algebra of continuous functions on this space. So categories of vector bundles can be regarded as infinite-dimensional 2-vector spaces. For the case that the underlying topological space is a _measure space_ such infinite dimensional K-V 2-vector spaces have been studied in
* John C. Baez, Aristide Baratin, [[Laurent Freidel]], Derek K. Wise, _Infinite-dimensional representations of 2-groups_ ([arXiv](https://arxiv.org/abs/0812.4969))
### Using a modular tensor category
The relevance of module categories as models for 2-vector spaces was apparently first realized in the context of [[conformal field theory]], where the monoidal category $V$ in question is a [[modular tensor category]]. A result by Victor Ostrik showed that _all_ $V$-module categories are equivalent to $Mod_A$ for $A$ some one-object $V$-enriched category (i.e., an algebra internal to $V$) in
* V. Ostrik, _Module Categories, weak Hopf Algebras and Modular Invariants_ ([arXiv](http://arxiv.org/abs/math.QA/0111139), [blog](http://golem.ph.utexas.edu/string/archives/000717.html))
## 2-Modules as modules over a 2-ring
One can go further and derive the identification of 2-modules and 2-linear maps with algebras and bimodules from a more fundamental notion of modules over [[2-rings]]. For the moment see there at
_[2-ring -- Compatibly monoidal presentable categories](2-rig#CompatiblyMonoidalPresentableCategories)_ for more details.
## Remark on the different notions of $2$-vector spaces
As the above list shows, there are 2-vector spaces of very different kind. There is not *the* notion of 2-vector space which is the universal right answer. Different notions of vector spaces are applicable and useful in different situations. This can be regarded as nothing but a more pronounced incarnation of the fact that already ordinary vector space appear in different flavors which are useful in different situations (real vector spaces, complex vector spaces, vector spaces over a [[finite field]], etc.)
For instance $Disc(k)$-module categories are crucial for higher [[Lie theory]] but 2-bundles with fibers $Disc(k)$-module categories are comparatively boring as far as general 2-bundles go, as they are essentially complexes of ordinary vector bundles. See
* [[Nils. A. Baas]], Marcel Bökstedt, Tore August Kro, _Two-Categorical Bundles and Their Classifying Spaces_, J. K-Theory, 10 (2012) 299 - 369, with a preliminary version at ([arXiv](http://arxiv.org/abs/math/0612549))
## $2$-Hilbert spaces##
2-vector spaces have to a large extent been motivated by and applied in (2-dimensional) [[quantum field theory]]. In that context it is usually not the concept of a plain vector space which needs to be categorified, but that of a Hilbert space.
2-Hilbert spaces as a $\Hilb$-[[enriched category|enriched categories]] with some extra properties were discribed in
* John Baez, _Higher-Dimensional Algebra II: 2-Hilbert Spaces_ ([arXiv](http://arxiv.org/abs/q-alg/9609018)) .
In applications one often assumes these 2-Hilbert spaces to be [[semisimple algebra|semisimple]] in which case such a 2-Hilbert space is a Kapranov--Voevodsky $2$-vector space equipped with extra structure.
A review of these ideas of 2-Hilbert spaces as well as applications of 2-Hilbert spaces to finite group representation theory are in
* Bruce Bartlett, _On unitary 2-representations of finite groups and topological quantum field theory_ ([arXiv](http://arxiv.org/abs/0901.3975))
## Properties
### Tannaka duality
[[!include structure on algebras and their module categories - table]]
## Related concepts
* [[vector space]]
* [[module category]], ("[[actegory]]")
* **2-vector space**, [[2-representation]]
* [[2-ring]]
* [[2Mod]]
* [[2-vector bundle]]
* _[[TwoVect]]_ is a Mathematica software package for computer algebra with 2-vector spaces
* [[n-vector space]]
## References
### As $n$-tuples of vector spaces
{#ReferencesAsnTuplesOfvectorSpaces}
The notion of 2-vector spaces as $n$-tuples of vector spaces is due to
* {#KapranovVoevodsky91} [[Mikhail Kapranov]], [[Vladimir Voevodsky]], *$2$-categories and Zamolodchikov tetrahedra equations* in *Algebraic groups and their generalization: quantum and infinite-dimensional methods*, University Park, PA (1991) (eds: W. J. Haboush and B. J. Parshall), Proc. Sympos. Pure Math. 56 (Amer. Math. Soc., Providence RI 1994), pp. 177-259 [[pdf](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/1994_Kapranov_Voevodsky.pdf)]
### As 2-term chain complexes
{#ReferencesAsTwoTermChainComplexes}
The notion of 2-vector spaces as 2-term [[chain complexes]] is due to
* {#BaezCrans04} [[John C. Baez]], [[Alissa S. Crans]], §3 of: *Higher-Dimensional Algebra VI: Lie 2-Algebras*, Theor. Appl. Categor. **12** (2004) 492-528 [[arXiv:math/0307263](https://arxiv.org/abs/math/0307263)]
and used in
* {#Forrester-Barker} [[Magnus Forrester-Barker]], *Representations of Crossed Modules and $Cat^1$-Groups*, PhD thesis U. Wales Bangor (2004) [[pdf](http://www.maths.bangor.ac.uk/research/ftp/theses/forrester-barker.pdf)]
### As algebras with bimodules between them
{#ReferencesAsAlgebrasWithBimodules}
The notion of 2-vector spaces with 2-linear maps between them as algebras with bimodules between them (subsuming the definition in [Kapranov & Voevodsky 1991](#KapranovVoevodsky91) as the special case of algebras that are [[direct sums]] of the [[ground field]]) is due to
* [[Urs Schreiber]], §A of: *AQFT from n-functorial QFT*, Commun. Math. Phys. **291** (2009) 357-401 [[arXiv:0806.1079](https://arxiv.org/abs/0806.1079), [doi:10.1007/s00220-009-0840-2](https://doi.org/10.1007/s00220-009-0840-2)]
following earlier discussion in
* [[Urs Schreiber]], *[2-vectors in Trondheim](https://golem.ph.utexas.edu/category/2006/10/topology_in_trondheim_and_kro.html)* (2006)
* [[Urs Schreiber]], *[Topology in Trondheim and Kro, Baas & Bökstedt on 2-vector bundles](https://golem.ph.utexas.edu/category/2007/11/2vectors_in_trondheim.html)* (2007)
which is picked up in
* [[Urs Schreiber]], [[Konrad Waldorf]], §4.4 of: *Connections on non-abelian Gerbes and their Holonomy*, Theory Appl. Categ., **28** 17 (2013) 476-540 [[arXiv:0808.1923](https://arxiv.org/abs/0808.1923), [tac:28-17](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html)]
and further developed into a theory of [[2-vector bundles]] (via algebra bundles with bundles of bimodules between them) in:
* {#KristelLudewigWaldorf22} [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *The insidious bicategory of algebra bundles* [[arXiv:2204.03900](https://arxiv.org/abs/2204.03900)]
Essentially the same notion also appears, apparently independently, in:
* {#FreedHopkinsTeleman09} [[Daniel S. Freed]], [[Michael J. Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], §7.1 with Ex. 2.13 in: *[[Topological Quantum Field Theories from Compact Lie Groups]]* [[arXiv:0905.0731](https://arxiv.org/abs/0905.0731)]
The notion is reviewed in a list of "standard" definitions in [BDSPV15](#BDSPV15), without however referencing it.
See also at
* _[[geometry of physics]]: [[geometry of physics - modules]]_
the section on 2-modules.
### Further discussion
Review includes
* {#BDSPV15} [[Bruce Bartlett]], [[Christopher L. Douglas]], [[Chris Schommer-Pries]], [[Jamie Vicary]], *A bestiary of 2–vector spaces*, Appendix A in: *Modular categories as representations of the 3-dimensional bordism 2-category* [[arXiv:1509.06811](https://arxiv.org/abs/1509.06811)]
Another definition of 2-modules over [[2-rings]] (see there for more) is in
* {#CJF} [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))
A treatment of [[2-representations]] of [[Lie 2-groups]] is in
* Zhen Huan, _2-Representations of Lie 2-groups and 2-Vector Bundles_ ([arXiv:2208.10042](https://arxiv.org/abs/2208.10042))
[[!redirects 2-vector spaces]]
[[!redirects 2-module]]
[[!redirects 2-modules]]
|
2008 changes | https://ncatlab.org/nlab/source/2008+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during 2008.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/20) and work backwards.
***
* [[Toby Bartels|Toby]] (2008-12-31): I have a precise definition of [[evil]], so I put that in.
* [[Urs Schreiber|Urs]] (Dec 31): started [[FQFT]], currently just a commented list of literature trying to give an impression of the idea. Also adjusted [[quantum field theory]] and created [[vertex operator algebra]].
* [[Urs Schreiber|Urs]] (Dec 30): started entries on [[groupoidification]] and [[quantum field theory]].
* [[John Baez|John]] (Dec 29): I'm going to make a little entry about the term [[evil]].
* [[Toby Bartels|Toby]] (Dec 28): I added a bunch of stuff about [[apartness relation|apartness relations]]. (Most of it is probably even correct.) I tried to say as much as I could in terms of enriched category theory, but the topic is really analysis.
* [[John Baez|John]] (Dec 23): I expanded the entry on [[action groupoid]] and started one for [[action]] and [[automorphism]]. In the process I noted that Urs seems to be using $Sets$ for the category of sets, while most mathematicians use [[Set]]. We should settle (pardon the pun) this issue. I hope that someday there will be lots of entries on specific important categories, listing some of the special features of these categories.
* [[Urs Schreiber|Urs]] (Dec 23): continued completing "lits of related entries" in the main entries linked to from the [[HomePage]]: did [[Lie theory]] and [[differential geometry]]
* [[Urs Schreiber|Urs]] (Dec 23): started creating [[nonabelian algebraic topology]] and [[crossed complex]] and expanded a bit related entries linked to from there. Am hoping we can eventually present more details here.
* [[Urs Schreiber|Urs]] (Dec 22): started replying to a question [[Eric Forgy|Eric]] raised at [[General Discussion]] about the shape of cells used in higher categories. Created [[geometric shapes for higher structures]] with a discussion of the general idea and linked to [[globular set]], [[simplicial set]] and [[cubical set]] from there. Still plenty of room to add details. Am planning to give details on the monoidal structure on cubical sets and the induced Crans-Gray tensor product on $\omega$-[[omega-category|categories]].
* [[Urs Schreiber|Urs]] (Dec 20): I started to add alphabetical lists of all related entries to the big headline entries that are linked to from the sidebar
of the [[HomePage]]. So far I did this in [[category theory]], [[sheaf and topos theory]], [[foundations]] and [[higher category theory]]. It might be good if each time we create a new entry we think about adding the correspondin link to one of these general lists.
* [[Urs Schreiber|Urs]] (Dec 20): I replied to the discussion at [[monoidal category]] on "Where does all this structure and coherence come from?" by adding a subsection on how all this comes from the general concept of $\infty$-[[lax functor]] for which one formalization can be given in terms of [[oriental]]s and Street's [[descent and codescent|descent]]. The pentagon identity is precisely the fourth [[oriental]]!
* [[Toby Bartels|Toby]] (Dec 20–23): I won\'t do anything, since I\'ll be travelling. (Just in case you wonder where I am.)
* [[Urs Schreiber|Urs]]: Thanks for the info, Toby. Myself, I will probably be online sporadically over the holidays.
* [[John Baez|John]] (Dec 20): I sketched the definitions of [[monoidal category]], [[braided monoidal category]], and [[symmetric monoidal category]] and gave links to full definitions and explanations. Is someone good enough at drawing diagrams in this environment to draw the necessary diagrams --- e.g. pentagon and hexagon identity?
* [[Toby Bartels|Toby]] (Dec 20): More on [[Grothendieck topos]] and [[well-pointed topos]], including definitions that are correct in constructive mathematics.
* [[Urs Schreiber|Urs]]: recently, after [[Eric Forgy|Eric]] had requested an "arrow-theoretic" description of [[category algebra]] and of [[graded vector space]]s, I had provided a description in terms of [[bi-brane]]s and remarked that this is part of a bigger picture. That bigger picture, together with these examples, is now described over in my private area at [[schreiber:Nonabelian cocycles and their quantum symmetries]]. For completeness I have added links to that to [[category algebra]] and [[graded vector space]]. Instead of giving the impression of forcing this relation to my work upon these entries, I would like you to understand this as an invitation to check and, if necessary, criticize.
* [[Urs Schreiber|Urs]]: I tried to flesh out [[tensor product]] and [[module]] a bit further, providing more details and attempting to bring the different points of view together. Check.
* [[Toby Bartels]]: I just expanded [[tensor product]], enough to include the example that links to it (without circularity).
* [[Urs Schreiber|Urs]]: I expanded on the entry [[action groupoid]] that [[Eric Forgy|Eric]] started.
***
**First list** --- No previous list --- [[2009 January changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 April changes | https://ncatlab.org/nlab/source/2009+April+changes | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during April 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/632) and work backwards.
***
# 2009-04-30 #
* [[Urs Schreiber|Urs]]:
* continued filling in material at [[geometric infinity-function theory]] -- am hoping that my co-journalists will eventually start helping me there
* replied to [[David Corfield|David]] at [[why (infinity,1)-categories?]]
* added
* [[(infinity,1)-category of (infinity,1)-presheaves]]
* [[(infinity,1)-category of (infinity,1)-functors]]
* [[(infinity,1)-category of (infinity,1)-categories]]
* [[David Corfield|David]]:
* asked question at [[why (infinity,1)-categories?]]
* [[Urs Schreiber|Urs]]:
* added one more paragraph to [[why (infinity,1)-categories?]]
* [[David Corfield|David]]:
* asked question at [[context]]
* [[Urs Schreiber|Urs]]:
* expanded [[presentable (infinity,1)-category]] by adding the theorem that every presentable $(\infty,1)$-category is indeed presented by a combinatorial simplicial model category.
* [[David Corfield|David]]:
* asked questions at [[exponential object]]
* [[Urs Schreiber|Urs]]:
* started [[localization of a simplicial model category]]
* created [[enriched model category]] and [[simplicial model category]]
# 2009-04-29 #
* [[Mike Shulman|Mike]]:
* Moved the discussion about comma categories from this page to the query box at [[comma category]].
* Will look at [[descent]] when I get a chance.
* Created a couple of firefox search plugins for the nLab and uploaded them to the [[HowTo]].
* [[Urs Schreiber|Urs]]:
* based on [[Zoran Skoda|Zoran]]'s references at [[enhanced triangulated category]] I created [[pretriangulated dg-category]] and [[twisted complex]], but then ran out of steam
* moved a bit of material from [[derived infinity-stack]] to [[derived stack]] and then made [[derived infinity-stack]] a redirect to [[derived stack]]
* finally wrote at least a blurb at [[stack]], only to make it look less orphaned in between [[sheaf]] and [[(infinity,1)-sheaf]].
* created [[Verdier site]]
* further fine-tuned the [DHI](http://arxiv.org/PS_cache/math/pdf/0205/0205027v2.pdf)-review at [[descent]]: now I dropped the discussion of homotopy limits entirely, as it's not really necessary; but I did include for a smoother presentation the assumption that we are on a [[Verdier site]], so that hypercovers "split" ([section 9](http://arxiv.org/PS_cache/math/pdf/0205/0205027v2.pdf#page=29)) which happens to be a Reedy fibrancy kind of condition after all ([page 11](http://arxiv.org/PS_cache/math/pdf/0205/0205027v2.pdf#page=11))
* I browsed a bit through Dominic Verity's work and created entries on [[stratified simplicial set]], [[complicial set]], [[weak complicial set]], [[simplicial weak omega-category]] and [[Verity-Gray tensor product]] -- my main motivation was the claim now recounted at [[stratified simplicial set]] that the $\omega$-nerve on strict $\omega$-categories with values in $Strat$ has a strong monoidal left adjoint
* [[Zoran Škoda]]: created [[enhanced triangulated category]]
* [[Urs Schreiber|Urs]]:
* filled the "details" section at [[descent for simplicial presheaves]] with the relevant material copy-and-pasted from [[descent]].
* keep polishing, expanding and rearranging [[descent]] -- when [[Mike Shulman|Mike]] comes back online I am hoping to discuss a bit more the relation between Street's descent for $Str \omega Cat$-valued presheaves and the standard descent for their SSet-valued image under the $\omega$-nerve. As the new version indicates: the homotopy limit may be a red herring and the lack of monoidalness of the left adjoint of the nerve might be fixed by recourse to stratified simplicial sets using Verity's results (?)
* added a pullback description to [[double comma object]]
* [[Mike Shulman|Mike]]:
* I've noticed that Urs (in particular) has been widely using the notation $(f,g)$ for [[comma category|comma categories]] that Toby tried to prevent me from denigrating. But I haven't heard any convincing arguments in favor of that notation, and I have lots of reasons to dislike it; see the discussion at [[comma category]]. If general opinion is against me, I'll shut up, but I want to hear from more people. If you use the notation $(f,g)$, do you have positive reasons to prefer it to $(f/g)$ or $(f\downarrow g)$?
# 2009-04-28 #
* [[Mike Shulman|Mike]]:
* I have a question about [[biproduct]]s.
* [[Urs Schreiber|Urs]]:
* created [[Bousfield-Kan map]]
* created [[category of simplices]]
* following [[Toby Bartels|Toby]]'s suggestion I moved [[descent and codescent]] to [[descent]] -- then I entriely rewrote it! Now it starts with very general nonsense on localization of $(\infty,1)$-presheaves and then derives descent conditions as concrete realizations of that localization. Currently where it ends I am planning to add discussion about how to further get from descent to gluing conditions (i.e. to $\Delta$- and oriental-weighted limits) following discussion that I am having with [[Mike Shulman|Mike]] on the blog [here](http://golem.ph.utexas.edu/category/2009/04/journal_club_geometric_infinit.html#c023460)
* added more details to [[ind-object]], relating the two different definitions
* added standard examples of presheaves on open subsets to [[inverse image]]
* started adding a list "properties" to [[colimit]] analogous to the one at [[limit]]
* [[Toby Bartels]]:
* Since I\'m making up a word at [[exponential object]], I decided that both terms should be adjectives, leaving the verb simpler.
* I included some lower-dimensional cases at [[associahedron]]. I managed to get down to $K_1$.
# 2009-04-27 #
* [[Urs Schreiber|Urs]]
* added a list of basic propeties to the end of [[limit]]
# 2009-04-26 #
* [[Mike Shulman|Mike]]:
* Added a couple new examples, and tried to uniformize the descriptions, at [[A-infinity-operad]].
* Created [[indiscrete category]].
# 2009-04-25 #
* [[Urs Schreiber|Urs]]:
* added a discussion of hom-objects in terms of homotopy limits at [[simplicial presheaf]] (in a new section "Properties")
* created [[reflective (infinity,1)-subcategory]] and [[localization of an (infinity,1)-category]] and [[local object]]
* created [[hypercompletion]] and [[descent for simplicial presheaves]]
* created [[associahedron]] (see the discussion with [[Jim Stasheff]] over at the blog, [here](http://golem.ph.utexas.edu/category/2009/04/categorification_and_topology.html#c023441))
* expanded [[A-infinity-operad]]: a bit about associahedra, but mostly more detailed links to references
# 2009-04-24 #
* [[Urs Schreiber|Urs]]
* created [[A-infinity operad]] and [[category over an operad]] and gave the operadic definition at [[A-infinity category]]
* [[Toby Bartels]]:
* Changed the numbering at [[k-tuply connected n-category]] but haven\'t moved the page yet.
* Created a new [category: lexicon](/nlab/list/lexicon) to find Tim\'s lexicon entries. (If these are incorporated into more mainstream $n$Lab entries, then we could get rid of this category, but for now it\'s nice to be able to track them down without running through the whole list.)
* I answered Mike\'s question at [[Boolean topos]].
* I coined a new term at [[exponential object]].
* I finished [[dependent product]].
* [[Urs Schreiber|Urs]]:
* came across the useful interrelation diagram and associated literature list on "enhancements of triangulated categories" [here](http://www.math.uni-bonn.de/people/schwede/EnhancedSeminar.pdf) and added this to [[stable (infinity,1)-category]], together with some links it suggests, to entries still to be created -- I am hoping we'll eventually be able to accumulate a good collection of material on this topic
* added Lie $\infty$-links to [[rational homotopy theory]]
* added missing links back and forth between [[Yoneda extension]] and [[free cocompletion]]
* added to [[generalized universal bundle]] the remark that it is a means to compute the "lax pullback" (really: [[comma object]]) of a point.
* added at [[category of elements]] the equivalent definition in terms of comma category and in terms of pullbacks of the universal Set-bundle -- and in terms of "lax pullback" (comma object) of the point
* added a link to [[category of elements]] at [[ind-object]] where we recently had a discussion about this point
* added the discussion and diagram at [[comma category]] characterizing it as a [[comma object]]
* added link and remark to [[comma object]] and [[comma category]] at [[co-Yoneda lemma]]
* made the comma category explicit on which a [[simplicial local system]] is a functor
* [[Tim Porter|Tim]]: I have created a few entries relating to the interaction of [[local system]] with ideas from rational homotopy theory, especially algebras of differential forms on simplicial sets, based on Sullivan and further back Thom and Whitney. These included [[simplicial local system]], see Urs comment below, to which I have started replying. Perhaps I will be able to add more shortly. These entries are not yet finished and do not yet deal with the Sullivan-Thom-Whitney stuff.
* [[Urs Schreiber|Urs]]: have some questions at [[simplicial local system]]
# 2009-04-23 #
* [[Mike Shulman|Mike]]: Since no one objected to my proposal on how to resolve the duplication between [[category of fractions]] and [[multiplicative system]], I implemented it. The relevant material is now at [[calculus of fractions]]. I deleted a bit of the material about derived functors because it was not really specific to calculi of fractions, belonging more at [[derived functor]].
* [[Toby Bartels]]: A question (not a dispute!, what do you know?) on terminology at [[exponential object]].
* [[Mike Shulman|Mike]]:
* Disagreed with Toby at [[k-tuply connected n-category]]: I think connectedness should be 0-connectedness, not '1-tuple' connectedness.
* [[Urs Schreiber|Urs]]:
* created [[path groupoid]] -- other realizations of that idea should be stated there, too
* started creating a random list of some examples at [[limits and colimits by example]] -- but not in the intended detailed form yet
* started reworking [[local system]] as we discussed there -- still lots of room for improvement left, of course!, in particular many references could use more details and links
* notice that I moved the discussion box to the end of the entry; _not_ to imply that the discussion is closed, but so that the box does not disturb the structure of the entry; I think this is reasonable for every discussion that concerns an entry as a whole more than a particular point in it
* [[Toby Bartels]]:
* From our first spam post (which I will not link to), I\'d like to preserve this: 'with a country like India, the [[limit]]s are [[end]]less' (links added).
* Wrote [[k-tuply connected n-category]]. I also made the numbering consistent on the other pages; connectedness should be $1$-tuple connectedness, not $0$-tuple connectedness.
* Tried my hand at a constructive version of [[cyclic order]]. Interestingly, the apartness relation doesn\'t seem to be recoverable from the cyclic order relation (which is related to the difficulty of doing antisymmetry and thus a reflexive version, which as well is not ---even classically--- equivalent to the negation of the irreflexive version).
* [[Urs Schreiber|Urs]]:
* have a reply at [[local system]]
* found time to work a bit more on [[Lie infinity-algebroid representation]]
* request to [[Tim Porter|Tim]] and others: there is naturally a kind of "twisting function" appearing there. One aim of my presentation is not to postulate it, but to derive that it arises from a (co)fibration of DGCAs. I am thinking that all the [[twisting function]]s and [[twisting cochain]]s should have such a _conceptual_ definition, from which one derives the usual component-wise definition by unravelling the component-wise mechanisms. I think here on the $n$Lab we should try to find and give conceptual categorical explanations as far as possible. My personal feelling is that all discussion of "twisting cochains" and related phenomena would become considerably clearer and less confusing if one had this.
* [[Toby Bartels]]: Please note that there are no actual links to [[Differential Nonabelian Cohomology]] (except this one just now). That there appear to be is actually a bug in Instiki (which I haven\'t bothered to report to Jacques yet).
* [[Tim Porter|Tim]]:
* I have added a request at [[local system]]. Basically the current entry reads as if it related to a relatively recent idea. I suggest we look at the origins of the idea, at least as old as 'Steenrod (1943)' if not before. It is central to much of the nLab work. Probably we need to be much less restrictive in the motivation of this entry.
* I have added some historical and motivational perspective in [[twisting function]] and would suggest that a similar section is needed for [[twisting cochain]]. The two threads of twisting the fibre and _deforming_ the local structure of a 'product' are at the origin of both concepts.
# 2009-04-22 #
* [[Urs Schreiber|Urs]]: created [[Lie infinity-algebroid representation]] -- but ran out of time before done with polishing
* [[Mike Shulman|Mike]]: Created [[cyclic order]] in order to propose a clean definition of the [[Connes' cyclic category|cycle category]].
* [[Zoran Škoda]]: [[quasicompact]]
# 2009-04-21 #
* [[Zoran Škoda]]: created [[quasicoherent sheaf]],[[kernel functor]], [[Gabriel composition of filters]], [[Gabriel filter]], [[uniform filter]], [[Serre subcategory]]. Corrected [[Gabriel multiplication]], thanks Toby. Created [[ringed space]] differing from [[ringed site]].
* [[Urs Schreiber|Urs]]: edited [[geometric infinity-function theory]] to go along with [this](http://golem.ph.utexas.edu/category/2009/04/journal_club_geometric_infinit.html#c023356) blog message
* [[Toby Bartels]]:
* Started to expand on [[dependent product]], but I need to leave now without finishing, so I\'ll let Mike do it. (^_^)
* A request at [[fibered n category]].
* There\'s an obvious typo at [[Gabriel multiplication]], but I don\'t know which way to fix it.
* [[Urs Schreiber|Urs]]:
* have two questions on examples at [[semi-abelian category]]
* created [[dependent product]] just to satisfy the link from [[universe in a topos]]
* added a list with a handful of general properties to [[adjoint functor]]
* moved the old discussion at [[representable functor]] to the bottom of the page
* [[Zoran Škoda]]: I have made [[D-module]] and [[local system]] somewhat more precise; actually I have put lots of more precise statements; the subtleties on wheather we work over a complex manifold, variety, variety in char zero, or nonsingular variety in char 0, may affect some of the statements. To suplement this I was forced to create a comprehensive entry [[regular differential operator]]. I see that for some reason people continue talking connections and
avoid going down to sheaves, resolutions of diagonal, de Rham site and regular differential operators, which are all necessary to cover this subject properly in my view; there are missing related items like [[holonomic D-module]], treatment of costratification, crystals and so on. I created [[coreflective subcategory]], just giving the definition and saying that the rest of abstract preoprteis are dual to [[reflective subcategory]] where more is written. But one should write specific examples which call specifically for coreflective subcategories. Created [[topologizing subcategory]], [[thick subcategory]] and [[Gabriel multiplication]] in the generality of abelian categories (one should add the proper discussion of thick, topologizing in triangulated and suspended categories, and Gabriel mult. for filters, but I run out of energy for today); one needs to add entry on Serre subcategory which is easy in module and Grothendieck categories, but more subtle in general abelian categories. There is a query under [[fibered n category]], I think we should have both entry [[fibered n category]] and [[n-fibration]]; the first entry dedicated to STRICT n-categories and consequently strict universal properties and the name due Grothendieck, Gabriel, Gray and Hermida; and the latter in weak version and with homotopy style nomenclature accordingly. My praise for creative expansion to Mike.
* [[David Roberts]]:
* Added examples to [[semi-abelian category]] - the opposite of the category of pointed objects in a topos, and the category of crossed modules.
# 2009-04-20 #
* [[Urs Schreiber|Urs]]:
* created stub for [[local system]] in the context of a comment I left [here](http://sbseminar.wordpress.com/2009/04/20/three-ways-of-looking-at-a-local-system-introduction-and-connection-to-cohomology-theories/)
* created [[derived stack]] to go along with our [Journal Club activity](http://golem.ph.utexas.edu/category/2009/04/journal_club_geometric_infinit.html)
* [[Toby Bartels]]: I think that the naming discussion at [[ind-object]] is still current until Eric is happy.
* [[Urs Schreiber|Urs]] responds: sure, I just thought for readability the discussion would be better had at the bottom of the page -- a big green box in the middle of the entry gives the reader the impression that he or she needs to beware of some urgent unsettled issue before reading on.
* _Toby_: Ah, whereas I see the big green box up top as simply indicating an unsettled issue, urgent or otherwise.
* _Urs_: okay, that makes sense, let's leave it this way -- maybe eventually we need some mechanism to decide when to move a disscussion box to the bottom of the page -- probably the mechanism should be that the one who started the discussion is the one decide
* [[Finn Lawler]]: Replied to Toby at [[minimal logic]], and slightly expanded [[paraconsistent logic]] to incorporate some of our discussion.
* [[Urs Schreiber|Urs]]:
* added to [[Yoneda lemma]] the following: a word on the proof, two further corollaries, a word on the meaning of the first two corollaries
* [[Toby Bartels]]: Put [[well-order|simulations]] everywhere. I need to think about this (and read Aczel) some more, but I\'m fairly sure that Mike is right about them.
* [[Urs Schreiber|Urs]]:
* started polishing the typesetting at [[bundle gerbe]], but there is still plenty of room for further improvement
* added a summary list to the section "Example: universes in SET" at [[universe in a topos]]
* [[Mike Shulman|Mike]]:
* Added the correct equivalent definition to [[semi-abelian category]] and deleted the discussion about what it might be.
* Added another version of [[Cantor's theorem]].
* Expanded [[extensional relation]] to discuss several possible notions of extensionality, reserving the term 'extensional' for the one that I think is most important (but feel free to disagree).
* Commented on the right notions of morphism for [[relation]], [[well-founded relation]], [[well-order]], and [[extensional relation]].
* [[David Corfield|David]]:
* imported Urs' material on [[bundle gerbe]]. nLab doesn't seem to like latex within lists. How do we fix this?
* [[Urs Schreiber|Urs]]:
* created [[limits and colimits by example]] for purposes discussed on the blog [here](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c023321) and [here](http://golem.ph.utexas.edu/category/2009/04/graphical_category_theory_demo.html#c023323)
* [[Toby Bartels]]:
* Yet more at [[pure set]]. See the pretty pictures! Or rather ... make them pretty if you know how, for I do not. (;_;)
* Another question for Finn, now at [[minimal logic]].
* More correct material at [[pure set]].
# 2009-04-19 #
* [[Toby Bartels]]: I\'ve started a dispute at [[paraconsistent logic]].
+--{: .query}
[[Finn Lawler|Finn]]: No, you haven't -- I was wrong.
=--
# 2009-04-18 #
* [[Finn Lawler]]:
* created [[sequent calculus]] and [[paraconsistent logic]]: just brief explanations and links to Wikipedia.
* [[Urs Schreiber|Urs]]:
* created a stub for [[differential cohomology]]
# 2009-04-17 #
* [[Zoran Škoda]]: created [[comodule]], [[flat module]], [[cotensor product]].
* [[Toby Bartels]]: Added information on morphisms to [[relation]], [[extensional relation]], [[well-founded relation]], and [[well-order]]. I hope that all of my claimed theorems are true, in which case I\'m sure that all of my proposed definitions are good. (^_^)
* [[David Corfield|David]]
* made a request at Mike's [categorified logic](http://ncatlab.org/michaelshulman/show/categorified+logic)
* created [[D-module]]
* created [[coherent logic]]
* added John's blog exposition to [[induced representation]]. Is the style OK?
* [[Urs Schreiber|Urs]]:
* expanded [[hom-set]] a bit
* remarked at [[semi-abelian category]] that the now deprecated second equivalent definition was taken directly from the (single) reference we give. I think instead of just removing it we should try to correct it.
* [[Toby Bartels]]:
* Wrote [[axiom of extensionality]] and [[transitive closure]] while writing the below.
* Added the more general definition to [[extensional relation]]. It seems right to me and I can prove that it\'s correct for well-founded relations, but I must admit that I made it up just now.
* More details about how to model [[pure set]]s in structural set theory, applying the above.
* Wrote [[Cantor's theorem]], including a constructive version from Paul Taylor.
# 2009-04-16 #
* [[Zoran Škoda]]: created [[Tohoku]], [[quasi-pointed category]], made changes to [[sheafification]],[[additive and abelian categories]], [[torsion]], [[torsion subgroup]]
* [[Mathieu Dupont|Mathieu]]:
* corrected [[semi-abelian category]] and answered to Mike
* [[Urs Schreiber|Urs]]:
* added some links to [[Tim Porter|Tim]]'s latest addition to [[rational homotopy theory]] -- in that context I created [[torsion]] and [[torsion subgroup]]
* finally created [[exact sequence]] which was requested by a bunch of entries
* added to the "Idea" section of [[Grothendieck category]] a statement suggesting that these are _precisely_ those additive categories for which sheafification exists. Is that right?
* [[Zoran Skoda|Zoran]] kindly looked up the definition of AB6-category which was still missing at [[additive and abelian categories]], I put it in now
* [[Tim Porter|Tim]]:
* I created the next entry in the rational homotopy lexicon series with the ungainly title [[differential graded algebras and differential graded Lie algebras-relationships]].
* I added another viewpoint to [[rational homotopy theory]] which is more in keeping with Quillen's 1969 paper.
* [[Urs Schreiber|Urs]]:
* created [[GUT]] and [[induced representation]] as places for collection of material currently discussed on the blog
* included [[Todd Trimble|Todd]]'s proof of MacLane's co-Yoneda into [[co-Yoneda lemma]]
* tried to bring [[A-infinity-category]] into some shape by adding more introductory discussion and ordering the references a bit -- also have a question
* [[Tim Porter|Tim]]:
* Replied to [[Urs Schreiber|Urs]] at [[bar and cobar construction]] (at least I hope the reply goes some way to answering the query).
* Created [[reduced suspension]] as I needed it for my 'reply' above.
* [[Urs Schreiber|Urs]]:
* replied to [[David Roberts|David]] at [[ind-object]]
* have a request and a question at [[bar and cobar construction]]
* [[David Roberts]]:
* Added a comment in section generalizations at [[ind-object]], re comma category.
# 2009-04-15 #
* [[Zoran Škoda]]: created [[fibered n category]], [[Karoubian category]], [[pseudo-abelian category]] (redirect), [[Koszul duality]], [[pure motive]], [[Voevodsky motive]], [[motives and dg-categories]] (there needs to be a separate big entry on [[mixed motive]], then [[motivic complexes]], [[standard conjectures]], [[Hodge filtration]] etc.), [[element in abelian category]]; created book-page [[Gray-adjointness-for-2-categories]]; additions and references to [[A-infinity category]], [[dg-category]], [[twisting cochain]].
* [[Urs Schreiber|Urs]]:
* created [[ind-object in an (infinity,1)-category]]
* created [[cofinal functor]] and [[cofinally small category]], but just the bare defintion so far
* expanded [[ind-object]]: added more motivation in "Idea" section, added examples, added list of properties
# 2009-04-14 #
* [[Urs Schreiber|Urs]]:
* added a discussion to [[universe in a topos]] with more details on how to get back the Grothendieck universe axioms in $SET$. Please check. I'd be grateful for improvement.
* to Andrew: I think we want here on the $n$Lab as much detail as we can get hold of -- if an entry becomes too long, though, it might be an option to split off entries from it "more details on xyz" or the like and link to them
* [[Andrew Stacey|Andrew Stacey]]: Added the basic definitions to [[Chen space]] and some of the other variants of [[generalized smooth space|generalised smooth space]]. How much detail do we want on these pages?
* [[Andrew Stacey|The English Pedant]]: I'm not convinced about the use of the word "heuristic" on the n-Lab. I've started a discussion on the [n-Forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=17) rather than force my views on the English language on the n-Lab. I realise that this is a little against the Wiki-spirit but I figured that if I went through changing all occurrences of the word "heuristic" then someone would object and we'd have a discussion about it; so to save a bit of agro, I'm instigating the discussion first.
* [[Toby Bartels]]: I started a new category, `foundational axiom`, in which I put the pages that contain axioms that one might (or might not) want in one [[foundations]] of mathematics. That way, anyone with opinions on the matter can check them to see that one\'s views are represented. (A few don\'t really have much in the way of an axiom right now, but one could be added or noted there.) This does not include things like the axioms for a [[group]], but rather axioms for [[set theory]] (or other foundational theory). (Although [[set theory]] itself I don\'t think should really be included, but [[ETCS]] is in there.)
#2009-04-13
* [[Mike Shulman|Mike]]: Did some massaging of [[well-order]], [[well-ordering theorem]], [[well-founded relation]], [[choice object]], and [[axiom of choice]], and created [[extensional relation]].
* [[Zoran Škoda]]: completed the definition of [[congruence]].
#2009-04-12
* [[Zoran Škoda]]: Today a Hopf algebraist resurrected in me...grrr...[[Hopf action]], [[measuring]], [[Heisenberg double]], [[bimonoid]], [[module algebra]], [[comodule algebra]]. Well, I am writing an article about a special case of a Heisenberg double, maybe that is why.
#2009-04-11
* [[Toby Bartels]]:
* Wrote [[axiom of foundation]], [[well-founded relation]], and [[well-ordering theorem]].
* Added to [[successor]], [[inaccessible cardinal]], [[well-order]], [[partial function]], [[ordinal number]], and [[cardinal number]].
* Commented on [[direct sum]], [[presentable (infinity,1)-category]], [[stable (infinity,1)-topos]], [[sequential convergence space]], and [[quantale]].
* Moved material from [[Cat]] to [[n-category]].
* Moved [[ordinal]] to [[ordinal number]], partly because it\'s more noun-like and partly to match [[cardinal number]].
* Moved $1$-categorial material from [[compact object in an (infinity,1)-category]] to [[compact object]].
* [[Mike Shulman|Mike]]: Thanks Finn! It was great to meet you too. I sliced up your paragraph mentioning fibrations and incorporated the material in other places where I thought it fit better.
* [[Zoran Škoda]]: Created [[Frobenius category]]; added material in [[dg-category]] on dg-modules, Yoneda functor and "pre-triangulated". One should also explain the pre-triangulated envelope functor.
#2009-04-10
* [[Finn Lawler]]:
* Created [[minimal logic]] and [[intuitionistic logic]], as very small stubs. Introduced yet more broken links there, which I'll fill in later.
* Slightly corrected Mike's new entry [[type theory]], and added a paragraph mentioning fibrations. I'm not sure it's in the right place, though. (PS. Mike: it was great to meet you in Cambridge! Also, Bruce: thanks for finding me a seat in the restaurant!)
* [[Toby Bartels]]: I aksed an idle question at [[combinatorial spectrum]] about Kan complexes and $\mathbf{Z}$-groupoids.
* [[Mike Shulman|Mike]]:
* Created [[type theory]] with an introduction for category-theorists. Additions and corrections are welcome.
* Some improvements and corrections to [[cardinal number]], [[ordinal]], [[well-order]] (what an ugly noun!), [[transitive set]], and [[inaccessible cardinal]], and created [[von Neumann hierarchy]]. In particular, I added the structural point of view to complement the naive and material ones.
* [[Urs Schreiber|Urs]]:
* added a reference with a remark to [[A-infinity category]]. Would be nice to have more literature here, and more discussion on the relation to [[dg-category]] and [[stable (infinity,1)-category]]
* [[Zoran Škoda]]: I have made additions to [[cardinal number]]: Urs uses a naive set definition where cardinals are equipotence classes of sets hence his cardinals are proper classes; I follow a choice of representative among ordinals ([[well-order]]ed [[transitive set]]s); to this aim I created [[transitive set]], [[ordinal]], [[successor]], [[inaccessible cardinal]]. In doing this I used some intro parts of my lectures on sheaf theory which I currently teach in Zagreb in Croatian (and which initially started with cardinals, universes and categories). I created [[twisting function]].
* [[Urs Schreiber|Urs]]:
* added a remark on the $(\infty,1)$-version at [[Dold-Kan correspondence]]
* created [[accessible (infinity,1)-category]] and [[accessible (infinity,1)-category]], but incomplete
* created [[compact object in an (infinity,1)-category]], but incomplete
* created [[cardinal number]] and [[well-order]], but experts should please check these
* created [[sigma-model]]
* created an entry [[geometric infinity-function theory]] to go along with the $n$Café entry [Journal Club -- Geometric Infinity-Function Theory](http://golem.ph.utexas.edu/category/2009/04/journal_club_geometric_infinit.html)
* created [[natural numbers object]] just to saturate links (and it should indeed be "natural numbers object", not "natural number object", agreed?)
* worked comment by [[David Ben-Zvi]] into [[why (infinity,1)-categories?]], but more needs to be done
#2009-04-09
* [[Urs Schreiber|Urs]]:
* I had been asked by students to say something about why they should care about learning about $(\infty,1)$-categories. I thought that would be a good thing to try to answer in an $n$Lab entry, so I started an entry [[why (infinity,1)-categories?]]. This is just a first attempt. Maybe somebody would enjoy adding his or her own points of views of correcting/improving mine.
* created [[Connes fusion]], but filled in only pointers to further references
* [[Zoran Škoda]]: created [[von Neumann algebra]] emphasising on sources of relations to category theory and low dimensional topology (particularly G. Segal's program on relations between CFT and elliptic cohomology). There is a good wikipedia entry on von Neumann algebras with lost of references and details, but neglecting the connection to the above topics which should be expanded on. Moreover somebody should mayve write entres on related topics as [[Connes fusion]], [[modular functor]] etc. as those are relevant for some of us.
* [[Toby Bartels]]: I came to some sort of decision at [[direct sum]].
#2009-04-08
* [[Urs Schreiber|Urs]]:
attempted a (long-winded) reply to [[David Corfield|David]]s question "What does it mean" at [[Coyoneda lemma]] (and would anyone mind if we renamed that to [[co-Yoneda lemma]]?)
* [[Mike Shulman|Mike]]:
* Following Toby's suggestion, moved [[subsequential space]] to [[sequential convergence space]]. Split [[convergence space]] and [[Cauchy space]] off from [[filter]], and added some stuff about pseudotopological spaces to [[convergence space]].
* Created [[Reedy category]].
* [[David Corfield|David]]: Created [[Coyoneda lemma]]. What does it mean?
#2009-04-07
* [[Mike Shulman|Mike]]: Created [[subsequential space]] with a bit of propaganda.
* [[Zoran Škoda]]: I created entries [[orbit category]], [[Dold-Thom theorem]] and [[satellite]] with most basic definitions, properties and references, but quickly run out of energy; one should at least add the definition of the morphism part of the satellite functors, the connecting morphism for long exact sequence of satellites, and connection to the Kan extensions. Made some additions to [[Dold-Kan correspondence]].
* [[Toby Bartels]]: I have a terminological question at [[direct sum]]. (It\'s a rather elementary question in universal algebra.)
#2009-04-06
* [[Urs Schreiber|Urs]]:
* created [[Deligne cohomology]]
* reacted to [[Bruce Bartlett|Bruce]]'s question at [[heuristic introduction to sheaves, cohomology and higher stacks]] by expanding further on the notion of morphisms of sheaves
* expanded slightly at [[AQFT]] -- still just a stub entry, though
* touched [[combinatorial spectrum]]: replied to [[Mike Shulman|Mike]], expanded the discussion of examples and changed the notation there a bit. But please check. I'll send a request about this to the blog.
* replied to [[Tim Porter|Tim]]'s comments on [[Zoran Skoda|Zoran]]'s comment at [[differential graded Lie algebra]]
* replied to the questions that were at [[restriction and extension of sheaves]] (on notation and existence) by adding more details
* [[Tim Porter|Tim]]:
* I have adjusted an addition to [[differential graded Lie algebra]]. This used the term 'simply', but it is often not clear that one definition is 'simpler' than another. 'Simplicity' is sometimes clear, but is often in the eye of the beholder. I would love to see some discussion on this as it is an easy trap to fall into (as I know to my cost!) and there are several other similar traps out there which are just as difficult to avoid! Perhaps on the café???
#2009-04-05
* [[Urs Schreiber|Urs]]:
* added a few links to examples at [[space and quantity]] (we have a general problem that many entries created eraly on don't currently point to entries created more recently which de facto they should point to)
* touched [[combinatorial spectrum]]: replied to [[Mike Shulman|Mike]], added a list of examples and have further questions
#2009-04-04
* [[Tim Porter|Tim]]:
* I have put another of the Lexicon series of entries up. It is [[bar and cobar construction]]. This looks at the differential algebra behind those constructions, and sketches the bar-cobar adjunction.
* I have tried to provide more links to and from this series of 'lexicon' entries. (soon will be finished!)
* [[Urs Schreiber|Urs]]:
* further polished [[nonabelian cohomology]]
* created [[combinatorial spectrum]]
* expanded [[abelian sheaf cohomology]]
* created [[cohomology]]
* created [[heuristic introduction to sheaves, cohomology and higher stacks]]
#2009-04-03
* [[Urs Schreiber|Urs]]:
* added a bit more details to [[abelian sheaf cohomology]]
* added discussion of the sheaf version to [[Dold-Kan correspondence]]
* started an entry [[abelian sheaf cohomology]], but have just the "Idea"-section so far (aiming to provide the right $\infty$-categorical perspective)
* provided, using [[Todd Trimble|Todd]]'s help, the details on the relations between the two definitions at [[closed monoidal structure on presheaves]] and created a supplementary entry [[functors and comma categories]] on properties of, well, functors on comma categories
#2009-04-02
* [[Tim Porter|Tim]]:
* I have put another of the Lexicon series of entries up. It is [[differential graded Hopf algebra]].
* [[Urs Schreiber|Urs]]:
* reorganized the entry [[mathematics]] a bit -- I am hoping that eventually this becomes a useful top of a small hierarchy of link-list entries which allow the reader to get an idea of the scope of topics covered (and not yet covered) by the $n$Lab, and possibly to facilitate searches by topic rather than by keyword
* as discussed with [[Timothy Porter]], I created a stub for [[rational homotopy theory]] whose main purpose at the moment is to contain the link list to his lexicon entries on concepts in differential graded algebra
* created [[closed monoidal structure on presheaves]] and [[closed monoidal structure on sheaves]], but am being dense: have a question at the former
* created [[direct image]] and [[inverse image]] and [[restriction and extension of sheaves]]
* moved discussion from [[semi-abelian category]] to [[Dold-Kan correspondence]] and added references
* added explicit formulas to [[Yoneda extension]] (not the [[end]]-yoga, though)
* added a question to [[Mike Shulman|Mike]]'s question at [[semi-abelian category]] (probably for [[Tim Porter|Tim]])
* polished [[infinity-topos]]
#2009-04-01
* [[Urs Schreiber|Urs]]:
* prodded by [[Tim Porter|Tim]]'s latest comment at [[Dold-Kan correspondence]] I looked up and then created an entry for [[semi-abelian category]]
* [[Tim Porter|Tim]]: (I seem to remember a request to put more recent changes at the top, even if you have one on today's page so ... .)
* I have put another of the Lexicon series of entries up. It is [[differential graded Lie algebra]].
* [[Urs Schreiber|Urs]]
* created [[infinity-stackification]]
* added a section "Idea" to [[abelian sheaf]]
* added a section "Idea" to [[Dold-Kan correspondence]]
* created [[injective object]]
* created [[complex]]
* created [[Grothendieck category]] -- it feels like this should make me say something about that axiom list at [[additive and abelian categories]]...
* tried to resolve/incorporate parts of the discussion at [[localization]] by reworking the entry a bit -- also left a comment there
* noticed that we have considerable overlap now between [[multiplicative system]] and [[category of fractions]]. Left a comment there to remind us. Somebody who knows the precise status of these two terms in the math community should please go ahead and merge the material in one entry, keeping a redirect page for the respective other term
* [[Tim Porter|Tim]]:
* I have added a comment on the terminology [[localization]]. Perhaps an algebraic geometric historical perspective could be useful here to help explain the terminology. (I'm not sure that I am competent to provide this however!)
* I have put another of the Lexicon series of entries up. It is [[differential graded coalgebra]].
* [[David Roberts]]:
* Added [[bicategory of fractions]], [[category of fractions]] and [[wide subcategory]].
* Started adding the construction of the [[localization]] of a category, as well as a speculative comment at that page on computing this as a [[fundamental category]].
* Migrated [[2009 March changes|March changes]]
* Continued discussion with Urs at my private page [[davidroberts:comments on chapter 2]].
***
[[2008 changes|First list]] --- [[2009 March changes|Previous list]] --- [[2009 May changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
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2009 August changes | https://ncatlab.org/nlab/source/2009+August+changes |
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### Archive
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[[!include all changes]]
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Archive of changes made during August 2009.
The substantive content of this page should **not** be altered.
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## 2009-08-31
* [[Urs Schreiber]]
* made [[infinitesimal space]] and [[infinitesimal quantity]] redirect to [[infinitesimal object]] -- this way we can use the links and still wait with deciding whether to split the latter entry or not
* created [[cosimplicial algebra]] (redirecting also [[cosimplicial ring]])
* created [[Chevalley-Eilenberg algebra in synthetic differential geometry]] where I aim to derive the [[Chevalley-Eilenberg algebra]] of a Lie group $G$ in terms of functions on the infinitesimal neighbourhood of the identity in a manner entirely analogous to how the deRham dgca is obtained at [[differential forms in synthetic differential geometry]]
* created [[infinitesimal singular simplicial complex]] -- it reiterates things said at [[differential forms in synthetic differential geometry]] but the motivation for creating it was to accomodate a reference that [[Zoran Skoda]] provided which gives a version of this notion in [[nonstandard analysis]]
* [[Jon Awbrey]] added a stub and a link for future development at [[Peirce's logic of information]].
* [[Urs Schreiber]]
* created an _Idea_ section at [[space and quantity]], mainly to list some links.
Given the pivotal role that this entry is playing I am unhappy with its rough appearance. Hope we can eventually improve on that.
* replied to Toby at [[infinitesimal object]] by agreeing that, yes, now that you mention it, it sounds like a good idea to rename that entry into [[infinitesimal space]] and create a parallel entry [[infinitesimal quantity]].
* thanks to Toby for the rephrasing, it's much better now, yes
* [[Toby Bartels]]:
* Rephrased (and tightened) the second warning above to what I think is the important point.
* Cross linked [[relation]] and [[relation theory]].
* An introductory question at [[infinitesimal object]].
* [[Urs Schreiber]]: various very minor edits:
* added a first sentence at [[twisted bundle]] that links the concept with [[twisted cohomology]] (but more details could be given on that)
* slightly edited the beginning of [[semifree dga]] (added the statement for the graded-commutative case)
* [[Urs Schreiber]]: there was some funny behaviour by the software ater I had restarted the server a few minutes ago, which resulted in the Lab-Elf announcement below to disappear for some mysterious reason. I have rolled back now, but that, too, involved some funny effects. It should all be restored now, but if anyone wonders where his or her comment disappeared to, it may have been eaten by the software in this process (but I don't think anything is lost, but let me know).
* [[Urs Schreiber]]
* added to [[differential forms in synthetic differential geometry]] references by Breen-Messing and by Kock with a few lines of remarks. Am hoping to eventually expand and polish this entry.
* slightly edited the first paragraphs of [[infinitesimal object]] and [[nonstandard analysis]] and inserted links back and forth
* [[Lab Elf|Lab Elf (public relations)]]: the migration will happen this week. I had hoped to get everything done before the semester started at the Cobblers School, but was not able to do so, which means it has to fit in around my schedule. What I don't want to do is do the migration, then go off for another violin lesson, and come back a few hours later to find that you've all crashed the site again.
Once that has been done, then the technical department can get on with figuring out how to improve things around here. However, whinging on this page is pretty pointless. If there's some feature you'd like implemented then you could bug the [[Jacques Distler|Chief Lab Elf]] himself (though rumour has it that he's quite busy at the moment) or [[Andrew Stacey|one of]] [[Toby Bartels|the minions]], but a __much better idea__ would be to start a discussion about it over on the [n-Forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/). Treat it as a _lab elf notice board_. If there's something you want __all__ denizens of the lab to see, put it here, but if it's primarily for the lab elves, put it on the forum.
(Note: the high honour of being a Lab Elf is one that anyone can have. There are various sub-departments, all with different roles and skills.)
* [[Urs Schreiber]]: replied to [[Rafael Borowiecki]] at [[category theory]]
following various suggestions I didn't quite delete the big discussion box there yet, but instead moved it to the very end of the entry, so that it doesn't interrupt the reading flow anymore where it used to be (since we reached agreement there).
* [[Urs Schreiber]]: replied to [[Rafael Borowiecki]] at [[Bousfield localization]] and asked for further details
* I second John's and Toby's comments: the reason why it is important to log changes here is because this is the only reasonable way that we alert each other of changes and thereby give us a chance to interact with each other. If nobody would log here, everybody would be bound to blindly edit the Lab on his or her own, whithout any interaction. I have added a remark to this extent now also to the blurb that appears on the top of this page.
And yes, we are all suffering from the slowness of the Lab. But the moment that Andrew Stacey decides he is ready, we will switch to a better server. He has already set up everything and is just waiting for a good moment to migrate.
* [[Toby Bartels]]:
* At [[relation theory]]:
* Some additions.
* Also some format changes, but revert what you don\'t like; I know that you (Jon)\'re still working out how you want to do things here.
* Comments to others:
* To [[Jon Awbrey]]: I may have some solutions for you at the [[Sandbox]].
* To [[Rafael Borowiecki]]: There is good reason to hope that the Lab will remain slow only for less than a month. Then Recently Revised will be able to return as well. (The reason that it got turned off is that it made things *really* slow!) There is a direct link to this page, incidentally; it\'s the link 'Recently Revised' up top! (That\'s only a temporary redirect, of course, until the real Recently Revised comes back.) Even when Recently Revised returns, the simple fact will be that, if you want people to know about what you\'ve done, then you\'ll probably need to log them here. But if you don\'t care whether anybody notices, then you don\'t have to log, as far as I personally am concerned; but don\'t be surprised if it\'s harder to have discussions and collaborate that way.
* [[Jon Awbrey]] added content to [[relation theory]]. Incidentally, does anyone know how to get an @ symbol in a math context?
* [[John Baez]]:
* Added another theorem to [[cartesian monoidal category]] and moved examples of semicartesian monoidal categories to a new page [[semicartesian monoidal category]].
* [[Rafael Borowiecki]]:
* Rewrote and renamed the subsection Generalizations and other structures in [[category theory]].
* Added that there are several equivalences in category theory at [[category theory]].
* Gave organismic supercategories as an application of category theory in biology at [[category theory]].
* Discussed with Toby manifold objects at [[manifold]].
* Suggested an answer to my question at [[Bousfield localization]].
I have been busy writing on the timeline of category theory and reading a prehistory of n-categorical physics. But i managed to write something in nLab.
To log everything just takes more time, and i am already tired of waiting while doing the normal editing. Neither is there a direct link to this page. And is not [[recently revised]] enough? I always used it. Since i don't see the wonderful thing with this page i won't have the energy and time to log everything here.
* [[John Baez]]: if you don't log your changes here, we probably won't read your work, since we won't know what you've done! That also means we're not likely to work on further developing your articles.
The first thing I do when visiting the $n$Lab is go to this page... and I'm probably not the only one! That's why this page is important: it's where we meet and point each other to what we've just done. The rest of us log our entries on this page for this reason... why not you?
* [[Rafael Borowiecki]]: OK, this is a good reason. But nLab is still too slow! Is nobody using [[recently revised]]? Except that it is broken now.
* [[John Baez]]:
* Edited [[cartesian monoidal category]] to add some nice examples of semicartesian monoidal categories; these examples should perhaps be moved someday.
## 2009-08-30
* [[Toby Bartels]]:
* Edited Zoran\'s latest additions. Note in particular:
* Moved [[nearness space]] to [[proximity space]] on the grounds that the former term is ambiguous.
* Replied to Zoran at [[topological vector space]].
* Defined non-Hausdorff [[locally convex spaces]] and [[proximity spaces]].
* Very brief references to nonclassical [[functional analysis]].
* Created links to [[circle group]] (but did not write it).
* Zoran, what did you have to change on [[topological vector space]]? Some technical advice:
* Often if you get an error when saving, the save did go through, and you should check that before going back. (Maybe you know this, and just didn\'t bother to mention it.)
* If you use the back button and submit again, then this can mess up the locking mechanism if somebody else starts to edit the page in the meantime. It is by starting a fresh 'edit' command that you tell people that you are still working on it. (If it helps, you can type in the URI directly ---`http://ncatlab.org/nlab/edit/whatever+the+page+title+is`--- without showing the page.)
- zoran: I know but with slow server and slow connection using back button speeds up the iterated quick fixes by over 100 percent. If I see that in this short fix interperiod the version changed it signals that somebody entered anyway. But I assume that most people won't touch the freshly changed entry which has the date stamp from 15 seconds ago and hence it is expected that it will be likely updated again by the author in next minute or two. Of course, this does not hold for frequent items like _latest changes_ where I won't risk this. Is this OK? When the server is fast and the connection is fats // not mobile // then I do not use back button. The future server will make that less necessary. But If I work have 15 minutes to finish and the server send 2 out of 3 tries empty and with delay of a minute, I will prefer to do my work with back button.
- _Toby_: As far as I\'m concerned, I\'d consider it OK as long as you check, when you\'re all done, whether there were any intermediate revisions by other authors. Also, I don\'t know if this works on your system, but perhaps you can open the edit page in a new tab just to set the locking mechanism? (I\'m not trying to tell you what you must do, just giving suggestion about how to tell other potential editors that you\'re still editing.)
* It\'s always a good idea to copy your material to the clipboard (or even a text file) before submitting it, just in case.
* Zoran: The error was not about the entry but the content. When I cut and paste some paragraphs to a sandbox it did not work either. Only after I changed some sentences, characters and formating in the text the text was allowed. The problem seemed to me with the two formulas in double dollar signs as they were the last which I changed before in about 40th attempt the thing worked.
- _Toby_: H\'m, I just put the displayed math back the way it was before your edit, and it seems OK now. What happens if you edit the page again?
* [[Zoran ?koda]] added a section on classical topological version in [[deformation retract]]. Corrected big chunk which [[Jon Awbrey]] has erased from _latest changes_ by an editing error. Added Pareigis classical reference to [[actegory]].
* [[Jon Awbrey]] is adding [[boolean domain]], [[boolean function]], and [[boolean-valued function]] --- according to his custom starting with a middling level of abstraction that comes up a lot in pedagogy and practice and is calculated to avoid scaring too many children and horses.
* [[Zoran ?koda]] created [[locally convex space]] and additions to [[topological vector space]]; the latter wasted an additional one hour or so because the server was not accepting the input but sending internal application error messages for some reason which according to many tries seemed to have originated in some characters in formulas in previous (Toby's text) it seems as only after changing his formulas it worked and removing my own paragraphs did not help, but who knows which combination of mine and earlier text is actually bugging the system. The repeated trials or removing and adding text took more time than necessary because of the old problem with nlab default that nlab rejects the submition if it is 4th subsequent trial without refreshing edit view from within show view (hence back button not allowed more than 3 times in a row). I think the default should be put to 10 or so, as I often hit over 3 submits from the text which I edit and do not want to wait (and also fight and pay slow internet when on wireless) to go back to show view and then again to ask for another edit: my mozilla keeps the old edit and I can go back to the text immeditaly if the submit was unsucessful or if I notice I do not like the outcome. Thus when server is slow I can reedit what I was editing 5 seconds ago instanteneously without repressing the edit and waiting for the server which may fail again. I think 10 resubmits within the frame could be a better deafult than 3. Created entries [[Fourier transform]] and [[Pontrjagin dual]]. Yesterday wrote few lines in [[functional analysis]] but on the submit the loss of connection lost it; I'll try again today.
* [[Jon Awbrey]] added [[multigrade operator]] and [[parametric operator]]. Added the missing Figures to [[minimal negation operator]] but didn't know how to scale them here --- 80% or 500px would probably be good.
* [[Toby Bartels]]: Edited [[nonstandard analysis]] a bit. Grammatically, I changed 'infinitesimally small' to 'infinitesimal', which already means 'infinitely small' (in absolute value).
## 2009-08-29
* [[Toby Bartels]]:
* Added a lot to [[graph]].
* Regarding [[SVGsandbox]], note also [[SVG Sandbox]] and [[Inclusion Sandbox]].
* Formatted [[minimal negation operator]], where [[Jon Awbrey]] has put material; note that there are two missing diagrams.
* Cross linked [[subfunctor]] and [[sieve]].
* [[Zoran ?koda]] created entries [[actegory]] (category with an action of a monoidal category), yet another sandbox [[SVGsandbox]], (mathematical) [[analysis]] and [[Weierstrass preparation theorem]] (the latter is in a basis of connections between analytic and algebraic geometry; I want eventually to say about algebraic approaches to [[analytic geometry]] like analytic local algebras and rigid analytic geometry; the theory is rather parallel to some aspects of modern algebraic geometry, e.g. in the way differential forms and regular differential operators are introduced); created [[subfunctor]] (needing your checking). Earlier this week much extended [[noncommutative geometry]] and did not have time to report it here. I do not see nonstandard analysis in usual sense in Moerdijk-Reyes (also at page 385 bottom they themselves also include that the relation of their axiomatics SIA with nonstanard analysis is not clear). The discussion in chapter 7 is in generalized sense kind of nonstandard analysis, in the general sense that it entails a version of the transfer principle. However the discussion relies on a very general setup of complicated axiomatics based on commutative rings the transfer refers to certain language using coherent formulas. Thus it is not about extending language $L(\mathbb{R})$; or about ultrafilter model of the nonstandard extension of $\mathbb{R}$. Of course, at the topos level, one can express everything in terms of internal language of the topos and there is a general transfer principle in that setup. This is much more abstract sense of [[nonstandard analysis]] than in the rather concrete and conservative article I started. Of course further discussions and contributions in both directions are necessary for us.
* [[Urs Schreiber]]: thanks, Zoran. By the way: in the later chapters of [[Models for Smooth Infinitesimal Analysis]] there is some detailed discussion of the relation between synthetic differential geometry and nonstandard analysis
* [[Zoran ?koda]] created [[nonstandard analysis]] containing basically an (incomplete) introduction to basic terms in the [[ultrafilter]] model of the nonstandard extension of real numbers. One of the reasons to create this right now is that Urs is talking about [[infinitesimal object]] in synthetic differential geometry, and it could be a moment to (hopefully usefully) compare with the notions of nonstandard analysis. Created [[Rouqier's cocovering]] (in subject of [[triangulated category]]).
* [[Jon Awbrey]] noticed that [[graph]] was unsaturated, so he whetted it. There are many definitions of _graph_ and many dialects of graph theory. I added one of my first and favorite. Read my …
* [[Tim Porter]]: I forgot.. I created [[braid group]] yesterday, as I needed them for examples and [[Toby]] already has improved the presentation! Thanks.
* [[Jon Awbrey]]: more almost ready for prime time $\nu$'s
* [[hypostatic abstraction]]
* [[inquiry driven system]]
* [[Peirce's law]]
* [[praeclarum theorema]]
* [[relation theory]]
## 2009-08-28
* [[Andrew Stacey]] solicits input about making the [[database of categories]] a real database; see [the Forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=57).
* Welcome, Jon! It would be ideal if you actually wrote about this material instead of just linking to discussion elsewhere; hopefully, you\'re already planning to do this. Assuming that you wrote the material linked elsewhere (or have permission from those who did write it), you could mostly copy and paste it here; we regulars would be happy to help with formatting and the like. Some more technical notes: we have a convention of lowercase page titles, so that people can make links to your pages from the middle of sentences; and speaking of making links, you should probably link this stuff from the main [[logic]] page. (I\'ll do some of this right now.) ---[[Toby Bartels]]
* Thanks, Toby, that's the plan. The beauty of interaction is that the same material injected into different environments tends to develop in different directions. ---[[Jon Awbrey]]
* [[Jon Awbrey]]: almost ready for prime time $\nu$'s
* [[cactus language]]
* [[differential logic]]
* [[logical graph]]
* [[minimal negation operator]]
* [[propositional equation reasoning system]]
* [[riffs & rotes]]
* [[Toby Bartels]]:
* A little geometry at [[complex number]].
* Cross-linked [[derivation]] and [[derivation on a group]].
* [[Tim Porter]]: I have created [[derivation on a group]] to provide some back up for [[Fox derivatives]]. This is also needed for the linearisation functor going from crossed complexes to chain complexes.
* [[David Roberts]]: further tinkering at [[history of cohomology with local coefficients]] - mentioned link of Reidemeister's original approach to crossed complexes. Forgive my ignorance Zoran, but I'm not sure what to do about your comment there. Is it meant to extend cohomological history generally or provide details on related topics?
* Zoran: Local cohomology is discovered by Steenrod and systematically used only around 1943 or so, but special cases like Reidemeister torsion can be listed as earlier if you decided to do so and if Steenrod liked to see the influence that way. But this is nonsense if other related cases of related OBSTRUCTIONS to those studied by Reidemeister, like Nielsen invariant, Whitehead torsion etc. are not listed as well. This is NOT intended to be generally about (co)homology, I pointed to the relation with specifically supposedly related to the things which you do list. But maybe my opinion about he connections is wrong, I just put the possibility that things apperaed earlier in a way not acknowledged by Steenrod, but possibly seen so nowdays.
## 2009-08-27
* [[Toby Bartels]]: I\'ll never be done with [[affine space]]. Never! Mwa-ha-ha-ha-ha!!!
* [[Tim Porter]] wrote a comment down at the bottom of August 26, so look there for it.
* [[Mike Shulman]]: I think I'm done with [[affine space]] for now.
* [[Alex Nelson]] commented at [[crossed complex]].
* [[Toby Bartels]]: Yes, always more at [[affine space]], such as morphisms.
* [[Mike Shulman]]: A bunch more at [[affine space]], including an "unbiased" definition. There's still a lot more to say.
## 2009-08-26
* [[Toby Bartels]]:
* Started [[complex number]], [[hypercomplex number]], [[dual number]], [[perplex number]], [[normed division algebra]].
* Edits to [[cobordism category]], particularly addition of #Idea# and noting that the alleged direct sum is just a coproduct.
* [[Urs Schreiber]]: created [[infinitesimal object]], with a bit of material, but haven't found yet the time to round this up and polish
* [[Zoran ?koda]]: created [[cobordism category]]; created [[Fox derivative]] (cute construction in [[combinatorial group theory]]). I think, it is the same Fox of the famous article Fox-Neuwirth on the topology of configuration spaces.
* [[David Corfield]]: And the Ralph Hartzler Fox of 'Crowell and Fox' about whom Ronnie Brown [wrote](http://www.bangor.ac.uk/~mas010/pdffiles/groupoidsurvey.pdf)
>Crowell and Fox in [43, p. 153] took the view that a few definitions 'like that of a group, or a topological space, have a fundamental importance for the whole of mathematics that can hardly be exaggerated. Others are more in the nature of convenient, and often highly specialised, labels which serve principally to pigeonhole ideas. As far as this book is concerned, the notions of category and groupoid belong in this latter class. It is an interesting curiosity that they provide a convenient systematisation of the ideas involved in developing the fundamental group.'
* [[Tim]]: The wikipedia article is of interest on Ralph Fox. He was the doctoral supervisor of Milnor, Stallings and Barry Mazur!
## 2009-08-25
* [[Toby Bartels]]:
* More at [[affine space]].
* You\'re welcome, Zoran; but to be honest, I\'m the one who invoked the problem. (I won\'t say that I *caused* the problem, since the server did make a mistake, probably timing out. But if I had been more careful, then I wouldn\'t have mindlessly hit reload and saved my edit again.)
* [[Mike Shulman]]:
* Did some editing of [[heap]], including adding two alternate definitions of the automorphism group, considering its functoriality, and incorporating the discussion into the main text.
* Claimed that Toby's fix of [[affine space]] contains superfluous data.
* [[Zoran ?koda]]: added links to [[heap]] and [[zoranskoda:affine space]] at [[affine space]]; and the original (B and PS) references to [[BPS-state]].
* created [[cluster algebra]] with overview but without strict definition yet.
* Thank you, Toby, for masterly fixing the connection inconsistency in [[dilogarithm]].
* [[Toby Bartels]]:
* A dropped connection caused some trouble at [[dilogarithm]], but I think that I fixed it.
* Fixed [[affine space]].
* Sorry about [[fine sheaf]]; I was going to rephrase things, decided not to, and didn\'t change that back.
* [[Zoran ?koda]]: created [[dilogarithm]].
* [[Urs Schreiber]]: in case anyone is waiting for reactions from me: I am currently on a small vacation with little internet access. Will be back at full speed next Sunday or else next Monday
* [[Zoran ?koda]]: added synonym flasque to [[flabby sheaf]]; query at [[history of cohomology with local coefficients]]. Created [[quantum dilogarithm]], but for now it consists only of references and links.
* [[Mike Shulman]]:
* Question/correction at [[affine space]].
* Comment at [[symmetric function]].
* Discovered, and restarted, the terminological discussion at [[lax natural transformation]] from back in June.
* [[Zoran ?koda]]: $A$ and $B$ in [[fine sheaf]] are already closed, so I removed Toby's correction taking closure.
* [[David Roberts]]: some dates at [[twisted cohomology]] on the earliest references, and added title, date and small clarification on Reidemeister's 1938 article at [[history of cohomology with local coefficients]].
## 2009-08-24
* [[Toby Bartels]]:
* Started [[Euclidean space]], [[Cartesian space]], [[inner product space]], [[affine space]], [[constant function]].
* Started [[ground ring]] (with redirects [[base field]] and interpolations, also terms with [[scalar]]). I see this as a place to talk about the issues related to this choice, although mostly it\'s a list of other articles now.
* Formatting edits at each of Zoran\'s new pages listed below.
* Discussion at [[database of categories]].
* Added a link to Andrew\'s request for comments below.
* [[Zoran ?koda]] created [[soft sheaf]], [[fine sheaf]], [[family of supports]], [[analytic geometry]].
* [[David Corfield]]: Mike's right at [[symmetric function]] isn't he? So the definition needs redoing. Would do it myself, but how does one put the bit about grading properly?
* [[Andrew Stacey]] pondered the format of [[database of categories]].
By the way, it would still be really useful if people could take a look at the migrated n-lab. The __main__ question I want to know is whether the pages look right. The migration involved a step or two that were pretty much guesswork and I want to know whether I guessed right or not. If a page looks horribly wrong, or something doesn't work how it ought, please let me know. The best ways to let me know are either by email or by [commenting over at the forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=52).
* [[David Roberts]]: trivial comment at [[synthetic differential geometry]] in response to Mike.
## 2009-08-23
* [[Gonçalo Marques]] has a report at [[symmetric function]].
* [[Toby Bartels]]: Tried to clean up the formatting at [[Jim Stasheff]]\'s new [[history of cohomology with local coefficients]]. In particular, this involved removing some stuff that seemed to refer to a bibliography that wasn\'t there, so please complain (or just put it back) if that was wrong.
* [[Mike Shulman]]:
* Wrote a bit about structured spaces at [[space]].
* Comments about terminology at [[compactly generated space]].
* Added $Mod_R$ to [[database of categories]].
* Continued discussions at [[symmetric function]] and [[synthetic differential geometry]].
## 2009-08-22:
* [[John Baez]]:
* added material to [[database of categories]] --- more on Rel and SimpSet --- and replied to Toby's and Rafael's comments there. What we want --- I think --- is not a massive list of all the categories we know (I could make up 5 a minute for the rest of my life), but a list of categories _including their basic categorical properties_. I don't want to worry too much about the format of this list until it's gotten a lot longer.
* tried to get Rafael to visit this 'latest changes' page (and the corresponding pages for many previous months), log his changes here, and interact with us a bit more here.
* replied to David's and Toby's comments on [[symmetric function]].
* wondered why each of these sub-entries is starting with an asterisk intead of a little circle, and tried in vain to correct it. Help!
* _Toby_: You need the spacing to match. In your line `*__[[John_Baez]]:` (spaces added), see how the `*__` takes up $3$ characters? Then every line that comes under it should begin with $3$ spaces. (Sometimes you can get away with fewer.)
* [[Toby Bartels]]: Slight edits to [[space]].
* [[Rafael Borowiecki]] has a question at [[Bousfield localization]].
* [[Urs Schreiber]]:
* expanded [[simplicial object]] a little, added Idea, Examples and mentioned cosimplicial objects
corrected at [[singular cohomology]] the alleged simplicial ring of functions to a cosimplicial ring
* replied to [[Ronnie Brown]] at [[singular cohomology]]
* thanks to [[Toby Bartels|Toby]] for [[space]] -- I have just one request for a change of wording there
* [[Toby Bartels]]: Thanks to [[Rafael Borowiecki]], I added a link at [[Higher Topos Theory]] to the published version on Lurie\'s MIT website.
## 2009-08-21:
* [[Vaughan Pratt]]: changed "category: categories" to "category: category" for Chu construction. (Rationale: The plural form was just my misremembering the name of that category, I didn't mean to start a new category. Chu(V,k) is a category and therefore should be findable as such, but it would take a while to write a separate page for every pair (V,k).) (I also edited "Editing latest changes" but presumably that goes without saying. :) -vp)
* _Toby_: That still doesn\'t fit in with the other pages in that category; it would be different if we had a page [[Chu]] with links like '[[Chu]]$(V,k)$' in other pages. *However*, I think that the fault is more in the peculiarity of the previous system (if I explained exactly why I created `category: category`, then you\'d see how silly it is), so I do *not* think that you should actually change anything (other than plural to singular, which you did).
* [[Vaughan Pratt]] has started [category: categories](/nlab/list/categories) and put [[Chu construction]] in it. [[Toby Bartels|I]]\'m not sure what this category is supposed to be for; the entry wouldn\'t fit into [category: category](/nlab/list/category) either, since it\'s not a specific category (although maybe that is not the best criterion).
* [[Ronnie Brown]]: Query (badly formatted) added to [[singular cohomology]].
* [[Toby Bartels]]:
* Wrote [[space]], with the idea that this should be very general and mostly point people to other, more specific, pages.
* Messed with [[Schur's lemma]] a bit.
* [[Urs Schreiber]]: further polished and edited [[∞-quantity]], gave more details on how the characterization of the deRham complex as the normalized Moore complex of functions on simplices follows from Anders Kock's results and provided a reference that provides the statements about the relation between functions on $\mathbf{B}G$ and the Chevalley-Eilenberg algebra in the algebraic context.
* [[Zoran ?koda]]: created [[Schur's lemma]].
* [[Toby Bartels]]: Some breaking up into paragraphs and things at [[hyperplane line bundle]], [[symplectic manifold]], and [[Kähler manifold]].
* [[Zoran ?koda]]: created [[Killing form]] with some words on Casimir operators. Eventually Casimir operators should be treated separately, but it is beneficial to develop the unique entry first to accumulate common facts, conventions and notation, because the two are closely related.
* [[Urs Schreiber]]:
* I am preparing some ground for a comprehensive discussion of the theorem -- which I think we have now, using the latest addition at [[monoidal Dold-Kan correspondence]] -- that Anders Kock's chacraterization of the deRahm DGCA is precisely nothing but the characterization of the image of the cosimplicial algebra of functionas on infinitesimal simplicies under the normalized Moore cochain complex.
I wanted to do that at the entry on $(\infty,1)$-quantity. It could also go at [[differential forms in synthetic differential geometry]], but as the $(\infty,1)$-quantity pages is globally marked as "research material" I thought it might be good to put it there and then just point to it from elsewhere.
But in the course of this I noticed first of all that my use "$(\infty,1)$-quantity" was a misnomer. It should be "$\infty$-quantity". To explain (to myself) why, I created the dual entry [[∞-space]]. Then I moved the material from the former [[(infinity,1)-quantity]] to the new [[∞-quantity]] and started editing a bit and put redirects.
* [[Zoran ?koda]]: created [[symplectic vector space]], [[symplectic manifold]], [[Kähler manifold]], [[hyperplane line bundle]] having not only $\mathcal{O}(1)$ but also the basics for $\mathcal{O}(-1)$, $\mathcal{O}(n)$.
* [[Urs Schreiber]]:
* moved the new discussion of references by [[David Roberts]] at [[topological T-duality]] to the References section, edited slightly and inserted some links
* expanded [[symmetric monoidal category]] which was very stubby (and still is): added an "Idea" section with pointers to the general context -- and started adding a list of examples
* [[David Corfield]]: asked a question at [[symmetric function]].
* [[David Roberts]]: links at [[topological T-duality]] and mention of early work on the topic.
* [[Omar Antolín-Camarena]]: added a definition to [[accessible category]], could use checking by someone who knows.
* [[Omar Antolín-Camarena]]: created [[small object]], whose definition was missing from [[locally presentable category]].
* [[John Baez]]: started a [[database of categories]]. This is very preliminary, but it could be very useful if we keep working on it. The idea is to list lots of categories and their categorical properties. If this list becomes long we can try to organize it somehow.
* Zoran: I was thinking of this idea long time ago and told many people including some of the nlab members. I am happy that somebody else also came up with this.
* [[David Roberts]]: question at [[topological T-duality]]. No mention has been made of the Adelaide school's treatement there!
* [[John Baez]]: so, please tell us what that school has done, or at least add some links to papers on the arXiv.
* [[David Roberts]]: I did provide a couple of references in my comment. But to quote Galois, \'I have not time!\' (but I'm not about to duel anyone). I'll do a little bit now.
* [[Toby Bartels]]:
* Started [[singular cohomology]] by copying the definition from [[cup product]].
* An important reference at [[database of categories]]. We\'re already using Instiki\'s category system here!
* Moved some stuff about the structure from [[topology]] to [[topological structure]] (which is really [[topological space]] now).
* Moved stuff from [[connection]] to [[connection on a bundle]]. It\'s possible that much of the latter could be put on a more specific page, something like [[parallel transport]] (which currently redirects).
* A bit more at [[continuous map]].
* A suggestion for [[Zoran ?koda]] at [[differential form]].
* Zoran: it seems you misunderstood the question, see my answer there. It is not about generalizations.
* Added links and such to [[nonabelian algebraic topology]]. Normally I don\'t log this sort of thing, but as much of this is a personal essay, I want [[Ronnie Brown]] to make sure that I didn\'t warp anything.
## 2009-08-20
* [[Roger Witte]] has joined us with an edit to [[category theory]].
* [[Zoran ?koda]]: added several (very carefully chosen, though I am not competent enough) references into [[BPS-state]].
* [[Urs Schreiber]]: split off [[monoidal Dold-Kan correspondence]] from [[Dold-Kan correspondence]] -- moved the material in the original section at the latter to the former and linked back and forth -- moreover I added a section _Lax monoidalness of the Moore cochain complex functor_ where I claim to prove the statement asserted by this headline. CHECK.
* [[Zoran ?koda]]: created an outline for the [[BPS-state]], expanded [[group theory]]. I side with Mike with long-surpressed urge to have nice conceptual explanation of the terminology semantics-structure adjunction.
* [[Mike Shulman]]: Corrected a handedness error I made in my initial reply at [[monadic adjunction]], and made a request: can anyone give a nice conceptual explanation of the terminology "semantics-structure adjunction"?
* [[Zoran ?koda]]: expanded slightly [[geometry]], [[diffeomorphism]] and much [[topology]] (with groupings of similar items, few hours of work), created [[Hurewicz connection]], [[continuous map]] and a stub for [[diffeity]] with few references. Additions to [[connection]].
* [[David Corfield]]: Are you sure it isn't 'diffiety'? That would match 'variety' better.
* Zoran: Oh, yes, thanks, [[diffiety]]!!
* [[Urs Schreiber]]: added the standard singular cohomology version to [[cup product]]
* [[Zoran ?koda]]: created [[projection measure]]. I agree with Toby that, in common convention in published literature, if one uses a German phrase involving word Satz (like in his Theorem example) one will still write it with capital letter. As a physicist, I witness and referree papers on daily basis which are using the word Ansatz in English physics texts, as rule capitalized. Mathematicians do not like the term and logic of using something as a means to solve the problem if it is not justified a priori (one mathematician was telling me words of disgust: "what is physics ? Nothing! Ansatz!").
I posted a query under [[differential form]] on interpretations as inner hom, i.e. functions on $\Pi TM$.
* [[Ronnie Brown]]: I have rewritten [[nonabelian algebraic topology]] to incorporate historical comments I made made on discussion lists, and so to show my experience of the relation with [[nonabelian cohomology]].
* [[Urs Schreiber]]: reacted to [[David Roberts]]' latest comment at [[category theory]] by expaning the item on "categories = 1d spaces" with a discussion of one way how this statement can be thought of as being made precise
* [[Mike Shulman]]: continued discussion at [[monotone function]].
* [[David Roberts]]: comment at [[category theory]] re category as a sort of directed space.
* [[Lab Elf|Lab Elf (wildebeest department)]]: We will be migrating the entire $n$-Category Lab to a new server soon. Please see [the announcement on the Café](http://golem.ph.utexas.edu/category/2009/08/nlab_migration.html) and report there all of the many problems!
* [[Toby Bartels]]: I did as David urged below; see [[category theory]] and [[universe]].
* [[David Roberts]]: Re: [[category theory]] - Toby, go ahead. In particular, I found last night the [[foundations and philosophy]] page about which some comments were directed (the people involved, including me, didn't know it already existed). There is a link to a paper on [[category theory]] about structuralism which might go there, as well as the (short) discussion surrounding it.
* [[David Corfield]]: re [[foundations and philosophy]], I can't do more than log its creation here (see 2009-08-17). Now to find time to write it up.
* [[Andrew Stacey]]: I'm hesitant to weigh in on this as I'm as guilty as everyone else, but merely flagging something here is not really enough. We should all think about how to organise the material here to make it easily findable. Of course, linking from related page to related page is good, but there should also be some hierarchical organisation. For example, there should be a philosophy index page and [[foundations and philosophy]] should be on it. Perhaps, appropriately enough, we should make more use of the [category](http://golem.ph.utexas.edu/instiki/show/Categories) features in Instiki. At the moment, we have the following categories: biography, category, delete, drafts, foundational axiom, lexicon, meta, people, place, redirect, reference, spam.
This comment is getting a little long, so I've started a [forum discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=53) with more details on how to implement this, so please reply there.
## 2009-08-19
* [[Toby Bartels]]: Reading Zoran\'s work reminds me that there\'s another way to interpretation notation for an integral, which I\'ve now recorded at [[measure space]]. (I think that capitalising the 'N' in 'Nullstellensatz' in English is like capitalising the 'T' in '[Cantor\'s Theorem](http://en.wikipedia.org/wiki/Cantor's_Theorem)', but I\'m not going to worry about it in a world with redirects.)
* [[Zoran ?koda]]: added standard references to [[commutative algebra]] and links to Murfet's online notes. [[Nullstellensatz]] as a German noun compound starts with a capital letter, and it is usually (though not always) quoted so in English references.
* [[Toby Bartels]]: Created [[algebraically closed field]].
* [[Zoran ?koda]]: created [[commutative algebra]] both for the notion of a commutative $k$-algebra and the subject of _the_ commutative algebra, which is one of the foundations of algebraic geometry. I have removed the redirect commutative algebra from [[associative unital algebra]]. Toby: a variety is an affine, quasiaffine, projective or quasiprojective variety. Hence if one talks about the category of all varieties, then all of those are objects simultaneously, and not by convention of choice. True enough quasiprojective includes all others as the affine space is a Zariski open subset of the projective space. I corrected nonlogical usage of the maximal compact to the maximal torus. The thing is that I usually use K for maximal compact in G which is complex, and now I used G for compact and $G^C$ for complexification, so in my normal notation it would be $K/T=G/B$ where $T$ is the maximal torus. Once I lost $K$ in the complexification notation I put it automatically at the place of the torus. Of course $SL(n,C)/B = SU(n)/T$, what is called Gram-Schmidt orthogonalization procedure :) Thanks for catching the inconsistency in language. I will use $T$ now and remove $K$ everywhere (better I used my notation).
* [[Toby Bartels]]:
* A question at [[flag variety]], and also a brief question at [[algebraic variety]].
* Responded to Mike at [[monotone function]]. What I have there is not quite right, but I\'ll let Mike react before I have another go.
* [[Zoran ?koda]]: created [[algebraic variety]], additions to [[integral scheme]] and lists [[mathematics]],[[geometry]]
* [[Toby Bartels]]: I\'m going to mess with [[category theory]] again, if somebody doesn\'t stop me.
* [[Mike Shulman]]:
* Definitional question at [[monotone function]].
* Agreed with Toby's edit at [[equipment]] and removed query box.
* Comment and question at [[homotopy hypothesis]].
* [[Zoran ?koda]]: after lots of work expanded the entry list for [[geometry]] (the first try was eaten up by internet explorer: nlab server did not accept my first submission and IE does not allow going back to the data (while never had this problem with firefox -- just can go back to the form with data still in)).
* [[Urs Schreiber]]: added a bit more on the monoidalness and the shuffle map at [[Dold-Kan correspondence]]
* [[Zoran ?koda]]: created [[flag variety]], [[Borel-Weil theorem]] and expanded [[coherent state]]
* [[Urs Schreiber]]: following [[David Roberts]]'s reaction below I took the following action at [[category theory]]
* I split the item on "categories as spaces" and "categories as mathematical universes" in two
* I edited a bit the statement about them being spaces
* then in the item on categories as mathematical universes I pasted in the paragraphs that me and David Roberts had suggested in the quesry box discussion below.
* [[David Roberts]]: responded to Urs' suggestion at [[category theory]] with one edited from his. We need to wrap up some of the discussion there, in particular, and, in my opinion, remove some points that would probably not be considered core to an introductory page on category theory - I hesitate to say \'points that are not generally supported by the category theory community\' because I know that there are some new and/or minority points of view that are enlightening. Certainly a couple of the extended discussions might be moved to a discussion section. I blame the lateness of the hour in Adelaide for not doing that particular task myself.
* [[Zoran ?koda]]: created [[coherent state]] (it will be much longer later)
* [[Urs Schreiber]]: replied at [[category theory]] by suggesting as an alternative an expanded version of the sentence on "mathematical universes" under discussion.
* [[Zoran ?koda]]: additions and changes to [[symplectic geometry]]
* [[Lab Elf|Lab Elf (golf department)]]: added an entry at the [[HowTo| How To]] on how to put parentheses (and other "unsafe" characters) in links.
* [[David Roberts]]: comment at [[category theory]] - nothing substantial, just adding my voice to Toby's comment.
* [[Omar Antolín-Camarena]]: added a section to [[adjoint functor theorem]] about the version for presentable (∞,1)-categories.
* [[Omar Antolín-Camarena]]: created [[solution set condition]] (has link from [[adjoint functor theorem]]).
## 2009-08-18
* [[Toby Bartels]]: Comments at [[equipment]] and [[category theory]].
* [[Urs Schreiber]] much as I regret it, I still have complaints -- but also constructive suggestions -- in the discussion at [[category theory]]
and i notice that a long discussion in a query box is getting cumbersome. Maybe we should move that to the blog.
* [[Mike Shulman]]
* commented on one discussion at [[category theory]]
* finally created [[equipment]]
* replied at [[monadic adjunction]]
* [[Urs Schreiber]]
* added "Idea" section to [[generalized complex geometry]]
* created [[geometric quantization]] collecting some links and importing John's old material from [here](http://math.ucr.edu/home/baez/quantization.html)
* created [[symplectic geometry]] with just some things to come back to later
* [[Zoran ?koda]]: created [[Leibniz algebra]]
* [[Urs Schreiber]]
* added to [[groupoid cardinality]] the general definition for [[infinity-groupoid]]s as well as a handful of further examples
* started working on the entry [[exercise in groupoidification - the path integral]]
* [[Toby Bartels]]: Wrote [[variety of algebras]].
* [[Urs Schreiber]]
* created a stub for [[quantization]]
* renamed [[Jim Stasheff]]'s entry [[Larmore]] into [[Larmore twisted cohomology]] and edited it; linked to it from the reference section at [[twisted cohomology]]
* made [[Chu space]] a redirect to [[Chu construction]]
* created entry for [[Michael Barr]]
* [[John Baez]]:
* added more references to [[Batanin omega-category]], especially references that discuss modelling homotopy types using Batanin omega-groupoids.
* added a reference to Simona Paoli's paper at [[homotopy hypothesis]]. This contains an answer to Urs' question about the functor that turns spaces into $cat^n$ groups. I also moved his question down to the 'Discussion' section near the end of the page.
* posed a question on [[monadic adjunction]]. The end of my question contains a formatting error of the sort Toby knows how to fix! I forget how!
* [[Jim Stasheff]] wrote [[Larmore]]; I\'m not sure if it\'s supposed to be about the person or about a cohomology theory.
* [[Toby Bartels]]:
* Started [[monadic adjunction]], doing only the definition in $Cat$.
* Since [[geometric function object]] didn\'t exist and at least one link to [[examples for geometric function objects]] was misdirected, I moved the latter to the former, at least for now, and put in redirects.
## 2009-08-17
* [[Urs Schreiber]]:
* found time to expand the discussion at [[examples for geometric function objects]] -- now this contains details on the simplest but somewhat archetypical example: that of over-categories. This is really the notion of geometric $\infty$-functions that is implicit in [[John Baez]]' notion of [[groupoidification]]. I am thinking that making its structure in the context of geometric $\infty$-function theory explicit is useful for putting this together with the discussion at [[geometric infinity-function theory]] into perspective. Will try to say more about the other examples indicated tomorrow.
* slightly edited and updated links at [[geometric function theory]] -- one of them now points to the new [[examples for geometric function objects]], which however is still to be written, but I have to interrupt for a moment
* created entry for [[Michael Batanin]]
* created stub for [[Batanin omega-category]]
* [[Toby Bartels]]:
* Cleaned up [[twisted cohomology]]; there\'s still a link to Urs\'s web that\'s broken.
* [[Urs Schreiber]]: yeah, sorry, I keep working on this at my private web with [[Jim Stasheff]] and then from time to time Jim makes me move the stuff to the public page -- but we'll try to stop doing that now and just edit the public page -- sorry again
* Noted the compact nature of the [[Gelfand spectrum]] (also at [[maximal spectrum]]).
* Mentioned $B^*$-algebras at [[operator algebra]]; but I really need to write [[C-star-algebra]].
* [[Urs Schreiber]]
* filled [[T-duality]] with a bit of content (motivation being to link to the Cavalcanti-Gualtieri article now linked to there)
* created stub entry for our esteemed new contributor professor [[Andrew Ranicki]]
* activity at [[twisted cohomology]]:
* [[Jim Stasheff]] added a long list with chronology of references on twisted cohomology
* [[Andrew Ranicki]] added a hyperlink to a reference (as far as I, [[Urs Schreiber]], can tell from the logs)
* [[Urs Schreiber]]
* edited Jim Staheff's chronology slightly
* and replaced the bulk of the entry with a newer version as it has evolved meanwhile in interaction between Urs and Jim at Urs' private web -- the new version has a longer motivational piece and includes details on the proof that and how the fibration-sequence definition of twisted cohomology that is used here reproduces the traditional one in terms of sections of bundles of spectra as a special case
* [[Urs Schreiber]]:
* created [[transversal maps]]
* expanded [[generalized smooth algebra]], adding more theorems, links and remarks (also corrected some minor mistakes in the previous version) -- but I also suffered a stupid data loss due to the system having an "internal error" and myself srewing up the backup copy of my edit and had to type everything twice. Hopefully I didn't delete unintentionally old content this way...
* created a stub entry for our new esteemed contributor, Prof. [[Charles Wells]]
* [[Zoran ?koda]]: created very incomplete entries [[operator algebra]], [[maximal ideal]], [[Gelfand spectrum]]. I have mentioned mostly just the unital case, as the nonunital case will be more tricky when completed.
* [[David Corfield]]: Started [[ideal completion]] and [[foundations and philosophy]].
* [[Charles Wells]] Lightly edited [[category theory]], notably introducing posets as another example of a category.
* Hey! We're honored to have you here, Prof. Wells! -- [[John Baez]]
## 2009-08-16
* [[Toby Bartels]]: More talk on [[category theory]]. Also note the existence of the article [[foundations]]; there\'s not a lot of philosophy there, but there could be and probably should be.
* [[Zoran ?koda]]: created [[coquasitriangular bialgebra]], [[cosemisimple coalgebra]]; added the redirect [[corepresentation]] and few words on this terminology to [[comodule]]
* [[David Roberts]]: More discussion at [[category theory]]. There are some statements that need unraveling, and I don't quite feel up to it. In particular, [[Rafael Borowiecki]] pointed out the need for a philosophical page on foundations, as he referenced a paper on structuralism, sets and categories I didn't feel fitted on [[category theory]] (or at least in the section where it was referenced). So maybe this is a plea to David Corfield, who is more qualified to talk about such things than I.
* [[David Corfield]] I'd be happy to help how I can. I've started a page [[foundations and philosophy]], to be expanded.
## 2009-08-15
* [[Toby Bartels]]: Started [[prime ideal theorem]] and [[maximal ideal theorem]]. Eventually I\'d like to have precise equivalences of these, constructively valid (preferably in any pretopos), to various forms of choice, with proofs. Now is just a list of very basic results, possibly valid only in a model of ETCS.
* [[Zoran ?koda]] created [[homotopical algebra]] as a rather terminologically-historical entry (as opposed to more descriptive and concrete [[homotopy theory]] and compared to [[homological algebra]]); created [[quasitriangular bialgebra]] with redirects [[quasitriangular Hopf algebra]] and [[universal R-element]]. Added links to [[mathematics]], [[algebra]] and maybe to some more entry(s).
* [[John Baez]]: Why does the main front page look so weird? Did the site get moved to a new host, or is it the result of an alien invasion?
* Hmm, now it's back to normal.
* [[Urs Schreiber]]: I can't tell what happened. Andrew is indeed preparing the migration and has set up the nLab on another server by now. That shouldn't affect what's going on here. Or might it be that the phenomenon was something just on your side? What was it actually?
* [[Toby Bartels]]: I was thinking of writing articles on $\Omega$-[[Omega-group|groups]] and [[Malcev variety|??????? varieties]] to talk about [[ideals]] in them. I agree that three articles, one on that subject, one on ideals in rings (and rigs, maybe even monoid objects in general) and one on lattices (and other partially ordered or preordered sets), would be a good idea.
## 2009-08-14
* [[Zoran ?koda]] created [[quantum group]], additions to [[fiber bundle]], [[Hopf-Galois extension]] and [[Timeline of category theory and related mathematics]] including a query discussion and 1970 Timeline entry for [[Benabou-Roubaud theorem]]. Some thoughts on [[ideal]] entry. Toby, I think we should eventually have an entry which would have a general and lattice notions of [[ideal]] separated from the entry for rings/algebras. Otherwise half of the entry is incomprehensible for ring/algebra theorists. For example for noncommutative rings commutative notion of prime ideal splits into nonequivalent notions of prime and completely prime ideals (just to start, compare also primitive etc.), which are now difficult even to list as the rest of the list is lattice-worded. So I would opt to have a general entry and specialized entries for lattices, rings/algebras. Another interesting context are [[$\Omega$-groups]] (additively written not necessarily commutative groups with a family of operations, not necessarily unary ones which distribute over group "addition"; I am going to add a stub now) where ideals correspond to quotient $\Omega$-groups; interesting is to compare those version of ideals to the categorical notions of normal subobject. Thus we have ideals for sheaves of rings (e.g. defining ideal of a subvariety) etc.
* [[Toby Bartels]]: Added principal ideals to [[ideal]].
* [[Zoran ?koda]] created [[flabby sheaf]]
* [[Toby Bartels]]:
* Wrote [[implication]].
* Pretty stubby biographies for [[Peter Johnstone]] and [[Paul Taylor]].
* [[David Roberts]]: More comments at [[category theory]], this time to statements in the section about the contrast with set theory due to [[Rafael Borowiecki]].
* [[Toby Bartels]]:
* Added some stuff on morphisms to [[partial order]], [[semilattice]], [[lattice]], [[Heyting algebra]], [[Boolean algebra]].
* Wrote [[monotone function]], with a question on terminology, the sort that could be answered through the literature for once!
* [[Rafael Borowiecki]]:
* wants more details for Zoran\'s entries at [[Timeline of category theory and related mathematics]].
* added some definitions to [[A-infinity-category]].
* [[David Roberts]] added to the discussion at [[category theory]].
## 2009-08-13
* [[Zoran ?koda]] created [[Benabou-Roubaud theorem]]
* [[Urs Schreiber]]
* created [[equivalence of quasi-categories]] -- at the moment just to record a useful lemma
* added various new links to the list at [[Higher Topos Theory]] -- also edited the paragraphs at the beginning a little
* removed the old and meanwhile highly incomplete link list at [[sheaf and topos theory]] and instead included links to our main link list pages on these topics -- would be nice if eventually we'd find the time to write a nice overview and exposition here on par with that at [[category theory]]
* created [[model structure on marked simplicial over-sets]] -- this is used to model the $(\infty,1)$-categorical [[Grothendieck construction]], so I edited the latter accordingly
* renamed [[right proper model category]] to [[proper model category]] and added the missing cases.
* [[Rafael Borowiecki]] is back at [[category theory]].
* [[Urs Schreiber]]
* added an "Idea" section to [[cartesian morphism]]
* started adding details on the relation between $(\infty,1)$[[Cartesian fibration]]s and [[(infinity,1)-presheaves]] to [[universal fibration of (infinity,1)-categories]] -- not sure yet where this material should ultimately go: this is really the $(\infty,1)$-Grothendieck construction. Maybe it should be at [[Cartesian fibration]].
## 2009-08-12
* [[Urs Schreiber]]:
* created [[marked simplicial set]]
* created [[(infinity,1)-category of cartesian sections]]
* [[Toby Bartels]]: Went through Section 1.1 of [[Stone Spaces]]; see the links from there.
* [[Urs Schreiber]]:
* added to [[Grothendieck construction]] the definition in terms of pullback of the "universal Cat-bundle" -- and added pointers to Grothendieck fibration and to category of elements at [[generalized universal bundle]]
* created [[fibrations of simplicial sets]]
* added to [[Kan fibration]] a sentence on right/left Kan fibrations and made [[inner Kan fibration]], [[left Kan fibration]], [[right Kan fibration]] and [[weak Kan fibration]] redirect to it
* made [[lifting property]] a redirect to [[weak factorization system]]
* created [[anodyne morphism]]
* started [[cartesian morphism]] in order to host, among the standard stuff, the $(\infty,1)$-categorical version
* corrected some minor points at [[limit in a quasi-category]] a bit
* created [[Karoubi envelope]] in order to collect references to the $\infty$-version of it
* expanded [[universal fibration of (infinity,1)-categories]]
* added a section "Idea" and a section "Generalization" to [[Grothendieck construction]]
* added an "Idea" section and started a section "Properties" at [[Cartesian fibration]]
* added some links to K-theoretic issues that have meanwhile come into existence at [[Grothendieck construction]].
Concerning the comment there: my feeling is that people pretty consistently use the term "Grothendieck construction" for the reconstruction of a fibration from a pseudofunctor, whereas for the K-theoretic aspect they say "Grothendieck group".
* added pointers to the entry [[Cartesian fibration]] at [[fibered category]] and [[Grothendieck fibration]]
* added to the discussion on universal fibrations at [[stuff, structure, property]] (in the section on logic) a comment about and pointer to the [[universal fibration of (∞,1)-categories]].
## 2009-08-11
* [[Toby Bartels]]:
* I also put up quite the stub at [[trivial bundle]].
* OK, now [[fiber bundle]] exists.
* [[Andreas Holmstrom]] has created a user page, including a link to a blog.
* [[Toby Bartels]]:
* Noticed that [[fiber bundle]] redirects to [[bundle]], but that page merely lists a fiber bundle (linking to itself!) as one of several kinds of bundle, so removed the redirect. So if you thought that that page existed, well, it doesn\'t!
* Noticed links to [[fiber]] and wrote a bit there.
* More edits to [[pullback stability]].
* [[Urs Schreiber]]: added to [[pullback stability]] that of right lifting property and hence that of fibrations -- restructured a bit.
* [[Toby Bartels]]:
* Added a note that there also other kinds of [[pullback stability]]; for example, the sense in which monomorphisms are stable under pullback.
* Some rephrasing and refactoring at [[pretopos]] too.
* [[Urs Schreiber]]: created [[pullback stability]] to satisfy links, but included so far just a pointer to [[commutativity of limits and colimits]]. Maybe we want to split that latter entry into the relevant subentries.
* [[Toby Bartels]]:
* Expanded and rephrased [[topos]].
* Decided to create [[Giraud's axioms]], realised that everything that I wanted to say was already at [[Grothendieck topos]], and so made a redirect instead. (This is not to say that one couldn\'t split it off later, however.)
* [[Urs Schreiber]]: created [[universal fibration of (∞,1)-categories]] and linked to it from [[generalized universal bundle]] and [[limit in a quasi-category]].
* [[Zoran ?koda]]: created [[normal variety]]
* [[Urs Schreiber]] added [[universal colimits]] as the last remaining entry on the four [[Giraud's axioms]] that was still missing at [[(infinity,1)-topos]]. The entry currently points to the relevant discussion at [[commutativity of limits and colimits]] for the content, but is supposed to serve for providing the particular terminology used here.
## 2009-08-10
* [[Toby Bartels]]:
* Contents at [[Stone Spaces]]. I hope to go through at least the first 4 chapters and relearn all of the topology there that I already know, only now in terms of locales (and hopefully constructively). As I do so, I\'ll write it up here.
* Added a note on $2$-[[2-colimit|colimits]] to [[2-limit]].
* [[Urs Schreiber]]: created [[Thomason model category]]
* [[Eric]]: Based on a discussion at the [nCafe](http://golem.ph.utexas.edu/category/2009/08/what_do_mathematicians_need_to.html#c025838), I created [[Online Resources]].
* [[Andrew Stacey]] marked his return with a small but significant observation about extensions of the category of smooth manifolds: if the inclusion of manifolds into an extension category preserves limits and colimits then the extension category _cannot_ be locally cartesian closed. At the moment, this is contained in a remark at the end of the third example of a [[Froelicher space]] - please check my reasoning!
(I'd like to be able to say that I figured this out whilst flying over Damascus, but I think it was actually Novosibirsk.)
## 2009-08-09
* [[Toby Bartels]]:
* Invited [[Hopfian group]] to be generalised to other categories.
* Started [[free group]].
* Noted that the [[free product]] is the coproduct in $Grp$.
* Added a note on the constructive validity of the [[Nielsen?Schreier theorem]].
* [[Zoran ?koda]]: created [[combinatorial group theory]], [[free product of groups]], [[Nielsen-Schreier theorem]], [[Hopfian group]]; quoted references for algebraic proofs of Nielsen-Schreier, somebody should add a reference with explanation of the topological proof (I think Massey's book would do but I do not have it at the moment).
## 2009-08-07
* [[Urs Schreiber]]: added to [[category algebra]] the description of (at least groupoid algebras) in terms of the weak colimit over the constand 2-functor to $Vect-Mod$. That's kind of remarkable. I have to admit to my shame that I wasn't aware of this fact before. It's extracted from Free-Hopkins-Lurie-Teleman's latest, where it is the starting point for a huge story.
* [[Zoran ?koda]]: I will most likely be on vacation for next about 10 days, what means mainly offline, though I hope to contribute with an item here and there within that period. To live safely (=with nlab) when offline I downloaded the whole html version of the site which I backed up online at my institute's server (which is pretty well working). Here is the today's file [nlab.tar.gz](http://www.irb.hr/korisnici/zskoda/nlab.tar.gz), only about 10Mb but with 77 Mb after gunzipping back to tar and about the same after untarring. Of course I will not update this file at least till full return back.
* [[Toby Bartels]]:
* Wrote [[Stone Spaces]] (about Johnstone\'s book); contents to come.
* Edited [[simple object]] and [[semisimple object]] to clarify that the zero object is semisimple but not simple.
* I was pretty sure, Zoran, but since it wasn\'t my field, I wanted to warn you.
* [[Zoran ?koda]] surely Toby, this is true for any fixed object $M$ (I wrote subobject?? hmm last night I entered a wrong building instead of the one in which my flat is). Just created [[coherent sheaf]]; because of severe time constraints it is rather short for a significant entry which will be large in future.
* [[Toby Bartels]]: Rewrote the definition at [[property sup]] slightly; it seems to me that $\Omega$ should be an ascending chain of subobjects of a fixed object $M$, rather than an ascending chain of a fixed subobject $M$ (which I can\'t even parse). That also fits in with a noetherian category\'s having the property, but I mention it here in case I\'m wrong.
* [[Aleks Kissinger]] has joined us, adding examples to [[dagger category]].
* [[Zoran ?koda]] created short entries [[simple object]], [[semisimple object]], [[socle]], [[nilpotent ideal]]; noticed that if I create a redirect ((apples)) for ((apple)) and used it in ((pear)), the entry ((apple)) is NOT listed in the list of "linked from" entries at the bottom of the page. Thus if I link by a redirect name, I will miss the backpointer. This happened with [[simple objects]] listed in [[semisimple object]], but [[simple object]] does not say that it is linked from [[semisimple object]]. Added more to [[artinian ring]]. Toby, I agree that $n Cat$ and $n$-$Cat$ are accepted synonyms, that hyphen looks better than minus, and that there is no <em>a priori</em> rational reason for $R$-$Mod$ as opposed to $R Mod$, however the tradition in math community and in professional typerighting (say in numerous journals of AMS) do not use $R Mod$, at least not noticably often, in favour of other versions, and if nlab has strange (even if abstractly correct) conventions in conventional part of math, it may be less attractive to students and professionals. Additional confusion may arise in confusion the name $R$ or so before $Mod$ with a modifier like $gr$, $dg$, $co$ or alike, which are more often (and with stronger arguments) in tradition written without hyphen.
* expanded [[germ]]; created (Gabriel's) [[property sup]] with redirect [[property (sup)]].
* [[Urs Schreiber]]: expanded and rearranged [[germ]] a bit
## 2009-08-06
* [[Toby Bartels]]: Added to [[germ]] the example that causes me to keep linking to it.
* [[Urs Schreiber]]: created [[germ]], added references to [[group cohomology]] on continuous/smooth case
* [[Toby Bartels]]:
* Moved [[double of algebra with involution]] to [[Cayley-Dickson construction]] to use the nice name, and added the reference where I first learnt about this stuff.
* The problem with '$R-Mod$' is that it contains a minus sign, not a dash at all. I\'ve been removing minus signs from '$R-Mod$', '$k-Vect$', '$n-Cat$' etc because they seem entirely wrong, and replacing them with nothing on the grounds that nothing is needed there (except a space, which should come automatically) in a mathematical formula. If you really want a minus sign, let me know, and I won\'t change it. If you want a dash, you can use `$R$-$Mod$` (which won\'t work inside a displayed equation), `$R\text{-}Mod$`, or `$R‐Mod$` (which is a little funky but arguably the most proper). I will change minus signs to one of those from now on, if that is what you prefer. (Of course, `$_R Mod$` also works.) Compare: '$R-Mod$', '$R Mod$', '$R$-$Mod$', '$R\text{-}Mod$', '$R‐Mod$', '$_R Mod$'.
* [[Urs Schreiber]]: added a section "Details" to the end of [[(infinity,1)-quantity]] to go with the blog discussion [here](http://golem.ph.utexas.edu/category/2009/08/question_on_synthetic_differen.html#c025770)
* [[Zoran ?koda]]: started [[Bredon cohomology]] and a stub for [[Mackey functor]] with few references.
* [[Urs Schreiber]]: created [[superconnection]]
* [[Zoran ?koda]]: created [[double of algebra with involution]],[[alternative algebra]],[[nilpotent element]], [[Azumaya algebra]]. We should have things much more detailed and explicit here, e.g. showing the correspondence between sheaves of Azumaya algebras and abelian gerbes...Though Duskin's construction of Azumaya complex deserves a separate entry of course and the Brower group does as well. Toby why do you use nonstandard $R Mod$ instead of $R-Mod$ or ${}_R Mod$ -- the standard algebraic literature places either dash or places $R$ in subscript...and it is even more weird for right modules with $Mod R$.
* [[Toby Bartels]]:
* Added a constructive bit to [[integral domain]].
* Split [[artinian ring]] (just a stub) off from [[noetherian ring]].
* [[David Roberts]]: added comment to [[fundamental groupoid]] about topology thereon and relation to local connectedness. In other news, I have a (non-academic) job, so as of next week my contributions will slow down a bit.
* [[Zoran ?koda]]: created several shorter entries [[open subscheme]], [[open immersion of schemes]], [[reduced scheme]], [[integral scheme]], [[integral domain]], [[noetherian scheme]] (warning: the listed proposition is not that obvious to prove) and made some changes to [[noetherian ring]].
## 2009-08-05
* [[Toby Bartels]]:
* Wrote [[compactum]].
* Expanded [[algebra for an endofunctor]] to actually include [[algebra for a monad]] (which has redirected there for some time).
* Wrote [[overt space]].
* Numbered and added to the definitions at [[compact space]].
* Wrote [[interval]].
* [[Eric]]: Created [[Hasse diagram]]. It may need polishing to make it technically (and morally) correct.
* [[Sridhar Ramesh]]: The article on [[Kleisli category]] erroneously made the remark that the free functor associated with an Eilenberg-Moore category would necessarily be faithful (as a simple counterexample, consider the monad on Set which sends everything to 1); I've reworded the line which stated this, as well as one other
* [[Zoran ?koda]]: thanks, I wrote this nonsense trying to quickly motivate the existence of two definitions of Kleisli category (I consider the one via morphisms $M\to TN$ more basic than as free algebras in fact) and while writing I said to myself this can't be (faithfulness) but quickly made up some ad hoc false arguments in my head to the opposite and continued writing this :) Somebody should write [[Kleisli object]] as well (following Street 1972 and Lack-Street 2002).
## 2009-08-04
* [[Zoran ?koda]]: created [[coherent module]], [[noetherian category]] (containing also some material on _locally noetherian_ abelian categories), [[Hilbert's basis theorem]], [[noetherian ring]], [[noetherian object]], [[irreducible topological space]], [[noetherian topological space]], [[integrable system]], [[differential topology]], [[general topology]], [[algebraic approaches to differential calculus]], [[skewfield]] (with redirect 'division ring'; I chose 1-word synonym as a main version however). Some users (outside of the contributor community) asked for more browsing access to nlab, namely to be able to descend to entries in specific subfields differently than via the long all pages list, or guessing and google search for notions; thus I gave small contribution to this by expanding [[algebra]], [[topology and geometry]] and [[mathematics]]; I do not think that we should care to ever have something like a complete tree, but having overview entries with lots of links in subfields will create more awareness of non-obvious entries. I also think that the descent from anything like top entries to concrete entries should not be unique in general and link overlap whenever logical and intelligent cross-referencing would not hurt. I have moved much of the material from [[reconstruction theorem]] to [[Lawvere's reconstruction theorem]] and added a link to latter and few more references, mainly for Tannakian and scheme cases.
* [[Toby Bartels]]:
* Discussion at [[directed graph]].
* Added [[characteristic morphism]] to [[characteristic function]].
* [[Zoran ?koda]]: Created [[exact sequence of Hopf algebras]] (well, for now short exact).
Oh John, you gave me an opportunity for recalling some sweet memories: I clumsily used word lazy for a lack of interaction (actually a barrier of shyness/fear between experts and non-experts) at a summer school on geometry and strings in 1999 and got excommunicated there for about a week :)
* [[John Baez]]: What, I'm the first one working on the $n$Lab and it's already past noon (in Paris)? The rest of you must be getting lazy! :-) I added a little bit about cosimplicial objects vs cochain complexes to [[Dold-Kan correspondence]].
## 2009-08-03
* [[Zoran ?koda]]: To assist [[John Baez]]'s links, created [[Eilenberg-Moore category]], [[Kleisli category]] and added details and redirects to [[monadic functor]]. To John's question: I think most basic facts in Tannaka reconstruction including sort of those on coalgebra level are in chapter 3 of [Bodo Pareigis' online notes](http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html); his emphasis is on [[end]]s and coends. However I hope you find Kazhdan's notes, they must be great. I created entry [[Tomasz Maszczyk]] with very interesting past seminar abstract at the bottom relating a new reconstruction theorem coming from nc geometry.
* [[John Baez]]: thanks for the reference to Pareigis' online notes! I'll look at them sometime --- I hadn't known about them.
* [[Urs Schreiber]]: expanded the example list at [[twisted cohomology]] -- in particular added "cohomology with local coefficients" as a special case. Added to the very beginning of [[local system]] a paragraph on how strictly speaking local systems were meant to be such "local systems of coefficients".
* [[Zoran ?koda]]: created [[smooth scheme]], [[flat morphism]], [[smooth morphism of schemes]], [[EGA IV]], [[locally affine space]], [[relativization in algebraic geometry]]. Notice that this is extending a series on regularity/smoothness/differential calculus in algebraic framework which included our earlier entries [[regular differential operator]], [[quasi-free dga]], [[formally smooth morphism]] and so on.
* John and Ramesh thank you for the nice detailed material on Lawvere reconstruction theorem, I'd suggest to create a separate lab entry [[Lawvere reconstruction theorem]] on it, moving the most of the material there (regarding the size and number of other reconstruction theorems waiting to be covered in detail the same way) with a link and short comment on the main [[reconstruction]] page.
* [[John Baez]]: that's fine --- go ahead and do it! We can talk about all the reconstruction theorems on the [[reconstruction]] page, but put details on separate pages. I'd been wanting to write a _book_ about reconstruction theorems, but this will either help me write it or help eliminate the need for doing so.
* [[Zoran ?koda]]: Done, now we have separate [[Lawvere's reconstruction theorem]] and a sentence on its content with the link at [[reconstruction theorem]].
* [[John Baez]] expanded [[reconstruction theorem]] by adding the example Lawvere theories.
* [[Sridhar Ramesh]] expanded this a little more, in particular discussing the infinitary analogue of Lawvere's reconstruction theorem and the monadic perspective on this, as well as cursorily noting how this is all nothing more and nothing less than the Yoneda embedding lemma cast into logical guise
* [[Urs Schreiber]] created [[Chevalley-Eilenberg cochain complex]]
* [[Zoran ?koda]]: posted _[[jibladzeCoeffLargeCats.djvu:file]]_ and linked it to [[crossed profunctor]]. Urs, how the integration approach to diff. forms fits with existance of classes of smooth, $C^1$-only, $L^1$-integrable etc. differential forms and the currents ("differential form-valued distributions"), and it seems it puts n-forms on n-manifolds in special position, than say k-forms on n-manifolds. There is a subject of geometric integration theory where integrability is related to geometric properties like rectifiability (Federer); how this fits with that. And finally with differential forms on singular varieties. It seems to me that this approach has advantages and applicability in some cases, while the easy approach via dualizing vector fields to get 1-forms and then proceeding algebraically in others. One should maybe also compare to Lurie's usage of cotangent bundle in expressing an alternative approach to higher descent.
* [[Urs Schreiber]]: there are many aspects to this, but what I wrote lives in the entirely smooth context. The integration map is that from Reyes-Moerdijk section 4 , which integrates "synthetic" forms over "synthetic" simplices. Technique-wise this is really rather conservative, the only new twist to it is that I am saying: its helpful to arrange some objects that people consider in the synthetic context into cosimplicial objects, that reveals some nice underlying structure.
* [[Urs Schreiber]]:
* created [[(infinity,1)-quantity]] -- some comments:
* this accompanies a blog question [here](http://golem.ph.utexas.edu/category/2009/08/question_on_synthetic_differen.html)
* among other things this aims to provide the full general nonsense $\infty$-version of the statement at the beginning of [[differential form]]
* I could move this to my private web. Let me know if you feel that would be better.
* [[Zoran ?koda]]: created [[crossed profunctor]] (of crossed modules) and [[butterfly]] (papillon). First after M. Jibladze (1990) and second after B. Noohi (2005). I did not draw the diagram but wrote equations explicitly.
* [[Eric]]:
* Gleefully responded to John's gleeful response on [[directed graph]]. Note: I need to learn about this 'resolution' stuff.
* [[Zoran ?koda]]: created [[reconstruction theorem]]. There should eventually be separate entries for each of the main classes and examples, which are listed in the main entry.
* [[John Baez]]: I love the idea of an article full of [[reconstruction theorems]]! I contributed one, perhaps not quite the sort Zoran was thinking --- but I just happened to have it on hand, and I think with some work one can see that it's part of the big family of Tannaka--Krein-like results. By the way, I took a course with Kazhdan in which he went through tons of Tannaka--Krein--like theorems, but I've been unable to find my notes from that course! He started with a very primitive (and thus important) result saying something like this: any $k$-linear abelian category with a faithful functor to $FinVect_k$ is equivalent to the category of finite-dimensional representations of some coalgebra. Does anyone know a reference to this sort of theorem? It's crucial that we use _co_algebras here.
* [[Urs Schreiber]]:
* added something on the dual Dold-Kan correspondence relating _co_ -chain complexes and co-simplicial abelian groups to [[Dold-Kan correspondence]].
* renamed differentials at [[chain complex]] from "$d$" to "$\partial$"
* expanded [[cochain complex]]
* created [[differential forms in synthetic differential geometry]] with two purposes: it reviews the definition found in the literature and then proposes a -- supposedly nicer -- reformulation
* [[Zoran ?koda]]: added a query in [[compact object]]: the stated characterization of categories of $R$-modules is known at least about 3 decades before Ginzburg's lectures. Maybe we should look into classical sources.
* [[John Baez]]:
* Added theorem characterizing categories of $R$-modules to [[compact object]].
* Added same theorem and also the Mitchell embedding theorem to [[abelian category]].
* Gleefully shot down an idea of Eric's over at [[directed graph]] --- but this idea can be resuscitated using the concept of 'resolution'.
* [[Sridhar Ramesh]] added a small note that [[Lawvere-Tierney topology|Lawvere-Tierney topologies]] are the same as (internal) closure operators on truth values (Being new here, I'm not exactly sure what the protocols are; in "Please log all your non-trivial changes", does "non-trivial" mean everything above, say, typo-correction?). I also observed, in the article on [[presheaf|presheaves]], that the representable presheaves on a category of presheaves are precisely those which turn colimits into limits (i.e., a functor of type $[C, Set]^{op} \to Set$ is representable just in case it is limit-preserving).
* [[Urs Schreiber]]: I added a remark to the text above on what "non-trivial" means. Generally, I think you can't log too much of your activity here, just too little. So if in doubt whether anyone else might be interested in your changes or not, drop a note here.
* [[John Baez]]: Welcome, Sridhar Ramesh! I'd like to add: we are not trying to force you to do lots of bureaucratic work, so please don't feel we are _demanding_ you to log changes here. If you make a change and feel too tired to log it here, we won't be mad at you. We just _like_ to see changes here, even smallish ones.
* [[Eric]]: Asked a question of [[Mike Shulman|Mike]] (or anyone else interested in SDG) at [[synthetic differential geometry]].
## 2009-08-01
* [[Sridhar Ramesh]] has joined, editing [[module]] and [[Mitchell-Benabou language]] so far.
* [[Urs Schreiber]]: replied and reacted at [[differential form]]
* [[Toby Bartels]]:
* Archived the [[2009 July changes]]. (Check there if you missed some ... which you did, I promise.)
* More discussion with [[Rafael Borowiecki]] at [[category theory]].
* Answered one of two un-logged questions by [[Jim Stasheff]] at the bottom of [[Timeline of category theory and related mathematics]].
* Asked Urs a question at [[differential form]].
***
[[2008 changes|First list]] --- [[2009 July changes|Previous list]] --- [[2009 September changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 February changes | https://ncatlab.org/nlab/source/2009+February+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during February 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/314) and work backwards.
***
# 2009-02-28
* [[Tim Porter|Tim]] Added new material to [[crossed square]].
# 2009-02-27
* [[Urs Schreiber|Urs]]
* edited [[action]] a bit more -- but that entry is still not nice
* extended [[geometric function theory]]
* continued replying to [[Mike Shulman|Mike]] at [[hyperstructure]]
# 2009-02-26
* [[Urs Schreiber|Urs]]
* created [[action of a category on a set]] in reply to a question raised over at _Secret Blogging Seminar_ [here](http://sbseminar.wordpress.com/2009/02/25/delignes-%e2%80%9cla-categorie-des-representations-du-groupe-symetrique-s_t-lorsque-t-n%e2%80%99est-pas-un-entier-naturel%e2%80%9d/#more-1487)
* [[Andrew Stacey|Andrew]]
* Created a [[request for help]] page. There are query boxes scattered all over the lab and I think that it ought to be a priority to answer these. A query on a page might be answerable by someone not an expert in what's on that page (or even not interested in what's on that page) - in fact, that's highly likely. Flagging questions on this page (i.e., Latest Changes) is also less than optimal as it's not clear who might be able to answer them (and probably not everyone checks this page). I freely admit that I'm doing this because no-one seems bothered in my questions on [[Froelicher space]]s which are, in fact, more general category theory than specifics to Frölicher spaces.
I've added it to the sidebar (and taken off [[General Discussion]]).
# 2009-02-25
* [[Urs Schreiber|Urs]]
* created [[enriched functor category]] for completeness
* added a bit more details and references concerning the SSet homotopy limit at [[weighted limit]]
* after checking with [[Mike Shulman|Mike]] added details of the definition of homotopy limits in terms of weighted limits in the [[SSet]]-enriched context to [[weighted limit]]; also edited accordingly the section _local definition_ at [[homotopy limit]] and created a section on weighted limits at [[representable functor]]
# 2009-02-24
* [[Urs Schreiber|Urs]]
* have a question at the end of [[weighted limit]] for the homotopy coherency experts
* created [[weighted limit]]
* added references to [[sheaf]] and [[simplicial set]]
# 2009-02-23
* [[Tim Porter|Tim]]:
* Created an entry on the Conduché decomposition of the group of $n$-simplices of a simplicial group: [[decomposition theorem for simplicial groups]]. (An idea for renaming this entry would be welcome!)
* Edited [[group T-complex]].
# 2009-02-19
* [[Mike Shulman|Mike]]: created [[n-fibration]].
+--{.query}
Why do I think of the Beach Boys every time I read the word $n$-fibrations? - Eric
=--
* [[Ronnie Brown|Ronnie]]: over the last two days, created [[compositions in cubical sets]] which gives the basic definitions and axioms for cubical sets with compositions and connections, (and inverses), with the example of the cubical singular complex in mind.
# 2009-02-18
* [[Mike Shulman|Mike]]:
* Created [[Euler characteristic]] in order to ask a question there.
# 2009-02-17
* [[Toby Bartels]]:
* Separated [[equivalence of categories]] from [[equivalence]].
* Basically copied [[2-category]] to [[3-category]], then added a bit to the end of each to make the difference nontrivial. Also created [[strict 3-category]] along similar lines, although it\'s still rather short.
* [[Mike Shulman|Mike]]:
* Introduced the terminology _strict [[homotopy limit]]_ by analogy with [[strict 2-limit]].
* Rearranged [[homotopy pullback]] to make it fit better with [[homotopy limit]].
# 2009-02-16
* [[Toby Bartels]]:
* Fixed links to [[equivalence]] that should really be to [[weak equivalence]].
* Separated [[Segal space]] from [[complete Segal space]].
* [[Mike Shulman|Mike]]: asked a question about [[horizontal categorification]].
# 2009-02-15
* [[Tim Porter|Tim]]: Added the construction of $\overline{W}$, the classifying space construction for simplicial groups to the entry for [[simplicial group]].
* [[Urs Schreiber|Urs]]
* created [[topological T-duality]]
* created [[homotopy pullback]]
* created [[weak equivalence]]
* replied at [[dg-category]]
# 2009-02-14
* [[Ronnie Brown|Ronnie]] Added entries on C. [[Ehresmann]] and on [[Grothendieck]]. The second probably should be expanded, but it has a link to a good account.
* [[Tim Porter|Tim]]: Asked a question in [[dg-category]] as to the degree of the differentials. We may need a convention on this.
* [[Urs Schreiber|Urs]]
* added Toën-Vezzosi proposal $2Vect := Ch(Vect)-Mod$ to [[2-vector space]] and created [[dg-category]] accompanying this
# 2009-02-13
* [[Mike Shulman|Mike]]:
* Incorporated the reference Tim suggested at [[homotopy limit]].
* Replied at [[strict 2-limit]] and [[comma category]].
* [[Tim Porter|Tim]]:
* Added something to [[homotopy limit]] in part to provide a possible 'old' reference in reply to Urs and Mike below.
* Started adding some summaries of results to [[group T-complex]].
* [[Toby Bartels]]:
* Imposed the convention for a group $G$, that $\mathcal{B}G$ is a space while $\mathbf{B}G$ is a groupoid, on [[classifying space]] and [[homotopy 1-type]] (although the groupoid has disappeared from that one), to hopefully reduce confusion.
* Linked [[FAQ]] and [[HowTo]] from one another.
* Terminological notes at [[strict 2-limit]] and [[comma category]].
* [[Mike Shulman|Mike]]: Urs, in my biased opinion, the reference for the [[homotopy limit]] comparison that I find easiest to understand is my own paper (reference added). Of course the fact has been known for a long time; I don't even know whether there's any "original" source one could cite.
# 2009-02-12
* [[Urs Schreiber|Urs]]: at [[homotopy limit]] I am asking for further details and/or literature on the "central theorem" which identified local and global definition
* [[Mike Shulman]]:
* Added what I think is a more standard viewpoint to [[classifying space]], although there are so many different meanings of that term.
* Created [[comma object]] and started messing around with using SVG for diagrams instead of `\array`. It's more work but it sure does look nicer (in the output, anyway).
+-- {: .query}
[[Urs Schreiber|Urs]]: looks good! We should have at [[FAQ]] a brief instruction for how to proceed to produce such diagrams (which software to download, how to get nicely typeset labels, etc.)
Mike: I gave it a [[FAQ|try]], but I don't have any great wisdom to impart; I just played around with the XML until it worked. If there is software that can do this for us that would be great (until someone manages to write an xypic-to-svg converter).
=--
# 2009-02-11
* [[Toby Bartels]]:
* Agreed with Mike at [[large site]].
* Replied to Andrew at [[Toby Bartels]].
* [[Mike Shulman|Mike]]
* I like the edits at [[2-limit]] and [[strict 2-limit]], so I removed the comments (but anyone else with opinions, feel free to restart the discussion).
* Naming question for [[large sheaf]].
* Answered at [[model structure on simplicial presheaves]]
* [[Urs Schreiber|Urs]]
* have a question for [[Mike Shulman|Mike]] at [[model structure on simplicial presheaves]]
* created [[Bousfield localization]]
* [[Toby Bartels]]:
* Comments (with related edits) at [[2-limit]] and [[strict 2-limit]].
* Expanded [[section]] and wrote [[retraction]] following it; similarly wrote [[split monomorphism]] following [[split epimorphism]].
* Separated [[k-tuply monoidal (n,r)-category]] from [[periodic table]].
* Linked to [[Pursuing Stacks]] from [[infinity-stack]] so it won\'t be an orphan. People more familiar with _Pursuing Stacks_ than I am may want to link to it in other ways.
# 2009-02-10
* [[Urs Schreiber|Urs]]
* created [[sheaves on large sites]] mainly to ask the question which I ask there
* created [[model structure on simplicial presheaves]], [[model structure on simplicial sheaves]]
* [[Mike Shulman|Mike]]:
* Wrote about effective-epimorphic sieves at [[subcanonical coverage]], by way of answering questions about [[anafunctor]]s.
* Wrote [[comma category]]. We should probably be consistent between [[under category]]/[[coslice category]] and [[slice category]]/[[over category]] about which is the page and which is the redirect. I'm pulling for slice and coslice myself.
* [[Andrew Stacey|Andrew]]
* Started the [[FAQ]]. (The link on [[HomePage]] could possibly be removed - I put it there to start the [[FAQ]] but then also put a link in the sidebar.)
* Okay, only a minor point: I corrected the syntax of the query block on the [[HowTo]] page. The syntax that was there works, but is not the "standard" syntax as defined by the [maruku filter](http://maruku.rubyforge.org/proposal.html).
* [[Toby Bartels]]
* Finally contributed to the discussion at [[anafunctor]].
* Asked a question at [[subcanonical coverage]].
* Wrote [[finite group]], since they\'re mentioned on [[profinite group]].
* Clarified [[essentially algebraic theory]] a bit.
* [[David Roberts]]
* Continued discussion with Mike at [[anafunctor]], and moved the discussion to its own section. I disagreed about the use of ''regular epimorphism'' for covers in categories without all pullbacks.
* Discussion at [[generalized smooth space]], and created (a very sketchy!) page [[profinite group]].
* [[Toby Bartels]]:
* Fixed the theorem environments at [[cat-1-group]] (you need six hash marks).
* Defined $(n,r)$-fold categories at [[n-fold category]], just because I could.
* [[Urs Schreiber|Urs]]
* created [[Courant Lie algebroid]]
* reacted to [[Toby Bartels|Toby]]'s remark at [[n-fold category]] by incorporating it and removing the query box
# 2009-02-09
* [[Mike Shulman|Mike]]:
* Replied at [[internal logic]].
* Merged the discussion at [[directed set]] into the entry and removed the discussion. Writing [[locally presentable category]] and [[accessible category]] is on my to-do list, if no one beats me to it.
* Raised an objection at [[entire relation]].
* [[Toby Bartels]]:
* Removed the discussion at [[internal logic]] and [[entire relation]] (which I thought I\'d already corrected!), since Mike is right and I\'m satisfied.
* Wrote [[entire relation]] and [[functional relation]] (but not yet [[relation]]!).
* Asked a question about $n$-fold groupoids at [[n-fold category]].
* [[Urs Schreiber|Urs]]
* fixed the mistake at [[interval object]] that [[Toby Bartels|Toby]] spotted and added a little bit of dicussion
* added Berger-Moerdijk's definition to [[interval object]]
* [[Toby Bartels]]:
* Made an objection at [[interval object]].
* Talked to Mike at [[directed set]], [[weak limit]], and [[internal logic]].
* [[Mike Shulman|Mike]]:
* Replied to Toby at [[directed set]], [[internal logic]], and [[familial regularity and exactness]].
* Created [[weak limit]].
# 2009-02-08
* [[Toby Bartels]]: Did some work on [[directed object]], which I think should have half spun off to [[undirected object]].
* [[Tim Porter|Tim]]:
* Created an entry on [[Dwyer-Kan loop groupoid]].
* [[Toby Bartels]]:
* A comment at [[directed space]], which led me to write [[directed set]] just to clarify terminology.
* Somewhat incorporated [[Ronnie Brown]]\'s remark on [[geometric shapes for higher structures]].
* Comments on [[pospace]] and [[cartesian monad]].
* Standardized the notation as $\Simp{C}$ for the category of [[simplicial object]]s in $C$, as on [[SimpSet]], [[simplicial set]] (where I dropped an unrelated note to Tim), [[simplicial group]], etc.
* Explained where the name of [[Galois connection]] comes from.
* Added some details to [[cobordism]].
* Wrote [[Practical Foundations]] about Paul Taylor\'s book and linked it from [[internal logic]] (where I also conversed with Mike).
* I really like [[familial regularity and exactness]], which confirms all my prejudices about regularity and exactness; I have a small, largely irrelevant, question.
* I separated [[join]] and [[meet]] from [[union]] and [[intersection]] (which are terms that I\'d use only for subobjects).
# 2009-02-07
* [[Ronnie Brown]]:
*I edited the discussion in [[fibration]] to give a reference to work of P.R. Heath on operations of the paths on the base on the fibres, a groupoid operation up to homotopy.
* [[Toby Bartels]]: I\'m back! I\'ll be slowly catching up. So far, I\'ve simply read **latest changes** and extended Andrew\'s navigation links for the archive. (So it looks like the sort of thing that daily updated websites create automatically using PHP, except that I did it all laboriously by hand!)
# 2009-02-06
* [[Jocelyn Paine|Jocelyn]]: edited the [[product]] page to link to an interactive demonstration written to use the same notation as in [[product]].
* [[Mike Shulman|Mike]]: responded to all of David's edits.
* [[David Roberts]]:
* Edited [[anafunctor]] to clean up conditions on covers, and some explanations of what each one is for.
* Created the page [[Grothendieck pretopology]]
* Added mention of Dold fibration on the page [[fibration]].
# 2009-02-05
* [[Mike Shulman|Mike]]:
* Continued discussion with Urs at [[hyperstructure]].
* Moved [[thin]] to [[thin element]] in line with the [[HowTo|naming convention]] that page names be singular nouns. Also added some other contexts in which "thinness" appears.
* [[Urs Schreiber|Urs]]
* replied to [[Mike Shulman|Mike]] at [[hyperstructure]]
# 2009-02-04
* [[Ronnie Brown]]
* added discussion of _strict versions of higher homotopy groupoids_ to [[fundamental infinity-groupoid]]
* [[Urs Schreiber|Urs]]:
* added a "laboratory section" to [[hyperstructure]] in which I am proposing a formalization of the concept
* created [[dagger category]]; started filling [[dagger compact category]] with content
# 2009-02-03
* [[Andrew Stacey|Andrew]]
* Implemented the archive of latest changes (scroll down to the bottom of this page to get the link)
* Added a fair bit more on Isbell Duality in [[Froelicher space]]s. A few more general queries for those a little more versed in the lore of categories than me.
# 2009-02-02
* [[Eric Forgy|Eric]]
* created [[Position, Velocity, and Acceleration]] to help augment the discussion going on at the [n-Cafe](http://golem.ph.utexas.edu/category/2009/01/the_third_time_is_the_charm.html#c021749)
* [[Andrew Stacey|Andrew]]: continued with [[Froelicher space]]s. In particular, laid out some of the details of the Isbell duality proof.
* [[Urs Schreiber|Urs]]
* added as a further example the computation of
$holim(\mathbf{B}H \to \mathbf{B}G \leftarrow \mathbf{B}H) \simeq H \backslash\backslash G // H$ to [[homotopy limit]]
* many thanks to [[Mike Shulman|Mike]] and [[Timothy Porter|Tim]] for all the further details at [[homotopy limit]]!
* re [[Mike Shulman|Mike]]'s suggestion: yes, I'd be in favor of archiving the latest changes list here more frequently
* [[Mike Shulman|Mike]]:
* Answered Urs's question in a very long-winded way by adding lots of stuff to [[homotopy limit]].
* What would people think about rotating (i.e. archiving) the [[latest changes]] every month, rather than every year? It's pretty big right now and it will probably be immense by next December.
# 2009-02-01
* [[Andrew Stacey|Andrew]]: After crashing the [[HomePage|nLab]] 3 times I've finally managed to upload something on [[Froelicher space|Frolicher spaces]]. Something is seriously wrong with my syntax - if no one's taken pity on me and cleaned it up I'll do so myself tomorrow. More importantly, I have a few queries for the more experienced categorists so I'd appreciate someone taking a look at those for me.
* [[Urs Schreiber|Urs]]: thanks, [[Mike Shulman|Mike]], very useful. Is anything similar known for 3-categories?
***
[[2008 changes|First list]] --- [[2009 January changes|Previous list]] --- [[2009 March changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 January changes | https://ncatlab.org/nlab/source/2009+January+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during January 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/217) and work backwards.
***
# 2009-01-31
* [[Mike Shulman|Mike]]
* Created [[retract]] and [[idempotent]].
* Clarified homotopy limits at [[model 2-category]] and [[strict 2-limit]].
* [[Urs Schreiber|Urs]]
* created [[model 2-category]] and tried to subsume our discussion about it at [[strict 2-limit]] -- would like ask [[Mike Shulman|Mike]] to check
* [[Ronnie Brown]]
* created [[higher dimensional algebra]]
# 2009-01-30
* [[Urs Schreiber|Urs]]
* added more examples to [[homotopy limit]] and [[span trace]] (see [blog comment](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021787))
* added _based loop object_ to _free loop object_ at [[loop space object]] and linked that with [[generalized universal bundle]]
* [[Tim Porter|Tim]]:
* Continued work on [[crossed module]], and [[crossed n-cube]]. I have added in the fibration example into [[crossed module]], and that may be of interest to others. It says, effectively, that a fibration induces a weak crossed module structure on the loop spaces.
# 2009-01-29
* [[Mike Shulman|Mike]]:
* Finally decided I need to get over my Australian training and make weak things the default, at least when writing on the nLab. So I split up [[2-categorical limit]] into [[2-limit]] for the weak notion and [[strict 2-limit]] for the strict one. The exposition here could probably use help, and I have a question about terminology for [[strict 2-limit]]s.
* Redirected [[Gray category]] to the already-existing [[Gray-category]]. Should we have an official policy on the use or non-use of hyphens? (-:
* I haven't done much here recently because I've been working in my [[michaelshulman:HomePage|own web]] on a project developing [[michaelshulman:2-categorical logic|2-categorical logic]].
* [[Eric Forgy|Eric]]:
* changed yet [[generalized universal bundle|another definition]] from _italic_ to **bold**. Recall that we agreed to follow that convention.
* [[David Corfield|David]]:
* Added a remark after Urs' remark on [[philosophy]].
* [[Tim Porter|Tim]]:
* Continued work on [[crossed module]], [[crossed n-cube]] and related entries.
* Linked the [[Dold-Kan correspondence]] to the entry on the [[Moore complex]] where there was already some discussion of this topic.
* [[Urs Schreiber|Urs]]
* have a question/request at [[2-limit]] about "homotopical enrichment". Can anyone say more about this? [[Timothy Porter|Tim]]?
* created [[loop space object]] and included at [[span trace]] as an example the statement that the [[loop space object]] is the [[homotopy limit|homotopy trace]] of the identity span on the object
* have a request at the end of [[2-categorical limit]] that we eventually give a formulation in terms of [[homotopy limit]]s.
* added the defintiion in terms of adjoints to the constant diagram functor to [[limit]]
* replied to [[Eric Forgy|Eric]] at [[generalized universal bundle]]
* created [[Dold-Kan correspondence]] and also [[Dold-Kan theorem]] as a redirect
* created [[simplicial presheaf]], [[SSet]]
* have a remark/question at one point in [[David Corfield|David]]'s text on [[philosophy]]
* started giving a discussion of the point of (higher) category theory in [[physics]]
# 2009-01-28
* [[Ronnie Brown]]
* created/worked on [[cat-n-group]], [[cat-2-group]], [[crossed square]], [[simplicial T-complex]], [[homotopy 3-type]], [[homotopy hypothesis]]
* [[Urs Schreiber|Urs]]
* added examples and a bit more to [[distributor]] and implemented [[Todd Trimble|Todd]]'s suggestion there
* have a question at [[homotopy theory]] on Loday's result
* created [[Gray category]]
* created [[2-vector space]], [[bimodule]], [[distributor]] (and [[profunctor]] as a redirect)
* [[Tim Porter|Tim]]:
* Continued adding stuff to various entries on $cat^n$-groups and references for Loday's theorem at [[cat-n-group]].
* Created [[2-crossed module]].
# 2009-01-27
* [[Urs Schreiber|Urs]]:
* have a question for [[Andrew Stacey|Andrew]] at [[Froelicher space]]
* added a reference to [[dualizing object]]
* created [[cat-n-group]]
* added a bunch of references to [[directed homotopy theory]] which [[Tim Porter]] kindly provided
* started adding/searching for references on $n$-fold groupoids to [[homotopy hypothesis]], added a respective remark to [[n-fold category]]
* [[Tim Porter|Tim]]:
* Continued [[Moore complex]], created [[homotopy 3-type]], included a list of some of the algebraic and categorical models for 3-types (with the intention of starting pages on the various types). (Request help on other models, please.)
# 2009-01-26
* [[Andrew Stacey|Andrew]]:
* Joined the $n$-Community. Hello.
* Modified [[HomePage]] to point to the $n$-Forum.
* Made list of generalised smooth spaces on the [[generalized smooth space|generalised smooth spaces]] page (with the intention of starting pages on the various types).
* [[Tim Porter|Tim]]:
* Created [[Moore complex]] which contains the definition of the homotopy groups of a simplicial group.
* Added query to [[homotopy theory]]: should a summary of the Baues approach to abstract homotopy theory be included somewhere?
* [[Urs Schreiber|Urs]]
* incorporated [[Todd Trimble|Todd]]'s remark into [[span trace]] and created a general entry on [[trace]] in monoidal categories; tried to add some clarification that the point of [[span trace]] and [[co-span co-trace]] was not to describe the concept of trace in general, but to describe how for spans it harmonizes with the interpretation of spans as linear maps in [[groupoidification]] and nicely matches with the fact that on co-spans regarded as [[cobordism]]s it realizes the idea that one glues the two ends of a cobordism together to get the trace -- I found that simple observation [noteworthy](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021486) in the context of the cobordisms hypothesis, in that all [[extended cobordism]]s seem to be generated by the interval just under cartesian product and co-span co-trace, so any map from extended cobordisms to [[multispan]]s which sends pushouts to pullbacks and regards the interval as weakly equivalent to the point should be fixed by its value on the point (that last statement should be scrutinized, but it is what made me want to make the (obvious) notion of co-span co-trace explicit)
* [[Mike Shulman|Mike]]:
* Started trying to bring some order to the treatment of coverages by creating [[coverage]] and working on [[Grothendieck topology]] and [[Lawvere-Tierney topology]]. I wasn't quite bold enough to try to excise the word "topology" from this subject entirely yet.
* Added relevant comments to [[anafunctor]] and [[folk model structure]].
# 2009-01-25
* [[Todd Trimble|Todd]]:
* added more to [[geometric shapes for higher structures]];
* gave an answer to Eric's question at [[globe]]
* took some issue with Urs about whether span traces (and implicitly, cospan traces) have or haven't appeared in the literature
* [[Tim Porter|Tim]]:
* Created [[pospace]] to help the entry on [[directed space]], at which I gave a link through to the new page.
* Created [[directed homotopy theory]]. This is at present a stub plus an inadequate list of references that do not do justice to the area ... as yet. I have built in some links but feel there should be others.
* Created [[group T - complex|group T - complex]], but more needs adding here.
# 2009-01-24
* [[Urs Schreiber|Urs]]
* somewhat hastily created [[fundamental infinity-groupoid]] (have the vague recollection that I wrote something like that before... but can't see it anymore...)
* created [[cobordism]] and [[extended cobordism]] and included a bunch of literature related to that and to [[multispan]], notably to [[Marco Grandis]]' work on [[Cospans in Algebraic Topology]] and to [[Jeffrey Morton]]'s work on [[multispan]]s ans [[extended cobordism]]s
* further details at [[multispan]] (or in the file linke to there)
* created an entry on Nils Baas' concept of [[hyperstructure]]s, mainly to point out how close it is at least in spirit to [[span]]s appearing in [[groupoidification]] -- and in fact to [[multispan]]s
* created [[span trace]] and [[co-span co-trace]]
(I am not sure about the (best) spelling convention!)
after I had checked with [[John Baez|John]] that this has not previously been said explicitly in the literature
* [[Tim Porter|Tim]]:
* Created [[simplicial group]]s which was needed by several entries.
* Commented in [[simplicial set]] about a notational problem that needs attention. The notation for face and degeneracies in [[simplicial set]] is at odds with the standard one in _the literature_. \(*Now fixed*\)
* Created [[simplicial identities]].
* Created [[simplicial object in Cat]] referring to [[simplicial object]] as it was a 'hanging link'in [[simplicial category]].
# 2009-01-23
* [[Mike Shulman|Mike]]:
* Replied at [[cartesian monad]], [[2-categorical limit]], [[internal logic]], and [[regular monomorphism]].
* Responded to David's question at [[internal logic]] by adding mention of [[sketch]]es and of disjunctive and geometric logic.
* Created [[familial regularity and exactness]].
* Added the internal version to [[anafunctor]].
* [[Urs Schreiber|Urs]]
* worked on [[directed object]]:
* added discussion of degenerate examples of interval objects
* suggested (and inserted) a formalization of the reparametrization axiom that [[Toby Bartels|Toby]] had left open
* replied to the discussion at [[directed space]] asking Toby to suggest how to deal with the slight terminological subtlety we're running into
* added references to Fahrenberg and Raussen at [[directed space]] (there must be more literature on _directed homotopy theory_! somebody should look it up)
* replied to [[Toby Bartels|Toby]]'s remark on the need for directed homotopies at [[directed space]].
* [[David Corfield|David]]:
* Asked a question at [[internal logic]] and another at [[regular monomorphism]]. Thanks for starting this page, Mike. There's still so much I don't understand about this topic.
* [[John Baez]]:
* Wrote [[pushout]] in the same gentle style as the previous article [[pullback]]. I want nice easy introductions to all our favorite limits and colimits!
* Wrote brief stubs for [[colimit]] and [[totally ordered set]].
* [[Toby Bartels]]:
* Content:
* Wrote [[context]], an idea that should be better appreciated.
* Tried a 'constructive' definition at [[directed object]], with relevant comments at [[directed space]].
* Wrote about the [[axiom of choice]] in superextensive sites.
* Discussed Banach spaces at [[concrete category]].
* Discussion:
* Asked a terminological question at [[cartesian monad]].
* Comments on comments at [[monoidal category]].
* Made a terminological suggestion at [[2-categorical limit]].
* Wiki structure:
* Set up [[ambimorphic object]] as a redirect.
* Separated [[finite object]] from [[finite set]].
* Moved [[Trimble's notion of weak n-category]] to [[Trimble n-category]].
* Other, more minor, edits.
# 2009-01-22
* [[Urs Schreiber|Urs]]
* created [[(infinity,n)-category of cobordisms]]
# 2009-01-21
* [[Mike Shulman|Mike]]:
* Finally got around to creating [[internal logic]], which coincidentally probably answers David's questions at [[regular category]].
* [[David Corfield|David]]
* Asked some questions at [[regular category]].
* [[Mike Shulman|Mike]]:
* Changed [[enriched homotopy theory]], to reflect my feeling that it is more general than [[homotopy coherent category theory]].
* Created [[enriched factorization system]], [[orthogonality]], and [[Galois connection]].
* Created [[split epimorphism]], [[strong epimorphism]], and [[extremal epimorphism]] just to satisfy links.
* [[Eric Forgy|Eric]]
* asked a question about "directed internalization" on [[directed space]]
* continues to have a brain aneurysm at [[directed space]] but is making progress
* [[Urs Schreiber|Urs]]
* added a bit of [[John Baez|John]]'s material to [[span]]
* created [[complete Segal space]] and [[(infinity,n)-category]]
* added diagrams to [[weak factorization system]] and created a stub entry [[monomorphism]] to saturate all the wanted links to it
# 2009-01-20
* [[John Baez|John]]:
* Since Urs and Eric didn't respond to my question about deleting the discussion in [[monoidal category]] (see 2009-01-19), I went ahead and deleted it --- restore it if you like, after reading the comments at the end of that entry!
* The picture of the pentagon identity in [[monoidal category]] has mysteriously disappeared, though the source code is still present. Help!
* I added [[endofunctor]] and [[strict monoidal category]].
* [[Mike Shulman|Mike]]:
* Replied at [[fibration]], [[directed object]], and [[homotopy hypothesis]].
* [[Urs Schreiber|Urs]]
* commented in the discussion at [[fibration]] on the use of the word _transport_
* [[Tim Porter|Tim]]
* I have added a question to [[homotopy hypothesis]] asking what criteria should be added to give 'good' categorical or groupoidal models for homotopy types.
* For [[directed object]] I have, similarly, tried to pose question about the criteria that should be 'directing' our search for good concepts in this case.
* I have also added a question about the 'optimal' definition of [[fibration]], as it seems to me that the lifting property is nearer the idea of fibration than the [[transport]] one that Mike has put forward.
* [[Mike Shulman|Mike]]:
* Created [[fibration]], [[Grothendieck fibration]], and [[pseudofunctor]].
# 2009-01-19
* [[Urs Schreiber|Urs]]
* separated [[directed object]] from [[directed space]] and included the definition of directed topological space by Grandis
* further reacted at [[directed space]] and created [[homotopy hypothesis]]
* following discussion by [[Eric Forgy|Eric]] at [[directed space]] I propose in the discussion section a formalization of the notion "an object $X$ is _directed_" and "an object $X$ is _undirected_" for the case that $X$ is an object in a category with [[interval object]].
* [[John Baez|John]]:
* expanded the entry on [[monoid]], giving lots of examples of monoid objects in monoidal categories. I think lists of examples like can be very useful and fun, and I want more! I would like a list of [[PROP|PROPs]], for example, saying that $FinSet$ is the PROP for commutative monoids, and so on.
* If Urs is happy with how the discussion at the end of [[monoidal category]] has been incorporated into the body of the article, maybe we can remove that discussion.
* slightly expanded the entry on [[braided monoidal category]] - but it really needs some diagrams!
# 2009-01-18
* [[Mike Shulman|Mike]]:
* Added detail, examples, and terminological comments to [[bicategory]].
* Created [[2-categorical limit]].
* Replied at [[finite set]].
* [[Todd Trimble]]
* added a bit of material on terminal coalgebras to [[coalgebra]].
* had an organizational thought at the end of [[coalgebra]].
* [[Toby Bartels]]
* Adjustments to [[source]], [[target]], and [[identity assigning morphism]].
* Comments and questions at [[finite set]], [[internal category]], and [[bicategory]].
* [[Urs Schreiber|Urs]]
* created [[bi-pointed object]] and started to describe what should be its closed monoidal structure to be used at [[interval object]], but are being interrupted now and have to run -- hope this is about right so far...
* [[Eric Forgy|Eric]]:
* Plagiarized the definition of [[internal category]] from [Baez and Crans](http://arxiv.org/abs/math/0307263) since the previous definition stated things should work in "the obvious way", which was not obvious to me
* Asked if [[directed space|directed spaces]] could be defined using [[interval object]] on the directed spaces page.
* [[Toby Bartels]]:
* Added other terms and meanings to [[decidable object]]; some of this should probably go to other (new) pages.
* Clarified my question at [[interval object]].
* Asked an organisational question at [[Trimble's notion of weak n-category]].
* Other minor edits.
* [[Mike Shulman|Mike]]:
* Added another argument in favor of using $B$ at [[category algebra]].
* Expanded the topos-theoretic discussion at [[finite set]], including some examples.
# 2009-01-17
* [[Todd Trimble|Todd]]
* responded to a plea of Eric at [[internal category]] to discuss an example; cf. Tim's latest change below.
* [[Tim Porter]]
* I created [[2-group]]. I have only as yet discussed the strict form. I also added in some discussion of the Brown-Spencer theorem and took the definition of 2-group apart a little partially in response to [[Eric]]'s comment on [[internal category]].
* I fixed the definition of [[internal category]]. Somehow the condition that the composites $s\cdot i$ and $t\cdot i$ had been omitted, as far as I could see at least.
* [[Emily Riehl]]
* created [[small object argument]]
* [[Urs Schreiber|Urs]]
* continued fiddling with [[interval object]]
* added references to [[Trimble's notion of weak n-category]]
# 2009-01-16
* [[Mike Shulman|Mike]]
* Added the internal version to [[group]].
* Created [[well-powered category]].
* [[Jim Stasheff]] added a word of caution at [[Hopf algebra]]
* [[Urs Schreiber|Urs]]
* added references to [[interval object]] which should be moved to [[Trimble's notion of weak n-category]] (which is however locked at the moment and I have to run now)
* meant to further develop [[interval object]] but got distracted -- the little I could get done is on [pages 5,6,7](http://ncatlab.org/schreiber/files/nacqJan15.pdf#page=5) [[schreiber:Nonabelian cocycles and their sigma model QFTs|here]]
* replied to discussion at [[interval object]]
# 2009-01-15
* [[Toby Bartels]]: Wrote [[algebraic theory]], mostly to distinguish it from [[Lawvere theory]].
* [[Mike Shulman|Mike]]:
* Toby and I appear to have finally reached a consensus on terminology at [[extensive category]]. I did some work on this page and [[coherent category]], [[pretopos]], and [[Grothendieck topos]] making them more consistent, and also created [[disjoint coproduct]] and [[coproduct]].
* Responded to Toby at [[Lawvere-Tierney topology]].
* [[Tim Porter]] I have added in a new entry on the notion of a [[ simplicial T-complex]], which should help provide background for the still needed entry on complicial sets.
* [[Todd Trimble]]:
* constructed an entry for "my" notion of weak $n$-category. Needs more revision and more work, but in part it was written in response to discussion between Urs and me (see e.g., [[interval object]]).
* [[Urs Schreiber]]
* to saturate links from [[Trimble's notion of weak n-category]] and [[interval object]] I created [[A-infinity algebra]] and [[A-infinity category]] and filled in a little bit of information
* started proposing a full definition at [[interval object]] following the discussion at that entry and at [[Trimble's notion of weak n-category]] -- please check!
* the entry [[smooth Lorentzian space]] is currently oscillating a bit between showing a desire to state a new nice definition for (possibly generalized) Lorentzian spaces and the aim to first establish some facts about the usual definition
* filled in stuff at [[causet]]
* added **questions** to [[Trimble's notion of weak n-category]]
* expanded on the examples in [[generalized universal bundle]]
* [[Toby Bartels]]: I added stuff that I\'m trying to understand to [[Lawvere-Tierney topology]]. It is probably correct, but I had some questions. (See also the relevant terminological discussion at [[Grothendieck topology]].)
# 2009-01-14
* [[Mike Shulman|Mike]]:
* Added detail about internal-homs and smash products to [[pointed object]], in response to Todd's comment.
* Created [[adjunct]].
* [[Urs Schreiber|Urs]]:
* started [[generalized universal bundle]] after all -- now I really need to run...
* created [[pointed set]] and [[under category]], am planning to expand on the bigger story indicated at [[pointed set]], but not today
* expanded the entry [[bi-brane]] in order to accompany some discussion about [[geometric function theory]] at [Ben-Zvi on geometric function theory](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html)
* said more at the beginning of [[interval object]] -- and are having a discussion with [[Todd Trimble|Todd]] about examples and the necessity to relax the structure, and his Trimblean definition of $\infty$-category -- which we ought to have an entry about
* in the entry [[category of fibrant objects]]
* added comments on relation to [[pointed object]] and [[pointed category]]
* made the _factorization lemma_ more explicit and related it to the notion of [[Morita equivalence]]
* created a `categories: reference`-entry for [[Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
* added a remark on the definition of kernel and cokernel of morphisms to [[pointed object]] -- some expert please have a look at that
* added [[pointed category]] and also a handful of trivial entries which are actually superfluous, since I hadn't had my first coffee yet and wasn't paying proper attention...
* [[Mike Shulman|Mike]]:
* Incorporated the discussion into [[k-tuply monoidal n-category]] and added more examples. Perhaps some of this page would better go at [[delooping hypothesis]]?
* Created [[pointed object]] and [[spectrum]].
# 2009-01-13
* [[Toby Bartels]]:
* Conversations, including a new one with [[Todd Trimble|Todd]] at [[balanced category]].
* Revised [[k-tuply monoidal n-category]]. Later I need to revise [[periodic table]] and write [[k-connected n-category]].
* Wrote [[Elephant]] and [[Categories Work]], creating `category: reference` for them.
+--{.query}
[[Urs Schreiber|Urs]]: concerning [[Toby Bartels|Toby]]'s
`category: reference`: I like that.
I was thinking about including separate entries on references, too. We have a couple of further references already which we refer to often enough to justify a separate entry for them eventually
=--
* [[Urs Schreiber|Urs]]
* created [[derived infinity-stack]], so far containing a sketch of the rough idea, a few references and pointers to current discussion on the blog
* [[Mike Shulman|Mike]]:
* Created [[bijection]] and [[subsingleton]].
* Answered Toby's three questions, and continued discussion at [[inhabited set]].
* [[Toby Bartels]]:
* I took part in conversations that are already listed here.
* I have questions for Mike at [[k-tuply monoidal n-category]], [[choice object]], and [[finite set]]. (Listed in order of decreasing importance.)
# 2009-01-12
* [[Mike Shulman|Mike]]:
* Joined the discussion at [[category algebra]].
* Removed the comments at [[constructivism]] that have now been incorporated into the main text (thanks for writing that detailed section, Toby).
* Created [[finite set]] and [[choice object]].
* [[Tim Porter|Tim]]
* I have asked on the page relating to [[action]] if the definition needs to be slightly wider for optimal later use.
* [[Todd Trimble|Todd]]
* added some discussion to [[classifying topos]], linking up to the entry [[Lawvere theory]]
* created entry on [[Cauchy complete category]]
* [[Toby Bartels]]:
* I\'m having a conversation with myself at [[local ring]].
* I\'m otherwise caught up and will rewrite [[k-tuply monoidal n-category]] before I do anything else.
* [[Mike Shulman|Mike]]:
* Removed the discussion at [[field]], since it had converged and been incorporated into the entry.
* Incorporated the apparent conclusion of the discussion at [[extensive category]] into the entry.
* Added some details to [[excluded middle]].
# 2009-01-11
* [[Todd Trimble|Todd]]
* began, at Urs's request, an entry for [[Yoneda reduction]]
* finally submitted draft of regular category, after having it locked for a few days and doing very little with it in the end.
* would like to point out that material relevant to entries like [[universal property]], [[universal construction]] and [[Yoneda lemma]] exists at [[representable functor]]; this could use further expansion
* [[Urs Schreiber]]:
* created a separate entry [[Yoneda embedding]], even though the information there is also recalled in [[Yoneda lemma]]
* [[Toby Bartels]]:
* I created [[local ring]] and [[excluded middle]].
* Discussions seem to be converging at [[constructivism]], [[extensive category]], [[field]], [[inhabited set]], [[k-tuply monoidal n-category]], [[predicativism]], and [[Grothendieck topology]] (whew!).
* However, the discussion at [[(-1)-groupoid]] has become [a Café post](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021235).
* I don\'t consider myself caught up yet, so if you\'re waiting for me to do something ... you\'ll have to wait another day.
# 2009-01-10
* [[Urs Schreiber|Urs]]:
* created [[Kan fibration]] and edited [[horn]], [[hypercover]] and [[Kan complex]]
* added remark on relation to [[FQFT]] at [[generalized tangle hypothesis]]
* have some questions for [[Toby Bartels|Toby]] (or other topos experts) at [[(-1)-groupoid]] -- see also my general discussion question at the blog: [enriched sheaf and topos theory?](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021224)
* [[Toby Bartels]]:
* I wrote [[Boolean topos]], [[COSHEP]], and [[finitism]].
* I expanded [[equivalence relation]] and [[kernel pair]].
* I wrote [[(-1)-groupoid]]; I\'ll try to do one of these a day, but they\'re dully repetitive.
* I talked with Mike at [[k-tuply monoidal n-category]] and [[inhabited set]].
# 2009-01-09
* [[Toby Bartels|Toby]]:
* I talked with Mike (and occasionally made substantive edits, too) at [[field]], [[Grothendieck topology]], [[extensive category]], [[predicativism]], and [[constructivism]].
* [[Mike Shulman|Mike]]:
* Created [[coherent category]], [[Boolean category]], [[Heyting category]], [[equivalence relation]], [[kernel pair]], [[exact category]], [[balanced category]], and [[pretopos]].
* Finally got fed up with the nonexistent links to it everywhere and created [[product]]. Also created [[generalized the]].
* Did some rephrasing at [[infinity-category]].
* [[David Corfield|David]]
* started a page on [[generalized tangle hypothesis]].
* [[Urs Schreiber|Urs]]:
* created [[directed space]]
* added a bit to [[David Corfield|David]]'s entry on the [[generalized tangle hypothesis]]
* reacted to [[Mike Shulman|Mike]]'s comments at [[infinity-stack homotopically]]
* [[Mike Shulman]]:
* Continued my discussions with Toby at [[Grothendieck topology]], [[extensive category]], and [[constructivism]], including one that should be moved to [[predicativism]].
* Started new discussions at [[infinity-stack homotopically]], [[k-tuply monoidal n-category]], and [[inhabited set]].
* Added the unit axiom to [[monoidal model category]].
# 2009-01-08
* [[Toby Bartels]]
* Mike and I are discussing terminology at [[Grothendieck topology]] and [[extensive category]] (possibly also elsewhere).
* I wrote [[regular epimorphism]] and [[epimorphism]].
* I prepared the wiki for an article [[closed category]] (as such) by moving most of the material to [[closed monoidal category]] and fixing the links. (But I didn\'t actually write it yet.)
* [[John Baez]]
* worked pertinent discussion into body of [[monoidal category]]
* [[Urs Schreiber]]
* reacted to the discussion at [[path object]]
* created [[cylinder functor]]
* added my 2cent and my questions to the discussion at [[homotopy]] on [[interval object]]s -- and added the entry [[interval object]] but filling it not with a statement but with discussion: I say what I came to think a well behaved interval object should satisfy and am asking for information on what is known in this regard
* created [[pushout-product axiom]] and linked to it from [[monoidal model category]]
* tried to provide a more unified perspective on [[infinity-stack]] and [[infinity-category]] and [[nonabelian cohomology]] at [[infinity-stack homotopically]]
* had the honor of creating [[Yoneda lemma]]
* created [[geometric function theory]] to accompany a [guest post](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html) by [[David Ben-Zvi]] -- maybe somebody of the [[groupoidification]] team can eventually add an entry on [[Hecke algebra]] and its relation to groupoidification
* [[Toby Bartels]]: I wrote about [[constructivism]] and the [[empty set]].
* [[Mike Shulman|Mike]]
* Created [[geometric morphism]], [[locale]], and [[sober space]].
* Added a more classical version to [[homotopy]].
# 2009-01-07
* [[Todd Trimble|Todd]]
* created [[regular category]]
* created [[image]]
* created [[Rel]]
* created [[Lawvere theory]]
* started some discussion at [[simplex category]]
* [[Timothy Porter]] created
* [[simplicial complex]]
* [[simplicially enriched category]]
* [[horn]]
* [[nerve]] (but see below)
* [[Urs Schreiber|Urs]]
* gave the bare definition of [[crossed module]]
* finally created [[algebraic definition of higher category]] and [[n-fold category]]. Just stubs so far, this deserves much (much) more discussion, clearly.
* did [[geometric realization]]
* started filling in [[nerve]] after I found an empty entry of that name -- now [[Timothy Porter]] sends me by email his version. Either he or I should merge the material...
* added details (definition and theorems) to [[category of fibrant objects]]
* restructured the index page [[mathematics]] slightly and tried to complete the lists of keywords at [[foundations and logic]]. Can I suggest that everybody who creates a new entry considers adding a link to that entry, if appropriate, to the lists of "related entries" at one of the big index pages? I am thinking it would be good to give readers a chance to browse our material by topic and get an impression for what is there and what not (yet).
* added references to [[Timothy Porter]] and Cordier at [[homotopy coherent category theory]]
* Expanded slightly on [[Timothy Porter]]'s entry on [[simplicially enriched category]].
* added a few remarks at [[higher category theory]]. I am not happy with that entry. Clearly we need a more comprehensive discussion there eventually.
* [[Mike Shulman|Mike]]: Created [[constructivism]] and imported the relevant discussion from [[apartness relation]].
# 2009-01-06
* [[Toby Bartels]]: [[Mike Shulman]] and I are having terminological discussions. Also, he fixed my theorem at [[extensive category]] (which will go in our paper, John).
* [[Urs Schreiber|Urs]]
* further expanded on a bit and harmonized a bit more the circle of entries [[globe category]], [[simplex category]], [[cube category]] and [[globe]], [[simplex]], [[cube]] linked to and summarized in [[geometric shapes for higher structures]]. At [[globe]] I give a reply to a question by Eric on how to think of globes as "pointed spheres" by stating a claim that the $(n+1)$-globe is the double cone over the $n$-globe in a precise sense. I believe this is true, but am not entirely happy with the proof I have. Would be great if somebody could check this.
* [[Mike Shulman|Mike]]
* Created [[power]] and [[copower]]. Possibly these should be just one page?
+--{.query}
I think it is good to have many separate entries for sub-concepts if they all link to each other, maybe with a brief comment. --[[Urs Schreiber|Urs]]
I agree. —Toby
=--
# 2009-01-05
* [[Toby Bartels]]: I reacted to everything that everybody did this year, and I interacted with everything that everybody did today. More specifically:
* I created [[parallel morphisms]], although perhaps I should have called it [[parallel pair]].
* I added material, some possible irrelevant, to [[global element]].
* I fixed typos in the stuff that I posted this year and copied some relevant bits to other pages, like [[2-category]].
* I added some false material to [[extensive category]].
* I added my opinion to [[Grothendieck topology]] and [[subcategory]].
* I followed Mike\'s changes to [[simplicial category]] and applied them to [[cubical category]], creating [[cube category]] and (for good measure) [[globe category]].
* I disambiguated links to [[simplicial category]], [[omega-category]], and [[internalization]].
* I gave my favourite example of the [[red herring principle]].
* I probably did some other stuff too, which I can no longer recall.
* [[Mike Shulman|Mike]]
* Split off [[internal category]] from [[internalization]]. Probably a lot of links need to be updated.
* Summarized the discussion at [[subcategory]] into a section called "Non-evil variants."
* Created the entry [[red herring principle]].
* [[Todd Trimble|Todd]]
* created the entry [[CW complex]]
* added an alternative definition of [[cube category]], linking it to $\Delta$ and string diagrams
* [[Urs Schreiber|Urs]]
* am reading [[Mike Shulman|Mike]]'s [Homotopy limits and colimits and enriched category theory](http://arxiv.org/abs/math.AT/0610194v1) and started adding some central definitions as entries here, such as [[closed monoidal homotopical category]] and [[enriched homotopical category]] and some related entries such as [[homotopical category]] and [[homotopy coherent category theory]]. Eventually I would like to see if some of my favorite monoidal homotopical categories are _examples_. See my request [here](closed monoidal homotopical category).
* added [[Crans-Gray tensor product]]
* started replying to [[Todd Trimble|Todd]]'s remarks at [[stack]] by addding a section _Descent in terms of pseudo-functors_ to [[descent and codescent]].
* following the discussion which I now moved to [[discussion on terminology -- omega-category]] I moved the original material on that to [[strict omega-category]] and kept at [[omega-category]] just a pointer to that entry. This means we should go through a bunch of entries and rename links to [[omega-category]] into links to [[strict omega-category]].
* [[David Corfield]]
* created [[philosophy]], [[Klein 2-geometry]]
* [[Emily Riehl]]
* created [[model structure on simplicial sets]]
* [[Mike Shulman]]:
* Added [[nice topological space]].
# 2009-01-04
* [[Mike Shulman]]:
* Added disambiguation comments to [[simplicial category]].
* Created [[Gray tensor product]] and [[Gray-category]].
* [[Toby Bartels]]:
* I posted [[k-tuply monoidal n-category]]. I\'ll correct typos and look at everything else tonight (when it\'s technically -05).
* Urs is right about $\infty$ vs $\omega$, and I\'ve changed the other appearances in [[(n,r)-category]] and [[periodic table]] (as well as fixing the new one!).
* [[Mike Shulman]]:
* Added a terminological suggestion to [[omega-category]].
* Refactored [[tensor product]], removing the discussion which prompted the refactoring. Created [[fc-multicategory]].
* [[Urs Schreiber|Urs]]:
* meant to follow up on [[Todd Trimble|Todd]]'s discussion of [[sheaf]] and [[stack]] in terms of [[sieve]]s and relate it to the entry on [[descent and codescent]] and [[infinity-stack]] but was distracted by other things. But I did write up a quick rough note on this into a LaTeX file, which you can see here for the time being: [[if.pdf:file]].
* expanded on the [[sheaf]]-condition, offering in parallel to the definition in terms of [[sieve]]s the definition in terms of [[oriental]]s with a a few remarks on how both are related and what the issue is with generalizing to [[stack]] and [[infinity-stack]].
* added, following email discussion with [[Todd Trimble|Todd]], a discussion to [[sieve]] of which sieves are the presheaf incarnations of which [[cover|covers]], and why
* adapted [[n-category]] and [[n-groupoid]] to [[Toby Bartels|Toby]]'s [[(n,r)-category]] (which is a great entry -- we should flesh that out further).
* took the liberty of changing in [[Toby Bartels|Toby]]'s entry [[(n,r)-category]] the "$\omega$-categories" appearing there to "$\infty$-categories", since I thought we did agree that "$\infty$-category" is the generic term while [[omega-category]] specifically means _strict globular $\infty$-category_.
* separated [[infinity-category]] from [[higher category theory]] and added some brief stub entries such as [[n-category]], [[n-groupoid]] and [[Kan complex]].
* created a _blog_ entry [nLab -- General Discussion](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html) and added a remark at the beginning of the original _wiki_ entry [[General Discussion]] asking to post further general discussion not to the wiki, but to the blog.
* rearranged [[tensor product]] following the comments and corrections by [[Toby Bartels|Toby]] and [[Mike Shulman|Mike]] (thanks!).
# 2009-01-03
* [[Mike Shulman]]:
* there is a discussion going on at [[subcategory]] over whether any faithful functor should be called a "subcategory." Input welcomed.
* added a comment on terminological objections to [[Grothendieck topology]].
* corrected the example of abelian groups at [[tensor product]].
* added examples and comment on non-monoidal closure to [[closed category]]
* [[Todd Trimble]]:
* added to (and edited) examples under [[closed category]]
* created [[dinatural transformation]]
* added examples to [[extensive category]]
* added to the discussion on [[Day convolution]]
* added some detail to the definition of [[Lawvere-Tierney topology]]
* [[Urs Schreiber]]:
* did [[closed category]] and linked to it from Mike's [[cartesian monoidal category]]
* expanded on [[Day convolution]]
* split the information previously contained in [[site]] into several separate entries: [[cover]], [[sieve]], [[Grothendieck topology]], [[Lawvere-Tierney topology]] and, of course, [[site]].
* [[Mike Shulman]] created [[extensive category]], [[cartesian monoidal category]]
* [[Todd Trimble]] expanded on [[stack]]
and connected it to the discussion of covering sieves at
[[site]]
* [[Toby Bartels]]: I posted [[periodic table]] and [[(n,r)-category]], which I wrote today offline. There are probably lots of typos right now; I don\'t have a lot of time online today. Also, the former one really only contains an appendix that I wanted to write, not so much the article itself. But at least you can read what\'s there now.
# 2009-01-02
* [[Todd Trimble]] created [[operad]]
* [[Urs Schreiber|Urs]]: started adding remarks to [[opposite category]] about the big story hidden here related to [[duality]].
* [[Todd Trimble]] has expanded on [[site]] and [[sheaf]]. I have started adding a bit more motivational detail to [[site]] and links to [[descent and codescent|descent]], [[stack]] and [[infinity-stack]]. There is still plenty of room here for saying more about this general story.
* [[Urs Schreiber|Urs]]: motivated by some [discussion](http://golem.ph.utexas.edu/category/2008/12/organizing_the_pages_at_nlab.html) we had on the blog, in the entry [[About]] I started giving some indications on what the $n$Lab is and what it is not.
* [[Owen Biesel]]: added [[diagram]], [[global element]], [[terminal object]], [[limit]].
# 2009-01-01
Apparently, we all took a break from the $n$Lab for the New Year!
***
[[2008 changes|First list]] --- [[2008 changes|Previous list]] --- [[2009 February changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 July changes | https://ncatlab.org/nlab/source/2009+July+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during July 2009.
The substantive content of this page should **not** be altered.
***
## 2009-07-31
* [[Mike Shulman]] fixed a mistake at [[axiom of foundation]].
* [[Ronnie Brown]]: Added in [[connection on cubical sets]] a reference to a recent preprint of Maltsiniotis showing that cubical sets with connections form a _strict test category in the sense of Grothendieck_, thus correcting a well known disadvantage of cubical sets in comparison with simplicial sets.
* [[Zoran ?koda]]: added a version (in my understanding) of main definitions in [[almost scheme]] and made a longer heuristical quote from Gabber-Romero Introduction chapter. Created stub [[Catégories Tannakiennes]] about the Deligne's seminal paper and [[Des catégories abéliennes]] about Pierre Gabriel's thesis.
* [[Urs Schreiber]]
* wrote a long "Idea" section at [[twisted cohomology]] and polished/expanded the rest, following comments by [[Jim Stasheff]]
* added to the "Idea" section at [[differential form]] a paragraph that gives some categorical or otherwise abstract nonsense description of how to make precise the statement that "a differential form is something that may be integrated".
* created an entry [[Models for Smooth Infinitesimal Analysis]] (so far containing just a summary and an incomplete link list) and linked to it from [[synthetic differential geometry]] and [[Ehresmann connection]]
I am thinking that section IV there, _Cohomology and Integration_ would be a good candidate for the high-brow abstract nonsense aspect section to be written at [[differential form]]
* added expositional text to [[Kan fibrant replacement]] prodded by the blog discussion [here](http://golem.ph.utexas.edu/category/2008/12/groupoidification_from_sigmamo.html#c025677)
* [[Toby Bartels]]:
* Created [[measure theory]], just because it was linked, to link other things from it.
* Additions to [[regular space]], [[separation axiom]], and [[Hausdorff space]].
## 2009-07-30
* [[Urs Schreiber]]:
* edited the intro to [[algebraic K-theory]] a bit more
* added to [[Waldhausen category]] the Weibel reference
* linked to [[Grothendieck group]] from [[K-theory]] and [[algebraic K-theory]] and [[Waldhausen category]]
* added to [[Waldhausen category]] an example section with the examples needed at [[Grothendieck group]]
* added various things to [[Grothendieck group]]
* [[Toby Bartels]]: More discussion at [[Grothendieck group]] and [[(infinity,1)-categorical hom-space]].
* [[Zoran ?koda]]: It looks correctly to me (I do not recall what wording I wrote and it looks like reading my mind). Bt it is late night and I should reread the whle entry rested at the day time. I created a stub version of [[schematic algebras]] (with references) and linked it at [[noncommutative algebraic geometry]]. It would be nice to have for comparison a more thorough entry on Gabber's notion of [[almost schemes]] (just created unfinished entry) in commutative geometry, but the reflective localizations are used to define 'exotic' affines to start with. Gabber-Lorenzo's book is an egregious sample of how a modern algebraic geometer of Grothendieck's school develops the theory of schemes and in a way it is a build ground-up with requiring sofistication level, but not many concrete details from the usual theory of schemes.
* [[Toby Bartels]]: Please check that I rephrased the definition correctly at [[noncommutative scheme]]. The original formulation did not make sense to me, but I think that I understood what was meant.
* [[Urs Schreiber]]
* replied at [[(infinity,1)-categorical hom-space]]
* have to think about the question at [[category with weak equivalences]] (which in any case shows that I phrased the sentence too carelessly)
* [[Zoran ?koda]]: created first draft of [[noncommutative scheme]] (after Rosenberg), and plan in few minutes to start [[schematic algebra]] (after van Oystaeyen). Somebody should also add [[noncommutative projective geometry]] (after Artin and Zhang). It is a pity that only few of the most stubborn contributors use summer to add new material...
* [[Urs Schreiber]] added some clauses to [[Grothendieck group]] and have some comments in a query box: I am thinking what the entry tries to define should be discussed at [[algebraic K-theory]] while "Grothendieck group" should be the definition of a group structure on $A \times A$ for a monoid $A$
* [[Toby Bartels]]:
* A question at [[category with weak equivalences]].
* A complaint at [[(infinity,1)-categorical hom-space]].
* Fixed the links to Sjoerd Crans\'s papers at [[strict omega-category]]; there may be more.
* Changes to [[symmetric function]].
* Added a bit to [[free object]] (yes, that bit).
* It\'s clear that the Grothendieck ring of a braided monoidal abelian category is commutative, so I rewrote the question at [[Grothendieck ring]].
* [[Zoran ?koda]]: Created [[cop]] (it should have been maybe created by the police department).
I thank people from Indiana math dept who discussed with me in Fall 2002 when I was searching for the appropriate name and discuraged me from using "mud" for that amorphous structure. BTW, does anybody know what to do on W XP when you loose permanently language bar ?
* [[John Baez]]: created [[representation ring]], [[Grothendieck group]], [[Grothendieck ring]], and [[symmetric function]]. Tidied up [[lambda-ring]] a little, but not enough.
* [[Zoran ?koda]]: Created [[coderivation]] and a new paragraph in [[derivation]].
* [[Urs Schreiber]]
* created [[(∞,1)-categorical hom-space]] with the Dwyer-Kan theorem, linked to it from [[simplicial localization]] and [[model category]] and [[category with weak equivalences]]
in the course of this I alsow expanded/rewrote the introductions to
* [[model category]]
* [[category with weak equivalences]]
* [[Tim]]: I changed the e to a é (easy on a Mac!) in [[Les Dérivateurs]] and created an entry for [[Georges Maltsiniotis]], note his first name has an 's' on the end.
* [[Urs Schreiber]]
replied to [[David Roberts|David]] at [[category theory]] -- am waiting for [[Rafael Borowiecki]] to get back to us before taking action outside of the query box
* [[David Roberts]]:
* Added bits to [[Pursuing Stacks]] and [[Les Derivateurs]] linking the two documents. Actually there should be an accent on the first e of Derivateurs, but I leave a polite request to the lab-elves to solve that piece of typographical trickery for me.
* [[Urs Schreiber]]
* created [[Sjoerd Crans]]
in that context I also added reference to his work to [[model structure on simplicial sheaves]] and [[model structure on presheaves of simplicial groupoids]]
also wrote a quick "Idea" section for the latter entry
* [[Toby Bartels]]: Moved [[the homotopy theory of Grothendieck]] to [[homotopy theory of Grothendieck]]. Normally, I don\'t mention that sort of thing here, but this time there\'s nothing in the naming conventions about this; it just feels right. Complaints are solicitied.
* [[David Roberts]]: added comment to discussion at [[category theory]] - possibly Rafael means homotopy types when he says spaces. Added point about Grothendieck's view on Cat as a category of models. Also a stub: [[the homotopy theory of Grothendieck]].
## 2009-07-29
* [[Urs Schreiber]]: more reactions in that discussion at [[category theory]] on that one paragraph by [[Rafael Borowiecki]] -- I suggest that this needs to be rewritten _somehow_. I make one suggestion, but chances are that I am still missing [[Rafael Borowiecki]]'s true intention.
* [[Toby Bartels]]: Noticed the many wanted links at [[differential form]] and started filling them out; a lot of basic stuff in differential topology. Many will be stubs, and many will be capable of unreported generalisation, internalisation, and categorification. So far:
* [[derivation]]
* [[skew-commutative algebra]] (definitely needs more)
* [[cotangent bundle]] (and [[tangent bundle]] to go with it)
* [[exterior algebra]]
* [[Rafael Borowiecki]] has figured out how categories are spaces at [[category theory]]. (I don\'t think that Rafael reads this page, so copy comments there.)
* [[Zoran ?koda]]: I have made changes to [[noncommutative algebraic geometry]]. Le Bruyn has kindly added a lot of material with ring-theoretic flavour (mainly references) and placed my unfinished text below his comment part. I have integrated his and my part, more chronologically and balancing categorical and ring-theoretic aspects; role of cyclic homology and many other directions (e.g. 30 years old subject of D-schemes of Beilinson) are missing. I would kindly invite Le Bruyn to write an adiditional separate entry on noncommutative projective geometry of Artin/Zhang flavour (I am not competent) as well on geometry at n, he is expert on. Quantum group aspects are planned to appear in entry [[equivariant noncommutative algebraic geometry]] which I just started.
* [[Urs Schreiber]]
* created a stub for [[Ehresmann connection]]
* [[Zoran ?koda]]: created [[formally smooth morphism]], [[quasi-free algebra]], [[universal differential envelope]], [[Kähler differential]].
* [[Toby Bartels]]
* Rewrote [[k-surjective functor]] a bit to give the non-evil ('essential') version equal time.
* Golf department! *Golf* department!! Bwahahahaha!!! Thanks, Andrew and David, I should have gotten that.
* Completed proofs at [[regular space]], sometimes by deciding not to prove anything obvious. (I wrote more too but forgot to save it when the Lab crashed last night).
* [[Urs Schreiber]]
* added a diagram and a sentence in the section "The homotopy category" at [[category of fibrant objects]] that makes the statement given there more explicit: that and how every cocycle out of a weak equivalence can be refined by a cocycle out of an acyclic fibration (i.e. by an "$\infty$-anafunctor").
* slightly expanded and polished the examples at [[Reedy model structure]] further
* [[David Roberts]]: cleaning up the [links](http://en.wiktionary.org/wiki/links#English) is clearly the job of the golf department
## 2009-07-28
* [[John Baez]]: warning: I believe "[[special lambda-ring]]" is an old-fashioned name for what almost everyone now calls a [[lambda-ring]]. This is explained by Hazewinkel in his article cited on [[lambda-ring]]. So, I do not believe we should have a separate article on [[special lambda-ring]].
* Zoran: created [[Dennis trace]] (experts please expand!)
* [[Toby Bartels]]:
* Incorporated results of discussion into [[effective epimorphism]] and [[regular epimorphism]].
* Everyone who uses '$\sqcup$' (`\sqcup`) should be aware of '$\amalg$' (`\amalg`) and '$\coprod$' (`\coprod`), as in '$A \amalg B$' and '$\coprod_i A_i$'.
* Interaction with [[Rafael Borowiecki]] at [[category theory]].
* Golf department? *Golf* department??
* [[Urs Schreiber]]: aha, I didn't get that either -- but figured it must be me missing some English thing
* [[Urs Schreiber]] created [[Kan fibrant replacement]]
* Zoran: created [[Grothendieck Festschrift]], and quoted it as an addition to the timeline entry.
* [[Urs Schreiber]]
* spelled out the simplest nontrivial example at [[Reedy model structure]]
* created [[global model structure on functors]] that was requested long ago as [[projective model structure]] and [[injective model structure]] which I made redirects to it
updated accordingly the list of examples at [[model category]] and interlinked with [[Reedy model structure]]
* I tracked down a reference that discusses enriched Reedy model structures for enriched functor categories from enriched Reedy categories to enriched model categories -- I made corresponding additions to [[Reedy model structure]] and also created
* [[enriched Reedy category]]
* thanks, Lab Elf. I just donated something to the _Society for the Promotion of Elfish Welfare_ for that.
* [[Andrew Stacey|Lab Elf (service department)]]: The problem was that there was a reference to a theorem in the section that you wanted to remove. By removing that section, you were trying to create a non-existent link. By reformatting that sentence, I was able to remove the section. Another [[Toby Bartels|Lab Elf (golf department)]] may be along later to clean up the links since I just did what was necessary to remove the section.
* [[Urs Schreiber]]
* split off [[Reedy model structure]] from [[Reedy category]] -- added an "Idea" section and started expanding on some of the technical details
by the way: it's funny I can't _remove_ the old section "Model structures" from [[Reedy category]] which is now reproduced and expanded at [[Reedy model structure]]: I always get an "Internal server error" when I try to do that. I am familiar with the occurence of this error when one _adds_ certain things (such as double-dollar included displyed math without line breaks before and after) but I couldn't figure out which problem the _removal_ of that paragraph causes. So that section is still sitting there, duplicated now.
* added clarifying remarks to the references at [[K-theory]]
* in a similar vein to the below comment i added to the beginning of [[delooping]] a remark how the one-object groupoid $\mathbf{B}G$ and the classifying space $B G$ are the same object under the homotopy hypothesis
* from private discussion with somebody it became clear to me that the entry [[homotopy hypothesis]] failed to get across one of the main points with the required emphasis. I now added the central theorem about the Quillen equivalence between Top and SSet right at the beginning. The disucssion of all the subtleties and generalizations should come after that.
* [[Andrew Stacey]]: Mathematical and non-mathematical stuff going on at [[Tall-Wraith monoids]]. Folded up the mathematical bit of the middle discussion (on what's so special about $AbGrp$) into the main text, but probably not in the nicest and clearest way. Also continued the discussion on fonts and the like further down.
* [[Urs Schreiber]]
* added a few words to [[(infinity,1)-topos]] (in particular pointed out that the equivalence of the two definitions given is a main theorem, and added the links to the entries on models)
## 2009-07-27
* [[Urs Schreiber]]
* replied a bit at [[effective epimorphism]] -- but don't trust me, it's way beyond my bed time -- see you tomorrow :-)
* moved [[Jim Stasheff]]'s insertion at [[cohomology]] one paragraph further down not to have it tear apart the main definition -- this should eventually be merged more with the rest of the entry, it overlaps in parts.
* [[Toby Bartels]]: More discussion at [[Tall-Wraith monoid]] (not actually about math) and [[effective epimorphism]] (actually about math!).
* [[Urs Schreiber]]: added to [[effective epimorphism]] a remark on how the 1-categorical case is a special case of the $(\infty,1)$-categorical case and replied to the discussion
* [[Toby Bartels]]:
* Reorganised [[integration]] (another new, unlogged page!).
* A question about [[special lambda-rings]] (a new, unlogged page!) at [[Lambda-ring]].
* Discussion at [[Tall-Wraith monoid]], with and without lab elves.
* A question about [[effective epimorphisms]] in an $(\infty,1)$-topos.
* Added the $1$-categorial concept to [[effective epimorphism]], with related material at [[regular epimorphism]] and [[quotient object]].
* Discussion with Zoran at [[quasigroup]] about the infestation of centipedes crawling around the Lab lately.
* [[Lab Elf|Lab Elf (service department)]]: 'Recently Revised' now gets redirected to this page. This redirection happens at the _server_ level (i.e. before anything gets to instiki) so shouldn't affect performance. This also means that the 'Recently Revised' pages for labs _other than_ then n-lab work as they ought to. If hitting those slows up the system then Other Steps Will Be Taken.
Please remember that this is hopefully a temporary problem and once we migrate to warmer climes, Normal Service can be resumed.
* [[Urs Schreiber]]
* created [[petit topos]] and made [[gros topos]] redirect to that
* added the links to both versions of David Spivak's work at [[derived smooth manifold]], [[structured generalized space]] and [[geometry (for structured (infinity,1)-toposes)]]
* [[Zoran ?koda]]: created [[heap]], [[quantum heap]].
* [[Urs Schreiber]]
* created [[geometry (for structured (infinity,1)-toposes)]]
in that context I also
* renamed [[structured generalized space]] to [[structured (infinity,1)-topos]]
* created a stub for [[effective epimorphism]]
* created [[simplicial resolution]]
* [[Andrew Stacey]] parried the latest Baezian riposte, and whinged about the lack of a <sarcasm> tag in XHTML.
PS Welcome back, Bruce.
* [[John Baez]]: answered Andrew Stacey's latest comments over at [[Tall-Wraith monoid]].
* [[Bruce Bartlett]]: Corrected a faulty link to the [nLab Stylish theme](http://userstyles.org/styles/17934) for FireFox at [[HowTo]]. Works now.
* [[Urs Schreiber]]:
* thanks to [[Mike Shulman]] for the comments at [[small object argument]] -- we should give both statements, for the non-locally presentable category, then with that other extra assumption, as well as for the locally presentable case -- I'll work that in later (my notation was following Lurie, by the way, but I agree that it is a bit weird)
* edited the references section at [[structured generalized space]]
* added a section with links to higher dimensional and homotopical generalizations to [[group]]
* started creating [[symmetric monoidal functor]] but then noticed that [[monoidal functor]] didn't even exist yet and postponed this to another time
* hyperlinked some more keywords at [[HQFT]] and [[sigma-model]].
* [[Andrew Stacey]]: spotted a sneaky paragraph at [[Tall-Wraith monoid]] and put a query for its originator ([[John Baez|John]]?).
* [[Toby Bartels]]:
* Finished what I wanted at [[regular space]], although there are still some proofs to fill out.
* Fixed an egregious error in [[normal space]] while translating the definition from closed sets to open sets.
* [[Tim]]:
* added a mention of inverse semigroup in [[semigroup]]. As these relate to partial symmetries and ordered groupoids this may merit being expanded but I cannot do it at present.
* [[Toby Bartels]]:
* Replied to an anonymous comment at [[evil]].
* Started [[regular space]], but I have to leave and have not finished it.
## 2009-07-26
* [[Toby Bartels]]: Lots of changes (mostly additions) to [[measure space]]. Please see if the notation is comprehensible. I have to check on a couple of things, but I left query boxes. There are several variations, but I only included things that people can actually get tenure by studying. No [[centipede mathematics]] just for the sake of it! (well, except for one comment, appropriately linked).
* [[Andrew Stacey]]: continued the sparring at [[Tall-Wraith monoid]] (and answered the serious query). I wish I'd known the fascination with centipedes earlier, we caught one today and I could have gotten a good picture of it.
* [[John Baez]]: inserted centipedes in [[quasigroup]], [[magma]], and the section on weakened definitions in [[group]]. Made a few other small changes in these.
* [[Toby Bartels]]:
* Rearranged the introduction to [[group]], including some stuff that John originally put at [[centipede mathematics]].
* Merged [[semigroups]] (now [[semigroups -- history]]) into [[semigroup]], as John describes below.
* [[John Baez]]: meddled a bit with [[centipede mathematics]]. Accidentally created a page called 'semigroups' --- sorry, Toby; it looks like you're merging it with [[semigroup]], which said a lot of the same stuff in a more sophisticated lingo.
* [[Mike Shulman|Mike]]: A couple of comments at [[small object argument]].
* [[Toby Bartels]]:
* Tricked up John\'s picture at [[centipede mathematics]], based on code found at [[Jacques Distler]].
* Objected to the charge of centipede mathematics at [[measurable space]], on the grounds of historical inaccuracy.
* [[John Baez]]
* asked a question about notation near the top of [[Tall-Wraith monoid]], and tried to polish the proof that a Tall-Wraith monoid in abelian groups is just a ring, and enjoyed bickering a bit more with Andrew Stacey in the big green box near the bottom.
* had some fun with [[centipede mathematics]] - see also my reply to Toby below.
* deleted query by Rafael over on [[category]], which had been answered by me and untouched for a while. He'd asked about 'categories as 1d CW complexes', but I think the item on categories as 'directed graphs with composition law' now answers that --- even for people who don't know what a CW complex is.
* reduced the number of appearances of the word 'isic' over on [[isomorphisms]]; while it's fun to make up new jargon, I don't think we should actually use 'isic' when explaining concepts when 'invertible' will do. We don't want to convey an impression of quirkiness, and we don't want to require the reader to look through the whole page to understand new jargon when well-known jargon already exists.
* deleted discussion by Toby and Tom over at [[regular monomorphism]], since Tom said it was okay to do so, and some time has passed.
* [[Todd Trimble]] wrote [[generalized multicategory]], and added a reference at [[Crans-Gray tensor product]] to Sjoerd Crans's papers. Guessed that his papers on teisi might be relevant to an inquiry Mike Shulman made there.
* [[Toby Bartels]] wrote [[separation axioms]] (and a stub at [[disjoint sets]]).
## 2009-07-25
* [[John Baez]] wrote comments on [[isomorphism]], [[bicategory]], [[measure space]], and under a July 24 comment of Eric\'s here. Toby has responded to all of them except the one at <span>bicategory</span>.
* [[Toby Bartels]]:
* As long as we\'re having a conversation, it should be safe for Eric to assume that I\'ve responded to him if I post something here, even if I don\'t mention him.
* A brief list of examples, mostly for the purposes of terminology, at [[isomorphism]].
* [[Eric]]: Responded to Toby at [[measure space]] and [[ericforgy:Densitized Pseudo Twisted Forms]].
* [[Tim]]: I have started an entry on [[HQFTs]]. Initially this will summarise Turaev's theory, but I hope to get a bit more daring later on. I hope someone will tell me (then) if I am talking through my hat. (I rarely wear one.)
* [[Toby Bartels]]: Comments for Eric at [[measure space]] and [[Eric Forgy:Densitized Pseudo Twisted Forms|on his web]].
## 2009-07-24
* [[Toby Bartels]]:
* Wrote [[differential form]]; also [[diffeomorphism]], [[restriction]], and [[characteristic function]], which I linked thence.
* Moved Eric\'s latest changes to the **top** so that they would be seen by Urs (and anybody else who checked this page after Urs added his changes below).
* Added a bit to [[decategorification]].
* Created [[locally discrete 2-category]].
* [[Eric]]:
* Remembered there IS a beautiful arrow theoretic way to think of measures, i.e. [[Leinster measure]]. Added a comment about it at [[measure space]].
* Created [[Leinster measure]] with, for now, just a [link to the n-Cafe](http://golem.ph.utexas.edu/category/2007/03/canonical_measures_on_configur_1.html).
* [[Urs Schreiber]]
* created [[K-theory spectrum]]
* created [[Waldhausen S-construction]] and edited [[Waldhausen category]] a bit, but needs still more work -- help is appreciated, I am not sure yet if I found the best literature
* [[Eric]]:
* Added more to questions on [[measure space]]. Whenever I see a long convoluted definition, e.g. [[measurable space]], I tend to think there should be some short, concise, arrow theoretic description that incorporates all the little factoids into one pretty picture. A wild guess (that I know is wrong, but hopefully inspires someone to write down what is right): a [[measurable space]] is some kind of [[presheaf]] or maybe a [[representation]] on ????.
* [[John Baez]]: the definition of measurable space is pretty darn simple and quick: it's a set with a collection of subsets that's closed under complements and countable unions. Such a collection is usually called a $\sigma$-algebra, and all this is explained pretty early on in the page [[measurable space]]. Whoever wrote the longer discussion below was just having fun analyzing the definition into little bite-sized pieces (I have my guess as to who this might be.)
* [[Toby Bartels]]: Guilty as charged.
* [[John Baez]]: The guilty conscience need not be accused by name. I think we should warn the reader when we go off on an excursion like this. Perhaps just a warning like: **The following passage might be considered [[centipede mathematics]]**, together with a small version of the following picture. I wish I knew how to center a picture!
* [[Tom Ellis]] created [[extremal monomorphism]]
* [[Urs Schreiber]]
* edited [[algebraic K-theory]] a bit
* created [[K-theory]] -- as opposed to my previous take on this which was then moved to the "Idea" section at [[topological K-theory]] this time this is aiming for the fully general bird's eye picture with indications how that produces all the special realizations in special cases
* created [[decategorification]] -- evidently much more can be said here, but it's a start
* expanded the "Idea" section at [[spectrum]] and effectively rewrote it -- added a link to [[combinatorial spectrum]] at the end, which probably should be thought of as a concrete realization of the idea of $\mathbb{Z}$-category -- accordingly I changed the title of the last section from "Conjectures" to "Combinatorial models".
* added a reference to Weibel's online book to [[algebraic K-theory]]
* [[Andrew Stacey]]: Tried answering John's questions over at [[Tall-Wraith monoid]]. Probably lots for the [[lab elf|lab elves]] to work on there.
(I confess that I did have the Hogwartian house elves uppermost in mind, but the shoemaker elves were not far off either. Being now in the Nordic realm I probably should have said 'lab troll' but trolls already have a place on the internet and it is Not Here)
* [[Urs Schreiber]]
* started putting material into [[brane]] -- as a preparation for something I plan at [[K-theory]]
* [[John Baez]]: I answered some remarks by Mike Stay and Eric over on [[free cocompletion]]. I also had an hour-long chat with Mike that should eventually push this exposition forward quite a bit: I explained coends to him, which is a lot easier in words than on paper. But I hope we get the explanation into the $n$Lab eventually!
* [[Toby Bartels]]: Answered Eric\'s first question; I\'m not ready to think about the second one yet.
* [[Eric]]: Asked some questions on [[measure space]].
## 2009-07-23
* [[Toby Bartels]]: Remarks on notation at [[measure space]].
* [[Ben Webster]] created [[Hecke algebra]]
* [[Zoran ?koda]]: created [[Dunkl operator]], [[double derivation]]; it is a start of a series which should include entries on Cherednik algebras, Knizhnik-Zamolodchikov connection, Calogero-Moser system, Gauss-Manin connection, Calogero-Moser space, deformed preprojective algebras and so on...
* [[Urs Schreiber]]:
* effectively rewrote [[infinity-stack]] -- expanding it considerably, adding the relevant pointers to all the new material that has come together since I first wrote this
* removed the discussion of costacks entirely. I'll turn that into a separate entry in its own right eventually
* John's question there had been about my notation $\to\gt$ -- that was a hack for the symbol for a _fibration_, an arrow with a double tip. In my new version this no longer appear, though it may still appear at [[hypercover]], which is linked to, and elsewhere.
[[Toby Bartels|Toby]] knows how to typeset such arrows correcty. Maybe he could add a section to [[HowTo]] with the relevant information and links to special symbol lists
* added a reference at [[(infinity,1)-functor]] to [[(infinity,1)-category of (infinity,1)-functors]] -- the discussion of this issue that I like most is currently at [[models for infinity-stack (infinity,1)-toposes]]. Eventually that should be discussed better at the relevant entries.
* [[Toby Bartels]]:
* Conversation with [[JCMcKeown]] at [[(n,k)-transformation]].
* Created [[lab elf]], just a bit of fun; credit [[Eric Forgy]] (I think) for the idea.
* [[Eric]]: Not me. I think Andrew gets credit for that. I added a link to [[lab elves]] from [[How to get started]] :)
* _Toby_: Ah yes, with his British references to Hogwarts. But the term made me think of den Brüdern Grimm instead; in fact, after reading their tale (an English version of which is linked from the new page) again, I see that Rowling\'s house elves owe more to it that I\'d remembered!
* Added my touch to [[How to get started]]; among other things, I didn\'t like the horizontal rules around the images, but maybe they were there for a reason?
* [[John Baez]]:
* Put in some gunk about [[Tall-Wraith monoid]], which Andrew Stacey improved. Later I put in two queries!
* Put in a query about [[D-modules]].
* Put in a query under [[infinity-stack]].
* [[Toby Bartels]]:
* I would like to make a plea for adding entries to the *top* of this list. There are some comments and entries that I see only because I check every single change in the history; while obsessive behaviour may be correlated with mathematical ability, you can\'t expect people to see things unless you add them to the top. If that means that you give yourself two entries in one day or even report one day\'s events in the next day\'s space, then so be it! (For conversations, I suppose that it depends on whether you think that the person that you\'re having a conversation with will see your comment; but be aware that they might not.)
* I wouldn\'t be very quick to get ban query boxes at [[How to get started]]; people need to be able to ask questions about how to get started! That doesn\'t really apply to discussion by the regulars about how to explain, of course, but I would want Jim Stasheff, for example, to ask questions there if he wanted to, without people trying to clean up too quickly.
* [[Urs Schreiber]]:
* created [[Dominic Verity]]
* created [[Verity on descent for strict omega-groupoid valued presheaves]]
* reworked the [[How to get started]] according to my opinions and our disucssions [here](http://golem.ph.utexas.edu/category/2009/07/nlab_how_to_get_started.html#c025573)
there are now two sections, one on how to paste source code of a comment one is about to submit to the blog, the other about how to paste non-source code material
in the course of this I have removed lots of the previous discussion on these points -- the goal is to keep that particular page clean of auxiliary discussion and as brief and to the point as possiblle, because that's the point of this page
if anyone feels I removed too much, please use Rollback to grab the deleted material and then cancel the rollback and insert the missing material in a suitable section at the main [[HowTo]] entry
* [[Andrew Stacey]]: I've banned 'Recently Revised' for the time being. My method of banning has probably blocked it for all the private webs as well. If that's really annoying then let me know and I'll try to find a more specific method of banning just the nlab one.
* [[Andrew Stacey]] was pleasantly pleased to stumble across [[Tall-Wraith monoid]]s and made a few minor alterations (mainly style, and added a couple of references). I'll shove this question over on the [forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=49) as well, but should we have a lab convention on fonts for categories, functors, objects, and the like?
* [[John Baez]]: especially given the large and opinionated group of contributors, we probably shouldn't fuss over fonts too much, but I'm in favor of the KISS philosophy: "keep it simple, sweetheart". Namely: use capital letters for big things, small letters for little things, and Greek letters when you run out of ordinary letters, or want to show off your erudition.
* [[David Corfield]]
* started [[P-ring]]
* [[Urs Schreiber]]
* renamed the new easy-basic-HowTo page to [[How to get started]]
then I reworked the formatting and edited pieces here and there
to [[Bruce Bartlett]]: I think on that particular page we don't want query boxes, as that page is supposed to provide quick unambiguous information that tries to deconfuse people instead of to confuse them -- please see my reply and check if you can work something into the paragraph right before the query box that allows to remove that query box
[[Andrew Stacey]] I concur, but couldn't delete the query box as I made a remark in it and so if I delete the box _now_ then that would permanently remove that remark. Someone else could do it (or I could in half an hour's time).
* [[David Corfield]]: Started [[Lambda-ring]] with some Baezian exposition and an abstract of James Borger. Hmm, is there a difference between $\lambda$-ring and $\Lambda$-ring? This [paper](http://wwwmaths.anu.edu.au/~borger/papers/03/lambda.pdf) uses both.
* [[Urs Schreiber]]: thanks, David, I was hoping you would
* [[Toby Bartels]] welcomed [[Sebastian Thomas]] at [[(n,k)-transformation]].
## 2009-07-22
* [[Tim Silverman]]: Answered a [request from the n-Cafe](http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025521) by creating [[How to Copy and Paste Material from the n-Cafe and Include Links Back and Forth]]
* [[Urs Schreiber]]
* slightly polsihed further at [[strong monad]] and removed the tentative-alert, now that [[Todd Trimble|Todd]] also approved of the statement
* filled in a bit of text and some references at [[conformal field theory]]
* added the notion of Frobenius lax-and-oplax functors to [[lax functor]] and provided pointers to their use in CFT
* added a remark by [[Todd Trimble]] to [[associahedron]] on their relation to [[oriental]]s that I asked him about by private email
* [[John Baez]]:
* Added more information to [[tensorial strength]]. Some of this should be checked.
* Added more examples to [[lax functor]]. I'm in a lax mood these days, and I really enjoyed it when Paul-André Melliès told me a definition of 'enriched category' in terms of lax functors. This works for categories enriched over a bicategory, not just a monoidal category. Do we have any entry on enrichment over bicategories? If so, maybe someone could add a link.
[[Urs Schreiber]]: we had some old discussion on the blog on this description of enriched categories -- I used to be interested in that in the context of [A Note on RCFT and Quiver Reps](http://golem.ph.utexas.edu/string/archives/000794.html) -- I'll maybe add something about this to the entry
* [[Urs Schreiber]]:
* created [[strong monad]]
* created [[lax functor]]
* added to [[monad]] the statement that a monad in $B$ is a lax functor ${*} \to B$
* replied at [[(n,k)-transformation]] -- I think that in principle this gives all the required information, but I am aware that eventually someone should describe that all explicitly in detail at that entry
* [[Toby Bartels]]: Copied to [[(n,k)-transformation]] a question that was sent to me by email, and partially answered it. (Urs could probably answer the rest.)
* [[Urs Schreiber]]:
* added references and links to [[vertex operator algebra]]
* created [[tensorial strength]] with material that [[Todd Trimble]] provided on the blog
* [[Toby Bartels]]:
* More discussion with Tom at [[regular monomorphism]].
* Formatting with `+-- {: .un_remark}`...`=--` at [[category theory]].
* Final comments to Urs at [[locally presentable category]].
* Added somewhat to [[(n,k)-transformation]] ... although still not a definition!
* [[Urs Schreiber]]:
* added links and references to [[tricategory]]
* worked on the _Idea_ section at [[category theory]]: I reformatted a bit the existing material, included lots of hyperlinks and filled in various further bits, such as a paragraph that lists the fundamental classes of examples and the quote from Barry Mitchell that Todd just mentioned on the blog
* I was surprised to find the entry in a much more developed and pleasant state than I remembered it -- maybe I missed the announcement here, or could it be that there was a major edit to the entry that wasn't logged here at Latest Changes? Please remember to alert us here.
* I am now hopeful that eventually we'll be able to turn what should be the pivotal $n$Lab entry into something decent, too: that on [[higher category theory]]. At the moment that one is not a good advertisement of the $n$Lab project.
* replied and reacted at [[locally presentable category]]
* [[Eric]]: Asked, "What is a 'component of a cocone'?" on
[[An Exercise in Kantization]].
* [[Urs Schreiber]] where did you see that term used? Maybe the question (or its answer) belongs at [[colimit]]. Do you have an idea what a cocone itself is? It consists of lots of morphisms from the objects of a diagram to the cocone tip. If we regard the cocone as a natural transformation to a constant functor, then the components of that natural transformation are these single morphism from objects to the tip of the cocone. These I would call "components of the cocone".
* [[Eric]]: I gave my answer at [[An Exercise in Kantization]] so that it will not get lost here.
* [[Zoran ?koda]]: Thank you Toby, your new clarifications in [[essential image]] and [[replete subcategory]] are pretty helpful and clear, and I agree with them. Still I would like to think of more clean scheme of thinking of various kinds of images internally, in connection to various kinds of factorization systems and even multistep factorizations like Postnikov systems. There is probably a framework where, despite the differences all the kinds of images including [[homotopy image]] belong. The crucial is choice of a sort of factorization system: a variety of an image is basically the second morphism in the factorization (or less precisely its domain). In higher categories sometimes multistep factorizations systems are interesting, like Postnikov towers in topology. This way it may satisfy the point of view of Urs, who was IMHO not precise at the beginning but eventually pointed in the right general direction, and the reference of Barwick which he found seems to be really useful.
* [[Toby Bartels]]:
* Tried to explain what sort of evil I meant at [[essential image]].
* Boldly put the default notion (in the higher-dimensional case) first at [[replete subcategory]].
* Edited the definition at [[transfinite composition]] to include the possibility that $\alpha = 0$ and also to allow the concept to be interepreted constructively.
* Reply to Tom (Tom who?) at [[regular monomorphism]].
* More discussion at [[locally presentable category]].
## 2009-07-21
* [[Urs Schreiber]]
* after discussion with [[Zoran Skoda]] I split off [[homotopy image]] from [[essential image]], reserving the latter for the essential image of a functor of categories -- I haven't touched the content of [[essential image]] otherwise
* updated link list at [[Higher Topos Theory]] (mostly under Appendix/Category Theory). For what that's worth, the appendix is now getting pretty close to being fully indexed.
* added reference to Richard Garner's _Understanding the small object argument_ to [[small object argument]]
* created [[transfinite composition]]
* made at [[small object argument]] the theorem a formal theorem (with theorem environment and all), added a list of references and -- in the paragraph that is now right before the theorem -- tweaked the former assumptions a bit, which I guess were taken by [[Mike Shulman]] from Hovey's book. My impression is that in the "modern" literature the ambient category is assumed to be locally presentable -- but it would be great if an expert checked my modifications (see also the further literature that I list)
* [[Andrew Stacey]] took the hint and started incorporating the discussion into the main thread at [[paracompact space]].
* [[Urs Schreiber]]
* added Jeff Smith's theorem to [[combinatorial model category]] and made [[Smith's theorem]] redirect to that
* added the Barwick reference also to [[Bousfield localization]], to [[combinatorial model category]] and to [[small object argument]]
* added an "Idea" section to [[essential image]], created subsections for different definitions and created one subsection with the definition of homotopy image as found in Clark Barwick's work and as kindly pointed out by [[John Baez]] on the blog [here](http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html#c025455)
* did some editing and have a discussion with [[David Corfield]] at [[group homotopy]]
* added to [[folk model structure]] a sentence that these model structures present $(\infty,1)$-categories of the collection of the given $n$-categories, as part of a reply to [[Rafael Borowiecki]]'s question to the Cat-theory mailing list that I just posted
* added to [[(infinity,n)-category]] the reference to Lurie's "Goodwillie"-article and a few remarks on some pertinent definitions there
* [[Zoran ?koda]]: I actually do not think that Toby's correction to [[essential image]] is correct. I mean that essential image is removing evil from image. No, [[image]] SUBcategory is just a specific and unique internal (subcategory in narrow sense) CHOICE of the (external) image of the functor within Cat as a category. Essential image subcategory is just a specific and unique choice of the bicategorical image of the functor considered as a 2-functor within Cat as a bicategory. The same with higher version. The homotopy image which Urs looks is just about image in external sense and not about the internal choice of which subset of k-cells for every k is chosen. Making a replete choice of subcategory is like taking a maximal atlas of a manifold to remove nonuniquness in the class of all atlases - so in a sense it is a maximal choice with respect to the target; the usual image of a functor is more calculated with respect to the domain of the functor. In bicategory Cat the two are equivalent; in category Cat they are not isomorphic.
* [[Urs Schreiber]]: I am not sure I know what you mean by external vs internal. But I supppose one point you are making is that an _essential image_ is/should be defined only up to the relevant notion of equivalence. Do you mean by "external" a characterization of essential image by a universal property, whereas by "internal" you mean a concrete representative of that (unique only up to equivalence)?
Do we agree on what the "external" definition should be? Is it the one I suggested it should be? In that case we might reorganize the entry by startiing it with the abstract nonsense definition and then taking the replete version as one concrete realization in Cat.
* [[Zoran Skoda]] No (I have the feeling that you are not reading what I wrote), we disagree on what external definition should be because the essential image is not a notion which is external. It is a CHOICE of literally a subcategory, not a choice of embedding of categories in abstract sense, it is a choice of a SUBSET of n-cells which is a n-subcategory which is replete. On the other hand there are two (three) DIFFERENT notions of image of a functor. One is the image in external sense, that is image in Cat taken as a category or as a 2-category. Another is image as a subcategory in literal sense. Image in literal sense is of course a very specific representative of an image in Cat 1-categorical sense and essential image is a very specifical choice of an image in 2-categorical sense; actually it is a specifical choice of such 2-categorical image that the embedding of essential image into the codomain is also literally surjective on objects. This is a bit strange from external point of view: you have something what is just equivalent to 1-categorical image, while it additional property is again of 1-categorical type. Thus it mixes the two. Hence it is by no means superimposable to homotopy limits in any case.
* [[Urs Schreiber]]:
* edited [[locally presentable category]] according to my discussion with [[Toby Bartels]] there -- and included links back and forth with [[presentable (infinity,1)-category]]
* [[Andrew Stacey]]: Responded at [[paracompact space]] and [[Froelicher space]]. Incidentally, if the start of a query box is indented for some reason (as on [[paracompact space]]) then it seems that all its contained paragraphs should be indented by at least the same amount.
* Thanks, Andrew. Maybe eventually somebody finds the time to move the insights gathered there out of the query box and distill them into proper entry text.
* [[David Corfield]]:
* started [[Moore space]]
* [[David Roberts]]:
* fixing up some statements at [[paracompact space]]. Added comment about existence partitions of unity being dependent on the category these will be constructed in.
## 2009-07-20
* [[Urs Schreiber]]:
* thinking about it, I followed Zoran's suggestion and moved the entire "Idea" part that I had t-yped into [[K-theory]] to [[topological K-theory]] -- also the query box with the discussion is now there, and [[K-theory]] is once again just a link list...
* brief reply and question at [[K-theory]]: what is the big global picture on K-theory that deserves to be put in the first sentences of the "Idea" section and really captures the full topic? Is there even any?
* have a question at [[essential image]]: we should consider the weak/homotopy version of the definition of limit as the equalizer of the cokernel pair of a morphism, is there any literature/knowledge about that?
* quick reply to Toby at [[locally presentable category]]: I didn't mean to leave out the "locally", but now that we are at it: what's the point of saying "locally" here in the first place?
* [[Toby Bartels]]: Added quite a bit to [[free monoid]].
* [[David Corfield]]:
* Carried out some tentative dualising at [[group homotopy]].
* [[Toby Bartels]]:
* Moved some discussion on terminology from [[cartesian monad]] to [[locally cartesian category]].
* Wrote [[locally cartesian category]], [[free monoid]], and [[identity monad]], all quite stubbily, because I linked to them from [[cartesian monad]].
* Asked a question on terminology at [[locally presentable category]].
* Wrote a brief **Idea** section at [[cartesian monad]] and made the previous one a **Motivation [...]** section.
* [[Eric]]
* Asked what was probably a very silly question on [[presheaf]] in an attempt to complete an Exercise on [[free cocompletion]], i.e. "Find a formula expressing every object in $\widehat{A}$ as a colimit of guys in the image of $Y$."
* [[Urs Schreiber]]:
* created [[presentable category]] for questionable reasons
* added to [[locally presentable category]] the explicit charascterization
* created [[cofibrantly generated model category]]
* started adding an "Application" section to [[models for infinity-stack (infinity,1)-toposes]]
* [[David Corfield]]: imported Patrick Schultz's cafe comment on cartesian monads to [[cartesian monad]], but now have doubts as to whether it ought to appear there first under 'Idea'. Are there other uses for cartesian monads? And anyway similar material appears at [[multicategory]].
* [[Toby Bartels]]: Answered an anonymous question at [[regular monomorphism]].
* [[Eric]]
* Added a small status update to efforts at [[An Exercise in Kantization]]
## 2009-07-19
* [[Toby Bartels]]:
* Replied to Rafael Borowiecki at [[category theory]] and [[Segal category]].
* Linked a blog comment from [[cartesian monad]].
* Comments to Eric at [[Note on Formatting]].
* Generalised [[refinement of a cover]] to [[refinement]].
* Replied to Andrew at [[paracompact space]].
* A brief comment on the latest counterexample at [[Frölicher space]].
* Referenced the [[adjoint functor theorem]] at [[cocontinuous functor]]. (More generally, there is much at [[continuous functor]] that might be brough there.)
* Removed some [[?]]s that were seen by the people that they were addressed to.
* [[Tim]]
* I have started [[Dowker's Theorem]], partially because it is useful for the entry on Cech methods, but also because it is relevant to Gavin's problem oposted on the Café. I have given Dowker's proof. It seems to me to be saying something about combinatorial duality. (actually duality in relations and Chu spaces.
* [[Eric]]
* Wonders aloud at [[An Exercise in Kantization]] about reformulating things on a [[double category]] and relating it to a [Feynman checkerboard](http://en.wikipedia.org/wiki/Feynman_checkerboard) somehow. Maybe even getting back to relating it to [[Position, Velocity, and Acceleration]].
## 2009-07-18
* [[Eric]]: Installed Cygwin so that I could convert Dugger's _Sheaves and Homotopy Theory_ from dvi to pdf. I uploaded the pdf to the nLab and added links to all references to the paper.
* [[Zoran ?koda]]: created [[comonad]], added more on [[connection for coring]] and [[semifree dga]]. I think David'd confusion might be genuine: not to call with dash or not, that is easy question of exact synonyms, but rather how to cleanly separate DIFFERENT but similar notions of say biadjunction and pseudoadjunction; setups in which they appear: strict and nonstrict 2-categories and Gray categories; and kinds of (pseudo/2)-monads they induce...to mention a few. The Memoirs booklet by Tom Fiore and some papers by Lack, Marmolejo, Vitale, Kelly...may be useful to compare and decide in this regard.
* [[David Corfield]]: Added a comment at [[free cocompletion]], which got me looking for "pseudoadjunction".
I would trigger a new page for it, but don't know optimal naming conventions.
* [[Eric]]: Hi David. Now that we have redirects, you can feel less concerned about naming conventions. For example, if you start a page [[pseudoadjunction]] and people come out with pitchforks saying it should be [[pseudo-adjunction]], we now have the capability to simply change the page name. Better than that, we can add [[redirects]] so that both [[pseudoadjunction]] and [[pseudo-adjunction]] point to the same page and then it doesn't matter. People can use either one when linking to your page.
* David: The worry was more about the name itself. I recall John Baez in TWF wishing to avoid the term pseudomonad, and I see [[2-monad]] covers various levels of weaknesses. Oh, I see we have [[lax 2-adjunction]].
* [[Eric]]: Would it make sense to add redirects for [[pseudoadjunction]] and [[pseudo-adjunction]] to [[lax 2-adjunction]]?
* [[Toby Bartels]]: We really need [[2-adjunction]]; that can link to more specialised pages or have more specialised titles redirected to it, as we find appropriate.
* [[Tim Porter|Tim]]: I have started an entry on [[dg-quiver]]. I have paused because I cannot decide whether this is the right version to put there or whether to use Peter May's discussion in the talk that is linked to from that page.
## 2009-07-17
* [[Urs Schreiber]] made a remark at [[free cocompletion]] in between the exchange between [[John Baez]] and [[Eric Forgy]]: the [[Yoneda extension]] discussed there is at least a special case of a left [[Kan extension]]
* Goncalo Marques replied at [[field]]
* [[Eric]]
* Added some words on [[Note on Formatting|?]] to reflect Toby's preference for the format "$L_\infty$-[[L-infinity-algebroid|algebroid]]". Also added a section with comments on specific pages and moved Toby's comment (and my response) from [[Lie infinity-algebroid representation]] to this section.
* Requested a PDF copy of Daniel Dugger, _Sheaves and Homotopy Theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html)) at [[free cocompletion]]. This reference appears several places and those without the ability to read DVIs could use a PDF copy. Instructions on how to upload files to the nLab are given [here](http://golem.ph.utexas.edu/instiki/show/File+Uploads). Once we have a PDF copy, I can go around and update links to this reference to make the PDF available.
* Added some very pedestrian stuff (to help me understand it) to the Decategorified Theorem section of [[free cocompletion]].
* [[John Baez]]
* wrote a lot more on [[free cocompletion]]. I hope other students of category theory, not just Mike Stay and David Corfield, ask questions or do some of the exercises!
* [[Eric]]: Thanks for doing this. I've been trying to understand [[Kan extension]] for [[An Exercise in Kantization]]. I promise to try not to stray off topic.
* [[David Corfield]]
* asked question of John's explanation at [[free cocompletion]].
* [[Urs Schreiber]]
* created [[refinement of a cover]]
* created [[groupoid of Lie-algebra valued forms]]
* [[Andrew Stacey]]
* Added an example at [[Froelicher space]] to show why it is not _locally_ cartesian closed. It's a simple example, and I think it sheds some light on what actually changes when you go from an "input only" to a "balanced" category (using neither term in any technical sense, just in case they have any).
* [[Urs Schreiber]]
* started list of gauge fields at [[gauge theory]], so far I have
* [[Yang-Mills field]]
* [[electromagnetic field]]
* [[Kalb-Ramond field]]
* [[supergravity C-field]]
* [[RR field]]
* puny start with [[Yang-Mills theory]] and [[Yang-Mills field]]
* created [[twisted K-theory]] presenting it as a special case of the discussion at [[twisted cohomology]] -- feeling slightly uneasy about making this public, though, maybe later I get scared and remove that content again, or move it to my private web. Or else, you tell me how obvious and well-known this is, then I can leave it there without further worries.
* created [[twisted bundle]] and [[bundle gerbe module]]
* added at [[homology]] an "Idea" section that introduces the concept as the image of homotopy under the Dold-Kan correspondence. Also added as an example explicitly the ordinary case of homology in chain complexes of abelian groups
* yes, the duality mentioned at [[cohomotopy]] is the one called [[Eckmann-Hilton duality]], at leat when the $(\infty,1)$-topos in question is [[Top]]. I have made that explicit now at [[cohomotopy]].
* yes, thanks for the improvement at [[RR field]]
* [[Toby Bartels]]:
* Removed the spurious hyphen from [[RR field]], in accordance with my intuition and the cited references.
* Is [[Eckmann?Hilton duality]] the sort of duality being referred to at [[cohomotopy]]?
* Used Eric\'s [[Note on Formatting]] method for the first time, at [[Lie infinity-algebroid representation]].
* Added terminological variations to [[normal space]]. At some point I need to write [[separation axiom]] like I did [at the English Wikipedia](http://secure.wikimedia.org/wikipedia/en/wiki/Separation_axiom).
## 2009-07-16
* [[Urs Schreiber]]:
* added links to [[gauge theory]] fields in the example section of [[differential cohomology]]
* created [[field strength]]
* created [[RR-field]]
* created [[differential K-theory]]
* added an "Idea" section to [[free cocompletion]], made John and Mike's central statement a standout box, gave the theorem a theorem environment and added various links. Noticed to my surprise that the entry [[decategorification]] is, as yet, missing.
* created [[concordance]]
* created [[vectorial bundle]] (notice the difference to [[vector bundle]])
* [[Eric]]:
* Created [[Note on Formatting]]. Now, when I make a minor cosmetic format change to a page, I leave a little signature at the bottom: [[Note on Formatting|?]].
* [[Urs Schreiber]]:
* created [[cohomotopy]] and linked to it from [[cohomology]]
* added "Idea"-section to [[K-theory]]
* expanded further the entry on [[Deligne cohomology]]: gave maps to underlying classes and characteristic forms and made chain maps explicit -- also reorganized slightly, making the perspective of the Deligne complex as the image under Dold-Kan of functors from the path $n$-groupoid the primary one.
* in order to link to the new article by Martins and Picken I created [[path n-groupoid]]
* then from that created [[Gray-groupoid]] and [[path infinity-groupoid]]
* added a pointer to some notes by Daniel Dugger to the discussion of free cocompletion at [[presheaf]] -- Dugger gives a nice pedagogical description
* added material about infinite-dimensional manifolds to [[paracompact space]], taken from private email discussion that [[David Roberts]] kindly provided -- but I notice that I still have a question, see there
* [[John Baez]]:
* started a pedagogical discussion of free cocompletion at [[presheaf]], then followed Urs' suggestion and moved it to the entry [[free cocompletion]]
* added remark and question at [[field]]
## 2009-07-15
* [[Urs Schreiber]]:
* Eric, why don't you make the material on electromagnetism in media that you added into a proper section at [[electromagnetic field]]? Then we could move what is proper discussion between us into a query box, after all, while having the genuine material visible in the netry and not hidden in the section Discussion.
* [[Eric]]: I've thought about this some more and something still bothers me about the idea. If electromagnetic properties, i.e. $\mu$, $\epsilon$, $\sigma$ can be geometrically incorporated into the Hodge star via the metric, this implies: 1.) Maxwell's equations in inhomogeneous media are _wrong_ (although in vacuum they reduce to the familiar form) and 2.) that the Hodge star should involve _convolution_, i.e. the metric should have a _memory_. Has anyone put forward any serious theories of a "metric with memory"? Asking that questions give me a sense of deja vu (getting old sucks).
* [[Eric]]:
* Started tinkering with a draft [[Discrete Causal Spaces]]. Help is more than welcome.
* Made a few comments on [[electromagnetic field]] and [[electric charge]].
* Asked a question on [[connection for a differential graded algebra]] (which Urs replied to).
* [[Urs Schreiber]]:
* added to [[twisted cohomology]] the May-Sigurdsson reference, mentioned their definition of twisted cohomology in terms of associated bundles of spectra and added a discussion on how that relates to the rest of the entry
* it seems to me that linking to a page via a redirect has as a consequence that the linking page is not listed at the bottom of the linked page under _Linked from_ . That's too bad.
* [[Eric]]: This is one of several issues that have arisen due to the new redirects feature that I would not worry about. In the future, this should work as desired. Redirects also produce unnecessary "Wanted Pages" on the "All Pages" page. See the [forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=43&page=1#Item_3).
* added the reference to Abad and Crainic at [[Lie infinity-algebroid representation]]
* realized only now that there is an entry [[semifree dga]], so I added to that entry a remark on [[Lie infinity-algebroid]] and conversely added there a pointer to the former
* replied at [[connection for a differential graded algebra]], remarking that this seems to be essentially the structure discussed at [[Lie infinity-algebroid representation]] -- this concept seems to be reinvented many times, just recently it seems that what Abad and Crainic describe in [0901.0319](http://arxiv.org/abs/0901.0319) is the same idea
* if indeed calling _recently revised_ should be avoided for the time being, it is all the more important that you indicate even small changes/additions here at _latest changes_
* yes, I am relieved to see [[Mike Shulman|Mike]] back, if only temporarily, I was worried that the $n$Lab had lost one of its most valuable contributors
* [[Toby Bartels]]:
* Welcome back, Mike!
* My experience is that Recently Revised, All Pages, and especially Search will degrade performance. I need them all, but I try to use them sparingly. Long pages (such as this one sometimes) can also degrade performance, but only temporarily.
* [[Mike Shulman|Mike]]:
* Replied to discussions at [[replete subcategory]] and [[pseudofunctor]].
* Sorry for suddenly disappearing; after graduating (thanks for the thoughts everyone!) I had immediate obligations to three coauthors that took priority, so I lost track of the nCommunity for a bit. (One of those papers, which people here might be interested in, should be appearing on the arXiv shortly.) Unfortunately I'll now be traveling and out of touch for the next month, but then I'll be back.
* [[Urs Schreiber]]:
* added three basic examples to [[metric space]]
* filled more information provided by [[Todd Trimble]] into the entry [[paracompact space]]
* created [[cup product]]
* checked by private email with [[Todd Trimble]] and probably see my confusion at [[paracompact space]] now -- replied there and added explicitly the example of second countable fin-dim manifolds
* created [[magnetic current]] and [[electric current]]
* I am getting the impression that the server runs much more smoothly when one avoids to call the "recently revised" page. This is a pity, because I used to go there all the time to see what's happening, but it would be helpful to figure out if maybe the cause of the performance problems we see can be narrowed down further. Maybe calling "recently revised" causes the software to go through the entire database in an inefficient way.
* [[Eric]]: When you mentioned this via email yesterday, I stopped viewing "Recently Revised" (which I had been viewing very frequently prior to that). The performance yesterday was MUCH better. It could very well be that performance is degraded when you view "Recently Revised" and possibly seriously degraded when several people attempt to access it simultaneously (which I'm sure happens frequently).
* have a question at [[paracompact space]] concerning what it says there about the "long line" compared to what it says at [[locally compact space]] -- this seems to be inconsistent to me
## 2009-07-14
* [[Urs Schreiber]]:
* added theorems about relation with abelian sheaf cohomology to [[Cech cohomology]]
* in the course of that created [[paracompact space]].
* [[Zoran ?koda]]: created [[Euler number]] (including Euler polynomial(s)) and expanded [[Legendre polynomial]]. Wasted part of the day browsing programming manuals about Ruby...interesting. Maybe something prompts me to be doing something about it :)
* [[Urs Schreiber]]: created [[Karoubi K-theory]]
* [[Andrew Stacey]]: Stumbled across the discussion at [[Timeline of category theory and related mathematics]] on bibliographies and realised that more people were keen for this to be sorted out than I'd thought. A few possibilities are laid out in the corresponding discussion on the [forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=40). Please stop by and let us know what you want from a bibliography system so that we can design it according to what everyone wants rather than just what a few of us want.
On that note, seeing as my mathematical skills are not in the mainstream of the current focus of the n-lab, I'm concentrating a bit more on technical support (stuff like the forum, bibliography, diagrams, useful little scripts like how to download the entire lab for offline browsing). There are lots of things that I (and the others who do a little hacking like this) _could_ do but only so much time in which to do it. So if there's something you'd like done, say a bibliography, that you think I could help with then the fact that someone actually wants it done pushes it up my priority list. However, _unless you tell me about it_ or mention it somewhere that I will _actually_ see it then I'm not going to do anything about it because I won't know about it!
* [[Urs Schreiber]]:
* created [[Kalb-Ramond field]]
* expanded the list of examples at [[model category]] and added at the beginning a sentence on combinatorial simplicial model categories
* after a little reflection I moved the previous content at [[electromagnetism]] to [[electromagnetic field]] and kept just a brief note at the former, for later expansion
then I worked on [[electromagnetic field]]
I renamed the section I was working on into _Mathematical model from physical input_ . This now starts with quick and concise derivation of the fact that the EM field is modeled by a Cech-Deligne cocycle based on a quick definition of Maxwell's equations and the quantization condition.
the following sections "the local picture" and "the global picture" are supposed to provide the remaining details and background. Still needs polishing.
* [[Toby Bartels]]:
* Fixed links at [[Timeline of category theory and related mathematics]] until I finally got tired (through 1969).
* Then had to deal with the agonisngly slow server while I checked this.
* And finally took a whole *hour* just to get access to this page so I could report on what I did ---nothing else!
* Having a discussion with [[Zoran ?koda]] about transliteration at [[M M Postnikov]].
* Removed the redirect from [[Cauchy colimit]] to [[Cauchy complete category]] on the grounds that they are not actually discussed there.
* Restored the link from [[direct sum]] accordingly (but maybe it should not link there?).
* Changed the example at [[redirect]] accordingly.
## 2009-07-13
* [[Urs Schreiber]]
* started working on [[electromagnetism]], but no nice entry yet -- will have to call it quits now -- won't mind if anyone feels like improving on the current situation, otherwise I'll continue tomorrow
* created [[gauge theory]] but only in order to create [[electromagnetism]]
* [[Zoran ?koda]]: created [[Otto Schreier]] and made some corrections and additions to [[Timeline of category theory and related mathematics|timeline]]. Many attributions give too late dates, e.g. [[Vladimir Voevodsky]]'s motives dated to 2000, while 1st versions emerged already around 1995-1996. When I saw his 1996 paper Homology of schemes in 1996 I immediately after reading about a page said to myself this is a Fields medal (and it was only in 2002 to my surprise); his preprint on K-theory arXiv on the solution of Milnor conjecture which was in that circle of methods is 1995 or 1996 as well. Note the usage of some concepts of homological algebra by Cayley before Hilbert.
* [[Urs Schreiber]]:
* added a remark about the general nonsense at [[nerve and realization]] to [[homotopy coherent nerve]]
* added some links to new entries to the link list at [[Higher Topos Theory]]
* [[Zoran ?koda]]: created [[scheme]], [[Nikolai Durov]], [[model stack]], [[pseudomodel stack]]. For timeline enthusiasts, I noticed that there is a big overlap (and some disagreements) with knowledgeable 40-page article
* Charles Weibel, [A history of homological algebra](http://www.math.rutgers.edu/~weibel/HA-history.dvi)
* [[Urs Schreiber]]:
* added references to [[John Baez|John]]'s lectures and TWFs to [[generalized (Eilenberg-Steenrod) cohomology]] and to [[Postnikov system]]
* brought [[models for infinity-stack (infinity,1)-toposes]] to a reasonable completion
* [[Eric]]: I just beautified this. By the way, every single instance of plural links on that page contained redirects so you can start saving time for yourself by typing links like <nowiki>[[pages]]</nowiki> instead of <nowiki>[[page]]s</nowiki> or even <nowiki>[[page|pages]]</nowiki>. Just FYI.
* added [[homotopy coherent nerve]] as a further example at [[nerve and realization]]
* [[Eric]]: Added a section to [[redirects]] on "Undoing a Redirect".
* [[Andrew Stacey]]: started fleshing out an example over at [[Frolicher space]].
* [[Urs Schreiber]]:
* added to [[limit]] the $(\infty,1)$-categorical definition with a pointer to the entry on limits in quasi-categories
* I am thinking (now) that generally we should in entries such as _adjoint_, _limit_, etc, list _all_ reasonable variations and generalizations, possibly just providing a link to a separate entry but still mentioning the generalized concept
* created [[Cartesian fibration]]
* added to [[adjoint functor]] the definition for $(\infty,1)$-categories
* created [[local equivalence]]
* expanded [[Bousfield localization]] by discussion of localization of model categories
* replied at [[chain homology and cohomology]] and have myself a comment -- but no time at the moment to work on this entry, will try to come back to it later, unless some helpful soul takes care of it in the meantime
* created [[models for infinity-stack (infinity,1)-toposes]] -- but still working
* [[David Roberts]]: Comment at [[chain homology and cohomology]] re: type of space represented by a positive-degrees chain complex. Also comment at [[sphere]] regarding topology on infinite sphere for the purposes of contractibility.
* [[Toby Bartels]]:
* Comments on [[Timeline of category theory and related mathematics]].
* Created [[opposite relation]], quite brief.
* [[David Roberts]]: Posted kernel of an idea at [[microbundle]] that has been sitting at the back of mind for a while, in the hope someone might be able to use it or do something interesting with it.
* [[Toby Bartels]]: Wrote [[subset collection]] (a [[foundations|foundational]] axiom of [[set theory]] intermediate between [[power set]]s and [[function set]]s, justified by [[type theory]] and strong enough to define the located Dedekind [[real number]]s).
## 2009-07-12
* [[Toby Bartels]]: Wrote [[sphere]] and [[pointed space]] to fill some gaps. The former has a reference to (yet unwritten) [[Whitehead's theorem]] with the provacative claim that this shows that [[generalised (Eilenberg?Steenrod) homotopy theory]] is unnecessary; I don\'t really intend to defend that, but maybe it will interest the people working on that subject?
* [[Andrew Stacey]]: Started a discussion on the n-forum about how to get a snapshot of the n-lab (since this is really an announcement page rather than a discussion page).
Discussion is [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=42).
* [[Toby Bartels]]: Added some illustrations to [[simplicial set]], based on those at [[cubical set]], as requested by an [[InterestedAnonymousCoward]].
* [[Zoran ?koda]] created [[microbundle]]. Note that classical references do not mention morphisms, just isomorphisms or equivalences of microbundles. Did anybody notice my update downstairs on the issue of export_html (answer to Urs/Toby answers) ? I suggested that once a week an export_html be posted as a file to be downloaded which is not up-to-date with a warning, as I think (maybe I should be corrected) that Jacques stopped serving export_html because of long generation/compilation time, while static file and new cimpilation once a week will do less harm. And leave generation of export_markup as it is, up-to-date.
## 2009-07-11
* [[Urs Schreiber|Urs]]:
* branched off [[chain homology and cohomology]] from [[cohomology]], prodded by the blog discussion [here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c025198) -- but left somewhat unfinished as I need to run
* [[Toby Bartels]]:
* Another tip for Zoran: If $\sqrt{\frac{a}{b}}$ looks bad, then the problem is on your end. To be sure, it looks bad to me too, but that problem is on my end; it looks good if I use the STIX fonts (as discussed [here](http://golem.ph.utexas.edu/instiki/show/Browsers/)), but I think that those are otherwise pretty ugly, so I don\'t use them. So the problem is that almost every font doesn\'t know how to do that sort of thing correctly ... but the MathML produced by Instiki is correct. (Update: Actually, that example looks just fine in DejaVu Serif too, but I remember that there are other examples that don\'t.)
* A question on terminology at [[replete subcategory]]. Possibly the answers should inform [[equivalence]].
## 2009-07-10
* [[Urs Schreiber|Urs]]:
* added model category version to [[local object]]
* added remark about relation of Quillen equivalences to the corresponding presented $(\infty,1)$-categories to [[presentable (infinity,1)-category]]
* created [[combinatorial simplicial model category]]
* created [[combinatorial model category]]
* [[Tim]]: I have started to reorganise some of the entries on Cech methods since David has started on [[homotopy (as an operation)]] and had an idea about [[Cech homotopy]]. I have encorporated a point made by [[Zoran ?koda|Zoran]] about the history of [[Cech methods]].
* [[Zoran ?koda]]: created a version (to be expanded) of [[Legendre polynomial]] and [[M M Postnikov]] and added references to [[Postnikov system]]. I think historically tower and system differed by inclusion of universal cohomological class representing the fibration into the notion of system (cf. Whitehead's big book, ch.9). This should be still noted: if one does not specify the cohomological class this is I think like missing the choice of isomorphism when the isomorphism exists. Technical issues: I encountered a problem that sqrt{fraction} puts sqrt only such that the numerator is under the root. I do not know how one should write correctly. Toby thank you for the tip for getting the SOURCE of old versions. I sometimes write some items partly motivated by need to have them for my students, and plan to incorporate something close to my version into student scripta. You are very knowledgable about wiki world. :) I was also trying to take the export of the whole nlab and succeeded to get the markupML version but not html: when asking nlab/export_html i get 403 FORBIDDEN message in my browser.
* [[Urs Schreiber]]: I also get this error message when trying to export the $n$Lab as html -- I remember the html export was particularly heavy on the server and maybe it was truned off in view of the server being a bit weak -- we are trying to move to a better host eventually
* _Toby_: Yes, Jacques turned that off because it was such a load on the server; I expect that we can turn it on again when we get a better host. In principle, you can get the HTML by exporting the source and compiling it on a local copy of Instiki, but of course you have to install Instiki to do that. (Also note that you\'ll need the CSS files if you want the HTML to look the same, including fonts, query boxes, etc.) And neither of these includes old versions; I think that Urs(?) is backing those up periodically in case the server crashes.
* _Zoran_: could then somebody make one copy of html zip file 2-3 times a month ? It would not be updated but still it would be useful for browsing math when offline. If I get the zip-file I can put it online on my homepage. Or simply could Jacques put one zip file of export_html weekly with link and warning that it is not up-to-date; and for editing we can anywy use markup version. Then the server does not get heavy with generation, I think that probably generating, compiling all takes time, the shear downloading from time to time would maybe not burden the server ?
* [[David Corfield|David]]: began an experiment on [[homotopy (as an operation)]] of dualizing the [[cohomology]] page. Began [[generalized (Eilenberg-Steenrod) homotopy]].
* [[Urs Schreiber]]:
* rewrote the intro to [[Cech cohomology]] (see there) and started adding a list of examples for the abelian case: the standard series line bundles, line bundle gerbes, with connection, etc. -- but not well written at the moment and no effort to get signs straight
* created [[nerve and realization]] in order to host Kan's general idea of how a functor into a category with colimits induces an adjoint pair of functors
* I think I know what I am doing, but I'd like to ask people like [[Tim Porter]] and [[Todd Trimble]] to have a critical look at it (where is [[Mike Shulman]]??) -- in particular at the moment I allowed myself to assume that we are copowered over the enriching category in order to get nice formulas, I wouldn't object if somebody finds the time to give the more general discussion
* then of course I adjusted links and made some comments accordingly at [[nerve]], [[geometric realization]] and [[Dold-Kan correspondence]]
* following [[Eric Forgy|Eric]]'s question I typed a quick reply into [[cohomology]] on how the ordinary notion of cohomology in cochain complexes is reproduced. In principle this gives all the necessary information, but I'll try to find the time later to give a long detailed exposition of how this basic important special case arises from the very general perspective
* added an "Idea" section to the beginning of
[[Postnikov system in triangulated category]]
* in the course of that I noticed that the sub-web of stable higher category links was lacking a bit of coherence. I tried to improve the link lists at [[stable (infinity,1)-category]] a bit accordingly and also added an "Idea" piece with links to [[enhanced triangulated category]].
* [[Toby Bartels]]: A quick note: I also have been changing `[[apple]]s` to `[[apples]]` but I will not do it now unless I have reason to think that it was written by someone who prefers `[[apples]]`. In general, I do not edit matters of taste; I didn\'t know that this was a matter of taste, but now I do know that. (Sometimes if I\'m changing something else, then I will change matters of taste at the same time, but only if I have substantially rewritten the sentence or if helps to standardise the notation and terminology. And this does not qualify as notation and terminology.) I\'m sorry if I caused offence, but please understand that I did not know that there was a difference of style.
* PS: If Eric adds `[[!redirects apples]]` to the bottom of `[[apple]]`, then both `http://ncatlab.org/nlab/show/apple` and `http://ncatlab.org/nlab/show/apples` should work. I will still add such redirects myself, for the benefit of the style preferred by Urs, Eric, and me; but I will no longer change styles in what Zoran has written.
* PPS: To see the source of an old version of a page, hit **Back in time** until you find the right version (or try **History** to get a list of versions), then hit **Rollback** to see the source of that version. After you\'ve copied the source, hit **Cancel** (or **Submit** if you really want to change it back to the old version).
* PPPS: You can type <code>`[[apple]]s`</code> to get `[[apple]]s` if you find it convenient to do so.
## 2009-07-09
* [[Zoran ?koda|Zoran ?k]]oda: but the plurals are NOT there -- if I write <nowiki>[[apple]]s</nowiki> I did not use the code for plural. Let me clear the issue (I will write round brackets): Eric is doing TWO things 1) he is taking entry ((apple)) and adding the redirection instruction inside to allow for ((apples)). This creates one new version, not too bad, you consider this robust, I can tolerate it. 2) he is changing every occurence of my reference ((apple))s which used to be correct usage from within entries ((banana)), ((pear)), ((ananas)) and ((strawberry)) to new format ((apples)). This amounts to not allowing me to use legitimate ((apple))s from within ((bananas)). This second thing, unlike the first, I can not tolerate, as it has no rational explanation. I do not know if it is [[evil|good :)) ]].
* [[Eric]]: The fact that many items appear as <nowiki>[[apple]]s</nowiki> on the nLab is an artifact of the period prior to having redirects. Prior to having redirects, we'd have to write that as <nowiki>[[apple|apples]]</nowiki> to get it to render correctly which gets old after a while, so people naturally gravitated to the easier <nowiki>[[apple]]s</nowiki>. If we'd had redirects from the beginning, there would be a redirect at <nowiki>[[apple]]</nowiki> for <nowiki>[[apples]]</nowiki> and no one in their right mind would ever write <nowiki>[[apple]]s</nowiki> again (which is distracting to look at) if they could just write <nowiki>[[apples]]</nowiki> instead. I'm at a total loss as to why you oppose this. Currently, I am trying to reverse the damage so that we can make things cleaner from here. Whenever, I see "]]s", I instinctively change this to "s]]" and add a redirect if it doesn't exist. In time, this should work itself out and we should have plural redirects for most links that are commonly used. It would work itself even faster if people stopped writing "]]s" and used the plural link form instead.
* Zoran: I disagree that for example I write <nowiki>[[apple]]s</nowiki> rather than <nowiki>[[apple|apples]]</nowiki> beause it is easier. I write it because the appearance of <nowiki>[[apple]]s</nowiki> I like more: it tells me by the color which part of the name is real URL (as I often type URL), plus I have no dislike for multicolored words. I will continue writing like that. If you like to write your way write, but leave my links the way they are. Otherwise I can not ENJOY writing. It takes sometimes the whole minute to reload the page and I often reload the page is somebody has changes it, and it disappoints if the change is a matter of taste, and is legitimate. Second there may be day when you will have no time to write redirects etc. and one will not memorize which redirects exist and which do not. If I write the way I do, I will never have problem with this. If I want a single-color appearance of the link for some reason, I do not mind writing it long way like <nowiki>[[apple|apples]]</nowiki>, it is about 2 seconds more, rather than spending far more time to check if there is a created redirect and wait half a minute to load the page, specially if in addition to slowness of the server I have my own internet connection problems what is very often.
* [[Urs Schreiber]]
* I started going a bit through the [[Timeline of category theory and related mathematics]] and added links to $n$Lab entries wherever I saw a possibility
* in that context: I like what [[Eric Forgy|Eric]] is doing. It makes the linkage of the Lab more robust. For instance quite a few of the Timeline's imported links _didn't_ break (while thousands broke) just because Eric had made sure that some plurals etc were recognized.
* [[Zoran Skoda]]
* created [[Postnikov system in triangulated category]]
* I STRONGLY disagree with creation of spurios PLURAL items when they are of NO use. Namely like Eric just created new version of [[bialgebra cocycle]] to say that it redirects [[bialgebra cocycles]], while it is simpler and better from memory point of view to write [[bialgebra cocycle]]s. Why having whole page archived and one more page to browse in history just to distiguish if s is before or after the brackets ?? Dear Eric we have thousands of items to enter and there will be thousands of new pages in future and why to increase entropy and spend yuor valuable time on this -- please take some book and help us entering something NEW and not messing with plurals and creating new versions for nothing. I have hard time browsing history when something is messed up and if I am going to spend it on such a nonsense than I will leave the idea to enter new items myself.
* [[Eric]]: Hi Zoran. By adding a redirect for plurals, we are not creating spurious pages (but we are creating spurious revisions, but I don't see that as a meaningful issue). In fact, my hope is that everyone will stop writing things like <nowiki>[[page]]s</nowiki> in favor of <nowiki>[[pages]]</nowiki> at which point I could stop correcting the links. Please see [[redirect]] for more information. Redirects are a very nice new feature of the nLab (partially motivated by your suggestions wrt symbolic links) and I hope they become a natural part of any contributors arsenal. PS: It is ironic that you complain about creating additional revisions when you just created a THIRD revision by changing my redirect BACK (???). PPS: A good place to discuss this and any other administrative issues is on the [n-forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/). That is a good place for such discussions and once a decision has been made about any issue, the conclusion will be placed on the brand new [nLab meta](http://ncatlab.org/nlabmeta/show/HomePage) site. We're working on decision making processes now and any feedback is more than welcome.
* Zoran: look at my explanation few paragraphs above: I accept your creating redirects, but do not accept not allowing me having my own format of calling links within my own text. I changed your redirect back because I want to assert my right to have the link called any possible way I like it. It functions, it correctly displays and it si even more informative: ((apple))s tells you even the information that s is not the name of the page, and I often TYPE the URL name of the page rather than clicking on the link, because often because of slowness of the server working on laptop and desktop simultaneously. Of course this kind of strange usage is useful just for the author, but if I am in process of improving the page which i largely created i think I have the right for the convenience.
* [[Eric]]: Hi Zoran. The nLab is a group effort, and as such, it has to have some rules and should probably have at least some (loose) sense of uniformity. I don't think it makes sense for Toby and I to be going around correcting links to have you change them back simply because you want assert yourself. The nLab is already MUCH looser than Wikipedia as far as standards and formats and I think that is a good thing, but I also think there should be some agreed upon "Matters of Style". I started a discussion on "Matters of Style" on the n-forum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=41). We should probably discuss it there. Whatever we all decide on, then we can add a "Matters of Style" page to [nLab meta](http://ncatlab.org/nlabmeta/show/HomePage). Anyone reading this is more than welcome to voice their opinion, but once a "quorum" is met, I think we should establish a rule. In a bit, we may also want to remove this discussion from [[latest changes]] since this isn't what the page was intended for.
* created [[bialgebra cocycle]], [[Drinfel'd twist]]
* by the way how does one download the source of a previous version ? I sometimes create a page and then there are changes after and I want to have my file for other purposes with what I wrote and I do not know how to access it.
* [[John Baez|John]]
* I did some work on the [[Timeline of category theory and related mathematics]], and proposed some new guidelines for how it should look.
* [[Urs Schreiber|Urs]]
* am working at [[Cech cohomology]] on the section "Abelian Cech cohomology" where my aim is to spell out a derivation of the standard Cech double complex from starting with the general definition of cohomology for the case that the coefficient object is in the image of the Dold-Kan map from chain complexes of sheaves to simplicial sheaves -- I am not really satisfied, but this is how far I got -- check critically
## 2009-07-08
* [[Urs Schreiber]]:
* added to [[nerve]] an "Idea" section and a further "examples"-section on nerves for chain complexes and the relation to the Dold-Kan correspondence
* after a bit of work and with a bit of luck, I found the old reference by Kan where the description of the Dold-Kan correspondence is given in terms of nerves -- this is the nice way to do it -- added that reference to the [[Dold-Kan correspondence]]
* that reminds me that it would be nice if we eventually had an entry on that verey general nonsense behind nerves (or do we already have that somewhere?)
* by coincidence I came across the old entry [[crossed module]] and noticed that there were meanwhile plenty of links to add to it -- so I did
* added two further references, by Birgit Richter, on the ($\infty$-)monoidal structure of the Dold-Kan correspondence to [[Dold-Kan correspondence]]
* added a small section on and a link to [[matching family]] at [[sheafification]]
* reorganized [[Dold-Kan correspondence]] in an attempt to make the material more systematic -- then I started adding a detailed def/lemma/theorem/proof list of the classical statement, but didn't get very far yet
* made a remark at [[Timeline of category theory and related mathematics]]
* pasted a blog comment by [[David Ben-Zvi]] into [[n-categorical physics]]
* [[Zoran ?koda]]: created [[essential image]], added n-category generalization to [[replete subcategory]], additions to [[image]], [[model structure on chain complexes]], remark due Leinster to [[Gray-category]]; created [[matching family]] following mainly conventions in MacLane-Moerdijk. This last item surely overlaps with [[sheafification]] but the approach and exposition is rather different; created micro-entry [[maximal sieve]].
## 2009-07-07
* [[Toby Bartels]]:
* Gave Tim a link at [[category theory]].
* Started [[suspension]] to fill links. (We already have [[reduced suspension]]. There\'s also a suspension defined at [[delooping]] which doesn\'t seem to be quite the same; what\'s the connection?)
* [[Urs Schreiber|Urs]]: when passing from topological spaces to spectra, suspension of spaces becomes the suspension mentioned at delooping: that in a [[stable (infinity,1)-category]], I'd say
* [[David Corfield|David]]
* added to [[co-H-space]]. It's not looking very much like a dual version of [[H-space]].
* [[Urs Schreiber|Urs]]
* created [[Cech cover]] and [[Cech model structure on simplicial presheaves]]
* in that context also rewrote and expanded the introduction at [[hypercover]] and included corresponding links at [[model structure on simplicial presheaves]].
* I notice that [[Timeline of category theory and related mathematics]] is a renamed version of the original Sandbox(!). (As you can see by clicking on its "history" link. But also the current [[Sandbox]]'s very first link claims to lead to the historical Sandbox, but takes on to the Sandbox poage with title "Timeline of category theory-...") I guess that's not intended.
* This came about because [[Rafael Borowiecki]] created the [[Timeline of category theory and related mathematics|Timeline]] by moving the [[Sandbox]] instead of as a new page. So I was faced with the choice to move it back and then create Rafael\'s page properly or to just recreate the Sandbox instead. So I recreated the Sandbox but then put a note at the top in case anybody wants to see the history of that page, which is now at the [[Timeline of category theory and related mathematics|Timeline]]. (But now that it\'s been more than half an hour, I can remove that note; anyone looking through the history will still see it.) ---Toby
* tried to improve the exposition at [[groupoid object in an (infinity,1)-category]]. More to be done eventually, though.
* [[Eric]]
* Reminds Urs that he no longer needs to type <nowiki>[[Urs Schreiber|Urs]]</nowiki> because there is now a redirect from [[Urs]] to [[Urs Schreiber]]., i.e. the link <nowiki>[[Urs]]</nowiki> points to [[Urs]]. Try it! :)
* Along similar lines, after seeing <nowiki>[[internalization|internal to]]</nowiki> about a million times, I just added a redirect so that typing <nowiki>[[internal to]]</nowiki> gets redirected automatically to [[internal to]]. Try it! :)
## 2009-07-06
* [[Urs]]
* replied to [[Toby Bartels|Toby]] at [[cogroup]] (yes, that wasn't so good) and to [[David Roberts|David R.]] at [[principal infinity-bundle]] (yes, that was a typo)
* [[Toby Bartels]]:
* I would like to suggest that the appearance of links to nonexistent pages is a *feature* that we should not break. Thus we should not create blank pages (or pages that are blank except for redirects) but instead create pages only when we have something to put there. Conversely, we shouldn\'t change links to go to redirected forms (as at [[geometric infinity-function theory]] currently) unless the redirects have actually been created. If this means that we have things like `[[∞-foo|infinity-foo]]` (when nobody has written about $\infty$-foos yet), then we\'re no worse off than before we had redirects, and the appearance of links to nonexistent pages still tells us something. (Note: there is some related discussion now hidden under July 3.)
* [[Eric]]: I agree that appearance of links to nonexistent pages is a **feature** that should not be broken. Currently, one of my self-assigned projects (a.k.a. a labor of love) is to pick a page and systematically "clean it". The last major cleaning was [[geometric ∞-function theory]]. I hope that for the most part (with some minor exceptions) most everyone would agree that it looks a lot better now. The issue that brought up your comment, however, came about as a result of me "cleaning" [[strict ∞-category]]. Part of my system for cleaning pages involves changing things like <nowiki>[[category|categories]]</nowiki> to simply <nowiki>[[categories]]</nowiki> and adding a redirect from [[categories]] to [[category]]. If I do this enough, then most n-Lab pages will have available redirects for plural forms of nouns. In fact, I would encourage everyone to stop including things like <nowiki>[[singular page name|plural page name]]</nowiki> and simply use the plural form <nowiki>[[plural page name]]</nowiki>. Then either leave the nonexistent link there for someone to add the redirect or add the [[redirect]] yourself. What happened on [[strict ∞-category]] was that there was a link <nowiki>[[exchange law|exchange laws]]</nowiki>. My "system" suggests that I change that to <nowiki>[[exchange laws]]</nowiki> and add a redirect from [[exchange laws]] to [[exchange law]]. In this rare case, [[exchange law]] did not exist yet, so I couldn't add a redirect without creating a blank page for [[exchange law]]. In this rare case, given that I am trying to systematically "clean" pages, I thought it was acceptable to break the rule about nonexistent pages. I still think it is justified as long as it doesn't become a habit, which I'm pretty sure it won't be.
* A question at [[cogroup]] about what we really want there. (Surely more than just the empty set?)
* [[Urs Schreiber|Urs]]: we want to say that pointed _spheres_ are co-groups, so that maps out of them, called homotopy groups, are, well, groups. Supposed to be dual to the statement that mapping _into_ a group coefficient object gives [[cohomology group]]s
* [[Urs Schreiber|Urs]]
* added to [[principal bundle]] a long detailed section called "the $G$-action from the homotopy pullback" -- this may look like weird overkill, but the point is that this serves as a warmup for an analogous discussion at [[principal infinity-bundle]]
* rediscovered that we had an entry [[Cech methods]] and added lots of links to that
* provided explicit details at [[Cech cohomology]] for the general (nonabelian) case in low degree
* [[Zoran ?koda]]: created [[small fibration]], added more general discussion on [[endomorphism monoids]].
* [[Urs Schreiber|Urs]] created [[Cech cohomology]]
* [[David Corfield|David]]
* more suggestions than changes, but it would be good to have entries for [[cogroup]] and [[co-H-space]]. Could [[homotopy group]] and [[cohomology group]] be made to resemble each other more? I.e., must the former be restricted to $n$-spheres as domain? Hmm, is something suboptimal about H-group and H-cogroup, whereas H-space and co-H-space? Perhaps 'co' and 'H' commute.
* [[Urs Schreiber|Urs]]: it would in principle be good to have expositions at [[cohomology]], [[cohomology group]] and [[homotopy]], [[homotopy group]] be more symmetric -- we just have to beware that we'd be going into untrodden territory and should indicate accordingly: while generalized cohomologies of all sorts are becoming familiar, I can't recall having seen mentioned the corresponding dual notion of generalized homotopy anywhere but in our discussion -- so we should maybe tag the corresponding discussion, if we implrement it, with a box saying "this is research material" or the like -- but I would enjoy it if we did so
* [[Urs Schreiber|Urs]]
* created [[groupoid object in an (infinity,1)-category]]
* added a discussion of this at [[delooping]], a brief reference to it at [[quotient object]] and a link to it in the fourth $\inft$-Giraud-axiom at [[(infinity,1)-topos]]
## 2009-07-05
* [[Toby Bartels]]:
* Mentioned [[endomorphism monoids]] too.
* Tried to give Tim some comfort at [[pseudofunctor]].
* [[Urs Schreiber]]
* added a tiny bit of discussion to [[principal infinity-bundle]] about how the homotopy pullback definition gives the $G$-action and vice versa. But need to say more here.
* By the way has anyone seen [[Mike Shulman|Mike]]? I know that he has thought about this, it's related to his [[michaelshulman:exactness hypothesis|exactness hypothesis]].
## 2009-07-04
* [[Toby Bartels]]: Mentioned [[automorphism groups]].
* [[Tim]]: I have asked a silly question at [[pseudofunctor]], but would appreciate an answer. Can I make a plea to someone to provide a more detailed treatment of the Grothendieck construction as well. (I mean the one which is related to the pseudocolimit. At least a general reference to that should be in the entry.)
* [[Toby Bartels]]:
* Put some stuff at [[equality]], but there is much more that could and should be said there.
* Added some pretty broad examples to [[evil]].
* [[Tim]]: I have added new references to [[distributor]] and [[Grothendieck fibration]].
* [[Toby Bartels]]: Added a bit to the scandalously short page [[isomorphism]].
## 2009-07-03
* [[Eric Forgy]] created [[exchange law]] blank (perhaps by accident?) and [[Toby Bartels]] wrote a very brief stub there.
* _Eric_: This was not an accident. I was cleaning some pages and wanted a redirect for [[exchange laws]], but [[exchange law]] did not exist yet, so I created it so I could insert a redirect assuming (correctly) that someone would fill in some content.
* _Toby_: I wouldn\'t agree with your assumption. What I wrote there is not very useful; if I\'d waited to write it by choice rather than by necessity, then I would have written rather more.
* [[Urs Schreiber|Urs]]
* edited and expanded the link list at [[generalized smooth space]]
* created [[smooth infinity-stack]] -- took the liberty of declaring this to be the term for "$\infty$-stacks on CartesianSpaces"
* see the example there to see the relevance of this, with an eye towards the discussion at [[principal 2-bundle]] and [[principal infinity-bundle]]
* created [[smooth space]] -- took the liberty of declaring this to be the term for "sheaves on Diff" = "sheaves on CartesianSpaces".
* this way a "diffeological space" is precisely a "concrete smooth space" and I
* edited [[diffeological space]]
* and created [[concrete sheaf]] (but [[concrete site]] still missing, and also maybe not the best definition at [[concrete sheaf]] yet)
to make this statement.
* [[Toby Bartels]]: Created [[Cauchy sequence]] and [[complete space]], including brief references at the end of each to the relationship with [[Cauchy complete categories]].
## 2009-07-02
* [[Urs Schreiber|Urs]]
* started a section "concrete realizations" at [[principal infinity-bundle]]. So far I recall the old result by Quillen on certain "1-categorica topological bundles" and their $\infty$-action groupoids. Then I start making some remarks on Jardine's approach using what he calls "diagrams" and have a remark on how that compares to the Bartels-Baković-etc-style. This requires eventually much more discussion, but I have to call it quits for today.
But the point is that Jardine works in the "petit topos" perspective where all bundles live over the fixed site. So the terminal object in his setup is not the point, but the underlying space over which one works. This means that the simple picture of a principal $\infty$-bundle as the homotopy pullback of the point no longer works and is the reason why he introduces the yoga of what he calls "diagrams".
On the other hand, when one works with simplicial sheaves in the context of a "gros topos" such as sheaves on $Diff$ or on open subsets of $\mathbb{R}^n$ or equivalent, then the simple conceptual picture remains valid.
Notice that placing oneself into the context of $n$-groupoids internal to diffeological spaces or the like is doing precissely this: working with simplicial sheaves on $Diff$.
Evidently i shouldn't be discussing this here but in some entry. But it's time for me to go to bed now.
* [[Zoran ?koda]] created [[grouplike element]]. It contains few words on Amitsur complex for a coring with a (semi-)grouplike. The aim is to soon (using the setup) introduce entries for connections for corings; and then correpsondence between falt connections and descent data in the comonadic and coring setups (after Menini et al; all coming back to the example which is the correspondence between 1-order costratifications and flat connections in the crystalline setup due Grothendieck).
* [[Urs Schreiber|Urs]]
* started adding the discussion of the model given by (pre)sheaves with values in _simplicial groupoids_ to [[model structure on simplicial presheaves]] (contained in a new big diagram containing all Quillen equivalent models for $\infty$-sheaves)
* in that context created [[model structure on presheaves of simplicial groupoids]]
* [[Zoran ?koda]] created [[coseparable coring]],[[Sweedler coring]], [[two dimensional sheaf theory]]; expanded [[stratifold]] (which was empty, but existing!), added a reference to [[fibration in a 2-category]] and somewhere else. I think that in K-theory delooping has a bit different multiple meanings which are related but are more procedures making from something what can not be delooped strictly in the sense of [[delooping]] to the delooping of something what is the best approximation of deloopable; there are procedures due to Quillen, Waldhausen, Karoubi etc.
* [[Urs Schreiber|Urs]]: the entry on [[two dimensional sheaf theory]] is motivated from some behind-the-scenes discussion Zoran and I are having -- Zoran rightly points out that the present characterization of [[derived stack]] may be wanting, as strictly speaking saying "derived stack" should be related to but not be regarded as equivalent to "higher stack on higher categorical site" -- what we really need eventually is more details on the Toën-Vezzosi [work](http://www.math.uiuc.edu/K-theory/0579/) on higher topos theory in the context of SSet-enriched categories,
* [[Urs Schreiber|Urs]]
* replied at [[delooping]]
* _[[Eric]]_: Added a redirect for [[Urs]] so that he no longer has to type <nowiki>[[Urs Schreiber|Urs]]</nowiki> and can simply type <nowiki>[[Urs]]</nowiki> and will be redirected to the correct page.
## 2009-07-01
* [[Urs Schreiber|Urs]]
* proudly presenting what is now the widest (horizontally speaking) $n$Lab entry as yet: at [[model structure on simplicial presheaves]] I added a section "Map" where I draw a big diagram that indicates at least part of the collection of model structures, their interrelation, definition and authors
* Yeah, I was thinking of reworking that map to run vertically .... ---Toby
* Would that really be better, though? Optimally, eventually we'd produce a LaTeXed pdf diagram. Here and elsewhere. That, however, inhibits joint editing a bit. ---Urs
* replied at [[delooping]], at [[group]], added a bit at [[Notation]]
* moved the key statement of [[Toby Bartels|Toby]]'s remark to the very beginning of [[group]]
* [[Toby Bartels]]:
* [[Andrew Stacey]] may be interested to see how I\'ve changed the formatting of the goal box at [[Froelicher space]].
* Incorporated Zoran\'s new references at [[principal 2-bundle]] into the text.
* Added a note at [[group]] how $Grp$ is a full sub-$2$-category of $Grpd_*$ (even though not of $Grpd$).
* [[Urs Schreiber|Urs]]
* expanded at [[group]] on the statement that "a group is a groupoid with a single object"
* added to [[loop space object]] the true definition, added also an "idea" section, links to related entries and an introductory blurb to the original material
* created [[delooping]] for the matter discussed by [[Eric Forgy|Eric]] and [[Toby Bartels|Toby]] at [[Dijkgraaf-Witten theory]]
* added Jardine-Luo reference to [[principal 2-bundle]] and [[principal infinity-bundle]]
* added to [[Froelicher space]] an "Idea" and a "Reference" section, prompoted by the blog entry [here](http://golem.ph.utexas.edu/category/2009/07/laubinger_on_lie_algebras_for.html)
* [[Toby Bartels]]:
* I archived the [[2009 June changes]] by the sneaky method of changing this page\'s name, copying the header and footer text (with appropriate changes) from the previous archive, and then creating this page anew. Maybe that will keep us from testing Instiki\'s tolerance for pages with thousands of edits in their history. (^_^)
* I\'ve completed migrating all old redirect pages to history pages; all links within the body of the Lab should now take you immediately to the target page. (There are a few victims of a bug, but I\'ll straighten that out with Jacques.) I have [more comments](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=39) on the Forum.
***
[[2008 changes|First list]] --- [[2009 June changes|Previous list]] --- [[2009 August changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 June changes | https://ncatlab.org/nlab/source/2009+June+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during June 2009.
The substantive content of this page should **not** be altered.
***
# 2009-06-30 #
* [[Urs Schreiber|Urs]]
* created [[Eilenberg-MacLane object]] prompted by the blog discussion [here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024902)
* replied at [[homotopy]]
* [[Toby Bartels]]:
* Responded to [[Ronnie Brown]] at [[groupoid]].
* Noted an incomplete thought at [[simplicial homotopy]].
* Complained a bit at [[homotopy]].
* [[Urs Schreiber|Urs]]
* created [[Quillen adjunction]]
* added definition to [[Quillen equivalence]]
* reorganized [[model structure on simplicial presheaves]]:
* expanded the idea/introduction-section
* moved the original material to
* [[global model structure on simplicial presheaves]] (and expanded that)
* [[local model structure on simplicial presheaves]]
* created [[local model structure on simplicial sheaves]]
* [[rectified infinity-stack]]
* fixed (hopefully) the nonsense paragraph (due to me) that [[Toby Bartels|Toby]] had pointed out at [[sieve]]
* [[Tim Porter|Tim]]:
* I have pointed out at [[simplicial homotopy]] that the combinatorial description has much greater validity than claimed there before. It works for simplicial objects in any finitely cocomplete category. (I think that is in Duskin's Memoir volume, but is not difficult to check directly.) It depends on having the [[copower]] with $\Delta[1]$, which brings me to
* We have two notations for simplices in $SSet$. We use both $\Delta[n]$ and $\Delta^n$. I do not like the second one. I have started a query on this at [[simplicial set]].
* [[Urs Schreiber|Urs]] that $\Delta^n$ is due to me -- I'll try to change it back to $\Delta[n]$ (same for horns) where I see it
* [[Urs Schreiber|Urs]]
* prompted by [[David Corfield|David]]'s remark at [[homotopy group]] I was led to all of the following:
I split [[homotopy]] into
* [[homotopy (as a transformation)]]
* [[homotopy (as an operation)]]
and moved the material originally found at [[homotopy]] to [[homotopy (as a transformation)]], kept only two commented links at [[homotopy]] and then wrote [[homotopy (as an operation)]] in the spirit of David's remark and in the light of the entry [[Eckmann-Hilton duality]] (but just some tentative abstract nonsense so far).
then finally I added to [[homotopy group]] a last section "some abstract nonsense" that indicates how the discussion of homtopy groups could be given analogous, just dual, to the discussion the way we have it (since recently) at [[cohomology group]]
* expanded at [[cohomology group]] (standard abelian examples, standard nonabelian examples)
* [[David Corfield]] has an interesting question there, I think he is right, will try to write something at [[homotopy group]]
* [[David Corfield|David]]
* added to [[Eckmann-Hilton duality]]
* [[Urs Schreiber|Urs]]
* continued polishing and adding statements and proofs to [[category of fibrant objects]]
* created a stub for [[simplicial skeleton]] -- much more to be said here eventually
# 2009-06-29 #
* [[John Baez]]: I polished up the entry about the book [[John Baez:Towards Higher Categories|Towards Higher Categories]] on my personal web, and modified the [[Towards Higher Categories|nLab entry]] accordingly. I expect someday to have lots of my papers and books on the $n$Lab. Somehow I want them to be easy to find from the main $n$Lab, while discouraging other people from changing them (yet perhaps not making this impossible). The search feature on the main portion of the $n$Lab doesn't search people's personal webs, does it?
* [[Urs Schreiber|Urs]]: I don't think it does
* [[Toby Bartels]]: People keep linking to [[left adjoint]] and [[right adjoint]] when talking about [[adjoint functor]]s, even though these redirect to the general concept of [[adjunction]]. This is very natural, so I\'ve now created those as pages in their own right. They consist of the definitions in a variety of contexts and then refer you to the other pages for more detail. While I was creating pages that collected information from other pages, I also wrote [[triangle identities]] and [[weak inverse]]. A picture in string diagrams would be very welcome at the former.
* [[Urs Schreiber]] continued filling in propositions and detailed proofs at [[category of fibrant objects]]
* [[Toby Bartels]] fleshed out [[discrete fibration]] a bit.
* [[Eric Forgy]] created [[interval category]].
* [[Zoran Škoda]]: created [[discrete fibration]], [[quasideterminant]] and made additions (please check for correctness) to [[orthogonal factorization system]], added Duskin's reference to [[gerbe (general idea)]] (btw it emphasises on internal point of view to gerbes -- as "bouquets", missing in division of gerbe entries in nlab).
* [[Urs Schreiber]]
* created reference entry [[Lectures on n-Categories and Cohomology]] in order to collect the existing $n$Lab entries on topics discussed there
# 2009-06-28 #
* [[Toby Bartels]] responded to Todd below.
* [[Todd Trimble|Todd]]: piped in on a discussion between [[Bruce Bartlett]] and [[Urs Schreiber]] at [[sieve]]; put in a related two cents at [[subobject]].
# 2009-06-27 #
* [[Todd Trimble|Todd]]: added more material to [[Trimble n-category]], outlining Leinster and Cheng's extension from $n$ to $\infty$.
* [[Eric Forgy]] has begun some pages related to administration of the Lab:
* [[Organization of the nLab]]
* [[On Scientific Contributions to the nLab]]
* [[AnonymousCoward]] (which is not new but has a new discussion of privacy issues)
* not so much administration but new suggestions for resources:
* [[Bibliography]]
* [[Notation]]
* [[Urs Schreiber]]
* created a section "Category theory for Trimble $n$-categories" at [[Trimble n-category]] and put in a query box with a list of questions -- I'd be interested in whatever partial answers and comments
* created [[Approaching Higher Category Theory]] (following blog discussion [here](http://golem.ph.utexas.edu/category/2009/06/this_book_needs_a_title.html#c024835) and [here](http://golem.ph.utexas.edu/category/2009/06/this_weeks_finds_in_mathematic_36.html#c024834))
* added to [[point of a topos]] a section "enough points" with discussion about what it means for a topos to have "enough points" and two examples
* created [[QFT with defects]]
* created [[2-category of 2-dimensional cobordisms]]
* added to [[cobordism hypothesis]] more details on formalization and proof by J. Lurie
* reworked [[(infinity,n)-category of cobordisms]]
* split off [[n-fold complete Segal space]] from [[(infinity,n)-category]]
* created [[(infinity,2)-category]]
# 2009-06-26 #
* [[Toby Bartels]]:
* Wrote [[upper set]] and [[lower set]] to satisfy more links (including the one from [[Chu construction]] that didn\'t mean what I thought that it meant!).
* Started to move stuff from [[generalized tangle hypothesis]] to [[cobordism hypothesis]] but realised that I don\'t really know enough to disentangle the material ---at least not without reading the references, which I don\'t have time to do right now.
* [[Urs Schreiber|Urs]]:
* created [[gerbe (in nonabelian cohomology)]] based on the blog discussion [here](http://golem.ph.utexas.edu/category/2008/08/connections_on_nonabelian_gerb.html#c024774)
* removed the former section "references" at [[cohomology]] and replaced it with a new section "History and references" by using the material of my blog comment [here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024778)
* added to [[Trimble n-category]] a section "basic idea" with a paragraph that [[John Baez|John]] suggested to include ([here](http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html#c024776))
* added discussion that Kan complexes form a category of fibrant objects to [[category of fibrant objects]] that is slightly more direct than the argument using the full model category structure (crucial point being the theorem now at [[model structure on simplicial sets]] that acyclic fibrations of simplicial sets are characterized by a right lifting property).
* added the example of a functor that is an equivalence of groupoids as inducing isomorphisms of simplicial homotopy groups under the nerve to [[simplicial homotopy group]]
* created [[right proper model category]]
* started filling in detailed proofs at [[category of fibrant objects]] -- am thinking that maybe eventually now the entry should be broken apart into a brief overview and one containing all the details
* [[Toby Bartels]]:
* Wrote [[up set]] and [[down set]] to satisfy some links.
* Inspired [[David Roberts]] to write [[numerable open cover]].
* Spun [[strict 2-group]] off of [[2-group]] and added a bit of material about weak and coherent $2$-groups.
* Started [[(n,j)-transformation]] based on discussion at [[modification]]; also wrote [[j-morphism]] and [[n-functor]] to fill links from it.
* [[Eric Forgy]] has a request at [[notation]].
* [[David Roberts]]: created [[fibration theory]]. Also a comment at [[gerbe (general idea)]] about the fibration theory axioms.
# 2009-06-25 #
* [[Todd Trimble|Todd]]: Recently [[loop space]] was created. Also, I added some examples to [[Chu construction]] and to [[star-autonomous category]].
* [[Urs Schreiber|Urs]]:
* added the example of principal bundles to [[fibration sequence]]
* added an example section to [[group cohomology]] with details of how the abstract-nonsense definition reproduces the familiar formulas
* created [[twisted cohomology]]
* created [[n-group]]
* created [[group cohomology]]
* the point of view adopted there is _almost_ fully explicit in Brown-Higgins-Sivera's book _Nonabelian algebraic topology_ , only that one needs to notice in addition that the morphisms out of free resolutions of crossed complexes discusses there are the [[anafunctor]]s that compute the morphisms in the corresponding homotopy category. Has this been made fully explicit anywhere in the existing literature?
* fiddled a bit with the entry [[higher category theory]] (added one more introductory sentence, created a hyperlinked list of definitions of higher categories) but I still feel that we should put more energy in this particular entry. It is sort of the single central entry one would expect an "$n$Lab" to be built around, but currently it doesn't even come close to living up to playing such a pivotal role. I am imagining that it should carry some paragraphs that highlight the powerful recent developments in view of [[Pursuing Stacks]], of the kind that I filled in today in the entry [[Carlos Simpson]]. Does any higher category theory expert out there feel like writing an expositional piece for the $n$Lab here?
* created [[simplicial localization]] but was then too lazy to draw the hammock. But main point here is the link to an article by [[Tim Porter]] that nicely collects all the relevant definitions and references
* there is plenty of further material by [[Tim Porter]] that should eventually be referenced at various $(\infty,1)$-categorical entries. By chance I just came across the slides [Weak categories](http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf) which I now linked to from [[weak omega-category]] and [[higher category theory]]
* created [[principal infinity-bundle]], just for completeness
* created [[principal 2-bundle]] -- this is just the result of what came to mind while typing, I am sure to have forgotten and misrepresented crucial aspects. [[Toby Bartels]] should please have a critical look and modify as necessary, as should [[Igor Bakovic]] and [[Christoph Wockel]] in case they are reading this.
* based on a reaction I received concerning my comment below on the entry [[gerbe]] I have split the material into entries [[gerbe (as a stack)]], [[gerbe (general idea)]], [[bundle gerbe]] and kept at [[gerbe]] only pointers to these entries -- let me know what you think
* created an entry for [[Carlos Simpson]] motivated by a link to a recent [pdf note](http://math.unice.fr/~carlos/slides/ihesAGjan09.pdf) -- that [[Zoran Skoda]] kindly pointed me to -- where Simpson briefly sketches the topic of higher stacks, old and recent progress and putting his own contribution into context. I thought that was a nice short comprehensive collection of keywords and so I reproduce that text now at the entry [[Carlos Simpson]] with all the keywords hyperlinked
* created [[fibration sequence]]
* added an "Idea"-section to [[gerbe]] supposed to be read as "general idea", where I try to describe the concept in a way independent of the notion of stack, relating it to princpal bundles, principal $\infty$-bundles, fibration sequences and cohomology . At the end I say "in the following we spell out concrete realization of this idea". Then I made [[Tim Porter]]'s material a section "Realization of gerbes as stacks". Eventually I'd like to add similarly "Realization of gerbes as Stasheff-Wirth fibrations", "Realization of gerbes as bundle gerbes" etc.
* I hope that this is okay with you all, in particular I hope that [[Tim Porter|Tim]] doesn't mind me putting such a chunk of material in front of his work. If anyone has the impression that the chunk I added should rather be separated to a different entry, I'll have no objections. But please have a look first to see what I am trying to get at.
* [[Tim Porter|Tim]]:
* I have added material to [[gerbe]], although it is still a long way from explaining the link with cohomology with integer coefficients that was requested.
* In the process of doing the above I have added a deconstruction section to [[torsor]], and created [[trivial torsor]].
* I created a brief entry on [[Jean-Luc Brylinski]], but this is really a stub with a link to Wikipedia.
# 2009-06-24 #
* [[Finn Lawler]]: Added a proof (sketch) of the 'lax Yoneda lemma' to [[lax natural transformation]]. Also replied re terminology at [[modification]] (thanks to Urs for a fantastic reply to my question there).
* [[Zoran Škoda]]: created [[Hochschild-Serre spectral sequence]].
* [[Eric Forgy|Eric]]: I've started the process of applying redirects for symbolic links. This maintains the original ascii titles and urls, but makes links look much better. I've also started adding redirects for plural nouns, e.g. you no longer have to type <nowiki>[[category|categories]]</nowiki>. Now you can simply type <nowiki>[[categories]]</nowiki> and will be redirected automatically to [[category]]. To see the changes in action, have a look at [[higher category theory]]. Feedback welcome!
* [[Zoran Škoda]]: Created [[Hopf envelope]], [[free Hopf algebra]] and [[matrix Hopf algebra]] simultaneously with posting a related comment to a discussion between Baez, Trimble and Vicary on free and cofree functors for bialgebras.
* [[Tim Porter|Tim]]: I have created a stub on [[gerbe|gerbes]]. There was a request on the Café for en entry and I thought this was a good way to remind me to start one. ... but feel free to do it for me!
* [[Zoran Škoda]]:
* Created [[class of adapted objects]]. It would be better to say class of objects adapted to a functor (we talk half exact additive functors between abelian categories). Created [[Grothendieck spectral sequence]] and [[spectral sequence]]. Both need more input/work...
* I have added a general paragraph at the beginning of [[descent]] because far not all the cases of descent theory fit into the framework of sheaf/stack theory, Grothendieck topologies and homotopical methods. Namely something is a sheaf when it satisfies the descent for all covers, but there are cases when one considers just the descent problem for a single morphism, not all morphisms of anything like a Grothendieck topology. There is also a descent along families of noncommutative localizations, when one needs to resort to noncommutative generalizations of Grothendieck topologies. So descent theory is more general than the geometry of sheaves and stacks, though the latter is surely he most important part.
* [[Urs Schreiber|Urs]]:
* created [[K-theory spectrum]] with just a question
* created a stub for [[tmf]] based on a recent Category Theory mailing list contribution -- am thinking that we should more generally try to move good stuff from the mailing list into $n$Lab entries
* created [[Eilenberg-MacLane space]], [[Eilenberg-MacLane spectrum]]
* am offering a reply to [[Finn Lawler]] at [[modification]]
* [[Toby Bartels]]: Just more questions, instead of answers, for Finn and Gavin.
# 2009-06-23 #
* [[Gavin Wraith]] wrote [[matrix theory]], [[tensor product theory]], and [[bimodel]], with a question at the last.
* [[Finn Lawler]]: created [[modification]] (and asked a question there) and [[lax natural transformation]], including a statement of what I think is the Yoneda lemma for them. I have a proof (I think), which I'll add a sketch of later, if no-one finds the statement obviously wrong in the meantime.
* [[Urs Schreiber|Urs]]
* added to [[weak factorization system]] the full details of the (elementary) proof that morphisms defined by a right lifting property are stable under pullback --
* added the theorem to [[Kan fibration]] that Kan fibrations between Kan complexes that are also weak equivalences are precisely those with the right lifting property with respect to simplex boundary inclusions
* added example of fundamental $\infty$-groupoid = singular simplicial complex to [[Kan complex]]
* edited [[Kan complex]] slightly and added two little propositions characterizing groupoids in terms of their Kan complex nerves
* added further illustrations to [[Kan fibration]]
# 2009-06-22 #
* [[Toby Bartels]]: A section on [[measurable space]] about the constructive theory due to Cheng. (This is *not* a straightforward variation on the classical theory but distinctly a generalisation, so may be of interest even to classical mathematicians.)
* [[Urs Schreiber|Urs]] linked to [[normal complex of groups]] by a little lemma now added to [[Moore complex]]
* [[Tim Porter]] added [[normal complex of groups]]
* [[Urs Schreiber|Urs]]:
* worked on [[simplicial group]]:
* added section about the adjunction between simplicial groups and simplicial sets, in particular mentioning the free simplicial abelian group functor (since that plays a role in the proof at [[abelian sheaf cohomology]], which now links to it)
* moved statement and proof that every simplicial group is a Kan complex into a formal theorem-proof environment
* added references to Goerss-Jardine, to [[Tim Porter]]'s [[Tim Porter:crossed menagerie|Crossed Menagerie]] and to Peter May's book "Simplicial objects in algebraic topology"
* added an "Idea" section
* added to [[Moore complex]]
* the theorem of how it relates to the other two complexes in the game (it is quasi-isomorphic to the alternating sum complex and isomorphic to that divided by degeneracies)
* illustrations of cells in low degree
* the reference to [[Tim Porter]]'s [[Tim Porter:crossed menagerie|The Crossed Menagerie]] notes
* a brief comment on use of terminology (after checking with [[Tim Porter]])
* [[Toby Bartels]]: Started more articles that I\'d linked earlier: [[exclusive disjunction]] (with a philosophical claim that I need to back up), [[countable choice]] (very much a stub), [[sigma-ideal]], [[Hartog's number]].
* [[Urs Schreiber|Urs]]:
* added the detailed formal proof to [[abelian sheaf cohomology]] that shows how it is sitting as a special case inside the more conceptual [[nonabelian cohomology]] (= hom-sets of [[infinity-stack]]s, really)
* in this context I also rewrote the "Idea" section in the desire to drive home the main point better, but without removing the previous material. As a result the text of the entry is now somewhat repetitive, I am afraid. Maybe a little later, when I have more distance to it, I'll try to trim it down again. Or else, maybe one of you feels like polishing it.
* [[Toby Bartels]]: Wrote [[Moore closure]], since I linked it and it\'s a neat idea. As I was writing it, I realised that it was more abstract than what I\'d been writing lately and must correspond to something very nice categorially; by the time I was done, I realised what it was: a special case of a [[monad]]. Then I saw that there were really no examples of monads on our page!, so I added a few, but the majority of examples of monads on the wiki now are Moore closures. (And believe me, Moore closures are everywhere.) Anyway, if you\'re trying to understand what monads are, why not try [[Moore closure]]s first?
* [[Urs Schreiber|Urs]]
* added an "Idea" section to [[Moore complex]], edited the section headers a bit -- and have a question onm terminology: isn't this really the "normalized chain complex" whereas the Moore complex is the one on all cells with differential the alternating sum of face maps?
* created [[organization of the nLab]] -- please see there for what this is about (or in fact, see the corresponding [nForum thread](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=36&page=1#Item_2) that is being linked to there)
* added references to Goerss-Jardine's _Simplicial homotopy theory_ to [[Dold-Kan correspondence]] and [[Moore complex]]
* reacted a bit at [[An Exercise in Kantization]] -- behind the scenes this is being developed further, I'd be happy to provide more detailed replies in a while, when things have stabilized a bit more
* added a mention of and a link to [[Kan lift]] to the "Idea" section of [[Kan extension]]
* replied to the discussion about pullback notation at [[Kan extension]]: I originally had "$p^*$" there. After somebody changed that to "$p_*$" I wrote the section "note on terminology". I'd be happy to have the $p^*$ reinstalled.
# 2009-06-21 #
* [[Toby Bartels]]:
* Over at [[An Exercise in Kantization]], [[Eric Forgy]] and [[Daniel de França MTd2]] would like confirmation or correction from somebody that understands the paper (Alm\'s _[[Kantization09May27.pdf:file]]_) better than I do.
* Some additions to [[Vect]], [[Hilb]], and [[measurable space]]. (I plan to do more on the last, then move to [[measure space]].)
* [[Todd Trimble|Todd]]:
* I have a notational comment at [[Kan extension]]. Spurred by [[David Corfield]]'s [post](http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html) at the Café, I hope to get started on [[Kan lift]] soon. (Done.)
* There's a running discussion between [[Toby Bartels]] and me at [[cyclic order]], centering on whether Connes' cycle category $\Lambda$ has been correctly characterized, or if not how to fix it. We agree now that a notion of "total cyclic order" is classically equivalent to the "linear cyclic order" notion used in the article, and equally feasible for purposes of trying to characterize $\Lambda$, but I'm currently perplexed by the apparent presence of a terminal object.
# 2009-06-20 #
* [[Tim Porter|Tim]]: Finally I have created [[2-crossed complex]], and in the process needed to create [[normal complex of groups]]. What would be nice is to work out exactly what $\infty$/$\omega$-groupoids correspond to 2-crossed complexes. Any ideas? (It probably is obvious viewed from the right perspective but I fear I do not yet have the right perspective to say 'aha!') I can characterise these objects in homotopy theoretic language, but really would like some neat way of describing them in $\infty$-cat terms.
* [[Todd Trimble]]: asked a question of [[Mike Shulman|Mike]] and [[Toby Bartels|Toby]] over at [[cyclic order]].
* [[Toby Bartels]]:
* Wrote [[countable set]], [[infinite set]], [[relative complement]], and [[preimage]].
* Has the 'Change page name.' feature broken for anybody else?
# 2009-06-19 #
* [[Tim Porter|Tim]]:
* I have added more material to [[2-crossed module]] including some exercises (at the foot of the 'page'! (Have fun!) I will not get around to doing an entry on 2-crossed complexes today.
* I have adjusted [[homotopy coherent nerve]] in an attempt to answer some of the points made there by [[Todd Trimble|Todd]],
* [[Urs Schreiber|Urs]]:
* almost missed [[Tim Porter]]'s addition about the [[Dwyer-Kan loop groupoid]] to [[simplicial homotopy group]] -- that sounds very good, I'd be happy if we make this the default point of view at that entry and derive the more traditional description only as a special case from that
* added the example of the bar construction of a group $G$ as the nerve of $\mathbf{B} G$ to [[nerve]]
* added illustration to [[Kan fibration]]
* added illustrative diagrams to [[boundary of a simplex]] and [[horn]]
* added a section with details on ordinary nerves of ordinary categories to [[nerve]]
* [[Toby Bartels]]:
* Expanded [[measurable space]] quite a bit, but I need to go, so it hasn\'t really been proofread.
* Split [[measure space]] off from [[measurable space]].
# 2009-06-18 #
* [[Todd Trimble]] keeps creating analysis pages that [[Toby Bartels|I]] link to; today\'s were [[measurable space]] and [[topological vector space]].
* [[Tim Porter|Tim]]:
* I finally added some stuff into [[crossed complex]] giving more information on the link with [[simplicial group]] and [[Moore complex]]. I hope to get around to creating 2-crossed complex in a day or two!
* The discussion on [[simplicial homotopy group]] possibly needs more opinions! I just added a bit more of my viewpoint to Urs's thoughts box, but I do find that I am not sure what the idea / context / viewpoint (or whatever) of this entry should 'optimally' be!
* [[Eric Forgy|Eric]]: In an attempt to understand Kan extension, I cooked up an example of a functor and added it to [[functor]]. Let me know if I made any mistake.
* [[Toby Bartels]]: Instead of changing links to [[path-connected space]] (which doesn\'t yet exist) to [[connected space]], I instead added `[[!redirects path-connected space]]` to [[connected space]]. That way, if we ever decide to separate out a new page [[path-connected space]], we don\'t need to go through all of the links and fix them!
* [[Urs Schreiber|Urs]]: added the defintion of the product to [[simplicial homotopy group]]
* [[Todd Trimble|Todd]]: wrote the beginnings of an apparently long-awaited article on [[weighted colimit]]s.
* [[Urs Schreiber|Urs]]
* filled in the definition at [[simplicial homotopy]] together with an intentionally pedestrian proof that simplicial homotopy in a Kan complex is an equivalence relation
* slightly polished the "remarks" section at [[simplicial set]]
* [[Todd Trimble|Todd]]:
* wrote [[measurable space]], including material on measures and basic integration theory.
* responded to queries of [[David Roberts]] at [[covering space]] and [[locally path-connected space]]. David wondered whether some material at [[covering space]] ought to be moved to [[universal covering space]]. I think we're covering similar material but with slightly different emphases; I'd like to see David put his imprimatur on the article on universal covering spaces, but leave [[covering space]] essentially intact while the material continues to settle into place; I'm thinking of adding more to it anyway (so that "to be continued" might still be appropriate).
* [[Urs Schreiber|Urs]]:
* added a reference (and minor comments on references) to [[model structure on simplicial presheaves]]
* added an opinion to [[simplicial homotopy group]]
* will add the definition of the group structure later today
* [[Andrew Stacey]]: Changed the example in [[topological concrete category]] for [[generalized smooth space]] since I think that all the examples listed there are topological over set.
* [[David Roberts]]:
* Edited [[covering space]] as requested by [[Todd Trimble|Todd]]. I still wonder whether some of the discussion would be better off at [[universal covering space]].
* Created stubs at [[semi-locally simply-connected space]] and [[locally path-connected space]].
# 2009-06-17 #
* [[Toby Bartels]]: Fixed mistakes at [[topological concrete category]], thanks to having a good online reference.
* [[Todd Trimble|Todd]]:
* Wrote [[approximation of the identity]], linked to by one thing I wrote at [[Hilbert space]].
* [[Tim Porter|Tim]]:
* Did my idea of a partial fix for the dilemma at [[simplicial homotopy group]]. Please edit! The entry still needs the definition of the group structure on the homotopy groups. I have included the link with the [[Moore complex]] of the [[Dwyer-Kan loop groupoid]]. I have also added a request for discussion here.
* [[Todd Trimble|Todd]]:
* Added a bit to the examples section at [[Hilbert space]], mentioning in particular the orthonormal basis theorem.
* [[Tim Porter|Tim]]:
* Added a useful reference to [[topological category]], plus a comment on terminology. It should be noted that the authors (Adamek, Herrlich and Strecker) use 'topological concrete category' in the index, which they shorten to 'topological category' in the text. Does that provide a solution to the terminological problem?
* [[Toby Bartels]]:
* Wrote [[Hilbert space]]. I finished the definitions and then got tired, so there\'s not much else. But we need this if we\'re ever going to discuss $2$-[[2-Hilbert space|Hilbert spaces]]!
* [[David Roberts]] and I would appreciate any terminological suggestions at the unfortunately-named [[topological category]].
# 2009-06-16 #
* [[Toby Bartels]]:
* Wrote [[topological category]], including a lot of vague stuff there, since I don\'t have good references here now. But I\'m pretty sure that everything that I said is at least true.
* Wrote [[core]], using a term that I learned from [[Mike Shulman]] for an [[underlying groupoid]].
* Wrote [[semigroup]] and [[magma]], since they\'re such basic topics, but I didn\'t say much.
* Added stuff to [[reflective subcategory]] about how obects of the ambient category can be seen as objects of the subcategory equipped with [[extra structure]]. Besides the examples there, see also [[core]].
* In reply to the last point in what Urs has said immediately below, if you want to put in something that you *expect* to be controversial, and hesitate for that reason, then put it in and just *label* it as controversial. Put in a [[HowTo|query box]] asking 'Is this right?' or saying 'This is my new definition.' or whatever. Or put it in a new section at the bottom of the page, marked '## Uncertain material' or whatever. The point is, get it down one way or another, and we will be very happy!
* [[Urs Schreiber|Urs]]: I have a comment and appeal at [[nInsights]]
* in this context I also want to ask again everybody:
* please don't forget to drop latest changes logs here alerting us of which changes you made where, if it's anything beyond fixing typos
* if you feel something needs to have a place in the nLab, don't feel hesitant to add it. I am getting the impression there is in parts some hesitation to insert material without checking with everybody else first. I'd say more efficient would be: add the material you would like to see, and if it should really turn out that there is serious disagreement with some other contributor, we can still roll back things and look for compromises.
* [[Urs Schreiber|Urs]]:
* expanded [[inverse image]]: added an "Idea" section, restructured discussion into presheaf and sheaf and sheaf on top-space bits, added the crucial lemmas and theorems and provided detailed proofs of some of them.
* replied again at [[simplicial homotopy group]]: [[Tim Porter|Tim]] or [[Toby Bartels|Toby]] should please feel free to implement the remaining terminology adjustment
* reacted and replied at [[simplicial homotopy group]] (also have a question) and added a few further bits
* [[Tim Porter|Tim]]:
* Discussion at [[simplicial homotopy group]].
# 2009-06-15 #
* [[Toby Bartels]]:
* Discussions at [[connected space]] and [[simplicial homotopy group]].
* New page at [[neighborhood]].
* [[Urs Schreiber|Urs]]
* created [[simplicial homotopy group]]
* added to [[model structure on simplicial sets]] a puny beginning of a discussion of what is supposed to eventually become a discussion of the relation to model structures on strict $\infty$-groupoids
* created [[omega-nerve]] and [[model structure on strict omega-groupoids]]
* expanded a bit on the discussion at the beginning of [[model structure on simplicial sets]]
* added detailed proof to [[geometric morphism]] of the fact that geometric morphisms $Sh(X) \to Sh(Y)$ are bijectively induced from continuous functions $X \to Y$
* [[Tim Porter|Tim]]: I have continued at [[homotopy coherent nerve]]. This ended up with several new links to not-yet-existing pages!
* [[Urs Schreiber]]:
* created [[homeomorphism]] and [[local homeomorphism]] just to satisfy links
* added theorem about stalkwise characterization of epi/mono/isos of sheaves to [[stalk]] and included an example
* added a bit more to [[string structure]] (discussion of description in terms of class on total space, also some references, but very incomplete) - but still not done
* split off remaining Friday session [[Oberwolfach Workshop, June 2009 -- Friday, June 12]] for [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] following [[Toby Bartels|Toby]]
* [[Todd Trimble]] created [[connected space]].
* [[Toby Bartels]]: [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] was getting too long for me to handle, so I removed the duplication of abstracts and broke the rest up by day.
# 2009-06-14 #
* [[Tim Porter|Tim]]: I have had a go at [[homotopy coherent nerve]]. Still more to add though.
* [[Urs Schreiber|Urs]]
* added more details on the sheaf-case to [[point of a topos]] and created [[skyscraper sheaf]]
# 2009-06-13 #
* [[Urs Schreiber|Urs]]
* started creating [[string structure]] but had to quit before fully finished -- will continue tomorrow
* created [[point of a topos]] and [[stalk]]
* created [[Oberwolfach]]
* pasted notes on Thomas Schick's talk on differential cohomology at [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] into [[differential cohomology]]
* created
* [[local net]]
* [[TQFT]]
* [[FFRS-formalism]]
* [[modular tensor category]]
but more like stubs to get some material in place, didn't find the leisure to really polish this
* started adding links to keywords in the list of abstracts that [[Bruce Bartlett|Bruce]] added to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] -- am hoping that eventually we'll fill these links with content, partly just by copy-and-pasting the relevant bits from the lecture notes (as a start)
* [[Tim Porter|Tim]]:
* Mike, ditto from me!
* Made some changes to [[simplex category]] and [[join of simplicial sets]]. A more explicit mention of ordinal sum has been made and some minor corrections done.
* [[Bruce Bartlett|Bruce]]:
* Congrats Mike!
* Added lots of talk summaries and notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]. Lots of editing help needed... what would be nice would be to have the links to the notes for the talks next to the talk titles --- both my notes as well as those of Gabriel Drummond-Cole (by linking to his [webpage](http://www.math.sunysb.edu/~blafard/notes/)).
* [[Urs Schreiber|Urs]]: I gather it is in order to express congratulations to [[Mike Shulman]] given [the latest changes](http://ncatlab.org/nlab/show/diff/Mike+Shulman) to his page
* [[Toby Bartels]]: I think that the (unordered) notion of _category with inverses_ at [[partially ordered category]] should be the same as that of _[[dagger category]]_.
# 2009-06-12 #
* [[Victor Porton]]: Some other newer terms added (notably [[partially ordered category]]) to [[Categories and Sheaves]].
* [[Todd Trimble]]: I finally found some time to write up more of what I had planned for [[covering space]]s, and tried to link up with the comments made by David Roberts. David and Urs, please take a look.
# 2009-06-11 #
* [[Urs Schreiber|Urs]]:
* created [[tetracategory]]
* added Thursday talks to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology
]]
* replied to [[Tim Porter]] at [[join of simplicial sets]]
* [[Bruce Bartlett|Bruce]]: Added some comments abotu query boxes in personal nLab pages in the [[HowTo]] page.
* [[Tim Porter|Tim]]: I have added a query to [[join of simplicial sets]]. Am I missing something, but I find the initial definition confusing as it does not mention the ordinal structure nor ordinal sum. (There may be a subtlety that I am missing so ...)
* New pages: [[doctrine]], [[pretopological space]].
# 2009-06-10 #
* New pages: [[Green-Schwarz mechanism]], [[quasigroup]].
* [[Urs Schreiber]] added Wednesday talk notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]
* [[David Roberts]]:
* Put next, very drafty, chapter of my thesis on my personal web. Just click on my name and follow the white rabbit.
* Added another comment to [[covering space]]. This time about the (total) tangent groupoid as applied to the non-basepoint version of the universal covering space.
# 2009-06-09 #
* More new pages: [[Freudenthal suspension theorem]], [[replete subcategory]], [[strictly full subcategory]].
* [[Zoran Škoda]]: much expanded [[orbifold]], created [[Lie group]], [[compact Lie group]], [[orbispace]].
* [[Urs Schreiber|Urs]]
* created [[orbifold]] and [[orientifold]]
* added further talk notes to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]
* added notes on first part of lecture by Thomas Schick on _differential cohomology_ to [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] -- am hoping to eventually move a polished version of that to [[differential cohomology]]
* [[David Roberts]]
* Created [[universal covering space]], but it's still pretty much a stub.
* Added some remarks to [[covering space]], but they will probably need to move to [[universal covering space]].
* [[Toby Bartels]]
* Started [[discrete space]].
* There seems to be a link missing from the [[HowTo]] on CSS in personal webs.
# 2009-06-08 #
* New pages:
* [[Mike Shulman]] wrote [[2-monad]].
* Besides asking people questions, [[Todd Trimble]] also started [[covering space]].
* [[Urs Schreiber]]: created [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]], linked to from [this](http://golem.ph.utexas.edu/category/2009/06/strings_fields_topology_in_obe.html) blog entry (am hoping to get around to add lots of further keyword links to existing $n$Lab entries, but maybe won't)
* [[Todd Trimble]]: asked Alex a question at [[Alex Hoffnung]].
* [[Bruce Bartlett|Bruce]]: Added link to notes to the strings, fields and topology workshop at Oberwolfach. Also added a "How To" section for creating CSS gismos like query boxes on your personal nLab wiki space.
# 2009-06-07 #
* [[John Baez|John]]: I created myself a [[John Baez:HomePage|new personal web]] since I lost the password for my old one --- and besides, the name of my old one was nonstandard. So far the only thrilling feature of this new web is the introduction to a paper I'm writing with [[James Dolan]], tentatively titled 'Doctrines of Algebraic Geometry'.
* [[Mike Shulman|Mike]]: Tried to distill a bit of the [cafe discussion](http://golem.ph.utexas.edu/category/2009/06/mathematical_principles.html#c024277) about the [[empty space]].
# 2009-06-04 #
* [[Mike Shulman|Mike]]: Added some exposition to [[Hopf algebra]].
# 2009-06-03 #
* [[Toby Bartels]]:
* Thanks for [[prometric space]], Mike. I didn\'t know about those, but they seem quite reasonable. I was thinking that I might want to figure out the categorial meaning of the definition of [[gauge space]], but maybe it\'s just that it\'s a halfway attempt at prometric space! (But do you have any good example of a nongaugeable prometric space?)
* I seem to have gotten into an edit conflict with you at [[Zorn's lemma]], Todd. I didn\'t tell it to override your edit, but something happened regardless. Anyway, I think that I\'ve fixed it.
* Also, our page is at [[Hausdorff maximal principle]], which seems to be more common, but maybe your name is better. But look!, now it redirects! And you can move it too!
* [[Todd Trimble]]: gave a proof of [[Zorn's lemma]]. Wouldn't mind expanding that entry to include the mutual equivalence between AC, Zorn, and well-ordering principle (assuming excluded middle). May get around to putting in something at [[Hausdorff maximality principle]].
* [[Mike Shulman|Mike]]: Inspired by [[gauge space]], created [[prometric space]].
* [[Toby Bartels]]:
* [[Eric Forgy|Eric]] and I have been testing the new move and redirect features at the *top* (for some reason) of the [[Sandbox]]. It seems possible to move and to create new redirects (make sure that you see how, it\'s backwards from MediaWiki), but not (yet) possible to regularise all of the current redirect pages.
* Created a few requested pages. The interesting ones are [[Bill Lawvere]], [[Boolean ring]], and [[ideal]].
# 2009-06-02 #
* [[Toby Bartels]]:
* An [[AnonymousCoward|anonymous coward]] created a blank version of [[Hausdorff maximal principle]], so I started that ... but it needs proofs! (at least).
* A new reference, [[HAF]], since I\'ve now cited it thrice.
* Created [[gauge space]]. A lot of this is original research; years ago, I wanted to work out the general theory (even constructively) of quasigauge spaces, and perhaps now I will. (Or perhaps I will stop here, with only results that are known or easy to prove from what is known.)
* [[Todd Trimble]]: Added some details on categorical operations on the category of [[Banach space]]s.
* [[Toby Bartels]]:
* Added the standard examples to [[uniform space]].
* Corrected Andrew\'s reply to Mike at [[Froelicher space]]. (I\'m pretty sure that this is what you meant, Andrew!)
* Replied to [[David Roberts]] at [[Atiyah Lie groupoid]].
* [[Andrew Stacey]]: replied to [[Mike Shulman]] at [[Froelicher space]]
* [[Toby Bartels]]:
* Archived [[2009 May changes]].
* Fixed some mistakes at [[uniform space]].
# 2009-06-01 #
* [[Andrew Stacey|Andrew]] proved a theorem about [[Froelicher space|Hausdorff Frölicher spaces]] and the relationship to limits and colimits of manifolds.
* [[Bruce Bartlett|Bruce]] added some stuff to Section 4 [[geometric infinity-function theory]] (with about query $QC(X)$ when $X$ is an $\omega$-groupoid internal to dg-manifolds) and ticked some things.
***
[[2008 changes|First list]] --- [[2009 May changes|Previous list]] --- [[2009 July changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 March changes | https://ncatlab.org/nlab/source/2009+March+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during March 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/459) and work backwards.
***
# 2009-03-31
* [[Urs Schreiber|Urs]]:
* started adding comments on embeddings at [[geometric morphism]], but ran out of time
* following [[Mike Shulman|Mike]]'s suggestion I moved the material that was at [[sheafification in a Lawvere-Tierney topos]] over to [[Lawvere-Tierney topology]]
* have (started) a discussion with [[David Roberts]] on the relation between internal [[anafunctor]]s and localization in higher sheaf categories over at his private area [[davidroberts:comments on chapter 2]]
* incorporated in part [[Zoran Skoda|Zoran]]'s comment at [[sheafification]] of non-set-valued presheaves and created an entry on the [[IPC-property]] in the course of that
* started [[Sheaves in Geometry and Logic]], on the MacLane-Moerdijk book
* created [[dense monomorphism]] (thanks to [[Mike Shulman|Mike]] for the relation to [[local isomorphism]]) and then [[sheafification in a Lawvere-Tierney topos]]
* created [[(infinity,1)-essentially surjective functor]] and [[(infinity,1)-fully faithful functor]]
* [[Mike Shulman|Mike]]:
* Mentioned at [[triangulated category]] that the definition is redundant. If I had time I would fix it myself.
* We are having an interesting discussion at [[derived functor]]. Unfortunately I am leaving town for a couple of weeks (starting with [PSSL88](http://www.cheng.staff.shef.ac.uk/pssl88/)) so will be less active for a while.
* [[Urs Schreiber|Urs]]:
* created [[category of open subsets]] and [[presheaf on open subsets]] to satisfy links at [[Categories and Sheaves]]
* fixed and expanded [[local isomorphism]] and used that to create [[category of sheaves]], [[(infinity,1)-category of (infinity,1)-sheaves]] and [[sheafification]]
* created [[base change]] and [[cobase change]]
* started listing [[Zoran Skoda|Zoran]]'s latest entries and previous ones on examples/refinements of additive and abelian categories at [[additive and abelian categories]].
* [[Tim Porter|Tim]]:
* Last night I continued my 'lexicon' with an entry on [[differential graded vector space]]s (yes I know these are just chain complexes, but my thought is to put the lexicon that I have on then prune and adjust!) (My connection crashed at that point as I was editing this page!)
* [[Zoran Škoda]]: Created [[suspended category]] and [[Quillen exact category]]. This is a continuation of our efforts to enter various classes of additive categories useful in homological algebra and K-theory.
# 2009-03-30
* [[David Roberts]]:
* Asked question about Grothendieck's tame topology at [[nice topological space]].
* [[Urs Schreiber|Urs]]:
* QUESTION: came across MacLane's
_Foundations for Categories and Sets_ where it it argued that neither standared set/class set theory nor [[Grothendieck universe]]s provide decent foundations for categories and a formalism of _schools_ is introduced instead -- can anyone comment on that in the light of our discussion at [[Grothendieck universe]]?
* added an "Idea" section to [[derived functor]] and split off [[derived functor on a derived category]] from that in order to discuss the special homological algebra aspects of derived functors separately -- but incomplete for the moment
* created [[null system]]
* created [[category of chain complexes]], but then didn't quite know where to go with this...
* agreed at [[homological algebra]] and [[filtrant category]]
* created [[homology]]
* expanded [[differential object]] and created [[differential]] just to satisfy links
* created [[matrix calculus]] and [[mapping cone]]
* [[Tim Porter|Tim]]:
* I have tried to edit [[homological algebra]] somewhat along the lines that my query suggested but taking in ideas from [[Zoran Škoda|Zoran's]] query at that entry.
* I have added some background material on [[dérivateur]]s on the [[triangulated category]] entry.
* I have a **lexicon** for the concepts needed for rational homotopy theory that may be useful. I have started adding some of this to [[graded vector space]]. My intention is to put quite a lot of routine stuff on this there and elsewhere and then to sort out links, conflicts of notation afterwards. Feedback, queries etc welcome. (They will encourage me to put more of the lexicon on there!)
* [[Toby Bartels]]:
* Wrote [[biproduct]], [[direct sum]], and [[direct product]].
* Added more versions to [[additive and abelian categories]].
* Questioned the purpose of [[filtrant category]].
* I hope that people saw Zoran\'s addition to a query box below.
* [[Urs Schreiber|Urs]]:
* concerning the entry [[homological algebra]]: [[Tim Porter]] and [[Zoran Skoda]] (or anyone else): please feel free to improve/revise the exposition
* created [[triangulated category]] -- have a question about _dérivateurs_ there
* created [[local epimorphism]]
* created [[multiplicative system]]
* [[David Roberts]]
* Uploaded notes on anafunctors to my [web](http://ncatlab.org/davidroberts/show/HomePage). Comments welcome and wanted.
# 2009-03-29
* [[Mike Shulman|Mike]]:
* A grammatical suggestion at [[stuff, structure, property]].
* Did some work on [[chain complex]] and all the [[additive and abelian categories]] pages.
* [[Urs Schreiber|Urs]]:
* began [[derived category]] before running out of time again...
* did some layout-editing for [[Mike Shulman|Mike]]'s additions to [[homotopy category of an (infinity,1)-category]]
* [[Finn Lawler]]: Continued experimenting with graphics for diagrams. Have a look and see what you think:
* Added some PNGs at [[Kan extension]]
* Replaced PNGs at [[adjunction]] with larger ones.
* [[David Corfield|David]]
* asked a question at [[differential object]]
* [[Urs Schreiber|Urs]]
* started preparing the ground for [[derived category]] and [[triangulated category]] by creating [[category with translation]], [[chain complex]] and [[differential object]] before running out of time...
* created [[homotopy category of an (infinity,1)-category]]
* created [[homological algebra]], mainly as a collection of links to the keywords listed there
* thanks to [[Mike Shulman|Mike]] for his polishing of my original [[filtrant category]] at [[filtered category]]
* in a first attempt to clean up the entries surrounding [[abelian category]] I created the overview entry [[additive and abelian categories]] and branched off [[Ab-enriched category]], made [[pre-additive category]] a commented redirect to that and "commented out" the respective discussion still to be found at [[additive category]]; also made [[pre-abelian category]] a separate entry, so that now there is in order of increasing structure/property
* [[pre-additive category]]
* [[additive category]]
* [[pre-abelian category]]
* [[abelian category]]
* am all in favor of [[Finn Lawler|Finn]]'s graphics! The only reason I don't include nice graphics myself a lot is that currently these take me longer to create than the MathML hacks
* [[Finn Lawler]]: Uploaded PNG images of the zig-zag identities and added them to [[adjunction]]. They're probably a bit too small, but what do people think of this approach as a work-around until there's an easy way to convert TeX to SVG? Any other suggestions?
(Note: I tried converting these diagrams to SVG as described [here](http://meta.wikimedia.org/wiki/Help:Displaying_a_formula#Commutative_diagrams) but the resulting files were huge and didn't display anyway when inserted into the markdown source. Instead I used `pdfcrop` and `convert` on the `pdflatex` output.)
# 2009-03-28
* [[Finn Lawler]]:
* Edited [[linear logic]] in response to Mike's question.
* Created [[star-autonomous category]].
* [[Mike Shulman|Mike]]: Prompted by discussion with Zoran, created [[strict epimorphism]] and added a lengthier discussion of types of epimorphism to [[epimorphism]].
* [[Zoran Škoda]]: Created [[etale space]]. The order of exposition is important, particularly in view of anticipated additional details. In [[Kan extension]] added a detailed paragraph on an example how left Kan extension pointwise formula has intuitive meaning in the case of constructing pullback for (pre)sheaves on topological spaces. Created [[torsor with structure category]] following the version in Moerdijk's book.
* [[Finn Lawler]]: Created [[linear logic]] -- just a short stub with
basic ideas on motivation and models, plus a couple of references.
Comments effusively welcomed. (Edit: also removed Thursday's query
box from [[context]]).
# 2009-03-27
* [[Mike Shulman|Mike]]:
* Zoran and I are having a discussion about definitions of [[abelian category]].
* Redirected [[filtrant category]] to [[filtered category]].
* Moved the discussion about the word "bimorphism" from [[balanced category]] to [[bimorphism]].
* Redirected [[parallel morphism]] to [[parallel morphisms]] (which _is_ in line with the naming conventions).
* [[Urs Schreiber|Urs]]:
* created [[exact functor]]
* created [[filtrant category]]
* added to [[Higher Topos Theory]] more introductory/overview remarks which are supposed to be helpful for the newbie
* created [[Yoneda extension]]
* added section to [[Kan extension]] on formulas in terms of limits and colimits over comma categories;
* added a section on the "local" computation of adjoint functors at [[adjoint functor]] and point out how this induces the local/global dichotomy at [[limit]], [[homotopy limit]] and [[Kan extension]] (see my previous modification below)
* if I noticed correctly, [[Mike Shulman|Mike]] had changed my original notation $p^* := - \circ p : [C',D] \to [C,D]$ for precomposition with a functor $p : C \to C'$ (pullback notation) at [[Kan extension]] to $p_*$ (pushforward notation). I have now added a section _Remark on terminology: pushforward vs. pullback_ which is supposed to clarify this terminology issue.
+--{: .query}
[[Mike Shulman|Mike]]: That wasn't me. I'm not sure that such a discussion belongs at [[Kan extension]]; it might belong somewhere but I would rather than the page [[Kan extension]] just pick one notation and possibly link to a discussion.
[[Zoran Škoda]]: It was me who changed, though I better did not. I am happy with the original notation as well. For as your discussion on pushfowards I am less happy. Namely, if one is not happy with the direction of maps between open sets, one just redefines what is a morphism of sites (opposite to the functor direction), so that the morphism of sites is always correct direction. So, unless one does not have strong feeling on the choice of pushfoward pullback meaning, what is not in this case, mayeb original notation just caring about covariant vs contravariant was better.
=--
* addressed [[Zoran Skoda|Zoran]]'s and [[Tim Porter|Tim]]'s remarks at [[Kan extension]]: I have added now to [[Kan extension]] as well as to [[limit]] -- in analogy to what we already had at [[homotopy limit]] -- an explicit discussion of the difference between _local_ and _global_ definitions of the universal constructions
* created [[universal construction]] -- but filled in just a question/query
* [[Tim Porter|Tim]]: I have raised a query at [[Kan extension]].
* [[Toby Bartels]]:
* Wrote [[hereditarily finite set]], which is more pretty than useful.
* Threatened to rewrite [[Grothendieck universe]] once again.
* Wrote about the $3$-way factorisation system at [[stuff, structure, property]].
* Accepted Mike\'s terminology ('moderate') at [[Grothendieck universe]].
* [[Zoran Škoda]]: Created [[abelian category]] with multiple equivalent definitions.
* [[Toby Bartels]]:
* If people don\'t like having several entries in one day per Mike\'s request, another option (hopefully good enough for Mike) is to move your entire list up to the top when you add to it (being sure to add to the top of your list too).
* Finn has nothing to apologise for at [[context]].
* Zoran and I are discussing terminology at [[projective limit]].
* Zoran and Mike are discussing terminology at [[representable functor]] (I only made a more philosophical comment).
* Compare [[nice category of spaces]] with [[convenient category of topological spaces]].
* I accept Mike\'s terminology at [[set theory]].
* I refactored [[kernel]] to use primarily the equaliser definition in any pointed-enriched category.
# 2009-03-26
* [[Mike Shulman|Mike]]:
* I have a question about the meaning of "large" at [[Grothendieck universe]].
* I'd like to request that people not add new sub-bullets under their own names on a given day if other people have since listed more changes above; rather, add a new bullet point at the top with your name. If that didn't make sense, it's what I'm doing now, rather than (what I could have done) adding a new bullet point below the other copy of my name today.
* Added some comments on syntactic categories to [[internal logic]], since Toby kindly saved me the work of defining them at [[context]].
* [[Finn Lawler]]: (Hello all -- long-time lurker, first-time editor.) For my first edit, I asked a silly question at [[context]] and then answered it myself a little later. I'll delete the query box if nobody has any comments. Apologies for noise.
* [[Mike Shulman|Mike]]:
* Split [[nice category of spaces]] from [[nice topological space]] and fixed all the links I could find.
* Wrote [[universe in a topos]] by way of responding to the query at [[Grothendieck universe]].
* Focused [[adjunction]] on the internal version in a 2-category, to distinguish it from [[adjoint functor]], which I reorganized and added a definition to. (The zig-zag identities are crying out for SVG!)
* Corrected [[generalized element]] to distinguish it from [[global element]].
* Made a terminological suggestion at [[set theory]].
* Commented about property-like structure at [[stuff, structure, property]]. It would be nice to move the examples earlier on this page.
* [[David Corfield|David]]
* created [[algebraic set theory]]
* [[Urs Schreiber|Urs]]
* started a stub-entry on [[Stable Infinity-Categories]] (Lurie's PhD part I) and advertized this little program of textbook $n$labification [here](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c022834) at the blog
* started an entry [[Higher Topos Theory]] (on Lurie's book) in a style analogous to [[Categories and Sheaves]] -- I included a link to [[Mike Shulman|Mike]]'s personal page [[michaelshulman:n-topos for large n|n-topos for large n]]; eventually it would be nice if we had an entry on the general idea and purpose of [[higher topos theory]]
* started expanding [[Kan extension]]
* created [[fiber product]], [[parallel morphism]], [[zero morphism]], [[kernel]], added example to [[pointed object]]
* good to see that [[Mike Shulman|Mike]] is back! Mike, there is a request for you at [[Grothendieck universe]]: can you say something about rephrasing that as a "topos object internal to the [[ETCS]]-version of $Set$"?
* added the globular zig-zag diagrams to [[adjunction]]
* started [[continuous functor]]
* added a link to [[David Corfield|David]]'s new entry [[algebraic set theory]] at [[set theory]]. It would be nice to put it into context there, eventually.
* started expanding [[limit]] (and also a bit [[colimit]]): more motivation, more details on definition, more examples
* created [[generalized element]]
* more or less completed the hyperlinked keyword list of chapter one of [[Categories and Sheaves]]
* filled in three equivalent definitions at [[adjoint functor]]
# 2009-03-25
* [[Urs Schreiber|Urs]]
* expanded a bit at [[natural transformation]]
* added simple remarks to [[contravariant functor]]
* added the illustrative diagram to [[over category]] and added a remark on over categories to [[subobject]];
* added an illustrative diagram to [[comma category]] and added a section there on how a comma category is a pullback;
* added a little bit of discussion that every presheaf is a colimit of representables to [[presheaf]];
* expanded a bit more at [[stuff, structure, property]]
# 2009-03-24
* [[Urs Schreiber|Urs]]
* have more stupid questions (for [[Toby Bartels|Toby]], probably) at [[Grothendieck universe]] and incorporated more of Toby's replies
* [[Toby Bartels]]:
* Gave a structural definition at [[Grothendieck universe]], but somebody should check it (Mike, this means you).
* Some more foundational material, including [[set theory]], [[pure set]], and [[point]] (which is a bit purple).
* Note that Urs also started [[stuff, structure, property]].
# 2009-03-23
* [[Urs Schreiber|Urs]]: created [[Grothendieck universe]] -- but have questions. Mentioned Grothendieck universes at [[small category]] and [[locally small category]].
# 2009-03-22
* [[Toby Bartels]]: I added a section on morphisms between [[context]]s (the substitutions, or interpretations), including (as an example) a complete description of the category of contexts of the theory of a group. There is an exercise (to describe that category in group-theoretic terms) whose formatting all authors might want to look at.
# 2009-03-21
* [[Tim Porter|Tim]]:
* I added a link to p-adic solenoid in [[shape theory]] as that example gives insights on the links between this area and dynamical systems.
# 2009-03-20
* [[Andrew Stacey]]: lifted the tangent/cotangent section from "Comparative Smootheology" to [[Froelicher space]]. I intend to remove this section from that paper and this seems like a good place to put and develop it.
* [[Zoran Škoda]]: created [[algebraic monad]], [[generalized ring]], [[compact object]], [[noncommutative algebraic geometry]], [[spectrum (geometry)]], [[Pierce spectrum]], [[filter]] (thanks Mike for an essential typographic correction), [[generator]], [[cogenerator]] (the latter were prompted by editing [[Morita equivalence]], paragraph on classical Morita). Zoran and Toby distributed the paragraph on ultrafilters from long growing entry [[filter]] partly to the new entry [[ultrafilter]]. Uploaded [[warsaw.gif:file]] with link within [[shape theory]].
# 2009-03-19
* [[Urs Schreiber|Urs]]
* followed [[Mike Shulman|Mike]]'s remark and moved the previous content in [[stable infinity-category]] to [[stable (infinity,1)-category]], keeping just a general nonsense statement at [[stable infinity-category]]
* added a remark on this and a link at [[spectrum]]
* [[Zoran Škoda]]: moved the earlier material from entry algebra to new entry [[associative unital algebra]], and put new material into [[algebra]]; one should have separate entry for any framework for algebras, and general entry [[algebra]] should have pointers to the major classes (like [[algebra over operad]]). Thanks Toby, we continue together on that: now there is an entry [[nonassociative algebra]] and so on. I have also addressed concerns of Mike in [[Connes' cyclic category]] which now has I hope correct definitions, plus more foundational issues and relevant literature and link to just uploaded file [[krasauskas.pdf:file]]
* [[Andrew Stacey|Andrew]]: Added some more to [[Froelicher space]]. I've started on a new project on this: adapting topological notions to Froelicher spaces.
* [[Urs Schreiber|Urs]]:
* created [[stable infinity-category]]
* implemented [[Zoran Skoda|Zoran]]'s remark below by rephrasing a bit at [[Categories and Sheaves]]
* added a bit more to [[large category]]
* [[Toby Bartels]]: Let [[specialization topology]] lead me to [[specialization order]].
* [[David Roberts]]:
* Created [[Grothendieck's Galois theory]]
* [[Mike Shulman|Mike]]:
* Removed the discussion about accented characters from [[fundamental group of a topos]] and put it in the [[FAQ]].
* Displayed my happiness at [[quotient object]] by removing the discussion.
# 2009-03-18
* [[Zoran Škoda]] has created [[dense subcategory]] (intentionally organized different than the entry for the entry for slight generalization, [[dense functor]]); created [[shape theory]] but needs much more work; I copied here references from [[fundamental group of a topos]] (plus to a Batanin's article) and in [[fundamental group of a topos]] I added the reference and link to Pataraia's article important for the abstract notion of fundamental groupoid in internal contexts.
* [[Urs Schreiber|Urs]]:
* created an entry on the book [[Categories and Sheaves]] (comment you say that "sheaf condition is localization" there; well the category of sheaves is a localization of the category of presheaves, but the sheaf condition...you really meant what you say? -- Zoran)
* in reaction to [[Toby Bartels|Toby]]'s discussion at [[large category]] I created entries for [[accessible category]], [[locally presentable category]] and [[sketch]]. But very incomplete.
* [[Toby Bartels]]: Started a discussion about [[large category]].
* [[Andrew Stacey|Andrew]]:
* More done on [[Froelicher space]]s. I think that I have finally figured out the relationship between Frölicher spaces and Isbell duality so if anyone else is interested in taking a look I'd appreciate your comments.
I also found the standard layout of the page a little hard to work with, in particular with regard to delimiting proofs and definitions (both of which could get quite long) so I've been experimenting with alternative ways of demarking them (on [[Froelicher space]]). Let me know if you like or dislike what you see.
* [[David Corfield|David]]:
* Added a reference in response to one of Mike's questions at [[fundamental group of a topos]].
* [[Mike Shulman|Mike]]:
* Created [[quantale]], [[adjoint functor theorem]], and [[total category]].
* Asked some questions at [[fundamental group of a topos]].
* [[Tim Porter|Tim]]:
* Created [[fundamental group of a topos]]. At present this is lifted / adapted from an article I wrote some time ago so needs some attention. (This necessary attention includes fixing some diacriticals on some names.)
# 2009-03-17
* [[Mike Shulman|Mike]]:
* Yes, homotopy theorists call a [[k-tuply groupal n-groupoid]] a _grouplike $E_k$-space_.
* Continued discussion at [[Crans-Gray tensor product]].
* [[Toby Bartels]]:
* Wrote [[congruence]] to mean an internal equivalence relation.
* Wrote [[k-tuply groupal n-groupoid]] based on [[k-tuply monoidal n-category]]. But I strongly suspect that homotopy theorists have something to say here in a completely different language. (Perhaps [[Mike Shulman]] knows.)
* Created [[wedge sum]] to link from the following, and [[smash product]] to go with it.
* Put in a general definition at [[homotopy group]], trying to show how $\pi_n$ has $n$ products that are all the same.
* [[Tim Porter|Tim]]: I have been trying to give an adequate categorical treatment of [[profinite completion of a group]].
* [[Urs Schreiber|Urs]]:
* made a little remark on [[Mike Shulman|Mike]]'s question at [[Crans-Gray tensor product]] and ask another question myself
* added a remark to [[Waldhausen category]] on its relation to [[category of fibrant objects]] and ask a [question](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c022572) about the precise statement to be made here over at the $n$Café
* [[Toby Bartels]]: Combined [[proset]] into [[preorder]], etc, as planned there.
* [[Andrew Stacey|Andrew]]:
* Shifted [[generalized object|generalised object]] to [[Isbell envelope]] in line with the comments from the categories mailing list. Mainly just altered the terminology in line with this nomenclature, but added a small section on how it relates to profunctors as well.
* Modified [[Froelicher space]] a little as well in line with the Isbell envelope nomenclature.
* [[Zoran Škoda]] created [[Waldhausen category]] and made a remark into entry [[cofibration category]].
* [[Toby Bartels]]: I created [[symmetric set]] out of material that [[Zoran Škoda]] added to [[FinSet]].
# 2009-03-16
* [[Toby Bartels]]: Tired of writing `[[constructivism|constructive mathematics]]`, I moved [[constructivism]] to [[constructive mathematics]] and fixed links. Similarly, I moved [[predicativism]] to [[predicative mathematics]]. After some thought, I also moved [[finitism]] to [[finite mathematics]] and expanded it a bit to fit the new name better. To go with this, I finally created [[FinSet]].
* [[Zoran Škoda|Zoran]]: Created [[Loday-Pirashvili category]], [[dense functor]] and [[equivariant object]]. There are two different notions of dense subcategory, first of which has two different definitions and is related to colimits and nerve functor, and second which is related to pro-objects. In the entry [[Bousfield localization]] I added a paragraph on Bousfield localization for triangulated categories; made changes to [[nerve]] (more to be done: one needs to clarify the example of geometric realization etc.).
* [[Tim Porter|Tim]]:
* I have put a request on [[Loday-Pirashvili category]] for some discussion of the links between the ideas here and those in [[2-vector space]], as there is overlap.
* I have started a discussion on the exposition of [[dg-algebra]]. My preferred approach is via graded object, differential algebra and chain complex. The present approach I find a bit confusing.
* In [[profinite completion of a group]], I have pointed out a difference in the use of the term [[profinite group]], that may need discussion.
# 2009-03-15
* [[Zoran Škoda|Zoran]]: Created (in last two days) several entries mainly related to co- Hopf- algebras and algebras in categories of chain complexes: [[Frechet-Uryson space]], [[Hopf module]], [[Hopf-Galois extension]], [[Maurer-Cartan equation]], [[category of elements]], [[compactly generated space]], [[coring]], [[dg-algebra]],[[distributive law]], [[torsor]] (very unfinished!), [[twisted module of homomorphisms]], [[twisted tensor product]], [[twisting cochain]] and made changes to few other entries including many changes in entry [[Hopf algebra]] and some in [[A-infinity-algebra]]. With the (Fukaya) convention used there $D_0$ should not exist.
* [[Mike Shulman|Mike]]: Created [[homotopy equivalence]] and [[weak homotopy equivalence]].
* [[Tim Porter|Tim]]:
* created [[I-category]] which includes an alternative axiomatisation of [[cylinder functor]]. This is needed for Baues' version of abstract homotopy theory. Some results and examples will need to be added later.
# 2009-03-14
* [[Bruce Bartlett]] has created [[nInsights]].
* [[Ronnie Brown]]: I've rewritten and expanded [[homotopy group]]s partly to clarify the operation of dimension 1 on higher dimensions and to emphasise the groupoid aspects.
* [[Toby Bartels]]: I\'ve written [[sequence]], [[net]], [[multi-valued function]], [[partial function]], and the long-delayed [[surjection]] and [[injection]]. Those interested in foundations may be particularly interested in my proposed alternative definition of [[sequence]].
* [[Tim Porter|Tim]]: I have included a discussion of the nerve of an [[internal category]] at that entry.
# 2009-03-13
* [[Tim Porter|Tim]]: I have changed the initial sentence of [[homotopy n-type]]. I think this is converging well thanks to the efforts of Mike and Toby.
* [[Toby Bartels]]: Since we already have [[fundamental groupoid]] and even [[fundamental infinity-groupoid]], I started [[fundamental group]] and [[homotopy group]]. But I only wrote #Idea# sections.
# 2009-03-12
* [[Mike Shulman|Mike]]:
* Responded at [[homotopy n-type]] and [[proset]].
* Did a little more bettering of [[profinite group]], and was inspired to create [[filtered category]], [[pro-object]], [[ind-object]], and (as stubs) [[free completion]] and [[topological group]].
* [[Tim Porter|Tim]]:
* I have edited [[homotopy type]] as recent changes seem to have altered the idea away from the usual one making it perhaps too narrow. A similar problem seems to occur with [[homotopy n-type]] and I have raised a terminological query there as well.
* In [[homotopy n-type]] I have also queried the use of 'nice space' and its relation with [[nice topological space]]. This latter entry is perhaps misnamed as it is really the category of these spaces that is nice rather than the spaces themselves.
* I have raised a question on *Disambiguation* at [[proset]]. Have we a policy as to how to handle terms with perfectly acceptable multiple meanings?
* I have tried to 'better' the previous entry on [[profinite group]]! (see the old version to see why I say it this way.)
* [[Mike Shulman]]: A question about the [[Crans-Gray tensor product]].
* [[Toby Bartels]]:
* Broke [[undirected object]] off of [[directed object]] as planned. Discussion stayed at [[directed object]], even though it discusses both concepts (and that of a [[directed space]]).
* Finally wrote [[full subcategory]] and directed links to it.
* I\'ve continued the conversation that [[Eric Forgy]] started at [[preorder]].
* I have my own terminological question at [[linear relation]].
# 2009-03-11
* [[Mike Shulman|Mike]]: I have a suggestion for a terminological change at [[k-tuply monoidal n-category]].
# 2009-03-10
* [[Tim Porter]]:
* Eventually we probably need a summary of some of the theory of algebraic homotopy that Baues has developed as if impinges on the [[homotopy hypothesis]] and on [[homotopical cohomology theory]]. To this end I have created a sort of historical entry on [[algebraic homotopy]].
* Created [[cofibration category]] as the first of the 'Bauesian' detailed entries.
* [[Toby Bartels]]: I\'ve written several more articles on very basic topics, such as those that used to be '?'-links below. You can see them on Recently Revised; I don\'t think that anything merits great attention.
# 2009-03-09
* [[Toby Bartels]]: I finally wrote [[relation]], which makes me realise that there is no [[subset]] yet .... Also [[order]], but that\'s just a list of links to more specific pages.
# 2009-03-08
* [[Toby Bartels]]: I tried to clarify the difference between a [[preorder]] (a structure on a given set that satisfies certain properties) and a [[proset]] (a set equipped with such a structure). I need to finish that for [[partial order]]/[[poset]] and [[total order]]/[[toset]], although I would also entertain the idea that these should all be redirected one way or the other. But I got sidetracked writing [[linear order]] and [[loset]] instead. (And then there\'s [[quasiorder]]; I don\'t think that [[quoset]] is necessary for reasons that I don\'t want to get into here.)
* [[John Baez]]:
* Attempted to answer Eric's plea for a category-theoretic definition of 'Hasse diagram', in the discussion at the bottom of [[preorder]]. Unfortunately I don't know the official definition of 'Hasse diagram' --- though I know one when I see one.
* Made a short page on [[proset]], since Toby seems to be using this as a synonym for [[preorder]].
# 2009-03-05
* [[Andrew Stacey]]:
* Continued debate on what to call "[[generalized object|generalised object]]" since there seem to be some generalised objections (sorry).
# 2009-03-03
* [[Andrew Stacey]]:
* Archived February
* Started [[generalized object]]
* [[Toby Bartels]]: Since Urs is using both 'over category' and 'over-category' (and not 'slice category'), I tried to standardise things as 'over category' to diminish the temptation to slip further into 'over-category'. Principally this means that I moved [[slice category]] to [[over category]] and [[over-category in quasi-categories]] to [[over quasi-category]].
# 2009-03-02
* [[Urs Schreiber|Urs]]
* created [[limit in quasi-categories]] and in the course of saturating links from there also created [[join of simplicial sets]], [[join of categories]], [[join of quasi-categories]], [[over-category in quasi-categories]], [[terminal object in a quasi-category]] and edited [[quasi-category]] itself a bit
# 2009-03-01
* [[Toby Bartels]]:
* Slightly revised [[Tim Porter]]\'s new introduction to [[homotopy 2-type]] and copied it (with appropriate changes) to [[homotopy 1-type]] and [[homotopy 3-type]]. Also combined it with what was already at [[homotopy n-type]], so that has the most complete discussion of the issues.
* Created a bunch of articles on specific notions of $n$-category, especially for $n \leq 3$: [[3-poset]], [[3-groupoid]], [[n-poset]], [[2-poset]], [[locally posetal 2-category]], [[2-groupoid]], [[1-category]], [[1-poset]], [[1-groupoid]], [[(-2)-groupoid]], [[(-1)-poset]]. Similarly, I moved some material from [[(-1)-groupoid]] to [[truth value]] to fit the purposes of the former page (as explained in old discussion that was also moved).
* Spun [[simplicial groupoid]] off of [[simplicially enriched category]]. Experts please check that the definitions there (based somewhat on a remark in [[Dwyer-Kan loop groupoid]]) are correct.
***
[[2008 changes|First list]] --- [[2009 February changes|Previous list]] --- [[2009 April changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 May changes | https://ncatlab.org/nlab/source/2009+May+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during May 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/revision/2009+June+changes/773) and work backwards.
***
# 2009-05-30 #
* [[Toby Bartels]]: I don\'t understand $diag$ at [[Atiyah Lie groupoid]].
* [[Urs Schreiber|Urs]]:
* started [[connection on a bundle]] in reply to discussion [here](http://sbseminar.wordpress.com/2009/05/28/local-systems-what-do-you-know-about-connections/#comment-5791)
* added to [[distributor]] the description in terms of colimit preserving functors on presheaf categories and added corresponding remarks at the beginning of [[symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]
* created [[Atiyah Lie-groupoid]] and [[principal bundle]] to link to from the blog discussion [here](http://sbseminar.wordpress.com/2009/05/28/local-systems-what-do-you-know-about-connections/#comment-5791)
* [[Toby Bartels]]:
* A new query at [[sieve]].
* Wrote [[finitely cocomplete category]] by copying [[finitely complete category]] and cutting it down.
* Responded to Bruce at [[coverage]], to Urs at [[sieve]], and to Eric at [[category of elements]].
# 2009-05-29 #
* [[Todd Trimble]] kindly created [[quasitopos]].
* [[Urs Schreiber|Urs]]
* reacted to [[Bruce Bartlett|Bruce]] at [[sieve]]
* [[Bruce Bartlett|Bruce]]
* added a question at [[coverage]]
* added comments to [[sieve]]
* [[Urs Schreiber|Urs]]
* added a long section "A detailed description of what's going on" to [[sieve]], meant to be pedagogical, veery small-stepped. I'd be interested to hear from non--experts to which extent they find this bit helpful
* replied (to [[Toby Bartels|Toby]], I presume?) at [[sieve]]
* created [[Dijkgraaf-Witten theory]]
# 2009-05-28 #
* [[Urs Schreiber|Urs]]
* created [[On the Classification of Topological Field Theories]], [[factorization algebra]]
* created [[category of generalized elements]] as a place for the material that was at [[Exploding a Category]] -- I am voting for not using that explosive terminology
* expanded [[path category]] and created [[Moore path category]] in the process
* [[David Roberts]]: Added comment to [[plus construction]] re disambiguation (recall the [[Quillen plus construction]]).
# 2009-05-27 #
* [[Toby Bartels]]: Last night, I made some lists of entries relevant to [[constructive mathematics]], starting with the back links and classifying them. (Hopefully the third list will become so large that we can\'t really use it, but for now I want to keep track of which entries have that material.)
* [[Urs Schreiber|Urs]]
* created [[Quantization as a Kan Extension]]
* created [[plus construction]] -- the material that should go there is currently at [[sheaf]]; should be extracted from there eventually
* added a link to the great Catsters video material to [[limits and colimits by example]] and created [[The Catsters]] and [[Eugenia Cheng]]
* [[David Corfield|David]] created [[extended natural numbers]], [[corecursion]], and [[coalgebra for an endofunctor]]. Moved proof of terminal coalgebra being fixed point to [[terminal coalgebra]]. 'Coalgebra' still needs disambiguation.
# 2009-05-26 #
* [[Bruce Bartlett|Bruce]] filled in a small thing at [[semisimple category]].
* [[Urs Schreiber|Urs]]:
* created [[path integral]]
* added to [[sieve]] a -- potentially long-winded -- lemma with detailed proof that spells out details of the fact that every subfunctor is the coimage of a morphism out of a coproduct of representables
* created [[separated presheaf]]
* further worked on the section "in terms of geometric embedding" at [[sheaf]]
# 2009-05-25 #
* [[Toby Bartels]]: Copied comments on Google and page names to the Café [here](http://golem.ph.utexas.edu/category/2009/05/nlab_more_general_discussion.html#c024021).
* [[Urs Schreiber|Urs]]
* expanded and restructured [[sieve]] -- introduced the explicit _distinction_ between the notion of _sieve_ and that of _subfunctor_ and made their _bijection_ explicit: this seems to be the standard way that "sieve" is used in the literature: authors don't seem to say "sieve" synonymously with "subfunctor", but say "sieve" if they explicitly mean the _set_ of elements of the subfunctor. There is a bijection, but not an equality of concepts.
* further expanded [[image]] and [[coimage]]
* edited and replied at [[semisimple category]]
* [[Bruce Bartlett|Bruce]] responded to comments at [[semisimple category]]...and then got stuck. Can anyone help? Update from Future Bruce to Past Bruce: solved.
* [[Zoran Skoda]]: *as far as google (and other search machine's) counterarguments of Toby (see down 22. may) I still disagree. I do care if our stuff is well indexed and hence better used by anybody looking for answer at the google including us; and I do not care about vanity issues of pro or contra google movements. If somebody gets directed to a less relevant page this is creating noise, and showing a less convenient side of our work. If effectivity is not important why are we doing this ?
* [[Urs Schreiber|Urs]]
* further developed the section "In terms of geometric embedding" at [[sheaf]], but will continue later. I am lacking (at least) one argument in one of the proofs currently...
* created [[Bertrand Toën]] (and [[Bertrand Toen]] as a redirect)
# 2009-05-24 #
* [[Toby Bartels]]:
* Cleaned up [[Cauchy space]] a bit.
* An answer for [[Mike Shulman]] at [[subsequential space]].
* A question for [[Bruce Bartlett]] at [[semisimple category]].
* [[Todd Trimble|Todd]]: Added a section "morphisms of manifolds" to [[manifold]] which attempts to enlarge the notion of pseudogroup, so as to get general notions of map for general notions of manifold. However, this is "original research" and should be vetted -- I don't see the enlarged notion (or an equivalent) in any of the literature I've read.
* [[Bruce Bartlett|Bruce]]: started/added material to some pages related to fusion categories, like [[fusion category]], [[Pivotal symbols]] (this name needs to be changed to fit the naming conventions), [[semisimple category]] and [[rigid monoidal category]].
* [[Urs Schreiber|Urs]]:
* started expanding [[sheaf]] -- my goal is (as now indicated) to give a full derivation of how a geometric morphism into a presheaf category defines a Grothendieck topology and sheaves with respect to that -- but my machine's battery runs low and will leave me offline any minute now, so will have to continue later (darn technology...)
* many thanks to [[Bruce Bartlett|Bruce]] for expanding [[fusion category]] and creating the relevant further entries
* created a stub for [[fusion category]]
* created a stub for [[String field theory]]
* created [[A Survey of Cohomological Physics]]
* added cross-links to [[monoidal category]] and [[manifold]]
# 2009-05-23 #
* [[Toby Bartels]]:
* Created [[Banach space]]s. I need to at least describe products and coproducts in $Ban$.
* Another example, Banach spaces, at [[internal hom]].
* Looking at other things that Urs has done, the problem may be this: If you want to put an equation within a bullet list, then you either have to put no blank lines between the equation and the surrounding text
$$
like this
$$
or put enough spacing before the equation to match the indentation of the list
$$
like this
$$
or, of course, both
$$
like this
$$
but it will fall outside of the bullet list if you do neither
$$
like this
$$
* [[Bruce Bartlett|Bruce]]:
* Added example of the internal hom in super vector spaces to [[internal hom]].
* Urs: I tried to understand what went wrong with your query box example for [[Tychonoff theorem]] but I think the roll-back has removed the original problem from the records.
# 2009-05-22 #
* [[Toby Bartels]]:
* Expanded [[centipede mathematics]] and [[negative thinking]], which [[David Corfield]] created yesterday.
* In response to Zoran immediately below, I don\'t think that we should change what works for us to fit Google. It\'s Google\'s job to give good search results, not ours to optimise Google\'s finding us, which Google tries to prevent anyway. That said, I agree with you about the page names, for other reasons. (Note to all: There is discussion [here](http://golem.ph.utexas.edu/category/2009/05/nlab_more_general_discussion.html#c023953).)
* [[Zoran Škoda]]: created [[comorphism]] (in sheaf theory); there is hard to find a sensible and comprehensive account in the literature. Note that the treatment is more general than in the usage for the case of ringed spaces. By the way, the server is very erratic tonight, and having sometimes responses delayed by 5-10 minutes. On the other hand I strongly disagree with the changes of the names of entries massively being done today by Eric: he moves infinity-category into $\infty$-category. Though this is graphically appealing, google and others put higher in search results items which have search name in the title, and the entries they index are index basically by the ascii. I want to see nlab entries high in the google search, this makes our effort more useful. Fancy graphics can be WITHIN the entry, and prefereably in this decade still not in the title. I would do the redirects in the symbol variant of the title instead!! What the others think ?
* [[Urs Schreiber|Urs]]
* I tried to fix the query box layout at [[Tychonoff theorem]] only to find that there is apparently some software clash between the query box syntax and some symbols used. So I "rolled back". Hopefully the entry is now again in the form that [[Todd Trimble|Todd]] and [[Toby Bartels|Body]] left it. Please check.
* started filling something into [[n-categorical physics]] just so the page looks less blank -- but didn't really have the energy or intention to produce anything of more than vaguely suggestive nature so far
* [[Zoran Škoda]] made changes to [[pure motives]]. Please do not use defined term [[algebraic spaces]] when it is not appropriate.
* _David_ added exposition references to [[pure motive]]
* [[Urs Schreiber|Urs]]
* added explicit description of colimits in $Set$ to [[limits and colimits by example]] -- I have a dumb question: frequently I want to use displayed math equations in a bullet list item. But more often than not when I try this the parser gets very mixed up and produces weirly formatted output -- what am I doing wrong??
# 2009-05-21 #
* [[Bruce Bartlett|Bruce]] added link to Evan Jenkins' notes of the [[Northwestern TFT Conference 2009]], and changed text+math font on the nLab to a serif theme by editing the CSS. This will only be for one day, just to create awareness of what's possible. Discussion on the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=23).
# 2009-05-20 #
* [[Chris Brav|Chris]] started work on section 5 of BZFN
for the journal club on geometric $\infty$-function theory.
* [[Todd Trimble|Todd]] wrote a longish reply to a query of David Roberts about tame topology over at [[nice topological space]].
* [[Toby Bartels]] wrote [[Hausdorff space]] and [[Zorn's lemma]].
* [[Todd Trimble]] wrote [[Tychonoff theorem]].
# 2009-05-19 #
* [[Todd Trimble|Todd]] added a bit more to [[compact space]], in particular linking back to [[quasicompact]]. There is so much one could say about compact spaces, it's quite the embarrassment of riches.
* [[Zoran Škoda]]: created [[algebraic geometry]], [[algebraic stack]], [[topological stack]]; updated [[quasicompact]] with the definition and facts about quasicompact morphisms.
* [[Urs Schreiber|Urs]]:
* created [[Northwestern TFT Conference 2009]]
* added cross-links between [[compact object]] and [[compact space]]
* [[Todd Trimble]] created [[compact space]]
* [[Urs Schreiber|Urs]]:
* added a short section _Basic idea in three words_ to the beginning of [[geometric infinity-function theory]], as I noticed that this simple introductory statement was not sufficiently amplified in the existing introductory section
* thanks to [[Toby Bartels|Toby]] for (in particular) his editorial work! it doesn't go unnoticed, is much appreciated
# 2009-05-18 #
* [[Todd Trimble]] has also created [[uniform space]] and [[ultrafilter theorem]].
* [[Chris Brav|Chris]]:
* Replied to Urs on [[fiber product]], suggesting that fiber products be computed using a cobar resolution and that we should eventually make a section about derived fiber product of spaces.
* [[Urs Schreiber|Urs]]:
* added an "Idea"-section to [[homotopy category]] and added the precise definition of the universality condition on $Q : C \to Ho(C)$
* replied to [[Chris Brav|Chris]] at [[fiber product]]
* expanded [[local isomorphism]]: more about sieves, more about pullbacks -- but needs polishing
* created [[commutativity of limits and colimits]] as a place to list results such as commutativity of filtered colimits with finite limits;
* added a further definition to [[image]], created [[coimage]] and [[strict morphism]]
# 2009-05-17 #
* [[Todd Trimble]] wrote [[metric space]], and there was much rejoicing.
* [[Urs Schreiber|Urs]]:
* restructured [[compact object]]
# 2009-05-16 #
* [[Toby Bartels]]:
* Created [[algebra for an endofunctor]] and [[monoid object]] just to be linked to from [[algebra]], but they are very much stubs; there is related discussion at [[coalgebra]].
* Changed the category of [[Anders Kock]] from [people](/nlab/list/people) to [biography](/nlab/list/biography), although frankly I can\'t remember why we keep these separate.
* Noted exactly which limits are needed at [[power object]].
* Noted on [[direct product]] just how trivial the direct coproduct is.
* Answered Mike\'s question at [[algebraic theory]] by incorporating actual theorems from an actual reference instead of using vague handwaving in the query box, for once.
# 2009-05-15 #
* [[Bruce Bartlett|Bruce]]:
* I created a Stylish extension for Firefox to make the edit box bigger and wider when editing pages (I always found this annoying). To install it, install [Stylish](https://addons.mozilla.org/en-US/firefox/addon/2108) and then [click here](http://userstyles.org/styles/17934).
* Started adding Todd's comments about compact objects in categories to the [[compact object]] page.
* [[David Corfield|David]]:
* began [[synthetic differential geometry]] and [[analytic versus synthetic]]
* [[Urs Schreiber|Urs]]:
* further refined the discussion of the relation to localization at [[geometric embedding]]
# 2009-05-14 #
* [[Urs Schreiber|Urs]]:
* added some bits to [[subobject classifier]]: a short intro, the realization in presheaf topoi and a paragraph on $n$-catgorical generalizations and the interpretation of $true$ as a $-1$-category.
* created [[power object]], which is mentioned in the alternative definition at [[topos]] -- guessed the definition from the _Axiom of power sets_ at [[Trimble on ETCS I]], but I am not really sure -- somebody please have a look and check
* expanded and restructured [[Grothendieck topos]]
* added what should be a detailed proof to [[geometric embedding]] that for every geometric embedding $f_* : F \hookrightarrow E$ the category $F$ is equivalent to the localization of $E$ at the morphisms that are sent by $f^*$ to isos
* there are two possibilities here: either I am mixed up or this is somewhere in the literature. If the latter, can anyone give me a pointer to a reference that mentions this explicitly?
* created [[geometric embedding]] -- am in the process of filling in a detailed discussion of the relation to [[localization]]
* added a paragraph to [[localization]] describing and emphasizing the point that lots of localizations one runs into in practice are [[reflective subcategory|reflective subcategories]] or actually [[geometric embedding|geometric embeddings]]. In particular [[Bousfield localization]] presents the corresponding [[reflective (infinity,1)-subcategory]]. I am still not really satisfied with the entry [[localization]], though, I am hoping we can eventually present the conceptual basis here more clearly. After all, the perspective on sheaves and sheafification in terms of localization/geometric embedding has been shown to be the workable road to $\infty$-stacks, which, as tradition has it, are well worth pursuing in general and on an $n$Lab in particular :-) It is curious that there seems to be a cultural divide in the literature here: the book by Kashiwara-Schapira for instance amplifies sheafification as localization, which paves the road for $\infty$-stacks presented by the [[model structure on simplicial presheaves]], while the book by MacLane-Moerdijk amplifies sheafification as geometric embedding, which paves the road for the simple definition of [[(infinity,1)-sheafification]] by [[reflective (infinity,1)-subcategory]]. Lurie's book effectively gives the unified perspective, which I think is worthwhile presenting very clearly here on the $n$Lab, since it is coceptually so simple and transparent and in practice so powerful. Over.
* added example to [[localization of a simplicial model category]]
* added a little proposition to [[dependent product]] and a little example to [[geometric morphism]]
* [[Tim Porter|Tim]]: I have given a partial reply to [[Urs Schreiber|Urs]] question at [[homotopy coherent nerve]].
# 2009-05-12 #
* [[Toby Bartels]]: An extensive rewrite and expansion of [[predicative mathematics]], including material on the non-[[constructive mathematics|constructive]] school (which rejects function sets), thanks to reading sparked by [one of Jonathan's posts](http://golem.ph.utexas.edu/category/2009/04/taming_the_boundless.html#c023753) to the Café.
* [[Urs Schreiber|Urs]]
* created [[derived smooth manifold]], but will have to continue tomorrow
* created [[category of local models]] and [[locally modeled monoid]]
* expanded a bit more on the "good limits"-motivation at [[derived stack]], alongside the [blog discussion](http://golem.ph.utexas.edu/category/2009/05/journal_club_geometric_infinit_3.html#c023770)
* added a few more details to [[generalized smooth algebra]] -- and have a question on that on the blog [here](http://golem.ph.utexas.edu/category/2009/05/journal_club_geometric_infinit_3.html#c023762)
* added a few bits and pieces to [[derived stack]], [[ind-object in an (infinity,1)-category]], [[perfect infinity-stack]]
* [[David Corfield|David]]
* began [[Eckmann-Hilton duality]]. Referred to it from [[duality]], but maybe that page needs some adjustment. By the way, I still don't think I have answers to questions posed [here](http://golem.ph.utexas.edu/category/2009/04/cohomology.html#c023057) and below.
* began [[initial algebra]] and [[terminal coalgebra]].
* [[Urs Schreiber|Urs]]
* added the coherence diagrams to [[braided monoidal category]]
* added the enriched versions to [[representable functor]] and [[opposite category]]
# 2009-05-11 #
* [[Toby Bartels]]: I moved all of the <abbr title="scalable vector graphics">SVG</abbr> out of article pages and into included subpages as described at [[HowTo]]. (Note that this is supposed to be a temporary fix until we get a working automated system to include somehthing like TikZ; see [the discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=20).)
* [[Urs Schreiber|Urs]]:
* added more details to and slightly restructured [[exact functor]]
* in particular added a remark on and a link to "[[flat functor]]", which overlaps in content
* I changed at [[flat functor]] the condition that $(d/F)$ be filtered to the condition that it be cofiltered. Please check that I am not hallucinating! But see for instance [[Categories and Sheaves|KashSch prop. 3.3.2]]
* after reading it I got the idea that [[Toby Bartels|Toby]]'s latest addition to [[stuff, structure, property]] is really about higher subobject classifiers as discussed at [[generalized universal bundle]]. I added corresponding remarks, but didn't find the time to look into this very carefully.
* [[David Corfield|David]]: Started [[locally finitely presentable category]]
* [[Toby Bartels]]:
* Added a sort of logical interpretation to [[stuff, structure, property]]. (This has some links that ought to be filled out.)
* There\'s more work going on at [[SVG Sandbox]] and being discussed in [a Forum post](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=20). See also the new [[Inclusion Sandbox]].
* There\'s been some work recently at [[generalized element]], in response to a reader\'s question.
# 2009-05-08 #
* [[Toby Bartels]]: Fixed the last 'heuristic', at [[local system]], among other such fixes. (You gotta open them up for editing and look at the source!)
* [[Urs Schreiber|Urs]]:
* tied up the loose end at [[descent]] using the expected result that Dominic Verity was kind enough to proof on request and then confirm by private communication
* [[Andrew Stacey|Andrew]]: (I forget whether I'm supposed to add this to my earlier comment, or here, or add a word of this to each of the other entries of the day). I've created [[SVG Sandbox]] expressly for the purpose of mucking about with SVGs to get them to look right. My rationale is explained at the top of that page, together with some suggestions on how it might work. The point is that one SVG can be rather large and I think that putting them in the regular Sandbox to test stuff is a bit anti-social. Hopefully we can get stuff in the [[FAQ]] and [[HowTo]] on good ways of importing SVGs as there seem to be a few "special features". I've shifted the recent SVG-related stuff from the original [[Sandbox]] to the [[SVG Sandbox]], but I shifted the discussion that Bruce started to the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).
* [[Urs Schreiber|Urs]]: I believe [[Mike Shulman|Mike]] had requested that we announce each change here always on the very top of the list, even if we had earlier logs the same day with other people's logs already on top of them. Seems to be a reasonable practice if the point of this page here is to alert others of changes. Which it is. So, yes, the way you did it should be the preferred way.
* [[Urs Schreiber|Urs]]:
* recalled [[Tim Porter]]'s comment at [[simplicial set]] and moved [[simplicial nerve of simplicial categories]] to [[homotopy coherent nerve]];
* started expanding, polishing and reorganizing [[simplicial set]] -- still not satisfied, though.
* [[Bruce Bartlett|Bruce]]: Added discussion to [[Sandbox]] about Andrew's TikZ->SVG method. Maybe we can get good and easy graphics going soon in Instiki!
* [[Urs Schreiber|Urs]]:
* added lots of further stuff to [[simplex category]]
* [[Andrew Stacey|Andrew]]: Finished off the heuristic shift. Bizarrely, one page, [[local system]], comes up in the Instiki search as having "heuristic" in it but I can't find it. The other instances of "heuristic" that are left are either correct or within discussions.
* [[Urs Schreiber|Urs]]:
* added explicitly the description in terms of weighted (co)limits and in terms of (co)ends to [[Kan extension]]; also reorganized the existing material somewhat.
* added the definition of composition to [[enriched functor category]];
* thanks to [[Andrew Stacey|Andrew]] for the thing about "heuristic". I'll be aware of that in the future.
* following [[Toby Bartels|Toby]] I made [[heuristic introduction to sheaves, cohomology and higher stacks|heuristic intro...]] a redirection to [[motivation for sheaves, cohomology and higher stacks]]
* [[Toby Bartels]]:
* I split [[identity]] into [[identity morphism]] and [[identity element]]. Maybe I\'ll do the same to [[inverse]] later, but not now.
* More details at [[extensional relation]].
# 2009-05-07 #
* [[Toby Bartels]]: Reply to Andrew below:
* Yeah, that\'s pretty much how to do it. Except that you mark the old page as `category: redirect` rather than as `category: delete`, since it contains edit history that we want to preserve. (Instiki has no cool page-move feature like MediaWiki does.) As you did, I also usually decline to change links from discussion.
* I would move [[heuristic introduction to sheaves, cohomology and higher stacks]] to [[motivation for sheaves, cohomology and higher stacks]], since Urs doesn\'t like [[introduction to sheaves, cohomology and higher stacks]].
* [[Andrew Stacey|Andrew]]
* Heuristically altered records containing the word "heuristic". The original list of records matching the search term "heuristic" is below (so anyone worried can check what I've done). Most actually were references to [[heuristic introduction to sheaves, cohomology and higher stacks]] which obviously needs to be renamed before the other pages are changed (what's the best way to rename a page? Is it: create a new one, copy over the content, put in a redirect, change all referring pages, mark old one for delete? Or is there a simpler way?). In a couple of the others the word was used in a discussion so I didn't change those. One I even left as I thought it was (almost) correctly used!
* [[2009 April changes]]
* [[Categories and Sheaves]]
* [[General Discussion]]
* [[Grothendieck's Galois theory]]
* [[cohomology]]
* [[full functor]]
* [[geometric infinity-function theory]]
* [[heuristic introduction to sheaves, cohomology and higher stacks]]
* [[hyperstructure]]
* [[induced representation]]
* [[limit]]
* [[local system]]
* [[sheaf]]
* [[sheaf cohomology]]
* [[space and quantity]]
* [[stuff, structure, property]]
* [[weighted limit]]
* [[why (infinity,1)-categories?]]
* [[Urs Schreiber|Urs]]
* as a reaction to the discussion taking place there I expanded the text at [[simplicial model for weak omega-categories]] -- also added references
# 2009-05-06 #
* [[Urs Schreiber|Urs]]
* reacted to the discussion at [[Day convolution]] and provided the requested further details in the main entry
* corrected the statement about the relation of $SetMod$ to $Span(Set)$ at [[distributor]] by adding the missing clause about discrete categories
* created [[structured generalized space]]
* created [[string theory]]
# 2009-05-05 #
* [[David Corfield|David]]:
* began [[things to be categorified]]
* [[Toby Bartels]]:
* Fixed up [[extensional relation]], although there seems to be a problem with one of the counterexamples.
* I also have an opinion at [[simplicial model for weak omega-categories]].
* [[Urs Schreiber|Urs]]:
* added a pedagogical example to [[enriched functor category]]
* [[David Corfield|David]]:
* began [[microcosm principle]]
* [[Urs Schreiber|Urs]]:
* added details to [[enriched functor]]
* added a bit more detail to [[closed monoidal structure on presheaves]]
* replied at [[cohomology]]
# 2009-05-04 #
* [[Urs Schreiber|Urs]]
* worked on [[end]]: added a section "End as equalizer" where I try to motivate the formula for the end over $V$-valued functors from the equalizer formula for limits, then give the equalizer formula for ends -- personally I find this a bit more helpful than dinatural transformations, but that is certainly just my ignorance -- I added an "Idea" and a "References" and an "Examples" section
* added the definition to [[enriched category]] (yes, that was still "left as an exercise")
# 2009-05-03 #
* [[Toby Bartels]]:
* Wrote [[FOLDS]] (and [[Michael Makkai]]), [[identity functor]], [[identity natural transformation]], [[anafunctor category]], [[identity anafunctor]], [[anabicategory]], and [[ananatural transformation]] to be linked from the below.
* Rewrote [[equivalence of categories]], moving one paragraph to [[Cat]] and otherwise expanding.
* Urs and I would like some advice from [[Tim Porter]] at [[simplicial nerve]].
* [[Urs Schreiber|Urs]]:
* further expanded section 2 of [[geometric infinity-function theory]], which I am to "report" on next Monday in our Journal Club.
* finally created [[simplicial nerve of simplicial categories]], but more details necessary...
* am trying to find out, for [[A-infinity ring]] and [[E-infinity ring]] to which extent the two definitions "algebra over an $\infty$-operad" and "algebra object in a monoidal $(\infty,1)$-category" are equivalent, as one would expect...
* created [[symmetric monoidal (infinity,1)-category]], [[commutative algebra in an (infinity,1)-category]], [[smash product of spectra]], [[associative ring spectrum]], [[A-infinity ring]], [[commutative ring spectrum]], [[E-infinity ring]]
* expanded the intro at [[higher algebra]] a bit
* thanks, [[Toby Bartels|Toby]], for your editorial help! Yes, right, "[[higher algebra]]" should be lower case, true.
# 2009-05-02 #
* [[Toby Bartels]]:
* Disambiguated [[simplicial category]], hopefully correctly, at [[(infinity,1)-category]], [[derived stack]], and [[pretriangulated dg-category]].
* Wrote [[proper class]].
* Moved [[Higher Algebra]] to [[higher algebra]] because it *seemed* to be about the subject rather than about just Lurie\'s two papers, but I may have been wrong about that.
* Formatted theorems at [[descent for simplicial presheaves]] and [[descent]].
* Answered questions at [[biproduct]], [[comma category]], [[exponential object]], and [[context]].
# 2009-05-01 #
* [[Finn Lawler]]
* Created [[lax 2-adjunction]], complete with spiffy PNGs.
* Added some links at [[Gray-adjointness-for-2-categories]] to entries on concepts in the book. Some of these are to non-existent pages like [[lax natural transformation]]. Do pages on 'elementary' stuff like this belong here on nLab, or should we add them to Wikipedia and link there?
+--{: .query}
[[Mike Shulman|Mike]]: If we have a page on [[category|categories]], we can certainly have a page on lax natural transformations!
[[Jonas Frey]]: The problem about lax natural transformations is that there is no consensus about the orientation of constraints in the literature. Johnstone in the elephant and Gurski in his PhD-thesis for example use an other orientation than Leinster and Borceux in their books.
_Toby_: All the more reason to write an article to explain the differences!
=--
* [[Urs Schreiber|Urs]]
* created new top-level link list [[Higher Algebra]] -- to be filled with content...
* considerably expanded [[(infinity,1)-category]] -- motivation was really to write [[relation between quasi-categories and simplicial categories]], but then ran out of steam
* removed all existing logs here and archived them as [[2009 April changes]]
***
[[2008 changes|First list]] --- [[2009 April changes|Previous list]] --- [[2009 June changes|Next list]] --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
2009 September changes | https://ncatlab.org/nlab/source/2009+September+changes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during September 2009.
The substantive content of this page should **not** be altered.
The announcement of the change to [the Forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) and the reasons for it have been [archived](http://ncatlab.org/nlab/revision/2009+September+changes/397).
***
## 2009-09-30
* [[Toby Bartels]]: The lab elves are going to try to convince people to use the Forum for latest changes in October; see the new section about this at the top of the page. (If the instructions there are unclear, you can edit them now; I will move the final version to a permanent post on the Forum in 23 hours, after which you won\'t be able to edit them.)
## 2009-09-29
* [[Urs Schreiber]]: created [[blob homology]]
* [[Jon Awbrey]]:
* Added a 'blink (= stub-link) at [[higher order proposition]].
* Added a quote to [[precursors]] in which Hilbert borrows an idea from Kant.
* [[Urs Schreiber]]:
* added the example of automorphism 2-groups to [[2-group]]
* added reference to the new article by Batanin, Cisinki and Weber to [[generalized Gray tensor product]] -- Mike, did you look at that?
* Not much yet, though it looks very neat. It looks like they are going the "non-closed monoidal structure" route?
* [[Toby Bartels]]: More at [[SEAR+?]].
## 2009-09-28
* [[Mike Shulman]]: Toby is [[SEAR+?|right]].
* [[Zoran ?koda]]: entry [[internal crossed module]].
* [[Urs Schreiber]] created [[differential crossed module]]
* [[Zoran ?koda]] created [[general linear group]].
* [[Toby Bartels]]: I\'m pretty sure that [[SEAR+?]] satisfies $COSHEP$.
* [[Todd Trimble]]: created [[PRO]], and began (re)adding material to [[cube category]], with a view toward incorporating material from the Grandis-Mauri [paper](http://www.emis.de/journals/TAC/volumes/11/8/11-08.pdf) on cubical sites. Discussion with [[David Corfield]] at [[cubical set]].
* [[Urs Schreiber]]: added the example of "derived schemes with $E_\infty$-ring"-valued structure sheaves to [[generalized scheme]].
* [[Zoran ?koda]] separated Lurie material, with a link from [[scheme]] to [[scheme as locally affine structured (infinity,1)-topos]].
[[Urs Schreiber]]: thanks, Zoran. I have renamed the entry to say $(\infty,1)$-topos, as I think we agreed to be (more) careful than Lurie is about this.
* [[Urs Schreiber]]: added the alternative definition in terms of sheaves on $Aff/X$ to [[quasicoherent sheaf]].
* [[Urs Schreiber]] expanded the definition at [[scheme]]:
added a word on the definition of morphisms, gave the sheaf-theoretic version and the definition from the derived scheme perspective. In that context I cited the full paragraph of Jacob Lurie where he argues that the standard definition is misleading in that it asserts an underlying topological space, and that one better thinks of a scheme as having an underlying [[locale]]/[[0-topos]].
* Urs why not separating this into another entry like scheme over a locale. There is zillion of things which should still enter into the entry of scheme and people needing basic info on schemes do not want hi-tech things which have no advantage in usual geometry, but rather as a motivation for generalizations ? Moreover this reinterpretation is missing the point of relative point of view. One usually have schemes over $S$, and being a scheme or being affine is a property of the morphism, and it can be done relative to anything. In that case, you do not care what is underlying. For example, nc schemes of Rosenberg have any category as a base while the exactness properties required for schemes are still strictly required.
[[Urs Schreiber]]: sure, let's split off sub-entries when it gets too long. You are currently locking the entry, otherwise I would have done it.
* [[Zoran ?koda]] changes to [[generalized scheme]].
* [[Urs Schreiber]]
* part 3 at [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]] -- in that context also created: [[Paul Goerss]], [[n-truncated structured (infinity,1)-topos]], [[Spectral Schemes]], [[Topological Algebraic Geometry - A Workshop]]
* stub entry for [[derived scheme]] and [[derived Deligne-Mumford stack]]
* have two questions (green query boxes) at [[algebraic stack]]
* moved the Deligne-Mumford reference from [[algebraic stack]] to [[Deligne-Mumford stack]]
* created brief entries on [[representable morphism of stacks]] and [[geometric stack]], but that needs attention
* added to [[affine space]] a section "Affine spaces as model spaces".
* more details at [[geometry (for structured (infinity,1)-toposes)]]
* [[David Corfield]]: Carried on the discussion at [[cubical set]]. Now we need to decide whether Pratt's use of the term is sufficiently widespread to justify disambiguation.
* [[Urs Schreiber]]:
* more details at [[generalized scheme]]
* edited [[derived algebraic geometry]] (a bit structuring by headlines and some paragraphs here and there)
* [[Mike Shulman]]: A reference at [[COSHEP]].
## 2009-09-27
* [[Mike Shulman]]: In response to the discussion at [[choice operator]], I started [[SEAR+?]] about whether and why adding a non-extensional choice operator is a conservative thing to do to a theory that lacks [[axiom of choice|AC]]. So far I can prove that it is conservative over [[COSHEP]].
* [[Jon Awbrey]]:
* Added potentially enlightening quotes from Hilbert and Ackermann to [[precursors]]. Better lights might be thrown by the original German or the first edition --- all I have on hand right now is the English translation of the second edition.
* Added a historical note to [[choice operator]].
* [[Toby Bartels]]:
* Maybe it\'s just my font choices (DejaVu all the way!), but the TOC at [[generalized scheme]] looks perfectly normal to me.
* Comments at [[choice operator]] and [[SEAR]].
* [[Bas Spitters]] joined to make a note at [[SEAR]] that I moved to [[ETCS]].
## 2009-09-26
* [[Zoran ?koda]] created entries [[affinoid algebra]], [[rigid analytic geometry]] (just started), [[derived noncommutative geometry]], [[unique factorization domain]].
* [[David Corfield]]
* a couple of queries at [[cubical set]].
[[Urs Schreiber]] I don't know the answer to the first question. Concerning the second: the answer to questions of the form "shouldn't we do ...?" is usually: "yes, we should, why don't you go ahead and do so?"
[[Todd Trimble]] took a crack at addressing queries.
* [[Urs Schreiber]]
* added a remark on Connes' program to [[noncommutative geometry]], but still no genuine content
* started adding details to [[generalized scheme]]
two formatting problems:
* the MathML in the headlines makes the TOC have too much vertical spacing
* the hyperlinks in the definition/theorem names don't appear as such (I think they used to work there in the old setup)
I leave it typeset this way anyway, trusting that we can eventually fix the software instead of working around it
* worked on [[noncommutative algebraic geometry]]: added lots of links (many to existing entries, some to entries that ought to be created eventually), added more sections and a table of contents, then I expanded the Idea section, trying to give a better idea to laymen. Please check!
* [[Jon Awbrey]] spun off the speculative archaeology of [[category theory]] to its own page at [[precursors]].
* [[Zoran ?koda]] changes to [[generalized scheme]].
* [[Urs Schreiber]] added material to [[cubical set]] (on relation to homotopy theory), that [[Ronnie Brown]] had posted to the Alg-Top mailing list
* [[Mike Shulman]]:
* talk at [[choice operator]], TOC at [[pure set]]
* added a sketch proof of the ZF collection axiom from the SEAR collection axiom to [[SEAR]]
## 2009-09-25
* [[Todd Trimble]]: added commentary to Rafael's suggestion at [[category theory]], at the end of the penultimate discussion box.
* [[Urs Schreiber]] created [[exterior differential system]] and related to that [[dg-ideal]] and [[vertical tangent Lie algebroid]]
* [[Toby Bartels]]: Sorry at [[natural numbers in SEAR]].
* [[Rafael Borowiecki]]: Replied at [[category theory]] and suggested changing one of the views what category theory is.
* [[Zoran ?koda]] created [[localized coinvariant]], [[universal localization]], [[noncommutative localization]], [[coinvariant]]; new links at [[Zoran ?koda]].
* [[David Roberts]]: question for Toby at [[natural numbers in SEAR]], in \'alternative approach\'.
* [[Toby Bartels]]: [[SEAR]], [[pure set]], [[natural numbers in SEAR]], [[category theory]], [[choice operator]].
* [[David Roberts]]: fixed up problem at [[natural numbers in SEAR]] pointed out by Mike, and included another, cleaner, definition.
* [[Zoran ?koda]]: created [[universal enveloping algebra]], [[enveloping algebra]].
## 2009-09-24
* [[Mike Shulman]]:
* Response/suggestion at [[natural numbers in SEAR]]
* Created [[choice operator]] in order to ask a question (the second one on the page, not the first).
* [[David Roberts]]: polishing up [[natural numbers in SEAR]]. The definition is in, now.
* [[Zoran ?koda]]: created [[Ore localization]].
* [[Urs Schreiber]]: worked on [[geometry (for structured (infinity,1)-toposes)]]
* expanded and improved "Idea" section
* in "References" section started commented list with explicit pointers to definitions
* needs expansion -- and warning: I think I missed some $(-)^{op}$s and $Ind(-)$s. But have to run now.
* [[Mike Shulman]]: Some discussion is continuing at [[pure set]].
* [[Urs Schreiber]] some rough notes on genera and the elliptic genus at [[elliptic cohomology]].
* [[Andrew Stacey]]: Who's the emacs guru who put the stuff on the [[HowTo]] about using Emacs?
* [[Mike Shulman]]: I guess that would be me.
* [[Andrew Stacey]]: Great! I have a [forum discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=85) for you. Of course, anyone else interested in using Emacs for editing nLab pages is welcome to join in.
* [[Ivo]] created [[free groupoid]] and [[Urs Schreiber]] went over it and added links, and typeset math.
* [[Mike Shulman]]: A different viewpoint on universes in [[SEAR]].
* [[Rafael Borowiecki]]: Replied at [[category theory]]. Now i understand less than i did before.
* [[David Roberts]]: A bit at [[Morita equivalence]], on the version for Lie groupoids, and a sketchy start to universes in [[SEAR]].
## 2009-09-23
* [[Zoran ?koda]] created a rough outline for [[Ore set]] and plan to have the entry on Ore localization separate.
* [[Urs Schreiber]]: created [[Euclidean supermanifold]] and worked slightly more on the stub for [[Clifford algebra]]
* [[Zoran ?koda]] created a rough outline for [[descent in noncommutative algebraic geometry]].
* [[Dmitri Pavlov]] Asked two questions at the [[Morita equivalence]] page.
* [[Andrew Stacey]] Reorganised the [[HowTo]] a little - hopefully haven't lost any information! - so that the automatic table of contents looks like the old one did (almost).
* [[Urs Schreiber]]: thanks, Andrew!!
* [[Urs Schreiber]]
* left some new entries unfinished, but have to resume finishing these later today
* added instructions for automatics TOCs to [[HowTo]]
* some content fed into [[supermanifold]] -- alse created [[SDiff]], [[SVect]], [[super vector space]], [[super algebra]], [[Grassmann algebra]], [[super Lie algebra]]
* [[Jon Awbrey]] is road-testing the "cylindricity" symbol $\text{⌭}$ (unicode ⌭) for composing functions and relations the _right_ way, that is, in arrow order. For example, see [[boolean-valued function]].
* I believe a semicolon $;$ is already commonly used by many people for composing things in "diagrammatic" order. -Mike
* Yes, there's a Z-notation semicolon ⨾ (unicode ⨾) that I tried for a while, but it always looks more like a (California) stop than a connector --- besides, Kurt Vonnegut says not to use 'em.
## 2009-09-22
* [[Urs Schreiber]]: added automatic TOCs to [[limit]], [[adjoint functor]], [[Kan extension]], [[cohomology]], [[FQFT]], [[category theory]], [[descent]] and maybe some other entries
this feature is great! we should insert it in most entries. Even though I was very much involved with all these entries, I was still surprised to see how long and detailed some of these tables of contents are. They really give an impression of a long entry that is not available otherwise.
one entry that really deserves a TOC is [[category of fibrant objects]] -- but I failed to insert one there. Encountered lots of strange behaviour like truncations of the entry (had to rollbck several times) or error messages after saving. Could it be that this entry exceeds some length limit?
[[Andrew Stacey|Lab Elf (carpentry division)]]: The length limit is _really long_ now, about 500 million characters. If we get that long we should contact the Guinness book of records. The problem was actually due to having a wiki-link in one of the headings. Oddly enough, there was no error message on this in the logs but changing it solved the problem (it now says "see also [[fibration sequence]]" after the heading). I imagine it's to do with the fact that things in the table of contents are links to the sections in the document so having a wiki-link would mean that one thing (the entry in the table of contents) was a link to two places (the section and the page the wiki-link points to). As the Maruku filter is independent of the wiki, it doesn't know of wiki-links and so doesn't know how to handle them.
* [[Mike Shulman]]: More at [[SEAR]], including how to eliminate equality entirely, and how to prove the SEAR Collection axiom from ZF.
* [[Urs Schreiber]]: many thanks to [[Jon Awbrey]] for providing help with automatic TOCs. Great that it works now, but why does it? What does that funny "tic" thing achieve? Let's discuss this on the forum -- [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=82&page=1#Item_1) -- and summarize the result at [[HowTo]] eventually
* [[Toby Bartels]]: Working the foundations at [[SEAR]], [[pure set]], and [[natural numbers in SEAR]].
* [[Zoran ?koda]] expanded [[proper map]].
* [[Urs Schreiber]]
* now a bit of real content at [[Brown representability theorem]]
* added link to Iglesias' PhD thesis to [[diffeological space]] -- the link to this thesis is hidden somewhere on his website, and it was pointed out to me that the thesis contains some noteworthy material which is not in the book (yet), such as discussion of diffeological principal bundles
* [[Zoran ?koda]] added references to [[Heisenberg double]] (including my own).
* [[Urs Schreiber]] wrote very stubby stub for [[Brown representability theorem]], in fact just recording a recent reference there
* [[Lab Elf|Lab Elf (carpentry department)]]: Maruku (the implementation of markdown used here) can do automatic tables of contents (for use _within_ a page, it can't do cross-page contents). For an example see [[Froelicher space|this page]] and for the syntax, see the [maruku extended syntax page](http://maruku.rubyforge.org/maruku.html#toc-generation).
* [[Urs Schreiber]]: ah, thanks Lab Elf! I wasn't aware of this. This will save me a few keystrokes. Should be mentioned at [[HowTo]].
* [[Jon Awbrey]] tried that at [[differential logic]] and couldn't get it to work. Is there some extra trick to that?
* [[Jon Awbrey]]: It looks like you have to start your headings with single #'s and use a flush left tag something like this:
* tic (any old list item will do the trick here)
{:toc}
* [[Urs Schreiber]]
* created [[functorial quantum field theory - contents]] and added it as a floating table of contents to relevant entries -- trusting that the related issues currently [discussed on the forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=61&page=1#Item_9) will eventually be solved by CSS means
* created [[bordism categories following Stolz-Teichner]]
* created stubs for [[Riemannian manifold]], [[Riemannian cobordism]], [[isometry]]
* added to [[Gray tensor product]] a link to Mike's new [[generalized Gray tensor product]]
* [[Mike Shulman]]:
* Replied to several comments at [[SEAR]], and rearranged the discussions to be (mostly) next to the text they are discussing. I'm happy about the interest this idea is generating.
* Wrote [[generalized Gray tensor product]] about a folklore no-go argument that I keep forgetting and having to re-produce.
* [[David Roberts]]: comments at [[SEAR]]:
* suggestion for renaming - SER (and SEPS for Toby's alternative)
* comment about the category of SEAR-sets
* the kick-start of a new project, [[natural numbers in SEAR]]. This is without assuming the axiom of infinity and otherwise assuming as few of the SEAR axioms as possible. A little over a stub at present, and maybe I'm making a big deal of nothing...
## 2009-09-21
* Heated discussion at [[SEAR]].
* $\mathbf{Cat}$ fight?
* [[Jon Awbrey]] is setting some tables at [[differential logic]].
* [[Urs Schreiber]]:
* created stub for [[supergroup]]
* created [[geometric models for elliptic cohomology]] with sub-entries so far:
* [[Axiomatic field theories and their motivation from topology]]
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
this comes with this blog entry [here](http://golem.ph.utexas.edu/category/2009/09/a_seminar_on_a_geometric_model.html)
* created stub for [[partition function]]
* created [[modular form]], but this needs more attention
* added the promised blurb to [[directed space]] under "homotopy theoretic perspective" that is supposed to indicate the idea of the relation to [[(n,r)-category]] without misleadingly sounding as if there were nothing left to do here. Check!
I also added links to Grandis' new book on [[Directed Algebraic Topology]] to [[directed space]] and to [[directed homotopy theory]]. Since I haven't had a chance to look at that book yet, I have a question at [[directed space]] about its content.
* [[David Corfield]]: Why call the page [[Directed Homotopy Theory]] when it's about a book called 'Directed Algebraic Topology'? Don't titles of pages for specific texts coincide with titles of the texts?
* [[Urs Schreiber]]: I just wondered about the same thing -- I made [[Directed Algebraic Topology]] now redirect to that entry, but if Zoran is okay with it I would also suggest to rename this. Book entries should carry the title of the book, at least up to abbreviation.
* Sorry, I am overloaded in last few days and made an error. Funny enough, there is a historical parallel with russian EGA. Russians have published a translation of the the introduction to 1971 EGAI edition. The external pages says Elements of algebraic geometry, while the title on the very first page is Elements of algebraic topology. I have a scan of this funny "typo".
* [[Urs Schreiber]]: restructured [[moduli space]], linked to it from [[classifying space]] and, notably, added a semilong discussion of the subtleties of the common slogan that " _Objects with automorphisms don't have fine moduli spaces_ . ", summing up some arguments that were exchanged in the blog discussion [here](http://golem.ph.utexas.edu/category/2009/09/a_seminar_on_gromovwitten_theo.html).
## 2009-09-20
* [[Toby Bartels]]: Reply to Mike at [[foundations]].
* [[Mike Shulman]]: Rafael, your clarification at [[category]] introduced another typo! (-: Now fixed.
* [[Urs Schreiber]]: quick reaction at [[(n,r)-category]] -- am still not quite back online, but trying...
* [[Todd Trimble]] responded to [[Rafael Borowiecki]] at [[category theory]].
* [[Toby Bartels]]: I don\'t think that those typos in [[(n,r)-category]] were typos, but I tried to clarify them.
* [[Zoran ?koda]]: new stub [[K-theory and physics]]. Book entry [[Directed Homotopy Theory]]. Added references to [[basic ideas of moduli stacks of curves and Gromov-Witten theory|basic ideas of GW]].
* [[Rafael Borowiecki]]:
* Finally took the time to answer at the discussion at [[category theory]]. Also moved some structures to structures that reduces to categories.
* Amazingly discovered a typo in the definition of a category at [[category]]. Added a clarification in the same definition.
* Corrected two presumable typos as [[(n,r)-category]].
* [[Zoran ?koda]]: new entry [[Vladimir Drinfel'd]]. Improvements to [[Q-category]], [[noncommutative algebraic geometry]].
* [[Mike Shulman]]: Incorporated the discussions at [[Crans-Gray tensor product]] into the entry, and deleted them.
* [[David Roberts]]: Comment/reference at [[directed homotopy theory]] - Grandis has a book out now on this stuff.
## 2009-09-19
* [Arrr!](http://www.talklikeapirate.com/piratehome.html)
## 2009-09-18
* [[Arnold Neumaier]] has joined to talk about [[SEAR]].
* [[Jon Awbrey]] added some (*hopefully*) motivating remarks and lots of pretty pictures to [[differential logic]].
* [[Andrew Stacey]] I've disabled the export features. They now redirect to a section of the [[HowTo]] which explains how to use `wget` to get a local copy of the n-Lab. If anyone wants to add instructions for other programs or OSes then, of course, feel free.
* [[David Corfield]]: chipped in at [[(n,r)-category]].
* [[Mike Shulman]]:
* Reply at [[SEAR]].
* Some cold water at [[(n,r)-category]].
* [[Jon Awbrey]] added more content to [[differential logic]].
* [[Zoran ?koda]]: [[Yuri Manin]], [[Arakelov geometry]]; small changes to [[moduli space]].
* [[David Roberts]]: Comments at [[SEAR]]. From what I've seen, I like it.
## 2009-09-17
* [[Urs Schreiber]]:
* branched off the seminar notes mentioned below to [[basic ideas of moduli stacks of curves and Gromov-Witten theory]]
in that context created [[moduli space]], currently also being the redirect for [[fine moduli space]], [[coarse moduli space]] and [[moduli stack]]
all still mighty rough, but I think valuable raw material to base further editorial developments on
* [[Andrew Stacey]] I have replied to Zoran's points over on the n-Forum at [this discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=64) since that seems a better place to have a discussion than here.
* [[Urs Schreiber]]:
* am taking notes in a seminar on [[Gromov-Witten theory]]
* started something at [[deRham theorem]]
* created [[simplicial sheaf]]
* Zoran: I am strongly against shifting latest changes to the forum. I do not know how to quickly link and do other features like here, it requires more downloading capacity when on expensive network like mobile, it may require account, it does not get recorded when downloading the whole nlab etc etc. Logging to forum is anyway pain when on mobile network. It logges you off for example if you are idle for 30 minutes. I will not do it simply. I quit logging changes if it is to the forum. I will edit nlab without logging changes in that case. Nlab is nlab, and it should be self contained. Forum is about general policies, it is complicated enough to explain to the new userts that there is latest changes notificatiopn, notg in addition now that they have to have an additional account and additional web page with different software. I never use RSS feeds nor want to use them: I do not check latest changes unles sI am generally interested what is there. If I work on the item "jabberwocky" I WORK on it. If I want to see latest changes I look at them. It is very important that I can download the whole nlab including the latest changes histories. Forum is different system and it should not be mandatory to use it. I also find useful that I can link and cut and paste formulas and nlab links easily within nlab latest changes the same way as I do the rest, the forum has a bit different formatting and makes it harder. Also nlab item jabberwocky has down there a link that it was mentioned with link in latest changes what I also find useful. I also do not find any argument in the saying that if I log to forum for looking at latest changes I will also see be "informed on the new things". Thjis is not a feature but a DISTRACTION when I work hard and follow references and try to format my mathematical text. Forum is about policies and politics, and software.- Nlab is about mathematics. I like to havce that CLEANLY separated. The state of my mind is the prerequisite for working on nlab. The alternative is that I work only in my personal nlab if you impose this new policy of mixing with forum.
* [[Jon Awbrey]]: i 2nd that e-motion.
* [[Urs Schreiber]]:
* added something to [[directed space]]. Check!
* added some $(n,r)$-topos cases to the "special cases" section at [[(n,r)-category]]
* added a paragraph at [[About]] in the part starting with "If you find yourself annoyed by the state some entry is in...". That paragraph is motivated by a recent reaction by somebody on some blog, who had indeed complained about some unfinished entry and after that needed some persuasion to help expand and improve it.
## 2009-09-16
* [[Urs Schreiber]]:
* created stub for [[(infinity,1)-operad]]
* added one and updated and commented another reference at [[Jacob Lurie]]
* am proposing an expanded introduction and a supposedly suggestive **slogan** at [[(n,r)-category]] -- check and see what you think
* That sort of thing doesn\'t give *me* any idea about what an $(n,r)$-category is ... although it does tell me what an $r$-directed $n$-type is! It\'s important, good stuff, but I wouldn\'t put it in the lede. ---Toby
* [[Zoran ?koda]]: created [[Q-category]] (clearly unfinished; e.g. somebody could type the cosieve and Grothendieck topology-induced $Q^\circ$-categories and $Q$-categories of thickenings from Kontsevich-Rosenberg preprint as examples), [[wave]], [[epipresheaf]]. New remark at [[formally smooth scheme]]. Updates to [[algebra]], [[mathematicscontents]] and to the discussion at [[classical mechanics]].
* [[Urs Schreiber]] created entry for [[Constantin Teleman]]
* [[David Corfield]]: created [[biology]].
* [[Urs Schreiber]]:
* noticed that despite all the blog discussion, nobody has so far created [[field with one element]]...
* created [[deRham space]]
* created [[formally smooth scheme]]
* [[Urs Schreiber]] expanded [[mathematicscontents]] and rearranged a bit -- notably I added [[category theory]], [[higher category theory]], [[topos theory]] and [[higher topos theory]] (the lower case version!) which clearly all deserve to be there. I added some entries to go as sub-entries under [[higher topos theory]] mainly to balance that [[topos theory]] has its natural sub-entries there, but maybe debateable. Then I moved [[homotopy theory]] and its special case [[stable homotopy theory]] from the "Geometry" bit up to the "Structural Foundations" bit, as that seems to better capture it (anything that cares about things only up to homotopy is not geometry nor even really topology, but is higher category theory in disguise). By this logic, also [[rational homotopy theory]] belongs there, so I moved it up.
* [[Urs Schreiber]]: worked on [[HomePage]]:
* added the floating tables of contents for math, physics and philosophy. They are sitting there now to the left of the $n$Lab-contents. I am thinking that here on the HomePage this is a good thing. Besides the introductory text we keep there, we want to make sure that the reader's attention is directed to tables of contents. I have heard of people who were pointed to the nLab, went to the HomePage for a minute and came back with the impression that there is nothing much to be found on the lab. While no table of contents can give an accurate impression of the full scope of the lab, the ideal would be that our main three top-level contents (math, physics, philosophy) will indicate the scope of topics and lure the reader further into the labyrinth. Optimally behind each of the links of the top-level toc the user finds another floating table of contents for the given sub-topic
* I edited the bit about the forum. With Andrew getting ready to make "latest changes" be on the discussion forum, the old material saying that the forum is just for meta-discussion is outdated.
* I included a link to the forum discussion on what the nLab's scope is. As long as noone finds the time to wrap things up, this ongoing discussion is probably the best idea that we can offer as to what we think the nLab is or might be.
* [[Jon Awbrey]] added articles or began content development at [[differential logic]], [[differential propositional calculus]], and [[universe of discourse]].
* [[Urs Schreiber]]: wrote a long bit at [[higher category theory]]. Rearranged some existing material in the process.
* [[David Corfield]]: Seeing "comparative $\infty$-categoriology" there, does anyone have thoughts on Borisov's [work](http://uk.arxiv.org/abs/0909.2534)? Perhaps we need to wait for the sequel.
* [[Jon Awbrey]]: Expanding on a [note](http://golem.ph.utexas.edu/category/2009/09/towards_a_computeraided_system.html#c026573) and responding to a [query](http://golem.ph.utexas.edu/category/2009/09/towards_a_computeraided_system.html#c026593) on the blog, I proposed several sources as "Precursors" to category theory. I have discovered a truly marvelous demonstration of this proposition that [[category theory|this margin]] is too narrow to contain, but I might try to elaborate on it elsewhere … elsewhen.
* [[Toby Bartels]]: Talk, talk, talk: [[SEAR]], [[classical mechanics]], [[category theory]].
* [[Mike Shulman]]: Motivated by recent discussions on the cafe, created [[SEAR]], which has been kicking around in my head for quite some time.
* [[Jon Awbrey]]: A bit too close to SOAR. But if you put your terms in alphabetical order you'd have ERAS, which would also be mnemonic for the fact that elementhood is a relation. Hm³, is there such a thing as mnepic?
* [[Zoran ?koda]]: created [[compact-open topology]].
## 2009-09-15
* [[Zoran ?koda]]: changes to [[A-infinity ring]]. I had to disagree at [[classical mechanics]]; created [[force]]. Minor changes to few other entries, like [[analysis]], and to contents pages ([[geometry]], [[mathematicscontents]]).
* [[Urs Schreiber]]:
* added a paragraph to [[enriched category theory]] (you can't miss it, its the only genuine content there so far)
I am dreaming that eventually we'll make the enties [[enriched category theory]] and especially [[higher category theory]] have content as nice as we now have at [[category theory]]. At time of this writing the entry [[higher category theory]] is a shame, given the nature of our project here. But I am growing fond of at least the first, stabilized part of [[category theory]]. It would be cool if we could do similar deeds at other entries of similarly fundamental nature
* further expanded [[category theory - contents]] and added it as floating table of contents to more entries
(any volunteers for similar floating tables of contents such as [[topos theory - contents]], [[enriched category theory contents]]?)
* [[Jon Awbrey]] added a subsection on categorical precursors to the References Section of [[category theory]], but that's already such a huge article that people may prefer to spin off a new page if it goes past a few links.
* [[Urs Schreiber]]:
* created [[category theory - contents]] and began adding it as a "floating" table of contents to some relevant entries.
* following [this blog discussion](http://golem.ph.utexas.edu/category/2009/09/towards_a_computeraided_system.html#c026593) I added a paragraph "Terminology" to [[category theory]]
I also tried to incorporate the content of the query box that used to be in the paragraph on "Abstract Nonsense" by [[Zoran Skoda]] and [[Todd Trimble]]. I moved that discussion box to the bottom of the entry now. But if you think more discussion is needed, we'll revive it.
* [[Mike Shulman]]: Thanks Urs. I expanded [[2-pullback]] a bit. Weirdly, the direct URL for [weak pullback](http://ncatlab.org/nlab/show/weak+pullback) currently takes me to the *old* disambiguation page, which doesn't seem to exist any more. Ordinary links like [[weak pullback]] are correctly redirected to [[weak limit]] however.
[[Lab Elf|Lab Elf (financial department)]]: weak pullback was still cached so a direct call got the cached page without a check to see if there was a corresponding page in the database. I've removed the cached page and it looks like it's correct now.
* [[Urs Schreiber]]
* added some links to the entry on [[Peter May]]
* created [[stable homotopy theory - contents]] and added it as a "floating" table of contents to some relevant entries
* rewrote [[A-infinity ring]] and [[E-infinity ring]]
* created [[higher algebra - contents]] and added is as a "floating" table of contents to some relevant entries
* created [[higher category theory - contents]] and added it as a "floating" table of contents to some relevant entries
* moved to [[monoidal (infinity,1)-category]] material that had long been sitting at [[geometric infinity-function theory]] on operadic definitions of higher monoidal structures
* [[Zoran Skoda]] alerts me that he has added corrections and caveats related to [[formal scheme]] and [[formal spectrum]] at [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* reacted to [[Mike Shulman]]'s remarks at [[graph of a functor]], [[cograph of a functor]] and at what used to be [[weak limit]] and is now renamed [[2-limit]] by trying to implement the suggested changes. Mike should please have a look and check if it looks better now. I have to admit that I had forgotten that "weak limit" meant something else than [[2-limit]] or other higher limits.
* added the link to [[stable homotopy theory]] to the main [[mathematicscontents|Mathematics Contents]]
* created [[coordinate-free spectrum]] and linked to it from [[spectrum]] (whose "Definition" section I slightly reorganized)
* created [[symmetric monoidal smash product of spectra]]
added small paragraph pointing to this new entry to [[stable homotopy theory]], [[smash product of spectra]]
* created an entry [[inbound citations]] and linked to it from the main table of contents. Let me know what you think of this suggestion. See also the blog entry on it [here](http://golem.ph.utexas.edu/category/2009/09/inbound_citations.html).
* [[Jon Awbrey]] added a [link to a Forum discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=71) at [[relation theory]].
* [[Mike Shulman]]: Two questions at [[graph of a functor]]. One of them spills over into [[cograph of a functor]] and [[weak pullback]].
## 2009-09-14
* [[Urs Schreiber]] slightly rephrased at [[formal spectrum]]: properly speaking that limit over rings gives the global sections of the structure sheaf, of course, not the structure sheaf itself
* [[Jon Awbrey]] replies to a query at [[relation theory]]. Hey! that rhymes with "weak and weary".
* [[Zoran ?koda]]: to support the development of entries on the basics of [[formal geometry]] created entry [[adic noetherian ring]] (I would like to warn Toby (who will likely like simpler generality) in advance that one needs to be very careful in treating adic situations without noetherianess, I do not feel competent to write without consulting literature such a more general entry). Thanks [[Urs]] for stimulating the expansion of the entries on the subject.
* [[Urs Schreiber]]:
* from more seminar notes filled in more material at
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
and started
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]] (not complete yet).
* many thanks to [[Zoran Skoda]] for fixing some nonsense that I committed in the "Idea" section at [[formal scheme]]
* [[Jon Awbrey]] added some stuff about finite dimensional boolean coordinate systems to [[boolean-valued function]].
* [[Toby Bartels]]: A little more explanation of terminological variations at [[direct limit]] and [[inverse limit]]; also [[projective limit]] and [[inductive limit]].
* [[Jon Awbrey]] added an epigraph to [[relation theory]]. Exercise for the Reader: Show that every functional graph factors into an epigraph and a monograph.
* [[Urs Schreiber]]:
* added a reference to Peter May et al.'s survey article to [[stable homotopy theory]] -- also added a link back to [[stable homotopy category]]
* filled in formal definition at [[elliptic curve]], also that of the corresponding formal group law and some examples -- but needs polishing/expanding
* filled in content at [[formal spectrum]]
* created [[Landweber exactness criterion]]
* created [[weakly periodic cohomology theory]]
* since I see entries pointing to "direct limit" and "inverse limit" I created entries [[direct limit]] and [[inverse limit]]. I know that we could just redirect to [[limit]] and [[colimit]] where the terminology is discussed, but maybe the reader following such a link will appreciated being quickly alerted to the terminological issue before being faced with a long entry on limits and colimits where this is hidden as a remark somewhere
* slightly edited [[formal scheme]]: added an "Idea" sentence and highlighted the definition of _formal spectrum_ a bit. In fact, created [[formal spectrum]] planning to have the definition there, but then hesitated. Maybe this needs someone more expert than me.
* stub for [[structure sheaf]]
* slightly edited the beginning of [[structured (infinity,1)-topos]]
* [[Toby Bartels]]:
* Turned [[set]] into an essay (*Was sind und was sollen die Mengen?*) on the issues involved, particularly trying to keep them (foundational style, smallness, skeletality) separate.
* Thanks, Urs!
* [[Urs Schreiber]]:
* created [[adjoint (∞,1)-functor]]
the essential information had before already been in the corresponding section at [[adjoint functor]], to which I added the relevant link.
* joined the discussion at [[graph of a function]]. I am thinking that this is best understood as a special case of a more general concept, for wich I created now entries
* [[graph of a functor]]
* [[cograph of a functor]]
This has some nice applications. For instance one has that two functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if they have the same cograph, up to reversal of arrows. Notice that, because the notion of cograph immediately generalizes to functors between higher categories, this is the basis for a definition of [[adjoint (infinity,1)-functor]].
I have expanded the material at [[graph of a function]] accordingly. Also I made [[cograph of a function]] redirect to that.
## 2009-09-13
* [[Toby Bartels]]: Not done at [[graph of a function]], Eric!
* [[Jon Awbrey]] expunged an assortment of ephemeral animadversions at [[graph]].
* [[Eric]]: After some discussion with [[Toby]], defined [[graph of a function]] as the [[category of elements]] of $F:\mathbf{2}\to Set$.
* [[Jon Awbrey]] added at comment to the discussion section at [[relation theory]].
* [[Toby Bartels]]: Added [[diagonal morphism]] and two ways to look at it in $Set$: [[diagonal subset]] and [[diagonal function]].
* [[Jon Awbrey]] added a bibliography on relations and related subjects to [[relation theory]].
## 2009-09-12
* [[Toby Bartels]]:
* Rejoined the discussion at [[category theory]].
* Removed the last links to [category: lexicon](http://ncatlab.org/nlab/list/lexicon) (now all listed at [[differential graded objects - contents]])
* Arguing with Mike at [[Categories Work]].
* A fact added to [[line bundle]]; we should still write [[line]].
* A very stubby [[classical mechanics]], grown out [[classical physics]].
* Generalised [[internal relation]] somewhat.
* [[Jon Awbrey]]:
* Finds his time too intermittent this weekend to do more than potter about in the $n$-garden, but was pleased to discover how to write the old Pascal "set-equal-to" as "$\:\text{:=}\:$" in a math context.
* Replies to a query at [[relation theory]].
* [[Toby Bartels]]:
* Added a bit to [[tree]].
* Answered questions from [[Eric Forgy]] at each of the below.
* Wrote [[graph of a function]] (split off from [[relation theory]]).
* Redid [[graph]]; please give feedback.
## 2009-09-11
* [[Andrew Stacey]] there's a [discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=64) going on at the forum about designing a better system for recording these latest changes. If you have an opinion, please contribute! At the moment, it's going on what Toby, Mike, and I think which may not be a representable sample.
Also, there was a brief glitch in the system that led to entities begin translated into their unicode counterparts (don't worry if that doesn't make sense). Unfortunately, this wasn't compatible with iTeX and there may be a few 'Unknown character's lurking around. If you spot one, [let me know](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=65) and I'll go catch it with my butterfly net. (It's important to let me know rather than just correcting it yourself as it _really_ messes up the `diff`s so I need to fix it properly rather than just papering over the cracks.)
* [[Toby Bartels]]:
* A bit at [[lax natural transformation]], which I really only mention since it\'s been discussed so much lately.
* Since Recently Revised works again, I\'ve restored the link to it up above, from what it had been in [June](http://ncatlab.org/nlab/revision/2009+June+changes/947).
* [[Zoran ?koda]]: Urs seems to take [[derived algebraic geometry]] (see my answer/note there) as a higher algebraic stacks, and forgets deriving on the other side. Nonabelian cohomology results from right derived picture (quotients = colimits), and the missing part is to take the limits in derived sense, that is taking equalizers, intersections of subschemes etc. in derived sense as well. We should discuss that, replace the paragraph with the better one and after agreeing and explaining, erase the critical paragraph. In another paragraph the things are in place:
> Where ordinary algebraic geometry uses schemes modeled on commutative rings, derived algebraic geometry uses structured (∞,1)-toposes modeled on E-∞ rings
Indeed, the higher stacks are about the (∞,1)-toposes, while the derived stacks ask also for the domain to be E-∞ rings. The "brave new algebraic geometry" on the other hand typically takes the second (alg geometry glued from spectra of infinity ring spectra), but not the first (higher stacks instead of schemes).
* [[Toby Bartels]]:
* Agreement is being reached at [[k-transfor]].
* Wrote [[2-functor]] as a portal to more precise definitions; also a little at [[semi-strict infinity-category]].
* [[Mike Shulman]]: For those who aren't reading at [[(n,k)-transformation]], the proposal is to replace that unlovely term with Sjoerd Crans' word **k-transfor** (so 0-transfor = functor, 1-transfor = transformation, 2-transfor = modification, etc.). Please comment!
* [[Zoran ?koda]]: stub [[derivator]] including few lines from [[triangulated category]]; maybe more discussion from there should be moved to [[derivator]] and just left a short notice at triangulated category on derivators (it seems in fact one wanted to talk about _triangulated derivators_!!), because t.cat. are a wide topic and the entry may expand in many different directions, while the motivation and the discussion may be useful at derivator. but have no time to decide and think of what is sensitive. Somebody should copy the axioms from Maltsionitis' notes for derivator.
* [[Jon Awbrey]] began watering his trans-plants at [[cactus language]] --- bit by bit, you have to be very incre-mental with cacti --- and started a [parallel (tangential?) discussion](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=69) at the $n$-forum.
* [[Zoran ?koda]]: additions to [[deformation theory]], [[derived algebraic geometry]]; created [[cotangent complex]].
* [[Toby Bartels]]: I\'m perfectly serious at [[(n,k)-transformation]].
## 2009-09-10
* [[Mike Shulman]]:
* Started trying to incorporate the results of the discussion into the entry at [[lax natural transformation]]. (Thanks very much -- _Todd_)
* Comment on alternative terminology at [[(n,k)-transformation]]
* [[Todd Trimble]]: commented at [[lax natural transformation]], and suggested a possible compromise at [[graph theory]] which I hope will be considered satisfying to all concerned. Thanks to Mike for creating [[icon]].
* Created a stub for [[elliptic curve]], in response to Urs.
* [[Urs Schreiber]] slightly edited [[A Survey of Elliptic Cohomology - formal groups and cohomology]] and added plenty of links
* [[Toby Bartels]]: More at [[graph]] and [[center]], since there is no [RSS feed](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=64).
* [[Jon Awbrey]] is leaving the fray at _The World According To [[graph|Graph]]_ --- which is clearly becoming more "productive" than "creative" --- and suggests, as an interim measure, that he be allowed to store a few standard definitions at [[graph theory]].
* [[Mike Shulman]]: Replied at [[center]] and [[lax natural transformation]], and created [[icon]].
* [[Toby Bartels]]: Thanks to David below! I have moved [[barycentric algebra]] to [[convex space]] after being more sure that this name is not already taken (but 'convex set' does conflict). Since the concept has been invented many times and has many names, let\'s use our own, which is a nice one. I forgot about the blog discussion before; I had a nagging feeling that this had come up once and I hadn\'t gotten around to commenting then, so I\'m glad that David remembered.
* [[Ryan Grady]] has given us [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[Todd Trimble]]: Finally got around to replying to [[Mike Shulman]] over at [[lax natural transformation]], with some responses of Ross Street and Steve Lack.
* [[Urs Schreiber]] started adding to [[infinitesimal object]] a discussion for how one can understand Lawvere's abstract definition intuitively as encoding infinitesimal extension. But am in a hurry and have to leave it unpolished for the moment.
* [[Toby Bartels]]: Replies at [[barycentric algebra]], [[evil]], [[graph]], [[boolean domain]], [[center]].
* [[David Corfield]]: convex spaces and barycentric algebras cropped up [here](http://golem.ph.utexas.edu/category/2009/04/convex_spaces.html#c023763). I don't know if there's anything useful there.
* [[Jon Awbrey]]: Further discussion and some data at [[graph]].
* [[Mike Shulman]]: Expanded a bunch at [[center]], and continued discussions at [[graph]] and [[evil]].
## 2009-09-09
* [[Toby Bartels]]: More commentary at [[boolean domain]] and [[graph]].
* [[Todd Trimble]]: added still more to the long discussion at the bottom of [[graph]].
* [[Toby Bartels]]: Created [[barycentric algebra]], mostly to explain the stuff about convex sets in the discussion at [[semicartesian monoidal category]].
* [[Urs Schreiber]]: motivated by [[Toby Bartels|Toby]]'s comment there I have now branched off the material at [[limits and colimits by example]] that describes a computer program to [[Paine on a Category Theory Demonstrations program]]. This is now linked to there from the section "Further resources". I have moved the pertaining discussion boxes to the bottom of the entry.
* [[Jon Awbrey]]: Piecewise discussion at [[boolean domain]], but it will be later before I can get to the rest of the pieces.
* [[Toby Bartels]]: Discussions at [[center]], [[evil]], [[graph]], [[boolean domain]], [[limits and colimits by example]], [[semicartesian monoidal category]].
* [[Mike Shulman]]:
* I have a different opinion about the [[center]] of a set.
* Discussions at [[evil]], [[graph]], and [[boolean domain]].
* [[Jon Awbrey]]: Continuing discussion at [[graph]]. I won't try to note each entry here, unless that's the rule.
* [[Jon Awbrey]]: I added an Idea section to [[relation theory]] with what I can recall of how I got into that. Incidentally, the x-tended code for the amphora symbol (@) now causes Ruby to go off the Rails, so I had to use \text{@} instead.
* [[Urs Schreiber]]: thanks, once more, to [[Todd Trimble|Todd]], for the discussion of monadicity et al at [[limits and colimits by example]]. We should eventually brach that kind of discussion off into an entry in its own right and expand
* [[Jon Awbrey]] entered the fray of discussion at [[graph]].
* [[Todd Trimble]]: Per Urs's request below, I checked the limits in under-category he wrote, and added a remark at the end.
* [[Zoran ?koda]]: last night my battery bailed out when I was editing several entries so did not log what I have done. Now I do not recall all entries which I updated, except that I created [[deformation theory]] (so far only references and links), expanded [[derived algebraic geometry]], added a reference to [[formal group]] and to [[quasicoherent sheaves]]. I thank lab elf for cograts for my entry 2000 which was however [[Grothendieck connection]] where I also made a small change last night (when saying $n$-costratification, $n$ refers to work with $n$-th infinitesimal neighborhood and not with $n$-categorical descent: the descent data were in Grothendieck 1-descent data). Today we have an excursion during the conference in Sevilla, so I will probably not be able to continue (have much to add to [[deformation theory]] etc.).
* [[Urs Schreiber]]
* added a section on limits in under categories to [[limits and colimits by example]] and a detailed proof of how they are computed by limits in the underlying category -- please check
* added a linked table of contents at [[limits and colimits by example]]
* added a query box at [[limits and colimits by example]] in the section titled "for programmers". I am suggesting that since the material doesn't actually use a computer program to explain limits and colimits but instead explains how to write such a computer program, the material should be moved elsewhere.
* added an "Idea" section to [[n-localic (infinity,1)-topos]]
* slightly expanded the remark under "Generalizations" at [[localic topos]], trying to indicate the pattern
* [[Urs Schreiber]]: re Toby's remark below:
yes, true, the $(\infty,1)$-topos material is "Grothendieck-Rezk-Lurie $(\infty,1)$-toposes of $(\infty,1)$-sheaves". If it weren't so cumbersome to say this, one should make this explicit more often. But maybe we should highlight it at least more at beginning of entries.
Another thing I noticed that maybe requires more emphasis is that the $\infty$-stack-$(\infty,1)$-toposes that one gets from the standard [[models for infinity-stack (infinity,1)-toposes]] are far from being the generic case. Many Grothendieck-Rezk-Lurie $(\infty,1)$-toposes are not equivalent to these. (Compare Lurie's discussion of [[topological localization]], [[hypercompletion]], [[n-localic (infinity,1)-topos]]es, etc.)
* [[Toby Bartels]]: I moved [[free field (algebra)]] to [[free field]] on the grounds that (as with [[field]] itself) the algebraic mathematical meaning is likely to be our default here. But I might be wrong. In a similar vein, I suggested [[physical field]] at [[field]].
* [[Jon Awbrey]] made some attempt to reorganize the discussion at [[boolean domain]]. This is one of my main stepping stones, so I'll need to keep the algæ at a minimum.
* [[Toby Bartels]]:
* The centre of a set at [[center]].
* The Hahn--Banach Theorem for separable spaces at [[locally convex space]].
* A note about the meanings of 'convex set' at [[semicartesian monoidal category]].
* Physics notation at [[Weyl algebra]].
## 2009-09-08
* [[Jon Awbrey]] finished up the basic definitions and expository examples at [[sign relation]].
* [[Toby Bartels]]: I do enjoy it, Urs! But remember that, for Lurie, an '$\infty$-topos' is not only an $(\infty,1)$-topos but in fact a *Grothendieck* $(\infty,1)$-topos. So when he says that a '$0$-topos' is a locale, he similarly means that a *Grothendieck* $(0,1)$-topos is a locale. (An *elementary* $(0,1)$-topos is a [[Heyting algebra]].) I also wrote [[(0,1)-category]], since you linked it; that\'s a p(r)oset.
* [[Urs Schreiber]]
* quick unformatted content filled in at [[n-localic (infinity,1)-topos]], to be continued later, have to run now...
* created stub for [[(0,1)-topos]], linked to it from [[locale]] with a small comment -- Toby might enjoy that -- this mainly to remind me to extract the essentce from section 6.4.2 in [[Higher Topos Theory|HTT]] later. Will also create [[(n,1)-topos]] then
* thanks to Todd for expanding at [[localic topos]]!
* [[Toby Bartels]]: More at [[evil]]; we might actually be working out some mathematical facts here before too long!
* [[Jon Awbrey]]: I wouldn't _count_ on it — cuz, y'know, that might be _evil_.
* [[Todd Trimble]] added some more to [[localic topos]].
* [[Jon Awbrey]] added a bit more content to [[sign relation]] and then broke for lunch. ($n$-tweet?)
* [[Urs Schreiber]]
* created stubs for [[localic topos]] and [[n-localic (∞,1)-topos]]
* added to [[Deligne-Mumford stack]] the alternative characterization as a 1-localic $G$-[[generalized scheme]] for $G$ the _etale geometry_ (defined there). Also added a brief "Idea"-section
* created [[affine scheme]]
* created _topic cluster floating table of contents_ [[(infinity,1)-topos - contents]] and included it in the linked entries
* created [[object classifier]] and linked to it from [[(∞,1)-topos]] and [[Higher Topos Theory]] and [[subobject classifier]]
* slightly reorganized and then expanded the _topic cluster_ floating table of contents [[cohomology - contents]]. in that context I
* created [[orientation]], [[Spin structure]], [[Fivebrane structure]] and slightly expanded [[String structure]]
* briefly discussed/mentioned these in the examples section at [[quantum anomaly]]
* and on that occasion created an entry for [[Ulrich Bunke]] whose recent article (yesterday!) I cite above, where he makes the old Killingback argument about how a String-structure makes the worldvolume anomaly of the heterotic string vanish rigorous
* corrected the mistake at [[rational topological space]] that [[David Corfield]] spotted below
* added to [[CW complex]] link to [[geometric realization]]
* started [[rational topological space]]
* [[David Corfield]]: Asked question there.
* [[Jon Awbrey]] added content to [[semiotic equivalence relation]].
* [[Mike Shulman]]:
* More discussion at [[evil]].
* Started some work on clarifying definitions at [[graph]].
* Spoke up in defense of the adverb at [[locally presentable category]].
* [[Jon Awbrey]]: Clarifying at [[graph theory]] and questying at [[evil]].
## 2009-09-07
* [[Toby Bartels]]: Replies to Mike at [[evil]] and [[graph]].
* [[Mike Shulman]]:
* Started a discussion -- at [[graph]] -- about what may be wrong with that page and how to fix it. I think that a page called [[graph theory]] should be more like the pages called [[topology]] and [[category theory]].
* Comments at [[evil]].
* [[Urs Schreiber]]:
* missed one train. Used the few minutes gained this way to quickly extract the following very brief entries from the notes mentioned below: [[multiplicative cohomology theory]], [[even cohomology theory]], [[periodic cohomology theory]], [[Bott element]] -- all these entries deserve to be greatly expanded, it particular by eyxamples, of course, but it should be a start
* started turning some talk notes of a seminar into entries, but requires polishing:
the full raw material is at [[A Survey of Elliptic Cohomology - cohomology theories]]. Using this I have so far split off [[cohomology theory]], [[Lazard ring]], [[complex cobordism cohomology theory]] and [[reduced cohomology]]
* wrote a somewhat revisionist "Idea" section at [[Grothendieck connection]]. Notice that it is in particular the Breen-Messing reference cited and discussed at [[infinitesimal singular simplicial complex]] that allows me to do that! (meaning: I am not making this up, but just putting the pieces together)
* [[Jon Awbrey]] added a page on [[graph theory]]. The page on [[graph]] has become too baroque to fix, but there needs to be a place to record the basic definitions of graph theory that are actually used by the larger schools of math folks who actually dare to call themselves graph theorists. This is stuff that is more in the back of my mind than the forefront of my attention, so I'll add to it over time as other work recurs to it.
* [[Rafael Borowiecki]]: Replied Todd Trimbles question at [[Timeline of category theory and related mathematics]].
* [[Todd Trimble|Todd]] replied back.
* [[Urs Schreiber]] started keyword-list entry [[Structured Spaces]] and in that context also started [[generalized scheme]]
* [[Andrew Stacey|Lab Elf (numerological department)]]: Just thought I'd congratulate Zoran on creating the 2000th page on the nLab. According to the database, it was [[smooth morphism of schemes]]. For those who prefer other coincidences, [[skewfield]] was our 2009th. Of course, in this more modern age we ought to really celebrate [[coherent sheaf]] instead. Whichever we celebrated, it would be Zoran that would get the metaphorical bottle of champagne as he created the lot.
* [[Urs Schreiber]]
* changed the coding of the floating table of contents [[cohomology - contents]] according to the recent discussion on the forum and added a few recent entries to it
* slightly expanded [[integral cohomology]]
* added a web reference by [[Nora Ganter]] (created stub fo that) on [[topological modular form]]s (created stub for that) kindly pointed out by [[David Roberts]] to [[A Survey of Elliptic Cohomology]] and to [[tmf]]
* it seems that yesterday I also created [[stable homotopy theory]]
* [[Jon Awbrey]] added an epigraph to [[evil]].
## 2009-09-06
* [[Mike Shulman]]:
* Created [[single-sorted definition of a category]] and [[collection]]. Afterwards I had the thought that perhaps the former should be called something more like "category (single-sorted definition)" -- any thoughts? Do we have a convention for this sort of thing yet?
* Asked a question at [[Categories Work]].
* [[Todd Trimble]]: responded to something Rafael wrote at [[category theory]], and asked a question at the bottom of the timeline page.
* [[Urs Schreiber]]
* created [[derived algebraic geometry]]
* wrote a bit of summary at [[A Survey of Elliptic Cohomology]]
* started [[elliptic cohomology]]
so far with quick stubs for [[line bundle]], [[cohomology ring]]
* expanded the content at [[Jacob Lurie]], fixed the links and added [[A Survey of Elliptic Cohomology]]
* merged the former material at [[∞-topos]] with that at [[higher topos theory]] and expanded and slightly rearranged the latter.
* coded the "floating tables of contents" for the topic clusters at [[mathematics]] and [[rational homotopy theory]] according to [[Andrew Stacey]]'s suggestion on the nForum -- see [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=61&page=1#Comment_743)
* [[Rafael Borowiecki]]:
* Replied at [[category theory]].
* Wrote a little more on [[extended topological quantum field theory]] but don't know the right references. Put that request in a query box.
* Debate at the timeline.
## 2009-09-05
* Note spelling: [[Saunders Mac Lane]].
* [[Zoran ?koda]] inserted and reminded of work of Rezk and Toën-Vezzosi on [[higher topos theory]] preceding the marvelous [[Higher Topos Theory|Lurie's work]].
* [[Andrew Stacey]]: without wishing to join in the fascinating debate as to whether the timeline should be kept in sync with Wikipedia, I'd just like to expand on my comment about asking the question on the forum. The specific question on timeline is technical in nature ("how do I deal with 1500 links?") and therefore most likely to be answered by one of the more technically minded people here. Some of those people do check every page on the lab for every revision, but others don't. However, they do check this page and they tend to check the forum as well. The best way to get your question seen is to put a brief note here and link it to a discussion on the forum. Also, the more detailed and precise you can make your question, the easier it is to understand and to answer and therefore the more likely it is to get an answer.
* [[Urs Schreiber]]
* created a stub entry for [[Samuel Eilenberg]] and put hyperlinks under the names of Eilenberg and MacLane at [[category theory]].
* expanded at [[infinity-stack]] the first paragraph of the "Definition" section and added a link to [[sheaf of n-types]].
* created [[sheaf of n-types]] which was requested in the "Timeline" entry. But I made this essentially just a commented redirect to [[∞-stack]], because it's just another word for that.
* expanded the introduction of [[AKSZ theory]] a little, added the original reference and linked Kontsevich's name there to the new entry on [[Maxim Kontsevich]] and added a paragraph that tries to briefly put this in context with related existing nLab entries. But the entry is still missing a discussion of its subject itself. I have some old blog material on this, but this deserves more spare time than I have at the moment.
* added links back and forth between the new [[higher topos theory]] and the old [[∞-topos]]. Probably some reorganization of the material over these two entries would eventually be reasonable.
* added plenty of hyperlinks to the entry on [[Maxim Kontsevich]] -- many of them point to existing nLab entries, many others point to nLab entries still to be created (and lots of them would be highly desireable)
* [[Toby Bartels]]: Created a stub [[higher topos theory]] (not to be confused with [[Higher Topos Theory]]).
* [[Benoit Jubin]]: asked a question at [[monoidal category]] about the necessity of requiring $\lambda_1=\rho_1$.
* [[Todd Trimble]] responded.
## 2009-09-04
* [[Tim Porter]]: On the issue of stacks, Deligne and Mumford explicitly (p. 97 of their famous paper) use the term 'stack' as an English translation of 'Champ' and attribute that to Giraud in _Cohomologie non-abelienne_. This latter source was published later than Deligne and Mumford's paper but is refered to by them as 'University of Columbia'. Giraud was a student of Grothendieck.
* [[Zoran ?koda]]: surely Grothendieck invented stacks in general categorical sense and had a picture in various setups; Deligne and Mumford did invent a particular kind of [[algebraic stack]] and tailored it toward a very specific application.
* [[Rafael Borowiecki]]:
* I was before talking about edit 143 of the timeline by Toby Bartels. I am now trying to figure out what it changed. The other edits i understand.
* Added a query box at [[Timeline of category theory and related mathematics]] to hear what others think about the first entry in the timeline regarding Cayleys paper.
* I must correct you Zoran at a point. I never said the entry must look good but i want the timeline to look good. This means links, easy to read, no stuff that should not be there and it should of course be correct. Those who know the years,names and category theory would see if it was not correct and say it looks bad.
* Since we are on the subject history. I Credited Deligne-Mumford for inventing stacks. But i recall rumours that it was Grothendieck that invented stacks, without any references.
* [[Zoran ?koda]]: added redirects and a reference to [[quasicoherent sheaf]]; expanded [[representation theory]]; created [[EGA]] (just an introduction to the entry, links and toc missing); created [[orbit]] wanted by [[coadjoint orbit]]; more importantly for the present needs, wrote [[Grothendieck connection]] (entry 2000 :)), required both by the timeline and by the current interests of the project pushed by Urs which I try to help and discuss; and it also refers to some things I was many time mentioning (and even to some ideas behind my preprint on cyclic comonads and related papers by Menini-Stefan and Böhm-Stefan). Wrote [[Poisson manifold]] and [[coadjoint orbit]] (unfinished).
No, Rafael, I was not trying to find oldest instance of category theory. But Hilbert's first entry and also things about Whitehead etc. nonabelian invariants require the earlier appearances of such ideas if they existed. Look, the main theorem of Cayley in modern language says that the resultant of polynomials is a determinant of certain Koszul complex. As far as Galois, I think it should absolutely be in timeline, definitely, not only because of the notion of the group, but in fact Galois theory itself is in a spirit and a stimulus of much of the modern category theory -- torsors, Grothendieck Galois theory, equivariant descent, Galois descent, Joyal-Tierney, Hopf-Galois to mention a few (On the other hand, the notion of an abelian group is essentially in Gauss' Disquisitiones Arithmeticae in pretty clean form, according to my friends who read it carefully; I can not judge). The Klein's Erlangen program is a related ideological item, but more disputable. I still do not understand what do you mean by that moving again the wikipedia and overwriting $n$lab version depends on ability to do links. If you change links to wikipedia somehow automatically to the format which finds the true wikipedia pages this will be good for most of the items, but wrong for those few items where we already do have nlab entries. So it is a problem of selection.
* [[Rafael Borowiecki]]: When i was writing i once by accident updated the timeline at wikipedia and undid the change. If you look at it from the history you will see that i have corrected some of what you wrote and i will with interest look at the rest. I find the dates usually on internet which mean they could be wrong. I did not include Poincares Analysis situs for progress on group theory but topology which is one of the related themes of the timeline. As for Cayleys paper i would like to hear what others think. But it looks to me you tried to find the earliest possible instance of category theory. Then Galois theory is earlier. I think of it of course as a precursor to modern Galois theory but did not include it.
* [[Toby Bartels]]: I have some opinions on the [[Timeline of category theory and related mathematics]], which I will put *there* where they belong.
* [[Zoran Skoda]]: created an entry on [[Maxim Kontsevich]] as his name is quoted in timeline and some other places in nlab.
* [[Zoran Skoda]]: The same way one could say that Poincare's papers done nothing on group theory. He does many things about groups completely in the language of manifolds, but in fact he proves the theorems on fundamental group, and these were transferred to group theory later. The homological algebra of Hilbert is equally linear algebra as is homological algebra of Cayley. Cohomologies in different language were used extensively by early Italian school of algebraic geometry; ask Japanese algebraic geometer Mukai to explain the 1899 paper (I think) of Castelnuovo with Castelnuovo's arguments literally, just with modern names for the quantities (it is all about cohomologies). I do not know why do you care that the entry LOOKS GOOD. I think it would be more important to be CORRECT, I listed once several instances of suspicious dates, and nobody cares about what is more important than syncing. To add new ones it is NOT true that localization is a special case of descent as it says in 1960 entry on descent within wikipedia timeline version; surely localization and descent can be combined in very nice ways, or sometimes phrased in the same language but the descent is descent and the localization is localization. Well descent data and the localized category can be both made comodules or modules over some comonad or a monad but this is a different issue and neither is special case of another. Rosenberg's proof of the reconstruction of scheme is not written first time in the 1998 paperon nc schemes, that paper has just appendix with a SKETCH of the proof which is in its full version in his earlier 1996 Max Planck Bonn preprint ([pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948)), some form of which is published in another journal also in 1998, with submission January 1997; in any case it is a different paper than [[noncommutative scheme]] paper. Similarly,
1960 date for Grothendieck's [[formal schemes]] is wrong, as the Bourbaki seminar paper for 1958/1959 is having a full article with already deep results (like Grothendieck existence theorem) on the subject. Look bibliography to [[formal scheme]]. 1960 for descent theory may also be too late though I am not sure at the moment, I am almost sure that one of the parts of FGA for descent has been published before 1960; just look for FGA (probably in Vistoli's survey you can find the year for the reference). For triangulated categories one should list (unlike wikipedia version of timeline) not only Verdier but also Grothendieck who was the true discoverer (and gave the list of chapters and main theorems to Verdier to prove and write up,a s he was usually doing with his students). You see these complaints I wrote only after 2 minutes of looking superficially at several lines of timeline. The timeline needs tens of such corrections historically and not the syncing,colors and look.
* [[Rafael Borowiecki]]: Zoran, the migration will ultimately depend on if i am able to do the linking. I could completely not care about the nLab timeline but i want it to look good. The long entries will be as short/long as they are now. I will try not to loose anything and discuss the entries i would remove. [AST](http://ncatlab.org/nlab/show/algebraic+set+theory) alteady exist as do pages for the other long entries. Regarding Cayley i have seen the paper long before and have it on my computer. Cayley calculates invariant theory in coordinates. The paper do not define any categorical concept or prove a categorical theorem. Nor do it introduce any method used later in category theory, it is just polynomial algebra. I would say it don't start homological algebra. You could put it as that it anticipated homological algebra but i don't think Cayley thought about this paper that abstractly.
* [[Zoran ?koda]]: Rafael, when you opened timeline in nlab you did NOT tell us, the conditions/plan of syncing, which is fundamentally incompatible with nlab and wikipedia as each of them has limitations AND advantages. For example, we like book entries in nlab; timeline has some entries in the tabel very short, some hugely long: e.g. the wikipedia has huge entry of about alf a page on a book of Joyal and Moerdijk on algebraic sets. Why not have a separate entry in nlab for that book with all the material and in nlab entry for that book just say book algebraic sets, yeas and that's it. Second new migration would take tens of hours of time to make links compatible with entries in nlab, some of which can not be automatic. How Cayley's paper benefited category theory ? Jee, you have tens of etries on homological algebra; including the Hilbert's which are about the SAME stuff, just much later. How do you expect a collaboration on an entry if you are going to just decide out of your taste what is important ? Even if we talk about papers which wree anticipating developments in other works by over half a century ? You are concerned about syncing and difficulties. There is a wikipedia and there is nlab. When it is easy to borrow and coopearte why not. When it is difficult and creates problem just the heck with it, let's develop aturally two versions. No sync...
* [[Rafael Borowiecki]]: Replied at the timeline of category theory. Zoran, my original plan was to have the timelines synced to optimize both timelines and i will try to do so. I will also try to have good links. Right now the wikipedia timeline contains almost all entries and information in the nLab timeline and very much more. So at the moment it would be nLab that benefits most. I will look into your changes once again and try to keep them. It is easier to only copy the links than to find them. You have not seen the new version but, as for Cayley i just don't see how his calculation benefited category theory so i removed it, but this is a discussion for the timeline page. I also recall removing the deRham theorem since it don't really fit with the structure of the timeline. I have not included dualities. But this one i can change back since it was a very important discovery about cohomology and there are no rules to follow any structure. I will check more now. Then i wish people added so much to the timelines it would be impractical to sync them. Now that nLab is fast enough it should go much better to edit the links.
* [[Zoran ?koda]]: created [[internal relation]]. I was once working on replacing tens (and spend hours on this) of wikipedia links in timeline with our own links, and even created new entries in nlab to support the new links in nlab. For example I created the entry for [[Otto Schreier]] in nlab to support the quotation for Schreier in nlab version of Timeline. I also do not understand why things like the Cayley 1843 entry are ignored (and hence will disappear in new migration); I spent hours of time looking into references which I recalled vaguely to confirm what I thought about it. I see no reason for wikipedia to overwrite our work on changes. 80 links changed in single day is possible done by me; I recall that I did work once on making many links either more functional (including making the blind link to wikipedia link) or update them with new biography entries in nlab. I would not say that in that particular day only the links changed.
* [[Jon Awbrey]] made a first pass at formatting [[Trimble on ETCS III]] and thinks to have earned himself a nice Labor Day vacation, so if it's messed up don't tell him till Tuesday.
* [[Rafael Borowiecki]]: Urs, i have taken care of that. I first updated the wikipedia timeline to match the nLab timeline. But a discussion might be at place since i have removed some of Zoran Škoda's entries. I was not able to see and do one change thought, there is one revision that changes something in most of the 80 first entries or so. Maybe only links. I only took a quick look, and i am now thinking about the link problem. I found nLabs way of comparing revisions often very hard to read. I think it colors much more than need to be.
* [[Zoran ?koda]]: oops, the link to Cartan Seminar at numdam was one char too long at Timeline; sorry. Regarding that the nlab has both different rules, format, link capacity, side resources etc. than wikipedia, that after so many updates are done in nlab version on our side, it woudl make no sense tooverwrite the nlab version of timeline with a new migration of wikipedia. One can update some particular items, but migrating it as a whole would mean relinking the part which is already relinked in nlab version. So, once the original bulk of timeline has once being moved the two timeline entries may live their separate lives and occasionally some individual additions or links could be manually added to borrow from nlab to wiki or other way around, by the criteria and will of nlab contributors while copying news to nlab and wikipedia contributors while copying news to wikipedia. It is too late to do bulk migration again; and wikipedia while better in some items, the nlab is better in some other additions to timeline which never made into wikipedia. Or I misunderstood something. I see no purpose into maintaining the two versions mutually equivalent; who wants to see another version can do this by clicking. The cooperation is rather to start the bulk and then to grow any way it likes.
* [[Urs Schreiber]] -- question to [[Rafael Borowiecki]]: below you write
> The same timeline at wikipedia has/will have soon about 1500 links. When i migrate it to nLab again, and better than the last time, i need to change the links
What do you mean by "migrate"? The nLab version of the timeline has now many entries and edits that don't seem to be reflected in the Wikipedia version. No?
* [[Rafael Borowiecki]]: Replied to Zoran's question.
* [[Zoran ?koda]]: put a query into the [[Timeline of category theory and related mathematics|Timeline]] entry: at numdam one can find the Cartan Sem from 1948, but I can not find there the write up of sheaf theory those some related notions in non-sheaf language can be easily found. Am I blind ? [numdam 1948 Cartan Sem](http://www.numdam.org/numdam-bin/feuilleter?id=SHC_1948-1949__1_). Timeline claims 1948 WRITE-UP. So where it is ?
* [[Urs Schreiber]]
* slightly expanded [[Lie algebroid]] and added several links to entries that didn't exist back when this was created but do exist now
in this context I made [[Atiyah Lie algebroid]] redirect to [[Atiyah Lie groupoid]]
* created [[tangent Lie algebroid]]
* [[Rafael Borowiecki]]:
* The [[Timeline of category theory and related mathematics]] is repaired thanks to Andrew Stacey. See the new question there about updating links in a new migration.
* Andrew, i am glad that you are interested in my big problem. But i don't really understand your question. Is it not enough to ask the question at the timeline page and say here that i have added a problem to solve to the page? Perhaps the question at the timeline was not clear. The same timeline at wikipedia has/will have soon about 1500 links. When i migrate it to nLab again, and better than the last time, i need to change the links to link to correct pages at nLab. It is only some of the names that need to be changed to link to correct nLabs pages, which often have different names than those in wikipedia.
* [[Zoran ?koda]]: created [[fibration of points]] following [[Borceux-Bourn]].
* [[Andrew Stacey]]: I have implemented all the little database tweaks that were needed and done my best to reverse all the truncations. Please see [this comment](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=52&page=2#Item_22) for details. Please also check if a long page is how it ought to be (thinking particularly of the timeline).
The main thing to note is that although the limits are larger and are sufficient to accommodate all that was on the old lab, _there are still limits_. In particular, page names and redirects are limited to 100 characters. Page contents is a little bigger!
Talking of the timeline, incidentally, now that I can see Rafael's question, could I ask him to ask the question again over on the Forum with at the very least a link to where I can see what it is that needs to be converted? Thanks.
* [[Zoran ?koda]]: additions to [[Ieke Moerdijk]]
* [[Tim Porter]]: I have removed the blue boxes as suggetsed by [[Urs]] (see below).
* [[Zoran ?koda]]: created the entry for the monograph [[Borceux-Bourn]] and extracted some material to add into [[Mal'cev category]]. Created [[Mal'cev variety]] including the definitions and redirects for [[Mal'cev operation]] and [[Mal'cev theory]].
* [[Urs Schreiber]]
* created entries for [[Saunders MacLane]], [[Gonzalo Reyes]] and [[Ieke Moerdijk]] and included links to them where we cite these people as authors (but I will have missed many pages where we do)
* am asking for discussion of my latest formatting decision concerning these floating tables of contents [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=61) on the forum
* expanded the entry [[dg-algebra]], moved the discussion there to the bottom, as I think it has been addressed (but Toby, let me know if not)
I find it kind of a pity that this entry exists in parallel with [[differential graded algebra]]. I understand that the motivation was that one entry gives the detailed component description while the other gives the abstract nonsense definition (monoid in chain complexes). But this is a general effect in the nLab and we should keep such things in different subsections on the same page. Maybe let me know what you think
* added floating table of contents to the "lexicon" entries on differential graded structures that [[Tim Porter]] created a while ago. See for instance starting at [[graded vector space]].
(these tables overlap with Tim's blue alert boxes. I am thinking we could remove these boxes now and let the table of comntents server their purpose, but before i do this I want to hear what Tim thinks about. I'll contact him)
* [[Jon Awbrey]] made a first pass at formatting [[Trimble on ETCS II]], but it will need to be checked.
* [[Urs Schreiber]]: thanks! That's very useful. i was hoping somebody would find the time to do that eventually. Great that you did it.
## 2009-09-03
* [[Todd Trimble]]: wrote article on [[tree]].
* [[Zoran ?koda]]: created [[formal group scheme]] also far from finished. Maybe [[John Baez|John]] would like to explain connections to the Witt ring ?
* [[Rafael Borowiecki]]: Wrote a hopefully not too long answer to the long discussion at [[category theory]].
* [[Zoran ?koda]]: created [[formal scheme]] but it is far from finished; small changes to few related items (e.g. [[Kähler differential]]).
* [[Benoit Jubin]] has kindly corrected some fonts at [[monoidal category]]. Welcome, Benoît!
* [[Toby Bartels]]: Answered the open question at [[An Exercise in Kantization]].
* [[Jon Awbrey]]: "Kantization" makes it sound like you are talking about Immanuel Kant --- I think you should call it "Kanonization".
* [[Jon Awbrey]] added a stub-link at [[semiotic equivalence relation]] --- and I see by the clock on the [TARDIS Wall](http://nlab.mathforge.org/nlab/recently_revised) that the Synchronoplastic Infundibulum has hic$\cup\partial$.
* [[Zoran ?koda]]: added some references to [[supermanifold]]. I agree with Urs that the <em>co</em>-things when the entries contain predominantly the definitions and non-specific information should be just under the things. However often the cothings are very unlike things in practice. For example, homology and cohomology in abelian categories is just the same and dual; however in geometry homology and cohomology of spaces are rather different; for example there are finiteness conditions in homology which are absent in cohomology. Naively defined Čech cohomology is a cohomology theory and Čech homology is not, as it fails exactness...but the coherent repair works. Or the rings. Artinian and noetherian are dual conditions, but for unital rings the unit breaks the symmetry, hence every unital artinian ring is noetherian but not by far other way around.
* [[lab elf|Lab Elf (Swiss department)]]: I _think_ I've fixed the timezone. I guess the real test is when I submit this page. Let's see now, it's about 12:50 UTC, so click 'Submit' and ... 12:51 is the reported time. Yippee!
* [[Andrew Stacey]]: The memory upgrade has happened. Our IP address has changed so if you can't access the nLab then you need to ... err ...
* [[Urs Schreiber]] made the following keywords all redirect to [[fibration sequence]]: [[cofibration sequence]], [[homotopy fiber]], [[homotopy cofiber]]
(this is supposed to be in line with what I think is a general strategy we should stick to: that all co-things are discussed in the same entry with things, since otherwise we get huge and unreasonable duplication)
* [[Zoran ?koda]]: created [[free field (algebra)]], [[perfect field]], [[algebraic group]].
* [[lab elf|Lab Elf (children's department)]]: We're getting a memory upgrade sometime soon. This will involve a downtime of approximately 35-40 minutes (they have to shift our "slice" to another machine to accommodate the upgrade). I don't know yet exactly when this will occur and I may not get notice in time to post it here (I've requested that it be ASAP). For obvious reasons, if I do get notice of when it will be then I'll put an announcement on the Forum and somewhere appropriate on the Cafe (I guess the 'nLab migration' thread seems most suitable).
* [[Urs Schreiber]]: expanded [[generalized (Eilenberg-Steenrod) cohomology]]
* added table of contents to [[cohomology]] and strated adding to related entries
* [[Toby Bartels]]: In answer to Jon Awbrey\'s question,
>What day is it?
it\'s still September 2 UTC, but for some reason the Lab is now on UTC+4, which makes it September 3. Hopefully we can get it back to UTC, which is the standard for international sites; but if not, then I\'ll probably just edit the 'UTC' up above to 'UTC+4' and leave it at that.
[[lab elf|Lab Elf (Swiss department)]]: On the TODO list ... Actually, maybe the TODO list ought to be a little more explicit. I'll stick it on the forum.
* [[Jon Awbrey]] finished formatting [[triadic relation]].
## 2009-09-02
* [[Urs Schreiber]] tried to usefully rearrange the table of contents at [[HowTo]] into subsections a bit, just a suggestion, probably not optimal yet, but I felt the reader might wish to have an easier overview
* [[Toby Bartels]]: Made a table of contents for [[HowTo]].
* [[Mike Shulman]]: A bit more on displaying MathML at [[HowTo]].
* [[Urs Schreiber]]
* expanded the beginning of [[extended topological quantum field theory]], also added further links to existing entries
* further expanded [[equivariant cohomology]] a bit: made the essential idea a standout box and added group cohomology and local systems as examples.
* [[Rafael Borowiecki]]:
* Replied at [[Bousfield localization]].
* Replied and did some minor changes on [[extended topological quantum field theory]]. I am waiting for the experts to improve what i have written before i write more.
* As for the timeline i am glad that it can be fixed but how about the long titles? Ah. the nForum.
* [[Urs Schreiber]]
* started an entry on [[equivariant cohomology]]
* I agree with Andrew: single long pages are not so good for the reader. An entry that becomes very long should be split into an overview page with a linked table of contents and sub-pages. We did this for instance for [[gerbe]].
* added to [[HomePage]] and to [[HowTo]] a note on software prerequisites for displaying MathML. At [[HowTo]] this discussion should eventually be expanded.
* I changed links throughout the Lab to the pages [[Chevalley-eilenberg algebra in synthetic differential geomet|Chevalley-eilenberg algebra in synthetic differential geometry]] and to [[Verity on descent for strict omega-groupoid valued presheave|Verity on descent for strict omega-groupoid valued presheaves]] to the new truncated title names, so that they will still work. There are more entries for which this is necessary. See the discussion at the forum.
* [[Andrew Stacey]] I've written enough on the Instiki/MediaWiki issues elsewhere and I'm not going to rehash them here (or even link to them). In short: it ain't gonna happen. The issues we're having is purely a matter of database differences - it's nothing to do with Instiki itself. I doubt many people do a database migration - certainly none of the nonsense (abstract or otherwise) that I've read has mentioned the problems we've had - so the issues aren't well known. I'm finding them out as we go along. I apologise that it's a live test, but there were only three serious beta testers and they didn't pick up on all of these issues as there were only three of them.
I don't know why splitting the timeline up wouldn't feel like 2009. I think that _long_ pages are a hangover from the _old days_. We should have shorter pages included in to bigger pages (which, incidentally, is what MediaWiki does only it does it without telling you). "Pages" should be layered so that a visitor gets a broad overview first, then clicks through to get finer and finer details. One long page seems to miss the point of hyperlinks.
However, as has been pointed out in elsewhere, this is a lousy place to have a discussion. These are interesting points to discuss, but distracting on this page, so if anyone would like to pursue them further, I suggest we shift it over to the [forum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/).
What is more important is to let me know (preferably on the forum) of any other issues with the migrated site. Now that my eyes have uncrossed, I've realised that I was misreading the MySQL article on storage limits and we _can_ have superlong pages so we will. Soon. I promise.
(I still say it's a bad idea)
* [[Mike Shulman]]
* Did some editing of [[center]] (added categorifications) and [[normalizer]], and created [[centralizer]].
* Tried to remove some of the duplication from [[dualizable object]], [[compact closed category]], and [[rigid monoidal category]], but some is still left. Perhaps the latter two should really be combined on one page?
* [[Urs Schreiber]]
* added links back and forth between the new [[extended topological quantum field theory]] and the old [[FQFT]] -- notice that there is a bit of overlap, we may want to rearrange material eventually over these two entries.
* replied at [[category theory]] (down in the query box at the bottom)
* [[Jon Awbrey]] added a stub and a few links on the subject of [[inquiry]].
## 2009-09-01
* [[Rafael Borowiecki]]: Wrote the beginning of the well deserved page [[extended topological quantum field theory]].
* [[Rafael Borowiecki]]: Just as an interesting fact the new timeline which is not finished is 113kb, then extract the 4 or so long entries that were moved to separate pages and add literature, references and the long discussions. So how are other sites doing it? It looks like it is just nLab that runs into different troubels (even before the migration). I could mention more such as line breaks that is not working smooth or at all, but not now. I can not compare it to many sites but one is wikipedia. Even for editing as Zoran Škoda mentioned, wikipedia has no problem in editing parts of a page. I don't know which technology nLab is using (except a part of it is called instiki) but how about an upgrade to such technology as wikipedia is using? In fact everything that i noticed don't work here work in wikipedia. In all, the timeline could be split in two year parts and one main page with discussions, but i don't really like it. Then it would not feel like 2009.
* [[Mike Shulman]]: Hmm, [deja vu](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=28).
* [[Zoran ?koda]]: Thanks, Andrew a lot for all you are doing. Good night! I am loggin here another concern about terminology abstract nonsense. I am moving much in (predominantly noncommutative and algebraic) geometric
community and my exprience is that when somebody says they proved a fact by abstract nonsense it is NOT confined to categorical methods only but to any CLEAN and GENERAL methods "from the book" as opposed to specific unclean improvizations tailored to a very specific circumstance in question.
* [[Andrew Stacey]]: My eyes are beginning to cross when reading the MySQL manual. I may have been misreading a couple of things to do with data storage and it may be possible to get around the timeline problem. However, I still think that long pages could be better split up.
I sincerely hope that the lab survives the night, but I'm going offline now so please be patient with it!
* [[Zoran ?koda]]: of course, this is not a solution to the database problem, but I anyway think the discussion part could be separated from the main table of _Timeline_ and the _Timeline_ could have a separate part till 1960 say, then 1960 till 1989 and then third since 1990 (for example, better estimates possible). That would be easier for editing, with big file it is difficult to scroll when editing anyway. I created [[normalizer]], [[center]] (with a word on and redirect [[centralizer]]) and [[holomorph]]. Please check, it is elementary, but it was a quick writing.
* [[Andrew Stacey]] is seriously considering finding The Doctor and borrowing the Tardis to go back and have a Serious Word with the designers of sqlite3. (Anyone who gets the reference, I have a great photograph of a bus seen here in Trondheim that proves that The Doctor's greatest enemies are sneakily planning their next invasion from Norway. But I digress.) The problem with the timeline is the same as that with the long page names (and with an old problem in the testing stage with stylesheets). It's down to a fundamental difference in design between sqlite3 (the old, slow database backend) and mysql (the newer, snappier model). Basically, while both allow you to declare certain entries to be a certain type, sqlite3 then proceeds to __ignore that type__. Mysql (and just about every other database) enforces it. So when Instiki says "page names should be at most 60 chars long", Mysql truncates them to 60 chars while sqlite3 merrily accepts page names as long as Lanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch and back again. Twice. Similarly, where MySQL has a limit on lengths of rows in its database (happens to be 65535 bytes), sqlite3 merrily goes on accepting data as long as it gets it.
You can see where this is going, can't you. Yup, timeline hit that latter limit. It's currently over 80000 _characters_, and some of those are unicode so I've no idea how many actual bytes it is!
Before anyone suggests going back to sqlite3, let me point out that we can't scale up with sqlite3. So that's a non-starter. There are complicated possibilities in which we have more than one row for a page, but the absolute simplest would be to split long pages into smaller ones and then include them from the main one. So we could split the timeline into, say, decades and then an extra bit for the discussion and simply include them all on the main timeline page. That would also make editing the page a bit slicker and quicker.
If that is acceptable, then I can load up the timeline in two segments for someone to carve up into more sensible pieces. It'll be tomorrow now before I get round to doing this (sorry).
I'll also have to figure out whether any more pages are affected by this. I'm afraid that I'll have to roll these back to how they were when the lab was migrated (but presumably no-one's actually tried to edit one of these truncated pages, otherwise there'd've been more bugs noted here).
* [[Todd Trimble]] added a teeny bit to [[locally convex topological vector space]]. I hope to be more in nLab editor mode soon. Congratulations on a successful migration (with big thanks to [[Andrew Stacey]]).
[[Andrew Stacey]]: I'd hold off on the champagne for a little bit ...
* [[Zoran ?koda]]: query in [[category theory]]: I think that blaming the terminology *abstract nonsense* to predominantly non-likers is misleading and that the wikipedia is this time more correct than nlab.
* [[Jon Awbrey]]: I have always understood the term "abstract nonsense" as a pun on the sense of the word "sense" that means "direction" --- hence "abstract nonsense" suggests something like the "formal path-independence" of commutative diagrams. I'm sure I mentioned this to several people back in the day, and they all said something like, "well, duh."
* [[Zoran ?koda]]: I never heard of such interpretation; plus this interpretation would not survive in other languages like Russian and French where the direction and sense/nonsense do not mix like that. Russian version of *abstract nonsense* is абстрактная чепуха.
* [[Jon Awbrey]] added stubs and links at [[sign relation]] and [[triadic relation]].
* [[Urs Schreiber]]: replied in the discussion at the bottom of [[category theory]] -- and have a question
* [[Todd Trimble]] chucked in two cents here.
* [[Urs Schreiber]] WATCH OUT WITH LONG ENTRY NAMES -- see the $n$forum discussion. Some long entry names got truncated in the migration. the entry "Chevalley-eilenberg algebra in synthetic differential geometry" for instance is now called [[Chevalley-Eilenberg algebra in synthetic differential geomet]]
See "all pages" to find out the truncated entry name of an entry that you know should be there but is missing.
* [[Rafael Borowiecki]]: To [[Andrew Stacey]]. The whole bottom part of the page [[Timeline of category theory and related mathematics]] is abruptly missing. Which is a lot, not only my question. Since i have the full version loaded in a browser i could try to reconstruct it but i would prefer if you did a rollback if you can.
* [[Jon Awbrey]] added content to [[hypostatic abstraction]].
* [[John Baez]]:
* answered Rafael's plea for a definition of 'CW complex' in that big discussion on [[category theory]], and also commented on Urs' remark about 'simplicial complexes';
* completed the definition of [[monoidal category]] by adding the triangle equation;
* added some remarks on right vs. left duals on [[rigid monoidal category]];
* noted that the $n$Lab crashed a couple of times while I was doing this. Unfortunately I did not get a screenshot of the fancy error message.
[[Andrew Stacey]]: Yes, we hit our memory ceiling a couple of times. I've lowered a couple of settings to try to ensure that we don't do this again but it's a bit experimental as to what the best settings are. It's all a bit of a learning experience for me! (Perhaps I shouldn't admit to that ...)
* [[Zoran ?koda]]: I restarted the system and now IE renders [[normal subgroup]] correctly. This is strange as it had problems only with new entries [[normal subgroup]] and [[normal closure]] (even after many reloads) and rendered correctly the other entries. Now after reboot even they appear correct.
[[Andrew Stacey]]: Okay, sounds like it was a cache bug. During the changeover, various addresses pointed all over the shop and there are redirects going all ways from Sunday, so it's not surprising that a browser get confused. Clearing caches is probably a good idea. It will take a while before ncatlab.org properly points here for everyone (for example, from my work machine it was working from about midday; now at home then it still resolves to the old lab). But if you type 'ncatlab.org' into a browser you will always end up _here_, it just might be via a slightly circuitous route.
* [[Zoran ?koda]]: I just created [[normal subgroup]], [[normal closure]], but they do not render correctly on my IE. Is this a new-system glitch? The letters and formulas are one across another. Did not make last night logging that I added a paragraph or so on the Jacobi matrix and the application (Alexander polynomial) into the [[Fox derivative]]. I also created [[derived affine scheme]] in the sense of Toen et al.
[[Andrew Stacey]]: No idea! Can you send me a screen shot? Unfortunately, the Windows machine that I have control over can't connect to the wireless network here (Oh, the irony!) and the windows machine that I don't have control over doesn't have MathML support.
* [[Andrew Stacey]]: The migrating eagle has landed.
There will inevitably be hiccoughs, hangups, and hassles. Please log them over at the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/). Also, if you notice weird behaviour then there may be an explanation of it over there (just because I know what causes it doesn't mean I've implemented the fix yet).
* [[Urs Schreiber]]
* replied at [[Bousfield localization]]
* [[Jon Awbrey]]:
* "A [[continuous predicate]], as described by [[Charles Sanders Peirce]], is a special type of relational predicate that arises as the limit of an iterated process of [[hypostatic abstraction]]."
* [[Rafael Borowiecki]]:
* Posed a question at [[Timeline of category theory and related mathematics]] regarding a new migration: How do i handle to update most of 1500 links!?
* [[Andrew Stacey]]: I've looked for your question and can't find it. I apologise if it's been lost in the migration, but can you ask it again? If it's technical, the forum might be a better place to ask it.
* Suggested that manifold objects should be treated or at least mentioned at [[manifold]].
* Provided references for my question at [[Bousfield localization]].
* Split the subsection what is category theory at [[category theory]] into two parts: In the narrow sense and In the wide sense. This makes sense.
* Added briefly how toposes and higher categories come into category theory as a foundations at [[category theory]].
* Added what it means for category theory to be a unifying tool and language in mathematics at [[category theory]].
* Replied to the discussion at [[category theory]].
I see the migration went well, at least so far. :)
([[Andrew Stacey]]: actually, it happened _after_ you posted this, but thanks all the same!
* [[Toby Bartels]]: Happy September!!!
* [[Toby Bartels]]: Now, you know it\'s not September for another half hour, right? (^_^)
* [[Urs Schreiber]]: oops, you are right, I forgot that I am not exactly at GMT +0.
* [[Urs Schreiber]]
* moved the accumulated latest changes of last month to [[2009 August changes]] (by renaming the last "latest changes" and then creating a new one)
***
[[2008 changes|First list]] --- [[2009 August changes|Previous list]] --- [Next list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) --- [Current list](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
***
category: meta
|
24 | https://ncatlab.org/nlab/source/24 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
24 is the [[natural number]] that follows 23 and precedes 25. It occurs in modern systems of measurement for historical reasons, thanks ultimately to its having many [[prime factors]]
$$
24
\,=\,
2 \cdot 2 \cdot 2 \cdot 3
$$
and it arises in many circumstances in [[mathematics]] and [[physics]], the relations between which are not always immediately apparent.
## Examples
### Modular arithmetic
Fix a [[natural number]] $n$, and consider those numbers $x$ whose squares are congruent to 1 [[modulo]] $n$:
\[ x^2 \equiv 1 (\mod n) \, . \]
A necessary condition for satisfying this is for $x$ and $n$ to be [[coprime integer|coprime]]. For some choices of $n$, this condition is sufficient, i.e., every $x$ coprime to $n$ satisfies this congruence. In fact, this congruence is equivalent to $gcd(x,n) = 1$ if and only if $n$ divides 24.
### Cannonball problem
The cannonball problem asks what size square pyramids can be made by stacking spheres such that the total number of spheres is a square number. Apart from the trivial solutions involving 0 spheres or 1 sphere, the only solution is a pyramid 24 layers tall:
\[ 0^2 + 1^2 + 2^2 + \cdots + 24^2 = 70^2 \, . \]
### Leech lattice
The [[Leech lattice]] is a "universally optimal" packing of hyperspheres in $\mathbb{R}^{24}$ originally discovered in [[coding theory]]. Surprisingly, it is closely related to the cannonball problem, since it can be constructed from a Lorentzian lattice in two higher dimensions by a method that uses the fact that
\[ v = (0, 1, 2, \ldots, 24, 70) \]
has zero norm under that metric.
### Mathieu groups
The largest [[Mathieu group]], $M_{24}$, is the [[automorphism group]] of a [[Steiner system]] containing 24 blocks. $M_{24}$ is also the automorphism group of the [[binary Golay code]], an [[abelian group]] which is a 12-dimensional subspace of the [[vector space]] $\mathbb{F}_2^{24}$.
The binary Golay code can be used to construct the Leech lattice. Essentially, each codeword defines a point in 24-dimensional space, and we can fill in and build out this set. In more detail: Let $c$ be a Golay codeword, scale it by a factor 2, and add either $4x$, where $x$ is a vector in $\mathbb{Z}^{24}$ whose components sum to an even number, or $1 + 4y$ where $y \in \mathbb{Z}^{24}$ and its components sum to an odd number.
### 24-cell and the binary tetrahedral group
The [[24-cell]] is a four-dimensional [[regular polytope]] with 24 vertices. Interpreting these vertices as [[quaternion]]s, they form a [[group]] under quaternion multiplication, and this group is [[isomorphism|isomorphic]] to the [[binary tetrahedral group]].
### Third stable homotopy group of spheres
The [[third stable homotopy group of spheres]] is the [[cyclic group]] of order 24.
### String theory
Critical [[bosonic string theory|Bosonic string theory]] requires 26 [[spacetime]] [[dimension of a manifold|dimensions]] of [[target spacetime]] (for vanishing [[dilaton]], at least), which can be broken down into 2 dimensions for the string [[worldsheet]] itself and 24 transversal directions for its oscillations.
This is also related to the conventional choice of normalization for the [[central charge]] of the [[Virasoro algebra]]:
\[ [L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0} \, . \]
Several classes of [[string theory vacua]] require the presence of exactly [[24 branes transverse to K3|24 branes of codimension 4]] transverse to a [[K3-surface]]-[[fiber]].
## Related concepts
Appearances of 24 and its factors, particularly 8 and 12, are a meta-pattern in mathematics; another such meta-pattern is [[ADE classification]]. Sometimes these meta-patterns overlap.
For example, [[tesselation|tessellating]] $\mathbb{R}^4$ with regular 24-cells creates the 24-cell honeycomb, in which the centers of the 24-cells are the points of the $D_4$ lattice. Meanwhile, the [[binary tetrahedral group]] corresponds to the $E_6$ [[Dynkin diagram]] (see *[[McKay correspondence]]*).
Families of two-dimensional [[conformal field theory|conformal field theories]], in which the [[Virasoro algebra]] plays a key role, are among the examples falling into an ADE classification.
* [[Bernoulli number]]
* [[moduli space of curves]]
* [[Dwyer-Wilkerson H-space]], which would be the automorphism group of a [[normed division algebra]] if one could exist in 24 dimensions
* [[Dedekind eta function]]
## References
Blog discussion:
* [More Mysteries of the Number 24](https://golem.ph.utexas.edu/category/2007/06/more_mysteries_of_the_number_2.html) at the $n$-Category Café
* [The Binary Octahedral Group](https://golem.ph.utexas.edu/category/2021/12/the_binary_octahedral_group.html) at the $n$-Category Café
* [John Baez on the Number 24](https://math.ucr.edu/home/baez/numbers/#24)
Discussion in relation to the [[Leech lattice]]:
* {#Sloane80} N. J. Sloane, *A note on the Leech lattice as a code for the Gaussian channel*. Information and Control, **46&** 3 (1980). 270--272 ([pdf](http://neilsloane.com/doc/Me75.pdf))
* {#Gannon06} [[Terry Gannon]], section 2.5.1 of: *Moonshine Beyond the Monster*, Cambridge University Press, 2006 ([doi:10.1017/CBO9780511535116](https://doi.org/10.1017/CBO9780511535116))
* {#CohnKumarMIllerRadschenkoViazovska19} H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, *Universal optimality of the $E_8$ and Leech lattices and interpolation formulas*, ([arXiv:1902.05438](https://arxiv.org/abs/1902.05438)).
Discussion in relation to the [[J-homomorphism]]:
* [Bernoulli Numbers and the J-homomorphism](https://golem.ph.utexas.edu/category/2020/12/bernoulli_numbers_and_the_jhom.html) at the $n$-Category Café
For an introduction to the [[third stable homotopy group of spheres]], see "Week 102" in
* J. Baez, *This Week's Finds in Mathematical Physics: Weeks 101 to 150*, ([arXiv:2303.02785](https://arxiv.org/abs/2303.02785)). |
24 branes transverse to K3 | https://ncatlab.org/nlab/source/24+branes+transverse+to+K3 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Several classes of [[string theory vacua]] require the presence of exactly [[24]] [[branes]] of [[codimension]] 4 transverse to a [[K3-surface]]-[[fiber]]; this happens notably:
* in [[F-theory]] on an [[elliptically fibered K3-surface]], which requires/implies the presence of 24 [[D7-branes]] ([Sen 96, p. 5](#Sen96)),
* in [[HET-theory]] [[KK-compactification|KK-compactified]] on a [[K3-surface]] with vanishing [[gauge field]] [[instanton number]], which requires/implies the presence of 24 [[NS5-branes]] ([Schwarz 96, p. 50](#Schwarz96)).
In both cases the condition arises as a kind of [[tadpole cancellation]]-condition, where the charge of the 24 branes in the [[compact topological space|compact]] [[K3]]-fiber space, which naively would be 24 in natural units, cancels out to zero, due to some subtle effect.
Despite the superficial similarity, this subtle effect is, at the face of it, rather different in the two cases:
* for [[F-theory]] it results from the [[Kodeira classification]] of [[elliptic fibrations]], which implies that an [[elliptically fibered K3]] has exactly 24 singular points, counted with multiplicity,
* for [[HET-theory]] it results, via the [[Green-Schwarz mechanism]], from the [[Euler characteristic]] of a [[K3-surface]] being 24, and hence vanishing, via the [[Poincare-Hopf theorem]], after the locus of 24 [[NS5-brane]] loci are cut out.
An argument that these two different-looking mechanisms are in fact equivalent, under suitable [[duality in string theory]], is given in [Braun-Brodie-Lukas-Ruehle 18, Sec. III](#BraunBrodieLukasRuehle18), following detailed analysis due to [Aspinwall-Morrison 97](#AspinwallMorrison97).
### In F-theory on K3
{#InFTheoryOnK3}
In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory "are" the singular locus of the elliptic fibration.
Now, considering [[F-theory on K3]], an [[elliptically fibered complex K3-surface]]
$$
\array{
T &\longrightarrow& K3
\\
&& \downarrow
\\
&& \mathbb{C}\mathbb{P}^1
}
$$
may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation
$$
y^2 = x^3 + f(z) x + g(z)
$$
for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is
$$
j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3}
\,.
$$
The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located.
\begin{imagefromfile}
"file_name": "K3CobordismBetween24ThreeSpheres.jpg",
"web": "nlab",
"width": 260,
"unit": "px",
"float": "right",
"margin": {
"top": -40,
"right": 0,
"bottom": 20,
"left": 20,
"unit": "px"
},
"alt": "homotopy pasting diagram exhibiting the homotopy Whitehead integral",
"caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}
Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$, see at _[[elliptically fibered K3-surface]] -- [singular points](elliptic+fibration+of+a+K3-surface#SingularPoints)_) there are necessarily _24 D7-branes_ ([Sen 96, page 5](#Sen96), [Lerche 99, p. 6](#Lerche99) , see also [Morrison 04, sections 8 and 17](#Morrison04), [Denef 08, around (3.41)](#Denef08), [Douglas-Park-Schnell 14](duality+between+M%2FF-theory+and+heterotic+string+theory#DouglasParkSchnell14)).
Notice that the _net charge_ of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. [Denef 08, below (3.41)](#Denef08)).
### In IIA-theory on K3
Under [[T-duality]] the [above](#InFTheoryOnK3) discussion in [[F-theory]] translates to 24 [[D6-branes]] in [[type IIA string theory]] on [[K3]] ([Vafa 96, Footnote 2 on p. 6](#Vafa96)).
### In HET-theory on K3
In [[heterotic string theory]] [[KK-compactification|KK-compactified]] on [[K3]] with vanishing [[gauge fields]]-[[instanton number]], the existence of exactly 24 [[NS5-branes]] is implied by the [[Green-Schwarz mechanism]]: This requires that the 3-flux density $H_3$ measuring the [[NS5-brane]] charge satisfies $ d H_3 = \trac{1}{2} p_1(\nabla)$; and using that on [[K3]] we have $\int_{K3} \tfrac{1}{2} p_1(\nabla) = \int_{K3} \chi_4(\nabla) = \chi_4[K3] = 24$ this implies, with [[Stokes' theorem]], that the $H_3$-flux through the [[3-spheres]] around transversal [[NS5-brane]]-punctures of the K3 equals 24
(e.g. [Schwarz 96, around p. 50](duality+in+string+theory#Schwarz97), [Aspinwall-Morrison 97, Sec. 4-5](#AspinwallMorrison97), [Johnson 98, p. 30](#Johnson98), [Braun-Brodie-Lukas-Ruehle 18, Section III.A](#BraunBrodieLukasRuehle18), [Choi-Kobayashi 19, Sec. 1.1](#ChoiKobayashi19)).
The [[duality in string theory|duality]] of this HET-phenomenon with that in F-theory [above](#InFTheoryOnK3) is discussed in [Braun-Brodie-Lukas-Ruehle 18, Section III](#BraunBrodieLukasRuehle18).
### Under Hypothesis H
The vanishing of the [[Euler characteristic]] of K3 after cutting out the [[complement]] of 24 points is precisely the mechanism which witnesses the [[order of a group|order]] 24 of the [[third stable homotopy group of spheres]], seen under [[Pontryagin's theorem]] as the existence of a [[framed manifold|framed]] [[cobordism]] $K3 \setminus 24 \cdot D^4$ between 24 [[3-spheres]]:
\begin{imagefromfile}
"web": "schreiber",
"file_name": "24BranesTransversalToK3ViaHypothesisH-20210207.jpg",
"width": 700,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}
This relates the number of 24 branes transverse on K3 to [[Hypothesis H]]:
\begin{imagefromfile}
"web": "schreiber",
"file_name": "KKOn24PuncturedK3ViaHypothesisH-20210207.jpg",
"width": 760,
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
},
"caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}
## Related concepts
* [[duality between M/F-theory and heterotic string theory]]
## References
### In F-theory
Discussion in [[F-theory]] via the [[Kodeira classification]] of [[elliptically fibered K3s]]:
* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_, Nucl. Phys. B475:562-578, 1996 ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))
* {#Lerche99} [[Wolfgang Lerche]], [p. 6](https://arxiv.org/pdf/hep-th/9910207.pdf#page=6) of: _On the Heterotic/F-Theory Duality in Eight Dimensions_, In: Baulieu L., Green M., Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_, NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer 2001 ([arXiv:hep-th/9910207](https://arxiv.org/abs/hep-th/9910207), [doi:10.1007/978-94-010-0852-5_2](https://doi.org/10.1007/978-94-010-0852-5_2))
* {#Morrison04} [[David Morrison]], _TASI Lectures on Compactification and Duality_ ([arXiv:hep-th/0411120](http://arxiv.org/abs/hep-th/0411120))
* {#Denef08} [[Frederik Denef]], [p. 34](https://arxiv.org/pdf/0803.1194.pdf#page=34) of: _Les Houches Lectures on Constructing String Vacua_, in: _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194), [spire:780946](https://inspirehep.net/literature/780946))
* {#DouglasParkSchnell14} [[Michael Douglas]], Daniel S. Park, Christian Schnell, _The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds_, JHEP05 (2014) 135 ([arXiv:1403.1595](https://arxiv.org/abs/1403.1595))
### In HET-theory
Discussion in [[heterotic string theory]] via the [[Green-Schwarz mechanism]] on K3 (see also at _[[small instantons]]_):
* {#Schwarz96} [[John Schwarz]], around [p. 50](https://arxiv.org/pdf/hep-th/9607201.pdf#page=51) of: _Lectures on Superstring and M Theory Dualities_, Nucl. Phys. Proc. Suppl. 55B:1-32, 1997 ([arXiv:hep-th/9607201](https://arxiv.org/abs/hep-th/9607201))
* {#Johnson98} [[Clifford Johnson]], _Études on D-Branes_, in: [[Mike Duff]] et. al. (eds.) _Nonperturbative aspects of strings, branes and supersymmetry_, Proceedings, Trieste, Italy, March 23-April 3, 1998 ([arXiv:hep-th/9812196](https://arxiv.org/abs/hep-th/9812196), [spire:481393](https://inspirehep.net/literature/481393))
* {#ChoiKobayashi19} Kang-Sin Choi, Tatsuo Kobayashi, _Transitions of Orbifold Vacua_, JHEP07 (2019) 111 ([arXiv:1901.11194](https://arxiv.org/abs/1901.11194))
### In F- and HET-theory
Joint discussion in F- and [[duality in string theory|dual]] HET-theory:
* {#AspinwallMorrison97} [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104), <a href="https://doi.org/10.1016/S0550-3213(97)00516-6">doi:10.1016/S0550-3213(97)00516-6</a>)
* {#BraunBrodieLukasRuehle18} [[Andreas Braun]], Callum Brodie, [[Andre Lukas]], [[Fabian Ruehle]], _NS5-Branes and Line Bundles in Heterotic/F-Theory Duality_, Phys. Rev. D 98, 126004 (2018) ([arXiv:1803.06190](https://arxiv.org/abs/1803.06190), [doi:10.1103/PhysRevD.98.126004](https://doi.org/10.1103/PhysRevD.98.126004))
> (in the context of [[heterotic line bundles]])
[[!include swampland cobordism conjecture -- references]]
|
24-cell | https://ncatlab.org/nlab/source/24-cell |
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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analog of the [[Platonic solids]]...
## Definition
The 24-cell is the [[regular polyhedron]] in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^4$ whose [[vertices]] are, under the identification $\mathbb{R}^4 \simeq_{\mathbb{R}} \mathbb{Q}$ with the space of [[quaternions]], the 8 unit [[quaternions]] $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by
$\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$.
(These [[24]] quaternions form a [[group]] under quaternion multiplication, and this group is [[isomorphism|isomorphic]] to the [[binary tetrahedral group]].)
## Properties
### Symmetry group
The [[finite rotation group]] inside [[orthogonal group|O(4)]] which is the [[symmetry group]] of the 24-cell is the [[Coxeter group]] [F4](https://ncatlab.org/nlab/revision/Coxeter+group/7#the_group_).
## Related concepts
* [[120-cell]]
* [[600-cell]]
## References
See also
* Wikipedia, _[24-cell](https://en.wikipedia.org/wiki/24-cell)_
|
2Cat | https://ncatlab.org/nlab/source/2Cat |
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###Context###
#### 2-Category theory
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[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_2Cat_ is the [[3-category]] of [[2-categories]].
It has
* [[objects]] are [[2-categories]];
* [[1-morphisms]] are [[2-functors]];
* [[2-morphisms]] are [[pseudonatural transformation|pseudo]]/[[lax natural transformations]];
* [[3-morphisms]] are [[modifications]].
## Related concepts
* [[Cat]], [[Operad]]
* **$2Cat$**
* [[(∞,1)Cat]], [[(∞,1)Operad]]
* [[(∞,2)Cat]]
* [[(∞,n)Cat]]
category: category
[[!redirects BiCat]]
[[!redirects Bicat]]
[[!redirects 2-Cat]]
|
2d Chern-Simons theory | https://ncatlab.org/nlab/source/2d+Chern-Simons+theory |
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###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 2.
## Examples
* [[Poisson sigma-model]]
* [[WZW-model|WZW-term]]/[[B-field]]-coupling of the [[string]]
## Properties
* [[extended geometric quantization of 2d Chern-Simons theory]]
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* **2d Chern-Simons theory**
* [[2d TQFT]]
* [[3d Chern-Simons theory]]
* [[2d Wess-Zumino-Witten theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[11d Chern-Simons theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[AKSZ sigma-model]]
[[!redirects 2d Chern-Simons theories]]
[[!redirects 2-dimensional Chern-Simons theory]]
[[!redirects 2-dimensional Chern-Simons theories]]
|
2d quantum gravity | https://ncatlab.org/nlab/source/2d+quantum+gravity |
#Contents#
* table of contents
{:toc}
## Idea
In the very low number of just 2 dimensions, [[quantum gravity]] is comparatively trivial and can be completely understood. While trivial as a theory of [[quantum gravity]] on a 2d spacetime, when that spacetime is regarded instead as the [[worldsheet]] of a string, then in fact the string's worldsheet [[sigma-model]] theory is a theory of 2d quantum gravity (before the usual gravitational [[gauge fixing]] in 2d, which then makes what remains a [[conformal field theory]], instead).
## Related concepts
* [[world sheets for world sheets]]
* [[3d quantum gravity]]
## References
* [[Paul Ginsparg]], [[Gregory Moore]], _Lectures on 2D gravity and 2D string theory (TASI 1992)_ ([arXiv:hep-th/9304011](http://arxiv.org/abs/hep--th/9304011)) |
2d SCFT | https://ncatlab.org/nlab/source/2d+SCFT |
+-- {: .rightHandSide}
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[SCFT|Super-conformal field theory]] in [[dimension]] $d = 2$, locally given by a [[super vertex operator algebra]].
For [[central charge]] 15 this is the [[worldsheet]] [[theory (physics)|theory]] of the [[superstring]].
May be regarded as a "[[2-spectral triple]]" (see there for more), the 2-dimensional generalization of [[spectral triples]] describing the quantum mechanics of [[spinning particles]] (super-particles).
## Examples
* [[superstring]] [[sigma-model]]
* [[spinning string]] [[superstring]]
* [[heterotic string]], [[type II superstring]]
* [[Gepner model]]
* [[Kazama-Suzuki model]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
## Properties
### Classification
See at [supersymmetry -- Classification -- Superconformal algebra -- In dimension 2](supersymmetry#ClassificationSuperconformalInDim2).
## Related concepts
* [[perturbative string theory vacuum]]
## References
### General
A basic but detailed exposition focusing on the super-[[WZW model]] (and the perspective of [[2-spectral triples]]) is in [Fröhlich & Gawedzki (1993)](#FroehlichGawedzki93).
Textbook account:
* [[Ralph Blumenhagen]], [[Erik Plauschinn]], *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]
Other accounts:
* [[Lance Dixon]], [[Paul Ginsparg]], [[Jeffrey Harvey]], _$\hat c = 1$ Superconformal field theory_ ([pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4515.pdf))
* [[Yasuyuki Kawahigashi]], _$\mathcal{N} = 2$ Superconformal Field Theory and Operator Algebras_ ([pdf](http://www.rci.rutgers.edu/~yzhuang/math/papers/kawahigashi.pdf))
Constructing 2d SCFTs from [[error-correcting codes]] and a hint for a relation to [[tmf]], vaguely in line with the lift of the [[Witten genus]] to the [[string orientation of tmf]]:
* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601 [[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)]
further on the resulting [[elliptic genera]]:
* Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* [[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)]
### Relation to 2-spectral triples
Discussion of 2d SCFTs as a higher analog of [[spectral triples]] ("[[2-spectral triples]]", see there for more) is in terms of [[vertex operator algebras]] in
* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawędzki]], _Conformal Field Theory and Geometry of Strings_, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))
* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)
and in terms of [[conformal nets]] in
* {#CHKL09} Sebastiano Carpi, Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral triples and the super-Virasoro algebra_, Commun.Math.Phys.295:71-97 (2010) ([arXiv:0811.4128](http://arxiv.org/abs/0811.4128))
[[!include D=2 CFT a functorial field theory -- references]]
[[!include elliptic genera as partition functions -- references]]
[[!redirects 2d SCFTs]]
[[!redirects 2d super conformal field theory]]
[[!redirects 2d super conformal field theories]]
[[!redirects 2d superconformal field theory]]
[[!redirects 2d superconformal field theories]]
[[!redirects 2d super-conformal field theory]]
[[!redirects 2d super-conformal field theories]]
|
2d TQFT | https://ncatlab.org/nlab/source/2d+TQFT |
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### Context
#### Quantum field theory
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#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
A _2-dimensional TQFT_ is a [[TQFT|topological quantum field theory]] on [[cobordisms]] of [[dimension]] 2.
## Classification
When formulated as an (only) "globally" as 1-[[functors]] on a 1-[[category of cobordisms]] (see at _[[FQFT]]_ for more), then 2d TQFTs have a comparatively simple classification: the [[bulk field theory]] is determined by a commutative [[Frobenius algebra]] structure on the [[finite dimensional vector space]] assigned to the circle ([Abrams 96](#Abrams96)).
However, such global 2d TQFTs with [[coefficients]] in [[Vect]]
do not capture the 2d TQFTs of most interest in [[quantum field theory]], which instead are "[[cohomological quantum field theories]]" ([Witten 91](#Witten91)) such as the [[topological string]] [[A-model]] and [[B-model]] that participate in [[homological mirror symmetry]].
These richer 2d TQFTs are instead local TQFTs in the sense of _[[extended TQFT]]_, i.e. they are [[(∞,2)-functors]] on a suitable [[(∞,2)-category of cobordisms]] (see at _[[FQFT]]_ for more), typically on "non-compact" 2-d cobordisms, meaning on those that have non-vanishing outgoing bounary. As such they are now classified by [[Calabi-Yau objects]] in an [[symmetric monoidal (infinity,2)-category]] ([Lurie 09, section 4.2](#Lurie09)). For coefficients in the [[(∞,2)-category]] of [[(∞,n)-vector space|(∞,2)-vector space]] (i.e. [[A-∞ algebras]] with [[(∞,1)-bimodules]] between them in the [[(∞,1)-category of chain complexes]]), these theories had been introduced under the name "[[TCFT]]" in ([Getzler 92](#Getzler92), [Segal 99](#Segal99)) following ideas of [[Maxim Kontsevich]], and have been classified in ([Costello 04](#Costello04)), see ([Lurie 09, theorem 4.2.11, theorem 4.2.14](#Lurie09)).
[[!include 2d TQFT -- table]]
## Filtrations of the moduli space of surfaces {#FiltrationOfModuliSpace}
The following study of the behaviour of 2-dimensional TQFTs in terms of the [[topology]] of the [[moduli spaces]] of marked hyperbolic surfaces is due to [[Ezra Getzler]]. It provides a powerful way to read off various classification results for 2d QFTs from the [[homotopy groups]] of the corresponding [[modular operad]].
### $A_\infty$-monoid objects
Let $Core$([[FinSet]]) be the [[core]] of the [[category]] of finite sets.
Under union of sets this is a [[symmetric monoidal category]]. Then for
$C$ any [[monoidal category]], a [[symmetric monoidal functor]]
$$
\Phi : Core(FinSet) \to C
$$
is a commutative [[monoid]] in $C$.
Let now $C$ be a [[category with weak equivalences]], then we can speak of a lax symmetric opmonoidal functor
$$
\Phi : Core(FinSet) \to C
$$
if the structure maps
$$
\Phi(n+m) \stackrel{\simeq}{\to} \Phi(m) \otimes \Phi(n)
$$
$$
\phi(m) \stackrel{\simeq}{\to} \Phi(1)^{\otimes m}
$$
are weak equivalences.
Segal called these "$\Delta$-objects". Since [[Carlos Simpson]] they are called
[[Segal object]]s.
There is also [[Jim Stasheff]]'s notion of an [[A-infinity algebra]], given in terms
of [[associahedra]] $K_n$, which are $(n-2)$-dimensional [[polytope]]s.
There is naturally a filtration on these guys with
$$
F_0 K_n \subset F_1 K_n \subset \cdots
\,,
$$
where $F_0 K_n$ is the set of vertices, $F_1 K_n$ the set of edges, etc.
The collection
$$
\{
S_\bullet(K_n)
\}
$$
of simplicial realizations of the $K_n$ form an [[sSet]]-[[operad]] $P$.
For $X$ a [[simplicial category]] that is symmetric monoidal, a
$P$-[[algebra over an operad]] $X$ in $C$ is an $A_\infty$-monoid object
$$
S_\bullet(K_n) \to C_\bullet(X^{\otimes n}, X)
$$
[[Saunders MacLane|MacLane]]'s [[coherence theorem]] says or uses that if $C$ is an [[n-category]],
we may replace $K_m$ here by the $n$-filtration $F_n K_m$.
### Closed 2d quantum field theory
#### Compactified moduli spaces of Riemann surfaces
Let
$$
(\Sigma, (z_1, \cdots, z_n))
$$
be a [[compact space|compact]] [[orientation|oriented]] surface
with $n$ distinct marked points. Write
$$
\mathcal{H}(\Sigma, (z_1, \cdots, z_n))
$$
for the [[moduli space]] of [[hyperbolic metric]]s with cusps at the $(z_i)$.
We have
$$
M(\Sigma, \vec z)
=
\mathcal{H}(\Sigma, \vec z)/Diff_+(\Sigma, \vec z)
$$
and
$$
M_{g,n} = \Tau_{g,n} / \Gamma^ng
\,,
$$
where $\Tau_{g,n}$ is the [[Teichmüller space]] and $\Gamma$ the [[mapping class group]].
Here we can assume that the [[Euler characteristic]] $\chi(\Sigma without \{z_i\}) \lt 0$
because otherwise this moduli space is empty.
#### Fenchel-Nielson coordinates on moduli space
We want to parameterize Teichmüller space by cutting surfaces into pieces with
geodesic boundaries and [[Euler characteristic]] $\xi = -1$. These building blocks
(of hyperbolic 2d geometry) are precisely
* the 3-holed sphere;
* the 2-holed cusp;
* the 1-holed 2-cusp;
* the 3-cusp
Each surface of [[genus]] $g$ with $n$ marked points will have
* $2g - 2 + n$ generalized pants;
* $3 g - 3 + n$ closed curves.
The boundary lengths $\ell_i \in \mathbb{R}_+$ and twists
$t_i \in \mathbb{R}$ of these pieces
for
$$
1 \leq i \leq 3g-3+n
$$
constitute the [[Fenchel-Nielsen coordinates]] on
[[Teichmüller space]] $\Tau$.
Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$
This constitutes is a real analytic [[atlas]] of Teichmüller space. On $M$ this
reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$, and these
constitute a real analytic atlas of moduli space.
Allow the lengths $\ell_i$ to go to 0, but keep the angles $\theta_i$.
The resulting space is a real analytic [[manifold with corners]]
$\hat \Tau$ (due to [[Bill Harvey]]) and this constitutes a Borel-Serre
[[bordification]] of $\Tau$.
The [[mapping class group]] $\Gamma$ still acts on $\hat \Tau$ and the
quotient $\hat M$ is an [[orbifold]] with corners, inside which still sits
our moduli space $M$.
Kimura-Stasheff-Voronov: add a choice of
directions at each nodal point in $\Sigma$. This removes
all automorphisms and hence we no longer have to deal with an [[orbifold]].
This yields the [[classifying stack]] $\mathcal{P}_{g,n}$ for $\Gamma_{g,n}$
Then the collection
$$
\{
\mathcal{P}_{g,n}
\}
$$
is a [[modular operad]]: the operad that describes gluing of marked surfaces at
marked points together with the informaiton on how to glue marked points of a single
surface to each other.
A 2-dimensional closed [[TQFT]] is an [[algebra over an operad]] over this in
a simplicial category, in the above sense.
This involves either the [[de Rham complex]] on $\mathcal{P}_{g,n}$ or
$S_\bullet(\mathcal{P}_{g,n})$.
Let
$$
F_k \mathcal{P}_{g,n}
:=
\left\{
[\Sigma] |
...
\right\}
$$
where $\Sigma$ has $\geq 2g-2+n-k$ spheres as components (after cutting along zero-length
closed curves).
So for instance
* $F_0 \mathcal{P}_{g,n}$ is the pants-decomposition;
* $F_1 \mathcal{P}_{g,n}$ is decompositions into pants and one piece being the
result of either gluing two pants to each other or of gluing two circles
of a single pant to each other.
This $F_1 ..$ is a connected space, due to a theorem by Hatcher-Thurston.
**Notice** This is equivalent to the familiar statement that a closed 2d TFT is
a commutative [[Frobenius algebra]].
* $F_2 \mathcal{P}_{g,n} $ is the decomposition into pieces as before
together with one two-holed torus or one five-holed sphere.
This space has the space [[fundamental group]] as $\mathcal{P}_{g,n}$.
This is equivalent to the theorem by Moore and Seiberg about categorified
2-d TFT.
+-- {: .un_theorem}
###### Theorem
([[Ezra Getzler]])
The inclusion
$$
F_k \mathcal{P}_{g,n} \hookrightarrow \mathcal{P}_{g,n}
$$
is $k$-[[connected]].
Here a map $X\to Y$ is $k$-connected if
* $\pi_0(X) \to \pi_0(Y)$ is surjective;
* $\pi_i(X,x) \to \pi_i(Y,f(x))$ is a bijection for $i \lt k$ and surjective $i = k$.
This means precisely that the [[mapping cone]] is $k$-connected.
=--
+-- {: .proof}
###### Proof
Use the cellular decomposition of moduli space $\mathcal{M}_{g,1}$
following Mumford, Thurston, Harer, Woeditch-Epstein, Penner.
=--
Some other versions of this:
$$
F_k \Tau_{g,n} \to \Tau_{g,n}
$$
is $k$-connected.
One can also use [[Deligne-Mumford compactification]]s
$$
F_k \bar \mathcal{M}_{g,n} \to \bar \mathcal{M}_{g,n}
$$
and this is also $k$-connected.
### Open-closed case
...
## Related concepts
* [[sewing constraint]]
* [[Cardy condition]]
* [[TQFT]]
* **2d TQFT**
* [[2d Chern-Simons theory]]
* [[TCFT]]
* [[A-model]], [[B-model]]
* [[Landau-Ginzburg model]]
* [[Levin-Wen model]]
* [[3d TQFT]]
* [[4d TQFT]]
## References
### Global
The [[folklore]] result that global closed 2d TQFTs with coefficients in [[Vect]] are equivalent to commutative [[Frobenius algebra]] structures is proven rigorously in
* {#Abrams96} [[Lowell Abrams]], *Two-dimensional topological quantum field theories and Frobenius algebra*, Jour. Knot. Theory and its Ramifications **5**, 569-587 (1996) [[doi:10.1142/S0218216596000333](https://doi.org/10.1142/S0218216596000333), [ps](http://home.gwu.edu/~labrams/docs/tqft.ps)]
The classification result for open-closed 2d TQFTs was famously announced and sketched in
* {#MooreSegal02} [[Greg Moore]], [[Graeme Segal]], _Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002), _ ([web](http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html))
* {#Lazaroiu00} [[Calin Lazaroiu]], _On the structure of open-closed topological field theory in two dimensions_, Nuclear Phys. B 603(3), 497–530 (2001), ([arXiv:hep-th/0010269](http://arxiv.org/abs/hep-th/0010269))
Textbook account:
* {#Kock2004} [[Joachim Kock]], *Frobenius Algebras and 2d Topological Quantum Field Theories*, Cambridge U. Press (2004) [[doi:10.1017/CBO9780511615443](https://doi.org/10.1017/CBO9780511615443), [webpage](http://mat.uab.cat/~kock/TQFT.html), [course notes pdf](http://mat.uab.es/~kock/TQFT/FS.pdf), [[Kock-FrobAlgTQFT-short.pdf:file]]]
A picture-rich description of what's going on:
* {#LaudaPfeiffer05} [[Aaron Lauda]], [[Hendryk Pfeiffer]], _Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras_ , Topology Appl. 155 (2008) 623-666. ([arXiv:math.AT/0510664](http://arxiv.org/abs/math.AT/0510664))
### Local
The local ([[extended TQFT]]) version of 2d TQFT which captures the [[topological string]] was mathematically introduced under the name "[[TCFT]]".
The concept is essentially a formalization of what used to be called [[cohomological field theory]] in
* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])
The definition was given independently by
* {#Getzler92} [[Ezra Getzler]], _Batalin-Vilkovisky algebras and two-dimensional topological field theories_ , Comm. Math. Phys. 159(2), 265–285 (1994) ([arXiv:hep-th/9212043](http://arxiv.org/abs/hep-th/9212043))
and
* {#Segal99} [[Graeme Segal]], _Topological field theory_ , (1999), Notes of lectures at Stanford university. ([web](http://www.cgtp.duke.edu/ITP99/segal/)). See in particular [lecture 5](http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf) ("topological field theory with cochain values").
The classification of [[TCFT]]s (i.e. "non-compact" local ([[extended TQFT|extended]] 2d TQFT)) by [[Calabi-Yau A-infinity categories]] is due to
* {#Costello04} [[Kevin Costello]], _Topological conformal field theories and Calabi-Yau categories_ Advances in Mathematics, Volume 210, Issue 1, (2007), ([arXiv:math/0412149](http://arxiv.org/abs/math/0412149))
* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([arXiv:math/0509264](http://arxiv.org/abs/math/0509264))
following conjectures by [[Maxim Kontsevich]], e.g.
* {#Kontsevich95} [[Maxim Kontsevich]], _Homological algebra of mirror symmetry_ , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.
The classification of local ([[extended TQFT|extended]]) 2d TQFT (i.e. the "compact" but fully local case) is spelled out in
* {#SchommerPries11} [[Chris Schommer-Pries]], _The Classification of Two-Dimensional Extended Topological Field Theories_ ([arXiv:1112.1000](http://arxiv.org/abs/1112.1000))
This classification is a precursor of the full [[cobordism hypothesis]]-theorem. This, and the reformulation of the original TCFT constructions in full generality is in
* {#Lurie09} [[Jacob Lurie]], section 4.2 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))
[[!redirects 2d TQFTs]]
[[!redirects 2-dimensional TQFT]]
[[!redirects 2-dimensional TQFTs]]
[[!redirects 2d topological field theory]]
[[!redirects 2d topological field theories]]
[[!redirects 2d topological quantum field theory]]
[[!redirects 2d topological quantum field theories]]
|
2d TQFT -- table | https://ncatlab.org/nlab/source/2d+TQFT+--+table |
| [[2d TQFT]] ("[[TCFT]]") | [[coefficients]] | [[algebra]] structure on [[space of quantum states]] | |
|-------------|-------------|--------------------|--------|
| [[open string|open]] [[topological string]] | [[Vect]]${}_k$ | [[Frobenius algebra]] $A$ | [[folklore]]+([Abrams 96](2d+TQFT#Abrams96)) |
| [[open string|open]] [[topological string]] with [[closed string]] [[bulk field theory|bulk theory]] | [[Vect]]${}_k$ | [[Frobenius algebra]] $A$ with [[trace]] map $B \to Z(A)$ and [[Cardy condition]] | ([Lazaroiu 00](#2d+TQFT#Lazaroiu00), [Moore-Segal 02](2d+TQFT#MooreSegal02)) |
| non-compact [[open string|open]] [[topological string]] | [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ algebra]] | ([Kontsevich 95](2d+TQFT#Kontsevich95), [Costello 04](2d+TQFT#Costello04)) |
| non-compact [[open string|open]] [[topological string]] with various [[D-branes]]| [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ category]] | " |
| non-compact [[open string|open]] [[topological string]] with various [[D-branes]] and with [[closed string]] [[bulk field theory|bulk]] sector | [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ category]] with [[Hochschild cohomology]] | " |
| [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]([[Vect]]${}_k$) over [[field]] $k$ | separable symmetric [[Frobenius algebras]] | ([SchommerPries 11](2d+TQFT#SchommerPries11)) |
| non-compact [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]([[category of chain complexes|Ch(Vect)]]) | [[Calabi-Yau A-∞ algebra]] | ([Lurie 09, section 4.2](2d+TQFT#Lurie09)) |
| non-compact [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]$(\mathbf{S})$ for a [[symmetric monoidal (∞,1)-category]] $\mathbf{S}$ | [[Calabi-Yau object]] in $\mathbf{S}$ | ([Lurie 09, section 4.2](2d+TQFT#Lurie09)) |
|
2d Wess-Zumino-Witten theory | https://ncatlab.org/nlab/source/2d+Wess-Zumino-Witten+theory |
* [[WZW model]]
## Related concepts
* [[2d Chern-Simons theory]]
|
2Mod | https://ncatlab.org/nlab/source/2Mod |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[2-category]] of [[2-modules]]/[[2-vector spaces]].
Fix some [[commutative ring]] or more generally an [[E-infinity ring]] $R$. Then a [[basis]] for a [[2-module]]/[[2-vector space]] over $R$ may be taken to be an $R$-[[associative algebra]] or more generally an $R$[[Mod]]-[[enriched category]] $\mathcal{A}$ (an [[algebras]]): the correspponding [[2-vector space]] is the [[category of modules]] $Hom_{R Mod}(\mathcal{A}, R Mod)$. Then $2Mod_R$ is equivalently the 2-category whose
* [[objects]] are $R$-algebras;
* [[1-morphisms]] are [[bimodules]],
* [[2-morphisms]] are [[intertwiners]].
In [[enriched category theory]] this is equivalently the 2-category of [[Mod]]${}_R$-[[enriched categories]] and [[profunctors]] between them. In this context one can write
$$
2Mod_R = Mod(Mod_R)
$$
or $2 Mod_R = $ [[Prof]]$(Mod_R)$ for this 2-category.
## Related constructions
### Algebra objects in $2Mod$
An [[algebra object]] in [[2Mod]]${}_3$ is equivalently a [[sesquiunital sesquialgebra]] over $R$. This may be taken to be a basis for a [[3-module]]/[[3-vector space]] over $R$.
### Line 2-bundles
(...) [[line 2-bundle]] (...)
## Related concepts
* [[module]], [[vector space]]
* [[2-module]]/[[2-vector space]], [[2-representation]]
* [[2-vector bundle]]
* _[[TwoVect]]_ is a Mathematica software package for computer algebra with 2-vector spaces
* [[(∞,1)-module]], [[(∞,1)Mod]]
* [[n-vector space]]
* [[nMod]]
[[!redirects 2Vect]]
|
2T relation | https://ncatlab.org/nlab/source/2T+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[knot theory]] the _2-term relations_ on [[horizontal chord diagrams]] are the following [[relations]] in the [[linear span]] of the set of [[horizontal chord diagrams]]:
For pairwise distinct indices $i,j,k,l$, the $(i,j)$-chord generator commutes with the $(k,l)$-chord generator, for instance:
<center>
<img src="https://ncatlab.org/nlab/files/HorizontalChordDiagram2TRelation.jpg" width="600">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
Together with the horizontal [[4T relations]] these are the relations respected by [[horizontal weight systems]].
## Related concepts
[[!include chord diagrams and weight systems -- table]]
## References
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))
[[!redirects 2T relations]]
[[!redirects 2T-relation]]
[[!redirects 2T-relations]]
[[!redirects 2-term relation]]
[[!redirects 2-term relations]]
|
3-brane in 6d | https://ncatlab.org/nlab/source/3-brane+in+6d |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
There is supposed to be a $(p=3)$-[[brane]] in 6-dimensional [[supersymmetry|super]]-[[spacetime]] given by the [[Green-Schwarz action functional]] induced by the exceptional super Lie algebra $(3+2)$-cocycle on $\mathfrak{siso}(5,1)$ ([Hughes-Liu-Polchinski 86](#HughesLiuPolchinski86)).
This is thought to be the [[intersecting brane|intersection]] locus of two [[M5-branes]] ([Papadopoulos & Townsend 1996](#PapadopoulosTownsend96), [Tseytlin 1996](#Tseytlin96), [Howe, Lambert & West 1998, p. 2](#HoweLambertWest98), [Kachru, Oz & Yin 1998](#KachruOzYin98)), hence the [[duality between M-theory and type IIA string theory|M-theory lift]] of [D4/NS5-brane intersection](NS5-brane#D4EndingOnNS5).
Since this brane has [[codimension]] 2, it is a *[[defect brane]]*.
## Related concepts
* [[D3-brane]], [[self-dual string]]
[[!include brane scan]]
## References
{#References}
The original construction is in
* {#HughesLiuPolchinski86} James Hughes, Jun Liu, [[Joseph Polchinski]], _Supermembranes_, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374 ([spire](http://inspirehep.net/record/20685))
Discussion building on that includes
* [[Martin Rocek]], [[Arkady Tseytlin]], _Partial breaking of global D=4 supersymmetry, constrained superfields, and 3-brane actions_, Phys.Rev.D59:106001, 1999 ([arXiv:hep-th/9811232](http://arxiv.org/abs/hep-th/9811232))
The relevant cocycle for discussion as a [[Green-Schwarz sigma-model]] is given in
* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], (III.7.18) of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific, 1991
(in [[D'Auria-Fré formulation of supergravity]])
Discussion of the [[3-brane in 6d]] explicitly as a [[black brane]] in an [[M5-brane]]/[[NS5-brane]] [[worldvolume]] is due to
* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Threebrane Soliton of the M-Fivebrane_, Phys. Lett. B419 (1998) 79-83 ([arXiv:hep-th/9710033](http://arxiv.org/abs/hep-th/9710033))
and the understanding of this configuration as resulting from two intersecting M5-branes is due to
* {#PapadopoulosTownsend96} [[George Papadopoulos]], [[Paul Townsend]], _Intersecting M-branes_, Phys. Lett. B380 (1996) 273 ([arXiv:hep-th/9603087](http://arxiv.org/abs/hep-th/9603087))
* {#Tseytlin96} [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl. Phys. B475 (1996) 149 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), <a href="https://doi.org/10.1016/0550-3213(96)00328-8">doi:10.1016/0550-3213(96)00328-8</a>)
* [[Hironori Mori]], *M-theory Perspectives on Codimension-2 Defects*, Osaka (2016) [[inspire:1519095](https://inspirehep.net/literature/1519095)]
* [[Hironori Mori]], Yuji Sugimoto, *Surface Operators from M-strings*, Phys. Rev. D **95** 026001 (2017) [[arXiv:1608.02849](https://arxiv.org/abs/1608.02849), [doi:10.1103/PhysRevD.95.026001](https://doi.org/10.1103/PhysRevD.95.026001)]
with a [[matrix model]]-description in:
* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))
For more on this see
* [[Joaquim Gomis]], [[David Mateos]], [[Joan Simón]], [[Paul Townsend]], _Brane-Intersection Dynamics from Branes in Brane Backgrounds_, Phys. Lett. B430 (1998) 231-236 ([arXiv:hep-th/9803040](http://arxiv.org/abs/hep-th/9803040))
See also
* S. Bellucci, N. Kozyrev, S. Krivonos, A Sutulin, *Component on-shell actions of supersymmetric 3-branes: I. 3-brane in $D = 6$, Class. Quantum Grav. **32** (2015) 035025 ([doi:10.1088/0264-9381/32/3/035025](https://iopscience.iop.org/article/10.1088/0264-9381/32/3/035025))
The relation to [[D=4 N=2 super Yang-Mills theory]] is discussed in
* {#HoweLambertWest98} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills_, Phys. Lett. B418 (1998) 85-90 ([arXiv:hep-th/9710034](https://arxiv.org/abs/hep-th/9710034))
* [[Neil Lambert]], [[Peter West]], _Gauge Fields and M-Fivebrane Dynamics_, Nucl. Phys. B524 (1998) 141-158 ([arXiv:hep-th/9712040](https://arxiv.org/abs/hep-th/9712040))
* {#LambertWest98} [[Neil Lambert]], [[Peter West]], _$N=2$ Superfields and the M-Fivebrane_, Phys. Lett. B424 (1998) 281-287 ([arXiv:hep-th/9801104](https://arxiv.org/abs/hep-th/9801104))
* [[Neil Lambert]], [[Peter West]], _Monopole Dynamics from the M-Fivebrane_, Nucl. Phys. B556 (1999) 177-196 ([arXiv:hep-th/9811025](https://arxiv.org/abs/hep-th/9811025))
and via [[F-theory]] in
* Robert de Mello Koch, Alastair Paulin-Campbell, Joao P. Rodrigues, _Monopole Dynamics in $\mathcal{N}=2$ super Yang-Mills Theory From a Threebrane Probe_, Nucl. Phys. B559 (1999) 143-164 ([arXiv:hep-th/9903207](https://arxiv.org/abs/hep-th/9903207))
On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and M3-[[defect branes]] inside [[M5-branes]]:
* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*, J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), <a href="https://doi.org/10.1007/JHEP03(2021)028">doi:10.1007/JHEP03(2021)028</a>]
As [[M5-brane|M5]]-[[probe branes]] in an [AdS7-CFT6](AdS-CFT+correspondence#AdS7CFT6) [[background field|background]] (i.e. in the [[near horizon limit]] of [[black brane|black]] [[M5-branes]]):
* [[Varun Gupta]], *Holographic M5 branes in $AdS_7 \times S^4$*, J. High Energ. Phys. **2021** 32 (2021) [[arXiv:2109.08551](https://arxiv.org/abs/2109.08551), <a href="https://doi.org/10.1007/JHEP12(2021)032">doi:10.1007/JHEP12(2021)032</a>]
* [[Varun Gupta]], *More Holographic M5 branes in $AdS_7 \times S^4$* [[arXiv:2301.02528](https://arxiv.org/abs/2301.02528)]
[[!redirects super 3-brane in 6d]]
[[!redirects M5-brane intersection]]
[[!redirects intersecting M5-branes]]
[[!redirects M3-brane]]
[[!redirects M3-branes]]
[[!redirects M3]]
|
3-category | https://ncatlab.org/nlab/source/3-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A $3$-category is any of several concepts that generalize $2$-[[2-category|categories]] one step in [[higher category theory]]. The original notion is that of a globular [[strict 3-category]], but the one most often used here is that of a [[tricategory]]. The concept generalizes to $n$-[[n-category|categories]].
## Definition
Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$3$-category__ is an $\infty$-category such that every 4-morphism is an [[equivalence]], and all parallel pairs of $j$-morphisms are equivalent for $j \geq 4$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-category, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as [[equality]].
## Specific versions
* globular [[strict 3-category]]
* [[Gray-category]]
* [[tricategory]]
* [[3-groupoid]]
## Examples
* [[3-category of fermionic conformal nets]] ([[2-Clifford algebras]])
## Related concepts
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]]
* **3-category**
* [[n-category]]
* [[(∞,0)-category]]
* [[(∞,1)-category]]
* [[(∞,2)-category]]
* [[(∞,n)-category]]
* [[(n,r)-category]]
[[!redirects 3-categories]]
|
3-category of fermionic conformal nets | https://ncatlab.org/nlab/source/3-category+of+fermionic+conformal+nets |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[categorification]] of the [[2-category]] of [[Clifford algebras]] and [[bimodules]] between them. The "2-Clifford algebra" objects are [[spinning string]] theories, modeled as fermionic [[conformal nets]].
This is supposedly related to the [[String 2-group]] and [[superstrings]] as Clifford algebras are related to the [[spin group]] and [[spinning particles]].
(...)
## References
A somewhat novel take on fermionic [[conformal nets]] is presented and studied in
* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets and local field theory_ ([arXiv:0912.5307](http://arxiv.org/abs/0912.5307))
* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets I: coordinate-free nets_ ([arXiv:1302.2604](http://arxiv.org/abs/1302.2604))
where fermionic conformal nets are arranged into a [[tricategory]] with [[symmetric monoidal category|symmetric monoidal structure]] that is a [[delooping]] of the [[bicategory]] of [[von Neumann algebra]]s and [[bimodule]]s between these. Among other things, this work connects the [[AQFT]] notion of conformal nets with the [[FQFT]] notion of [[cobordism]] representations.
More on this in
* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_
* {#DouglasHenriquesGeometric} [[Chris Douglas]], [[André Henriques]], _Geometric string structures_ ([pdf](http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf))
[[!redirects 2-Clifford algebra]]
[[!redirects 2-Clifford algebras]]
|
3-category with contravariance | https://ncatlab.org/nlab/source/3-category+with+contravariance | # 3-categories with contravariance
* table of contents
{: toc}
## Idea
A *3-category with contravariance* is a [[categorification]] of a [[2-category with contravariance]]. It is an abstract structure that generalizes the structure present on the collection of [[2-categories]] and all four sorts of [[contravariant 2-functor]].
## No mixed-variance transformations
As for [[2-category with contravariance]], the simplest case is when we only consider transformations between functors of the same variance. Thus for each pair of objects $x,y$ we have four disjoint hom-2-categories $hom^{++}(x,y)$, $hom^{+-}(x,y)$, $hom^{-+}(x,y)$, and $hom^{--}(x,y)$, with sixteen composition 2-functors that act like $\mathbb{Z}/2\times \mathbb{Z}/2$ on the gradings and are appropriately variant on their inputs.
Like [[2-categories with contravariance]], these can be described abstractly (at least in the strict case of [[strict 3-categories]], or the [[semistrict n-category|semistrict]] case of [[Gray-categories]]) as [[enriched categories]], in terms of an action of $\mathbb{Z}/2\times \mathbb{Z}/2$ on $2Cat$. See [Shulman 2016](#Shulman2016) for a few details. Depending on whether we use the strict cartesian product on $2 Cat$, the usual "pseudo" Gray tensor product, or its lax or colax versions, we would obtain a structure including strict, pseudo, lax, or colax [[2-natural transformations]] (all between 2-functors of the same variance only).
## Mixed-variance transformations
We can also attempt to include transformations between 2-functors of different variance. In this case our hom-2-categories will have a $\mathbb{Z}/2\times\mathbb{Z}/2$-grading on their objects, but include morphisms between objects of different variance, yielding sixteen different kinds of natural transformation. We could also choose to make these transformations strict, lax, colax, or pseudo. All these kinds of transformation can also be regarded as special cases of [[2-dinatural transformations]]; see there for an example.
It is natural to then conjecture that there is a tensor product on an appropriate category of $\mathbb{Z}/2\times\mathbb{Z}/2$-graded 2-categories giving rise to a notion of enriched category that described this structure.
## References
* [[Mike Shulman]], *Contravariance through enrichment*, [arXiv](https://arxiv.org/abs/1606.05058), 2016
{#Shulman2016}
[[!redirects 3-categories with contravariance]]
|
3-group | https://ncatlab.org/nlab/source/3-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A _3-group_ is equivalently
1. a [[2-truncated]] [[∞-group]];
1. a [[2-groupoid]] $G$ equipped with the structure of a [[loop space object]] of a connected [[3-groupoid]] $\mathbf{B}G$ (its [[delooping]]);
1. a [[monoidal 2-category]] in which every [[object]] has an weak inverse under the tensor product, every 1-morphism has a weak inverse, and every 2-morphism has an inverse.
## Properties
### Presentation by crossed complexes
Some classes of 3-groups are modeled by [[2-crossed modules]] or [[crossed squares]].
## Related concepts
* [[group]]
* [[2-group]], [[crossed module]], [[differential crossed module]]
* **3-group**, [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]
* [[braided 3-group]], [[sylleptic 3-group]], [[symmetric 3-group]]
* [[n-group]]
* [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]
[[!include homotopy n-types - table]]
[[!redirects 3-groups]]
|
3-groupoid | https://ncatlab.org/nlab/source/3-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of **3-groupoid** is the next higher generalization in [[higher category theory]] of [[groupoid]] and [[2-groupoid]].
## Definition
A **3-groupoid** is an [[∞-groupoid]] such that all [[parallel pair]]s of [[k-morphism]] are [[equivalence|equivalent]] for $k \geq 4$: a 3-[[truncated]] [[∞-groupoid]].
Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as [[equality]].
See also [[n-groupoid]].
## Models
A general 3-groupoid is [[geometric definition of higher categories|geometrically modeled]] by a 4-[[simplicial skeleton|coskeletal]] [[Kan complex]]. Equivalently -- via the [[homotopy hypothesis]]-theorem -- by a [[homotopy 3-type]].
A small model of this is a 3-[[hypergroupoid]], where all [[horn]]-filelrs in dimension $\geq 4$ are _unique_ .
A 3-groupoid is [[algebraic definition of higher categories|algebraically modeled]] by a [[tricategory]] in which all morphisms are invertible, and by a 3-[[truncated]] [[algebraic Kan complex]].
A [[semi-strict infinity-category|semistrict]] [[algebraic definition of higher categories|algebraic model]] for 3-groupoids is provided by the notion of [[Gray-groupoid]]. These in turn are encoded by [[2-crossed module]]s.
An entirely strict algebraic model for 3-groupoids (which no longer models all [[homotopy 3-type]]s) is a 3-[[truncated]] [[strict omega-groupoid]].
## Related concepts
[[!include homotopy n-types - table]]
## References ##
* [[Simona Paoli]], Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids, _Journal of Pure and Applied Algebra_ **211** (2007), 801-820. ([arXiv](http://arxiv.org/abs/math/0607330))
* [[Carlos Simpson]], _Homotopy types of strict 3-groupoids_ ([arXiv](http://arxiv.org/abs/math/9810059))
On the [[homotopy hypothesis]] for [[Grothendieck infinity-groupoid|Grothendieck 3-groupoids]]:
* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)].
[[!redirects 3-groupoids]]
|
3-groupoid of Lie 3-algebra valued forms | https://ncatlab.org/nlab/source/3-groupoid+of+Lie+3-algebra+valued+forms |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given a [[smooth manifold]] $U$ and a [[Lie 3-algebra]] $\mathfrak{g}$, the 3-groupoid of Lie 3-algebra valued forms over $U$ has as objects [[∞-Lie algebroid valued differential forms]] with values in $\mathfrak{g}$, as morphisms gauge transformations of these, as 2-morphisms 2-gauge transformations and so on.
This can be understood as the 3-groupoid of _trivial_ $G$-[[principal ∞-bundle|principal 3-bundle]]s over $U$ with nontrivial connection, for $G$ the [[∞-Lie groupoid|3-Lie group]] related to $\mathfrak{g}$ by [[Lie integration]].
Regarded as a presheaf of 3-groupoids over all suitable manifolds $U$, this is a non-concrete [[∞-Lie groupoid|3-Lie groupoid]].
A [[cocycle]] with coefficients in this 3-groupoid is a [[connection on a 3-bundle]].
## Related concepts
* [[groupoid of Lie-algebra valued forms]]
* [[2-groupoid of Lie 2-algebra valued forms]]
* **3-groupoid of Lie 3-algebra valued forms**
* [[∞-groupoid of ∞-Lie-algebra valued forms]]
## References
For Lie 3-algebras coming from [[differential 2-crossed module]]s, at least parts of this data have been discussed in
* {#MartinsPicken11} [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_, Differential Geometry and its Applications **29** 2 (2011) 179-206 [[arXiv:0907.2566](http://arxiv.org/abs/0907.2566), [doi:10.1016/j.difgeo.2010.10.002](https://doi.org/10.1016/j.difgeo.2010.10.002)]
|
3-manifold | https://ncatlab.org/nlab/source/3-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _3-manifold_ is a [[manifold]] of [[dimension]] 3. (Our default meaning of "manifold" is [[topological manifold]], unless a qualifier is added, e.g., *smooth* manifold.)
## Properties
### Triangulability and smoothing
The following is taken from [Hatcher](#Hatcher):
> A pleasant feature of 3-manifolds, in contrast to higher dimensions, is that there is no essential difference between smooth, piecewise linear, and topological manifolds. It was shown by Bing and [Moise](#Moise52) in the 1950s that [[triangulation theorem|every topological 3-manifold can be triangulated]] as a simplicial complex whose combinatorial type is unique up to subdivision. And every triangulation of a 3-manifold can be taken to be a smooth triangulation in some differential structure on the manifold, unique up to diffeomorphism. Thus every topological 3-manifold has a unique smooth structure, and the classifications up to diffeomorphism and homeomorphism coincide.
Thus it makes no essential difference if we consider 3-manifolds as mere topological manifolds, or as [[piecewise-linear manifolds]] or [[smooth manifolds]]. It's often technically convenient to work in the smooth category.
### Poincaré conjecture
+-- {: .num_theorem}
###### Theorem
**([[Poincaré conjecture]])**
Every [[simply connected]] [[compact space|compact]] 3-manifold without boundary is [[homeomorphism|homeomorphic]] to the 3-sphere.
=--
+-- {: .proof}
###### Proof
A proof strategy was given by [[Richard Hamilton]]: imagine the manifold is equipped with a [[metric]]. Follow the [[Ricci flow]] of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.
The hard technical part of this program is to show that the passage through the singularities can be controlled. This was finally shown by [[Grigori Perelman]].
=--
### Geometrization conjecture
The _[[geometrization conjecture]]_ says that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.
### Virtually fibered conjecture
The _[[virtually fibered conjecture]]_ says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
## Related concepts
* [[irreducible 3-manifold]]
* [[atoroidal 3-manifold]]
* [[Seifert 3-manifold]]
* [[hyperbolic 3-manifold]]
* [[Dehn surgery]]
* [[Kirby calculus]]
* [[low dimensional topology]]
* [[knot theory]]
* [[arithmetic topology]]
* [[Chern-Simons theory]]
* [[Atiyah 2-framing]]
* [[lens space]]
* [[associative submanifold]]
* [[2-manifold]], [[4-manifold]], [[8-manifold]]
## References
### General
Review:
* {#Thurston80} [[William Thurston]], _Geometry and topology of three-manifolds_ (1980), electronic version 1.1 (2002) available from MSRI ([web](http://library.msri.org/books/gt3m/))
* {#Hatcher} [[Allen Hatcher]], _The classification of 3-manifolds -- a brief overview_, ([pdf](https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf)).
* {#Martelli16} Bruno Martelli, _An Introduction to Geometric Topology_ ([arXiv:1610.02592](https://arxiv.org/abs/1610.02592))
The [[triangulation theorem]] for [[3-manifolds]]:
* {#Moise52} [[Edwin E. Moise]], *Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung*, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 ([doi:10.2307/1969769](https://doi.org/10.2307/1969769), [jstor:1969769](https://www.jstor.org/stable/1969769))
3-manifolds as [[branched covers]] of the [[3-sphere]]:
* J. Montesinos, _A representation of closed orientable 3-manifolds as 3-fold branched coverings of $S^3$_, Bull. Amer. Math. Soc. 80 (1974), 845-846 ([Euclid:1183535815](https://projecteuclid.org/euclid.bams/1183535815))
See also
* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))
### Hyperbolic 3-manifolds
On [[hyperbolic 3-manifolds]]:
* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds_, Annals of Math, 124 (1986), 203--246 ([jstor:1971277](https://www.jstor.org/stable/1971277), [arXiv:math/9801019](https://arxiv.org/abs/math/9801019))
* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle_ ([arXiv:math/9801045](https://arxiv.org/abs/math/9801045))
* [[William Thurston]], _Three dimensional manifolds, Kleinian groups and hyperbolic geometry_, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 ([euclid.bams/1183548782](https://projecteuclid.org/euclid.bams/1183548782))
### Vafa-Witten theory
Computations of Vafa-Witten invariants of 3-manifolds are given in
* [[Sergei Gukov]], Artan Sheshmani, [[Shing-Tung Yau]], _3-manifolds and Vafa-Witten theory_ ([arXiv:2207.05775](https://arxiv.org/abs/2207.05775)).
[[!redirects 3-manifolds]]
|
3-morphism | https://ncatlab.org/nlab/source/3-morphism |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* [[object]]
* [[morphism]]
* [[2-morphism]]
* **3-morphism**
* [[k-morphism]]
***
#Contents#
* table of contents
{:toc}
## Definition
A _3-morphism_ in an [[n-category]] is a morphism between [[2-morphism]]s.
## Examples
For $A$ an [[abelian group]], the 3-fold [[delooping]] $\mathbf{B}^3 A$ is a [[3-groupoid]] whose 3-morphisms are given by $A$.
[[!redirects 3-morphisms]]
|
3-poset | https://ncatlab.org/nlab/source/3-poset | A $3$-poset is any of several concepts that generalize $2$-[[2-poset|posets]] one step in [[higher category theory]]. One does not usually here about $3$-posets by themselves but instead as special cases of $3$-[[3-category|categories]].
$3$-posets can also be called $(2,3)$-[[(n,r)-category|categories]]. The concept generalizes to $n$-[[n-poset|posets]].
# Definition
Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$3$-poset__ is an $\infty$-category such that all parallel pairs of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 3$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $3$-morphisms, not even whether two given parallel $3$-morphisms are equivalent. |
3-sphere | https://ncatlab.org/nlab/source/3-sphere |
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###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The $n$-[[sphere]] for $n = 3$.
## Properties
### Isomorphisms
The underlying manifold of the [[special unitary group]] [[SU(2)]] happens to be [[isomorphism|isomorphic]] to the 3-sphere, hence also that of [[spin group|Spin(3)]].
The [[quotient]] of that by the [[binary icosahedral group]] is the [[Poincaré homology sphere]].
### Homotopy groups
{#HomotopyGroups}
The first few [[homotopy groups]] of the 3-sphere:
| $n =$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
|-----|---|---|---|--|--|---|---|---|--|--|--|--|--|
| $\pi_n(S^3) =$ | $\ast$ | $0$ | $0$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_3$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ |
e.g. [Calabrese 16](#Calabrese16), for more see at _[[homotopy groups of spheres]]_.
## Related concepts
* [[Hopf fibration]]
* [[fuzzy 3-sphere]]
* [[2-sphere]]
* [[6-sphere]]
* [[7-sphere]]
* [[n-sphere]]
## References
Discussion of [[homotopy groups of spheres]] for the 3-sphere:
* {#Calabrese16} John Calabrese, _The fourth homotopy group of the sphere_, 2016 ([pdf](https://math.rice.edu/~jrc9/stuff/sss.pdf))
Discussion of [[3-manifolds]] as [[branched covers]] of the 3-sphere:
* J. Montesinos, _A representation of closed orientable 3-manifolds as 3-fold branched coverings of $S^3$_, Bull. Amer. Math. Soc. 80 (1974), 845-846 ([Euclid:1183535815](https://projecteuclid.org/euclid.bams/1183535815))
Classification of [[Riemannian orbifolds]] whose coarse underlying topological space is a 3-sphere:
* [[William Dunbar]], _Geometric orbifolds_, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)
[[!redirects 3-spheres]] |
3-strand-braid-1-SVG | https://ncatlab.org/nlab/source/3-strand-braid-1-SVG |
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|
3-theory | https://ncatlab.org/nlab/source/3-theory |
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###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
* table of contents
{: toc}
## Idea
In the context of an account of [[logic]] described in terms of $n$-theories ([Shulman 18a](#Shulman18a)), a *3-theory* specifies a kind of [[2-category]].
It does so by prescribing the allowable judgment structures of the [[2-theories]] that can be expressed within it.
## Examples
1. The 3-theory of (totally unstructured) 2-categories. Syntactically, this corresponds to "unary type (2-)theories" with judgments such as $x:A \vdash b:B$ with exactly one type to both the left and the right of the turnstile ([LS 15](#LS15)).
1. The 3-theory of cartesian monoidal 2-categories (or cartesian 2-multicategories). Syntactically, this corresponds to "simple type (2-)theories" with judgments such as $x:A, y:B, z:C \vdash d:D$ with a finite list of types to the left but exactly one type to the right, and no dependent types ([LSR 17](#LSR17)).
1. The 3-theory of first-order logic. Syntactically, this corresponds to type 2-theories with one layer of dependency: there is one layer of types which cannot depend on anything, and then there is a second layer of types (“propositions” in the logical reading) which can depend on types in the first layer, but not on each other. 2-theories here are for hyperdoctrines and indexed monoidal categories, and include (typed) first-order classical logic, first-order intuitionistic logic, first-order linear logic, the type theory of indexed monoidal categories, first-order modal logic, etc. Semantic 2-theories in this 3-theory should be a sort of “2-hyperdoctrine”. A [[higher-order logic]] is a 2-theory which adds rules specifying a universe of propositions.
1. The 3-theory that corresponds syntactically to "dependent type (2-)theories" with judgments such as $x:A, y:B(x), z:C(x, y) \vdash d:D(x, y, z)$. The semantic version of this should be some kind of "comprehension 2-category" ([Shulman 18b](#Shulman18b)). [[Homotopy type theory]] is a 2-theory defined in this 3-theory.
1. The 3-theory that corresponds syntactically to "classical simple type (2-)theories" with judgments such as $A,B,C \vdash D,E$ that allow finite lists of types on both sides of the turnstile. For instance, the 2-theory of $\ast$-autonomous categories (classical linear logic), and also of Boolean algebras (classical nonlinear logic), live in this 3-theory, as does the 2-theory for symmetric monoidal categories described in ([Shulman 19](#Shulman19)). Variants would allow composition along multiple types at once (corresponding to a prop), composition only along one type at a time (corresponding to a polycategory), or both.
1. The classical 3-theory that extends the first-order 3-theory by allowing multiple sequents. (The term 'classical' here and above owes its origin to the contrast between systems LK and LJ of [[Gentzen's]] [[sequent calculus]], where LJ provides a proof system for [[intuitionistic logic]] via restriction to single sequents of the rules of LK for classical logic. This naming practice leads to a certain confusion as it is quite possible to represent classical first-order logic in a single sequent system such as Gentzen's NJ.)
A 3-theory may be said to subsume another 3-theory, in the sense that a [[2-theory]] written in the latter may be formulated in the former. Here the first four 3-theories are ordered in terms of increasing expressivity. (5) subsumes (2), and (6) subsumes (3), but (4) and (5) are not comparable.
Modal variants of 2-theories may be defined in 3-theories where the types appearing in judgments are indexed by a system (2-category) of modes ([Shulman 18b](#Shulman18b)).
## References
* {#Shulman18a} [[Mike Shulman]], _What Is an n-Theory?_, ([blog post](https://golem.ph.utexas.edu/category/2018/04/what_is_an_ntheory.html))
* {#LS15} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_, in _[Logical Foundations of Computer Science 2016](http://lfcs.info/lfcs-2016/)_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf), [slides](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint-lfcs-slides.pdf))
* {#LSR17} [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))
* {#Shulman18b} [[Mike Shulman]], _Type 2-theories_, ([HoTTEST seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html))
* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/current-trimester-program/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))
* {#Shulman19} [[Mike Shulman]], _A practical type theory for symmetric monoidal categories_, ([arXiv:1911.00818](https://arxiv.org/abs/1911.00818)) |
3d Calabi-Yau object | https://ncatlab.org/nlab/source/3d+Calabi-Yau+object |
#Contents#
* table of contents
{:toc}
## Idea
A [[Calabi-Yau object]] ([[Calabi-Yau manifold]], [[Calabi-Yau category]]) of (complex) [[dimension]] 3.
## Properties
### As exceptional geometry
[[!include Spin(8)-subgroups and reductions -- table]]
### Moduli spaces of line bundles
[[!include moduli of higher lines -- table]]
## Related concepts
* [[Calabi-Yau cohomology]]
## References
### Intermediate Jacobian
Discussion of [[intermediate Jacobians]] of Calabi-Yau 3-folds includes
* C. Herbert Clemens, [[Phillip Griffith]], _The intermediate Jacobian of the cubic threefold_, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 ([JSTOR](http://www.jstor.org/stable/1970801))
* [[Claire Voisin]] ([pdf](http://www.math.polytechnique.fr/~voisin/Articlesweb/griffithsgroup.pdf))
* [[Andreas Höring]], _Minimal classes on the intermediate Jacobian of a generic cubic threefold_, 2008 ([pdf](http://math.unice.fr/~hoering/articles/a5-intermediate.pdf))
Discussion of the [[Artin-Mazur groups]] of CY3s in [[positive number|positive]] [[characteristic]]:
* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_ ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))
### Hall algebra
Discussion of [[motivic Hall algebras]] of CY 3-folds is in
* {#KontsevichSoibelman08} [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_ ([arXiv:0811.2435](http://arxiv.org/abs/0811.2435))
[[!redirects 3d Calabi-Yau space]]
[[!redirects Calabi-Yau 3-fold]]
[[!redirects Calabi-Yau 3-folds]]
|
3d Chern-Simons theory | https://ncatlab.org/nlab/source/3d+Chern-Simons+theory |
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###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 3.
## Examples
* ordinary [[Chern-Simons theory]]
* [[Dijkgraaf-Witten theory]]
* [[Courant sigma-model]]
* [[3d quantum gravity]]
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* **3d Chern-Simons theory**
* [[3d TQFT]]
* [[3d quantum gravity]]
* [[2d Wess-Zumino-Witten theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[11d Chern-Simons theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[string field theory]]
* [[AKSZ sigma-model]]
[[!redirects 3d Chern-Simons theories]]
[[!redirects 3-dimensional Chern-Simons theory]]
[[!redirects 3-dimensional Chern-Simons theories]]
|
3d gravity and Chern-Simons theory -- references | https://ncatlab.org/nlab/source/3d+gravity+and+Chern-Simons+theory+--+references | ### 3d Gravity and Chern-Simons theory
{#3dGravityAndChernSimonsTheoryReferences}
On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:
The original articles on [[3d gravity]], discussing its formulation as a [[Chern-Simons theory]] and discovering its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):
* [[Stanley Deser]], [[Roman Jackiw]], [[Gerard 't Hooft]], _Three-dimensional Einstein gravity: Dynamics of flat space_, Ann. Phys. 152 (1984) 220 (<a href="https://doi.org/10.1016/0003-4916(84)90085-X">doi:10.1016/0003-4916(84)90085-X</a>)
* [[Stanley Deser]], [[Roman Jackiw]], _Three-dimensional cosmological gravity: Dynamics of constant curvature_, Annals of Physics, Volume 153, Issue 2, 1 April 1984, Pages 405-416 (<a href="https://doi.org/10.1016/0003-4916(84)90025-3">doi:10.1016/0003-4916(84)90025-3</a>, [spire:192694](http://inspirehep.net/record/192694))
* [[Roman Jackiw]], _Lower dimensional gravity_, Nuclear Physics B Volume 252, 1985, Pages 343-356 (<a href="https://doi.org/10.1016/0550-3213(85)90448-1">doi:10.1016/0550-3213(85)90448-1</a>, [spire:204694](http://inspirehep.net/record/204694))
* J. D. Brown, [[Marc Henneaux]], _Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity_, Commun. Math. Phys. (1986) 104: 207 ([doi:10.1007/BF01211590](https://doi.org/10.1007/BF01211590))
* A. Achucarro, [[Paul Townsend]], _A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories_, Phys. Lett. B180 (1986) 89 (<a href="https://doi.org/10.1016/0370-2693(86)90140-1">doi:10.1016/0370-2693(86)90140-1</a>, [spire:21208](http://inspirehep.net/record/21208))
* [[Steven Carlip]], _Inducing Liouville theory from topologically massive gravity_, Nuclear Physics B Volume 362, Issues 1–2, 16 September 1991, Pages 111-124 (<a href="https://doi.org/10.1016/0550-3213(91)90558-F">doi:10.1016/0550-3213(91)90558-F</a>)
* O. Coussaert, [[Marc Henneaux]], P. van Driel, _The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant_, Class. Quant. Grav. 12 (1995) 2961-2966 ([arXiv:gr-qc/9506019](https://arxiv.org/abs/gr-qc/9506019))
The corresponding [[non-perturbative QFT|non-perturbative]] [[quantization]] of 3-dimensional gravity, via [[quantization of 3d Chern-Simons theory]]:
* {#Witten88} [[Edward Witten]], _(2+1)-Dimensional Gravity as an Exactly Soluble System_ Nucl. Phys. B311 (1988) 46. ([web](http://adsabs.harvard.edu/abs/1988NuPhB.311...46W))
* [[Herman Verlinde]], _Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space_, Nuclear Physics B
Volume 337, Issue 3, 25 June 1990, Pages 652-680 (<a href="https://doi.org/10.1016/0550-3213(90)90510-K">doi:10.1016/0550-3213(90)90510-K</a>)
Review:
* [[Steven Carlip]], _Lectures in (2+1)-Dimensional Gravity_, J. Korean Phys. Soc. 28: S447-S467, 1995 ([arXiv:gr-qc/9503024](https://arxiv.org/abs/gr-qc/9503024))
* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions_, Cambridge Monographs on Mathematical Physics (2003) ([publisher](http://www.cambridge.org/de/academic/subjects/physics/cosmology-relativity-and-gravitation/quantum-gravity-21-dimensions))
* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe_, Living Rev. Rel. 8:1, 2005 ([arXiv:gr-qc/0409039](https://arxiv.org/abs/gr-qc/0409039))
* [[Steven Carlip]], _[My Research -- (2+1)-Dimensional quantum gravity](http://www.physics.ucdavis.edu/Text/Carlip.html#2+1)_
* [[Laura Donnay]], _Asymptotic dynamics of three-dimensional gravity_, Proceedings of *Eleventh Modave Summer School in Mathematical Physics*, POS **271** (2016) $[$[arXiv:1602.09021](http://arxiv.org/abs/1602.09021), [doi:10.22323/1.271.0001 ](https://doi.org/10.22323/1.271.0001 )$]$
* [[Wout Merbis]], _Chern-Simons-like Theories of Gravity_ ([arXiv:1411.6888](https://arxiv.org/abs/1411.6888))
* Chen-Te Ma, *$AdS_3$ Einstein Gravity and Boundary Description: Pedagogical Review* $[$[arXiv:2310.04665](https://arxiv.org/abs/2310.04665)$]$
Further developments:
* {#Witten} [[Edward Witten]], _Three-dimensional gravity revisited_, (2007) [arxiv/0706.3359](http://arxiv.org/abs/0706.3359)
* [[Paul Townsend]], _Massive 3d (super)gravity_, slides, ([pdf](http://superfields.web.cern.ch/Superfields/docs/Seminars/Townsend.pdf))
* Gaston Giribet, _Black hole physics and AdS3/CFT2 correspondence_, lectures at [[Croatian Black Hole School]] 2010
* Alan Garbarz, Gaston Giribet, Yerko Vásquez, _Asymptotically AdS$_3$ solutions to topologically massive gravity at special values of the coupling constants_, [arxiv/0811.4464](http://arxiv.org/abs/0811.4464)
* Rudranil Basu, Samir K Paul, _Consistent 3D Quantum Gravity on Lens Spaces_ ([arXiv:1109.0793](http://arxiv.org/abs/1109.0793))
* [[Marc Henneaux]], [[Wout Merbis]], Arash Ranjbar, _Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions_ ([arXiv:1912.09465](https://arxiv.org/abs/1912.09465))
* Viraj Meruliya, [[Sunil Mukhi]], Palash Singh, _Poincaré Series, 3d Gravity and Averages of Rational CFT_ ([arXiv:2102.03136](https://arxiv.org/abs/2102.03136))
* [[Scott Collier]], [[Lorenz Eberhardt]], [[Mengyang Zhang]], *3d gravity and Chern-Simons theory -- references* $[$[arXiv:2304.13650](https://arxiv.org/abs/2304.13650)$]$
See also:
* Nathan Benjamin, Scott Collier, Alexander Maloney, *Pure Gravity and Conical Defects*, Journal of High Energy Physics **2020** 34 (2020) $[$[arXiv:2004.14428](https://arxiv.org/abs/2004.14428), <a href="https://doi.org/10.1007/JHEP09(2020)034">doi:10.1007/JHEP09(2020)034</a>$]$
|
3d mirror symmetry | https://ncatlab.org/nlab/source/3d+mirror+symmetry |
## Related concepts
* [[symplectic duality]]
* [[duality in physics]]
## References
The original article is
* [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_ ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))
based on the [[ADE classification]] of [[ALE spaces]] due to
* [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))
Review includes
* Wikipedia, _[3d Mirror symmetry](https://en.wikipedia.org/wiki/3D_mirror_symmetry)_
* Giulia Ferlito, _Coulomb branch and the moduli space of instantons_, 2015 ([pdf](http://www.maths.liv.ac.uk/TheorPhys/RESEARCH/STRING_THEORY/2015/Slides/GiuliaFerlito.pdf))
Further developments include
* [[Mina Aganagic]], [[Kentaro Hori]], [[Andreas Karch]], [[David Tong]], _Mirror Symmetry in 2+1 and 1+1 Dimensions_, JHEP 0107:022,2001 ([arXiv:hep-th/0105075](http://arxiv.org/abs/hep-th/0105075))
|
3d quantum gravity | https://ncatlab.org/nlab/source/3d+quantum+gravity |
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## Idea
Solving the problem of [[quantization]] of systems including [[gravity]] -- [[quantum gravity]] -- is notoriously hard. It does however simplify drastically in very low [[dimensions]]. While the theory in dimension 4 ([[Kaluza-Klein mechanism|or higher]]) is evidently relevant for the phenomenology of the world that we perceive, the [[Einstein-Hilbert action]] and its variants that defines the theory of [[gravity]] (at least as an [[effective QFT]]) makes sense in any dimension.
The case of dimension 3 is noteworthy, because in this case the quantum theory can be and has been fairly completely understood and is nevertheless non-trivial. Informally, this is due to the fact that behaviour of [[gravity]] in 3-dimensions is much simpler than in higher dimensions: there cannot be [[gravitational waves]] in 3-dimensions, hence no "local excitations". Accordingly, the theory turns out to have a _finite dimensional_ [[covariant phase space]].
More formally, one finds that in 3-dimensions, the [[Einstein-Hilbert action]], in the [[first order formulation of gravity]] ([[Cartan connection]]), becomes equivalent to the [[action functional]] of a very well studied 3-dimensional field theory, namely [[Chern-Simons theory]] for [[gauge group]] the [[Poincaré group]] $Iso(2,1)$ (for vanishing [[cosmological constant]]) or the [[de Sitter group]]/[[anti de Sitter group]] $Iso(2,2)$ or $Iso(3,1)$ (for non-vanishing cosmological constant). See at _[[de Sitter gravity]]_ and at _[Chern-Simons Gravity](http://ncatlab.org/nlab/show/Chern-Simons+gravity)_.
This means that one can take the [[quantization]] of $Iso(2,1)$-[[Chern-Simons theory]] as the _definition_ of 3d [[quantum gravity]]. This has first been noticed and successfully carried out in ([Witten88](#Witten88)). One should note here that this means that one allows degenerate [[vielbein]]/[[pseudo-Riemannian metric]] tensors as field configurations of gravity. In fact, as ([Witten88](#Witten88)) discusses in detail, this [[compactification]] of configuration space can be seen as the source of the reasons why 3d quantum gravity makes exists.
Based on this striking situation in 3-dimensions it is natural to wonder if it makes sense to consider $Iso(n,1)$-[[higher dimensional Chern-Simons theory]], or variants thereof, as theories of [[quantum gravity]] in higher dimensions. The for $n \gt 2$ the corresponding [[action functional]]s differ from the [[Einstein-Hilbert action]] by higher curvature terms, but in suitable limits of the theory these can be argued to play a negligible role. For more on this see the entry _[[Chern-Simons gravity]]_ .
## Variants
One can add additional terms arriving at what is called _massive 3d gravity models_ . Very relevant for its study is the [[holographic principle|AdS3/CFT2 correspondence]].
## Related concepts
* [[BTZ black hole]]
* [[3d TQFT]]
* [[3-dimensional supergravity]]
* [[quantum gravity]]
* [[Chern-Simons gravity]]
* [[Boulatov model]]
* [[2d quantum gravity]]
* [[volume conjecture]]
* [[piecewise flat spacetime]]
## References
* Nathan Benjamin, Scott Collier, Alexander Maloney, *Pure Gravity and Conical Defects*, Journal of High Energy Physics **2020** 34 (2020) [[arXiv:2004.14428](https://arxiv.org/abs/2004.14428), <a href="https://doi.org/10.1007/JHEP09(2020)034">doi:10.1007/JHEP09(2020)034</a>]
* Mauricio Leston, et al., *3d Quantum Gravity Partition Function at 3 Loops: a brute force computation* [[arXiv:2307.03830](https://arxiv.org/abs/2307.03830)]
Survey:
* [[Steve Carlip]], *Quantum Gravity in 2+1 Dimensions*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12596](https://arxiv.org/abs/2312.12596)]
[[!include 3d gravity and Chern-Simons theory -- references]]
[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]
[[!redirects 3d gravity]]
[[!redirects D=3 gravity]]
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3d superconformal gauge field theory | https://ncatlab.org/nlab/source/3d+superconformal+gauge+field+theory |
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## Idea
According to the [classification of superconformal symmetry](supersymmetry#ClassificationSuperconformal), there should exists [[superconformal field theories]] in 3 dimensions
[[!include superconformal symmetry -- table]]
Such superconformal [[gauge field theories]] involving [[Chern-Simons theory]] coupled to matter exist and are thought to describe the [[worldvolume]] theory of [[black brane|black]] [[M2-branes]] located at [[ADE-singularities]]:
[[!include 7d spherical space forms -- table]]
The corresponding story for the [[M5-brane]] is the [[6d superconformal gauge field theory]].
## Related concepts
* [[D=3 N=2 super Yang-Mills theory]]
* [[D=3 N=4 super Yang-Mills theory]]
## References
The original article on the $N=8$-case (the [[BLG model]]):
* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).
* [[Jonathan Bagger]], [[Neil Lambert]], Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).
The original article on the $N=6$-case (the [[ABJM model]]):
* {#ABJM08} [[Ofer Aharony]], [[Oren Bergman]], Daniel Louis Jafferis, [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_, JHEP 0810:091,2008, [DOI:10.1088/1126-6708/2008/10/091](http://iopscience.iop.org/article/10.1088/1126-6708/2008/10/091/meta;jsessionid=FCE6764D4E19F3038C9530E50B057A56.c3.iopscience.cld.iop.org) ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218))
The $\mathcal{N}=5$-case is discussed in
* {#HLLLP08} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds_, JHEP 0809:002, 2008 ([arXiv:0806.4977](https://arxiv.org/abs/0806.4977))
* {#BHRSS08} [[Eric Bergshoeff]], [[Olaf Hohm]], Diederik Roest, [[Henning Samtleben]], [[Ergin Sezgin]], _The Superconformal Gaugings in Three Dimensions_, JHEP0809:101, 2008 ([arXiv:0807.2841](https://arxiv.org/abs/0807.2841))
* {#AharonyBergmanJafferis08} [[Ofer Aharony]], Oren Bergman, Daniel Louis Jafferis, _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))
The $N=4$-case is discussed in
* {#HLLLP08b} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=4$ Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets_, JHEP 0807:091,2008 ([arXiv:0805.3662](https://arxiv.org/abs/0805.3662))
* {#ChenWu10} Fa-Min Chen, Yong-Shi Wu, _Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories_, Phys.Rev.D82:106012, 2010 ([arXiv:1007.5157](https://arxiv.org/abs/1007.5157))
Review includes
* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350, 2010 ([arXiv;0909.1580](https://arxiv.org/abs/0909.1580))
* Neil B. Copland, _Introductory Lectures on Multiple Membranes_ ([arXiv:1012.0459](https://arxiv.org/abs/1012.0459))
* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 2013 ([arXiv:1203.3546](https://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))
For more see at _[[ABJM model]]_.
[[!redirects D=3 SYM]]
[[!redirects D=3 super Yang-Mills theory]]
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3d toric code | https://ncatlab.org/nlab/source/3d+toric+code |
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## Idea
What has come to be called the "3d toric code" is a [[4d TQFT]] (i.e. with 3 spatial dimensions, whence the name) which is the higher-dimensional analog of the [[3d TQFT]] known as the *[[toric code]]*.
## References
The model is due to:
* Alioscia Hamma, Paolo Zanardi, [[Xiao-Gang Wen]], *String and Membrane condensation on 3D lattices*, Phys. Rev. B72:035307, 2005 ([arXiv:cond-mat/0411752](https://arxiv.org/abs/cond-mat/0411752), [doi:10.1103/PhysRevB.72.035307](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.035307))
Further discussion of [[defect QFT|defects]] in the model (such as [[defect brane|defect strings]]):
* {#KongTianZhang20} [[Liang Kong]], Yin Tian, Zhi-Hao Zhang, *Defects in the 3-dimensional toric code model form a braided fusion 2-category*, J. High Energ. Phys. 2020, 78 (2020) ([arXiv:2009.06564](https://arxiv.org/abs/2009.06564), <a href="https://doi.org/10.1007/JHEP12(2020)078">doi:10.1007/JHEP12(2020)078</a>)
[[!redirects 3d toric codes]]
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3d TQFT | https://ncatlab.org/nlab/source/3d+TQFT |
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## Idea
[[TQFT]] of [[dimension]] 3.
## Constructions
From [[3-modules]] realized as [[sesquialgebra]], [[bialgebras]], [[Hopf algebra]], [[quantum groups]], [[fusion categories]]:
* [[Reshetikhin-Turaev model]]
* [[Turaev-Viro model]]
## Examples
* [[3d Chern-Simons theory]]
* [[Dijkgraaf-Witten theory]]
* [[analytically continued Chern-Simons theory]]
* [[Courant sigma-model]]
* [[Levin-Wen model]]
## Related concepts
* [[TQFT]]
* [[2d TQFT]]
* [[2d Chern-Simons theory]]
* [[3d TQFT]]
* [[3d Chern-Simons theory]]
* [[4d TQFT]]
* [[4d Chern-Simons theory]]
## References
{#References}
* [[Kevin Walker]], _On Witten's 3-manifold invariants_ (1991) [pdf](http://tqft.net/other-papers/KevinWalkerTQFTNotes.pdf)
{#Walker1991}
* {#CarterKauffmanSaito96} [[Scott Carter]], [[Louis Kauffman]], Masahico Saito, _Diagrams, Singularities, and Their Algebraic Interpretations_, in "10th Brazilian Topology Meeting, S. Carlos, July 22 26, 1996 ([pdf](http://homepages.math.uic.edu/~kauffman/cksBrasil.pdf) (dead))
Discussion via the [[cobordism hypothesis]] with [[fusion categories]] as [[fully dualizable objects]] is in
* {#DSPS} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _The Structure of Fusion Categories via 3D TQFTs_ ([talk pdf](https://sites.google.com/site/chrisschommerpriesmath/Home/recent-and-upcoming-talks/UPennTalk.pdf?attredirects=0))
* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))
* [[Bruce Bartlett]], [[Christopher Douglas]], [[Christopher Schommer-Pries]], [[Jamie Vicary]], _Extended 3-dimensional bordism as the theory of modular objects_ ([arXiv:1411.0945](http://arxiv.org/abs/1411.0945))
Constructions in the generality of non-[[semisimple category|semisimple]] [[tensor categories]]:
* Christian Blanchet, Marco De Renzi, _Modular Categories and TQFTs Beyond Semisimplicity_ ([arXiv:2011.12932](https://arxiv.org/abs/2011.12932))
[[!redirects 3d TQFTs]]
[[!redirects 3d topological field theory]]
[[!redirects 3d topological field theories]]
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3d-3d correspondence | https://ncatlab.org/nlab/source/3d-3d+correspondence |
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## Idea
A [[duality in string theory]]: relation between the
[[D=3 N=2 super Yang-Mills theory|D=3 N=2 SYM]]
[[worldvolume]] [[quantum field theory]] on [[M5-brane]] [[wrapped brane|wrapped]] on a ([[hyperbolic 3-manifold|hyperbolic]]) [[3-manifold]] and 3d [[Chern-Simons theory]]/[[analytically continued Chern-Simons theory]].
## Properties
### Relation to volume conjecture
{#RelationToTheVolumeConjecture}
In [Gang-Kim-Lee 14b, 3.2](#GangKimLee14b), [Gang-Kim 18 (21)](#GangKim18) it is argued that the [[volume conjecture]] for [[Chern-Simons theory]] on [[hyperbolic 3-manifolds]] $\Sigma^3$ is the combined statement of two [[dualities in string theory]]
1. [[AdS/CFT duality]]
1. [[3d-3d correspondence]]
for the situation of [[M5-branes]] [[wrapped brane|wrapped on]] $\Sigma^3$ ([DGKV 10](#DGKV10)):
<center>
<img src="https://ncatlab.org/nlab/files/VolumeConjectureAsAdSCFTPlus3d3dDualityII.jpg" width="650">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
## Related concepts
[[wrapped brane|Wrapping]] the [[M5-brane]] on a [[torsion subgroup|torsion]] 3-cycle yields: [[fractional M2-brane]].
[[wrapped brane|Wrapping]] the [[M5-brane]] instead on a 2-manifold yields: [[AGT correspondence]].
* [[McKay correspondence]]
* [[AGT correspondence]]
* [[knots-quivers correspondence]]
## References
### General
Original articles include
* [[Yuji Terashima]], [[Masahito Yamazaki]], _$SL(2,\mathbb{R})$ Chern-Simons, Liouville, and Gauge Theory on Duality Walls_, JHEP 1108:135, 2011 ([arXiv:1103.5748](https://arxiv.org/abs/1103.5748))
* [[Yuji Terashima]], [[Masahito Yamazaki]], _Semiclassical Analysis of the 3d/3d Relation_, Phys.Rev.D88:026011, 2013 ([arXiv:1106.3066](https://arxiv.org/abs/1106.3066))
* [[Tudor Dimofte]], [[Davide Gaiotto]], [[Sergei Gukov]], _Gauge Theories Labelled by Three-Manifolds_ ([arXiv:1108.4389](https://arxiv.org/abs/1108.4389))
* [[Tudor Dimofte]], [[Davide Gaiotto]], [[Sergei Gukov]], _3-Manifolds and 3d Indices_ ([arXiv:1112.5179](https://arxiv.org/abs/1112.5179))
* [[Dongmin Gang]], Eunkyung Koh, Sangmin Lee, [[Jaemo Park]], _Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus_, JHEP01(2014)063 ([arXiv:1305.0937](https://arxiv.org/abs/1305.0937))
* [[Junya Yagi]], _3d TQFT from 6d SCFT_, JHEP08(2013)017 ([arXiv:1305.0291](https://arxiv.org/abs/1305.0291))
* Sungjay Lee, Masahito Yamazaki, _3d Chern-Simons Theory from M5-branes_, JHEP1312:035, 2013 ([arXiv:1305.2429](https://arxiv.org/abs/1305.2429))
* [[Clay Cordova]], [[Daniel Jafferis]], _Complex Chern-Simons from M5-branes on the Squashed Three-Sphere_ ([arXiv:1305.2891](https://arxiv.org/abs/1305.2891))
* [[Du Pei]], Ke Ye, _A 3d-3d appetizer_, JHEP 11 (2016) 008 ([arXiv:1503.04809](https://arxiv.org/abs/1503.04809))
* Sungbong Chun, [[Sergei Gukov]], Sunghyuk Park, Nikita Sopenko, _3d-3d correspondence for mapping tori_ ([arxiv:1911.08456](https://arxiv.org/abs/1911.08456))
* Hee-Joong Chung, _Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds_ ([arXiv:1912.13486](https://arxiv.org/abs/1912.13486))
Specifically for [[Seifert 3-manifolds]] (such as [[lens spaces]]):
* [[Du Pei]], _3d-3d correspondence for Seifert manifolds_, 2016 ([spire:1469350](https://inspirehep.net/literature/1469350), [pdf](https://inspirehep.net/files/d509ff9e32448da3a5674f286b93224a))
Review is in
* {#Dimofte14} [[Tudor Dimofte]], _3d Superconformal Theories from Three-Manifolds_, In: [[Jörg Teschner]] (ed.), *Exact Results on $\mathcal{N} = 2$ Supersymmetric Gauge Theories*, Springer 2015 ([arXiv:1412.7129](https://arxiv.org/abs/1412.7129), [doi:10.1007/978-3-319-18769-3_11](https://doi.org/10.1007/978-3-319-18769-3_11))
Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]] and the [[3d-3d correspondence]]:
* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))
* Sam van Leuven, Gerben Oling, _Generalized Toda Theory from Six Dimensions and the Conifold_, J. High Energ. Phys. (2017) 2017: 50 ([arxiv:1708.07840](https://arxiv.org/abs/1708.07840))
See also:
* Julius Eckhard, Heeyeon Kim, [[Sakura Schafer-Nameki]], Brian Willett, _Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence_ ([arxiv:1910.14086](https://arxiv.org/abs/1910.14086))
* Yale Fan, *3D-3D Correspondence from Seifert Fibering Operators* ([arXiv:2008.13202](https://arxiv.org/abs/2008.13202))
### Black 5-branes and AdS/CFT
Discussion of [[D=11 N=1 supergravity]] solutions for the [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] $\Sigma^3 = H^3/\Gamma$:
* {#DGKV10} Aristomenis Donos, [[Jerome Gauntlett]], [[Nakwoo Kim]], Oscar Varelam, _Wrapped M5-branes, consistent truncations and AdS/CMT_, JHEP 1012:003, 2010 ([arXiv:1009.3805](https://arxiv.org/abs/1009.3805))
* {#GangKimLee14a} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of Wrapped M5-branes and Chern-Simons theory_, Physics Letters B, Volume 733, 2 June 2014, Pages 316-319 ([arXiv:1401.3595](https://arxiv.org/abs/1401.3595))
* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))
* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))
* {#GangKimPandoZayas19} [[Dongmin Gang]], [[Nakwoo Kim]], Leopoldo A. Pando Zayas, _Precision Microstate Counting for the Entropy of Wrapped M5-branes_ ([arXiv:1905.01559](https://arxiv.org/abs/1905.01559))
### Relation to perturbative CS-observables
Relation to [[perturbative quantization of 3d Chern-Simons theory]]:
specifically to [[Reidemeister torsion]]:
* [[Dongmin Gang]], Seonhwa Kim, Seokbeom Yoon, _Adjoint Reidemeister torsions from wrapped M5-branes_ ([arXiv:1911.10718](https://arxiv.org/abs/1911.10718))
### Relation to volume conjecture
Discussion of the [[volume conjecture]] by combining the 3d/3d correspondence with [[AdS/CFT]] in these backgrounds:
* [Gang-Kim-Lee 14b, Section 3.2](#GangKimLee14b)
* [Gang-Kim 18, around (21)](#GangKim18)
Enhanced to a [[defect field theory]]:
* [[Dongmin Gang]], [[Nakwoo Kim]], Mauricio Romo, Masahito Yamazaki, _Aspects of Defects in 3d-3d Correspondence_, J. High Energ. Phys. (2016) ([arXiv:1510.05011](https://arxiv.org/abs/1510.05011))
More in:
* Jin-Beom Bae, [[Dongmin Gang]], Jaehoon Lee, _3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes_, JHEP 08 (2017) 118 ([arXiv:1610.09259](https://arxiv.org/abs/1610.09259))
### Knot invariants
For discussion of [[knot invariants]]:
* [[Sergei Gukov]], Pavel Putrov, [[Cumrun Vafa]], _Fivebranes and 3-manifold homology_, J. High Energ. Phys. (2017) 2017: 71 ([arXiv:1602.05302](https://arxiv.org/abs/1602.05302))
### Entropy computation
Applied to computation of [[Bekenstein-Hawking entropy]] for [[black holes in string theory]]:
* [Gang-Kim-Pando-Zayas 19](#GangKimPandoZayas19)
* Francesco Benini, Dongmin Gang, Leopoldo A. Pando Zayas, _Rotating Black Hole Entropy from M5 Branes_ ([arXiv:1909.11612](https://arxiv.org/abs/1909.11612))
[[!redirects 3d-3d correspondences]]
[[!redirects 3d-3d relation]]
[[!redirects 3d-3d relations]]
[[!redirects 3d/3d correspondence]]
[[!redirects 3d/3d correspondences]]
[[!redirects 3d/3d relation]]
[[!redirects 3d/3d relations]]
|
3x3 lemma | https://ncatlab.org/nlab/source/3x3+lemma |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Diagram chasing lemmas
+-- {: .hide}
[[!include diagram chasing lemmas - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _$3 \times 3$-lemma_ or _nine lemma_ is one of the basic [[diagram chasing lemmas]] in [[homological algebra]].
## Statement
+-- {: .num_lemma}
###### Lemma
Let
$$
0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0
$$
be a [[short exact sequence]] of [[chain complexes]]. Then
if two of the three complexes $A_\bullet, B_\bullet, C_\bullet$
are [[long exact sequence|exact]], so is the remaining third.
=--
+-- {: .num_lemma}
###### Lemma
Let
$$
\array{
&& 0 && 0 && 0 &&
\\
&& \downarrow && \downarrow && \downarrow &&
\\
0 &\to& A' &\to& B' &\to& C' &\to& 0
\\
&& \downarrow && \downarrow && \downarrow &&
\\
0 &\to& A &\to& B &\to& C &\to& 0
\\
&& \downarrow && \downarrow && \downarrow &&
\\
0 &\to& A'' &\to& B'' &\to& C'' &\to& 0
\\
&& \downarrow && \downarrow && \downarrow &&
\\
&& 0 && 0 && 0 &&
}
$$
be a [[commuting diagram]] in some [[abelian category]] such that each of the three columns is an [[exact sequence]]. Then
1. If the two bottom rows are exact, then so is the top.
1. If the top two rows are exact, then so is the bottom.
1. If the top and bottom rows are exact _and_ $A \to C$ is the [[zero morphism]], then also the middle row is exact.
=--
A proof by way of the [[salamander lemma]] is spelled out in detail at _[Salamander lemma - Implications - 3x3 lemma](http://ncatlab.org/nlab/show/salamander+lemma#3x3Lemmas)_.
## Related concepts
* [[salamander lemma]]
* [[snake lemma]], [[5-lemma]]
* [[horseshoe lemma]]
## References
### In abelian categories
An early appearance of the $3 \times 3$-lemma is as lemma (5.5) in
* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))
In
* [[Charles Weibel]], _[[An introduction to homological algebra]]_
it appears as exercise 1.3.2.
The sharp $3 \times 3$-lemma appears as lemma 2 in
* Temple Fay, [[Keith Hardie]], [[Peter Hilton]], _The two-square lemma_, Publicacions Matemàtiques, Vol 33 (1989) ([pdf](http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf))
Also lemma 3.2-3.4 of
* [[Saunders MacLane]], _Homology_, Grundlehren der math. Wissenshaften vol 114, Springer (1995)
### In non-abelian categories
Discussion of generalization to non-abelian categories is in
* Marino Gran, Diana Rodelo, _Goursat categories and the $3 \times 3$-lemma_, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. ([journal](http://www.google.com/url?q=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%252Fs10485-010-9236-x&sa=D&sntz=1&usg=AFQjCNG1kEggBxeeDfQlkWOa4ShUo8vwLA), [pdf slides](http://ct2010.disi.unige.it/slides/Rodelo_CT2010.pdf))
* Marino Gran, Zurab Janelidze and Diana Rodelo, _$3 \times 3$ lemma for star-exact sequences_, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. ([journal](http://www.intlpress.com/HHA/v14/n2/a1/))
* [[Dominique Bourn]], _$3 \times 3$-lemma and protomodularity_, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)
* [[Dominique Bourn]], _The denormalized $3 \times 3$ lemma_, Journal of Pure and Applied Algebra, Volume 177, Issue 2, 24 January 2003, Pages 113-129, doi:[10.1016/S0022-4049(02)00143-3](https://doi.org/10.1016/S0022-4049%2802%2900143-3)
[[!redirects nine lemma]]
[[!redirects 9 lemma]]
[[!redirects 3x3-lemma]]
[[!redirects sharp 3x3 lemma]]
|
4-3-2 8-7-6 | https://ncatlab.org/nlab/source/4-3-2+8-7-6 |
The talk
* [[Daniel Freed]],
**4-3-2 8-7-6**
talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012
[pdf](https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf), [[Freed432876.pdf:file]]
discusses aspects and open problems of understanding
* [[3d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[2d WZW model]]
* [[7d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[6d (2,0)-supersymmetric QFT]]
in the context of [[extended TQFT]].
See also
* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_
(Other talks at this conference include [[David Ben-Zvi]]: _[[Loop Groups, Characters and Elliptic Curves]]_.)
One explicit version of
* [[3d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[2d WZW model]]
is the relation [[Turaev-Viro model]] $\leftrightarrow$ [[Crane-Yetter model]] due to theorem 2 in
* [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))
{#BarrettGarciIslasMartins04}
category: reference |
4-manifold | https://ncatlab.org/nlab/source/4-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
a [[manifold]] of [[dimension]] 4.
## Examples
* [[4-sphere]]
* [[spacetime]]
## Properties
#### Cohomotopy
Let $X$ be a [[4-manifold]] which is [[connected topological space|connected]] and [[orientation|oriented]].
The [[Pontryagin-Thom construction]] as [above](cohomotopy#RelationToCobordismGroup) gives for $n \in \mathbb{Z}$ the [[commuting diagram]] of sets
$$
\array{
\pi^n(X)
&\overset{\simeq}{\longrightarrow}&
\mathbb{F}_{4-n}(X)
\\
{}^{ \mathllap{h^n} }
\downarrow
&&
\downarrow^{ h_{4-n} }
\\
H^n(X,\mathbb{Z})
&\underset{\simeq}{\longrightarrow}&
H_{4-n}(X,\mathbb{Z})
\,,
}
$$
where $\pi^\bullet$ denotes [[cohomotopy]] sets, $H^\bullet$ denotes [[ordinary cohomology]], $H_\bullet$ denotes [[ordinary homology]] and $\mathbb{F}_\bullet$ is [[normal bundle|normally]] [[framing|framed]] [[cobordism classes]] of [[normal bundle|normally]] [[framing|framed]] [[submanifolds]]. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of [[Eilenberg-MacLane spaces]]):
$$
h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z}
\,.
$$
Now
* $h^0$, $h^1$, $h^4$ are [[isomorphisms]]
* $h^3$ is an isomorphism if $X$ is "odd" in that it contains at least one closed oriented [[surface]] of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-[[quotient group]] of $\pi^3(X)$ (which is a group via the [[group]]-[[structure]] of the [[3-sphere]] ([[special unitary group|SU(2)]]))
([Kirby-Melvin-Teichner 12](#KirbyMelvinTeichner12))
## Related concepts
* [[Yang-Mills instanton]]
* [[Donaldson-Thomas invariants]]
* [[2-manifold]], [[3-manifold]], [[8-manifold]]
## References
### General
All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the [[4-sphere]]:
* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))
* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))
On [[cohomotopy]] of 4-manifolds:
* [[Daniel Freed]], [[Karen Uhlenbeck]], Appendix B of: _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))
* {#KirbyMelvinTeichner12} [[Robion Kirby]], [[Paul Melvin]], [[Peter Teichner]], _Cohomotopy sets of 4-manifolds_, GTM 18 (2012) 161-190 ([arXiv:1203.1608](https://arxiv.org/abs/1203.1608))
### Relation to 2d CFTs via 6d CFT
On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:
* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]
* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I* (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]
* [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)]
In relation to [[M5-brane elliptic genus]]:
* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*, J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), <a href="https://doi.org/10.1007/JHEP05(2021)084">doi:10.1007/JHEP05(2021)084</a>, [spire:1704312](https://inspirehep.net/literature/1704312)]
and in relation to [[QFT with defects|defects]]:
* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>]
[[!redirects 4-manifolds]] |
4-sphere | https://ncatlab.org/nlab/source/4-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 4.
## Properties
### Basic differential geometry
{#Differential geometry}
An embedded radius-$R$ 4-sphere inherits a [[volume form]] (a degree-4 [[differential form]] from the ambient $5$-dimensional [[Euclidean space]], namely
$$
\omega = \frac{1}{R} \sum_{j=1}^{5} (-1)^{j-1} x_j d x_1 \wedge \ldots \wedge \widehat{d x_j} \wedge \ldots \wedge d x_{5}
$$
where the hat means omit that factor. This is equal to $\ast d r$, where $\ast$ is the [[Hodge star operator]] in $\mathbb{R}^5$ for the Euclidean [[Riemannian metric|metric]], and $d r$ is the [[exterior derivative]] of the radius function.
The volume of the manifold $S^4$ with this volume form is then given by $8\pi^2R^4/3$.
### Coset space structure
{#CosetSpaceStructure}
As any [[sphere]], the [[4-sphere]] has the [[coset space]] [[structure]]
$$
S^4 \simeq O(5)/O(4) \simeq SO(5)/SO(4) \simeq Spin(5)/Spin(4)\simeq Pin(5)/Pin(4).
$$
There is also this:
+-- {: .num_example #Sp2Sp1BySp1Sp1Sp1IsS4}
###### Example
The [[coset space]] of [[Sp(2).Sp(1)]] ([this Def.](SpnSp1#SpnSp1)) by [[Sp(1)Sp(1)Sp(1)]] ([this Def.](SpnSp1#Spin4Spin3)) is the [[4-sphere]]:
$$
\frac{
Sp(2)\cdot Sp(1)
}
{
Sp(1)Sp(1)Sp(1)
}
\;\simeq\;
S^4
\,.
$$
This follows essentially from the [[quaternionic Hopf fibration]] and its $Sp(2)$-[[equivariant function|equivariance]]...
=--
(e.g. [Bettiol-Mendes 15, (3.1), (3.2), (3.3)](#BettiolMendes15))
### Homotopy groups
{#HomotopyGroups}
The [[homotopy groups]] of the 4-sphere in low degree are:
| $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-----|---|---|---|---|---|---|---|---|---|---|----|-----|----|
| $\pi_k(S^4)$ | $\ast$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z} \times \mathbb{Z}_{12}$ | $\mathbb{Z}_2^2$ | $\mathbb{Z}_2^2$ | $\mathbb{Z}_{24} \times \mathbb{Z}_3$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_2$ |
For more see at *[[Serre finiteness theorem]]* and at *[[homotopy groups of spheres]]*.
### Bundles over the 4-sphere
#### The quaternionic Hopf fibration
The 4-sphere participates in the [[quaternionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[quaternions]] or Hamiltonian numbers $\mathbb{H}$.
$$
\array{
S^3 &\hookrightarrow& S^7
\\
&& \downarrow^\mathrlap{p}
\\
&& S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2)
}
$$
Here the idea is that $S^7$ may be construed as
$$
\array{
S^7
&\simeq
S(\mathbb{H}^4)
\\
& \simeq
\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\},
}
$$
with $p$ mapping $(x, y)$
to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each [[fiber]] a [[torsor]] parameterized by quaternionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^3$). This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.
{#HopfParameterization} There are other useful ways to parameterize the quaternionic Hopf fibration, such as the original _[[Hopf construction]]_, see there the section _[Hopf fibrations](Hopf+construction#HopfFibrations)_. By this parameterization $S^4$ is identified as $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$.
#### The Calabi-Penrose fibration
See at _[[Calabi-Penrose fibration]]_.
#### The complex projective plane
{#AsAQuotientOfTheComplexProjectivePlane}
+-- {: .num_prop}
###### Proposition
**([[Arnold-Kuiper-Massey theorem]])**
The 4-sphere is the [[quotient space]] of the [[complex projective plane]] by the [[action]] of [[complex conjugation]] (on homogeneous coordinates):
$$
\mathbb{C}P^2 / (-)^*
\simeq
S^4
$$
=--
### Exotic smooth structures
It is open whether the 4-sphere admits an [[exotic smooth structure]]. See ([Freedman-Gompf-Morrison-Walker 09](#FreedmanGompfMorrisonWalker09)) for review.
### $SU(2)$ action
{#QuaternionAction}
If we identify $\mathbb{R}^5 \simeq_{\mathbb{Q}} \mathbb{R} \oplus \mathbb{H}$ with the [[direct sum]] of the [[real line]] with the [[real vector space]] underlying the [[quaternions]], so that
$$
S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})
$$
as in the discussion of the quaternionic Hopf fibration [above](#HopfParameterization), then there is induced an [[action]] of the group [[special unitary group|SU(2)]] on the 4-sphere, by identifying
$$
SU(2) \simeq S(\mathbb{Q})
$$
and then acting by left multiplication.
#### Circle action
{#CircleAction}
+-- {: .num_prop}
###### Proposition
Given an continuous [[action]] of the [[circle group]] on the [[topological space|topological]] [[4-sphere]], its [[fixed point]] space is of one of two types:
1. either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$
1. or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere.
=--
([Félix-Oprea-Tanré 08, Example 7.39](#FelixOpreaTanre08))
For more see at _[[group actions on spheres]]_.
As a special case of the $SU(2)$-action from [above](#QuaternionAction), we discuss the induced circle action via the embedding
$$
S^1 \simeq U(1) \hookrightarrow SU(2)
\,.
$$
Consider the following [[circle group|circle]] [[group action on an n-sphere|group action on the 4-sphere]]:
+-- {: .num_defn #CircleActionOn4Sphere}
###### Definition
**($SU(2)$-action on 4-sphere)**
Regard
$$
S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})
$$
as the [[unit sphere]] inside the [[direct sum]] (as [[real vector spaces]]) of the [[real numbers]] with the [[quaternions]], and regard the [[special unitary group]] $SU(2)$ as the group of unit-norm quaternions
$$
SU(2) \simeq S(\mathbb{H},\cdot)
$$
In particular this restricts to an [[action]] of the [[circle group]]
$$
S^1 \simeq U(1) \hookrightarrow SU(2)
$$
(as the [[diagonal matrices]] inside $SU(2)$) on the 4-sphere.
=--
The resulting ordinary [[quotient]] is $S^4/_{ord} S^1 \simeq S^3$ and the [[projection]] $S^4 \to S^3$ is the [[suspension]] of the [[complex Hopf fibration]] $S^3 \to S^2$.
The [[fixed point]] set of the action is the two poles
$$
S^0 \;=\; \{(\pm 1, 0,0,0,0)\} \;\in\; \mathbb{R} \oplus \mathbb{H}
$$
introduced by the suspension, hence forms the [[0-sphere]] space. Since this is not the [[empty set]], the [[homotopy quotient]] $S^4 // S^1$ of the [[circle action]] differs from $S^3$, but there is still the canonical [[projection]]
$$
S^4//S^1 \longrightarrow S^4 / S^1 \simeq S^3
\,.
$$
Hence both $S^4$ and $S^4 // S^1$ are canonically [[homotopy types]] over $S^3$.
A [[minimal dg-module]] presentation in [[rational homotopy theory]] for these projections is given in [Roig & Saralegi-Aranguren 00, second page](#RoigSaralegiAranguren00):
+-- {: .num_prop #FourSphereOverThreeSphereMinimalDgModel}
###### Proposition
**([Roig & Saralegi-Aranguren 00, p. 2](#RoigSaralegiAranguren00))**
Write
$$
CE(\mathfrak{l}(S^3))) = Sym^\bullet \langle \underset{\text{deg 3}}{\underbrace{h_3}} \rangle
$$
for the [[minimal Sullivan model]] of the [[3-sphere]]. Then [[rational homotopy theory|rational]] [[minimal dg-modules]] for the maps (via Def. \ref{CircleActionOn4Sphere})
$$
\array{
S^4
\\
\downarrow
\\
S^3
}
\,,\phantom{AA}
\array{
S^4//S^1
\\
\downarrow
\\
S^3
}
\,,\phantom{AA}
\array{
S^0
\\
\downarrow
\\
S^3
}
$$
as [[dg-modules]] over $CE(\mathfrak{l}(S^3))$ are given as follows, respectively:
$$
\label{FourSphereAndRelatedOverThreeSphereMinimalDGModels}
\array{
\text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} }
\\
\array{
S^4
\\
\downarrow
\\
S^3
}
&
Sym^\bullet \langle
\underset{
deg = 3
}{
\underbrace{
h_3
}}
\rangle
\otimes
\left\langle
\underset{deg = 2p}{
\underbrace{
\tilde\omega_{2p}
}},
\underset{deg = 2p + 4}{
\underbrace{
\omega_{2p + 4}
}} \,\vert\, p \in \mathbb{N}
\right\rangle
&
d \colon
\left\{
\begin{aligned}
\tilde\omega_0 & \mapsto 0
\\
\tilde\omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
\\
\omega_4 & \mapsto 0
\\
\omega_{2p+6} & \mapsto h_3 \wedge \omega_{2p + 4}
\end{aligned}
\right.
\\
\array{
S^0
\\
\downarrow
\\
S^3
}
&
Sym^\bullet \langle
\underset{
deg = 3
}{
\underbrace{
h_3
}}
\rangle
\otimes
\left\langle
\underset{deg = 2p}{
\underbrace{
\tilde \omega_{2p}
}},
\underset{
deg = 2p
}{
\underbrace{
\omega_{2p}
}} \,\vert\, p \in \mathbb{N}
\right\rangle
&
d \colon
\left\{
\begin{aligned}
\tilde \omega_0 & \mapsto 0
\\
\tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
\\
\omega_0 & \mapsto 0
\\
\omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p}
\end{aligned}
\right.
\\
\array{
S^4//S^1
\\
\downarrow
\\
S^3
}
&
Sym^\bullet \langle
\underset{
deg = 3
}{
\underbrace{
h_3
}}
,
\underset{
deg = 2
}{
\underbrace{
\omega_2
}}
\rangle
\otimes
\left\langle
\underset{deg = 2p}{
\underbrace{
\tilde \omega_{2p}
}},
\underset{
deg =2p + 4
}{
\underbrace{
\omega_{2p + 4}
}} \,\vert\, p \in \mathbb{N}
\right\rangle
&
d \colon
\left\{
\begin{aligned}
\tilde \omega_0 & \mapsto 0
\\
\tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
\\
\omega_2 & \mapsto 0
\\
\omega_{2p+4} & \mapsto h_3 \wedge \omega_{2p + 2}
\end{aligned}
\right.
}
$$
=--
Beware that in the model for $S^4//S^2$ the element $\omega_2$ induces its entire polynomial algebra as generator of the dg-module.
Notice that we changed the notation of the generators compared to [Roig & Saralegi-Aranguren 00, second page](#RoigSaralegiAranguren00), to bring out the pattern:
| $\phantom{A}$Roig$\phantom{A}$ | $\phantom{A}$here$\phantom{A}$ |
|------------|-----------------|
| $\phantom{A}a\phantom{A}$ | $\phantom{A}h_3\phantom{A}$ |
| $\phantom{A}1\phantom{A}$ | $\phantom{A}\tilde\omega_0\phantom{A}$ |
| $\phantom{A}c_{2n}\phantom{A}$ | $\phantom{A}\tilde\omega_{2n+2}\phantom{A}$ |
| $\phantom{A}c_{2n+1}\phantom{A}$ | $\phantom{A}\omega_{2n+4}\phantom{A}$ |
| $\phantom{A}e\phantom{A}$ | $\phantom{A}\omega_2\phantom{A}$ |
| $\phantom{A}\gamma_{2n}\phantom{A}$ | $\phantom{A}\tilde\omega_{2n}\phantom{A}$ |
| $\phantom{A}\gamma_{2n+1}\phantom{A}$ | $\phantom{A}\omega_{2n}\phantom{A}$ |
#### M5-brane orbifolds
The [[supersymmetry|supersymmetric]] [[Freund-Rubin compactifications]] of [[11-dimensional supergravity]] which are [[Cartesian products]] of 7-dimensional [[anti-de Sitter spacetime]] with a compact 4-dimensional [[orbifold]]
$$
AdS_7 \times X_4
$$
(the [[near horizon geometry]] of a [[black brane|black]] [[M5-brane]]) are all of the form
$$
X_4 \simeq S^4//G
$$
where $G \subset SU(2)$ is a [[finite group|finite]] [[subgroup]] of $SU(2)$ (i.e. an [[ADE classification|ADE group]]), [[action|acting]] via the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ as [above](#QuaternionAction), and where the double slash denotes the [[homotopy quotient]] ([[orbifold quotient]]).
See ([AFHS 98, section 5.2](#AFHS98), [MF 12, section 8.3](#MF12)).
### Free and cyclic loop space
{#FreeLoopSpace}
We discuss the [[rational homotopy theory]] of the [[free loop space]] $\mathcal{L}(S^4)$ of $S^4$, as well as the [[cyclic loop space]] $\mathcal{L}(S^4)/S^1$ using the results from _[[Sullivan models of free loop spaces]]_:
+-- {: .num_example}
###### Example
Let $X = S^4$ be the [[4-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model
$$
(\wedge^\bullet \langle g_4, g_7 \rangle, d)
$$
with
$$
d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4
\,.
$$
Hence [this prop.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace) gives for the rationalization of
$\mathcal{L}S^4$ the model
$$
( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )
$$
with
$$
\begin{aligned}
d_{\mathcal{L}S^4} h_3 & = 0
\\
d_{\mathcal{L}S^4} \omega_4 & = 0
\\
d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4
\\
d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
\\
\end{aligned}
$$
and [this prop](Sullivan+model+of+free+loop+space#ModelForS1quotient)
gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model
$$
( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} )
$$
with
$$
\begin{aligned}
d_{\mathcal{L}S^4 / / S^1} h_3 & = 0
\\
d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0
\\
d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2
\\
d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4
\\
d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6
\end{aligned}
\,.
$$
=--
+-- {: .num_prop}
###### Proposition
Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a [[central extension|central]] [[Lie algebra extension]] by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding [[L-∞ algebra cohomology|L-∞ 2-cocycle]] with coefficients in the [[line Lie n-algebra|line Lie 2-algebra]] $b \mathbb{R}$, hence ([[schreiber:The brane bouquet|FSS 13, prop. 3.5]]) so that there is a [[homotopy fiber sequence]] of [[L-∞ algebras]]
$$
\hat \mathfrak{g}
\longrightarrow
\mathfrak{g}
\overset{\omega_2}{\longrightarrow}
b \mathbb{R}
$$
which is dually modeled by
$$
CE(\hat \mathfrak{g})
=
( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\; d_{\hat \mathfrak{g}} e = \omega_2)
\,.
$$
For $X$ a space with [[Sullivan model]] $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding [[L-∞ algebra]], i.e. for the $L_\infty$-algebra whose [[Chevalley-Eilenberg algebra]] is $(A_X,d_X)$:
$$
CE(\mathfrak{l}X) = (A_X,d_X)
\,.
$$
Then there is an [[isomorphism]] of [[hom-sets]]
$$
Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) )
\;\simeq\;
Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) )
\,,
$$
with $\mathfrak{l}(S^4)$ from [this prop.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace)
and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient),
where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient)
Moreover, this isomorphism takes
$$
\hat \mathfrak{g}
\overset{(g_4, g_7)}{\longrightarrow}
\mathfrak{l}(S^4)
$$
to
$$
\array{
\mathfrak{g}
&&
\overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow}
&&
\mathfrak{l}( \mathcal{L}X / S^1 )
\\
&
{}_{\mathllap{\omega_2}}\searrow
&&
\swarrow_{\mathrlap{\omega_2}}
\\
&& b \mathbb{R}
}
\,,
$$
where
$$
\omega_4 = g_4 - h_3 \wedge e
\;\,,
\;\;\;
h_7 = g_7 + \omega_6 \wedge e
$$
with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.
=--
This is observed in ([FSS 16](Sullivan+model+of+free+loop+space#FiorenzaSatiSchreiber16), [FSS 16b](#FSS16b)), where it serves to formalize, on the level of [[rational homotopy theory]], the [[double dimensional reduction]] of [[M-branes]] in [[M-theory]] to [[D-branes]] in [[type IIA string theory]] (for the case that $\mathfrak{g}$ is type IIA [[super Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of [[schreiber:The brane bouquet]]).
+-- {: .proof}
###### Proof
By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.
Hence it is sufficient to observe that under this decomposition the defining equations
$$
d g_4 = 0
\,,\;\;\;
d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4
$$
for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:
$$
\begin{aligned}
& d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0
\\
\Leftrightarrow \;\;\;\;
&
d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0
\;\;\; and \;\;\;
d_{\mathfrak{g}} h_3 = 0
\\
\Leftrightarrow \;\;\;\;
&
d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2
\;\;\; and \;\;\;
d_{\mathfrak{g}} h_3 = 0
\end{aligned}
$$
as well as
$$
\begin{aligned}
& d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e )
= -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e)
\\
\Leftrightarrow \;\;\;\;
&
d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2
=
-\tfrac{1}{2}\omega_4 \wedge \omega_4
\;\;\; and \;\;\;
- d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4
\\
\Leftrightarrow \;\;\;\;
&
d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2
\;\;\; and \;\;\;
d_\mathfrak{g} h_6 = h_3 \wedge \omega_4
\end{aligned}
$$
=--
The [[unit of an adjunction|unit]] of the [[double dimensional reduction]]-[[adjunction]]
$$
\infty Grpd
\underoverset
{\underset{\mathcal{L}(-)/S^1}{\longrightarrow}}
{\overset{hofib}{\longleftarrow}}
{\bot}
\infty Grpd_{/B S^1}
$$
([this prop.](double+dimensional+reduction#GeneralReduction)) applied to the $S^1$-[[principal infinity-bundle]]
$$
\array{
S^4
\\
{}^{\mathllap{hofib(c)}}\downarrow
\\
S^4 / S^1
&\underset{c}{\longrightarrow}&
B S^1
}
$$
is a natural map
$$
S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1
$$
from the [[homotopy quotient]] by the [[circle action]] (def. \ref{CircleActionOn4Sphere}), to the [[cyclic loop space]] of the 4-sphere.
### Diffeomorphism group
Counterexamples (via [[graph complexes]]) to the analogue of the [[Smale conjecture]] for the 4-sphere are claimed in [Watanabe 18](diffeomorphism+group#Watanabe18), reviewed in [Watanabe 21](diffeomorphism+group#Watanabe21).
## Related entries
[[spheres -- contents]]
* [[fuzzy 4-sphere]]
* [[2-sphere]]
* [[3-sphere]]
* [[5-sphere]]
* [[6-sphere]]
* [[7-sphere]]
* [[n-sphere]]
## References
### General
* {#FreedmanGompfMorrisonWalker09} [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], _Man and machine thinking about the smooth 4-dimensional Poincaré conjecture_, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 ([arXiv:0906.5177](http://arxiv.org/abs/0906.5177))
* {#RoigSaralegiAranguren00} [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], _Minimal Models for Non-Free Circle Actions_, Illinois Journal of Mathematics, volume 44, number 4 (2000) ([arXiv:math/0004141](https://arxiv.org/abs/math/0004141))
* {#AFHS98} [[Bobby Acharya]], [[José Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_ , Adv. Theor. Math. Phys. 2 (1998) 1249–1286
* {#FelixOpreaTanre08} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Algebraic Models in Geometry_, Oxford University Press 2008
* {#MF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))
* {#BettiolMendes15} Renato G. Bettiol, Ricardo A. E. Mendes, _Flag manifolds with strongly positive curvature_, Math. Z. 280 (2015), no. 3-4, 1031-1046 ([arXiv:1412.0039](https://arxiv.org/abs/1412.0039))
* Selman Akbulut, _Homotopy 4-spheres associated to an infinite order loose cork_ ([arXiv:1901.08299](https://arxiv.org/abs/1901.08299))
* Akio Kawauchi, _Smooth homotopy 4-sphere_ ([arXiv:1911.11904](https://arxiv.org/abs/1911.11904))
* David T. Gay, _Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins_ ([arXiv:2102.12890](https://arxiv.org/abs/2102.12890))
### Branched covers
All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the [[4-sphere]]:
* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))
* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))
Speculative remarks on the possible role of maps from [[spacetime]] to the [[4-sphere]] in some kind of [[quantum gravity]] via [spectral geometry](spectral+triple) (related to the [[Connes-Lott-Chamseddine-Barrett model]]) are in
* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471))
* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))
* {#Connes17} [[Alain Connes]], section 4 of _Geometry and the Quantum_, in _Foundations of Mathematics and Physics One Century After Hilbert_, Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125))
* [[Alain Connes]], from 58:00 to 1:25:00 in _Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG_, talk at IHES, 2017 ([video recording](https://www.youtube.com/watch?v=qVqqftQ92kA))
[[!redirects 4-spheres]] |
4d Chern-Simons theory | https://ncatlab.org/nlab/source/4d+Chern-Simons+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 4.
## Examples
* [[Yetter model]]
* [[BF theory]]
* [[semi-holomorphic 4d Chern-Simons theory]]
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* **4d Chern-Simons theory**
* [[4d TQFT]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[11d Chern-Simons theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[AKSZ sigma-model]]
## References
* Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, *Higher Chern-Simons based on (2-)crossed modules*, JHEP **2023** 207 (2023) [[arXiv:2212.04667](https://arxiv.org/abs/2212.04667), <a href="https://link.springer.com/article/10.1007/JHEP07(2023)207">doi:10.1007/JHEP07(2023)207</a>]
(...)
[[!redirects 4d Chern-Simons theories]]
[[!redirects 4-dimensional Chern-Simons theory]]
[[!redirects 4-dimensional Chern-Simons theories]]
[[!redirects D=4 Chern-Simons theory]]
|
4d superconformal gauge field theory | https://ncatlab.org/nlab/source/4d+superconformal+gauge+field+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
There exists [[superconformal field theories]]/[[super Yang-Mills theories]] in 4 dimensions:
* [[N=1 D=4 super Yang-Mills theory]]
* [[N=2 D=4 super Yang-Mills theory]]
* [[N=4 D=4 super Yang-Mills theory]]
and by [[topological twist|topological twisting]] these give rise to
* [[topologically twisted D=4 super Yang-Mills theory]]
according to the [classification of superconformal symmetry](supersymmetry#ClassificationSuperconformal)...
[[!include superconformal symmetry -- table]]
[[!redirects D=4 SYM]]
[[!redirects 4d superconformal gauge field theories]]
[[!redirects 4d super gauge field theory]]
[[!redirects 4d super gauge field theories]]
[[!redirects D=4 super Yang-Mills theory]]
[[!redirects D=4 super Yang-Mills theories]]
|
4d supergravity Lie 2-algebra | https://ncatlab.org/nlab/source/4d+supergravity+Lie+2-algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[4d supergravity]] is often formulated as a theory of [[gravity]] coupled to scalar-[[vector multiplets]] only, i.e. 1-form [[gauge fields]]. On the other hand, when the theory is thought of as obtained by [[KK-compactification]] from [[11-dimensional supergravity]], then it naturally contains also 2-form fields ([[tensor multiplets]]), i.e. [[higher gauge fields]]. If these are massless, then in 4-dimensions they may be dualized (via [[Hodge duality]] of their [[field strengths]]) to scalars, which is why they often do not explicitly appear in the formulation. However, when they are massive, which happens for instance when the higher dimensional theory is reduced via a [[flux compactification]], then such dualization does not apply (at least not so directly) and the 2-form [[higher gauge fields]] need to be made explicit (see the introduction of [Gunyadin-McReynolds-Zagerman 05](#GunyadinMcReynoldsZagerman05)).
In the [[D'Auria-Fré formulation of supergravity]] such higher form field contributions are reflected by [[L-infinity algebra]] [[L-infinity extensions|extensions]] of the [[super Minkowski spacetime]] [[supersymmetry]] [[super Lie algebra]] (traditionally displayed in terms of dual [[Chevalley-Eilenberg algebras]], called "FDA"s in the supergravity literature).
For the 2-form fields of $N = 2$ [[4d supergravity]] this yields a [[Lie 2-algebra]] ([Andrianopoli-D'Auria-Sommovigo 07 (4.1)-(4.7)](#AndrianopoliDAuriaSommovigo07)), which hence might be called the "4d supergravity Lie 2-algebra". In fact, including the [[moduli]] fields, this is a [[L-infinity algebroid|Lie 2-algebroid]] ([Andrianopoli-D'Auria-Sommovigo 07 (4.8)-(4.9)](#AndrianopoliDAuriaSommovigo07)). See also ([AAST 11, (4.1)-(4.9)](#AAST11)).
This Lie 2-algebra is a non-abelian variant of the [[L-infinity extension]] classified by the 3-cocycle $\propto \overline{\psi} \wedge \Gamma^a \psi \wedge e_a$ ([Andrianopoli-D'Auria-Sommovigo 07 (4.5)](#AndrianopoliDAuriaSommovigo07)), which is the [[WZW term]] for the [[Green-Schwarz superstring]] in 4d (see the notation and conventions at _[[super-Minkowski spacetime]]_).
But the [[brane scan]] says that there is also a [[super 2-brane in 4d]] whose [[WZW-term]] is the 4-cocycle $\propto \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$. Accordingly there should also be a 4d supegravity [[Lie 3-algebra]]. This is left to "future investigations" in ([Andrianopoli-D'Auria-Sommovigo 07, p. 19](#AndrianopoliDAuriaSommovigo07)), but the relevant extension formula by the 4-cocycle is shown in ([Andrianopoli-D'Auria-Sommovigo 07 (2.24)](#AndrianopoliDAuriaSommovigo07))
## Related concepts
* [[supergravity Lie 3-algebra]]
* [[supergravity Lie 6-algebra]]
## References
Background discussion (and further pointers) for $p$-form fields in [[4d supergravity]] is in the introduction of
* {#GunyadinMcReynoldsZagerman05} [[Murat Gunaydin]], S. McReynolds, M. Zagermann, _Unified $N=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions_, JHEP 0509:026,2005 ([arXiv:hep-th/0507227](https://arxiv.org/abs/hep-th/0507227))
The 4d supergravity Lie 2-algebra was given in
* {#AndrianopoliDAuriaSommovigo07} Laura Andrianopoli, [[Riccardo D'Auria]], Luca Sommovigo, _$D=4$, $N=2$ Supergravity in the Presence of Vector-Tensor Multiplets and the Role of higher p-forms in the Framework of Free Differential Algebras_, Adv.Stud.Theor.Phys.1:561-596,2008 ([arXiv:0710.3107](http://arxiv.org/abs/0710.3107))
Further discussion is in
* {#AAST11} Laura Andrianopoli, [[Riccardo D'Auria]], Luca Sommovigo, Mario Trigiante, _$D=4$, $N=2$ Gauged Supergravity coupled to Vector-Tensor Multiplets_, Nucl.Phys.B851:1-29,2011 ([arXiv:1103.4813](http://arxiv.org/abs/1103.4813))
[[!redirects 4d supergravity Lie 2-algebras]]
[[!redirects 4d supergravity Lie 3-algebra]]
[[!redirects 4d supergravity Lie 3-algebras]]
[[!redirects 4d supergravity Lie 2-algebroid]]
[[!redirects 4d supergravity Lie 2-algebroids]]
[[!redirects 4d supergravity Lie 3-algebroid]]
[[!redirects 4d supergravity Lie 3-algebroids]]
|
4d TQFT | https://ncatlab.org/nlab/source/4d+TQFT |
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#### Physics
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#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
[[TQFT]] of [[dimension]] 4.
## Constructions
In analogy to how [[3d TQFTs]] are induced from [[quantum groups]]/[[Hopf algebras]]/ and generally [[bialgebras]] (hence [[3-modules]], the [[higher geometry|higher]] [[space of quantum states]] assigned to the point which by the [[cobordism theorem]] defines the theory) one may build 4d TQFTs from higher analogs of these, namely models of [[4-modules]] given by algebraic structures such as _[[trialgebras]]_ and _[[Hopf categories]]_.
## Examples
* [[Yetter model]]
* [[Kapustin-Witten TQFT]]
* [[4d Chern-Simons theory]]
* [[Walker-Wang model]]
* [[3d toric code]]
## Related concepts
* [[loop braid group]]
* [[TQFT]]
* [[2d TQFT]]
* [[3d TQFT]]
* **4d TQFT**
* [[4d Chern-Simons theory]]
## References
Original references:
* [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997) ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))
* {#CraneFrenkel} [[Louis Crane]], [[Igor Frenkel]], _Four dimensional topological quantum field theory, Hopf categories, and the canonical bases_, J. Math. Phys. 35 (1994) 5136-5154, ([arXiv:hep-th/9405183](http://arxiv.org/abs/hep-th/9405183))
See also
* {#CarterKauffmanSaito96} [[Scott Carter]], [[Louis Kauffman]], Masahico Saito, _Diagrams, Singularities, and Their Algebraic Interpretations_, in "10th Brazilian Topology Meeting, S. Carlos, July 22 26, 1996 ([pdf](http://homepages.math.uic.edu/~kauffman/cksBrasil.pdf))
Construction via [[factorization homology]] from [[braided monoidal category|braided]] [[tensor categories]] (with relation to [[double affine Hecke algebras]]) is discussed in
* [[David Ben-Zvi]], [[Adrien Brochier]], [[David Jordan]], _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_ ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))
From [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:
* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))
As descriptions of [[topological insulators]]/[[topological phases of matter]]:
* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))
> (see [[Walker-Wang model]])
* [[Clement Delcamp]], *Excitation basis for (3+1)d topological phases*, Journal of High Energy Physics **2017** (2017) 128 ([arXiv:1709.04924](https://arxiv.org/abs/1709.04924) <a href="https://doi.org/10.1007/JHEP12(2017)128">doi:10.1007/JHEP12(2017)128</a>)
\linebreak
Many more references should eventually go here...
[[!redirects 4-dimensional TQFT]]
[[!redirects 4d TQFTs]]
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4d WZW model | https://ncatlab.org/nlab/source/4d+WZW+model |
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#### $\infty$-Wess-Zumino-Witten theory
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}
## Idea
The 4-dimensional version of [[higher dimensional WZW theory]], the boundary theory of the [[higher dimensional Chern-Simons theory|5-dimensional Chern-Simons theory]] that is induced from the third [[Chern class]]. Also called _Nair-Schiff theory_ or _Kähler Chern-Simons theory_ .
## Properties
### Equations of motion
The [[Euler-Lagrange equations]] of the 4d WZW model are those of [[self-dual Yang-Mills theory]].
### Finiteness
The model is finite at 1-loop ([Ketov](#Ketov)) and expected to be finite to all orders.
## References
* [[Ali Chamseddine]], Phys. Lett. B233, 291 (1989);
* V.P.Nair, _Kähler-Chern-Simons Theory_ ([arXiv:hep-th/9110042](http://arxiv.org/abs/hep-th/9110042))
* A. Losev, [[Greg Moore]], [[Nikita Nekrasov]], Samson Shatashvili, _Four-Dimensional Avatars of Two-Dimensional RCFT_ ([arXiv:hep-th/9509151](http://arxiv.org/abs/hep-th/9509151))
* Takeo Inami, Hiroaki Kanno, Tatsuya Ueno, Chuan-Sheng Xiong, _Two-toroidal Lie Algebra as Current Algebra of Four-dimensional Kähler WZW Model_ ([arXiv:hep-th/9610187](http://arxiv.org/abs/hep-th/9610187))
* A. V. Ketov, Phys.Lett. B383 (1996) 390-396
{#Ketov}
* T. Inami, H. Kanno and T. Ueno, Mod. Phys. Lett. A12, 2757 (1997).
* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional AdS Spacetime_ ([arXiv:hep-th/9905228](http://arxiv.org/abs/hep-th/9905228))
* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional Chern-Simons Gravity_ ([arXiv:hep-th/0010020](http://arxiv.org/abs/hep-th/0010020))
[[!redirects 4d WZW models]]
[[!redirects 4d Wess-Zumino-Witten model]]
[[!redirects 4d Wess-Zumino-Witten models]]
[[!redirects 4d WZW theory]] |
4T relation | https://ncatlab.org/nlab/source/4T+relation |
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#### Knot theory
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#Contents#
* table of contents
{:toc}
## Idea
### For round chord diagrams
{#ForCircularChordDiagrams}
In [[knot theory]] by the _4-term relations_ or _4T-relations_, for short, one means the following [[relations]] in the [[linear span]] of [[chord diagrams]]:
<center>
<img src="https://ncatlab.org/nlab/files/4TRelationsForRoundChordDiagrams.jpg" width="340">
</center>
([Bar-Natan 95, Def. 1.6](#BarNatan95))
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
These are the relations respected by [[weight systems]] on chord diagrams.
### For horizontal chord diagrams
{#ForHorizontalChordDiagrams}
For [[horizontal chord diagrams]] the 4T relations is the following:
<center>
<img src="https://ncatlab.org/nlab/files/HorizontalChordDiagram4TRelation.jpg" width="600">
</center>
([Bar-Natan 96, p. 3](#BarNatan96))
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
When the [[linear span]] of [[horizontal chord diagrams]] is regarded as an [[associative algebra]] under concatenation of strands ([here](horizontal+chord+diagram#AlgebraOfHorizontalChordDiagrams)), this relation is the [[infinitesimal braid relation]]
$$
\big[
t_{i k} + t_{j k}
\,,\,
t_{i j}
\big]
\;=\;
0
\,.
$$
## Properties
### Relation between horizontal and round 4T relations
{#RelationBetweenHorizontalAndRound4TRelations}
The 4T-relations for [[round chord diagrams]] are the image of the
4T relations for [[horizontal chord diagrams]] under [[tracing horizontal to round chord diagrams]]:
<center>
<img src="https://ncatlab.org/nlab/files/4TRelationsHorizontalToRound.jpg" width="800">
</center>
### Relation to STU-relations
Under the embedding of the set of [[round chord diagrams]] into the set of [[Jacobi diagrams]], the [[STU-relations]] imply the [[4T relations]] on [[round chord diagrams]]:
<center>
<img src="https://ncatlab.org/nlab/files/STURelationImplies4TRelation.jpg" width="280">
</center>
Using this, one finds that [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]:
<center>
<img src="https://ncatlab.org/nlab/files/ChordDiagModulo4TAreJAcobiDiagModuloSTU.jpg" width="840">
</center>
> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]
## Related concepts
* [[infinitesimal braid relation]]
* [[STU relation]]
[[!include chord diagrams and weight systems -- table]]
## References
Original articles
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
(for [[horizontal chord diagrams]])
Textbook accounts
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv/1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* [[David Jackson]], [[Iain Moffat]], Section 11 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
Lecture notes:
* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))
The graphics above are taken from [Sati-Schreiber 19]().
[[!redirects 4T relations]]
[[!redirects 4T-relation]]
[[!redirects 4T-relations]]
[[!redirects 4 term relation]]
[[!redirects 4 term relations]]
[[!redirects 4-term relation]]
[[!redirects 4-term relations]]
[[!redirects horizontal 4T relation]]
[[!redirects horizontal 4T relations]]
[[!redirects horizontal 4T-relation]]
[[!redirects horizontal 4T-relations]]
[[!redirects horizontal 4 term relation]]
[[!redirects horizontal 4 term relations]]
[[!redirects horizontal 4-term relation]]
[[!redirects horizontal 4-term relations]]
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5-manifold | https://ncatlab.org/nlab/source/5-manifold |
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### Context
#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
A [[manifold]] of [[dimension]] 5
## Related concepts
* [[Sasakian manifold]], [[SU(2)-structure]]
[[!include low dimensional manifolds -- table]]
## References
* {#CadekVanzura93} [[Martin Čadek]], [[Jiří Vanžura]], _On the classification of oriented vector bundles over 5-complexes_, Czechoslovak Mathematical Journal, Vol. 43 (1993), No. 4, 753–764 ([dml:128427](https://dml.cz/handle/10338.dmlcz/128427))
[[!redirects 5-manifolds]]
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5-sphere | https://ncatlab.org/nlab/source/5-sphere |
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#### Spheres
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[[!include spheres -- contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] for $n = 5$.
## Properties
### Free group actions
\begin{example}
There are, up to [[isomorphism]] of smooth $\mathbb{Z}/2$-actions, 4 distinct free involutions on the [[5-sphere|$S^5$]].
\end{example}
([López de Medrano, Sec. V.6.1](group+actions+on+spheres#LopezdeMedrano71)).
See at *[[group actions on spheres]]*.
## Related concepts
* [[AdS5/CFT4]]
[[!redirects 5-spheres]] |
6-sphere | https://ncatlab.org/nlab/source/6-sphere |
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#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] of [[dimension]] $n = 6$.
## Properties
### Coset structure
The 6-sphere, as a [[smooth manifold]] is [[diffeomorphism|diffeomorphic]] to the [[coset space]]
$$
S^6 \simeq G_2/ SU(3)
$$
of [[G2]] ([[automorphism group]] of the [[octonions]]) by [[special unitary group|SU(3)]] ([Fukami-Ishihara 55](#FukamiIshihara55)).
For more see at _[[G2/SU(3) is the 6-sphere]]_.
The induced [[action]] of [[G2]] on $S^6$ induces an [[almost Hermitian structure]] which makes it a [[nearly Kaehler manifold]].
Review in is in [Agrikola-Borowka-Friedrich 17](#AgrikolaBorowkaFriedrich17)
[[!include coset space structure on n-spheres -- table]]
### Complex structure
A famous open problem is the question whether the 6-sphere admits an actual [[complex structure]]. For review see [Bryant 14](#Bryant14).
## Related entries
* [[fuzzy 6-sphere]]
* [[2-sphere]]
* [[3-sphere]]
* [[4-sphere]]
* [[7-sphere]]
* [[n-sphere]]
## References
* {#FukamiIshihara55} T. Fukami, S. Ishihara, _Almost Hermitian structure on $S^6$_ , Tohoku Math J. 7 (1955), 151–156.
* {#AgrikolaBorowkaFriedrich17} [[Ilka Agricola]], Aleksandra Borówka, [[Thomas Friedrich]], _$S^6$ and the geometry of nearly Kähler 6-manifolds_ ([arXiv:1707.08591](https://arxiv.org/abs/1707.08591))
* {#Bryant14} [[Robert Bryant]], _S.-S. Chern's study of almost-complex structures on the six-sphere_ ([arXiv:1405.3405](https://arxiv.org/abs/1405.3405))
* [[Robert Bryant]], _Remarks on the geometry of almost complex 6-manifolds_ ([arXiv:math/0508428](https://arxiv.org/abs/math/0508428))
[[!redirects 6-spheres]] |
600-cell | https://ncatlab.org/nlab/source/600-cell |
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analog of the [[Platonic solids]]...
The 120 vertices of the 600-cell form the [[binary icosahedral group]].
## Related concepts
* [[24-cell]]
* [[120-cell]]
## References
* [[John Baez]], _The 600-Cell_ ([Part I](https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/), [Part II](https://johncarlosbaez.wordpress.com/2017/12/24/the-600-cell-part-2/), [Part III](https://johncarlosbaez.wordpress.com/2017/12/28/the-600-cell-part-3/))
See also
* Wikipedia, _[600-cell](https://en.wikipedia.org/wiki/600-cell)_
|
6d Chern-Simons theory | https://ncatlab.org/nlab/source/6d+Chern-Simons+theory |
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[[!include infinity-Chern-Simons theory - contents]]
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Physics
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[[!include physicscontents]]
=--
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=--
#Contents#
* table of contents
{:toc}
## Idea
The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 6.
## Examples
(...)
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* **6d Chern-Simons theory**
* [[7d Chern-Simons theory]]
* [[11d Chern-Simons theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[AKSZ sigma-model]]
[[!redirects 6d Chern-Simons theories]]
[[!redirects 6-dimensional Chern-Simons theory]]
[[!redirects 6-dimensional Chern-Simons theories]]
|
6j symbol | https://ncatlab.org/nlab/source/6j+symbol | [[!redirects Wigner 6-j symbol]]
[[!redirects Wigner 6-j symbol]]
#Contents#
* table of contents
{:toc}
## Idea
$6j$ symbols are pieces of combinatorial data which can be associated to [[fusion categories]], encoding their [[associator]] with a finite amount of algebraic data. The $6j$ symbols along with the [[fusion ring]] is enough data to uniquely reconstruct the category.
$6j$ symbols are only defined for [[multiplicity free fusion category|multiplicity free fusion categories]]. That is, fusion categories in which all of the fusion coefficients are $0$ or $1$. For fusion categories with larger coefficients, one has to replace $6j$ symbols (which are complex numbers) with higher-dimensional matrices.
The name "$6j$ symbol" comes from the fact that the symbol has $6$ indicies associated to it.
The $6j$ symbols depend on a local choice of basis of hom-spaces of your fusion category, and hence they are not bona-fide invariants. Choosing a different choice of local basis corresponds to a [[gauge transformation]].
## Definition
Let $\mathcal{C}$ be a [[multiplicity free fusion category]]. Let
$$\alpha: (- \otimes -)\otimes - \to - \otimes (-\otimes -)$$
be its associator. We wish to encode the associator with a finite amount of algebraic data. That is, with a finite quantity of complex numbers. We do this by studying the action of $\alpha$ with respect to the (finite) set $\mathcal{L}$ of isomorphism classes of simple objects of $\mathcal{C}$.
For every triple $A,B,C$ of simple objects of $\mathcal{C}$, choose a distinguished generator
\begin{imagefromfile}
"file_name": "distinguished-basis-element.jpg",
"width": 300
\end{imagefromfile}
This will be a non-zero morphism whenever $\dim \text{Hom}(A\otimes B,C)\geq 1$ and $0$ otherwise. Here, we are using the graphical language of [[string diagrams]]. There generators are chosen arbitrarily, except for the conditions that
\begin{imagefromfile}
"file_name": "basis-element-conditions.jpg",
"width": 300
\end{imagefromfile}
where $\rho_A$ denotes the right [[unitor]] and $\lambda_A$ denotes the left unitor.
Note the key use of the fact that $\mathcal{C}$ is multiplicity free. If fusion coefficients could be $\geq 2$ then we would not have one-element bases of the spaces $\text{Hom}(A\otimes B,C)$. Instead, we would have to choose larger bases of these spaces are get matrices for our $6j$ symbols instead of complex numbers.
For all quadruples $A,B,C,D$ of simple objects, there is a basis of $\text{Hom}((A\otimes B)\otimes C,D)$ given by the symbols
\begin{imagefromfile}
"file_name": "6j-basis-1.jpg",
"width": 300
\end{imagefromfile}
and there is a basis of $\text{Hom}(A\otimes (B\otimes C),D)$.
\begin{imagefromfile}
"file_name": "6j-basis-2.jpg",
"width": 300
\end{imagefromfile}
where in both cases $N$ ranges over representatives of isomorphism classes of simple objects.
\begin{definition} The associator $\alpha$ induces a map
$$\alpha_{A,B,C}:\text{Hom}((A\otimes B)\otimes C,D)\to \text{Hom}(A\otimes (B\otimes C),D)$$
for all simple objects $A,B,C,D\in\mathcal{C}$. Expressing the source and target of this map in the distinguished bases above, we can canonically identify $\alpha_{A,B,C}$ with a matrix.
The __6j symbol__ $F^{a,b,c}_{d;n,m}$ is defined to be the $(N,M)$th coefficient of this matrix, where $a,b,c,d,n,m$ are isomorphism classes of simple objects. That is, the $6j$ symbols are the unique complex numbers making the identity
\begin{imagefromfile}
"file_name": "6j-identity.jpg",
"width": 300
\end{imagefromfile}
holds.
\end{definition}
## Use of Yoneda perspective
It is a key observation that one does not use the $6j$ symbols to directly encode $\alpha$. Instead, the $6j$ symbols are used to describe the action of the hom-space $\text{hom}(-,D)$.
They key point is that objects are not associated vector spaces in a linear category: hom-spaces are. Hence, when encoding maps as matrices one has to take a dual perspective from objects to morphisms.
The [[Yoneda lemma]] exactly guarantees that the data encoded is enough to recover $\alpha$. Namely, knowing the action of $\alpha$ on the hom-space $\text{hom}((A\otimes B)\otimes C,D)\to \text{hom}(A\otimes (B\otimes C),D)$ for all simple objects $D$ one gets the full data of a natural transformation $\text{hom}((A\otimes B)\otimes C,-)\to \text{hom}(A\otimes (B\otimes C),-)$ by the [[semisimple category | semisimplicity]] of $\mathcal{C}$. Because the Yoneda embedding is fully faithful, we can uniquely determine what map $\alpha$ must have induces this natural transformation.
## Related concepts
* [[Wigner 3-j symbol]] ([[Clebsch-Gordan coefficients]])
* [[Wigner 9j symbol]]
* [[Yoneda lemma]]
[[!redirects Wigner 6-j symbols]]
[[!redirects Wigner 6j symbol]]
[[!redirects Wigner 6j symbols]]
[[!redirects 6-j symbol]]
[[!redirects 6-j symbols]]
[[!redirects 6j symbols]] |
7-sphere | https://ncatlab.org/nlab/source/7-sphere |
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[[!include spheres -- contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 7.
This is one of the [[parallelizable manifold|parallelizable]] spheres, as such corresponds to the [[octonions]] among the [[division algebras]], being the manifold of unit octonions, and is the only one of these which does not carry ([[Lie group|Lie]]) [[group]] structure but just [[Moufang loop]] structure.
## Properties
### Quaternionic Hopf fibration
{#QuaternionicHopfFibration}
The 7-sphere participates in the [[quaternionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[quaternions]] or Hamiltonian numbers $\mathbb{H}$.
$$
\array{
S^3 &\hookrightarrow& S^7
\\
&& \downarrow^\mathrlap{p}
\\
&& S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2)
}
$$
Here the idea is that $S^7$ can be construed as $\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each [[fiber]] a [[torsor]] parametrized by quaternionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^3$). This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.
### Coset space realizations
{#CosetSpaceRealization}
+-- {: .num_prop #QuotientOfSpin7ByG2IsS7}
###### Proposition
**([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])**
Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]),
1. This action is is [[transitive action|transitive]];
1. the [[stabilizer group]] of any point on $S^7$ is [[G2]];
1. all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other.
Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]]
$$
S^7
\;\simeq_{diff}\;
Spin(7)/G_2
\,.
$$
=--
(e.g [Varadarajan 01, Theorem 3](#Varadarajan01))
Other coset realizations of the usual [[differentiable manifold|differentiable]] 7-sphere ([Choquet-Bruhat, DeWitt-Morette 00, p. 288](#Choquet-Bruhat+DeWitt-Morette00)):
* $S^7 \simeq_{diff} $ [[Spin(6)]]$/SU(3) \simeq_{iso} SU(4)/SU(3)$ (by [this Prop.](sphere#OddDimSphereAsSpecialUnitaryCoset));
* $S^7 \simeq_{diff} Spin(5)/SU(2)$ ([Awada-Duff-Pope 83](#AwadaDuffPope83), [Duff-Nilsson-Pope 83](#DuffNilssonPope83))
These three coset realizations of 'squashed' 7-spheres together with a fourth
* $S^7 \simeq_{diff} Spin(8)/Spin(7)$,
the realization of the 'round' 7-sphere, may be seen jointly as resulting from the 8-dimensional representations of even [[Clifford algebras]] in 5, 6, 7, and 8 dimensions (see [Baez](#Baez)) and as such related to the four [[normed division algebras]]. See also [Choquet-Bruhat+DeWitt-Morette00, pp. 263-274](#Choquet-Bruhat+DeWitt-Morette00).
[[!include coset space structure on n-spheres -- table]]
The following gives an [[exotic 7-sphere]]:
* $S^7 \simeq_{homeo} Sp(1)\backslash Sp(2)/Sp(1)$ ([[Gromoll-Meyer sphere]])
\linebreak
### Exotic 7-spheres
A celebrated result of Milnor is that $S^7$ admits [[exotic smooth structures]] (see at _[[exotic 7-sphere]]_), i.e., there are [[smooth manifold]] structures on the [[topological manifold]] $S^7$ that are not [[diffeomorphism|diffeomorphic]] to the standard smooth structure on $S^7$. More structurally, considering smooth structures up to [[orientation|oriented]] diffeomorphism, the different smooth structures form a [[monoid]] under a (suitable) operation of [[connected sum]], and this monoid is isomorphic to the [[cyclic group]] $\mathbb{Z}/(28)$. With the notable possible exception of $n = 4$ (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension $7$. This phenomenon is connected to the [[h-cobordism theorem]] (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented [[homotopy spheres]]).
One explicit construction of the smooth structures is given as follows (see [Milnor 1968](#Mil2)). Let $W_k$ be the algebraic variety in $\mathbb{C}^5$ defined by the equation
$$z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0$$
and $S_\epsilon \subset \mathbb{C}^5$ a sphere of small radius $\epsilon$ centered at the origin. Then each of the $28$ smooth structures on $S^7$ is represented by an intersection $W_k \cap S_\epsilon$, as $k$ ranges from $1$ to $28$. These manifolds sometimes go by the name _Brieskorn manifolds_ or _[[Brieskorn spheres]]_ or _[[Milnor spheres]]_.
### $G_2$-structure
{#G2Structure}
Let $\phi_0 \in \Omega^3(\mathbb{R}^7)$ be the [[associative 3-form]] and let
$$
\Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7)
$$
be given by
$$
\Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0
$$
(where $x_0$ denotes the canonical coordinate on the first factor of $\mathbb{R}$ and $\phi_0$ is pulled back along the projection to $\mathbb{R}^7$)
.
By construction this is its own [[Hodge dual]]
$$
\Phi = \star \Phi
\,.
$$
This implies that as we restrict $\Phi_0$ to
$$
\mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7
$$
then there is a unique 3-form
$$
\phi \in \Omega^3(S^7)
$$
on the 7-sphere such that
$$
\Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi
\;\;\;\; (on \; \mathbb{R}^8 - \{0\})
\,.
$$
This 3-form $\phi$ defines a [[G2-structure]] on $S^7$. It is _nearly parallel_ in that
$$
d \phi = 4 \star \phi
\,.
$$
(e.g. [Lotay 12, def.2.4](#Lotay12))
## Related concepts
[[!include spheres -- contents]]
## References
* [[Martin Cederwall]], Christian R. Preitschopf, _The Seven-sphere and its Kac-Moody Algebra_, Commun. Math. Phys. 167 (1995) 373-394 ([arXiv:hep-th/9309030](http://arxiv.org/abs/hep-th/9309030))
* Takeshi Ôno, _On the Hopf fibration $S^7 \to S^4$ over $Z$_, Nagoya Math. J.
Volume 59 (1975), 59-64. ([Euclid](http://projecteuclid.org/euclid.nmj/1118795554))
Relation to the [[Milnor fibration]]:
* [[Kenneth Intriligator]], [[Hans Jockers]], [[Peter Mayr]], [[David Morrison]], M. Ronen Plesser, _Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes_, Adv.Theor.Math.Phys. 17 (2013) 601-699 ([arXiv:1203.6662](http://arxiv.org/abs/1203.6662))
An [[ADE classification]] of finite subgroups of $SO(8)$ [[free action|acting freely]] on $S^7$ (see at _[[group action on an n-sphere]]_) such that the quotient is [[spin structure|spin]] and has at least four [[Killing spinors]] (see also at [[ABJM model]]) is in
* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))
* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_ ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761))
Discussion of [[subgroups]]:
* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf))
Discussion of [[exotic smooth structures]] on 7-spheres includes
* Wikipedia, _Exotic sphere_, [link](https://en.wikipedia.org/wiki/Exotic_sphere).
The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in
* {#Mil2} [[John Milnor]], "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968).
Discussion of (nearly) [[G2-structures]] on $S^7$ and [[calibrated submanifolds]] includes
* {#Lotay12} [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](http://arxiv.org/abs/1006.0361), [talk slides](http://www.homepages.ucl.ac.uk/~ucahjdl/JDLotay_KIAS2011_slides.pdf))
On [[coset]]-realizations:
* {#Kramer98} Linus Kramer, _Octonion Hermitian quadrangles_, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 ([euclid:1103409015](https://projecteuclid.org/euclid.bbms/1103409015))
* {#Choquet-Bruhat+DeWitt-Morette00} [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, Part II, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$
* {#AwadaDuffPope83} M. A. Awada, [[Mike Duff]], [[Christopher Pope]], _$N=8$ Supergravity Breaks Down to $N=1$_, Phys. Rev. Lett. 50, 294 – Published 31 January 1983 ([doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294))
* {#DuffNilssonPope83} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043))
* {#Baez} [[John Baez]], _Rotations in the 7th Dimension_, ([blog post](https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html)), and _TWF 195_, ([webpage](http://math.ucr.edu/home/baez/week195.html))
[[!redirects 7-spheres]]
[[!redirects seven sphere]]
[[!redirects seven spheres]]
|
7d Chern-Simons theory | https://ncatlab.org/nlab/source/7d+Chern-Simons+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The case of [[higher dimensional Chern-Simons theory]] in [[dimension]] 7.
## Examples
We discuss
1. [Abelian 7d CS theory](#AbelianTheory) of an abelian 3-form connection;
1. [7d p2 theory on String 2-connections](#OnString2Connections)
1. [2-species cup-product theory on a G2 manifold](#TwoSpeciesCupProductTheoryOnG2Manifold)
### Abelian theory
{#AbelianTheory}
A basic 7d [[higher dimensional Chern-Simons theory]] is the abelian theory, whose [[extended Lagrangian]] $\mathbf{L}$ is the [[diagonal]] of the [[cup product in ordinary differential cohomology]]
$$
\mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}
\colon
\mathbf{B}^3 U(1)_{conn}
\stackrel{\Delta}{\to}
\mathbf{B}^3 U(1)_{conn}
\times
\mathbf{B}^3 U(1)_{conn}
\stackrel{\widehat {\cup}}{\to}
\mathbf{B}^7 U(1)_{conn}
\,.
$$
The [[transgression]] of this to [[codimension]] 0 hence for $\Sigma_7$ a [[closed manifold]] of [[dimension]] 7 is the [[action functional]]
$$
\exp\left(
2 \pi i \int_{\Sigma_7}
[\Sigma_7, \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}]
\right)
\;\colon\;
[\Sigma_7, \mathbf{B}^3 U(1)_{conn}]
\to
U(1)
\,.
$$
A [[gauge field]] configuration
$$
\phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}
$$
here is a [[circle n-bundle with connection|circle 3-bundle with connection]]. In the special case that the underlying [[circle 3-group]] [[principal 3-bundle]] is trivializable and trivialized, this is equivalently a [[differential 3-form]] $C \in \Omega^3(\Sigma_7)$ and the above [[action functional]] takes this to the simple expression
$$
C
\mapsto
\exp\left(
2 \pi i \int_{\Sigma_7} C \wedge d C
\right)
\in U(1)
\,,
$$
where in the [[exponent]] we have the [[integration of differential forms]] over the [[wedge product]] of $C$ with its [[de Rham differential]]. On general field configurations the action functional is the suitable globalization of this expression.
In ([Witten 97](#Witten97)), ([Witten 98](#Witten98)) a slight refinement of this construction (a [[quadratic refinement]] induced by an [[integral Wu structure]]) was argued to be the [[holographic principle|holographic dual]] to the [[self-dual higher gauge theory]] of the abelian self-dual 2-form gauge field in the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of the [[M5-brane]]. The issue of the quadratic refinement was discussed in more detail in ([HopkinsSinger](#HopkinsSinger)). A refinement to [[extended Lagrangians]] as above is discussed in ([FSSII](#FSSII)).
By the argument in ([Witten98](#Witten98)) the above relation holds when we interpret the fields $\phi \colon : \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}$ as the [[supergravity C-field]] after [[Kaluza-Klein mechanism|compactification]] on a 4-[[sphere]] in the [[AdS-CFT|AdS7-CFT6]] setup. By the discussion at [[11-dimensional supergravity]] this field is in general not simply a 3-connection as above but receives corrections by a [[Green-Schwarz mechanism]] and "flux quantization" which give it non-abelian components. This, and the resulting non-abelian generalization of the above extended Lagrangian is discussed in ([FSSI](#FSSI), [FSSII](#FSSII)).
The nonabelian 7d action functional this obtained contains the following two examples as summands.
### Nonabelian $p_2$ theory on String 2-connections
{#OnString2Connections}
The [[second fractional Pontryagin class]]
$$
[\tfrac{1}{6}p_2] \in H^8(B String, \mathbb{Z})
$$
has a smooth and differential refinement (see at _[[twisted differential fivebrane structure]]_) to an [[extended Lagrangian]]
$$
\tfrac{1}{2}\hat \mathbf{p}_2
\;\colon\;
\mathbf{B}String_{conn}
\to
\mathbf{B}^7 U(1)_{conn}
\,.
$$
where the domain is the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 2-stack]] of [[String 2-group]] [[connection on a 2-bundle|principal 2-connections]] (see at _[[differential string structure]]_ for more). This modulates the [[Chern-Simons circle 7-bundle]] with connection on $\mathbf{B}String_{conn}$.
The [[transgression]] of this to codimension 0 yields an [[action functional]]
$$
\exp\left(
2 \pi i \int_{\Sigma_7}
[\Sigma_7, \tfrac{1}{6}\hat \mathbf{p}_2]
\right)
\;\colon\;
[\Sigma_7, \mathbf{B}String_{conn}]
\to
U(1)
$$
on string 2-connection fields. This is part of the quantum-corrected and flux-quantized extended action functional of the [[supergravity C-field]] in [[11-dimensional supergravity]] by the analysis in ([FSSII](#FSSII)).
### Two-species cup-product theory on a $G_2$ manifold
{#TwoSpeciesCupProductTheoryOnG2Manifold}
For $X$ a [[G2-manifold]] with characteristic [[differential forms]]
$$
\omega_3 \in \Omega^3(X)
$$
and
$$
\omega_4 = \star \omega_3\in \Omega^4(X)
$$
and for $G$ a simply connected compact [[semisimple Lie group]] with [[invariant polynomial]] $\langle -,-\rangle$, consider the [[action functional]] on the space of $\mathfrak{g}$-[[Lie algebra valued 1-forms]] $A$ given by the [[integration of differential forms]]
$$
A
\mapsto
\exp\left(
2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right)
\right)
\,,
$$
where $CS(A) \in \Omega^3(X)$ is the [[Chern-Simons form]] of $A$. This, or some suitable globalization of this, has been considered as an [[action functional]] for 7-dimensional Chern-Simons-type theory in ([Donaldson-Thomas](#DonaldsonThomas)) and ([Baulieu-Losev-Nekrasov](#BaulieuLosevNekrasov)). This appears as an action functional in [[topological M-theory]] ([deBoer et al](#deBoerEtAl)).
To refine this to an [[extended Lagrangian]] and then fully globalize the action functional we can ask for a [[higher geometric quantization|higher geometric prequantization]] of $\omega_4$, regarded as a [[n-plectic structure|3-plectic structure]], by a [[prequantum n-bundle|prequantum 3-bundle]] $\hat \mathbf{G}_2$
$$
\array{
&& \mathbf{B}^3 U(1)_{\mathrm{conn}}
\\
& {}^{\mathllap{\hat \mathbf{G}_2}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}
\\
X
&\stackrel{\omega_4}{\to}&
\Omega^4_{cl}
}
\,,
$$
where $\mathbf{B}^3 U(1)_{conn} \in $ [[Smooth∞Grpd]] is the smooth [[moduli ∞-stack]] of [[circle n-bundle with connection|circle 3-bundles with connection]].
If moreover we write
$$
\hat \mathbf{c} \;:\;
\mathbf{B}G_{conn}
\to
\mathbf{B}^3 U(1)_{conn}
$$
for the universal differential characteristic map which is the [[Lie integration]] of $\langle-,-\rangle$ (as discussed at _[[differential string structure]]_), hence the [[extended Lagrangian]] for ordinary [[3d Chern-Simons theory|3d]] $G$-[[Chern-Simons theory]], then an [[extended Lagrangian]] for the above [[action functional]] is given by the [[cup product in ordinary differential cohomology]]
$$
\exp\left(
2 \pi i \int_{\Sigma_7} [\Sigma_7, \hat {\mathbf{G}}_4 \hat \cup \hat \mathbf{S}]
\right)
\;\colon\;
X
\times
\mathbf{B}G_{conn}
\stackrel{(\hat \mathbf{G}_2, \hat \mathbf{c})}{\to}
\mathbf{B}^3 U(1)_{conn}
\times
\mathbf{B}^3 U(1)_{conn}
\stackrel{\hat \cup}{\to}
\mathbf{B}^7 U(1)_{conn}
\,.
$$
(This is an cup product extended Lagrangian of the kind considered in ([FSSIII](#FSSIII)).)
Notice that the prequantization lift to [[differential cohomology]] is entirely demanded by the interpretation of $\omega_4$ as the [[field strength]] of the [[supergravity C-field]] in interpretations of this setup in [[M-theory on G2-manifolds]].
Moreover, the above considerations do not really need $X$ to be a [[G2-manifold]] to go through, a manifold with [[weak G2 holonomy]] is just as well, hence equipped with $\phi \in \Omega^3(X)$ such that
$$
\omega_4 = \lambda \star \phi
$$
and
$$
d \phi = \omega_4
\,.
$$
This arises from [[Freund-Rubin compactifications]] with [[cosmological constant]] $\lambda$ ([Bilal-Derendinger-Sfetsos](#BilalDerendingerSfetsos)).
## Properties
### Moduli of fields (abelian case)
[[!include moduli of higher lines -- table]]
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* **7d Chern-Simons theory**
* [[11d Chern-Simons theory]]
* [[AKSZ sigma-models]]
* [[string field theory]]
* [[infinite-dimensional Chern-Simons theory]]
* [[M-theory on G2-manifolds]], [[topological M-theory]]
* [[Hitchin functional]]
## References
### Abelian theory
The abelian 7d [[higher dimensional Chern-Simons theory]] of a [[circle n-bundle with connection|circle 3-bundle with connection]] was considered in
* {#Witten97} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ J. Geom. Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
* {#Witten98} [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))
and argued to be the [[holographic principle|holographic dual]] to the [[self-dual higher gauge theory]] of an abelian 2-form connection on a _single_ [[M5-brane]] in its [[6d (2,0)-supersymmetric QFT]] on the worldvolume.
The precise formulation of this functional in terms of [[differential cohomology]] and [[integral Wu structure]] was given in
* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_
{#HopkinsSinger}
### Nonabelian theories
{#ReferencesNonabelianTheories}
In
* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_
{#FSSI}
* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_
{#FSSII}
the 7d Chern-Simons action obtained by [[Kaluza-Klein reduction|compactifying]] [[11-dimensional supergravity]] including the quantum corrections of the [[supergravity C-field]] on a 4-sphere (the [[AdS-CFT|AdS7/CFT6]] setup) is considered and refined to an [[extended Lagrangian]]. It contains the [Donaldson-Thomas](#DonaldsonThomas)-functional $\int_X CS(A) \wedge G_4$ as one summand and the [Witten 97](#Witten97)-functional as another.
Further discussion of [[extended Lagrangians]] for 7d CS theories is in
* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_
{#FSSIII}
### On $G_2$-manifolds
The Chern-Simons type action functionals $A \mapsto \int_X CS(A) \wedge \omega_4$ on a 7d [[G2-manifold]] $(X, \omega_3)$ was first considered in
* [[S. Donaldson]], R. Thomas, _Gauge theory in higher dimensions_ ([pdf](http://www2.imperial.ac.uk/~rpwt/skd.pdf))
{#DonaldsonThomas}
and around (3.23) of
* L. Baulieu, A. Losev, [[Nikita Nekrasov]], _Chern-Simons and Twisted Supersymmetry in Higher Dimensions_, Nucl.Phys. B522 (1998) 82-104 ([arXiv:hep-th/9707174](http://arxiv.org/abs/hep-th/9707174))
{#BaulieuLosevNekrasov}
In
* [[Jan de Boer]], [[Paul de Medeiros]], Sheer El-Showk, Annamaria Sinkovics, _Open $G_2$ Strings_ ([arXiv:hep-th/0611080](http://arxiv.org/abs/hep-th/0611080)) {#deBoerEtAl}
this is put into the context of [[topological M-theory]] (see around equation (2) in the introduction).
Discussion for [[weak G2-holonomy]] is in
* A. Bilal, J.-P. Derendinger, K. Sfetsos, _(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry_, Nucl.Phys. B628 (2002) 112-132 ([arXiv:hep-th/0111274](http://arxiv.org/abs/hep-th/0111274))
{#BilalDerendingerSfetsos}
### Formulation in extended TQFT
Formulation in [[extended TQFT]] is discussed in
* [[Dan Freed]], _[[4-3-2 8-7-6]]_, talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012
[[!redirects D=7 Chern-Simons theory]]
[[!redirects 7-dimensional Chern-Simons theory]]
[[!redirects 7d Chern-Simons theories]]
[[!redirects 7-dimensional Chern-Simons theories]] |
7d spherical space forms -- table | https://ncatlab.org/nlab/source/7d+spherical+space+forms+--+table |
| $N$ [[Killing spinors]] on <br/> [[spherical space form]] $S^7/\widehat{G}$ | $\phantom{AA}\widehat{G} =$ | [[spin group|spin]]-lift of [[subgroup]] of <br/>[[isometry group]] of [[7-sphere]] | [[3d superconformal gauge field theory]] <br/> on [[black brane|back]] [[M2-branes]] <br/> with [[near horizon geometry]] $AdS_4 \times S^7/\widehat{G}$ |
|---------|----------------------------------------------|------|----|
| $\phantom{AA}N = 8\phantom{AA}$ | $\phantom{AA}\mathbb{Z}_2$ | [[cyclic group of order 2]] | [[BLG model]] |
| $\phantom{AA}N = 7\phantom{AA}$ | --- | --- | --- |
| $\phantom{AA}N = 6\phantom{AA}$ | $\phantom{AA}\mathbb{Z}_{k\gt 2}$ | [[cyclic group]] | [[ABJM model]] |
| $\phantom{AA}N = 5\phantom{AA}$ | $\phantom{AA}2 D_{k+2}$ <br/> $2 T$, $2 O$, $2 I$ | [[binary dihedral group]], <br/> [[binary tetrahedral group]], <br/> [[binary octahedral group]], <br/> [[binary icosahedral group]] | ([HLLLP 08a](ABJM+model#HLLLP08), [BHRSS 08](ABJM+model#BHRSS08)) |
| $\phantom{AA}N = 4\phantom{AA}$ | $\phantom{A}2 D_{k+2}$ <br/> $2 O$, $2 I$ | [[binary dihedral group]], <br/> [[binary octahedral group]], <br/> [[binary icosahedral group]] | ([HLLLP 08b](3d+superconformal+gauge+field+theory#HLLLP08b), [Chen-Wu 10](#ChenWu10)) |
* [[José Figueroa-O'Farrill]] et al. 2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))
|
8-manifold | https://ncatlab.org/nlab/source/8-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[manifold]] of [[dimension]] 8.
## Properties
### Signature
{#Signature}
Let $X$ be a [[compact topological space|compact]] [[orientation|oriented]] [[smooth manifold|smooth]] [[8-manifold]]. Then its [[signature genus|signature]] is related to the [[second Pontryagin class]] $p_2$ and the [[cup product]] of the [[first Pontryagin class]] $p_1$ with itself, both evaluated on the [[fundamental class]] of $X$, by
\[
\label{SignatureFormula}
\sigma[X]
\;=\;
\tfrac{1}{45}
\big(
7 p_2
-
(p_1)^2
\big)[X]
\,.
\]
### G-Structures on 8-Manifolds
{#GStructuresOn8Manifolds}
We state results on [[cohomology|cohomological]] [[obstructions]] to and characterization of various [[G-structures]] on [[closed manifold|closed]] [[8-manifolds]].
+-- {: .num_prop #Spin5StructureOnConnected8Manifolds}
###### Proposition
**([[Spin(5)]]-[[G-structure|structure]] on [[8-manifolds]])**
Let $X$ be a [[closed manifold|closed]] [[connected topological space|connected]] [[8-manifold]]. Then $X$ has [[G-structure]] for $G =$ [[Spin(5)]] if and only if the following conditions are satisfied:
1. The second and sixth [[Stiefel-Whitney classes]] (of the [[tangent bundle]]) vanish
$$
w_2 \;=\; 0
\,,
\phantom{AAA}
w_6 \;=\; 0
$$
1. The [[Euler class]] $\chi$ (of the [[tangent bundle]]) evaluated on $X$ (hence the [[Euler characteristic]] of $X$) is proportional to [[I8]] evaluated on $X$:
$$
\begin{aligned}
8 \chi[X]
&=
192 \cdot I_8[X]
\\
& =
4
\Big(
p_2 - \tfrac{1}{2}\big(p_1\big)^2
\Big)[X]
\end{aligned}
$$
1. The [[Euler characteristic]] is divisible by 4:
$$
\tfrac{1}{4}\chi[X] \;\in\; \mathbb{Z}
\,.
$$
=--
([Čadek-Vanžura 97, Corollary 5.5](#CadekVanzura97))
\linebreak
+-- {: .num_prop #Spin4StructureOnClosed8dSpinManifolds}
###### Proposition
**([[Spin(4)]]-[[G-structure|structure]] on [[8-manifolds]])**
Let $X$ be a [[closed manifold|closed]] [[connected topological space|connected]] [[spin structure|spin]] [[8-manifold]]. Then $X$ has [[G-structure]] for $G =$ [[Spin(4)]]
\[
\label{Spin4Structure}
\array{
&&
B Spin(4)
\\
&
{}^{\mathllap{ \widehat{T X} }}
\nearrow
&
\big\downarrow
\\
X
&
\underset{T X}{\longrightarrow}
&
B Spin(8)
}
\]
if and only if the following conditions are satisfied:
1. the sixth [[Stiefel-Whitney class]] of the [[tangent bundle]] vanishes
$$
w_6(T X) \;=\; 0
$$
1. the [[Euler class]] of the [[tangent bundle]] vanishes
$$
\chi_8(T X)
\;=\;
0
$$
1. the [[I8]]-term evaluated on $X$ is divisible as:
$$
\tfrac{1}{32}
\Big(
p_2
-
\big(
\tfrac{1}{2}
\big(
p_1
\big)^2
\big)
\Big)
\;\in\;
\mathbb{Z}
$$
1. there [[existential quantifier|exists]] an [[integer]] $k \in \mathbb{Z}$ such that
1. $p_2 = (2k - 1)^2 \left( \tfrac{1}{2} p_1 \right)^2$;
1. $\tfrac{1}{3} k (k+2) p_2[X] \;\in\; \mathbb{Z}$.
Moreover, in this case we have for $\widehat{T X}$ a given [[Spin(4)]]-[[G-structure|structure]] as in (eq:Spin4Structure) and setting
\[
\label{TildeG4}
\widetilde G_4
\;\coloneqq\;
\tfrac{1}{2}
\chi_4(\widehat{T X})
+
\tfrac{1}{4}p_1(T X)
\]
for $\chi_4$ the [[Euler class]] on $B Spin(4)$ (which is an integral class, by [this Prop.](Spin4#IntegralCohomologyOfClassifyingSpace))
the following relations:
1. $\tilde G_4$ (eq:TildeG4) is an integer multiple of the [[first fractional Pontryagin class]] by the factor $k$ from above:
$$
\widetilde G_4
\;=\;
k \cdot \tfrac{1}{2}p_1
$$
1. The (mod-2 reduction followed by) the [[Steenrod operation]] $Sq^2$ on $\widetilde G_4$ (eq:TildeG4) vanishes:
$$
Sq^2
\left(
\widetilde G_4
\right)
\;=\;
0
$$
1. the shifted square of $\tilde G_4$ (eq:TildeG4) evaluated on $X$ is a multiple of 8:
$$
\tfrac{1}{8}
\left(
\left(
\widetilde G_4
\right)^2
-
\widetilde G_4 \big( \tfrac{1}{2} p_1\big)[X]
\right)
\;\in\;
\mathbb{Z}
$$
1. The [[I8]]-term is related to the shifted square of $\widetilde G_4$ by
$$
4
\Big(
\left(
\widetilde G_4
\right)^2
-
\widetilde G_4
\left(
\tfrac{1}{2}p_1
\right)
\Big)
\;=\;
\Big(
p_2
-
\big(
\tfrac{1}{2}p_1
\big)^2
\Big)
$$
=--
([Čadek-Vanžura 98a, Cor. 4.2 with Cor. 4.3](#CadekVanzura98a))
\linebreak
## Examples
### With exotic boundary 7-spheres
{#ExoticBoundary7Spheres}
Consider $S^4$ the [[4-sphere]] and let $D^4$ denote the 4-[[disk]] regarded as a [[manifold with boundary]]. Then a $D^4$-[[fiber bundle]] over $S^4$ is an 8-dimensional [[manifold with boundary]].
By the [[clutching construction]], such bundles are classified by [[homotopy classes]] of maps
$$
f_{(m,n)}
\;\colon\;
S^3
\longrightarrow
SO(4)
$$
from a [[3-sphere]] (the [[equator]] of $S^4$) to [[SO(4)]]. By [this Prop.](SO4#TheExceptionalIso) such maps are classified by [[pairs]] of [[integers]] $(m,n) \in \mathbb{Z} \times \mathbb{Z}$.
If here $m+n = \pm 1$ then the [[boundary]] of the corresponding [[8-manifold]] is [[homotopy equivalent]] to a [[7-sphere]], and in fact [[homeomorphism|homeomorphic]] to the [[7-sphere]].
Assuming this 8-manifold is a [[smooth manifold]], then plugging in the numbers into the [[signature genus|signature]] formula (eq:SignatureFormula) yields the relation
$$
p_2[X]
\;\coloneqq\;
\tfrac{1}{7}
\big(
4(2m -1)^2 + 45
\big)
$$
Here the left hand side must be an [[integer]], while the right hand side is not an integer for all choices of [[pairs]] $(m,n)$. This means that for these choices the [[boundary]] [[7-sphere]] is not [[diffeomorphism|diffeomorphic]] to the standard smooth 7-sphere -- it is instead an [[exotic 7-sphere]].
(see [Joachim-Wraith, p. 2-3](#JoachimWraith))
From the point of view of [[M-theory on 8-manifolds]], these [[8-manifolds]] $X$ with (exotic) [[7-sphere]] [[boundaries]] correspond to [[near horizon limits]] of [[black brane|black]] [[M2 brane]] spacetimes $\mathbb{R}^{2,1} \times X$, where the [[M2-branes]] themselves would be sitting at the center of the [[7-spheres]] (if that were included in the spacetime, see also [[Dirac charge quantization]]).
([Morrison-Plesser 99, section 3.2](#MorrisonPlesser99))
\linebreak
## Related concepts
* [[M-theory on 8-manifolds]]
* [[Cayley 4-form]]
[[!include low dimensional manifolds -- table]]
## References
### General
* Anand Dessai, _Topology of positively curved 8-dimensional manifolds with symmetry_ ([arXiv:0811.1034](https://arxiv.org/abs/0811.1034))
### $G$-Structure
* {#CadekVanzura95} [[Martin Čadek]], [[Jiří Vanžura]], _On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes_, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 ([dml-cz:118764](https://dml.cz/handle/10338.dmlcz/118764))
* {#CadekVanzura97} [[Martin Čadek]], [[Jiří Vanžura]], _On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles_, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 ([jstor:43737249](https://www.jstor.org/stable/43737249))
* {#CadekVanzura98a} [[Martin Čadek]], [[Jiří Vanžura]], _On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes_, Colloquium Mathematicum (1998), 76 (2), pp 213-228 ([web](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-cmv76z2p213bwm))
* {#CadekVanzura98b} [[Martin Čadek]], [[Jiří Vanžura]], _Almost quaternionic structures on eight-manifolds_, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 ([euclid:1200787905](https://projecteuclid.org/euclid.ojm/1200787905))
* {#CadekVanzura98c} [[Martin Čadek]], [[Jiří Vanžura]], _Various structures in 8-dimensional vector bundles over 8-manifolds_, Banach Center Publications (1998) Volume: 45, Issue: 1, page 183-197 ([dml:208903](https://eudml.org/doc/208903))
* {#CadekCrabbVanzura08} [[Martin Čadek]], Michael Crabb, [[Jiří Vanžura]], _Obstruction theory on 8-manifolds_, manuscripta math. 127 (2008), 167-186 ([arXiv:0710.0734](https://arxiv.org/abs/0710.0734))
* {#CadekCrabbVanzura10} [[Martin Čadek]], Michael Crabb, [[Jiří Vanžura]], _Quaternionic structures_, Topology and its Applications Volume 157, Issue 18, 1 December (2010), Pages 2850-2863 ([doi:10.1016/j.topol.2010.09.005](https://doi.org/10.1016/j.topol.2010.09.005))
### Exotic boundary 7-spheres
* {#JoachimWraith} [[Michael Joachim]], D. J. Wraith, _Exotic spheres and curvature_ ([pdf](https://ivv5hpp.uni-muenster.de/u/joachim/exo.pdf))
* {#MorrisonPlesser99} [[David Morrison]], [[M. Ronen Plesser]], section 3.2 of _Non-Spherical Horizons, I_, Adv. Theor. Math. Phys.3:1-81, 1999 ([arXiv:hep-th/9810201](https://arxiv.org/abs/hep-th/9810201))
[[!redirects 8-manifolds]]
|
8-sphere | https://ncatlab.org/nlab/source/8-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] for $n = 8$.
## Properties
+-- {: .num_prop}
###### Proposition
There is a [[homeomorphism]]
$$
\mathbb{O}P^1 \,\simeq\, S^8
$$
between the [[octonionic projective line]] and the [[8-sphere]].
=--
### Exotic 8-sphere
There is a unique [exotic smooth structure on the 8-sphere](https://ncatlab.org/nlab/show/exotic+smooth+structure#Exotic8Sphere). |
9-dimensional supergravity | https://ncatlab.org/nlab/source/9-dimensional+supergravity |
#Contents#
* table of contents
{:toc}
## Idea
[[supergravity]] in [[dimension]] 9
## Properties
### U-duality
[[!include U-duality -- table]]
## References
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _Supermembrane origin of type II gauged supergravities in 9D_, JHEP 1209 (2012) 063 [[arXiv:1203.2767](http://arxiv.org/abs/1203.2767)]
[[!redirects 9d supergravity]]
[[!redirects D=9 supergravity]]
|
A Categorical Manifesto | https://ncatlab.org/nlab/source/A+Categorical+Manifesto |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
[_A Categorical Manifesto_]( http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.362 ) is an article written by [[Joseph Goguen]], encouraging the use of [[category theory]] in [[computer science]].
To quote from his introduction:
> This paper tries to explain why [[category theory]] is useful in [[computer science|computing science]]. The basic answer is that computing science is a young field that is growing rapidly, is poorly organised, and needs all the help it can get, and that [[category theory]] can provide help with at least the following:
> * Formulating definitions and theories. In computing science, it is often more difficult to formulate concepts and results than to give a proof. The seven guidelines of this paper can help with formulation; the guidelines can also be used to measure the elegance and coherence of existing formulations.
> * Carrying out [[proofs]]. Once basic concepts have been correctly formulated in a categorical language, it often seems that proofs "just happen": at each step, there is a "natural" thing to try, and it works. [[diagram chasing|Diagram chasing]] provides many examples of this. It could almost be said that the purpose of category theory is to reduce all proofs to such simple calculations.
> * Discovering and exploiting relations with other fields. Sufficiently abstract formulations can reveal surprising connections. For example, an analogy between [[Petri nets]] and the [[lambda-calculus|$\lambda$-calculus]] might suggest looking for a [[closed category]] structure on the category of Petri nets.
> * Dealing with abstraction and representation independence. In computing science, more abstract viewpoints are often more useful, because of the need to achieve independence from the often overwhelmingly complex details of how things are represented or implemented. A corollary of the first guideline is that two objects are "abstractly the same" if they are [[isomorphic]]. Moreover, [[universal constructions]] (i.e., [[adjoints]]) define their results uniquely up to isomorphism, i.e., "abstractly" in just this sense.
> * Formulating conjectures and research directions. Connections with other fields can suggest new questions in your own field. Also the seven guidelines can help to guide research. For example, if you have found an interesting [[functor]], then you might be well advised to investigate its [[adjoints]].
> * Unification. Computing science is very fragmented, with many different sub-disciplines having many different schools within them. Hence, we badly need the kind of conceptual unification that category theory can provide.
### Abstract ###
> This paper tries to explain why and how [[category theory]] is useful in [[computer science|computing science]], by giving guidelines for applying seven basic categorical concepts: [[category]], [[functor]], [[natural transformation]], [[limit]], [[adjoint]], [[colimit]] and [[comma category]]. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion.
## Related entries
* [[computational trinitarianism]]
## Reference
* [[Joseph A. Goguen]], _A Categorical Manifesto_. In _Mathematical Structures in Computer Science_, Vol. 1, No. 1. (1991), pp. 49-67, doi:[10.1017/S0960129500000050](https://doi.org/10.1017/S0960129500000050), [CiteSeerX](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.362).
category: reference, computer science |
A Concise Course in Algebraic Topology | https://ncatlab.org/nlab/source/A+Concise+Course+in+Algebraic+Topology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
This page collects material related to:
* [[Peter May]]:
\linebreak
**A Concise Course in Algebraic Topology**
\linebreak
University of Chicago Press (1999)
[ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html)
[pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf)
a textbook on [[algebraic topology]] and [[homotopy theory]].
See also:
* [[Peter May]], [[Kate Ponto]]:
\linebreak
**More concise algebraic topology -- Localization, Completion, and Model Categories**
\linebreak
University of Chicago Press (2012)
[ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html)
[pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)
{#Problem} Beware that the latter has an issue in [Lem. 17.1.7](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf#page=371) (following an analogous problematic statement in [Cole 2006, Prop. 5.3](Strøm+model+structure#Cole06)) where it comes to establishing a [[Strøm model structure]]. This was noticed by [[Richard Williamson]], see [Barthel & Riehl, p. 2 and Rem 5.12 and Sec. 6.1](Strøm+model+structure#BarthelRiehl13) for details.
category: reference
[[!redirects Concise course]]
[[!redirects Concise Course]]
[[!redirects A concise course in algebraic topology]]
[[!redirects Concise]]
[[!redirects More concise algebraic topology]]
[[!redirects More Concise Algebraic Topology]]
|
A crash course in topos theory -- The big picture | https://ncatlab.org/nlab/source/A+crash+course+in+topos+theory+--+The+big+picture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
* [[André Joyal]],
_A crash course in topos theory: The big picture_,
lecture series
at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris
three parts
1) [[locale]] theory
2) [[topos theory]]
3) [[(infinity,1)-topos theory]]
([recordings with time annotations](https://sites.google.com/site/logiquecategorique/autres-seminaires/ihes/ihestopos/cours/joyal))
category: reference |
A Cubical Approach to Synthetic Homotopy Theory | https://ncatlab.org/nlab/source/A+Cubical+Approach+to+Synthetic+Homotopy+Theory |
* [[Dan Licata]], [[Guillaume Brunerie]]:
**A Cubical Approach to Synthetic Homotopy Theory**
LICS 2015,
30th Annual ACM/IEEE Symposium on Logic in Computer Science
[doi:10.1109/LICS.2015.19](https://doi.org/10.1109/LICS.2015.19)
[PDF from Dan's homepage](http://dlicata.web.wesleyan.edu/pubs/lb15cubicalsynth/lb15cubicalsynth.pdf),
[PDF from Guilluame's homepage](https://guillaumebrunerie.github.io/pdf/lb15cubicalsynth.pdf)
on [[cubical type theory|cubical]] [[homotopy type theory]].
> **Abstract.** [[homotopy theory|Homotopy theory]] can be developed [[synthetic homotopy theory|synthetically]] in homotopy type theory, using [[types]] to describe spaces, the [[identity type]] to describe paths in a space, and iterated identity types to describe higher-dimensional paths. While some aspects of homotopy theory have been developed synthetically and [[Formalized Homotopy Theory|formalized]] in [[proof assistants]], some seemingly easy examples have proved difficult because the required manipulations of paths becomes complicated. In this paper, we describe a [[cubical]] approach to developing homotopy theory within type theory. The [[identity type]] is complemented with higher-dimensional cube types, such as a type of [[squares]], dependent on four points and four lines, and a type of three-dimensional [[cubes]], dependent on the boundary of a cube. Path-over-a-path types and higher generalizations are used to describe cubes in a fibration over a cube in the base. These higher-dimensional cube and path-over types can be defined from the usual identity type, but isolating them as independent conceptual abstractions has allowed for the formalization of some previously difficult examples.
category: reference |
A first idea of quantum field theory -- Feynman diagrams | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Feynman+diagrams | ## Feynman diagrams
{#FeynmanDiagrams}
So far we considered only the [[axioms]] on a consistent perturbative S-matrix /[[time-ordered products]]
and its formal consequences. Now we discuss the actual construction of [[time-ordered products]],
hence of perturbative S-matrices, by the process called _[[renormalization]] of [[Feynman diagrams]]_.
We first discuss how [[time-ordered product]], and hence the perturbative S-matrix [above](#PerturbativeSMatrixAndTimeOrderedProducts),
is uniquely determined away from the locus where interaction points coincide (prop. \ref{TimeOrderedProductAwayFromDiagonal} below).
Moreover, we discuss how on that locus the time-ordered product is naturally expressed as a sum
of [[products of distributions]] of [[Feynman propagators]] that are labeled by [[Feynman diagrams]]: the _[[Feynman perturbation series]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below).
This means that the full [[time-ordered product]] is an [[extension of distributions]] of these
_[[scattering amplitudes]]- to the locus of coinciding vertices. The space of possible such
extensions turns out to be finite-dimensional in each order of $g/\hbar, j/\hbar$, parameterizing the choice of
[[point-supported distributions]] at the interaction points whose [[degree of a distribution|scaling degree]]
is bounded by the given Feynman propagators.
+-- {: .num_defn #TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport}
###### Definition
For $k \in \mathbb{N}$, write
$$
\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}
\hookrightarrow
\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}
$$
for the subspace of the $k$-fold [[tensor product]] of the space of compactly supported polynomial
local densities (def. \ref{CompactlySupportedPolynomialLocalDensities}) on those [[tuples]]
which have pairwise disjoint spacetime [[support]].
=--
+-- {: .num_prop #TimeOrderedProductAwayFromDiagonal}
###### Proposition
**([[time-ordered product]] away from the diagonal)**
Restricted to $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$
(def. \ref{TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport})
there is a unique [[time-ordered product]] (def. \ref{TimeOrderedProduct}),
given by the [[star product]] that is induced by the [[Feynman propagator]] $\omega_F$
$$
F \star_{\omega_F} G
\;\coloneqq\;
prod \circ
\exp\left(
\hbar
\left\langle
\omega_F , \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi}
\right\rangle
\right)
(F \otimes G)
$$
in that
$$
T( L_1 \cdots L_k )
=
L_1 \star_{\omega_F} L_2 \star_{\omega_F} \cdots \star_{\omega_F} L_k
\,.
$$
=--
+-- {: .proof}
###### Proof
Since the [[singular support]] of the [[Feynman propagator]] is on the [[diagonal]],
and since the support of elements in $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$
is by definition in the complement of the diagonal, the star product $\star_{\omega_F}$ is well defined.
By construction it satisfies the axioms "peturbation" and "normalization" in def. \ref{TimeOrderedProduct}.
The only non-trivial point to check is that it indeed satisfies "[[causal factorization]]":
Unwinding the definition of the [[Hadamard state]] $\omega$ and the [[Feynman propagator]] $\omega_F$, we have
$$
\begin{aligned}
\omega & = \tfrac{i}{2}( \Delta_R - \Delta_A ) + H
\\
\omega_F & = \tfrac{i}{2}( \Delta_R + \Delta_A ) + H
\end{aligned}
$$
where the propagators on the right have, in particular, the following properties:
1. the [[advanced propagator]] vanishes when its first argument is not in the causal past of its second argument:
$$
(supp(F) \geq supp(G))
\;\Rightarrow\;
\left(
\left\langle \Delta_A , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = 0
\right)
\,.
$$
1. the [[retarded propagator]] equals the [[advanced propagator]] with arguments switched:
$$
\left\langle \Delta_R , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle
=
\left\langle \Delta_A , \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle
$$
1. $H$ is symmetric:
$$
\left\langle H, \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle
=
\left\langle H, \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle
$$
It follows for causal ordering $supp(F) \geq supp(G)$ (def. \ref{CausalOrdering}) that
$$
\begin{aligned}
F \star_{\omega_F} G
& =
prod \circ \exp\left(
\hbar
\left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ
\exp\left(
\hbar
\left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ
\exp\left(
\hbar
\left\langle \tfrac{i}{2}\Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ
\exp\left(
\hbar
\left\langle \tfrac{i}{2}( \Delta_R - \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ
\exp\left(
\hbar
\left\langle \omega , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
F \star_{\omega} G
\end{aligned}
$$
and for $supp(G) \geq supp(F)$ that
$$
\begin{aligned}
F \star_{\omega_F} G
& =
prod \circ \exp\left(
\hbar
\left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ \exp\left(
\hbar
\left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ \exp\left(
\hbar
\left\langle \tfrac{i}{2} \Delta_A + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( F \otimes G )
\\
& =
prod \circ \exp\left(
\hbar
\left\langle \tfrac{i}{2} \Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( G \otimes F )
\\
& =
prod \circ \exp\left(
\hbar
\left\langle \tfrac{i}{2} (\Delta_R - \Delta_A) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle
\right)
( G \otimes F )
\\
& =
G \star_{\omega} F
\,.
\end{aligned}
$$
This shows that $\star_F$ is a consistent time-ordered product on the subspace of functionals
with disjoint support. It is immediate from the above that it is the unique solution on this subspace.
=--
+-- {: .num_remark #TimeOrderedProductAssociative}
###### Remark
**([[time-ordered product]] is [[associativity|assocativative]])**
Prop. \ref{TimeOrderedProductAwayFromDiagonal} implies in particular that the
time-ordered product is [[associativity|associative]], in that
$$
T( T(V_1 \cdots V_{k_1}) \cdots T(V_{k_{n-1}+1} \cdots V_{k_n} ) )
=
T( V_1 \cdots V_{k_1} \cdots V_{k_{n-1}+1} \cdots V_{n_n} )
\,.
$$
=--
It follows that the problem of constructing time-ordered products, and hence (by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}) the
perturbative S-matrix, consists of finding compatible [[extension of distributions|extension]]
of the distribution
$ prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right)$ to the diagonal.
Moreover, by the nature of the exponential expression, this means in each order to
extend [[product of distributions|products]] of Feynman propagators labeled by
[[graphs]] whose [[vertices]] correspond to the polynomial factors in $F$ and $G$ and whose
[[edges]] indicate over which variables the Feynman propagators are to be multiplied.
+-- {: .num_defn #ScalarFieldFeynmanDiagram}
###### Definition
**([[scalar field]] [[Feynman diagram]])**
A _[[scalar field]] [[Feynman diagram]]_ $\Gamma$ is
1. a [[natural number]] $v \in \mathcal{N}$ (number of [[vertices]]);
1. a $v$-[[tuple]] of elements $(V_r \in \mathcal{F}_{loc} \langle g,j\rangle)_{r \in \{1, \cdots, v\}}$ (the interaction and external field vertices)
1. for each $a \lt b \in \{1, \cdots, v\}$ a natural number $e_{a,b} \in \mathbb{N}$ ("of [[edges]] from the $a$th to the $b$th vertex").
For a given [[tuple]] $(V_j)$ of interaction vertices we write
$$
FDiag_{(V_j)}
$$
for set of scalar field Feynman diagrams with that tuple of vertices.
=--
+-- {: .num_prop #FeynmanPerturbationSeriesAwayFromCoincidingPoints}
###### Proposition
**([[Feynman perturbation series]] away from coinciding vertices)**
For $v \in \mathbb{N}$ the $v$-fold
[[time-ordered product]] away from the diagonal,
given by prop. \ref{TimeOrderedProductAwayFromDiagonal}
$$
T_v
\;\colon\;
\left(\mathcal{F}_{loc}\langle g,j\rangle\right)_{pds}^{\otimes^{v}}
\longrightarrow
\mathcal{W}[ [ g/\hbar, j/\hbar] ]
$$
is equal to
$$
T_k(V_1 \cdots V_v)
\;=\;
prod
\circ
\underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum}
\underset{ r \lt s \in \{1, \cdots, v\} }{\prod}
\tfrac{1}{e_{r,s}!}
\left\langle
\hbar \omega_F
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} }
\right\rangle
(V_1 \otimes \cdots \otimes V_v)
\,,
$$
where the edge numbers $e_{r,s} = e_{r,s}(\Gamma)$ are those of the given Feynman diagram $\Gamma$.
=--
([Keller 10, IV.1](Feynman+diagram#Keller10))
+-- {: .proof}
###### Proof
We proceed by [[induction]] over the number of [[vertices]].
The statement is trivially true for a single vertex.
Assume it is true for $v \geq 1$ vertices. It follows that
$$
\begin{aligned}
T(V_1 \cdots V_v V_{v+1})
& =
T( T(V_1 \cdots V_v) V_{v+1} )
\\
&=
prod \circ
\exp\left(
\left\langle
\hbar \omega_F, \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi}
\right\rangle
\right)
\left(
prod
\circ
\underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum}
\underset{ r \gt s \in \{1, \cdots, v\} }{\prod}
\frac{1}{e_{r,s}!}
\left\langle
\hbar \omega_F
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} }
\right\rangle
(V_1 \otimes \cdots \otimes V_v)
\right)
\;\otimes\;
V_{v+1}
\\
& =
prod
\circ
\underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum}
\underset{ r \gt s \in \{1, \cdots, v\} }{\prod}
\tfrac{1}{e_{r,s}!}
\left\langle
\hbar \omega_F
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} }
\right\rangle
\left(
\underset{e_{1,{v+1}}, \cdots e_{v,v+1} \in \mathbb{N}}{\sum}
\underset{t \in \{1, \cdots v\}}{\prod}
\tfrac{1}{e_{t,v+1} !}
\left(
\frac{\delta^{e_{1,v+1}} V_1 }{\delta \phi_{1}^{e_{1,v+1}}}
\otimes
\cdots
\otimes
\frac{ \delta^{e_{v,v+1}} V_v}{ \delta \phi_{v}^{e_{v,v+1}} }
\right)
\;\otimes\;
\frac{\delta^{e_{1,v+1} + \cdots + e_{v,v+1}} V_{v+1}}{\delta \phi_{v-1}^{e_{1,v+1} + \cdots + e_{v,v+1}}}
\right)
\\
&=
prod
\circ
\underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v+1}}}{\sum}
\underset{ r \lt s \in \{1, \cdots, v+1\} }{\prod}
\tfrac{1}{e_{r,s}!}
\left\langle
\hbar \omega_F
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} }
\right\rangle
(V_1 \otimes \cdots \otimes V_{v+1})
\end{aligned}
$$
Here in the first step we use the [[associativity]] of the time-ordered product
(remark \ref{TimeOrderedProductAssociative}), in the second step we use the induction assumption,
in the third we pass the outer functional derivatives through the pointwise product
using the [[product rule]], and in the fourth step we recognize that this amounts to summing
in addition over all possible choices of sets of edges from the first $v$ vertices to the new $v+1$st vertex,
which yield in total the sum over all diagrams with $v+1$ vertices.
=--
+-- {: .num_remark}
###### Remark
**([[loop order]] and powers of [[Planck's constant]])**
From prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} one deduces that the
order in [[Planck's constant]] that a ([[planar graph|planar]]) [[Feynman diagram]] contributes to the
S-matrix is given (up to a possible offset due to external vertices) by the "number of loops" in the diagram.
In the computation of [[scattering amplitudes]] for [[field (physics)|fields]]/[[particles]] via [[perturbative quantum field theory]] the [[scattering matrix]] ([[Feynman perturbation series]]) is a [[formal power series]] in (the [[coupling constant]] and) [[Planck's constant]] $\hbar$ whose contributions may be labeled by [[Feynman diagrams]]. Each Feynman diagram $\Gamma$ is a finite labeled [[graph]], and the order in $\hbar$ to which this graph contributes is
$$
\hbar^{ E(\Gamma) - V(\Gamma) }
$$
where
1. $V(\Gamma) \in \mathbb{N}$ is the number of [[vertices]] of the graph
1. $E(\Gamma) \in \mathbb{N}$ is the number of [[edges]] in the graph.
This comes about, according to the above, because
1. the explicit $\hbar$-dependence of the [[S-matrix]] is
$$
S\left(\tfrac{g}{\hbar} L_{int} \right)
=
\underset{k \in \mathbb{N}}{\sum} \frac{g^k}{\hbar^k k!} T( \underset{k \, \text{factors}}{\underbrace{L_{int} \cdots L_{int}}} )
$$
1. the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is
$$
T(L_{int} L_{int}) = prod \circ \exp\left( \hbar \int \omega_{F}(x,y) \frac{\delta}{\delta \phi(x)} \otimes \frac{\delta}{\delta \phi(y)} \right) ( L_{int} \otimes L_{int} )
\,,
$$
where $\omega_F$ denotes the [[Feynman propagator]] and $\phi(x)$ the field observable at point $x$ (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as [[scalar fields]], because this is all that affects the counting of the $\hbar$ powers).
The resulting terms of the S-matrix series are thus labeled by
1. the number of factors of the [[interaction]] $L_{int}$, these are the [[vertices]] of the corresponding Feynman diagram and hence each contibute with $\hbar^{-1}$
1. the number of integrals over the Feynman propagator $\omega_F$, which correspond to the edges of the Feynman diagram, and each contribute with $\hbar^1$.
Now the formula for the [[Euler characteristic of planar graphs]] says that the number of regions in a plane that are encircled by edges, the _faces_ here thought of as the number of "loops", is
$$
L(\Gamma) = 1 + E(\Gamma) - V(\Gamma)
\,.
$$
Hence a planar Feynman diagram $\Gamma$ contributes with
$$
\hbar^{L(\Gamma)-1}
\,.
$$
So far this is the discussion for internal edges. An actual scattering matrix element is of the form
$$
\langle \psi_{out} \vert S\left(\tfrac{g}{\hbar} L_{int} \right)
\vert \psi_{in} \rangle
\,,
$$
where
$$
\vert \psi_{in}\rangle
\propto
\tfrac{1}{\sqrt{\hbar^{n_{in}}}}
\phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{in}}) \vert vac \rangle
$$
is a state of $n_{in}$ free field quanta and similarly
$$
\vert \psi_{out}\rangle
\propto
\tfrac{1}{\sqrt{\hbar^{n_{out}}}}
\phi^\dagger(k_1) \cdots \phi^\dagger(k_{n_{out}}) \vert vac \rangle
$$
is a state of $n_{out}$ field quanta. The normalization of these states, in view of the commutation relation $[\phi(k), \phi^\dagger(q)] \propto \hbar$, yields the given powers of $\hbar$.
This means that an actual [[scattering amplitude]] given by a [[Feynman diagram]] $\Gamma$ with $E_{ext}(\Gamma)$ external vertices scales as
$$
\hbar^{L(\Gamma) - 1 + E_{ext}(\Gamma)/2 }
\,.
$$
(For the analogous discussion of the dependence on the actual [[quantum observables]] on $\hbar$ given by [[Bogoliubov's formula]], see [there](Bogoliubov's+formula#PowersInPlancksConstant).)
=--
|
A first idea of quantum field theory -- Field variations | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Field+variations | ## Field variations
{#FieldVariations}
In this chapter we discuss these topics:
* _[Jet bundles](##JetBundles)_
* _[Differential operators](#DifferentialOperators)_
* _[Variational calculus and the Variational bicomplex](#VariationaCalculusAndVariationalBicomplex)_
$\,$
Given a [[field bundle]] as in def. \ref{FieldsAndFieldBundles} above, then we know what [[type]] of quantities the corresponding
[[field histories]] assign to a given spacetime point (a given [[event]]). Among all consistent such field histories, some are to qualify as those that "may occur in reality" if we think of the field theory as a means to describe parts of the [[observable universe]]. Moreover, if the reality to be described does not exhibit "action at a distance" then admissibility of its field histories should be determined over arbitrary small spacetime regions, in fact over the [[infinitesimal neighbourhood]] of any spacetime point (remark \ref{JetBundleInTermsOfSyntheticDifferentialGeometry} below). This means equivalently that the realized field histories should be those that satisfy a given _[[differential equation]]_, namely an [[equation]] between the [[partial derivatives]] of the field history at any spacetime point. This is called the _[[equation of motion]]_ of the field theory (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} below).
In order to formalize this, it is useful to first collect all the possible partial derivatives that a field history may have at any given point into one big space of "field derivatives at spacetime points". This collection is called the _[[jet bundle]]_ of the [[field bundle]], given as def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below.
Moving around in this space means to change the possible value of fields and their derivatives, hence to _vary_ the fields. Accordingly _[[variational calculus]]_ of fields is just [[differential calculus]] on the [[jet bundle]] of the [[field bundle]], this we consider in def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime} below.
$\,$
**[[jet bundles]]**
{#JetBundles}
+-- {: .num_defn #JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}
###### Definition
**([[jet bundle]] of a [[trivial vector bundle]] over [[Minkowski spacetime]])**
Given a [[field fiber]] [[super vector space]] $F = \mathbb{R}^{b\vert s}$ with [[linear basis]] $(\phi^a)$,
then for $k \in \mathbb{N}$ a natural number, the _order-$k$ [[jet bundle]]_
$$
\array{
J^k_{\Sigma}( E )
\\
\downarrow^{\mathrlap{jb_k}}
\\
\Sigma
}
$$
over [[Minkowski spacetime]] $\Sigma$ of the [[trivial vector bundle]]
$$
E \coloneqq \Sigma \times F
$$
is the [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) which is spanned by coordinate functions to be denoted as follows:
$$
\left(
(x^\mu)
\,,\,
(\phi^a )
\,,\,
( \phi^a_{,\mu} )
\,,\,
( \phi^a_{,\mu_1\mu_2} )
\,,\,
\cdots
\,,\,
( \phi^a_{,\mu_1 \cdots \mu_k} )
\,,\,
\cdots
\right)
$$
where the indices $\mu, \mu_1, \mu_2, \cdots$ range from 0 to $p$, while the index $a$ ranges from $1$ to $b$ for the
even field coordinates, and then from $b+1$ to $b+s$ for the odd-graded field coordinates and the lower indices are
symmetric:
$$
\label{JetCoodinatesSymmetry}
\phi^a_{\mu_1 \cdots \mu_{i} \cdots \mu_j \cdots \mu_k}
\;=\;
\phi^a_{\mu_1 \cdots \mu_{j} \cdots \mu_i \cdots \mu_k}
\,.
$$
In terms of these coordinates the [[bundle]] [[projection]] map $jb_k$ is just the one that remembers the spacetime coordinates $x^\mu$ and forgets the values of the field $\phi^a$ and its derivatives $\phi_{\mu}$. Similarly there are intermediate projection maps
$$
\array{
\cdots
&\overset{jb_{3,2}}{\longrightarrow}&
J^{2}_\Sigma(E)
&\overset{jb_{2,1}}{\longrightarrow}&
J^1_\Sigma(E)
&\overset{jb_{1,0}}{\longrightarrow}&
E
\\
&& &{}_{\mathllap{jb_2}}\searrow& {}^{\mathllap{jb_1}}\downarrow &\swarrow_{\mathrlap{fb}}&
\\
&& && \Sigma &&
}
$$
given by forgetting coordinates with more indices.
The _infinite-order [[jet bundle]]_
$$
J^\infty_\Sigma(E) \in SuperSmoothSet
$$
is the [[direct limit]] of [[super formal smooth sets|super smooth sets]] (def. \ref{SuperFormalSmoothSet}) over these finite order jet bundles. Explicitly this means that it is the [[smooth set]] which is defined
by the fact that a smooth function (a plot, by prop. \ref{SuperCartSpYpnedaLemma})
$$
U \overset{f}{\longrightarrow} J^\infty_\Sigma(E)
$$
from some [[super Cartesian space]] $U$ is equivalently a system of ordinary smooth functions
into all the finite-order jet spaces
$$
\left(
U \overset{f_k}{\longrightarrow} J^k_\Sigma(E)
\right)_{k \in \mathbb{N}}
\,,
$$
such that this system is compatible with the
above projection maps, i.e. such that
$$
\underset{k \in \mathbb{N}}{\forall} \left(
jb_{k+1,k} \circ f_{k+1} = f_k
\right)
\phantom{AAAAAAA}
\array{
&& && U &&
\\
&& & {}^{\mathllap{f_2}}\swarrow& {}_{\mathllap{f_1}}\downarrow &\searrow^{f_0}&
\\
\cdots
&\overset{jb_{3,2}}{\longrightarrow}&
J^{2}_\Sigma(E)
&\overset{jb_{2,1}}{\longrightarrow}&
J^1_\Sigma(E)
&\overset{jb_1}{\longrightarrow}&
E
\\
&& &{}_{\mathllap{jb_2}}\searrow& {}^{\mathllap{jb_1}}\downarrow &\swarrow_{\mathrlap{fb}}&
\\
&& && \Sigma &&
}
$$
=--
The coordinate functions $\phi^a_{\mu_1 \cdots \mu_k}$ on a [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
are to be thought of as [[partial derivatives]] $\frac{\partial}{\partial x^{\mu_1}} \cdots \frac{\partial}{\partial x^{\mu_k}} \Phi^a$ of components $\Phi^a$ of would-be [[field histories]] $\Phi$. The power of the jet bundle is that it allows
to disentangle relations between would-be partial derivatives of field history components in themselves
from consideration of actual [[field histories]]. In traditional physics texts this is often done implicitly.
We may make it fully explit by the operation of _[[jet prolongation]]_ which reads in a [[field history]]
and records all its partial derivatives in the form of a section of the jet bundle:
+-- {: .num_defn #JetProlongation}
###### Definition
**([[jet prolongation]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) which happens to be a [[trivial vector bundle]]
over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}.
There is a [[smooth function]] from the [[space of sections]] of $E$, the [[space of field histories]] (example \ref{SupergeometricSpaceOfFieldHistories}) to the space of sections of the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb^\infty}{\to} \Sigma$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) which records the field $\Phi$ and all its spacetimes [[derivatives]]:
$$
\array{
\Gamma_\Sigma(E)
&\overset{j^\infty_\Sigma}{\longrightarrow}&
\Gamma_\Sigma(J^\infty_\Sigma(E))
\\
(\Phi^a)
&\mapsto&
\left(
\left( \Phi^a \right)
\,,\,
\left( \frac{\partial \Phi^a}{\partial x^\mu} \right)
\,,\,
\left( \frac{\partial^2 \Phi^a}{\partial x^{\mu_1} \partial x^{\mu_2}} \right)
\,,\,
\cdots
\right)
}
\,.
$$
This is called the operation of _[[jet prolongation]]_: $j^\infty_\Sigma(\Phi)$ is the jet prolongation of $\Phi$.
=--
+-- {: .num_remark #JetBundleInTermsOfSyntheticDifferentialGeometry}
###### Remark
**([[jet bundle]] in terms of [[synthetic differential geometry]])**
In terms of the [[synthetic differential geometry|infinitesimal geometry]] of [[formal smooth sets]]
(def. \ref{FormalSmoothSet}) the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb_\infty}{\to} \Sigma$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
of a [[field bundle]] $E \overset{fb}{\to}\Sigma$ has the following incarnation:
A [[section]] of the [[jet bundle]] over a point $x \in \Sigma$ of [[spacetime]] (an [[event]]),
is equivalently a section of the original [[field bundle]] over the [[infinitesimal neighbourhood]]
$\mathbb{D}_x$ of that point (example \ref{InfinitesimalNeighbourhood}):
$$
\left\{
\array{
&& J^\infty_\Sigma(E)
\\
& \nearrow & \downarrow^{\mathrlap{jb_\infty}}
\\
\{x\}
&\hookrightarrow&
\Sigma
}
\phantom{AA}
\right\}
\phantom{AA}
\simeq
\phantom{AA}
\left\{
\array{
&& E
\\
& {}^{\mathllap{}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
\mathbb{D}_x &\hookrightarrow& \Sigma
}
\phantom{AA}
\right\}
\,.
$$
Moreover, given a [[field history]] $\Phi$, hence a [[section]] of the [[field bundle]],
then its [[jet prolongation]] $j^\infty(\Phi)$ (def. \ref{JetProlongation}) is that [[section]]
of the [[jet bundle]] which under the above identification is simply the restriction of
$\Phi$ to the [[infinitesimal neighbourhood]] of $x$:
$$
\array{
&& E
\\
& {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
\Sigma & = & \Sigma
}
\phantom{AAAA}\overset{j^\infty_\Sigma}{\mapsto} \phantom{AAAA}
\array{
&& J^\infty_\Sigma(E)
\\
& {}^{\mathllap{j^\infty_\Sigma(\Phi)}}\nearrow & \downarrow^{\mathrlap{jb_\infty}}
\\
\Sigma &=& \Sigma
}
\phantom{AAAA}
\overset{(-)\vert_{\{x\}} }{\mapsto}
\phantom{AAAA}
\array{
&& E
\\
& {}^{\mathllap{\Phi\vert_{\mathbb{D}_x}}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
\mathbb{D}_x &\hookrightarrow& \Sigma
}
\,.
$$
This follows with an argument as in example \ref{InfinitesimalNeighbourhoodsInTheRealLine}.
Hence in [[synthetic differential geometry]] we have:
_The jet of a section $\Phi$ at $x$ is simply the restriction of that section to the [[infinitesimal neighbourhood]] of $x$._
=--
([[schreiber:Synthetic variational calculus|Khavkine-Schreiber 17, section 3.3]])
So the canonical [[coordinates]] on the jet bundle are the spacetime-point-wise _possible_ values of fields and field derivates,
while the [[jet prolongation]] picks the actual collections of field derivatives that may occur for an actual field
history.
+-- {: .num_example #JetFaraday}
###### Example
**(universal [[Faraday tensor]]/[[field strength]] on [[jet bundle]])**
Consider the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) of the [[electromagnetic field]] (example \ref{Electromagnetism})
over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), i.e. the [[cotangent bundle]] $E = T^\ast \Sigma$ (def. \ref{Differential1FormsOnCartesianSpaces}) with jet coordinates
$((x^\mu), (a_\mu), (a_{\mu,\nu}), \cdots )$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Consider the functions on the [[jet bundle]] given by the linear combinations
$$
\label{FaradayTensorJet}
\begin{aligned}
f_{\mu \nu}
& \coloneqq
a_{[\nu,\mu]}
\\
& \coloneqq
\tfrac{1}{2}\left(
a_{\nu,\mu} - a_{\mu,\nu}
\right)
\end{aligned}
$$
of the first order jets.
Then for an [[electromagnetism|electromagnetic]] [[field history]] ("[[vector potential]]"), hence a [[section]]
$$
A \in \Gamma_\Sigma(T^\ast \Sigma) = \Omega^1(\Sigma)
$$
with components $A^\ast (a_\mu) = A_\mu$, its [[jet prolongation]] (def. \ref{JetProlongation})
$$
j^\infty_\Sigma(A) \in \Gamma_\Sigma(J^\infty_\Sigma(T^\ast \Sigma))
$$
has components
$$
\left(
(A_\mu),
\left(
\frac{d A_\mu}{d x^\nu}
\right)
,
\cdots
\right)
\,.
$$
The [[pullback of differential forms|pullback]] of the functions $f_{\mu \nu}$ (eq:FaradayTensorJet)
along this jet prolongation are the components of the [[Faraday tensor]] of the field (eq:TensorFaraday):
$$
\begin{aligned}
\left(j^\infty_\Sigma(A)\right)^\ast(f_{\mu \nu})
& =
F_{\mu \nu}
\\
& =
(d A)_{\mu \nu}
\,.
\end{aligned}
$$
More generally, for $\mathfrak{g}$ a [[Lie algebra]] and
$$
E \coloneqq T^\ast \Sigma \otimes \mathfrak{g}
$$
the [[field bundle]] for [[Yang-Mills theory]] from example \ref{YangMillsFieldOverMinkowski}, consider the functions
$$
f^\alpha_{\mu \nu}
\;\in \;
\Omega^{0,0}_\Sigma(E)
=
C^\infty(J^\infty_\Sigma(E))
$$
on the [[jet bundle]] given by
$$
\label{YangMillsJetFieldStrengthMinkowski}
\begin{aligned}
f^\alpha_{\mu \nu}
& \coloneqq
\tfrac{1}{2}
\left(
a^\alpha_{\nu,\mu}
-
a^\alpha_{\mu,\nu}
+
\gamma^{\alpha}{}_{\beta \gamma}
a^\beta_{\mu} a^\gamma_{\nu}
\right)
\end{aligned}
$$
where $(\gamma^\alpha{}_{\beta \gamma})$ are the structure constants of the Lie algebra as in (eq:LieAlgebraStructureConstants),
and where the square brackets around the indices denote anti-symmetrization.
We may call this the _universal [[Yang-Mills theory|Yang-Mills]] [[field strength]]_, being the
_[[covariant exterior derivative]]_ of the universal Yang-Mills field history.
For $\mathfrak{g} = \mathbb{R}$ the [[line Lie algebra]] and $k$ the canonical [[inner product]] on $\mathbb{R}$ the expression (eq:YangMillsJetFieldStrengthMinkowski) reduces to the universal [[Faraday tensor]] (eq:FaradayTensorJet) for
the [[electromagnetic field]] (example \ref{JetFaraday}).
For $A \in \Gamma_\Sigma(T^\ast \Sigma \otimes \mathfrak{g}) = \Omega^1(\Sigma,\mathfrak{g})$ a field history of
[[Yang-Mills theory]], hence a [[Lie algebra-valued differential 1-form]], then the value of this function on that field
are called the components of the _[[covariant exterior derivative]]_ or _[[field strength]]_
$$
\begin{aligned}
F_{\mu \nu}
& \coloneqq
A^\ast(D_{[\mu} a_{\nu]})
\\
& =
(d_A A)_{\mu \nu}
\end{aligned}
$$
=--
+-- {: .num_example #BFieldJetFaraday}
###### Example
**(universal [[B-field|B-]][[field strength]] on [[jet bundle]])**
Consider the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) of the [[B-field]] (example \ref{BField})
over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}) with jet coordinates
$((x^\mu), (b_{\mu \nu}), (b_{\mu \nu,\rho}), \cdots )$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Consider the functions on the [[jet bundle]] given by the linear combinations
$$
\label{BFieldFaradayTensorJet}
\begin{aligned}
h_{\mu_1 \mu_2 \mu_3}
& \coloneqq
\tfrac{1}{2}
b_{[\mu_1 \mu_2, \mu_3]}
\\
& \coloneqq
\tfrac{1}{6}
\left(
\underset{ \sigma \atop \text{permutation} }{\sum}
(-1)^{ {\vert \sigma \vert} }
b_{\mu_{\sigma_1} \mu_{\sigma_2}, \mu_{\sigma_3}}
\right)
\\
& =
b_{\mu_1 \mu_2, \mu_3}
+
b_{\mu_2 \mu_3, \mu_1}
+
b_{\mu_3 \mu_1, \mu_2}
\,,
\end{aligned}
$$
where in the last step we used that $b_{\mu \nu} = - b_{\nu \mu}$.
=--
$\,$
While the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) is not [[finite number|finite]] [[dimension|dimensional]],
reflecting the fact that there are arbitrarily high orders of spacetime derivatives of a field histories,
it turns out that it is only very "mildly [[infinite dimensional manifold|infinite dimensional]]"
in that
[[smooth functions]] on jet bundles turn out to _locally_ depend on only finitely many of the jet coordinates
(i.e. only on a finite order of spacetime derivatives). This is the content
of the following prop. \ref{JetBundleIsLocallyProManifold}.
This reflects the _locality_ of [[Lagrangian field theory]] defined over [[jet bundles]]: If
functions on the jet bundle could depend on infinitely many jet coordinates, then by [[Taylor series]]
expansion of fields the function at one point over spacetime could in fact depend on field
history values at a _different_ point of spacetime. Such non-local dependence is ruled out by prop. \ref{JetBundleIsLocallyProManifold} below.
In practice this means that the situation is very convenient:
1. Any given [[local Lagrangian density]] (which will define a field theory, we come to this in def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below)
will locally depend on some finite number $k$ of derivatives and may hence locally
be treated as living on the ordinary manifold $J^k_\Sigma(E)$.
1. while at the same time all formulas (such as for the [[Euler-Lagrange equations]], def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) work uniformly
without worries about fixing a maximal order of derivatives.
+-- {: .num_prop #JetBundleIsLocallyProManifold}
###### Proposition
**([[jet bundle]] is a [[locally pro-manifold]])**
Given a [[jet bundle]] $J^\infty_\Sigma(E)$ as in def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime},
then a [[smooth function]] out of it
$$
J^\infty_\Sigma(E) \longrightarrow X
$$
is such that around each point of $J^\infty_\Sigma(E)$ there is a [[neighbourhood]] $U \subset J^\infty_\Sigma(E)$
on which it is given by a function on a smooth function on $J^k_\Sigma(E)$ for some finite $k$.
=--
(see [Khavkine-Schreiber 17, section 2.2 and 3.3](locally+pro-manifold#KhavkineSchreiber17))
$\,$
**[[differential operators]]**
{#DifferentialOperators}
Example \ref{JetFaraday} shows that the [[de Rham differential]] (def. \ref{deRhamDifferential}) may be encoded in terms of
composing [[jet prolongation]] with a suitable function on the [[jet bundle]].
More generally, jet prolongation neatly encodes (possibly non-linear) [[differential operators]]:
+-- {: .num_defn #DifferentialOperator}
###### Definition
**([[differential operator]])**
Let $E_1 \overset{fb_1}{\to} \Sigma$ and $E_2 \overset{fb_2}{\to} \Sigma$
be two smooth [[fiber bundles]] over a common base space $\Sigma$. Then a
(possibly non-linear) _[[differential operator]]_ from [[sections]] of $E_1$
to sections of $E_2$ is a [[bundle morphism]] from the [[jet bundle]] of $E_1$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
to $E_2$:
$$
\array{
J^\infty_\Sigma(E_1)
&& \overset{\tilde D}{\longrightarrow} &&
E_2
\\
& \searrow && \swarrow
\\
&& \Sigma
}
$$
or rather the function $D$ between the [[spaces of sections]] of these bundles
which this induces after [[composition]] with [[jet prolongation]] (def. \ref{JetProlongation}):
$$
D
\;\colon\;
\Gamma_\Sigma(E_1)
\overset{j^\infty_\Sigma}{\longrightarrow}
\Gamma_\Sigma(J^\infty_\Sigma(E_1))
\overset{\tilde D \circ (-)}{\longrightarrow}
\Gamma_\Sigma(E_2)
\,.
$$
If both $E_1$ and $E_2$ are [[vector bundles]] (def. \ref{VectorBundle})
so that their [[spaces of sections]] canonically are [[vector spaces]], then $D$
is called a _[[linear differential operator]]_ if it is a [[linear function]]
between these vector spaces. This means equivalently that $\tilde D$ is a [[linear function]]
in jet coordinates.
=--
+-- {: .num_defn #NormallyHyperbolicDifferentialOperator}
###### Definition
**([[normally hyperbolic differential operator]] on [[Minkowski spacetime]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) which is a [[vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}). Write $E^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle})
A [[linear differential operator]] (def. \ref{DifferentialOperator})
$$
P
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\Gamma_{\Sigma}(E^\ast)
$$
is of _second order_ if it has a coordinate expansion of the form
$$
(P \Phi)_a
\;=\;
P^{\mu \nu}_{a b} \frac{\partial^2 \Phi^b}{\partial x^\mu \partial x^\nu}
+
P^\mu_{a b} \frac{\partial \Phi^b}{\partial x^\mu}
+
P_{a b} \Phi^b
$$
for $\{(P^{\mu \nu}_{a b}), (P^\mu_{a b}), P_{a b}\}$ [[smooth functions]] on $\Sigma$.
This is called a _[[normally hyperbolic differential operator]]_ if its _[[principal symbol]]_
$(P^{\mu \nu}_{a b})$ is proportional to the inverse [[Minkowski metric]] (prop./def. \ref{SpacetimeAsMatrices}) $(\eta^{\mu \nu})$,
i.e.
$$
P^{\mu \nu}_{a b} = \eta^{\mu \nu} Q_{a b}
\,.
$$
=--
+-- {: .num_defn #FormallyAdjointDifferentialOperators}
###### Definition
**([[formally adjoint differential operators]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]]
$\Sigma = \mathbb{R}^{p,1}$ (def. \ref{MinkowskiSpacetime}) and write $E^\ast \to \Sigma$ for the [[dual vector bundle]] (def. \ref{DualVectorBundle}).
Then a [[pair]] of [[linear differential operators]] (def. \ref{DifferentialOperator}) of the form
$$
P, P^\ast
\;\colon\;
\Gamma_\Sigma(E_1)
\longrightarrow
\Gamma_\Sigma(E^\ast)
$$
are called _[[formally adjoint differential operators]]_ via a [[bilinear map|bilinear]] [[differential operator]]
$$
\label{FormallyAdjointDifferentialOperatorWitness}
K \;\colon\;
\Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E)
\longrightarrow
\Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)
$$
with values in [[differential n-form|differential p-forms]] (def. \ref{DifferentialnForms})
such that for all [[sections]] $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have
$$
\left(
P(\Phi_1) \cdot \Phi_2
-
\Phi_1 \cdot P^\ast(\Phi_2)
\right) dvol_\Sigma
\;=\;
d K(\Phi_1, \Phi_2)
\,,
$$
where $dvol_\Sigma$ is the [[volume form]] on [[Minkowski spacetime]] (eq:MinkowskiVolume) and
where $d$ denoted the [[de Rham differential]] (def. \ref{deRhamDifferential}).
This implies by [[Stokes' theorem]] (prop. \ref{StokesTheorem}) in the case of [[compact support]] that
under an [[integral]] $P$ and $P^\ast$ are related via [[integration by parts]].
=--
([Khavkine 14, def. 2.4](Green+hyperbolic+partial+differential+equation#Khavkine14))
$\,$
**[[variational calculus]] and the [[variational bicomplex]]**
{#VariationaCalculusAndVariationalBicomplex}
+-- {: .num_remark #ReplacingBundleMorphismsByDifferentialOperators}
###### Remark
**([[variational calculus]] -- replacing plain [[bundle morphisms]] by [[differential operators]])**
Various concepts in [[variational calculus]], especially the concept of _[[evolutionary vector fields]]_ (def. \ref{EvolutionaryVectorField} below)
and _[[gauge parameter|gauge parameterized]] implicit [[infinitesimal gauge symmetries]]_ (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation} below) follow from
concepts in plain [[differential geometry]] by systematically replacing
plain [[bundle morphisms]] by bundle morphisms out of the [[jet bundle]], hence by [[differential operators]] $\tilde D$
as in def. \ref{DifferentialOperator}.
=--
+-- {: .num_defn #VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}
###### Definition
**([[variational derivative]] and [[total derivative|total spacetime derivative]] -- the [[variational bicomplex]])**
On the [[jet bundle]] $J^\infty_\Sigma(E)$ of a [[trivial vector bundle|trivial]] [[super vector space]]-[[vector bundle]] over [[Minkowski spacetime]] as in def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} we may consider its [[de Rham complex]] of
[[super differential forms]] (def. \ref{DifferentialFormOnSuperCartesianSpaces}); we write its [[de Rham differential]] (def. \ref{deRhamDifferential}) in boldface:
$$
d \;\colon\; \Omega^\bullet(J^\infty_\Sigma(E)) \longrightarrow \Omega^{\bullet+1}(J^\infty_\Sigma(E))
\,.
$$
Since the jet bundle unifies spacetime with field values, we want to decompose this differential into
a contribution coming from forming the [[total derivatives]] of fields along spacetime ("[[horizontal derivatives]]"), and
actual _variation_ of fields at a fixed spacetime point ("[[vertical derivatives]]"):
The _[[total derivative|total spacetime derivative]]_ or _[[horizontal derivative]]_ on $J^\infty_\Sigma(E)$
is the map on [[differential forms]] on the jet bundle of the form
$$
d
\;\colon\;
\Omega^\bullet( J^\infty_\Sigma(E) )
\longrightarrow
\Omega^{\bullet+1}( J^\infty_\Sigma(E) )
$$
which on functions $f \colon J^\infty_\Sigma(E) \to \mathbb{R}$ (i.e. on 0-forms) is defined by
$$
\label{SpacetimeTotalDerivativeOnSmoothFunctions}
\begin{aligned}
d f
& \coloneqq
\frac{d f}{d x^\mu}
\mathbf{d} x^\mu
\\
& \coloneqq
\left(
\frac{\partial f}{\partial x^\mu}
+
\frac{\partial f}{\partial \phi^a}
\phi^a_{,\mu}
+
\frac{ \partial f }{ \partial \phi^a_{,\nu}}
\phi^a_{,\nu \mu }
+
\cdots
\right)
\mathbf{d} x^\mu
\end{aligned}
$$
and extended to all forms by the graded [[Leibniz rule]], hence as a nilpotent [[derivation]] of degree +1.
The _[[variational derivative]]_ or _[[vertical derivative]]_
$$
\label{VariationalDerivative}
\delta
\;\colon\;
\Omega^\bullet( J^\infty_\Sigma(E) )
\longrightarrow
\Omega^{\bullet+1}( J^\infty_\Sigma(E) )
$$
is what remains of the full [[de Rham differential]] when the total spacetime derivative ([[horizontal derivative]]) is subtracted:
$$
\label{VerticalDerivative}
\delta \coloneqq \mathbf{d} - d
\,.
$$
We may then extend the [[horizontal derivative]] from functions on the jet bundle to all
[[differential forms]] on the jet bundle by declaring that
$$
d \circ \mathbf{d} \;\coloneqq\; - \mathbf{d} \circ d
$$
which by (eq:VerticalDerivative) is equivalent to
$$
\label{HorizontalAndVerticalDerivativeAnticommute}
d \circ \;\delta\; = - \delta \circ d
\,.
$$
For example
$$
\begin{aligned}
d \delta \phi
& =
- \delta d \phi
\\
& =
- \delta \left( \phi_{,\mu} d x^\mu \right)
\\
& = - \delta \phi_{,\mu} \wedge d x^\mu
\,.
\end{aligned}
$$
This defines a bigrading on the [[de Rham complex]] of $J^\infty_\Sigma(E)$, into horizontal degree $r$ and vertical degree $s$
$$
\Omega^\bullet\left( J^\infty_\Sigma(E) \right)
\;\coloneqq\;
\underset{r,s}{\oplus} \Omega^{r,s}(E)
$$
such that the horizontal and vertical derivative increase horizontal or vertical degree, respectively:
$$
\label{VariationalBicomplexDiagram}
\array{
C^\infty(J^\infty_\Sigma(E)) = & \Omega^{0,0}(E)
&\overset{d}{\longrightarrow}&
\Omega^{1,0}_\Sigma(E)
&\overset{d}{\longrightarrow}&
\Omega^{2,0}_\Sigma(E)
&\overset{d}{\longrightarrow}&
\cdots
&\overset{d}{\longrightarrow}&
\Omega^{p+1,0}_\Sigma(E)
\\
& \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}}
&& \cdots && \downarrow^{\mathrlap{\delta}}
\\
&
\Omega^{0,1}_\Sigma(E)
&\overset{d}{\longrightarrow}&
\Omega^{1,1}_\Sigma(E)
&\overset{d}{\longrightarrow}&
\Omega^{2,1}_\Sigma(E)
&\overset{d}{\longrightarrow}&
\cdots
&\overset{d}{\longrightarrow}&
\Omega^{p+1,1}_\Sigma(E)
\\
& \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}}
&& \cdots && \downarrow^{\mathrlap{\delta}}
\\
&
\Omega^{0,2}(E)
&\overset{d}{\longrightarrow}&
\Omega^{1,2}(E)
&\overset{d}{\longrightarrow}&
\Omega^{2,2}(E)
&\overset{d}{\longrightarrow}&
\cdots
&\overset{d}{\longrightarrow}&
\Omega^{p+1,2}_\Sigma(E)
\\
& \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}}
&& \cdots && \downarrow^{\mathrlap{\delta}}
\\
& \vdots && \vdots && \vdots
}
\,.
$$
This is called the _[[variational bicomplex]]_.
Accordingly we will refer to the differential forms on the jet bundle often as _variational differential forms_.
=--
$\,$
**[[derivatives]] on [[jet bundle]]**
| def. | symbols | name in physics | name in mathematics |
|--|--|---|--|
| def. \ref{DifferentialFormOnSuperCartesianSpaces} | $\; \mathbf{d}$ | [[de Rham differential]] | [[de Rham differential]] |
| \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime} | $\; d \coloneqq d x^\mu \frac{d}{d x^\mu}$ | [[total derivative|total spacetime derivative]] | [[horizontal derivative]] |
| \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime} | $ \; \frac{d}{d x^\mu} \coloneqq \frac{\partial}{\partial x^\mu} + \phi^a_{,\mu} \frac{\partial}{\partial \phi^a} + \cdots $ | [[total derivative|total spacetime derivative]] <br/> along $\partial_\mu$ | [[horizontal derivative]] <br/> along $\partial_\mu$ |
| \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime} | $\; \delta \coloneqq \mathbf{d} - d$ | [[variational derivative]] | [[vertical derivative]] |
| \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} | $\; \delta_{EL} \mathbf{L} \coloneqq \mathbf{d}\mathbf{L} + d \Theta_{BFV}$ | [[Euler-Lagrange variational derivative|Euler-Lagrange variation]] | [[Euler-Lagrange operator]] |
| \ref{BVComplexOfOrdinaryLagrangianDensity} | $\; s_{BV}$ | [[BV-differential]] | [[Koszul complex|Koszul differential]] |
| \ref{LocalOffShellBRSTComplex} | $\; s_{BRST} $ | [[BRST differential]] | [[Chevalley-Eilenberg complex|Chevalley-Eilenberg differential]] |
| \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} | $\; s $ | [[BV-BRST differential]] | [[Chevalley-Eilenberg complex|Chevalley-Eilenberg]]-[[Koszul-Tate complex|Koszul-Tate differential]] |
| \ref{BVVariationalBicomplex} | $\; s - d $ | [[local BV-BRST differential]] | |
$\,$
+-- {: .num_example #BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}
###### Example
**(basic facts about [[variational calculus]])**
Given the jet bundle of a [[field bundle]] as in def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}, then
in its [[variational bicomplex]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
we have the following:
* The spacetime [[total derivative]] ([[horizontal derivative]]) of a spacetime coordinate function $x^\mu$
coincides with its ordinary de Rham differential
$$
\begin{aligned}
d x^\mu
& =
\frac{\partial x^\mu}{ \partial x^\nu} \mathbf{d}x^\nu
\\
& = \mathbf{d} x^\mu
\end{aligned}
$$
which hence is a horizontal 1-form
$$
\mathbf{d}x^\mu \;\in\; \Omega^{1,0}_\Sigma(E)
\,.
$$
* Therefore the variational derivative ([[vertical derivative|vertical derivative]]) of a spacetime coordinate function vanishes:
$$
\label{VariationalDerivativeOfSpacetimeCoordinateVanishes}
\delta x^\mu = 0
\,,
$$
reflective the fact that $x^\mu$ is not a field coordinate that could be varied.
* In particular the given [[volume form]] on $\Sigma$ gives a horizontal $p+1$-form on the jet bundle,
which has the same coordinate expression (and which we denote by the same symbol)
$$
dvol_\Sigma = d x^0 \wedge d x^1 \wedge \cdots \wedge d x^p
\;\in\;
\Omega^{p+1,0}
\,.
$$
* Generally any horizontal $k$-form is of the form
$$
f_{\mu_1 \cdots \mu_k} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_k}
\;\in\;
\Omega^{k,0}_{\Sigma}(E)
$$
for
$$
f_{\mu_1 \cdots \mu_k} = f_{\mu_1 \cdots \mu_k}\left((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots\right)
\in C^\infty(J^\infty_\Sigma(E))
$$
any smooth function of the spacetime coordinates and the field coordinates (locally depending only on a finite
order of these, by prop. \ref{JetBundleIsLocallyProManifold}).
* In particular every horizontal $(p+1)$-form $\mathbf{L} \in \Omega^{p+1,0}(E)$ is proportional to the
above volume form
$$
\mathbf{L} = L \, dvol_\Sigma
$$
for $L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots)$ some smooth function that may depend on all the
spacetime and field coordinates.
* The spacetimes [[total derivatives]] /horizontal derivatives) of the variational derivative (vertical derivative) $\delta \phi$ of a field variable is the differential 2-form of horizontal degree 1 and vertical degree 1 given by
$$
\begin{aligned}
d (\delta \phi^a)
& =
-
\delta (d \phi_a)
\\
& =
- (\delta \phi^a_{,\mu}) \wedge \mathbf{d} x^\mu
\end{aligned}
\,.
$$
In words this says that "the spacetime derivative of the variation of the field is the variation of its spacetime derivative".
=--
The following are less trivial properties of variational differential forms:
+-- {: .num_prop #PullbackAlongJetProlongationIntertwinesHorizontalDerivative}
###### Proposition
**([[pullback of differential forms|pullback]] along [[jet prolongation]] compatible with [[total derivative|total spacetime derivatives]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{FieldsAndFieldBundles}),
with induced [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Then for $\Phi \in \Gamma_\Sigma(E)$ any field history, the [[pullback of differential forms]] (def. \ref{PullbackOfDifferentialForms})
$$
j^\infty_\Sigma(\Phi)^\ast
\;\colon\;
\Omega^\bullet(J^\infty_\Sigma(E))
\longrightarrow
\Omega^\bullet(\Sigma)
$$
along the [[jet prolongation]] of $\Phi$ (def. \ref{JetProlongation})
1. intertwines the [[de Rham differential]] on [[spacetime]] (def. \ref{Differential1FormsOnCartesianSpaces}) with the [[total spacetime derivative]] ([[horizontal derivative]]) on the [[jet bundle]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):
$$
d \circ j^\infty_\Sigma(\Phi)^\ast
\;=\;
j^\infty_\Sigma(\Phi)^\ast \circ d
\,.
$$
1. annihilates all [[vertical differential forms]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):
$$
j^\infty_\Sigma(\Phi)^\ast\vert_{\Omega^{r, \geq 1}_\Sigma(E)} = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
The operation of [[pullback of differential forms]] along any [[smooth function]] intertwines the full [[de Rham differentials]]
(prop. \ref{PullbackOfDifferentialForms}). In particular we have that
$$
d \circ j^\infty_\Sigma(\Phi)^\ast
=
j^\infty_\Sigma(\Phi)^\ast \circ \mathbf{d}
\,.
$$
This means that the second statement immediately follows from the
first, by definition of the variational (vertical) derivative as the difference between
the full de Rham differential and the horizontal one:
$$
\begin{aligned}
j^\infty_\Sigma(\Phi)^\ast \circ \delta
& =
j^\infty_\Sigma(\Phi)^\ast \circ (\mathbf{d} - d)
\\
& =
(d - d) \circ j^\infty_\Sigma(\Phi)^\ast
\\
& =
0
\end{aligned}
$$
It remains to see the first statement:
Since the [[jet prolongation]] $j^\infty_\Sigma(\Phi)$ preserves the spacetime coordinates $x^\mu$
(being a [[section]] of the [[jet bundle]]) it is immediate that the claimed relation
is satisfied on the
horizontal [[linear basis|basis]] 1-forms $\mathbf{d}x^\mu = d x^\mu$ (example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}):
$$
d j^\infty_\Sigma(\Phi)^\ast( \mathbf{d}x^\mu )
=
d^2 x^\mu = 0
\phantom{AAAAA}
j^\infty_\Sigma(\Phi)^\ast d \mathbf{d} x^\mu
=
j^\infty_\Sigma(\Phi)^\ast d^2 x^\mu
\,.
$$
Therefore it finally remains only to check the first statement on smooth functions (0-forms).
So let
$$
f
=
f\left(
(x^\mu)
\,,\,
(\phi^a)
\,,\,
( \phi^a_{,\mu} )
\,,\,
\cdots
\right)
$$
be a smooth function on the jet bundle. Then by the [[chain rule]]
$$
\begin{aligned}
d j^\infty_\Sigma(\Phi)^\ast
f\left(
(x^\mu)
\,,\,
(\phi^a)
\,,\,
( \phi^a_{,\mu} )
\,,\,
\cdots
\right)
& =
d
f\left(
(x^\mu)
\,,\,
(\Phi^a)
\,,\,
\left( \frac{\partial \Phi^a}{\partial x^\mu} \right)
\,,\,
\cdots
\right)
\\
& =
\left(
\frac{\partial f}{\partial x^\mu}
+
\frac{\partial f}{\partial \phi^a}
\frac{\partial \Phi^a}{\partial x^\mu}
+
\frac{\partial f}{\partial \phi^a_{,\nu}}
\frac{\partial^2 \Phi^a}{\partial x^\nu \partial x^\mu}
+
\cdots
\right)
d x^\mu
\end{aligned}
$$
That this is equal to $j^\infty_\Sigma(\Phi)^\ast d f$ follows by the very definition
of the total spacetime derivative of $f$ (eq:SpacetimeTotalDerivativeOnSmoothFunctions).
=--
+-- {: .num_prop #HorizontalVariationalComplexOfTrivialFieldBundleIsExact}
###### Proposition
**([[horizontal variational complex]] of [[trivial vector bundle|trivial]] [[field bundle]] is [[exact sequence|exact]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] which is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}). Then the [[chain complex]] of [[horizontal differential forms]] $\Omega^{s,0}_\Sigma(E)$
with the [[total spacetime derivative]] ([[horizontal derivative]]) $d$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
$$
\label{ExactSequenceTotalSpacetimeDerivative}
\mathbb{R}
\overset{}{\hookrightarrow}
\Omega^{0,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{1,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{2,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\cdots
\overset{d}{\longrightarrow}
\Omega^{p,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{p+1,0}_\Sigma(E)
$$
is [[exact sequence|exact]]: for all $0 \leq s \leq p$ the [[kernel]] of $d$ coincides with the [[image]] of $d$
in $\Omega^{s,0}_\Sigma(E)$.
More explicitly, this means that not only is every horizontally exact differential form $\omega = d \alpha$ horizontally closed
$d \omega = 0$ (which follows immediately from the fact that we have a [[cochain complex]] in the first place, hence that $d^2 = 0$),
but, conversely, if $\omega \in \Omega^{0 \leq s \leq p,0}_\Sigma(E)$ satisfies $d \omega = 0$, then there exists
$\alpha \in \Omega^{s-1,0}_\Sigma(E)$ with $\omega = d \alpha$.
=--
(e.g. [Anderson 89, prop. 4.3](variational+bicomplex#Anderson89))
+-- {: .num_remark #EulerLagrangeComplex}
###### Remark
**([[Euler-Lagrange complex]])**
In fact the exact sequence (eq:ExactSequenceTotalSpacetimeDerivative) from prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} continues further to the right, as such called the _[[Euler-Lagrange complex]]_. The next differential is the _[[Euler-Lagrange operator]]_ and then then next is the _[[Helmholtz operator]]_.
Here we do not discuss this in detail, but we encounter aspects of the exactness further to the right below in example \ref{TrivialLagrangianDensities} and in prop. \ref{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}.
=--
$\,$
This concludes our discussion of [[variational calculus]] on the [[jet bundle]] of the [[field bundle]].
In the [next chapter](#Lagrangians) we apply this to [[Lagrangian densities]] on the [[jet bundle]], defining [[Lagrangian field theories]].
|
A first idea of quantum field theory -- Fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Fields | ## Fields
{#Fields}
In this chapter we discuss these topics:
* _[Field bundles](#FieldBundles)_
* _[Spaces of field histories](#NonFiniteDimensionalGeometry)_
* _[Infinitesimal geometry](#InfinitesimalGeometry)_
* _[Fermion fields and Supergeometry](#Supergeometry)_
$\,$
A [[field history]] on a given [[spacetime]] $\Sigma$ (a history of spatial [[field configurations]], see remark \ref{FieldHistoriesAsHistoriesOfFieldConfigurations} below) is a [[quantity]] assigned to each point of spacetime (each [[event]]), such that this assignment varies smoothly with spacetime points. For instance an _[[electromagnetic field]] [[field history|history]]_ (example \ref{Electromagnetism} below) is at each point of spacetime a collection of [[vectors]] that encode the direction in which a [[charged particle]] passing through that point would feel a [[force]] (the "[[Lorentz force]]", see example \ref{Electromagnetism} below).
This is readily formalized (def. \ref{FieldsAndFieldBundles} below): If $F$ denotes the [[smooth manifold]] of "values" that the given kind of field may take at any spacetime point, then a field history $\Phi$ is modeled as a [[smooth function]] from spacetime to this space of values:
$$
\Phi
\;\colon\;
\Sigma
\longrightarrow
F
\,.
$$
It will be useful to unify [[spacetime]] and the space of [[field (physics)|field]] values (the [[field fiber]]) into a single manifold, the [[Cartesian product]]
$$
E \;\coloneqq\; \Sigma \times F
$$
and to think of this equipped with the [[projection]] map onto the first factor as a [[fiber bundle]] of spaces of field values over spacetime
$$
\array{
E &\coloneqq& \Sigma \times F
\\
{}^{\mathllap{fb}}\downarrow & \swarrow_{\mathrlap{pr_1}}
\\
\Sigma
}
\,.
$$
This is then called the _[[field bundle]]_, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the [[fiber]] of this [[fiber bundle]] (def. \ref{FiberBundle}), it is sometimes also called the _[[field fiber]]_. (See also at _[[fiber bundles in physics]]_.)
Given a [[field bundle]] $E \overset{fb}{\to}\Sigma$, then a _[[field history]]_ is a [[section]] of that bundle
(def. \ref{Sections}). The discussion of [[field theory]] concerns the [[space of field histories|space of all possible field histories]],
hence the [[space of sections]] of the [[field bundle]] (example \ref{DiffeologicalSpaceOfFieldHistories} below).
This is a very "large" [[generalized smooth space]], called a _[[diffeological space]]_ (def. \ref{DiffeologicalSpace} below).
Or rather, in the presence of [[fermion fields]] such as the [[Dirac field]] (example \ref{DiracFieldBundle} below),
the [[Pauli exclusion principle]] demands that the [[field bundle]] is a [[supermanifold|super-manifold]],
and that the fermionic [[space of field histories]] (example \ref{DiracSpaceOfFieldHistories} below)
is a [[supergeometry|super-geometric]] [[generalized smooth space]]: a _[[super smooth set]]_ (def. \ref{SuperFormalSmoothSet} below).
This smooth structure on the [[space of field histories]] will be crucial when we discuss [[observables]]
of a [[field theory]] [below](#Observables), because these are smooth functions on the [[space of field histories]].
In particular it is this smooth structure which allows to derive that _linear_ observables of a [[free field theory]]
are given by [[distributions]] (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}) below.
Among these are the point evaluation observables ([[delta distributions]]) which are traditionally denoted
by the same symbol as the [[field histories]] themselves.
Hence there are these aspects of the concept of "[[field (physics)|field]]" in [[physics]], which are closely related, but crucially different:
$\,$
**aspects of the concept of [[field (physics)|fields]]**
| aspect | [[term]] | [[type]] | description | def. |
|--|------------|-|---------|----|
| [[field bundle|field component]] | $\phi^a$, $\phi^a_{,\mu}$ | $J^\infty_\Sigma(E) \to \mathbb{R}$ |[[coordinate function]] on [[jet bundle]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} |
| [[field history]] | $\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$ | $\Sigma \to J^\infty_\Sigma(E)$ | [[jet prolongation]] of [[section]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetProlongation} |
| [[observable|field observable]] | $\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x), $ | $\Gamma_{\Sigma}(E) \to \mathbb{R}$ | [[derivative of a distribution|derivatives]] of [[delta-distribution|delta]]-[[functional]] on [[space of sections]] | def. \ref{Observable}, example \ref{PointEvaluationObservables} |
| averaging of [[observable|field observable]] | $\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$ | $\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$ | [[operator-valued distribution|observable-valued distribution]] | def. \ref{RegularLinearFieldObservables} |
| [[algebra of quantum observables]] | $\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$ | $\mathbb{C}Alg$ | [[non-commutative algebra]] [[structure]] on [[observable|field observables]] | def. \ref{WickAlgebraOfFreeQuantumField}, def. \ref{GeneratingFunctionsForCorrelationFunctions} |
$\,$
**[[field bundles]]**
{#FieldBundles}
+-- {: .num_defn #FieldsAndFieldBundles}
###### Definition
**([[field (physics)|fields]] and [[field histories]])**
Given a [[spacetime]] $\Sigma$, then a _[[type]] of [[field|fields]]_ on $\Sigma$
is a [[smooth set|smooth]] [[fiber bundle]] (def. \ref{FiberBundle})
$$
\array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }
$$
called the _[[field bundle]]_,
Given a [[type]] of [[field|fields]] on $\Sigma$ this way, then a _[[field history]]_ of that type on $\Sigma$
is a [[term]] of that [[type]], hence is a smooth [[section]] (def. \ref{Sections}) of this [[bundle]], namely a [[smooth function]] of the form
$$
\Phi \;\colon\; \Sigma \longrightarrow E
$$
such that composed with the [[projection]] map it is the [[identity function]], i.e. such that
$$
fb \circ \Phi = id
\phantom{AAAAAAA}
\array{
&& E
\\
& {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
\Sigma & = & \Sigma
}
\,.
$$
The set of such [[sections]]/[[field histories]] is to be denoted
$$
\label{SetOfFieldHistories}
\Gamma_\Sigma(E)
\;\coloneqq\;
\left\{
\array{
&& E
\\
& {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
\Sigma &=& \Sigma
}
\phantom{fb}
\right\}
$$
=--
+-- {: .num_remark #FieldHistoriesAsHistoriesOfFieldConfigurations}
###### Remark
**([[field histories]] are histories of spatial [[field configurations]])**
Given a [[section]] $\Phi \in \Gamma_\Sigma(E)$ of the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and given a [[spacelike]] (def. \ref{SpacelikeTimelikeLightlike}) [[submanifold]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{SmoothManifoldInsideDiffeologicalSpaces}) of [[spacetime]] in [[codimension]] 1, then the [[restriction]] $\Phi\vert_{\Sigma_p}$ of $\Phi$ to $\Sigma_p$ may be thought of as a _[[field configuration]]_ in space. As different spatial slices $\Sigma_p$ are chosen, one obtains such field configurations _at different times_. It is in this sense that the entirety of a section $\Phi \in \Gamma_\Sigma(E)$ is a _history_ of field configurations, hence a [[field history]] (def \ref{FieldsAndFieldBundles}).
=--
+-- {: .num_remark #PossibleFieldHistories}
###### Remark
**([[possible worlds|possible]] field histories)**
After we give the set $\Gamma_\Sigma(E)$ of field histories (eq:SetOfFieldHistories) [[differential geometry|differential geometric]] structure, below in example \ref{DiffeologicalSpaceOfFieldHistories} and example \ref{SupergeometricSpaceOfFieldHistories},
we call it the _[[space of field histories]]_. This should be read as space of _[[possibility|possible]]_ field histories;
containing all field histories that qualify as being of the [[type]] specified by the [[field bundle]] $E$.
After we obtain [[equations of motion]] below in def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime},
these serve as the "laws of nature" that field histories should obey, and they define the subspace of those
field histories that do solve the equations of motion; this will be denoted
$$
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}= 0}
\overset{\phantom{AAA}}{\hookrightarrow}
\Gamma_\Sigma(E)
$$
and called the _[[on-shell]] [[space of field histories]]_ (eq:InclusionOfOnShellSpaceOfFieldHistories).
=--
For the time being, not to get distracted from the basic idea of
[[quantum field theory]], we will focus on the following simple special case of field bundles:
+-- {: .num_example #TrivialVectorBundleAsAFieldBundle}
###### Example
**([[trivial vector bundle]] as a [[field bundle]])**
In applications the [[field fiber]] $F = V$ is often a [[finite dimensional vector space]].
In this case the [[trivial bundle|trivial]] [[field bundle]] with [[fiber]] $F$ is
of course a _[[trivial vector bundle|trivial]] [[vector bundle]]_ (def. \ref{VectorBundle}).
Choosing any [[linear basis]] $(\phi^a)_{a = 1}^s$ of the field fiber, then over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) we have canonical [[coordinates]] on the total space of the field bundle
$$
( (x^\mu), ( \phi^a ) )
\,,
$$
where the index $\mu$ ranges from $0$ to $p$, while the index $a$ ranges from 1 to $s$.
If this trivial vector bundle is regarded as a [[field bundle]] according to def. \ref{FieldsAndFieldBundles}, then
a field history $\Phi$ is equivalently an $s$-[[tuple]] of [[real number|real]]-valued [[smooth functions]]
$\Phi^a \colon \Sigma \to \mathbb{R}$ on spacetime:
$$
\Phi = ( \Phi^a )_{a = 1}^s
\,.
$$
=--
+-- {: .num_example #RealScalarFieldBundle}
###### Example
**([[field bundle]] for [[real scalar field]])**
If $\Sigma$ is a [[spacetime]] and if the [[field fiber]]
$$
F \coloneqq \mathbb{R}
$$
is simply the [[real line]], then the corresponding trivial [[field bundle]] (def. \ref{FieldsAndFieldBundles})
$$
\array{
\Sigma \times \mathbb{R}
\\
{}^{\mathllap{pr_1}}\downarrow
\\
\Sigma
}
$$
is the _[[trivial fiber bundle|trivial]] [[real line bundle]]_ (a special case of example \ref{TrivialVectorBundleAsAFieldBundle}) and the corresponding [[field (physics)|field]] type (def. \ref{FieldsAndFieldBundles}) is called the _[[real scalar field]]_ on $\Sigma$. A configuration of this field is simply a [[smooth function]] on $\Sigma$ with values in the [[real numbers]]:
$$
\label{SpaceOfFieldHistoriesOfRealScalarField}
\Gamma_\Sigma(\Sigma \times \mathbb{R})
\;\simeq\;
C^\infty(\Sigma)
\,.
$$
=--
+-- {: .num_example #Electromagnetism}
###### Example
**([[field bundle]] for [[electromagnetic field]])**
On [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) be given by the
[[cotangent bundle]]
$$
E \coloneqq T^\ast \Sigma
\,.
$$
This is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with canonical [[field (physics)|field]]
coordinates $(a_\mu)$.
A [[section]] of this bundle, hence a [[field history]], is a [[differential 1-form]]
$$
A \in \Gamma_\Sigma(T^\ast \Sigma) = \Omega^1(\Sigma)
$$
on [[spacetime]] (def. \ref{Differential1FormsOnCartesianSpaces}). Interpreted as a [[field history]] of the [[electromagnetic field]] on $\Sigma$, this is often called the _[[vector potential]]_.
Then the [[de Rham differential]] (def. \ref{deRhamDifferential}) of the [[vector potential]] is a [[differential 2-form]]
$$
F \coloneqq d A
$$
known as the _[[Faraday tensor]]_. In the canonical coordinate basis 2-forms this may be expanded as
$$
\label{TensorFaraday}
F
=
\underoverset{i = 1}{p}{\sum}
E_i d x^0 \wedge d x^i
+
\underset{1 \leq i \lt j \leq p}{\sum}
B_{i j} d x^i \wedge d x^j
\,.
$$
Here $(E_i)_{i = 1}^p$ are called the components of the _[[electric field]]_, while $(B_{i j})$ are called the
components of the _[[magnetic field]]_.
=--
+-- {: .num_example #YangMillsFieldOverMinkowski}
###### Example
**([[field bundle]] for [[Yang-Mills field]] over [[Minkowski spacetime]])**
Let $\mathfrak{g}$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]] with [[linear basis]]
$(t_\alpha)$, in terms of which the [[Lie bracket]] is given by
$$
\label{LieAlgebraStructureConstants}
[t_\alpha, t_\beta]
\;=\;
\gamma^\gamma{}_{\alpha \beta} t_\gamma
\,.
$$
Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}),
consider then the [[field bundle]] which is the [[cotangent bundle]] [[tensor product|tensored]] with the [[Lie algebra]] $\mathfrak{g}$
$$
E \coloneqq T^\ast \Sigma \otimes \mathfrak{g}
\,.
$$
This is the [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with induced [[field (physics)|field]] [[coordinates]]
$$
( a_\mu^\alpha )
\,.
$$
A [[section]] of this bundle is a [[Lie algebra-valued differential 1-form]]
$$
A \in \Gamma_\Sigma(T^\ast \Sigma \otimes \mathfrak{g}) = \Omega^1(\Sigma, \mathfrak{g})
\,.
$$
with components
$$
A^\ast(a_\mu^\alpha) = A^\alpha_\mu
\,.
$$
This is called a [[field history]] for _[[Yang-Mills theory|Yang-Mills]] [[gauge theory]]_
(at least if $\mathfrak{g}$ is a _[[semisimple Lie algebra]]_, see example \ref{YangMillsLagrangian} below).
For $\mathfrak{g} = \mathbb{R}$ is the [[line Lie algebra]], this reduces to the case of the [[electromagnetic field]] (example \ref{Electromagnetism}).
For $\mathfrak{g} = \mathfrak{su}(3)$ this is a [[field (physics)|field]] history for the [[gauge field]] of the [[strong nuclear force]] in [[quantum chromodynamics]].
=--
For readers familiar with the concepts of _[[principal bundles]]_ and _[[connections on a bundle]]_
we include the following example \ref{YangMillsFieldInInstantonSector} which generalizes the
[[Yang-Mills field]] over [[Minkowski spacetime]] from example \ref{YangMillsFieldOverMinkowski}
to the situation over general [[spacetimes]].
+-- {: .num_example #YangMillsFieldInInstantonSector}
###### Example
**(general [[Yang-Mills field]] in fixed [[instanton|topological sector]])**
Let $\Sigma$ be any [[spacetime]] [[manifold]] and let $G$ be a [[compact Lie group]]
with [[Lie algebra]] denoted $\mathfrak{g}$. Let $P \overset{is}{\to} \Sigma$ be a
$G$-[[principal bundle]] and $\nabla_0$ a chosen [[connection on a bundle|connection]] on it,
to be called the [[background field|background]] $G$-[[Yang-Mills theory|Yang-Mills]] field.
Then the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) for $G$-[[Yang-Mills theory]] _in the [[instanton|topological sector]]_ $P$
is the [[tensor product of vector bundles]]
$$
E \coloneqq \left(P \times^{ad}_G \mathfrak{g}\right) \otimes_\Sigma \left( T^\ast \Sigma \right)
$$
of the [[adjoint bundle]] of $P$ and the [[cotangent bundle]] of $\Sigma$.
With the choice of $\nabla_0$, every (other) connection $\nabla$ on $P$ uniquely decomposes as
$$
\nabla = \nabla_0 + A
\,,
$$
where
$$
A \in \Gamma_\Sigma(E)
$$
is a [[section]] of the above [[field bundle]], hence a [[Yang-Mills field|Yang-Mills]] [[field history]].
=--
The [[electromagnetic field]] (def. \ref{Electromagnetism}) and the [[Yang-Mills field]] (def. \ref{YangMillsFieldOverMinkowski}, def. \ref{YangMillsFieldInInstantonSector}) with [[differential 1-forms]] as [[field histories]] are the basic examples of _[[gauge fields]]_ (we consider this in more detail below in _[Gauge symmetries](#GaugeSymmetries)_). There are also _[[higher gauge fields]]_
with [[differential n-forms]] as [[field histories]]:
+-- {: .num_example #BField}
###### Example
**([[field bundle]] for [[B-field]])**
On [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) be given by the
skew-symmetrized [[tensor product of vector bundles]] of the [[cotangent bundle]] with itself
$$
E \coloneqq \wedge^2_\Sigma T^\ast \Sigma
\,.
$$
This is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with canonical [[field (physics)|field]] coordinates $(b_{\mu \nu})$ subject to
$$
b_{\mu \nu} \;=\; - b_{\nu \mu}
\,.
$$
A [[section]] of this bundle, hence a [[field history]], is a [[differential 2-form]] (def. \ref{DifferentialnForms})
$$
B \in \Gamma_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) = \Omega^2(\Sigma)
$$
on [[spacetime]].
=--
$\,$
**[[space of field histories]]**
{#NonFiniteDimensionalGeometry}
Given any [[field bundle]], we will eventually need to regard the set of all [[field histories]] $\Gamma_\Sigma(E)$
as a "[[smooth set]]" itself, a smooth _[[space of sections]]_, to which
constructions of [[differential geometry]] apply (such as for the discussion of [[observables]] and [[states]] [below](#Observables) ). Notably we need to be talking about [[differential forms]] on $\Gamma_\Sigma(E)$.
However, a [[space of sections]] $\Gamma_\Sigma(E)$ does not in general carry the structure of a [[smooth manifold]];
and it carries the correct smooth structure of an [[infinite dimensional manifold]]
only if $\Sigma$ is a [[compact space]] (see at _[[manifold structure of mapping spaces]]_).
Even if it does carry [[infinite dimensional manifold]] structure, inspection shows that
this is more [[structure]] than actually needed for the discussion of [[field theory]].
Namely it turns out below that all we need to know is what counts as
a _smooth family_ of [[sections]]/[[field histories]], hence which [[functions]] of [[sets]]
$$
\Phi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(E)
$$
from any [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) into $\Gamma_\Sigma(E)$ count as [[smooth functions]],
subject to some basic consistency condition on this choice.
This [[structure]] on $\Gamma_\Sigma(E)$ is called the structure of a _[[diffeological space]]_:
+-- {: .num_defn #DiffeologicalSpace}
###### Definition
**([[diffeological space]])**
A _[[diffeological space]]_ $X$ is
1. a [[set]] $X_s \in $ [[Set]];
1. for each $n \in \mathbb{N}$ a choice of [[subset]]
$$
X(\mathbb{R}^n) \subset Hom_{Set}(\mathbb{R}^n_s, X_s) = \left\{ \mathbb{R}^n_s \to X_s \right\}
$$
of the [[function set|set of functions]] from the underlying set $\mathbb{R}^n_s$ of $\mathbb{R}^n$ to $X_s$,
to be called the _smooth functions_ or _plots_ from $\mathbb{R}^n$ to $X$;
1. for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$
between [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) a choice of function
$$
f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
$$
to be thought of as the precomposition operation
$$
\left(
\mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
\right)
\;\overset{f^\ast}{\mapsto}\;
\left(
\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X
\right)
$$
such that
1. ([[constant functions]] are smooth)
$$
X(\mathbb{R}^0) = X_s
\,,
$$
1. ([[functor|functoriality]])
1. If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the [[identity function]] on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the identity function on the set of plots $X(\mathbb{R}^n)$;
1. If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two [[composition|composable]] [[smooth functions]] between [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), then pullback of plots along them consecutively equals the pullback along the [[composition]]:
$$
f^\ast \circ g^\ast
=
(g \circ f)^\ast
$$
i.e.
$$
\array{
&& X(\mathbb{R}^{n_2})
\\
& {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
\\
X(\mathbb{R}^{n_1})
&& \underset{ (g \circ f)^\ast }{\longleftarrow} &&
X(\mathbb{R}^{n_3})
}
$$
1. ([[sheaf|gluing]])
If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a [[differentiably good open cover]] of a [[Cartesian space]]
(def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts
$\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots
$$
X(\mathbb{R}^n)
\overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow}
\underset{i \in I}{\prod} X(U_i)
$$
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots,
which are "[[matching families]]" in that they agree on [[intersections]]:
$$
\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j}
\phantom{AAAAAA}
\array{
&& U_i \cap U_j
\\
& \swarrow && \searrow
\\
U_i && && U_j
\\
& {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
\\
&& X
}
$$
Finally, given $X_1$ and $X_2$ two diffeological spaces, then a [[smooth function]] between them
$$
f \;\colon\; X_1 \longrightarrow X_2
$$
is
* a [[function]] of the underlying sets
$$
f_s \;\colon\; (X_1)_s \longrightarrow (X_2)_s
$$
such that
* for $\Phi \in X(\mathbb{R}^n)$ a plot of $X_1$, then the [[composition]]
$f_s \circ \Phi_s$ is a plot $f_\ast(\Phi)$ of $X_2$:
$$
\array{
&& \mathbb{R}^n
\\
& {}^{\mathllap{\Phi}}\swarrow && \searrow^{\mathrlap{f_\ast(\Phi)}}
\\
X_1 && \underset{f}{\longrightarrow} && X_2
}
\,.
$$
(Stated more [[category theory|abstractly]], this says simply that [[diffeological spaces]]
are the [[concrete sheaves]] on the [[site]] of [[Cartesian spaces]] from def. \ref{DifferentiablyGoodOpenCover}.)
=--
For more background on [[diffeological spaces]] see also _[[geometry of physics -- smooth sets]]_.
+-- {: .num_example #SmoothManifoldsAreDiffeologicalSpaces}
###### Example
**([[Cartesian spaces]] are [[diffeological spaces]])**
Let $X$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) Then it becomes a [[diffeological space]] (def. \ref{DiffeologicalSpace})
by declaring its plots $\Phi \in X(\mathbb{R}^n)$ to the ordinary [[smooth functions]]
$\Phi \colon \mathbb{R}^n \to X$.
Under this identification, a function $f \;\colon\; (X_1)_s \to (X_2)_s$ between the underlying sets
of two [[Cartesian spaces]] is a [[smooth function]] in the ordinary sense
precisely if it is a smooth function in the sense of [[diffeological spaces]].
Stated more [[category theory|abstractly]], this statement is an example of the _[[Yoneda embedding]]_
over a _[[subcanonical site]]_.
More generally, the same construction makes every [[smooth manifold]] a [[smooth set]].
=--
+-- {: .num_example #DiffeologicalSpaceOfFieldHistories}
###### Example
**([[diffeological space|diffeological]] [[space of field histories]])**
Let $E \overset{fb}{\to} \Sigma$ be a smooth [[field bundle]] (def. \ref{FieldsAndFieldBundles}).
Then the set $\Gamma_\Sigma(E)$ of [[field histories]]/[[sections]] (def. \ref{FieldsAndFieldBundles})
becomes a [[diffeological space]] (def. \ref{DiffeologicalSpace})
$$
\label{SpaceOfFieldHistories}
\Gamma_\Sigma(E) \in DiffeologicalSpaces
$$
by declaring that a smooth family $\Phi_{(-)}$ of field histories,
parameterized over any [[Cartesian space]] $U$ is a smooth function out of the [[Cartesian product]]
manifold of $\Sigma$ with $U$
$$
\array{
U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
\\
(u,x) &\mapsto& \Phi_u(x)
}
$$
such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.
$$
\array{
&& E
\\
& {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
}
\,.
$$
=--
The following example \ref{FrechetManifoldsAreDiffeologicalSpaces} is included only for readers who wonder how [[infinite-dimensional manifolds]]
fit in. Since we will never actually use [[infinite-dimensional manifold]]-structure, this example is may be ignored.
+-- {: .num_example #FrechetManifoldsAreDiffeologicalSpaces}
###### Example
**([[Fréchet manifolds]] are [[diffeological spaces]])**
Consider the particular type of [[infinite-dimensional manifolds]] called _[[Fréchet manifolds]]_.
Since ordinary [[smooth manifolds]] $U$ are an example, for $X$ a [[Fréchet manifold]] there is
a concept of [[smooth functions]] $U \to X$. Hence we may give $X$ the structure of a
[[diffeological space]] (def. \ref{DiffeologicalSpace}) by declaring the plots over
$U$ to be these smooth functions $U \to X$, with the evident postcomposition action.
It turns out that then that for $X$ and $Y$ two [[Fréchet manifolds]],
there is a [[natural bijection]] between the [[smooth functions]]
$X \to Y$ between them regarded as [[Fréchet manifolds]] and [regarded as [[diffeological spaces]].
Hence it does not matter which of the two perspectives we take (unless of course a [[diffeological space]]
more general than a [[Fréchet manifolds]] enters the picture, at which point the second definition generalizes,
whereas the first does not).
Stated more [[category theory|abstractly]], this means that [[Fréchet manifolds]] form a [[full subcategory]]
of that of [[diffeological spaces]] ([this prop.](Fréchet+manifold#FFEmbeddingOfFrechetInDiffeological)):
$$
FrechetManifolds \hookrightarrow DiffeologicalSpaces
\,.
$$
If $\Sigma$ is a [[compact space|compact]] [[smooth manifold]] and $E \simeq \Sigma \times F \to \Sigma$
is a [[trivial fiber bundle]] with [[fiber]] $F$ a [[smooth manifold]], then the set of [[sections]] $\Gamma_\Sigma(E)$ carries
a standard structure of a [[Fréchet manifold]] (see at _[[manifold structure of mapping spaces]]_).
Under the above inclusion of [[Fréchet manifolds]] into [[diffeological spaces]], this [[smooth structure]] agrees with
that from example \ref{DiffeologicalSpaceOfFieldHistories} (see [this prop.](Fréchet+manifold#CompatibilityWithDiffeologicalMappingSpaces))
=--
Once the step from [[smooth manifolds]] to [[diffeological spaces]] (def. \ref{DiffeologicalSpace}) is made,
characterizing the [[smooth structure]] of the space entirely by how we may probe it by mapping
smooth Cartesian spaces into it, it becomes clear that the underlying set $X_s$ of a diffeological space $X$
is not actually crucial to support the concept: The space is already entirely defined [[structuralism|structurally]]
by the system of smooth plots it has, and the underlying set $X_s$ is recovered from these as the set of plots
from the point $\mathbb{R}^0$.
This is crucial for [[field theory]]: the [[spaces of field histories]] of [[fermionic fields]] (def. \ref{FermionicBosonicFields} below)
such as the _[[Dirac field]]_ (example \ref{DiracSpaceOfFieldHistories} below) do not have underlying
sets of points the way [[diffeological spaces]] have. Informally, the reason is that a point
is a [[bosonic]] object, while and the nature of [[fermionic fields]] is [[antimodal type|the opposite of]]
bosonic.
But we may just as well drop the mentioning of the underlying set $X_s$ in the definition of [[generalized smooth spaces]].
By simply stripping this requirement off of def. \ref{DiffeologicalSpace} we obtain the following
more general and more useful definition (still "bosonic", though, the [[supergeometry|supergeometric]] version
is def. \ref{SuperFormalSmoothSet} below):
+-- {: .num_defn #SmoothSet}
###### Definition
**([[smooth set]])**
A _[[smooth set]]_ $X$ is
1. for each $n \in \mathbb{N}$ a choice of [[set]]
$$
X(\mathbb{R}^n) \in Set
$$
to be called the set of _smooth functions_ or _plots_ from $\mathbb{R}^n$ to $X$;
1. for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$
between [[Cartesian spaces]] a choice of function
$$
f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
$$
to be thought of as the precomposition operation
$$
\left(
\mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
\right)
\;\overset{f^\ast}{\mapsto}\;
\left(
\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X
\right)
$$
such that
1. ([[functor|functoriality]])
1. If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the [[identity function]] on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n)$.
1. If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two [[composition|composable]] [[smooth functions]] between [[Cartesian spaces]], then consecutive pullback of plots along them equals the pullback along the [[composition]]:
$$
f^\ast \circ g^\ast
=
(g \circ f)^\ast
$$
i.e.
$$
\array{
&& X(\mathbb{R}^{n_2})
\\
& {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
\\
X(\mathbb{R}^{n_1})
&& \underset{ (g \circ f)^\ast }{\longleftarrow} &&
X(\mathbb{R}^{n_3})
}
$$
1. ([[sheaf|gluing]])
If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a [[differentiably good open cover]] of a [[Cartesian space]] (def. \ref{DifferentiablyGoodOpenCover})
then the function which restricts
$\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots
$$
X(\mathbb{R}^n)
\overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow}
\underset{i \in I}{\prod} X(U_i)
$$
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots,
which are "[[matching families]]" in that they agree on [[intersections]]:
$$
\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j}
\phantom{AAAA}
\text{i.e.}
\phantom{AAAA}
\array{
&& U_i \cap U_j
\\
& \swarrow && \searrow
\\
U_i && && U_j
\\
& {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
\\
&& X
}
$$
Finally, given $X_1$ and $X_2$ two [[smooth sets]], then a [[smooth function]] between them
$$
f \;\colon\; X_1 \longrightarrow X_2
$$
is
* for each $n \in \mathbb{N}$ a [[function]]
$$
f_\ast(\mathbb{R}^n)
\;\colon\;
X_1(\mathbb{R}^n) \longrightarrow X_2(\mathbb{R}^n)
$$
such that
* for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between [[Cartesian spaces]] we have
$$
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2})
=
f_\ast(\mathbb{R}^{n_1}) \circ g^\ast_1
\phantom{AAAAA}
\text{i.e.}
\phantom{AAAAA}
\text{i.e.}
\phantom{AAAAA}
\array{
X_1(\mathbb{R}^{n_2})
&\overset{f_\ast(\mathbb{R}^{n_2})}{\longrightarrow}&
X_2(\mathbb{R}^{n_2})
\\
\mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
\\
X_1(\mathbb{R}^{n_1})
&\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
X_2(\mathbb{R}^{n_1})
}
$$
Stated more [[category theory|abstractly]], this simply says that [[smooth sets]] are the _[[sheaves]]
on the [[site]] of [[Cartesian spaces]] from def. \ref{DifferentiablyGoodOpenCover}.
=--
Basing [[differential geometry]] on [[smooth sets]] is an instance of the general approach to [[geometry]] called _[[functorial geometry]]_
or _[[topos theory]]_. For more background on this see at _[[geometry of physics -- smooth sets]]_.
First we verify that the concept of smooth sets is a consistent generalization:
+-- {: .num_example #SmoothSetsDiffeologicalSpaces}
###### Example
**([[diffeological spaces]] are [[smooth sets]])**
Every [[diffeological space]] $X$ (def. \ref{DiffeologicalSpace}) is a [[smooth set]] (def. \ref{SmoothSet})
simply by [[forgetful functor|forgetting]] its underlying set of points and remembering only its sets of plot.
In particular therefore each [[Cartesian space]] $\mathbb{R}^n$ is canonically a [[smooth set]]
by example \ref{SmoothManifoldsAreDiffeologicalSpaces}.
Moreover, given any two [[diffeological spaces]], then the [[morphisms]] $f \colon X \to Y$
between them, regarded as diffeological spaces, are [[natural bijection|the same]] as the morphisms
as [[smooth sets]].
Stated more [[category theory|abstractly]], this means that we have [[full subcategory]] inclusions
$$
CartesianSpaces
\overset{\phantom{AAA}}{\hookrightarrow}
DiffeologicalSpaces
\overset{\phantom{AAA}}{\hookrightarrow}
SmoothSets
\,.
$$
=--
Recall, for the next proposition \ref{CartSpYpnedaLemma}, that
in the definition \ref{SmoothSet} of a [[smooth set]] $X$ the sets $X(\mathbb{R}^n)$
are abstract sets which are _to be thought of_ as would-be smooth functions "$\mathbb{R}^n \to X$".
Inside def. \ref{SmoothSet} this only makes sense in quotation marks, since inside that definition
the smooth set $X$ is only being defined, so that inside that definition there is not yet an actual
concept of smooth functions of the form "$\mathbb{R}^n \to X$".
But now that the definition of [[smooth sets]] and of [[morphisms]] between them has been stated,
and seeing that [[Cartesian space]] $\mathbb{R}^n$ are examples of [[smooth sets]], by
example \ref{SmoothSetsDiffeologicalSpaces}, there is now an actual concept of
smooth functions $\mathbb{R}^n \to X$, namely as smooth sets.
For the concept of smooth sets to be consistent, it ought to be true that this
_a posteriori_ concept of smooth functions from [[Cartesian spaces]] to [[smooth sets]]
coincides wth the _a priori_ concept, hence that we "may remove the quotation marks" in the above.
The following proposition says that this is indeed the case:
+-- {: .num_prop #CartSpYpnedaLemma}
###### Proposition
**(plots of a [[smooth set]] really are the [[smooth functions]] into the smooth set)**
Let $X$ be a [[smooth set]] (def. \ref{SmoothSet}). For $n \in \mathbb{R}$, there is a [[natural transformation|natural]] [[function]]
$$
Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} X(\mathbb{R}^n)
$$
from the set of homomorphisms of smooth sets from $\mathbb{R}^n$ (regarded as a smooth set via example \ref{SmoothSetsDiffeologicalSpaces})
to $X$, to the set of plots of $X$ over $\mathbb{R}^n$, given by evaluating on the [[identity function|identity]] plot $id_{\mathbb{R}^n}$.
This function is a _[[bijection]]_.
This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the _would-be_ smooth functions
into $X$, end up being the _actual_ smooth functions into $X$.
=--
+-- {: .proof}
###### Proof
This elementary but profound fact is called the _[[Yoneda lemma]]_, here in its incarnation over the [[site]] of [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).
=--
A key class of examples of [[smooth sets]] (def. \ref{SmoothSet}) that are not [[diffeological spaces]] (def. \ref{DiffeologicalSpace})
are universal smooth [[moduli spaces]] of [[differential forms]]:
+-- {: .num_example #UniversalSmoothModuliSpaceOfDifferentialForms}
###### Example
**(universal [[smooth set|smooth]] [[moduli spaces]] of [[differential forms]])**
For $k \in \mathbb{N}$ there is a [[smooth set]] (def. \ref{SmoothSet})
$$
\mathbf{\Omega}^k \;\in\; SmoothSet
$$
defined as follows:
1. for $n \in \mathbb{N}$ the set of plots from $\mathbb{R}^n$ to $\mathbf{\Omega}^k$ is the
set of smooth [[differential forms|differential k-forms]] on $\mathbb{R}^n$ (def. \ref{DifferentialnForms})
$$
\mathbf{\Omega}^k(\mathbb{R}^n) \;\coloneqq\; \Omega^k(\mathbb{R}^n)
$$
1. for $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
the operation of pullback of plots along $f$ is just the [[pullback of differential forms]] $f^\ast$ from prop. \ref{PullbackOfDifferentialForms}
$$
\array{
\mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1})
\\
\downarrow^{\mathrlap{f}} && \uparrow^{\mathrlap{f^\ast}}
\\
\mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2})
}
$$
That this is [[functor|functorial]] is just the standard fact (eq:PullbackOfDiffereentialFormsCompatibleWithComposition) from prop. \ref{PullbackOfDifferentialForms}.
For $k = 1$ the smooth set $\mathbf{\Omega}^0$ actually is a [[diffeological space]], in fact
under the identification of example \ref{SmoothSetsDiffeologicalSpaces} this is just the [[real line]]:
$$
\mathbf{\Omega}^0 \simeq \mathbb{R}^1
\,.
$$
But for $k \geq 1$ we have that the set of plots on $\mathbb{R}^0 = \ast$ is a [[singleton]]
$$
\mathbf{\Omega}^{k \geq 1}(\mathbb{R}^0) \simeq \{0\}
$$
consisting just of the zero differential form. The only diffeological space with this property is
$\mathbb{R}^0 = \ast$ itself. But $\mathbf{\Omega}^{k \geq 1}$ is far from being that trivial:
even though its would-be underlying set is a single point, for all $n \geq k$ it admits
an infinite set of plots. Therefore the smooth sets $\mathbf{\Omega}^k$ for $k \geq$ are not
diffeological spaces.
That the [[smooth set]] $\mathbf{\Omega}^k$ indeed deserves to be addressed as the _universal [[moduli space]] of [[differential n-forms|differential k-forms]]_ follows from prop. \ref{CartSpYpnedaLemma}: The universal moduli space of $k$-forms
ought to carry a universal differential $k$-forms $\omega_{univ} \in \Omega^k(\mathbf{\Omega}^k)$
such that every differential $k$-form $\omega$ on any $\mathbb{R}^n$ arises as the [[pullback of differential forms]]
of this universal one along some _[[modulating morphism]]_ $f_\omega \colon X \to \mathbf{\Omega}^k$:
$$
\array{
\{\omega\} &\overset{(f_\omega)^\ast}{\longleftarrow}& \{\omega_{univ}\}
\\
\\
X &\underset{f_\omega}{\longrightarrow}& \mathbf{\Omega}^k
}
$$
But with prop. \ref{CartSpYpnedaLemma} this is precisely what the definition of the plots of $\mathbf{\Omega}^k$ says.
Similarly, all the usual operations on differential form now have their universal archetype on the
universal [[moduli spaces]] of differential forms
In particular, for $k \in \mathbb{N}$ there is a canonical [[morphism]] of [[smooth sets]] of the form
$$
\mathbf{\Omega}^k \overset{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1}
$$
defined over $\mathbb{R}^n$ by the ordinary [[de Rham differential]] (def. \ref{deRhamDifferential})
$$
\label{deRhamDifferentialUniversal}
\Omega^k(\mathbb{R}^n) \overset{d}{\longrightarrow} \Omega^{k+1}(\mathbb{R}^n)
\,.
$$
That this satisfies the compatibility with precomposition of plots
$$
\array{
\mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1}) &\overset{d}{\longrightarrow}& \Omega^{k+1}(\mathbb{R}^{n_1})
\\
{}^{\mathllap{f}}\downarrow && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}}
\\
\mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2}) &\underset{d}{\longrightarrow}& \Omega^k( \mathbb{R}^{n_2} )
}
$$
is just the compatibility of [[pullback of differential forms]] with the [[de Rham differential]] of from prop. \ref{PullbackOfDifferentialForms}.
=--
The upshot is that we now have a good definition of [[differential forms]] on any [[diffeological space]]
and more generally on any [[smooth set]]:
+-- {: .num_defn #DifferentialFormsOnDiffeologicalSpaces}
###### Definition
**([[differential forms]] on [[smooth sets]])**
Let $X$ be a [[diffeological space]] (def. \ref{DiffeologicalSpace})
or more generally a [[smooth set]] (def. \ref{SmoothSet}) then a [[differential form|differential k-form]]
$\omega$ on $X$ is equivalently a [[morphism]] of [[smooth sets]]
$$
X \longrightarrow \mathbf{\Omega}^k
$$
from $X$ to the universal [[smooth set|smooth]] [[moduli space]] of differential froms from
example \ref{UniversalSmoothModuliSpaceOfDifferentialForms}.
Concretely, by unwinding the definitions of $\mathbf{\Omega}^k$ and of [[morphisms]] of smooth sets, this
means that such a differential form is:
* for each $n \in \mathbb{N}$ and each plot $\mathbb{R}^n \overset{\Phi}{\to} X$
an ordinary [[differential form]]
$$
\Phi^\ast(\omega) \in \Omega^\bullet(\mathbb{R}^n)
$$
such that
* for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between [[Cartesian spaces]]
the ordinary [[pullback of differential forms]] along $f$ is compatible with these choices, in that
for every plot $\mathbb{R}^{n_2} \overset{\Phi}{\to} X$ we have
$$
f^\ast\left(\Phi^\ast(\omega)\right)
=
( f^\ast \Phi )^\ast(\omega)
$$
i.e.
$$
\array{
\mathbb{R}^{n_1} && \overset{f}{\longrightarrow} && \mathbb{R}^{n_2}
\\
& {}_{\mathllap{f^\ast \Phi}}\searrow && \swarrow_{\mathrlap{\Phi}}
\\
&& X
}
\phantom{AAAA}
\array{
\Omega^\bullet( \mathbb{R}^{n_1} ) && \overset{f^\ast}{\longleftarrow} && \Omega^\bullet(\mathbb{R}^{n_2})
\\
& {}_{\mathllap{(f^\ast \Phi)^\ast}}\nwarrow && \nearrow_{\mathrlap{\Phi^\ast}}
\\
&& \Omega^\bullet(X)
}
\,.
$$
We write $\Omega^\bullet(X)$ for the set of differential forms on the smooth set $X$
defined this way.
Moreover, given a [[differential form|differential k-form]]
$$
X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^k
$$
on a [[smooth set]] $X$ this way, then its [[de Rham differential]] $d \omega \in \Omega^{k+1}(X)$ is
given by the [[composition|composite]]
of [[morphisms]] of [[smooth sets]] with the universal de Rham differential from (eq:deRhamDifferentialUniversal):
$$
\label{FormsOnSmoothSetDeRhamDifferential}
d \omega
\;\colon\;
X
\overset{\omega}{\longrightarrow}
\mathbf{\Omega}^k
\overset{d}{\longrightarrow}
\mathbf{\Omega}^{k+1}
\,.
$$
Explicitly this means simply that for $\Phi \colon U \to X$ a plot, then
$$
\Phi^\ast (d\omega)
\;=\;
d\left( \Phi^\ast \omega\right)
\;\in\;
\Omega^{k+1}(U)
\,.
$$
=--
The usual operations on ordinary [[differential forms]] directly generalize plot-wise
to differential forms on [[diffeological spaces]] and more generally on [[smooth sets]]:
+-- {: .num_defn #ExteriorCalculusOnDiffeologicalSpaces}
###### Definition
**([[exterior differential]] and [[exterior product]] on [[smooth sets]])**
Let $X$ be a [[diffeological space]] (def. \ref{DiffeologicalSpace}) or more generally a [[smooth set]] (def. \ref{SmoothSet}). Then
1. For $\omega \in \Omega^n(X)$ a [[differential form]] on $X$ (def. \ref{DifferentialFormsOnDiffeologicalSpaces})
its [[exterior differential]]
$$
d \omega \in \Omega^{n+1}(X)
$$
is defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as the ordinary [[exterior differential]]
of the pullback of $\omega$ along that plot:
$$
\Phi^\ast(d \omega) \coloneqq d \Phi^\ast(\omega)
\,.
$$
1. For $\omega_1 \in \Omega^{n_1}$ and $\omega_2 \in \Omega^{n_2}(X)$ two differential forms on $X$ (def. \ref{DifferentialFormsOnDiffeologicalSpaces}) then their [[exterior product]]
$$
\omega_1 \wedge \omega_2 \;\in\; \Omega^{n_1 + n_2}(X)
$$
is the differential form defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as
the ordinary exterior product of the pullback of th differential forms $\omega_1$ and $\omega_2$
to this plot:
$$
\Phi^\ast(\omega_1 \wedge \omega_2)
\;\coloneqq\;
\Phi^\ast(\omega_1) \wedge \Phi^\ast(\omega_2)
\,.
$$
=--
$\,$
**Infinitesimal geometry**
{#InfinitesimalGeometry}
It is crucial in [[field theory]] that we consider [[field histories]] not only over
all of [[spacetime]], but also restricted to [[submanifolds]] of spacetime. Or rather,
what is actually of interest are the restrictions of the field histories to the _[[infinitesimal neighbourhoods]]_
(example \ref{InfinitesimalNeighbourhood} below)
of these submanifolds. This appears notably in the construction of _[[phase spaces]]_ [below](#PhaseSpace).
Moreover, [[fermion fields]] such as the [[Dirac field]] (example \ref{DiracFieldBundle} below)
take values in _[[graded object|graded]]_ [[infinitesimal]] spaces, called _[[super spaces]]_
(discussed [below](#Supergeometry)). Therefore "infinitesimal geometry", sometimes called
_[[formal geometry]]_ (as in "[[formal scheme]]") or _[[synthetic differential geometry]]_
or _[[synthetic differential supergeometry]]_, is a central aspect of [[field theory]].
In order to mathematically grasp what _[[infinitesimal neighbourhoods]]_ are, we appeal to the first magic algebraic property of differential
geometry from prop. \ref{AlgebraicFactsOfDifferentialGeometry}, which says that we may recognize
[[smooth manifolds]] $X$ [[formal dual|dually]] in terms of their [[commutative algebras]] $C^\infty(X)$
of [[smooth functions]] on them
$$
C^\infty(-) \;\colon\; SmoothManifolds \overset{\phantom{AAA}}{\hookrightarrow} (\mathbb{R} Algebras)^{op}
\,.
$$
But since there are of course more [[associative algebras|algebras]] $A \in \mathbb{R}Algebras$ than arise
this way from smooth manifolds, we may turn this around and try to regard any algebra $A$
as _defining_ a would-be [[space]], which would have $A$ as its [[algebra of functions]].
For example an _[[infinitesimally thickened point]]_ should be a space which is "so small"
that every smooth function $f$ on it which vanishes at the origin takes values so
tiny that some finite power of them is not just even more tiny, but actually vanishes:
+-- {: .num_defn #InfinitesimallyThickendSmoothManifold}
###### Definition
**([[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]])**
An _[[infinitesimally thickened point]]_
$$
\mathbb{D} \coloneqq Spec(A)
$$
is represented by a [[commutative algebra]]
$A \in \mathbb{R}Alg$ which as a [[real vector space]] is a [[direct sum]]
$$
A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
$$
of the 1-dimensional space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1 with
a [[finite dimensional vector space]] $V$ that is a [[nilpotent ideal]] in that for
each element $a \in V$ there exists a [[natural number]] $n \in \mathbb{N}$ such that
$$
a^{n+1} = 0
\,.
$$
More generally, an [[infinitesimally thickened manifold|infinitesimally thickened Cartesian space]]
$$
\mathbb{R}^n \times \mathbb{D} \;\coloneqq\; \mathbb{R}^n \times Spec(A)
$$
is represented by a [[commutative algebra]]
$$
C^\infty(\mathbb{R}^n) \otimes A \;\in\; \mathbb{R} Alg
$$
which is the [[tensor product of algebras]] of the algebra of smooth functions
$C^\infty(\mathbb{R}^n)$ on an actual [[Cartesian space]] of some [[dimension]] $n$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of an infinitesimally thickened point, as above.
We say that a _smooth function between two [[infinitesimally thickened manifolds|infinitesimally thickened Cartesian spaces]]_
$$
\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
$$
is by definition [[formal dual|dually]] an $\mathbb{R}$-algebra [[homomorphism]] of the form
$$
C^\infty(\mathbb{R}^{n_1}) \otimes A_1
\overset{f^\ast}{\longleftarrow}
C^\infty(\mathbb{R}^{n_2}) \otimes A_2
\,.
$$
=--
+-- {: .num_example #InfinitesimalNeighbourhoodsInTheRealLine}
###### Example
**([[infinitesimal neighbourhoods]] in the [[real line]] )**
Consider the [[quotient ring|quotient algebra]] of the [[formal power series algebra]] $\mathbb{R}[ [\epsilon] ]$
in a single parameter $\epsilon$ by the ideal
generated by $\epsilon^2$:
$$
(\mathbb{R}[ [\epsilon] ])/(\epsilon^2)
\;\simeq_{\mathbb{R}}\;
\mathbb{R} \oplus \epsilon \mathbb{R}
\,.
$$
(This is sometimes called the _[[algebra of dual numbers]]_, for no good reason.) The underlying [[real vector space]]
of this algebra is, as show, the [[direct sum]] of the multiples of 1 with the multiples of $\epsilon$. A general element
in this algebra is of the form
$$
a + b \epsilon \in (\mathbb{R}[\epsilon])/(\epsilon^2)
$$
where $a,b \in \mathbb{R}$ are [[real numbers]]. The product in this algebra is given by
"multiplying out" as usual, and discarding all terms proportional to $\epsilon^2$:
$$
\left(
a_1 + b_1 \epsilon
\right)
\cdot
\left(
a_2 + b_2 \epsilon
\right)
\;=\;
a_1 a_2 + ( a_1 b_2 + b_1 a_2 ) \epsilon
\,.
$$
We may think of an element $a + b \epsilon$ as the truncation to first order
of a [[Taylor series]] at the origin of a [[smooth function]] on the [[real line]]
$$
f \;\colon\; \mathbb{R} \to \mathbb{R}
$$
where $a = f(0)$ is the value of the function at the origin, and where $b = \frac{\partial f}{\partial x}(0)$ is its first [[derivative]] at the origin.
Therefore this algebra behaves like the algebra of smooth function on an [[infinitesimal neighbourhood]] $\mathbb{D}^1$
of $0 \in \mathbb{R}$ which is so tiny that its [[generalized element|elements]] $\epsilon \in \mathbb{D}^1 \hookrightarrow \mathbb{R}$
become, upon squaring them, not just tinier, but actually zero:
$$
\epsilon^2 = 0
\,.
$$
This intuitive picture is now made precise by the concept of [[infinitesimally thickened points]] def. \ref{InfinitesimallyThickendSmoothManifold}, if we simply set
$$
\mathbb{D}^1
\;\coloneqq\;
Spec\left(
\mathbb{R}[ [\epsilon] ]/(\epsilon^2)
\right)
$$
and observe that there is the [[monomorphism|inclusion]] of infinitesimally thickened Cartesian spaces
$$
\mathbb{D}^1 \overset{\phantom{AA}i\phantom{AA} }{\hookrightarrow} \mathbb{R}^1
$$
which is dually given by the algebra homomorphism
$$
\array{
\mathbb{R} \oplus \epsilon \mathbb{R}
&\overset{i^\ast}{\longleftarrow}&
C^\infty(\mathbb{R}^1)
\\
f(0) + \frac{\partial f}{\partial x}(0) &\longleftarrow& \{f\}
}
$$
which sends a [[smooth function]] to its value $f(0)$ at zero plus $\epsilon$ times its
[[derivative]] at zero. Observe that this is indeed a [[homomorphism]] of algebras due to the [[product law]]
of [[differentiation]], which says that
$$
\begin{aligned}
i^\ast(f \cdot g)
& =
(f \cdot g)(0) + \frac{\partial f \cdot g}{\partial x}(0) \epsilon
\\
& =
f(0) \cdot g(0)
+
\left(
\frac{\partial f}{\partial x}(0) \cdot g(0) + f(0) \cdot \frac{\partial g}{\partial x}(0)
\right) \epsilon
\\
& =
\left(
f(0) + \frac{\partial f}{\partial x}(0) \epsilon
\right)
\cdot
\left(
g(0) + \frac{\partial g}{\partial x}(0) \epsilon
\right)
\end{aligned}
$$
Hence we see that restricting a smooth function to the infinitesimal neighbourhood of a point is
equivalent to restricting attention to its [[Taylor series]] to the given order at that point:
$$
\array{
\mathbb{D}^1 &\overset{i}{\hookrightarrow}& \mathbb{R}^1
\\
& {}_{\mathllap{(\epsilon \mapsto f(0) + \frac{\partial f}{\partial x}(0) \epsilon) }}\searrow & \downarrow_{\mathrlap{f}}
\\
&& \mathbb{R}^1
}
$$
Similarly for each $k \in \mathbb{N}$ the algebra
$$
(\mathbb{R}[ [ \epsilon ] ])/(\epsilon^{k+1})
$$
may be thought of as the algebra of [[Taylor series]] at the origin of $\mathbb{R}$ of [[smooth functions]] $\mathbb{R} \to \mathbb{R}$,
where all terms of order higher than $k$ are discarded. The corresponding [[infinitesimally thickened point]] is
often denoted
$$
\mathbb{D}^1(k) \;\coloneqq\; Spec\left( \left(\mathbb{R}[ [\epsilon] ]\right)/(\epsilon^{k+1}) \right)
\,.
$$
This is now the [[subobject]] of the [[real line]]
$$
\mathbb{D}^1(k) \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{R}^1
$$
on those elements $\epsilon$ such that $\epsilon^{k+1} = 0$.
=--
([Kock 81](synthetic+differential+geometry#Kock81), [Kock 10](synthetic+differential+geometry#Kock10))
The following example \ref{UniquePointOfInfinitesimalLine} shows that infinitesimal thickening is invisible for ordinary spaces when
mapping _out_ of these. In contrast example \ref{SyntheticTangentVectorFields} further below shows that
the morphisms _into_ an ordinary space out of an infinitesimal space are interesting: these are [[tangent vectors]]
and their higher order infinitesimal analogs.
+-- {: .num_example #UniquePointOfInfinitesimalLine}
###### Example
**([[infinitesimal]] line $\mathbb{D}^1$ has unique [[global point]])**
For $\mathbb{R}^n$ any ordinary [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
and $D^1(k) \hookrightarrow \mathbb{R}^1$ the order-$k$ [[infinitesimal neighbourhood]] of the
origin in the [[real line]] from example \ref{InfinitesimalNeighbourhoodsInTheRealLine},
there is exactly only one possible morphism of [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] from $\mathbb{R}^n$ to $\mathbb{D}^1(k)$:
$$
\array{
\mathbb{R}^n && \overset{\exists !}{\longrightarrow} &6 \mathbb{D}^1(k)
\\
& {}_{\mathllap{\exists !}}\searrow && \nearrow_{\mathrlap{\exists !}}
\\
&& \mathbb{R}^0 = \ast
}
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition such a morphism is [[formal duality|dually]] an algebra homomorphism
$$
C^\infty(\mathbb{R}^n)
\overset{f^\ast}{\longleftarrow}
\left(
\mathbb{R}[ [\epsilon] ])/(\epsilon^{k+1}
\right)
\simeq_{\mathbb{R}}
\mathbb{R} \oplus \mathcal{O}(\epsilon)
$$
from the higher order "[[algebra of dual numbers]]" to the [[algebra of functions|algebra of]] [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}).
Now this being an $\mathbb{R}$-algebra homomorphism, its action on the multiples $c \in \mathbb{R}$ of the identity
is fixed:
$$
f^\ast(1) = 1
\,.
$$
All the remaining elements are proportional to $\epsilon$, and hence are nilpotent.
However, by the [[homomorphism]] property of an algebra homomorphism it follows that it must send
nilpotent elements $\epsilon$ to nilpotent elements $f(\epsilon)$, because
$$
\begin{aligned}
\left(f^\ast(\epsilon)\right)^{k+1}
& = f^\ast\left( \epsilon^{k+1}\right)
\\
& = f^\ast(0)
\\ & = 0
\end{aligned}
$$
But the only nilpotent element in $C^\infty(\mathbb{R}^n)$ is the zero element, and hence it follows that
$$
f^\ast(\epsilon) = 0
\,.
$$
Thus $f^\ast$ as above is uniquely fixed.
=--
+-- {: .num_example #SyntheticTangentVectorFields}
###### Example
**([[synthetic differential geometry|synthetic]] [[tangent vector fields]])**
Let $\mathbb{R}^n$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}),
regarded as an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) and consider $\mathbb{D}^1 \coloneqq Spec( (\mathbb{R}[ [\epsilon] ])/(\epsilon^2) )$
the first order infinitesimal line from example \ref{InfinitesimalNeighbourhoodsInTheRealLine}.
Then homomorphisms of [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]]
of the form
$$
\array{
\mathbb{R}^n \times \mathbb{D}^1
&& \overset{\tilde v}{\longrightarrow} &&
\mathbb{R}^n
\\
& {}_{\mathllap{pr_1}}\searrow && \swarrow_{\mathrlap{id}}
\\
&& \mathbb{R}^n
}
$$
hence smoothly $X$-parameterized collections of morphisms
$$
\tilde v_x \;\colon\; \mathbb{D}^1 \longrightarrow \mathbb{R}^n
$$
which send the unique base point $\Re(\mathbb{D}^1) = \ast$ (example \ref{UniquePointOfInfinitesimalLine}) to $x \in \mathbb{R}^n$,
are in [[natural bijection]] with [[tangent vector fields]] $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ (example \ref{TangentVectorFields}).
=--
+-- {: .proof}
###### Proof
By definition, the morphisms in question are [[formal duality|dually]] $\mathbb{R}$-[[associative algebra|algebra]]
[[homomorphisms]] of the form
$$
(C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
\longleftarrow
C^\infty(\mathbb{R}^n)
$$
which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to
$$
f + (\partial f) \epsilon
$$
for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$ we have
$$
(f_1 f_2 + (\partial (f_1 f_2))\epsilon )
\;=\;
(f_1 + (\partial f_1) \epsilon)
\cdot
(f_2 + (\partial f_2) \epsilon)
\,.
$$
Multiplying this out and using that $\epsilon^2 = 0$, this is equivalent to
$$
\partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
\,.
$$
This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a [[derivation]].
With this the statement follows with the third magic algebraic property of smooth functions (prop. \ref{AlgebraicFactsOfDifferentialGeometry}):
[[derivations of smooth functions are vector fields]].
=--
We need to consider infinitesimally thickened spaces more general than the
thickenings of just Cartesian spaces in def. \ref{InfinitesimallyThickendSmoothManifold}. But just as [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) serve as the local
test geometries to induce the general concept of [[diffeological spaces]] and [[smooth sets]] (def. \ref{SmoothSet}),
so using infinitesimally thickened Cartesian spaces as test geometries immediately induces the corresponding
generalization of smooth sets with infinitesimals:
+-- {: .num_defn #FormalSmoothSet}
###### Definition
**([[formal smooth set]])**
A _[[formal smooth set]]_ $X$ is
1. for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]]
$\mathbb{R}^n \times Spec(A)$ (def. \ref{InfinitesimallyThickendSmoothManifold}) a [[set]]
$$
X(\mathbb{R}^n \times Spec(A)) \in Set
$$
to be called the set of _[[smooth functions]]_ or _plots_ from $\mathbb{R}^n \times Spec(A)$ to $X$;
1. for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$
between [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] a choice of function
$$
f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
$$
to be thought of as the precomposition operation
$$
\left(
\mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
\right)
\;\overset{f^\ast}{\mapsto}\;
\left(
\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X
\right)
$$
such that
1. ([[functor|functoriality]])
1. If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the [[identity function]] on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n \times Spec(A))$;
1. If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$ are two [[composition|composable]] [[smooth functions]] between infinitesimally thickened Cartesian spaces, then pullback of plots along them consecutively equals the pullback along the [[composition]]:
$$
f^\ast \circ g^\ast = (g \circ f)^\ast
$$
i.e.
$$
\array{
&& X(\mathbb{R}^{n_2} \times Spec(A_2))
\\
& {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
\\
X(\mathbb{R}^{n_1} \times Spec(A_1))
&& \underset{ (g \circ f)^\ast }{\longleftarrow} &&
X(\mathbb{R}^{n_3} \times Spec(A_3))
}
$$
1. ([[sheaf|gluing]])
If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that
$$\{ U_i \overset{f_i }{\to} \mathbb{R}^n \}_{i \in I}$$
in a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts
$\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots
$$
X(\mathbb{R}^n \times Spec(A))
\overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow}
\underset{i \in I}{\prod} X(U_i \times Spec(A))
$$
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots,
which are "[[matching families]]" in that they agree on [[intersections]]:
$$
\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
$$
i.e.
$$
\array{
&& (U_i \cap U_j) \times Spec(A)
\\
& \swarrow && \searrow
\\
U_i\times Spec(A) && && U_j \times Spec(A)
\\
& {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
\\
&& X
}
$$
Finally, given $X_1$ and $X_2$ two [[formal smooth sets]], then a [[smooth function]] between them
$$
f \;\colon\; X_1 \longrightarrow X_2
$$
is
* for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] $\mathbb{R}^n \times Spec(A)$ (def. \ref{InfinitesimallyThickendSmoothManifold}) a function
$$
f_\ast(\mathbb{R}^n \times Spec(A))
\;\colon\;
X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))
$$
such that
* for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between
infinitesimally thickened Cartesian spaces we have
$$
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
=
f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
$$
i.e.
$$
\array{
X_1(\mathbb{R}^{n_2} \times Spec(A_2))
&\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}&
X_2(\mathbb{R}^{n_2} \times Spec(A_2))
\\
\mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
\\
X_1(\mathbb{R}^{n_1} \times Spec(A_1))
&\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
X_2(\mathbb{R}^{n_1} \times Spec(A_1))
}
$$
=--
([Dubuc 79](Cahiers+topos#Dubuc79))
Basing [[synthetic differential geometry|infinitesimal geometry]] on [[formal smooth sets]] is an instance of the general approach to [[geometry]] called _[[functorial geometry]]_ or _[[topos theory]]_. For more background on this see at _[[geometry of physics -- manifolds and orbifolds]]_.
We have the evident generalization of example \ref{SmoothManifoldsAreDiffeologicalSpaces} to smooth geometry with [[infinitesimals]]:
+-- {: .num_example #YonedaLemmaForFormalSmoothSets}
###### Example
**([[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] are [[formal smooth sets]])**
For $X$ an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}), it becomes a [[formal smooth set]] according to def. \ref{FormalSmoothSet}
by taking its plots out of some $\mathbb{R}^n \times \mathbb{D}$ to be the homomorphism of infinitesimally
thickened Cartesian spaces:
$$
X(\mathbb{R}^n \times \mathbb{D})
\;\coloneqq\;
Hom_{FormalCartSp}( \mathbb{R}^n \times \mathbb{D}, X )
\,.
$$
(Stated more [[category theory|abstractly]], this is an instance of the _[[Yoneda embedding]]_ over a _[[subcanonical site]]_.)
=--
+-- {: .num_example #FormalSmoothSetsIncludedSmoothSet}
###### Example
**([[smooth sets]] are [[formal smooth sets]])**
Let $X$ be a [[smooth set]] (def. \ref{SmoothSet}).
Then $X$ becomes a [[formal smooth set]] (def. \ref{FormalSmoothSet}) by declaring the
set of plots $X(\mathbb{R}^n \times \mathbb{D})$ over
an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold})
to be [[equivalence classes]] of [[pairs]]
$$
\mathbb{R}^n \times \mathbb{D} \longrightarrow \mathbb{R}^{k}
\,,
\phantom{AA}
\mathbb{R}^k \longrightarrow X
$$
of a [[morphism]] of infinitesimally thickened Cartesian spaces and of a plot of $X$, as shown,
subject to the [[equivalence relation]] which identifies two such pairs if there exists
a smooth function $f \colon \mathbb{R}^k \to \mathbb{R}^{k'}$ such that
$$
\array{
&& \mathbb{R}^n \times \mathbb{D}
\\
& \swarrow && \searrow
\\
\mathbb{R}^k && \overset{f}{\longrightarrow} && \mathbb{R}^{k'}
\\
\mathbb{R}^k && \underset{f}{\longrightarrow} && \mathbb{R}^{k'}
\\
& \searrow && \swarrow
\\
&& X
}
$$
Stated more [[category theory|abstractly]] this says that $X$ as a [[formal smooth set]]
is the _[[left Kan extension]]_ (see [this example](Kan+extension#CoendFormulaForPresheavesOfSets)) of $X$ as a [[smooth set]] along the [[functor]]
that [[full subcategory|includes]] [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
into [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold}).
=--
+-- {: .num_defn #ReductionAndInfinitesimalShape}
###### Definition
**([[reduction modality|reduction]] and [[infinitesimal shape]])**
For $\mathbb{R}^n \times \mathbb{D}$ an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]]
(def. \ref{InfinitesimallyThickendSmoothManifold}) we say that the underlying ordinary [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is its _[[reduced object|reduction]]_
$$
\Re\left(
\mathbb{R}^n \times \mathbb{D}
\right)
\;\coloneqq\;
\mathbb{R}^n
\,.
$$
There is the canonical inclusion morphism
$$
\Re\left(
\mathbb{R}^n \times \mathbb{D}
\right)
=
\mathbb{R}^n
\overset{\phantom{AAAA}}{\hookrightarrow}
\mathbb{R}^n \times \mathbb{D}
$$
which [[formal dual|dually]] corresponds to the [[homomorphism]] of [[commutative algebras]]
$$
C^\infty(\mathbb{R}^n)
\longleftarrow
C^\infty(\mathbb{R}^n)
\otimes_{\mathbb{R}}
A
$$
which is the identity on all smooth functions $f \in C^\infty(\mathbb{R}^n)$ and is zero on all elements $a \in V \subset A$
in the nilpotent ideal of $A$ (as in example \ref{UniquePointOfInfinitesimalLine}).
Given any [[formal smooth set]] $X$, we say that its _[[infinitesimal shape]]_ or _[[de Rham shape]]_ (also: _[[de Rham stack]]_)
is the [[formal smooth set]] $\Im X$ (def. \ref{FormalSmoothSet}) defined to have as plots the [[reduction modality|reductions]]
of the plots of $X$, according to the above:
$$
(\Im X)( U ) \;\coloneqq\: X(\Re(U))
\,.
$$
There is a canonical morphism of formal smooth set
$$
\eta_X
\;\colon\;
X
\longrightarrow
\Im X
$$
which takes a plot
$$
U = \mathbb{R}^n \times \mathbb{D} \overset{f}{\longrightarrow} X
$$
to the [[composition]]
$$
\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times \mathbb{D} \overset{f}{\hookrightarrow} X
$$
regarded as a plot of $\Im X$.
=--
+-- {: .num_example #MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}
###### Example
**([[mapping space]] out of an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]])**
Let $X$ be an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) and let $Y$ be a [[formal smooth set]] (def. \ref{FormalSmoothSet}).
Then the _[[mapping space]]_
$$
[X,Y] \;\in\; FormalSmoothSet
$$
of smooth functions from $X$ to $Y$ is the [[formal smooth set]]
whose $U$-plots are the morphisms of [[formal smooth sets]]
from the [[Cartesian product]] of [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] $U \times X$ to $Y$,
hence the $U \times X$-plots of $Y$:
$$
[X,Y](U) \;\coloneqq\; Y(U \times X)
\,.
$$
=--
+-- {: .num_example #TangentBundleSynthetic}
###### Example
**([[synthetic differential geometry|synthetic]] [[tangent bundle]])**
Let $X \coloneqq \mathbb{R}^n$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
regarded as an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (\ref{InfinitesimallyThickendSmoothManifold}) and thus regarded as a [[formal smooth set]] (def. \ref{FormalSmoothSet})
by example \ref{YonedaLemmaForFormalSmoothSets}.
Consider the infinitesimal line
$$
\mathbb{D}^1
\hookrightarrow
\mathbb{R}^1
$$
from example \ref{InfinitesimalNeighbourhoodsInTheRealLine}. Then the [[mapping space]]
$[\mathbb{D}^1, X]$ (example \ref{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace})
is the total space of the [[tangent bundle]] $T X$ (example \ref{TangentVectorFields}).
Moreover, under restriction along the [[reduced object|reduction]] $\ast \longrightarrow \mathbb{D}^1$,
this is the full [[tangent bundle]] [[projection]], in that there is a [[natural isomorphism]] of
[[formal smooth sets]] of the form
$$
\array{
T X &\simeq& [\mathbb{D}^1, X]
\\
{}^{\mathllap{tb}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{D}^1, X ] }}
\\
X &\simeq& [\ast, X]
}
$$
In particular this implies immediately that smooth [[sections]] (def. \ref{Sections}) of the tangent bundle
$$
\array{
&& [\mathbb{D}^1, X] & \simeq T X
\\
& {}^{\mathllap{v}}\nearrow & \downarrow
\\
X &=& X
}
$$
are equivalently morphisms of the form
$$
\array{
&& X
\\
& {}^{\mathllap{\tilde v}}\nearrow & \downarrow^{\mathrlap{id}}
\\
X \times \mathbb{D}^1 &\underset{pr_1}{\longrightarrow}& X
}
$$
which we had already identified with [[tangent vector fields]] (def. \ref{TangentVectorFields})
in example \ref{SyntheticTangentVectorFields}.
=--
+-- {: .proof}
###### Proof
This follows by an analogous argument as in example \ref{SyntheticTangentVectorFields},
using the [[Hadamard lemma]].
=--
While in [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold})
only [[infinitesimals]] to any [[finite number|finite]] order may exist, in [[formal smooth sets]] (def. \ref{FormalSmoothSet})
we may find infinitesimals to any arbitrary finite order:
+-- {: .num_example #InfinitesimalNeighbourhood}
###### Example
**([[infinitesimal neighbourhood]])**
Let $X$ be a [[formal smooth sets]] (def. \ref{FormalSmoothSet}) $Y \hookrightarrow X$
a sub-formal smooth set. Then the _[[infinitesimal neighbourhood]]_ to arbitrary infinitesimal
order of $Y$ in $X$
is the [[formal smooth set]] $N_X Y$ whose plots are those plots of $X$
$$
\mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
$$
such that their [[reduced object|reduction]] (def. \ref{ReductionAndInfinitesimalShape})
$$
\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
$$
factors through a plot of $Y$.
=--
This allows to grasp the restriction of [[field histories]] to the [[infinitesimal neighbourhood]]
of a [[submanifold]] of [[spacetime]], which will be crucial for the discussion of [[phase spaces]]
[below](#PhaseSpace).
+-- {: .num_defn #FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}
###### Definition
**([[field histories]] on [[infinitesimal neighbourhood]] of [[submanifold]] of [[spacetime]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles})
and let $S \hookrightarrow \Sigma$ be a [[submanifold]] of [[spacetime]].
We write $N_\Sigma(S) \hookrightarrow \Sigma$ for its [[infinitesimal neighbourhood]] in $\Sigma$ (def. \ref{InfinitesimalNeighbourhood}).
Then the _set of field histories restricted to $S$_, to be denoted
$$
\label{SpaceOfFieldHistoriesInHigherCodimension}
\Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}
$$
is the set of section of $E$ restricted to the [[infinitesimal neighbourhood]] $N_\Sigma(S)$ (example \ref{InfinitesimalNeighbourhood}).
=--
$\,$
We close the discussion of [[synthetic differential geometry|infinitesimal differential geometry]] by explaining how we may recover the concept of _[[smooth manifolds]]_ inside the more general [[formal smooth sets]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below).
The key point is that the presence of [[infinitesimals]] in the theory allows an intrinsic definition of
[[local diffeomorphisms]]/[[formally étale morphism]] (def. \ref{FormalSmoothSetLocalDiffeomorphism} and example \ref{AbstractLocalDiffeomorphismsOfCartesianSpaces} below). It is noteworthy that the only role this concept plays in the development
of [[field theory]] below is that [[smooth manifolds]] admit [[triangulations]] by smooth [[singular simplices]] (def. \ref{SingularSimplicesInCartesianSpaces})
so that the concept of [[fiber integration|fiber]] [[integration of differential forms]] is well defined over manifolds.
+-- {: .num_defn #FormalSmoothSetLocalDiffeomorphism}
###### Definition
**([[local diffeomorphism]] of [[formal smooth sets]])**
Let $X,Y$ be [[formal smooth sets]] (def. \ref{FormalSmoothSet}). Then a [[morphism]] between them is called a _[[local diffeomorphism]]_ or _[[formally étale morphism]]_, denoted
$$
f \;\colon\; X \overset{et}{\longrightarrow} Y
\,,
$$
if $f$ if for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) $\mathbb{R}^n \times \mathbb{D}$ we have a natural identification between the $\mathbb{R}^n \times \mathbb{D}$-plots of $X$ with those $\mathbb{R}^n n\times \mathbb{D}$-plots of $Y$ whose [[reduction modality|reduction]] (def. \ref{ReductionAndInfinitesimalShape}) comes from an $\mathbb{R}^n$-plot of $X$, hence if we have a [[natural transformation|natural]] [[fiber product]] of [[sets]] of plots
$$
X(\mathbb{R}^n \times \mathbb{D})
\;\simeq\;
Y(\mathbb{R}^n \times \mathbb{D})
\underset{Y(\mathbb{R}^n)}{\times^f}
X(\mathbb{R}^n)
$$
i. e.
$$
\array{
&& X(\mathbb{R}^n \times \mathbb{D})
\\
& \swarrow && \searrow
\\
Y(\mathbb{R}^n \times \mathbb{D}) && \text{(pb)} && X(\mathbb{R}^n)
\\
& \searrow && \swarrow
\\
&& Y(\mathbb{R}^n )
}
$$
for all [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] $\mathbb{R}^n \times \mathbb{D}$.
Stated more [[category theory|abstractly]], this means that the
[[naturality square]] of the [[unit of a monad|unit]] of the [[infinitesimal shape]] $\Im$ (def. \ref{ReductionAndInfinitesimalShape}) is a [[pullback square]]
$$
\array{
X &\overset{\eta_X}{\longrightarrow}& \Im X
\\
{}^{\mathllap{f}}\downarrow &\text{(pb)}& \downarrow^{\mathrlap{\Im f}}
\\
Y &\underset{\eta_Y}{\longrightarrow}& \Im Y
}
$$
=--
([Khavkine-Schreiber 17, def. 3.1](local+diffeomorphism#KhavkineSchreiber17))
+-- {: .num_example #AbstractLocalDiffeomorphismsOfCartesianSpaces}
###### Example
**([[local diffeomorphism]] between [[Cartesian spaces]] from the general definition)**
For $X,Y \in CartSp$ two ordinary [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}),
regarded as [[formal smooth sets]] by example \ref{YonedaLemmaForFormalSmoothSets} then a [[morphism]]
$f \colon X \to Y$ between them is a [[local diffeomorphism]] in the general sense of def. \ref{FormalSmoothSetLocalDiffeomorphism}
precisely if it is so in the ordinary sense of def. \ref{LocalDiffeomorphismBetweenCartesianSpaces}.
=--
([Khavkine-Schreiber 17, prop. 3.2](geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17))
+-- {: .num_defn #SmoothManifoldInsideDiffeologicalSpaces}
###### Definition/Proposition
**([[smooth manifolds]])**
A _[[smooth manifold]]_ $X$ of [[dimension]] $n \in \mathbb{N}$ is
* a [[diffeological space]] (def. \ref{DiffeologicalSpace})
such that
1. there exists an [[indexed set]] $\{ \mathbb{R}^n \overset{\phi_i}{\to} X\}_{i \in I}$
of morphisms of [[formal smooth sets]] (def. \ref{FormalSmoothSet}) from [[Cartesian spaces]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) (regarded as [[formal smooth sets]] via example \ref{SmoothManifoldsAreDiffeologicalSpaces}, example \ref{SmoothSetsDiffeologicalSpaces} and example \ref{FormalSmoothSetsIncludedSmoothSet})
into $X$, (regarded as a [[formal smooth set]] via example \ref{SmoothSetsDiffeologicalSpaces} and example \ref{FormalSmoothSetsIncludedSmoothSet})
such that
1. every point $x \in X_s$ is in the [[image]] of at least one of the $\phi_i$;
1. every $\phi_i$ is a [[local diffeomorphism]] according to def. \ref{FormalSmoothSetLocalDiffeomorphism};
1. the [[final topology]] induced by the set of plots of $X$ makes $X_s$ a [[paracompact Hausdorff space]].
=--
([Khavkine-Schreiber 17, example 3.4](geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17))
For more on [[smooth manifolds]] from the perspective of [[formal smooth sets]] see at _[[geometry of physics -- manifolds and orbifolds]]_.
$\,$
**[[fermion fields]] and [[supergeometry]]**
{#Supergeometry}
Field theories of interest crucially involve [[fermionic fields]] (def. \ref{FermionicBosonicFields} below), such as the [[Dirac field]] (example \ref{DiracFieldBundle} below),
which are subject to the "[[Pauli exclusion principle]]", a key reason for the [[stability of matter]].
Mathematically this principle means that these
[[field (physics)|fields]] have [[field bundles]] whose [[field fiber]]
is not an ordinary [[manifold]], but an odd-graded _[[supermanifold]]_ (more on this in remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature} and remark \ref{SupergeometricNatureOfDiracEquation} below).
This "[[supergeometry]]" is an immediate generalization of the
[[synthetic differential geometry|infinitesimal geometry]] [above](#InfinitesimalGeometry), where now the [[infinitesimal]]
elements in the [[algebra of functions]] may be equipped with a [[graded object|grading]]: one speaks of _[[superalgebra]]_.
The "super"-terminology for something as down-to-earth as the mathematical principle behind the [[stability of matter]]
may seem unfortunate. For better or worse, this terminology has become standard
since the middle of the 20th century. But the concept that today is called _[[supercommutative superalgebra]]_
was in fact first considered by [[Ausdehnungslehre|Grassmann 1844]] who got it right ("[[Ausdehnungslehre]]") but apparently
was too far ahead of his time and remained unappreciated.
Beware that considering [[supergeometry]] does _not_ necessarily involve considering "[[supersymmetry]]".
Supergeometry is the geometry of [[fermion fields]] (def. \ref{FermionicBosonicFields} below), and that all [[matter]] fields in the [[observable universe]] are fermionic
has been [[experiment|experimentally]] established since the [[Stern-Gerlach experiment]] in 1922. Supersymmetry,
on the other hand, is a hypothetical extension of [[spacetime]]-[[symmetry]]
within the context of [[supergeometry]]. Here we do not discuss supersymmetry; for details see instead at _[[geometry of physics -- supersymmetry]]_.
+-- {: .num_defn #SupercommutativeSuperalgebra}
###### Definition
**([[supercommutative superalgebra]])**
A _[[real number|real]] $\mathbb{Z}/2$-[[graded algebra]]_ or _[[superalgebra]]_ is an [[associative algebra]] $A$ over the [[real numbers]]
together with a [[direct sum]] decomposition of its underlying [[real vector space]]
$$
A \simeq_{\mathbb{R}} A_{even} \oplus A_{odd}
\,,
$$
such that the product in the algebra respects the multiplication in the [[cyclic group|cyclic]] [[group of order 2]] $\mathbb{Z}/2 = \{even, odd\}$:
$$
\left.
\array{
A_{even} \cdot A_{even}
\\
A_{odd} \cdot A_{odd}
}
\right\}
\subset A_{even}
\phantom{AAAA}
\left.
\array{
A_{odd} \cdot A_{even}
\\
A_{even} \cdot A_{odd}
}
\right\}
\subset A_{odd}
\,.
$$
This is called a _[[supercommutative superalgebra]]_ if for all elements $a_1, a_2 \in A$ which are of homogeneous degree
${\vert a_i \vert} \in \mathbb{Z}/2 = \{even, odd\}$ in that
$$
a_i \in A_{{\vert a_i\vert}} \subset A
$$
we have
$$
a_1 \cdot a_2 = (-1)^{{\vert a_1 \vert \vert a_2 \vert}} a_2 \cdot a_1
\,.
$$
A _[[homomorphism]] of [[superalgebras]]_
$$
f \;\colon\; A \longrightarrow A'
$$
is a [[homomorphism]] of [[associative algebras]] over the [[real numbers]] such that the $\mathbb{Z}/2$-[[graded object|grading]]
is respected in that
$$
f(A_{even}) \subset A'_{even}
\phantom{AAAAA}
f(A_{odd}) \subset A'_{odd}
\,.
$$
=--
For more details on superalgebra see at _[[geometry of physics -- superalgebra]]_.
+-- {: .num_example #GrassmannAlgebra}
###### Example
**(basic examples of [[supercommutative superalgebras]])**
Basic examples of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra})
include the following:
1. Every [[commutative algebra]] $A$ becomes a [[supercommutative superalgebra]] by declaring it to be all in even degree:
$A = A_{even}$.
1. For $V$ a [[finite dimensional vector space|finite dimensional]] [[real vector space]], then the
[[Grassmann algebra]] $A \coloneqq \wedge^\bullet_{\mathbb{R}} V^\ast$ is a supercommutative superalgebra with
$A_{even} \coloneqq \wedge^{even} V^\ast$ and $A_{odd} \coloneqq \wedge^{odd} V^\ast$.
More explicitly, if $V = \mathbb{R}^s$ is a [[Cartesian space]] with canonical dual [[coordinates]] $(\theta^i)_{i = 1}^s$
then the [[Grassmann algebra]] $\wedge^\bullet (\mathbb{R}^s)^\ast$ is the real algebra which is [[generators and relations|generated]]
from the $\theta^i$ regarded in odd degree and hence subject to the relation
$$
\theta^i \cdot \theta^j = - \theta^j \cdot \theta^i
\,.
$$
In particular this implies that all the $\theta^i$ are [[infinitesimal]] (def. \ref{InfinitesimallyThickendSmoothManifold}):
$$
\theta^i \cdot \theta^i = 0
\,.
$$
1. For $A_1$ and $A_2$ two [[supercommutative superalgebras]], there is their _[[tensor product of algebras|tensor product]]_
supercommutative superalgebra $A_1 \otimes_{\mathbb{R}} A_2$. For example for $X$ a [[smooth manifold]]
with ordinary algebra of smooth functions $C^\infty(X)$ regarded as a supercommutative superalgebra
by the first example above, the tensor product with a [[Grassmann algebra]] (second example above) is the
supercommutative superalgebta
$$
C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet((\mathbb{R}^s)\ast)
$$
whose elements may uniquely be expanded in the form
$$
f + f_i \theta^i + f_{i j} \theta^i \theta^j + f_{i j k} \theta^i \theta^j \theta^k + \cdots + f_{i_1 \cdots i_s} \theta^{i_1} \cdots \theta^{i_s}
\,,
$$
where the $f_{i_1 \cdots i_k} \in C^\infty(X)$ are smooth functions on $X$ which are skew-symmetric in their indices.
=--
The following is the direct super-algebraic analog of
the definition of [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold}):
+-- {: .num_defn #SuperCartesianSpace}
###### Definition
**([[super Cartesian space]])**
A _[[superpoint]]_ $Spec(A)$ is represented by a [[super-commutative superalgebra]]
$A$ (def. \ref{SupercommutativeSuperalgebra}) which as a $\mathbb{Z}/2$-[[graded vector space]] ([[super vector space]]) is a [[direct sum]]
$$
A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
$$
of the 1-dimensional even vector space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1, with
a [[finite dimensional vector space|finite dimensional]] [[super vector space]]
$V$ that is a [[nilpotent ideal]] in $A$ in that for
each element $a \in V$ there exists a [[natural number]] $n \in \mathbb{N}$ such that
$$
a^{n+1} = 0
\,.
$$
More generally, a [[super Cartesian space]]
$\mathbb{R}^n \times Spec(A)$ is represented by a [[super-commutative algebra]]
$C^\infty(\mathbb{R}^n) \otimes A \in \mathbb{R} Alg$ which is the [[tensor product of algebras]] of the algebra of smooth functions
$C^\infty(\mathbb{R}^n)$ on an actual [[Cartesian space]] of some [[dimension]] $n$, with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of a [[superpoint]] (example \ref{GrassmannAlgebra}).
Specifically, for $s \in \mathbb{N}$, there is the superpoint
$$
\label{StandardSuperpoints}
\mathbb{R}^{0 \vert s}
\;\coloneqq\;
Spec\left( \wedge^\bullet (\mathbb{R}^s)^\ast \right)
$$
whose [[algebra of functions]] is by definition the [[Grassmann algebra]] on $s$ generators $(\theta^i)_{i = 1}^s$ in odd degree (example
\ref{GrassmannAlgebra}).
We write
$$
\begin{aligned}
\mathbb{R}^{b\vert s}
& \coloneqq
\mathbb{R}^b \times \mathbb{R}^{0 \vert s}
\\
& =
\mathbb{R}^b \times Spec( \wedge^\bullet(\mathbb{R}^s)^\ast )
\\
& =
Spec\left( C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^s)^\ast \right)
\end{aligned}
$$
for the corresponding super Cartesian spaces whose algebra of functions is as in example \ref{GrassmannAlgebra}.
We say that a _smooth function_ between two [[super Cartesian spaces]]
$$
\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
$$
is by definition [[formal dual|dually]] a [[homomorphism]] of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra}) of the form
$$
C^\infty(\mathbb{R}^{n_1}) \otimes A_1
\overset{f^\ast}{\longleftarrow}
C^\infty(\mathbb{R}^{n_2}) \otimes A_2
\,.
$$
=--
+-- {: .num_example #SuperpointInducedByFiniteDimensionalVectorSpace}
###### Example
**([[superpoint]] induced by a [[finite dimensional vector space]])**
Let $V$ be a [[finite dimensional vector space|finite dimensional]] [[real vector space]]. With $V^\ast$
denoting its [[dual vector space]] write $\wedge^\bullet V^\ast$ for the [[Grassmann algebra]]
that it generates. This being a [[supercommutative algebra]], it defines a [[superpoint]] (def. \ref{SuperCartesianSpace}).
We denote this superpoint by
$$
V_{odd} \simeq \mathbb{R}^{0 \vert dim(V)}
\,.
$$
=--
All the [[differential geometry]] over [[Cartesian space]] that we discussed [above](#Geometry)
generalizes immediately to [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace})
if we stricly adhere to the [[signs in supergeometry|super sign rule]] which says that
whenever two odd-graded elements swap places, a minus sign is picked up. In particular
we have the following generalization of the [[de Rham complex]] on [[Cartesian spaces]] discussed [above](#DifferentialFormsAndCartanCalculus).
+-- {: .num_defn #DifferentialFormOnSuperCartesianSpaces}
###### Definition
**([[super differential forms]] on [[super Cartesian spaces]])**
For $\mathbb{R}^{b\vert s}$ a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}), hence the [[formal dual]]
of the [[supercommutative superalgebra]] of the form
$$
C^\infty(\mathbb{R}^{b\vert s})
\;=\;
C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^s
$$
with canonical even-graded [[coordinate functions]] $(x^i)_{i = 1^b}$ and odd-graded coordinate functions $(\theta^j)_{j = 1}^s$.
Then the _[[de Rham complex]] of [[super differential forms]] on $\mathbb{R}^{b\vert s}$_
is, in super-generalization of def. \ref{DifferentialnForms}, the $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded commutative algebra]]
$$
\Omega^\bullet(\mathbb{R}^{b|s})
\;\coloneqq\;
C^\infty(\mathbb{R}^{b|s})
\otimes_{\mathbb{R}}
\wedge^\bullet \langle
d x^1, \cdots, d x^b,
\;
d \theta^1, \cdots, d\theta^s
\rangle
$$
which is generated over $C^\infty(\mathbb{R}^{b\vert s})$ from new generators
$$
\underset{
\text{deg} = (1,even)
}{\underbrace{ d x^i }}
\phantom{AAAAA}
\underset{
\text{deg} = (1,odd)
}{
\underbrace{
d \theta^j
}
}
$$
whose [[differential]] is defined in degree-0 by
$$
d f
\;\coloneqq\;
\frac{\partial f}{\partial x^i} d x^i
+
\frac{\partial f}{\partial \theta^j} d \theta^j
$$
and extended from there as a bigraded [[derivation]] of bi-degree $(1,even)$,
in super-generalization of def. \ref{deRhamDifferential}.
Accordingly, the operation of contraction with [[tangent vector fields]] (def. \ref{ContractionOfFormsWithVectorFields})
has bi-degree $(-1,\sigma)$ if the tangent vector has super-degree $\sigma$:
| generator | bi-degree |
|--|--|
| $\phantom{AA} x^a$ | (0,even) |
| $\phantom{AA} \theta^\alpha$ | (0,odd) |
| $\phantom{AA} dx^a$ | (1,even) |
| $\phantom{AA} d\theta^\alpha$ | (1,odd) |
| derivation | bi-degree |
|------------|-----------|
| $\phantom{AA} d$ | (1,even) |
| $\phantom{AA}\iota_{\partial x^a}$ | (-1, even) |
| $\phantom{AA}\iota_{\partial \theta^\alpha}$ | (-1,odd) |
This means that if $\alpha \in \Omega^\bullet(\mathbb{R}^{b\vert s})$ is in bidegree $(n_\alpha, \sigma_\alpha)$,
and $\beta \in \Omega^\bullet(\mathbb{R}^{b \vert \sigma})$ is in bidegree $(n_\beta, \sigma_\beta)$, then
$$
\alpha \wedge \beta
\;
=
\;
(- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
\;
\beta \wedge \alpha
\,.
$$
Hence there are _two_ contributions to the sign picked up when exchanging two super-differential forms in the wedge product:
1. there is a "cohomological sign" which for commuting an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;
1. in addition there is a "super-grading" sign which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.
For example:
$$
x^{a_1} (dx^{a_2}) \;=\; + (dx^{a_2}) x^{a_1}
$$
$$
\theta^\alpha (dx^a) \;=\; + (dx^a) \theta^\alpha
$$
$$
\theta^{\alpha_1} (d\theta^{\alpha_2})
\;=\;
- (d\theta^{\alpha_2}) \theta^{\alpha_1}
$$
$$
dx^{a_1}
\wedge
d x^{a_2}
\;=\;
-
d x^{a_2}
\wedge
d x^{a_1}
$$
$$
dx^a
\wedge
d \theta^{\alpha}
\;=\;
-
d\theta^{\alpha}
\wedge
d x^a
$$
$$
d\theta^{\alpha_1}
\wedge
d \theta^{\alpha_2}
\;=\;
+
d\theta^{\alpha_2}
\wedge
d \theta^{\alpha_1}
$$
=--
(e.g. [Castellani-D'Auria-Fré 91 (II.2.106) and (II.2.109)](signs+in+supergeometry#CastellaniDAuriaFre91), [Deligne-Freed 99, section 6](signs+in+supergeometry#DeligneFreed99))
Beware that there is also _another_ sign rule for [[super differential forms]]
used in the literature. See at _[[signs in supergeometry]]_ for further discussion.
$\,$
It is clear now by direct analogy with the definition of [[formal smooth sets]] (def. \ref{FormalSmoothSet})
what the corresponding [[supergeometry|supergeometric]] generalization is. For definiteness we
spell it out yet once more:
+-- {: .num_defn #SuperFormalSmoothSet}
###### Definition
**([[super formal smooth set|super smooth set]])**
A _[[super formal smooth set|super smooth set]]_ $X$ is
1. for each [[super Cartesian space]]
$\mathbb{R}^n \times Spec(A)$ (def. \ref{SuperCartesianSpace}) a [[set]]
$$
X(\mathbb{R}^n \times Spec(A)) \in Set
$$
to be called the set of _[[smooth functions]]_ or _plots_ from $\mathbb{R}^n \times Spec(A)$ to $X$;
1. for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$
between [[super Cartesian spaces]] a choice of function
$$
f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
$$
to be thought of as the precomposition operation
$$
\left(
\mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
\right)
\;\overset{f^\ast}{\mapsto}\;
\left(
\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X
\right)
$$
such that
1. ([[functor|functoriality]])
1. If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the [[identity function]] on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n \times Spec(A))$.
1. If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$
are two [[composition|composable]] [[smooth functions]] between infinitesimally thickened Cartesian spaces, then
pullback of plots along them consecutively equals the pullback along the [[composition]]:
$$
f^\ast \circ g^\ast
=
(g \circ f)^\ast
$$
i.e.
$$
\array{
&& X(\mathbb{R}^{n_2} \times Spec(A_2))
\\
& {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
\\
X(\mathbb{R}^{n_1} \times Spec(A_1))
&& \underset{ (g \circ f)^\ast }{\longleftarrow} &&
X(\mathbb{R}^{n_3} \times Spec(A_3))
}
$$
1. ([[sheaf|gluing]])
If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that
$$\{ U_i \overset{f_i }{\to} \mathbb{R}^n \}_{i \in I}$$
is a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts
$\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots
$$
X(\mathbb{R}^n \times Spec(A))
\overset{( (f_i)^\ast )_{i \in I} }{\hookrightarrow}
\underset{i \in I}{\prod} X(U_i \times Spec(A))
$$
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots,
which are "[[matching families]]" in that they agree on [[intersections]]:
$$
\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
$$
i.e.
$$
\array{
&& (U_i \cap U_j) \times Spec(A)
\\
& \swarrow && \searrow
\\
U_i\times Spec(A) && && U_j \times Spec(A)
\\
& {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
\\
&& X
}
$$
Finally, given $X_1$ and $X_2$ two [[super formal smooth sets]], then a [[smooth function]] between them
$$
f \;\colon\; X_1 \longrightarrow X_2
$$
is
* for each [[super Cartesian space]] $\mathbb{R}^n \times Spec(A)$ a function
$$
f_\ast(\mathbb{R}^n \times Spec(A))
\;\colon\;
X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))
$$
such that
* for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between
super Cartesian spaces we have
$$
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
=
f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
$$
i.e.
$$
\array{
X_1(\mathbb{R}^{n_2} \times Spec(A_2))
&\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}&
X_2(\mathbb{R}^{n_2} \times Spec(A_2))
\\
\mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
\\
X_1(\mathbb{R}^{n_1} \times Spec(A_1))
&\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
X_2(\mathbb{R}^{n_1} \times Spec(A_1))
}
$$
=--
([Yetter 88](synthetic+differential+supergeometry#Yetter88))
Basing [[supergeometry]] on [[super formal smooth sets]] is an instance of the general approach to [[geometry]] called _[[functorial geometry]]_ or _[[topos theory]]_. For more background on this see at _[[geometry of physics -- supergeometry]]_.
In direct generalization of example \ref{SmoothManifoldsAreDiffeologicalSpaces} we have:
+-- {: .num_example #SuperSmoothSetSuperCartesianSpaces}
###### Example
**([[super Cartesian spaces]] are [[super formal smooth sets|super smooth sets]])**
Let $X$ be a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) Then it becomes a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet})
by declaring its plots $\Phi \in X(\mathbb{R}^n \times \mathbb{D})$ to the algebra homomorphisms
$ C^\infty(\mathbb{R}^n \times \mathbb{D}) \leftarrow C^\infty(\mathbb{R}^{b\vert s})$.
Under this identification, morphisms between [[super Cartesian spaces]] are in [[natural bijection]]
with their morphisms regarded as [[super formal smooth set|super smooth sets]].
Stated more [[category theory|abstractly]], this statement is an example of the _[[Yoneda embedding]]_
over a _[[subcanonical site]]_.
=--
Similarly, in direct generalization of prop. \ref{CartSpYpnedaLemma} we have:
+-- {: .num_prop #SuperCartSpYpnedaLemma}
###### Proposition
**(plots of a [[super formal smooth set|super smooth set]] really are the [[smooth functions]] into the smooth smooth set)**
Let $X$ be a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet}).
For $\mathbb{R}^n \times \mathbb{D}$ any [[super Cartesian space]] (def. \ref{SuperCartesianSpace})
there is a [[natural transformation|natural]] [[function]]
$$
Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\simeq}{\longrightarrow} X(\mathbb{R}^n)
$$
from the set of homomorphisms of super smooth sets from $\mathbb{R}^n \times \mathbb{D}$ (regarded as a super smooth set via example \ref{SuperSmoothSetSuperCartesianSpaces})
to $X$, to the set of plots of $X$ over $\mathbb{R}^n \times \mathbb{D}$, given by evaluating on the [[identity function|identity]] plot $id_{\mathbb{R}^n \times \mathbb{D}}$.
This function is a _[[bijection]]_.
This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the _would-be_ smooth functions
into $X$, end up being the _actual_ smooth functions into $X$.
=--
+-- {: .proof}
###### Proof
This is the statement of the _[[Yoneda lemma]]_ over the [[site]] of [[super Cartesian spaces]].
=--
We do not need to consider here [[supermanifolds]] more general than the [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}).
But for those readers familiar with the concept we include the following direct analog of
the characterization of [[smooth manifolds]] according to def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces}:
+-- {: .num_defn #SuperSmoothManifolds}
###### Definition/Proposition
**([[supermanifolds]])**
A _[[supermanifold]]_ $X$ of [[dimension]] super-dimension $(b,s) \in \mathbb{N} \times \mathbb{N}$ is
* a [[super smooth set]] (def. \ref{SuperFormalSmoothSet})
such that
1. there exists an [[indexed set]] $\{ \mathbb{R}^{b\vert s} \overset{\phi_i}{\to} X\}_{i \in I}$
of morphisms of [[super formal smooth sets|super smooth sets]] (def. \ref{SuperFormalSmoothSet}) from [[super Cartesian spaces]] $\mathbb{R}^{b\vert s}$ (def. \ref{SuperCartesianSpace}) (regarded as [[super formal smooth set|super smooth sets]] via example \ref{SuperSmoothSetSuperCartesianSpaces}
into $X$, such that
1. for every plot $\mathbb{R}^n \times \mathbb{D} \to X$ there is a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) restricted to which the plot factors through the $\mathbb{R}^{b\vert s}_i$;
1. every $\phi_i$ is a [[local diffeomorphism]] according to def. \ref{FormalSmoothSetLocalDiffeomorphism},
now with respect not just to [[infinitesimally thickened points]], but with respect to [[superpoints]];
1. the [[bosonic modality|bosonic]] part of $X$ is a [[smooth manifold]] according to def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces}.
=--
Finally we have the evident generalization of the smooth moduli space $\mathbf{\Omega}^\bullet$ of [[differential forms]]
from example \ref{UniversalSmoothModuliSpaceOfDifferentialForms} to [[supergeometry]]
+-- {: .num_example #ModuliOfSDifferentialForms}
###### Example
**(universal [[smooth set|smooth]] [[moduli spaces]] of [[super differential forms]])**
For $n \in \mathbf{M}$ write
$$
\mathbf{\Omega}^n \;\in\; SuperSmoothSet
$$
for the [[super smooth set]] (def. \ref{SuperSmoothSetSuperCartesianSpaces}) whose set of plots
on a [[super Cartesian space]] $U \in SuperCartSp$ (def. \ref{SuperCartesianSpace})
is the set of [[super differential forms]] (def. \ref{DifferentialFormOnSuperCartesianSpaces})
of cohomolgical degree $n$
$$
\mathbf{\Omega}^n(U) \;\coloneqq\; \Omega^n(U)
$$
and whose maps of plots is given by [[pullback of differential forms|pullback]] of super differential forms.
The [[de Rham differential]] on [[super differential forms]] applied plot-wise yields a morpism
of super smooth sets
$$
\label{SuperUniversalDeRhamDifferential}
d \;\colon\; \mathbf{\Omega}^n \longrightarrow \mathbf{\Omega}^{n+1}
\,.
$$
As before in def. \ref{DifferentialFormsOnDiffeologicalSpaces} we then define
for any [[super smooth set]] $X \in SuperSmoothSet$ its set of differential $n$-forms to be
$$
\Omega^n(X)
\;\coloneqq\;
Hom_{SuperSmoothSet}(X,\mathbf{\Omega}^n)
$$
and we define the [[de Rham differential]] on these to be given by postcomposition with (eq:SuperUniversalDeRhamDifferential).
=--
$\,$
+-- {: .num_defn #FermionicBosonicFields}
###### Definition
**([[bosonic fields]] and [[fermionic fields]])**
For $\Sigma$ a [[spacetime]], such as [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime})
if a [[fiber bundle]] $E \overset{fb}{\longrightarrow} \Sigma$ with total space a [[super Cartesian space]] (def. \ref{SuperCartesianSpace})
(or more generally a [[supermanifold]], def./prop. \ref{SuperSmoothManifolds}) is regarded as a super-[[field bundle]] (def. \ref{FieldsAndFieldBundles}),
then
* the even-graded [[sections]] are called the _[[bosonic field|bosonic]]_ [[field histories]];
* the odd-graded [[sections]] are called the _[[fermionic field|fermionic]]_ [[field histories]].
In components, if $E = \Sigma \times F$ is a [[trivial bundle]] with [[fiber]] a [[super Cartesian space]] (def. \ref{SuperCartesianSpace})
with even-graded [[coordinates]] $(\phi^a)$ and odd-graded [[coordinates]] $(\psi^A)$, then the $\phi^a$ are called the
_[[bosonic field|bosonic]]_ field coordinates, and the $\psi^A$ are called the _[[fermionic field|fermionic]]_ field coordinates.
=--
What is crucial for the discussion of [[field theory]] is the following immediate [[supergeometry|supergeometric]] analog of the smooth structure on the [[space of field histories]] from example \ref{DiffeologicalSpaceOfFieldHistories}:
+-- {: .num_example #SupergeometricSpaceOfFieldHistories}
###### Example
**([[super smooth set|supergeometric]] [[space of field histories]])**
Let $E \overset{fb}{\to} \Sigma$ be a super-[[field bundle]] (def. \ref{FieldsAndFieldBundles}, def. \ref{FermionicBosonicFields}).
Then the _[[space of sections]]_, hence the _[[space of field histories]]_, is the [[super formal smooth set]] (def. \ref{SuperFormalSmoothSet})
$$
\Gamma_\Sigma(E) \in SuperSmoothSet
$$
whose plots $\Phi_{(-)}$ for a given [[Cartesian space]] $\mathbb{R}^n$ and
[[superpoint]] $\mathbb{D}$ (def. \ref{SuperCartesianSpace}) with the [[Cartesian products]]
$U \coloneqq \mathbb{R}^n \times \mathbb{D}$ and $U \times \Sigma$ regarded as
[[super formal smooth set|super smooth sets]] according to example \ref{SuperSmoothSetSuperCartesianSpaces} are defined to be the [[morphisms]]
of [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet})
$$
\array{
U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
}
$$
which make the following [[commuting diagram|diagram commute]]:
$$
\array{
&& E
\\
& {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
\\
U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
}
\,.
$$
Explicitly, if $\Sigma$ is a [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and $E = \Sigma \times F$
a [[trivial bundle|trivial]] [[field bundle]] with [[field fiber]] a [[super vector space]] (example \ref{TrivialVectorBundleAsAFieldBundle}, example \ref{FermionicBosonicFields}) this means [[formal duality|dually]] that a plot $\Phi_{(-)}$
of the super smooth set of field histories is a
[[homomorphism]] of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra})
$$
\array{
C^\infty(U \times \Sigma) &\overset{\left(\Phi_{(-)}(-)\right)^\ast}{\longleftarrow}& C^\infty(E)
}
$$
which make the following [[commuting diagram|diagram commute]]:
$$
\array{
&& C^\infty(E)
\\
& {}^{\mathllap{\left( \Phi_{(-)}(-) \right)^\ast }}\nearrow & \uparrow^{\mathrlap{fb^\ast}}
\\
C^\infty(U \times \Sigma) &\underset{pr_2^\ast}{\longleftarrow}& C^\infty(\Sigma)
}
\,.
$$
=--
We will focus on discussing the [[supergeometry|supergeometric]] [[space of field histories]] (example \ref{SupergeometricSpaceOfFieldHistories}) of the _[[Dirac field]]_ (def. \ref{DiracFieldBundle} below). This we consider below in example \ref{DiracFieldBundle};
but first we discuss now some relevant basics of general [[supergeometry]].
Example \ref{SupergeometricSpaceOfFieldHistories} is really a special case of a general
relative [[mapping space]]-construction as in example \ref{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}.
This immediately generalizes also to the [[supergeometry|supergeometric]] context.
+-- {: .num_defn #MappingSpaceOutOfASuperCartesianSpace}
###### Definition
**([[supergeometry|super]]-[[mapping space]] out of a [[super Cartesian space]])**
Let $X$ be a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) and let $Y$ be a [[super formal smooth sets|super smooth set]] (def. \ref{SuperFormalSmoothSet}).
Then the _[[mapping space]]_
$$
[X,Y] \;\in\; SuperSmoothSet
$$
of super smooth functions from $X$ to $Y$ is the [[super formal smooth set]]
whose $U$-plots are the morphisms of [[super formal smooth set|super smooth set]]
from the [[Cartesian product]] of [[super Cartesian space]] $U \times X$ to $Y$,
hence the $U \times X$-plots of $Y$:
$$
[X,Y](U) \;\coloneqq\; Y(U \times X)
\,.
$$
=--
In direct generalization of the [[synthetic differential geometry|synthetic]] [[tangent bundle]]
construction (example \ref{TangentBundleSynthetic}) to supergeometry we have
+-- {: .num_defn #TangentBundleOdd}
###### Definition
**([[odd tangent bundle]])**
Let $X$ be a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet})
and $\mathbb{R}^{0 \vert 1}$ the [[superpoint]] (eq:StandardSuperpoints) then the [[supergeometry|supergeometry]]-[[mapping space]]
$$
\array{
T_{odd} X & \coloneqq& [\mathbb{R}^{0\vert 1}, X]
\\
{}^{\mathllap{tb_{odd}}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{R}^{0 \vert 1}, X ] }}
\\
X & = & X
}
$$
is called the _[[odd tangent bundle]]_ of $X$.
=--
+-- {: .num_example #SuperpointsMapping}
###### Example
**([[mapping space]] of [[superpoints]])**
Let $V$ be a [[finite dimensional vector space|finite dimensional]] [[real vector space]] and
consider its corresponding [[superpoint]] $V_{odd}$ from exampe \ref{SuperpointInducedByFiniteDimensionalVectorSpace}.
Then the [[mapping space]] (def. \ref{MappingSpaceOutOfASuperCartesianSpace}) out of the
[[superpoint]] $\mathbb{R}^{0\vert 1}$ (def. \ref{SuperCartesianSpace}) into $V_{odd}$ is
the [[Cartesian product]] $V_{odd} \times V$
$$
[\mathbb{R}^{0\vert 1}, V_{odd}]
\;\simeq\;
V_{odd} \times V
\,.
$$
By def. \ref{TangentBundleOdd} this says that $V_{odd} \times V$ is the "[[odd tangent bundle]]" of $V_{odd}$.
=--
+-- {: .proof}
###### Proof
Let $U$ be any [[super Cartesian space]]. Then by definition we have the
following sequence of [[natural bijections]] of sets of plots
$$
\begin{aligned}
\left[
\mathbb{R}^{0\vert 1}, V_{odd}
\right](U)
& =
Hom_{SuperSmoothSet}( \mathbb{R}^{0\vert 1} \times U, V_{odd} )
\\
&
\simeq
Hom_{\mathbb{R}sAlg}( \wedge^\bullet(V^\ast)\,,\, C^\infty(U)[\theta]/(\theta^2) )
\\
&
\simeq
Hom_{\mathbb{R}Vect}( V^\ast \,,\, (C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta\rangle )
\\
& \simeq
Hom_{\mathbb{R}Vect}( V^\ast\,,\, C^\infty(U)_{odd} ) \,\times\, Hom_{\mathbb{R}Vect}( V^\ast, C^\infty(U)_{even} )
\\
& \simeq
V_{odd}(U) \times V(U)
\\
& \simeq
(V_{odd} \times V)(U)
\end{aligned}
$$
Here in the third line we used that the [[Grassmann algebra]] $\wedge^\bullet V^\ast$ is [[free construction|free]]
on its generators in $V^\ast$, meaning that a homomorphism of [[supercommutative superalgebras]] out of the Grassmann algebra is uniquely
fixed by the underlying degree-preserving [[linear function]] on these generators. Since in a [[Grassmann algebra]]
all the generators are in odd degree, this is equivalently a linear map from $V^\ast$ to the odd-graded
[[real vector space]] underlying $C^\infty(U)[\theta](\theta^2)$,
which is the [[direct sum]] $C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta \rangle$.
Then in the fourth line we used that [[finite coproduct|finite]] [[direct sums]] are [[Cartesian products]],
so that linear maps into a direct sum are [[pairs]] of linear maps into the direct summands.
That all these [[bijections]] are [[natural bijection|natural]] means that they are compatible with
morphisms $U \to U'$ and therefore this says that $[\mathbb{R}^{0\vert 1}, V_{odd}]$ and $V_{odd} \times V$
are the same as seen by super-smooth plots, hence that they are [[isomorphism|isomorphic]] as
[[super formal smooth set|super smooth sets]].
=--
With this [[supergeometry]] in hand we finally turn to defining the [[Dirac field]] species:
+-- {: .num_example #DiracFieldBundle}
###### Example
**([[field bundle]] for [[Dirac field]])**
For $\Sigma$ being [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}),
of dimension $2+1$, $3+1$, $5+1$ or $9+1$, let $S$ be the [[spin representation]] from prop. \ref{SpinorRepsByNormedDivisionAlgebra},
whose underlying [[real vector space]] is
$$
S \;=\;
\left\{
\array{
\mathbb{R}^2 \oplus \mathbb{R}^2 & \vert & p + 1 = 2+1
\\
\mathbb{C}^2 \oplus \mathbb{C}^2 &\vert& p + 1 = 3 + 1
\\
\mathbb{H}^2 \oplus \mathbb{H}^2 &\vert& p + 1 = 5 + 1
\\
\mathbb{O}^2 \oplus \mathbb{O}^2 &\vert& p + 1 = 9 + 1
}
\right.
$$
With
$$
S_{odd} \simeq \mathbb{R}^{0 \vert dim(S)}
$$
the corresponding [[superpoint]] (example \ref{SuperpointInducedByFiniteDimensionalVectorSpace}),
then the [[field bundle]] for the _[[Dirac field]]_ on $\Sigma$ is
$$
E \;\coloneqq\; \Sigma \times S_{odd} \overset{pr_1}{\to} \Sigma
\,,
$$
hence the [[field fiber]] is the [[superpoint]] $S_{odd}$. This is the corresponding [[spinor bundle]]
on [[Minkowski spacetime]], with fiber in odd super-degree.
The traditional two-component [[spinor]] basis from remark \ref{TwoComponentSpinorNotation}
provides [[fermionic field]] coordinates (def. \ref{FermionicBosonicFields}) on the [[field fiber]] $S_{odd}$:
$$
\left(
\psi^A
\right)_{A = 1}^4
\;=\;
\left(
(\chi_a), (\xi^{\dagger \dot a})
\right)_{a,\dot a = 1,2}
\,.
$$
Notice that these are $\mathbb{K}$-valued odd functions: For instance if $\mathbb{K} = \mathbb{C}$ then each $\chi_a$ in turn has
two components, a [[real part]] and an [[imaginary part]].
A key point with the [[field bundle]] of the [[Dirac field]] (example \ref{DiracFieldBundle})
is that the field fiber coordinates $(\psi^A)$ or $\left((\chi_a), (\xi^{\dagger \dot a})\right)$
are now odd-graded elements in the function algebra on the field fiber, which is the
[[Grassmann algebra]] $C^\infty(S_{odd}) = \wedge^\bullet(S^\ast)$. Therefore they anti-commute
with each other:
$$
\label{DiracFieldCoordinatesAnticommute}
\psi^\alpha \psi^{\beta} = - \psi^{\beta} \psi^\alpha
\,.
$$
<img src="https://ncatlab.org/nlab/files/DermisekAnticommutingSpinorFieldCoordinates.jpg" width="400">
> snippet grabbed from ([Dermisek 09](Dirac+field#DermisekI8))
=--
We analyze the special nature of the [[supergeometry|supergeometry]] [[space of field histories]] of the [[Dirac field]] a little (prop. \ref{DiracSpaceOfFieldHistories}) below
and conclude by highlighting the crucial role of [[supergeometry]] (remark \ref{DiracFieldSupergeometric} below)
+-- {: .num_defn #DiracSpaceOfFieldHistories}
###### Proposition
**([[space of field histories]] of the [[Dirac field]])**
Let $E = \Sigma \times S_{odd} \overset{pr_1}{\to} \Sigma$ be the super-[[field bundle]] (def. \ref{FermionicBosonicFields}) for the [[Dirac field]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$ from example \ref{DiracFieldBundle}.
Then the corresponding [[supergeometry|supergeometric]] [[space of field histories]]
$$
\Gamma_\Sigma(\Sigma \times S_{odd})
\;\in\;
SuperSmoothSet
$$
from example \ref{SupergeometricSpaceOfFieldHistories} has the following properties:
1. For $U = \mathbb{R}^n$ an ordinary [[Cartesian space]] (with no super-geometric thickening, def. \ref{SuperCartesianSpace})
there is only a single $U$-parameterized collection of [[field histories]], hence a single plot
$$
\Psi_{(-)}\;\colon\;\mathbb{R}^n \overset{ 0 }{\longrightarrow} \Gamma_\Sigma(\Sigma \times S_{odd})
$$
and this corresponds to the [[zero section]], hence to the trivial [[Dirac field]]
$$
\Psi^A_{(-)} = 0
\,.
$$
1. {#OddParameterizedFieldHistories} For $U = \mathbb{R}^{n \vert 1}$ a [[super Cartesian space]] (\ref{SuperCartesianSpace}) with a single super-odd dimension, then $U$-parameterized collections of field histories
$$
\Phi_{(-)} \;\colon\; \mathbb{R}^{n\vert 1} \longrightarrow \Gamma_\Sigma(\Sigma \times S_{odd})
$$
are in [[natural bijection]] with plots of sections of the [[bosonic field|bosonic]]-field bundle with field fiber $S_{even} = S$ the [[spin representation]] regarded as an ordinary vector space:
$$
\Psi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(\Sigma \times S_{even})
\,.
$$
Moreover, these two kinds of plots determine the fermionic field space completely: It is in fact [[isomorphism|isomorphic]], as a [[super vector space]], to the bosonic field space shifted to odd degree (as in example \ref{SuperpointInducedByFiniteDimensionalVectorSpace}):
$$
\Gamma_\Sigma(\Sigma \times S_{odd})
\;\simeq\;
\left(
\Gamma_\Sigma(E\times S_{even})
\right)_{odd}
\,.
$$
=--
+-- {: .proof}
###### Proof
In the first case, the plot is a morphism of [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}) of the form
$$
\mathbb{R}^n \times \mathbb{R}^{p,1} \longrightarrow S_{odd}
\,.
$$
By definitions this is [[formal duality|dually]] homomorphism of real [[supercommutative superalgebras]]
$$
C^\infty(\mathbb{R}^n \times \mathbb{R}^{p,1})
\longleftarrow
\wedge^\bullet S^\ast
$$
from the [[Grassmann algebra]] on the [[dual vector space]] of the [[spin representation]] $S$ to the ordinary algebras of [[smooth functions]] on $\mathbb{R}^n \times \mathbb{R}^{p,1}$.
But the latter has no elements in odd degree, and hence all the Grassmann generators need to be
send to zero.
For the second case, notice that a morphism of the form
$$
\mathbb{R}^{n\vert 1}
\overset{\Phi_{(-)}}{\longrightarrow}
S_{odd}
$$
is by def. \ref{TangentBundleOdd} [[natural bijection|naturally identified]] with a morphism of the form
$$
\mathbb{R}^n
\overset{\Psi_{(-)}}{\longrightarrow}
[\mathbb{R}^{0 \vert 1}, S_{odd}]
\simeq
S_{odd} \times S_{even}
\,,
$$
where the identification on the right is from example \ref{SuperpointsMapping}.
By the [[universal property|nature]] of [[Cartesian products]] these morphisms in turn
are [[natural bijection|naturally identified]] with [[pairs]] of morphisms of the form
$$
\left(
\array{
\mathbb{R}^n &\overset{}{\longrightarrow}& S_{odd}\,,
\\
\mathbb{R}^n &\overset{}{\longrightarrow}& S_{even}
}
\right)
\,.
$$
Now, as in the first point above, here the first component is uniquely fixed
to be the [[zero morphism]] $\mathbb{R}^n \overset{0}{\to} S_{odd}$;
and hence only the second component is free to choose. This is precisely the claim to be shown.
=--
+-- {: .num_remark #DiracFieldSupergeometric}
###### Remark
**([[supergeometry|supergeometric]] nature of the [[Dirac field]])**
Proposition \ref{DiracSpaceOfFieldHistories} how two basic facts about the [[Dirac field]],
which may superficially seem to be in tension with each other, are properly unified by [[supergeometry]]:
1. On the one hand a [[field history]] $\Psi$ of the [[Dirac field]] is _not_ an ordinary section of an ordinary [[vector bundle]].
In particular its component functions $\psi^A$ anti-commute with each other, which is not the case for
ordinary functions, and this is crucial for the [[Lagrangian density]] of the Dirac field to be
well defined, we come to this below in example \ref{LagrangianDensityForDiracField}.
1. On the other hand a [[field history]] of the [[Dirac field]] is supposed to be a [[spinor]], hence
a [[section]] of a [[spinor bundle]], which is an ordinary [[vector bundle]].
Therefore prop. \ref{DiracSpaceOfFieldHistories} serves to shows how, even though a Dirac field is not defined to be an ordinary section of an ordinary vector bundle, it is nevertheless encoded by such an ordinary section:
One says that this ordinary section is a "[[superfield]]-component" of the Dirac field,
the one linear in a Grassmann variable $\theta$.
=--
$\,$
This concludes our discussion of the concept of _[[field (physics)|fields]]_ itself.
In the [following chapter](#FieldVariations) we consider the [[variational calculus]] of fields.
|
A first idea of quantum field theory -- Free quantum fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Free+quantum+fields | ## Free quantum fields
{#FreeQuantumFields}
In this chapter we discuss the following topics:
* _[Wick algebra](#WickAlgebraAbstract)_
* _[Time-ordered product](#AbstractTimeOrderedProduct)_
* _[Operator product notation](#OperatorProductAndNormalOrderedProduct)_
* _[Hadamard vacuum state](#HadamardVacuumStatesOnWickAlgebras)_
* _[Free quantum BV-differential](#FreeQuantumBVDifferential)_
* _[Schwinger-Dyson equation](#SchwingerDysonEquation)_
$\,$
In the [previous chapter](#Quantization) we discussed _[[quantization]]_ of linear [[phase spaces]], which turns the [[algebra of observables]] into a [[noncommutative algebra]] of [[quantum observables]]. Here we apply this to the [[covariant phase spaces]] of [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theories]] (as discussed in the chapter _[Gauge fixing](#GaugeFixing)_), obtaining genuine [[quantum field theory]] for [[free fields]].
For this purpose we first need to find a sub-algebra of all observables which is large enough to contain all [[local observables]] (such as the [[phi^n interaction]], example \ref{InWickAlgebraphinInteraction} below, and the [[electron-photon interaction]], example \ref{InWickAlgebraElectronPhotonInteraction} below) but small enough for the [[star product]] [[deformation quantization]] to meet [[Hörmander's criterion]] for absence of [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}). This does exist (example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal} below): It is called the algebra of _[[microcausal polynomial observables]]_ (def. \ref{MicrocausalObservable} below).
[[!include perturbative observables -- table]]
While the [[star product]] of the [[causal propagator]] still violates [[Hörmander's criterion]] for absence of [[UV-divergences]] on [[microcausal polynomial observables]], we have seen in the [previous chapter](#Quantization) that qantization freedom allows to shift this [[Poisson tensor]] by a symmetric contribution. By prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} such a shift is provided by passage from the [[causal propagator]] to the [[Wightman propagator]], and by prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} this reduces the [[wave front set]] and hence the UV-singularities "by half".
This way the [[deformation quantization]] of the [[Peierls-Poisson bracket]] exists on [[microcausal polynomial observables]] as the [[star product]] algebra induced by the [[Wightman propagator]]. The resulting [[non-commutative algebra|non-commutative]] [[algebra of observables]] is called the _[[Wick algebra]]_ (prop. \ref{MoyalStarProductOnMicrocausal} below).
Its algebra structure may be expressed in terms of a commutative "[[normal-ordered product]]" (def. \ref{NormalOrderedProductNotation} below) and
the [[vacuum expectation values]] of [[field observables]] in a canonically induced [[vacuum state]] (prop. \ref{WickAlgebraCanonicalState} below).
The analogous [[star product]] induced by the [[Feynman propagator]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below) acts by first [[causal ordering]] its arguments and then multiplying them with the [[Wick algebra|Wick algebra product]] (prop. \ref{CausalOrderingTimeOrderedProductOnRegular} below) and hence is called the _[[time-ordered product]]_ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below). This is the key structure in the discussion of [[interacting field theory]] discussed in the next chapter _[Interacting quantum fields](#InteractingQuantumFields)_.
Here we consider this on [[regular polynomial observables]] only, hence for averages of [[field observables]] that evaluate
at distinct [[spacetime]] points. The [[extension]] of the [[time-ordered product]] to
[[local observables]] is possible, but requires making choices: This is called _[[renormalization]]_, which we turn to in
the chapter _[Renormalization](#Renormalization)_ below.
[[!include Wick algebra -- table]]
While the [[Wick algebra]] with its [[vacuum state]] provides a [[quantization]] of the [[algebra of observables]]
of [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theories]], the possible existence of
[[infinitesimal gauge symmetries]] implies that the physically relevant observables are
just the [[gauge invariance|gauge invariant]] [[on-shell]] ones, exhibited by the [[cochain cohomology]] of the [[BV-BRST differential]] $\{-S' + S'_{BRST}, (-)\}$. Hence to complete [[quantization]]
of [[gauge theories]], the [[BV-BRST differential]] needs to be lifted to the [[noncommutative algebra]] of [[quantum observables]] --
this is called _[[BV-BRST quantization]]_.
To do so, we may regard the [[gauge fixing|gauge fixed]] [[BRST complex|BRST]]-[[action functional]] $S'_{BRST}$ as an [[interaction]] term,
to be dealt with later via [[scattering theory]], and hence consider quantization of just the free BV-differential $\{-S',(-)\}$.
One finds that this is equal to its [[time-ordered product|time-ordered]] version $\{-S',(-)\}_{\mathcal{T}}$ (prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} below) plus a quantum correction, called the _[[BV-operator]]_ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} below) or _[[BV-Laplacian]]_ (prop. \ref{ComponentsBVOperator} below).
Applied to [[observables]] this relation is the _[[Schwinger-Dyson equation]]_ (prop. \ref{DysonSchwinger} below), which expresses
the [[quantum physics|quantum]]-correction to the [[equations of motion]] of the [[free field theory|free]] [[gauge fixing|gauge field]] [[Lagrangian field theory]] as seen by [[time-ordered products]] of [[observables]] (example \ref{SchwingerDysonDistributional} below.)
After introducing [[field (physics)|field]]-[[interactions]] via [[scattering theory]] in the [next chapter](#InteractingQuantumFields)
the quantum correction to the [[BV-differential]] by the [[BV-operator]] becomes the "[[quantum master equation]]"
and the [[Schwinger-Dyson equation]] becomes the "[[master Ward identity]]". When choosing [[renormalization]] these
identities become _conditions_ to be satisfied by [[renormalization]] choices in order for the
interacting quantum BV-BRST differential, and hence for [[gauge invariance|gauge invariant]] quantum observables, to be well defined in
[[perturbative quantum field theory]] of [[gauge theories]]. This we discuss below in _[Renormalization](#Renormalization)_.
$\,$
**[[Wick algebra]]**
{#WickAlgebraAbstract}
The abstract [[Wick algebra]] of a [[free field theory]] with [[Green hyperbolic differential equation]] is directly analogous to the [[star product]]-algebra induced by a [[finite dimensional vector space|finite dimensional]] [[Kähler vector space]] (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace}) under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]:
+-- {: .num_remark #WightmanPropagatorAsKaehlerVectorSpaceStructure}
###### Remark
**([[Wightman propagator]] as [[Kähler vector space]]-[[structure]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[Green hyperbolic differential equation]]. Then the corresponding [[Wightman propagator]] is analogous to the rank-2 tensor on a [[Kähler vector space]] as follows:
| [[covariant phase space]] <br/> of [[free field theory|free]] [[Green hyperbolic differential equation|Green hyperbolic]] <br/> [[Lagrangian field theory]] | [[finite dimensional vector space|finite dimensional]] <br/> [[Kähler vector space]] |
|-------------------|-------------------------|
| [[space of field histories]] <br/> $\Gamma_\Sigma(E)$ | $\mathbb{R}^{2n}$ |
| [[symplectic form]] <br/> $\tau_{\Sigma_p} \Omega_{BFV}$ | [[Kähler form]] $\omega$ |
| [[causal propagator]] $\Delta$ | $\omega^{-1}$ |
| [[Peierls-Poisson bracket]] <br/> $\{A_1,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)$ | [[Poisson bracket]] |
| [[Wightman propagator]] <br/> $\Delta_H = \tfrac{i}{2} \Delta + H$ | [[Hermitian form]] <br/> $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ |
=--
([Fredenhagen-Rejzner 15, section 3.6](pAQFT#FredenhagenRejzner15), [Collini 16, table 2.1](pAQFT#Collini16))
+-- {: .num_defn #MicrocausalObservable}
###### Definition
**([[microcausal observable|microcausal]] [[polynomial observables]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] which is a [[vector bundle]], over some [[spacetime]] $\Sigma$.
A [[polynomial observable]] (def. \ref{PolynomialObservables})
$$
\begin{aligned}
A
& = \phantom{+}
\alpha^{(0)}
\\
&
\phantom{=}
+
\int_{\Sigma} \mathbf{\Phi}^a(x) \alpha^{(1)}_a(x) \, dvol_\Sigma(x)
\\
&
\phantom{=}
+
\int_{\Sigma^2}
\mathbf{\Phi}^{a_1}(x_1) \cdot \mathbf{\Phi}^{a_2}(x_2)
\alpha^{(2)}_{a_1 a_2}(x_1, x_2) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
\\
&
\phantom{=}
+
\int_{\Sigma^3}
\mathbf{\Phi}^{a_1}(x_1)
\cdot
\mathbf{\Phi}^{a_2}(x_2)
\cdot
\mathbf{\Phi}^{a_3}(x_3)
\alpha^{(3)}_{a_1 a_2 a_3}(x_1,x_2,x_3)
\, dvol_\Sigma(x_1) dvol_\Sigma(x_2) dvol_\Sigma(x^3)
\\
&
\phantom{=}
+
\cdots
\,.
\end{aligned}
$$
is called _[[microcausal polynomial observable|microcausal]]_ if each [[distribution|distributional]] [[coefficient]]
$$
\alpha^{(k)}
\;\in\;
\Gamma'_{\Sigma^k}(E^{\boxtimes^k})
$$
as above has [[wave front set]] (def. \ref{WaveFrontSet}) _not_ containing those elements $(x_1, \cdots x_k, k_1, \cdots k_k)$ where the $k$ [[wave vectors]] are all in the [[closed future cone]] or all in the [[closed past cone]] (def. \ref{CausalPastAndFuture}).
We write
$$
\array{
PolyObs(E)_{mc}
&\hookrightarrow&
PolyObs(E)
\\
PolyObs(E,\mathbf{L})_{mc}
\simeq
PolyObs(E)_{mc}/im(P)
&\hookrightarrow&
PolyObs(E,\mathbf{L})
}
$$
for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal polynomial observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]].
=--
The important point is that [[microcausal polynomial observables]] still contain all [[regular polynomial observables]] but also all polynomial [[local observables]]:
+-- {: .num_example #MicrocausalRegularObservables}
###### Example
**([[regular polynomial observables]] are [[microcausal observables|microcausal]])**
Every [[regular polynomial observable]] (def. \ref{PolynomialObservables}) is [[microcausal polynomial observable|microcausal]] (def. \ref{MicrocausalObservable}).
=--
+-- {: .proof}
###### Proof
By definition of regular polynomial observables, their [[coefficients]] are [[non-singular distributions]] and because the [[wave front set]] of [[non-singular distributions]] is [[empty set|empty]] (example \ref{NonSingularDistributionTrivialWaveFrontSet})
=--
+-- {: .num_example #PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}
###### Example
**(polynomial [[local observables]] are [[microcausal polynomial observables|microcausal]])**
Every polynomial [[local observable]] (def. \ref{LocalObservables}) is a [[microcausal polynomial observable]] (def. \ref{MicrocausalObservable}).
=--
+-- {: .proof}
###### Proof
For notational convenience, consider the case of the [[scalar field]] with $k = 2$; the general case is directly analogous. Then the [[local observable]] coming from $\phi^2$ (a [[phi^n interaction]]-term), has, regarded as a [[polynomial observable]], the [[delta distribution]] $\delta(x_1-x_2)$ as [[coefficient]] in degree 2:
$$
\begin{aligned}
A(\Phi)
& =
\underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x)
\\
& =
\underset{\Sigma \times \Sigma}{\int}
\underset{ = \alpha^{(2)}}{
\underbrace{
g(x_1)
\delta(x_1 - x_2)
}}
\,
\Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
\end{aligned}
\,.
$$
Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a [[chart]] around this point, the [[Fourier transform of distributions]] of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors:
$$
\begin{aligned}
(k_1, k_2)
& \mapsto
\underset{\mathbb{R}^{2n}}{\int}
g(x_1)
\delta(x_1, x_2)
e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )}
\, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
\\
& \propto \hat g(k_1 + k_2)
\end{aligned}
\,.
$$
Since $g$ is a plain [[bump function]], its [[Fourier transform]] $\hat g$ is quickly decaying (according to prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) along $k_1 + k_2$, as long as $k_1 + k_2 \neq 0$. Only on the [[cone]] $k_1 + k_2 = 0$ the Fourier transform is [[constant function|constant]], and hence in particular not decaying.
<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/WaveFrontSetOfDeltaDistributionInTwoVariables.png" width="200"/>
</div>
{#WaveFrontOfdeltaxy} This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the [[closed future cone]] or both in the [[closed past cone]] precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is [[microcausal observable|microcausal]].
> (graphics grabbed from [Khavkine-Moretti 14, p. 45](microcausal+polynomial+observable#KhavkineMoretti14))
=--
+-- {: .num_prop #MoyalStarProductOnMicrocausal}
###### Proposition
**([[Hadamard distribution|Hadamard]]-[[Moyal star product]] on [[microcausal observables]] -- [[abstract Wick algebra]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$. Write $\Delta$ for the [[causal propagator]] and let
$$
\Delta_H
\;=\;
\tfrac{i}{2}\Delta + H
$$
be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[star product]] induced by $\Delta_H$
$$
A \star_H A
\;\coloneqq\;
prod
\circ
\exp\left(
\int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)}
\otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g
\right)
(P_1 \otimes P_2)
$$
on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ (def. \ref{MicrocausalObservable}) is well defined in that the [[wave front sets]] involved in the [[products of distributions]] that appear in expanding out the [[exponential]] satisfy [[Hörmander's criterion]].
Hence by the general properties of [[star products]] (prop. \ref{AssociativeAndUnitalStarProduct}) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]]
$$
\left(
PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H
\right)
\,.
$$
This is the _[[off-shell]] [[Wick algebra]]_ corresponding to the choice of [[Wightman propagator]] $H$.
Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the [[on-shell]] [[microcausal observables]] to yield the _[[on-shell]] [[Wick algebra]]_
$$
\left(
PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H
\right)
\,.
$$
Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that
$$
\left(
A_1 \star_H A_2
\right)^\ast
=
A_2^\ast \star_H A_1^\ast
\,.
$$
=--
(e.g. [Collini 16, p. 25-26](Wick+algebra#Collini16))
+-- {: .proof}
###### Proof
By prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} the [[wave front set]] of $\Delta_H$ has all cotangents on the first variables in the [[closed future cone]] (at the given base point, which itself is on the [[light cone]])
<center>
<img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60">
</center>
and hence all those on the second variables in the [[closed past cone]].
The first variables are integrated against those of $A_1$ and the second against $A_2$. By definition of [[microcausal observables]] (def. \ref{MicrocausalObservable}), the wave front sets of $A_1$ and $A_2$ are disjoint from the subsets where all components are in the [[closed future cone]] or all components are in the [[closed past cone]]. Therefore the relevant sum of of the wave front covectors never vanishes and hence [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) for partial [[products of distributions|products of]] [[distributions of several variables]] (prop. \ref{PartialProductOfDistributionsOfSeveralVariables}).
It remains to see that the star product $A_1 \star_H A_2$ is itself again a [[microcausal observable]]. It is clear that it is again a [[polynomial observable]] and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the [[wave front set]] follows directly from the fact that the wave front set of a [[product of distributions]] is inside the fiberwise sum of elements of the factor wave front sets (prop. \ref{WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}, prop. \ref{PartialProductOfDistributionsOfSeveralVariables}).
Finally the [[star algebra]]-structure via [[complex conjugation]] follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in prop. \ref{StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}.
=--
+-- {: .num_remark #WickAlgebraIsFormalDeformationQuantization}
###### Remark
**([[Wick algebra]] is [[formal deformation quantization]] of [[Poisson-Peierls bracket|Poisson-Peierls algebra of observables]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$ with [[causal propagator]] $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$ be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).
Then the [[Wick algebra]] $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. \ref{MoyalStarProductOnMicrocausal} is a [[formal deformation quantization]] of the [[Poisson algebra]] on the [[covariant phase space]] given by the [[on-shell]] [[polynomial observables]] equipped with the [[Poisson-Peierls bracket]] $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have
$$
A_1 \star_H A_2
\;=\;
A_1 \cdot A_2
\;mod\;
\hbar
$$
and
$$
A_1 \star_H A_2
-
A_2 \star_H a_1
\;=\;
i \hbar \{A_1, A_2\}
\;mod\;
\hbar^2
\,.
$$
=--
([Dito 90](Wick+algebra#Dito90), [Dütsch-Fredenhagen 00](Wick+algebra#DuetschFredenhagen00), [Dütsch-Fredenhagen 01](Wick+algebra#DuetschFredenhagen01), [Hirshfeld-Henselder 02](Wick+algebra#HirschfeldHenselder02))
+-- {: .proof}
###### Proof
By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] (example \ref{MoyalStarProductIsFormalDeformationQuantization}).
Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then
$$
\begin{aligned}
& A_1 \star_H A_2
\\
& =
\phantom{+} A_1 A_2
\\
& \phantom{=}
+
\hbar
\int
\left(
\tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2)
\right)
\frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)}
\frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)}
\;mod\;
\hbar^2
\\
& =
\phantom{+} A_1 A_2
\\
& \phantom{=}
+
\hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\left(
\tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2)
+ H^{a_1 a_2}(x_1, x_2)
\right)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\end{aligned}
$$
Now since $\Delta$ is skew-symmetric while $H$ is symmetric (prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) it follows that
$$
\begin{aligned}
A_1 \star_H A_2
-
A_2 \star_H A_1
& =
i \hbar
\left(
\int
(\alpha_1)_{a_1}(x_1)
\Delta^{a_1 a_2}(x_1, x_2)
(\alpha_2)_{a_2}(x_2)
\right)
\;mod\;
\hbar^2
\\
& =
i \hbar \, \left\{ A_1, A_2\right\}
\end{aligned}
\,.
$$
The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line.
=--
$\,$
**[[time-ordered product]]**
{#AbstractTimeOrderedProduct}
+-- {: .num_defn #OnRegularPolynomialObservablesTimeOrderedProduct}
###### Definition
**([[time-ordered product]] on [[regular polynomial observables]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].
Then the _[[time-ordered product]]_ on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via prop. \ref{PropagatorStarProduct}):
$$
\array{
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
\\
(A_1, A_2)
&\mapsto&
\phantom{\coloneqq} A_1 \star_F A_2
}
$$
hence
$$
A_1 \star_F A_2
\;
\coloneqq
\;
((-)\cdot(-))
\circ
\exp\left(
\underset{\Sigma \times \Sigma}{\int}
\Delta_F^{a b}(x,y)
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}^b(y)}
\,
dvol_\Sigma(x)
\,
dvol_\Sigma(y)
\right)
$$
(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the _homogeneous_ [[equations of motion]].)
=--
+-- {: .num_prop #CausalOrderingTimeOrderedProductOnRegular}
###### Proposition
**([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].
Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]]
$$
A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
$$
with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have
$$
A_1 \star_F A_2
\;=\;
\left\{
\array{
A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\,.
$$
Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation ("$S_1$ does not intersect the [[past cone]] of $S_2$"). Beware that for general [[pairs]] $(S_1, S_2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$.
=--
+-- {: .proof}
###### Proof
Recall the following facts:
1. the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively
$$
supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
$$
1. they turn into each other under exchange of their arguments (cor. \ref{CausalPropagatorIsSkewSymmetric}):
$$
\Delta_\pm(y,x) = \Delta_{\mp}(x,y)
\,.
$$
1. the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}):
$$
H(x,y) = H(y,x)
$$
Using this we compute as follows:
$$
\begin{aligned}
A_1 \underset{\Delta_{F}}{\star} A_2
\;
& =
A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\\
& =
\left\{
\array{
A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
\\
A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
}
\right.
\end{aligned}
$$
=--
+-- {: .num_prop #IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}
###### Proposition
**([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)
The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is [[isomorphism|isomorphic]] to the pointwise product of [[observables]] (def. \ref{Observable}) via the [[linear isomorphism]]
$$
\mathcal{T}
\;\colon\;
PolyObs(E)_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E)_{reg}[ [\hbar] ]
$$
given by
$$
\label{OnregularPolynomialObservablesPointwiseTimeOrderedIsomorphism}
\mathcal{T}A
\;\coloneqq\;
\exp\left(
\tfrac{1}{2}
\hbar
\underset{\Sigma}{\int}
\Delta_F(x,y)^{a b}
\frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
\right)
A
$$
in that
$$
\begin{aligned}
T(A_1 A_2)
& \coloneqq
A_1 \star_{F} A_2
\\
& =
\mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) )
\end{aligned}
$$
hence
$$
\array{
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{(-)\cdot (-)}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
\\
{}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow
&&
\Big\downarrow{}^{\mathrlap{\mathcal{T}}}_\simeq
\\
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg}[ [\hbar] ]
&\overset{(-) \star_F (-)}{\longrightarrow}&
PolyObs(E)_{reg}[ [\hbar] ]
}
$$
=--
([Brunetti-Dütsch-Fredenhagen 09, (12)-(13)](time-ordered+product#BrunettiDuetschFredenhagen09), [Fredenhagen-Rejzner 11b, (14)](time-ordered+product#FredenhagenRejzner11b))
+-- {: .proof}
###### Proof
Since the [[Feynman propagator]] is symmetric (prop. \ref{SymmetricFeynmanPropagator}), the statement is a special case of prop. \ref{SymmetricContribution}.
=--
+-- {: .num_example #RegularObservablesExponentialTimeOrdered}
###### Example
**([[time-ordered product|time-ordered]] [[exponential]] of [[regular polynomial observables]])**
Let $V \in PolyObs_{reg, deg = 0}[ [ \hbar ] ]$ be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero, and write
$$
\exp(V)
=
1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots
$$
for the [[exponential]] of $V$ with respect to the pointwise product (eq:ObservablesPointwiseProduct).
Then the [[exponential]] $\exp_{\mathcal{T}}(V)$ of $V$ with respect to the [[time-ordered product]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equal to the [[conjugation]] of the exponential with respect to the
pointwise product by the time-ordering isomorphism $\mathcal{T}$ from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}:
$$
\begin{aligned}
\exp_{\mathcal{T}}(V)
&
\coloneqq
1 + V + \tfrac{1}{2} V \star_F V + \tfrac{1}{3!} V \star_F V \star_F V + \cdots
\\
& =
\mathcal{T} \circ \exp(-) \circ \mathcal{T}^{-1}(V)
\,.
\end{aligned}
$$
=--
+-- {: .num_remark }
###### Remark
**([[renormalization]] of [[time-ordered product]])**
The [[time-ordered product]] on [[regular polynomial observables]] from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]] (def. \ref{LocalObservables}), then taking values in [[microcausal observables]] (def. \ref{MicrocausalObservable}):
$$
T
\;\colon\;
PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
\longrightarrow
PolyObs(E)_{mc}[ [\hbar] ]
\,.
$$
This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of _[[renormalization scheme]]_ for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory, namely an extension of the time-ordered exponential $\exp_{\mathcal{T}}$ (example \ref{RegularObservablesExponentialTimeOrdered}) from regular to local observables.
This construction of [[perturbative quantum field theory]] is called _[[causal perturbation theory]]_. We discuss this below in the chapters _[Interacting quantum fields](#InteractingQuantumFields)_ and _[Renormalization](#Renormalization)_.
=--
$\,$
**operator product notation**
{#OperatorProductAndNormalOrderedProduct}
+-- {: .num_defn #NormalOrderedProductNotation}
###### Definition
**(notation for [[operator product]] and [[normal-ordered product]])**
It is traditional to use the following alternative notation for the product structures on [[microcausal polynomial observables]]:
1. The [[Wick algebra]]-product, hence the [[star product]] $\star_H$ for the [[Wightman propagator]] (def. \ref{MoyalStarProductOnMicrocausal}), is rewritten as plain juxtaposition:
$$
\text{"operator product"}
\phantom{AAA}
A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1
\phantom{AAAA}
\array{
\text{star product of}
\\
\text{Wightman propagator}
}
\,.
$$
1. The pointwise product of observables (def. \ref{Observable}) $A_1 \cdot A_2$
is equivalently written as plain juxtaposition enclosed by colons:
$$
\array{
\text{"normal-ordered}
\\
\text{product"}
}
\phantom{AAAA}
:A_1 A_2:
\phantom{AA}\coloneqq\phantom{AA}
A_1 \cdot A_2
\phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}
$$
1. The [[time-ordered product]], hence the [[star product]] for the [[Feynman propagator]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equivalently written as plain juxtaposition prefixed by a "$T$"
$$
\array{
\text{"time-ordered}
\\
\text{product"}
}
\phantom{AAAA}
T(A_1 A_2)
\phantom{AA}\coloneqq\phantom{AA}
A_1 \star_F A_2
\phantom{AAAA}
\array{
\text{star product of}
\\
\text{Feynman propagator}
}
$$
Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]]
the first product becomes the _[[operator product]]_, while the second becomes the operator poduct applied after
suitable re-ordering, called "[[normal-ordered product|normal odering]]" of the factors.
Disregarding the [[Fock space]]-representation, which is [[faithful representation|faithful]], we may still
refer to these "abstract" products as the "operator product" and the "normal-ordered product", respectively.
=--
$\,$
+-- {: .num_example #InWickAlgebraphinInteraction}
###### Example
**([[phi^n interaction]])**
Consider [[phi^n theory]] from example \ref{phintheoryLagrangian}.
The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[phi^n interaction]] is the following [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable:
$$
\begin{aligned}
S_{int}
& =
\underset{\Sigma}{\int}
\underset{
n \, \text{factors}
}{
\underbrace{
\mathbf{\Phi}(x)
\cdot
\mathbf{\Phi}(x)
\cdots
\mathbf{\Phi}(x)
\cdot
\mathbf{\Phi}(x)
}
}
\,
dvol_\Sigma(x)
\\
& =
\underset{\Sigma}{\int}
:
\underset{
n \, \text{factors}
}{
\underbrace{
\mathbf{\Phi}(x)
\mathbf{\Phi}(x)
\cdots
\mathbf{\Phi}(x)
\mathbf{\Phi}(x)
}
}
:
\,
dvol_\Sigma(X)
\end{aligned}
\,,
$$
Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}.
=--
+-- {: .num_example #InWickAlgebraElectronPhotonInteraction}
###### Example
**([[electron-photon interaction]])**
Consider the [[Lagrangian field theory]] defining [[quantum electrodynamics]] from example \ref{LagrangianQED}.
The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[electron-photon interaction]] is the [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable
$$
\begin{aligned}
S_{int}
& \coloneqq
i
\underset{\Sigma}{\int}
g_{sw}(x)
\,
(\Gamma^\mu)^\alpha{}_\beta
\,
\overline{\mathbf{\Psi}}_\alpha(x)
\cdot
\mathbf{\Psi}^\beta(x)
\cdot
\mathbf{A}_\mu(x)
\,
dvol_\Sigma(x)
\\
& =
i
\underset{\Sigma}{\int}
g_{sw}(x)
\,
(\Gamma^\mu)^\alpha{}_\beta
\,
:
\overline{\mathbf{\Psi}}_\alpha(x)
\mathbf{\Psi}^\beta(x)
\mathbf{A}_\mu(x)
:
\,
dvol_\Sigma(x)
\end{aligned}
\,,
$$
Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}.
=--
(e.g. [Scharf 95, (3.3.1)](electron-photon+interaction#Scharf95))
$\,$
**[[Hadamard vacuum state]]**
{#HadamardVacuumStatesOnWickAlgebras}
+-- {: .num_prop #WickAlgebraCanonicalState}
###### Proposition
**(canonical [[Hadamard vacuum state|vacuum]] [[state on a star-algebra|states]] on abstract [[Wick algebra]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].
For
$$
\Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$
any [[on-shell]] [[field history]] (i.e. solving the [[equations of motion]]), consider the function from the [[Wick algebra]] to [[formal power series]] in $\hbar$ with [[coefficients]] in the [[complex numbers]]
which evaluates any [[microcausal polynomial observable]] on $\Phi_0$
$$
\array{
PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
&\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}&
\mathbb{C}[ [\hbar] ]
\\
A &\mapsto& A(\Phi_0)
}
$$
Specifically for $\Phi_0 = 0$ (which is a solution of the [[equations of motion]] by the assumption that $(E,\mathbf{L})$ defines a [[free field theory]]) this is the function
$$
\array{
PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
&\overset{\langle -\rangle_0}{\longrightarrow}&
\mathbb{C}[ [\hbar] ]
\\
\left.
\begin{aligned}
A & = \alpha^{(0)}
\\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
\\
& \phantom{=} + \cdots
\end{aligned}
\right\}
&\mapsto&
A(0) = \alpha^{(0)}
}
$$
which sends each [[microcausal polynomial observable]] to its value $A(\Phi = 0)$ on the zero [[field history]], hence to the constant contribution $\alpha^{(0)}$ in its [[polynomial]] expansion.
The function $\langle -\rangle_0$ is
1. [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$;
1. real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$
$$
\langle A^\ast \rangle = \langle A \rangle^\ast
$$
1. positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that
$$
\langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A
\,,
$$
1. normalized, in that
$$
\langle 1\rangle_H = 1
$$
where $(-)^\ast$ denotes componet-wise [[complex conjugation]].
This means that $\langle -\rangle_{0}$ is a [[state on a star-algebra|state]] on the [[Wick algebra|Wick]] [[star-algebra]] $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. \ref{MoyalStarProductOnMicrocausal}). One says that
* $\langle - \rangle_0$ is a _[[Hadamard vacuum state]]_;
and generally
* $\langle - \rangle_{\Phi_0}$ is called a _[[coherent state]]_.
=--
([Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)](#Duetsch18))
+-- {: .proof}
###### Proof
The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a [[representation]] of the Wick algebra on a [[Fock space|Fock]] [[Hilbert space]] (this algebra [[homomorphism]] is _[[Wick's lemma]]_), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the [[inner product]] of the [[Hilbert space]].
=--
+-- {: .num_example #HadamardMoyalStarProductOfTwoLinearObservables}
###### Example
**([[operator product]] of two [[linear observables]])**
Let
$$
A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}
$$
for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by
$$
A_i
\;=\;
\int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i)
\,.
$$
Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product
with their value
$$
\label{EvaluatingLinearObservablesInWightmanPropagator}
\langle A_1 \star_H A_2 \rangle_0
\;\coloneqq\;
i \hbar \int \int
(\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
\,dvol_\Sigma(x_1)
\,dvol_\Sigma(x_2)
$$
in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}:
$$
A_1 \star_H A_2
\;=\;
A_1 \cdot A_2
\;+\;
\langle A_1 \star_H A_2 \rangle_0
$$
In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads
$$
A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle
\,.
$$
=--
+-- {: .num_example #WeylRelations}
###### Example
**([[Weyl relations]])**
Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] and with [[Wightman propagator]] $\Delta_H$.
Then for
$$
A_1, A_2
\;\in\;
LinObs(E,\mathbf{L})_{mc}
\hookrightarrow
PolyObs(E,\mathbf{L})_{mc}
$$
two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]:
$$
e^{A_1}
\star_H
e^{A_2}
\;=\;
e^{A_1 + A_2} \; e^{\langle A_1 \star_H A_2\rangle_0}
$$
where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
=--
(e.g. [Dütsch 18, exercise 2.3](Wick+algebra#Duetsch18))
+-- {: .num_example }
###### Example
**([[Wightman propagator]] is [[2-point function]] in the [[Hadamard vacuum state]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].
With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Wightman propagator]] $\Delta_H(x,y)$ itself is the _[[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]:
$$
\left\langle
\mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y)
\right\rangle_0
\;=\;
\underset{ = 0 }{
\underbrace{
\left\langle
\mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
\right\rangle
}}
+
\underset{ = \hbar \Delta^{a b}_H(x,y) }{
\underbrace{
\left \langle
\hbar
\underset{\Sigma \times \Sigma}{\int}
\delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y')
\right\rangle
}}
$$
by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:
$$
\left\langle
\mathbf{\Phi}^a(x)
\mathbf{\Phi}^b(y)
\right\rangle_0
\;=\;
\hbar \Delta_H(x,y)
\,.
$$
=--
Similarly:
+-- {: .num_example }
###### Example
**([[Feynman propagator]] is time-ordered [[2-point function]] in the [[Hadamard vacuum state]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]] with induced [[Feynman propagator]] $\Delta_F$.
With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Feynman propagator]] $\Delta_F(x,y)$ itself is the _time-ordered [[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]:
$$
\left\langle
T\left(
\mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y)
\right)
\right\rangle_0
\;=\;
\underset{ = 0 }{
\underbrace{
\left\langle
\mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
\right\rangle
}}
+
\underset{ = \hbar \Delta^{a b}_H(x,y) }{
\underbrace{
\left \langle
\hbar
\underset{\Sigma \times \Sigma}{\int}
\delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y')
\right\rangle
}}
$$
analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.
Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:
$$
\left\langle
T\left(
\mathbf{\Phi}^a(x)
\mathbf{\Phi}^b(y)
\right)
\right\rangle_0
\;=\;
\hbar \Delta_F(x,y)
\,.
$$
=--
[[!include propagators - table]]
$\,$
**[[BV-operator|free quantum BV-differential]]**
{#FreeQuantumBVDifferential}
So far we have discussed the plain (graded-commutative) [[algebra of quantum observables]] of a [[gauge fixing|gauged fixed]] [[free field theory|free]] [[Lagrangian field theory]], [[deformation quantization|deforming]] the commutative pointwise product of [[observables]]. But after [[gauge fixing]], the algebra of observables is not just a (graded-commutative) algebra, but carries also a [[differential]] making it a [[differential graded-commutative superalgebra]]: the global [[BV-differential]] $\{-S' + S_{BRST}, -\}$ (def. \ref{ComplexBVBRSTGlobal}). The [[gauge invariance|gauge invariant]] [[on-shell]] [[observables]] are (only) the [[cochain cohomology]] of this differential. Here we discuss what becomes of this differential as we pass to the non-commutative [[Wick algebra|Wick]]-[[algebra of quantum observables]].
+-- {: .num_prop #OnMicrocausalObservablesGlobalBVDifferential}
###### Proposition
**(global [[BV-differential]] on [[Wick algebra]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). Let $\Delta_H$ be a compatible [[Wightman propagator]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}).
Then the global [[BV-differential]] $\{-S',(-)\}$ (def. \ref{ComplexBVBRSTGlobal}) restricts from [[polynomial observables]] to a linear map on [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable})
$$
\{-S',(-)\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
$$
and as such is a [[derivation]] not only for the pointwise product, but also for the product in the [[Wick algebra]] (the [[star product]] induced by the [[Wightman propagator]]):
$$
\{-S', A_1 \star_H A_2\}
\;=\;
\{-S', A_1\} \star_H A_2
+
A_1 \star_H \{-S', A_2\}
\,.
$$
We call $\{-S,(-)\}$ regarded as a nilpotent derivation on the [[Wick algebra]] this way the _free quantum [[BV-differential]]_.
=--
([Fredenhagen-Rejzner 11b, below (37)](BV-differential#FredenhagenRejzner11b), [Rejzner 11, below (5.28)](BV-differential#Rejzner11))
+-- {: .proof}
###### Proof
By example \ref{BVDifferentialGlobal} the action of $\{-S',(-)\}$ on polyomial observables is to replace [[antifield]] [[field observables]] by
$$
\mathbf{\Phi}^\ddagger_a(x)
\;\mapsto\;
\pm
(P_{A B}\mathbf{\Phi}^A)(x)
\,,
$$
where $P$ is a [[differential operator]]. By [[partial integration]] this translates to $\{-S',(-)\}$ acting by the [[formally adjoint differential operator]] $P^\ast$ (def. \ref{FormallyAdjointDifferentialOperators}) via [[derivative of distributions|distributional derivative]] on the [[distribution|distributional]] [[coefficients]] of the given polynomial observable.
Now by prop. \ref{RetainsOrShrinksWaveFrontSetDifferentialOperator} the application of $P^\ast$ retains or shrinks the [[wave front set]] of the distributional coefficient, hence it preserves the microcausality condition (def. \ref{MicrocausalObservable}). This makes $\{-S',(-)\}$ restrict to microcausal polynomial observables.
To see that $\{-S',(-)\}$ thus restricted is a [[derivation]] of the Wick algebra product, it is sufficient to see that its [[commutators]] with the [[Wightman propagator]] vanish in each argument:
$$
\left[
\{-S',(-)\} \otimes id
\;,\;
\Delta_H
\left(
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\right]
\;=\;
0
$$
and
$$
\left[
id \otimes \{-S',(-)\}
\;,\;
\Delta_H
\left(
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\right]
\;=\;
0
\,.
$$
Because with this we have:
$$
\begin{aligned}
\{-S', A_1 \star_H A_2\}
& =
\{-S',(-)\}
\circ
((-)\cdot(-))
\circ
\exp\left(
\hbar
\Delta_H\left(
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\right)
(A_1 \otimes A_2)
\\
& =
((-)\cdot(-))
\circ
\left(
\phantom{a \atop a}
\{-S',-\} \otimes id
+
id \otimes \{-S',(-)\}
\right)
\circ
\exp\left(
\hbar
\Delta_H\left(
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\right)
(A_1 \otimes A_2)
\\
& =
((-)\cdot(-))
\circ
\exp\left(
\hbar
\Delta_H\left(
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\right)
\circ
\left(
\phantom{a \atop b}
\{-S',-\} \otimes id
+
id \otimes \{-S',(-)\}
\right)
(A_1 \otimes A_2)
\\
& =
\{-S',A_1\} \star_H A_2
+
A_1 \star_H \{-S', A_2\}
\end{aligned}
$$
Here in the first step we used that $\{-S',(-)\}$ is a derivation with respect to the pointwise product, by construction (def. \ref{ComplexBVBRSTGlobal}) and then we used the vanishing of the above commutators.
To see that these commutators indeed vanish, use that by example \ref{BVDifferentialGlobal} we have
$$
\begin{aligned}
&
\left[
\{-S',(-)\} \otimes id
\;,\;
\Delta_H\left( \frac{\delta }{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right)
\right]
\\
& =
\left[
\underset{A}{\sum} (-1)^{deg(\phi^A)}
\underset{\Sigma}{\int}
(P_{A B}\mathbf{\Phi}^A)(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \otimes id
\,
dvol_\Sigma(x)
\;\,\;
\underset{\Sigma \times \Sigma}{\int}
\Delta_H^{A B}(x,y)
\frac{\delta}{\delta \mathbf{\Phi}^A(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}^B(y)}
\, dvol_\Sigma(x)
\, dvol_\Sigma(y)
\right]
\\
& =
-\underset{a}{\sum} (-1)^{deg(\phi^A)}
\underset{\Sigma \times \Sigma}{\int}
\underset{
= 0
}{
\underbrace{
(P_x \Delta_H)_A{}^B(x,y)
}
}
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}^B(y)}
\, dvol_\Sigma(x)
\, dvol_\Sigma(y)
\\
& = 0
\end{aligned}
$$
and similarly for the other order of the tensor products. Here the term over the brace vanishes by the fact that the Wightman propagator is a solution to the homogeneous equations of motion by prop. \ref{OnMinkowskiWightmanIsDistributionalSolutionToKleinGordon}.
=--
To analyze the behaviour of the free quantum BV-differential in general and specifically after passing to [[interacting field theory]] (below in chapter _[Interacting quantum fields](#InteractingQuantumFields)_) it is useful to re-express it in terms of the incarnation of the global [[antibracket]] with respect not to the pointwise product of observables, but the [[time-ordered product]]:
+-- {: .num_defn #AntibracketTimeOrdered}
###### Definition
**([[time-ordered product|time-ordered]] [[antibracket]])
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).
Then the _time-ordered global [[antibracket]]_ on [[regular polynomial observables]]
$$
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\overset{\{-,-\}_{\mathcal{T}}}{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
is the [[conjugation]] of the global [[antibracket]] (def. \ref{ComplexBVBRSTGlobal}) by the time-ordering operator $\mathcal{T}$ (from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}):
$$
\{-,-\}_{\mathcal{T}}
\;\coloneqq\;
\mathcal{T}\left(\left\{ \mathcal{T}^{-1}(-), \mathcal{T}^{-1}(-)\right\}\right)
$$
hence
$$
\array{
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
&\overset{\{-,-\}}{\longrightarrow}&
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\\
{}^{\mathllap{\mathcal{T}}}_{\mathllap{\simeq}}\Big\downarrow
&&
\Big\downarrow{}^{\mathrlap{\mathcal{T}}}_\simeq
\\
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
&\overset{ \{-,-\}_{\mathcal{T}} }{\longrightarrow}&
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
}
$$
=--
([Fredenhagen-Rejzner 11, (27)](#FredenhagenRejzner11), [Rejzner 11, (5.14)](#Rejzner11))
+-- {: .num_prop #GaugeFixedActionFunctionalTimeOrderedAntibracket}
###### Proposition
**([[time-ordered product|time-ordered]] [[antibracket]] with [[gauge fixing|gauge fixed]] [[action functional]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).
Then the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the gauge fixed BV-[[action functional]] $-S'$ (def. \ref{ComplexBVBRSTGlobal}) equals the [[conjugation]] of the global [[BV-differential]] with the [[isomorphism]] $\mathcal{T}$ from the pointwise to the [[time-ordered product]] of [[observables]] (from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise})
$$
\{-S',-\}_{\mathcal{T}}
\;=\;
\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
\,,
$$
hence
$$
\array{
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
&\overset{ \{-S',-\} }{\longrightarrow}&
PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\\
{}^{\mathllap{\mathcal{T}}}\Big\downarrow
&&
\Big\downarrow{}^{\mathrlap{\mathcal{T}}}
\\
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
&\overset{ \{-S',-\}_{\mathcal{T}} }{\longrightarrow}&
PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
}
$$
=--
+-- {: .proof}
###### Proof
By the assumption that $(E,\mathbf{L})$ is a [[free field theory]] its [[Euler-Lagrange equations]] are linear in the fields, and hence $S'$ is quadratic in the fields. This means that
$$
\mathcal{T}^{-1}S' = S' + const
\,,
$$
where the second term on the right is independent of the fields, and hence that
$$
\{\mathcal{T}^{-1}(-S'),-\}
=
\{-S', - \}
\,.
$$
This implies the claim:
$$
\begin{aligned}
\{-S',-\}_{\mathcal{T}}
& \coloneqq
\mathcal{T}\left(\{ \mathcal{T}^{-1}(-S'), \mathcal{T}^{-1}(-) \}\right)
\\
& =
\mathcal{T}\left(\{ -S', \mathcal{T}^{-1}(-) \}\right)
\\
& =
\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
\,.
\end{aligned}
$$
=--
+-- {: .num_defn #ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}
###### Definition
**([[BV-operator]] for [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}) and with corresponding [[gauge fixing|gauge-fixed]] global [[BV-BRST differential]] on graded [[regular polynomial observables]]
$$
\{-S' + S'_{BRST}, -\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
(def. \ref{GaugeFixingLagrangianDensity}).
Then the corresponding _[[BV-operator]]_
$$
\Delta_{BV}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
on [[regular polynomial observables]] is, up to a factor of $i \hbar$, the difference between the free component $\{-S',-\}$ of the gauge fixed global BV differential and its time-ordered version (def. \ref{AntibracketTimeOrdered})
$$
\Delta_{BV}
\;\coloneqq\;
\tfrac{1}{i \hbar}
\left(
\left\{
-S',-
\right\}_{\mathcal{T}}
-
\left\{ -S',(-) \right\}
\right)
\,,
$$
hence
$$
\label{BVOperatorDefiningRelation}
\{-S',-\}_{\mathcal{T}}
\;=\;
\{-S',-\} + i \hbar \Delta_{BV}
\,.
$$
=--
+-- {: .num_prop #ComponentsBVOperator}
###### Proposition
**([[BV-operator]] in components)**
If the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates collectively denoted $(\phi^A)$ then
the [[BV-operator]] $\Delta_{BV}$ from prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} is given explicitly by
$$
\Delta_{BV}
\;=\;
\underset{a}{\sum} (-1)^{deg(\Phi^A)}
\underset{\Sigma}{\int}
\frac{\delta}{\delta \Phi^A(x)}
\frac{\delta}{\delta \Phi^{\ddagger}_A(y)}
dvol_\Sigma
$$
Since this formula exhibits a graded [[Laplace operator]], the BV-operator is also called the _BV-Laplace operator_ or _BV-Laplacian_, for short.
=--
([Fredenhagen-Rejzner 11, (29)](#FredenhagenRejzner11), [Rejzner 11, (5.20)](#Rejzner11))
+-- {: .proof}
###### Proof
By prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} we have equivalently
$$
i \hbar \Delta_{BV}
\;=\;
\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
\,-\,
\{-S',-\}
$$
and by example \ref{BVDifferentialGlobal} the second term on the right is
$$
\begin{aligned}
\left\{ -S', -\right\}
& =
\underset{\Sigma}{\int}
j^{\infty}\left(\mathbf{\Phi}\right)^\ast
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A}
\right)(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\\
& =
\underset{a}{\sum}
(-1)^{deg(\phi^A)}
\underset{}{\int}
(P\mathbf{\Phi})_A(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\end{aligned}
$$
With this we compute as follows:
$$
\label{AAA}
\begin{aligned}
\{-S',-\}_{\mathcal{T}}
& =
\mathcal{T} \circ \left\{ -S,-\right\} \circ \mathcal{T}^{-1}
\\
& =
\exp\left(
\left[
\hbar \tfrac{1}{2} \Delta_F
\left(
\frac{\delta}{\delta \mathbf{\Phi}},
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\,,\,
-
\right]
\right)
\left(
\{-S',-\}
\right)
\\
& =
\{-S',-\}
+
\left[
\hbar \tfrac{1}{2}
\Delta_F
\left(
\frac{\delta}{\delta \mathbf{\Phi}},
\frac{\delta}{\delta \mathbf{\Phi}}
\right)
\,,
\{-S',-\}
\right]
+ \underset{ = 0 }{\underbrace{\hbar^2(...)}}
\\
& =
\phantom{+}
\left\{ -S' , -\right\}
\\
&
\phantom{=}
+
\left[
\tfrac{1}{2}\hbar
\underset{\Sigma \times \Sigma}{\int}
\Delta_F^{A B}(x,y)
\frac{\delta^2}{\delta \mathbf{\Phi}^A(x) \delta \mathbf{\Phi}^B(y)}
\, dvol_\Sigma(x)
\, dvol_\Sigma(y)
\;,\;
\underset{a}{\sum}
(-1)^{deg(\phi^A)}
\underset{\Sigma}{\int}
(P\mathbf{\Phi})_A(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\right]
\\
& =
\left\{ -S', -\right\}
\\
& \phantom{=}
+
\underset{A}{\sum}
(-1)^{deg(\phi^A)}
\underset{\Sigma \times \Sigma}{\int}
\underset{ = i \delta(x-y) }{\underbrace{P_x \Delta_F(x,y)}}
\frac{\delta}{\delta \mathbf{\Phi}^A(x)}
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(y)}
\, dvol_\Sigma(x)
\, dvol_\Sigma(y)
\\
& = \left\{ -S', -\right\}
+ i \hbar
\underset{A}{\sum}
(-1)^{deg(\phi^A)}
\underset{\Sigma}{\int}
\frac{\delta}{\delta \mathbf{\Phi}^A(x)}
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\end{aligned}
$$
Here we used
1. under the first brace that by assumption of a [[free field theory]], $\{-S',-\}$ is linear in the fields, so that the first [[commutator]] with the [[Feynman propagator]] is independent of the fields, and hence all the higher commutators vanish;
1. under the second brace that the [[Feynman propagator]] is $+i$ times the [[Green function]] for the [[Green hyperbolic differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (cor. \ref{GreenFunctionFeynmanPropagator}).
=--
+-- {: .num_prop #AntibracketBVOperatorRelation}
###### Proposition
**(global [[antibracket]] exhibits failure of [[BV-operator]] to be a [[derivation]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$
The [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) and the global [[antibracket]] $\{-,-\}$ (def. \ref{ComplexBVBRSTGlobal}) satisfy for all [[polynomial observables]] (def. \ref{PolynomialObservables}) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ the relation
$$
\label{GlobalAntibracketInteractingWithBVOperator}
\{A_1, A_2\}
\;=\;
(-1)^{deg(A_2)}
\,
\Delta_{BV}(A_1 \cdot A_2)
-
(-1)^{deg(A_2)}
\,
\Delta_{BV}(A_1) \cdot A_2
-
A_1 \cdot \Delta_{BV}(A_2)
$$
for $(-) \cdot (-)$ the pointwise product of observables (def. \ref{Observable}).
Moreover, it commutes on [[regular polynomial observables]] with the [[time-ordered product|time-ordering operator]] $\mathcal{T}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise})
$$
\Delta_{BV} \circ \mathcal{T}
=
\mathcal{T} \circ \Delta_{BV}
\phantom{AAA}
\text{on}
\,\,
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
and hence satisfies the analogue of relation (eq:GlobalAntibracketInteractingWithBVOperator) also for the time-ordered antibracket $\{-,-\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) and the [[time-ordered product]] $\star_F$ on regular polynomial observables
$$
\{A_1, A_2\}_{\mathcal{T}}
\;=\;
(-1)^{deg(A_2)}
\,
\Delta_{BV}(A_1 \star_F A_2)
-
(-1)^{dag(A_2)}
\Delta_{BV}(A_1) \star_F A_2
-
A_1 \star_F \Delta_{BV}(A_2)
\,.
$$
=--
(e.g. [Henneaux-Teitelboim 92, (15.105d)](antibracket#HenneauxTeitelboim92))
+-- {: .proof}
###### Proof
With prop. \ref{ComponentsBVOperator} the first statement is a graded version of the analogous relation for an ordinary [[Laplace operator]] $\Delta \coloneqq g^{a b} \partial_a \partial_b$ acting on [[smooth functions]] on [[Cartesian space]], which on [[smooth functions]] $f,g$ satisfies
$$
\Delta(f \cdot g)
\;=\;
(\nabla f, \nabla g) - \Delta(f) g - f \Delta(g)
\,,
$$
by the [[product law]] for [[differentiation]], where now $\nabla f \coloneqq (g^{a b} \partial_b f)$ is the [[gradient]] and $(v,w) \coloneqq g_{a b} v^a w b$ the [[inner product]]. Here one just needs to carefully record the relative signs that appear.
That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by [[partial derivative|partial]] [[functional derivatives]] with _[[constant function|constant]]_ [[coefficients]]. This immediately implies the last statement from the first.
=--
+-- {: .num_example #TimeOrderedExponentialBVOperator}
###### Example
**([[BV-operator]] on [[time-ordered product|time-ordered]] [[exponentials]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.
Let moreover $V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ]$ be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero. Then the application of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to the [[time-ordered product|time-ordered]] [[exponential]] $\exp_{\mathcal{T}}(V)$ (example \ref{RegularObservablesExponentialTimeOrdered}) is the [[time-ordered product]] of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global [[antibracket]] $\tfrac{1}{2}\{V,V\}$ of $V$ with itself:
$$
\Delta_{BV}
\left(
\exp_{\mathcal{T}}(V)
\right)
\;=\;
\left(
\Delta_{BV}(V)
+
\tfrac{1}{2}\{V,V\}
\right)
\star_F
\exp_{\mathcal{T}}(V)
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{AntibracketBVOperatorRelation} $\Delta_{BV}$ acts as a [[derivation]] on the [[time-ordered product]] up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of [[exponentials]].
$$
\begin{aligned}
& \Delta_{BV}
\left(
1 + V + \tfrac{1}{2}V \star_F V + \cdots
\right)
\\
& =
\Delta_{BV}(V)
+
\tfrac{1}{2}\left( \Delta_{BV}(V) \star_F V + V \star_F \Delta_{BV}(V) \right)
+
\tfrac{1}{2}\{V,V\}
+
\cdots
\\
& =
\Delta_{BV}(V)
+
\tfrac{1}{2}\{V,V\}
\;+\;
\Delta_{BV}(V) \star_F V
+ \cdots
\end{aligned}
$$
=--
$\,$
**[[Schwinger-Dyson equation]]**
{#SchwingerDysonEquation}
A special case of the general occurrence of the [[BV-operator]] is the following important property of [[on-shell]] [[time-ordered products]]:
+-- {: .num_prop #DysonSchwinger}
###### Proposition
**([[Schwinger-Dyson equation]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).
Let
$$
\label{SchwingerDysonTestObservable}
A
\;\coloneqq\;
\underset{\Sigma}{\int}
A^a(x) \cdot \mathbf{\Phi}^\ddagger_a(x)
\, dvol_\Sigma(x)
\;\in\;
PolyObs_{reg}(E_{\text{BV-BRST}})
$$
be an [[off-shell]] [[regular polynomial observable]] which is [[linear map|linear]] in the [[antifield]] [[field observables]] $\mathbf{\Phi}^\ddagger$. Then
$$
\label{EquationSchwingerDyson}
\mathcal{T}^{\pm 1}
\left(
\underset{\Sigma}{\int}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
\cdot
A^a(x)
\, dvol_\Sigma(x)
\right)
\;=\;
\pm i \hbar
\,
\mathcal{T}^{\pm}
\left(
\underset{\Sigma}{\int}
\frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
\,
dvol_\Sigma(x)
\right)
\phantom{A}
\in
\underset{
\text{on-shell}
}{
\underbrace{
PolyObs_{reg}(E_{\text{BV-BRST}}, \mathbf{L'})
}}
\,.
$$
This is called the _[[Schwinger-Dyson equation]]_.
=--
The following proof is due to ([Rejzner 16, remark 7.7](#Rejzner16)) following the informal traditional argument ([Henneaux-Teitelboim 92, (15.108b)](#HenneauxTeitelboim92)).
+-- {: .proof}
###### Proof
Applying the inverse time-ordering map $\mathcal{T}^{-1}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) to equation (eq:BVOperatorDefiningRelation) applied to $A$ yields
$$
\underset{
\mathcal{T}^{-1}
\underset{\Sigma}{\int}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
\cdot
A^a(x)
dvol_\Sigma(x)
}{
\underbrace{
\mathcal{T}^{-1}\left\{ -S', A\right\}
}
}
\;=\;
-
\underset{
i \hbar
\mathcal{T}^{-1}
\underset{\Sigma}{\int}
\frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
\, dvol_\Sigma
}{
\underbrace{
i \hbar \mathcal{T}^{-1}\Delta_{BV}(A)
}
}
+
\underset{
\{-S',\mathcal{T}^{-1}(A)\}
}{
\underbrace{
\mathcal{T}^{-1}\left\{ -S',A\right\}_{\mathcal{T}}
}
}
$$
where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from prop. \ref{BVDifferentialGlobal}, 2) prop. \ref{ComponentsBVOperator} and 3) prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}.
The last term is manifestly in the [[image]] of the [[BV-differential]] $\{-S',-\}$ and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] (eq:OnShellPolynomialObservablesAsBVCohomology)
$$
\underset{
\text{on-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
}}
\;\simeq\;
\underset{
\text{off-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
}}/im(\{-S',-\})
$$
(by example \ref{BVDifferentialGlobal}).
The same argument with the replacement $\mathcal{T} \leftrightarrow \mathcal{T}^{-1}$ throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the $\hbar$-term reversed).
=--
+-- {: .num_remark }
###### Remark
**("Schwinger-Dyson operator")**
The proof of the [[Schwinger-Dyson equation]] in prop. \ref{DysonSchwinger} shows that, up to [[time-ordered product|time-ordering]], the [[Schwinger-Dyson equation]] is the on-shell vanishing of the "quantized" [[BV-differential]] (eq:BVOperatorDefiningRelation)
$$
\{-S',-\}_{\mathcal{T}}
\;=\;
\{-S', -\}
\,+\,
i \hbar \, \Delta_{BV}
\,,
$$
where the [[BV-operator]] is the quantum correction of order $\hbar$. Therefore this is also called the _Schwinger-Dyson operator_ ([Henneaux-Teitelboim 92, (15.111)](#HenneauxTeitelboim92)).
=--
+-- {: .num_example #SchwingerDysonDistributional}
###### Example
**([[distribution|distributional]] [[Schwinger-Dyson equation]])**
Often the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}) is displayed before spacetime-smearing of [[field observables]] in terms of [[operator products]] of [[operator-valued distributions]], taking the observable $A$ in (eq:SchwingerDysonTestObservable) to be
$$
A^a(x)
\;\coloneqq\;
\delta(x-x_0) \delta^a_{a_0}
\mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
\,.
$$
This choice makes (eq:EquationSchwingerDyson) become the [[distribution|distributional]] [[Schwinger-Dyson equation]]
$$
\begin{aligned}
& T
\left(
\frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
\cdot
\mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
\right)
\\
&
\underset{\text{on-shell}}{=}
- i \hbar
\underset{k}{\sum}
T
\left(
\mathbf{\Phi}^{a_1}(x_1)
\cdots
\mathbf{\Phi}^{a_{k-1}}(x_{k-1})
\cdot
\delta(x_0 - x_k) \delta^{a_0}_{a_k}
\cdot
\mathbf{\Phi}^{a_{k+1}}(x_{k+1})
\cdots
\mathbf{\Phi}^{a_n}(x_m)
\right)
\end{aligned}
$$
(e.q. [Dermisek 09](Schwinger-Dyson+equation#Dermisek09)).
In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then
$$
T
\left(
\frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
\cdot
\mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
\right)
\;=\;
0
\phantom{AAA}
\text{on-shell}
$$
Since by the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction}) the equation
$$
\frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
\;=\;
0
$$
is the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (for the [[classical field theory]]) "at $x_0$", this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for [[time-ordered product|time-ordered]] [[quantum theory|quantum]] [[expectation values]], as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction measured by the [[BV-operator]].
=--
$\,$
This concludes our discussion of the [[algebra of quantum observables]] for [[free field theories]]. In the [next chapter](#InteractingQuantumFields) we discuss the [[perturbative QFT]] of [[interacting field theories]] as [[deformations]] of such free quantum field theories.
$\,$
$\,$
|
A first idea of quantum field theory -- Gauge fixing | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Gauge+fixing |
## Gauge fixing
{#GaugeFixing}
In this chapter we discuss the following topics:
1. _[Quasi-isomorphisms between local BV-BRST complexes](#QuasiIsomorphismsBetweenBVBRSTComplexes)_
1. [gauge fixing chain maps](#GaugeFixingChainMaps);
1. [adjoining contractible complexes of auxiliary fields](#AdjoiningAuiliaryFields)
1. _[Example: gauge fixed electromagnetic field](#GaugeFixingExamples)_
$\,$
While in the [previous chapter](#ReducedPhaseSpace) we had constructed the [[reduced phase space]] of a [[Lagrangian field theory]], embodied by the [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}), as the [[homotopy quotient]] by the [[infinitesimal gauge symmetries]] of the [[homotopy intersection]] with the [[shell]], this in general still does not yield a [[covariant phase space]] of [[on-shell]] [[field histories]] (prop. \ref{CovariantPhaseSpace}), since [[Cauchy surfaces]] for the [[equations of motion]] may still not exist (def. \ref{CauchySurface}).
However, with the [[homological resolution]] constituted by the [[BV-BRST complex]] in hand, we now have the freedom to adjust the [[field (physics)|field]]-content of the theory without changing its would-be [[reduced phase space]], namely without changing its [[BV-BRST cohomology]]. In particular we may adjoin further "[[auxiliary fields]]" in various degrees, as long as they contribute only a [[contractible chain complex|contractible cochain complex]] to the [[BV-BRST complex]]. If such a _[[quasi-isomorphism]]_ of [[BV-BRST complexes]] brings the [[Lagrangian field theory]] into a form such that the [[equations of motion]] of the combined [[field (physics)|fields]], [[ghost fields]] and potential further [[auxiliary fields]] are [[Green hyperbolic differential equations]] after all, and thus admit a [[covariant phase space]], then this is called a _[[gauge fixing]]_ (def. \ref{GaugeFixingLagrangianDensity} below), since it is the [[infinitesimal gauge symmetries]] which [[obstruction|obstruct]] the existence of [[Cauchy surfaces]] (by prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} and remark \ref{GaugeParametrizedInfinitesimalGaugeTransformation}).
The archetypical example is the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] of the [[electromagnetic field]] (example \ref{NLGaugeFixingOfElectromagnetism} below) which reveals that the gauge-invariant content of [[electromagnetic waves]] is only in their transversal [[wave polarization]] (prop. \ref{GaugeInvariantPolynomialOnShellObservablesOfFreeElectromagneticField} below).
The tool of [[gauge fixing]] via [[quasi-isomorphisms]] of [[BV-BRST complexes]] finally brings us in position to consider, in the following chapters, the [[quantization]] also of [[gauge theories]]: We use [[gauge fixing]] [[quasi-isomorphisms]]
to bring the [[BV-BRST complexes]] of the given [[Lagrangian field theories]] into a form that admits degreewise [[quantization]] of a [[graded manifold|graded]] [[covariant phase space]] of [[fields (physics)|fields]], [[ghost fields]] and possibly further [[auxiliary fields]], compatible with the gauge-fixed [[BV-BRST differential]]:
$\,$
$$
\array{
\underline{\mathbf{\text{pre-quantum geometry}}}
&&
\underline{\mathbf{\text{higher pre-quantum geometry}}}
\\
\,
\\
\left\{
\array{
\text{Lagrangian field theory with}
\\
\text{infinitesimal gauge transformations}
}
\right\}
&\overset{ \text{homotopy quotient by} \atop \text{gauge transformations} }{\longrightarrow}&
\left\{
\array{
\text{dg-Lagrangian field theory with}
\\
\text{quotiented by gauge transformations}
\\
\text{embodied by BRST complex }
}
\right\}
\\
&& \Big\downarrow{}^{\mathrlap{ \text{pass to} \atop \text{derived critical locus} }}
\\
&&
\left\{
\array{
\text{dg-reduced phase space}
\\
\text{ embodied by BV-BRST complex }
}
\right\}
\\
&& {}^{\mathllap{\simeq}}\Big\downarrow{}^{\mathrlap{\text{fix gauge} }}
\\
\left\{
\array{
\text{ decategorified }
\\
\text{ covariant }
\\
\text{ reduced phase space }
}
\right\}
&\underset{\text{pass to cohomology}}{\longleftarrow}&
\left\{
\array{
\text{ dg-covariant}
\\
\text{reduced phase space }
}
\right\}
\\
&& \Big\downarrow{}^{\mathrlap{
\array{
\text{ quantize }
\\
\text{degreewise}
}
}}
\\
\left\{
\array{
\text{gauge invariant}
\\
\text{quantum observables}
}
\right\}
&\underset{\text{pass to cohomology}}{\longleftarrow}&
\left\{
\array{
\text{quantum}
\\
\text{BV-BRST complex}
}
\right\}
}
$$
Here:
| term | meaning |
|-------|-------------|
| "phase space" | [[derived critical locus]] of [[Lagrangian density|Lagrangian]] equipped with [[Poisson bracket]] |
| "reduced" | [[gauge transformations]] have been [[homotopy quotient|homotopy-quotiented]] out |
| "covariant" | [[Cauchy surfaces]] exist degreewise |
$\,$
**[[quasi-isomorphisms]] between [[local BV-BRST complexes]]**
{#QuasiIsomorphismsBetweenBVBRSTComplexes}
Recall (prop. \ref{CochainCohomologyOfBVBRSTComplexInDegreeZero}) that given a [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) with [[BV-BRST differential]] $s$, then the space of
[[local observables]] which are [[on-shell]] and [[gauge invariance|gauge invariant]] is the [[cochain cohomology]]
of $s$ in degree zero:
$$
H^0(s \vert d)
\;=\;
\left\{
\array{
\text{gauge invariant on-shell}
\\
\text{local observables}
}
\right\}
$$
The key point of having [[resolution|resolved]] (in chapter _[Reduced phase space](#ReducedPhaseSpace)_) the naive [[quotient]] by [[infinitesimal gauge symmetries]] of the naive [[intersection]] with the [[shell]] by the [[L-infinity algebroid]] whose [[Chevalley-Eilenberg algebra]] is called the _[[local BV-BRST complex]]_, is that placing the [[reduced phase space]] into the [[(infinity,1)-category|context]]
of [[homotopy theory]]/[[homological algebra]] this way provides the freedom of changing the choice of [[field bundle]] and of
[[Lagrangian density]] without actually changing the [[Lagrangian field theory]] _up to [[equivalence]]_, namely without changing the
[[cochain cohomology]] of the [[BV-BRST complex]].
A [[homomorphism]] of [[differential graded-commutative superalgebras]] (such as [[BV-BRST complexes]]) which
induces an [[isomorphism]] in [[cochain cohomology]] is called a _[[quasi-isomorphism]]_. We now discuss two
classes of [[quasi-isomorphisms]] between [[BV-BRST complexes]]:
1. _[[gauge fixing]]_ (def. \ref{GaugeFixingLagrangianDensity} below)
1. _adjoining [[auxiliary fields]]_ (def. \ref{AuxiliaryFields} below).
$\,$
**[[gauge fixing]] [[chain maps]]**
{#GaugeFixingChainMaps}
+-- {:.num_prop #ExponentialOfLocalAntibracket}
###### Proposition
**([[local antibracket|local anti-]][[Hamiltonian flow]] is [[automorphism]] of [[local antibracket]])**
Let
$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
-
\right\}
}
}
\;+\;
d
\right)
$$
be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}).
Then for
$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1,0}
\left(
T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1]
\right)
$$
a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) on the [[graded manifold|graded]] [[field bundle]]
$$
\mathbf{L}_{gf} \;=\; L_{gf} \ dvol_\Sigma
$$
of degree
$$
deg(L) = (-1, even)
$$
then the [[exponential]] of forming the [[local antibracket]] (def. \ref{LocalAntibracket}) with $\mathbf{L}_{gf}$
$$
\array{
\Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right)
&
\overset{
e^{\left\{ \mathbf{L}_{gf} \,,\, -\right\}}(-)
}{\longrightarrow}
&
\Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right)
\\
\mathbf{K}
&\mapsto&
\left\{ \mathbf{L}_{gf} , \mathbf{K} \right\}
+
\tfrac{1}{2}
\left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\, \mathbf{K} \right\} \right\}
+
\tfrac{1}{6}
\left\{ \mathbf{L}_{gf} \,,\,\left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\,\mathbf{K} \right\} \right\} \right\}
+
\cdots
}
$$
is an [[endomorphism]] of the [[local antibracket]] (def. \ref{LocalAntibracket}) in that
$$
e^{
\left\{ \mathbf{\psi} \,,\, - \right\}
}
\left(
\left\{ \mathbf{A} \,,\, \mathbf{B} \right\}
\right)
\;=\;
\left\{
e^{
\left\{ \mathbf{\psi} \,,\, - \right\}
}
\left(\mathbf{A}\right)
\,,\,
e^{
\left\{ \psi \,,\, - \right\}
}
\left(\mathbf{B}\right)
\right\}
$$
and in fact an [[automorphism]], with [[inverse morphism]] given by
$$
\left(e^{\left\{ \psi \,,\, -\right\}}(-)\right)^{-1}
\;=\;
e^{\left\{ -\psi \,,\, -\right\}}(-)
\,.
$$
We may think of this as the _[[Hamiltonian flow]]_ of $\mathbf{L}_{gf}$ under the [[local antibracket]].
In particular when applied to the [[BV-Lagrangian density]]
$$
s_{gf}
\;\coloneqq\;
\left\{
e^{\left\{ \mathbf{L}_{gf},-\right\}}\left(- \mathbf{L} + \mathbf{L}_{BRST}\right)
\,,\,
-
\right\}
$$
this yields another [[differential]]
$$
\left( s_{gf}\right)^2
\;=\;
0
$$
and hence another [[differential graded-commutative superalgebra]] (def. \ref{differentialgradedcommutativeSuperalgebra})
$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s_{gf}}{
\underbrace{
\left\{
e^{\left\{ \mathbf{L}_{gf}, - \right\}}\left( - \mathbf{L} + \mathbf{L}_{BRST} \right)
\,,\,
-
\right\}
}
}
\;+\;
d
\right)
$$
Finally, $e^{\left\{\mathbf{L}_{gf},-\right\}}$ constitutes a [[chain map]]
from the [[local BV-BRST complex]] to this deformed version, in fact a
[[homomorphism]] of [[differential graded-commutative superalgebras]], in that
$$
s_{gf} \circ e^{ \left\{ \mathbf{L}_{gf}\,,\, - \right\} }
\;=\;
e^{ \left\{ \mathbf{L}_{gf}\,,\, - \right\} } \circ s
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{BasicPropertiesOfTheLocalAntibracket}
the [[local antibracket]] $\left\{ -,-\right\}$ is a graded [[derivation]] in its second argument, of degree one more than the degree of its first argument (eq:LocalAntibracketGradedDerivationInSecondArgument). Hence for the first argument of degree -1 this implies
that $e^{\{\mathbf{L}_{gf}, - \}}$ is an automorphism of the local antibracket. Moreover, it is clear from the definition that
$\left\{ \mathbf{L}_{gf},-\right\}$ is a [[derivation]] with respect to the pointwise product of smooth functions,
so that $e^{\{\mathbf{L}_{gf},-\}}$ is also a homomorpism of graded algebras.
Since $e^{\{\mathbf{L}_{gf}, -\}}$ is an automorphism of the local antibracket, and since $s$ and $s_{gf}$
are themselves given by applying the local antibracket in the second argument, this implies that
$e^{\{\mathbf{L}_{gf},-\}}$ respects the differentials:
$$
\array{
\mathbf{A}
&\overset{e^{\{\mathbf{L}_{gf},-\}}}{\longrightarrow}&
e^{\{\mathbf{L}_{gf},-\}}\left( \mathbf{A} \right)
\\
{}^{\mathllap{s}}\downarrow && \downarrow^{\mathrlap{s_{gf}}}
\\
\left\{ \left(-\mathbf{L} + \mathbf{L}_{BRST}\right)\,,\, \mathbf{A}\right\}
&\underset{ e^{\{\mathbf{L}_{gf}\,,\,-\}} }{\longrightarrow}&
\left\{ e^{\{\mathbf{L}_{gf},-\}}\left(-\mathbf{L} + \mathbf{L}_{BRST}\right)
\,,\,
e^{\{\mathbf{L}_{gf},-\}}(\mathbf{A})
\right\}
}
$$
=--
+-- {: .num_defn #GaugeFixingLagrangianDensity}
###### Definition
**([[gauge fixing Lagrangian density]])**
Let
$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
-
\right\}
}
}
\;+\;
d
\right)
$$
be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) and let
$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1,0}
\left(
T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1]
\right)
$$
be a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) on the [[graded manifold|graded]] [[field bundle]] such that
$$
deg(L_{gf}) = -1
\,.
$$
If the [[quasi-isomorphism]] of [[BV-BRST complexes]] given by the [[local antibracket|local anti-]][[Hamiltonian flow]]
$\mathbf{L}_{gf}$ via prop. \ref{ExponentialOfLocalAntibracket}
$$
e^{\left\{ \mathbf{L}_{gf},-\right\}}
\;\colon\;
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
\overset{\phantom{A}\simeq_{qi}\phantom{A}}{\longrightarrow}
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
$$
is such that for the transformed graded [[Lagrangian field theory]]
$$
\label{GaugeFixedLagrangianDensity}
-\underset{deg_{af} = 0}{\underbrace{\mathbf{L}' }} + \mathbf{L}'_{BRST}
\;\coloneqq\;
e^{\{\mathbf{L}_{gf},-\}}(-\mathbf{L} + \mathbf{L}_{BRST})
$$
(with [[Lagrangian density]] $\mathbf{L}'$ the part independent of [[antifields]])
the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) admit [[Cauchy surfaces]] (def. \ref{CauchySurface}), then we call $\mathbf{L}_{gf}$ a _[[gauge fixing Lagrangian density]]_
for the original Lagrangian field theory, and $\mathbf{L}'$ the corresponding _gauge fixed_ form of the original [[Lagrangian density]] $\mathbf{L}$.
=--
+-- {: .num_remark}
###### Remark
**(warning on terminology)**
What we call a _[[gauge fixing Lagrangian density]]_ $\mathbf{L}_{gf}$ in def. \ref{GaugeFixingLagrangianDensity} is traditionally called a _[[gauge fixing fermion]]_ and denoted by "$\psi$" ([Henneaux 90, section 8.3, 8.4](BRST+complex#Henneaux90)).
Here "fermion" is meant as a reference to the fact that the cohomological degree $deg(L_{gf}) = -1$, which is reminiscent of the odd [[supergeometry|super-degree]] of [[fermion]] fields such as the [[Dirac field]] (example \ref{DiracFieldBundle}); see at _[[signs in supergeometry]]_ the section _[The super odd sign rule](signs+in+supergeometry#SuperOddConvention)_.
=--
+-- {: .num_example #GaugeFixingViaAntiLagrangianSubspaces}
###### Example
**([[gauge fixing]] via [[local antibracket|anti-]][[Lagrangian subspaces]])**
Let $\mathbf{L}_{gf}$ be a [[gauge fixing Lagrangian density]] as in def. \ref{GaugeFixingLagrangianDensity}
such that
1. its [[local antibracket]]-square vanishes
$$
\left\{ \mathbf{L}_{gf},\, \left\{ \mathbf{L}_{gf}, \, -\right\} \right\} = 0
$$
hence its [[local antibracket|anti-]][[Hamiltonian flow]] has at most a linear component in its argument $\mathbf{A}$:
$$
e^{\left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\}}
\;=\;
\mathbf{A} + \left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\}
$$
1. it is independent of the [[antifields]]
$$
deg_{af}\left( L_{gf} \right) \;=\; 0
\,.
$$
Then with
* $(\phi^A)$ collectively denoting all the [[field (physics)|field]] coordinates
(including the actual fields $\phi^a$, the [[ghost fields]] $c^\alpha$ as well as possibly further [[auxiliary fields]])
* $(\phi^\ddagger_A)$ collectively denoting all the [[antifield]] coordinates
(includion the antifields $\phi^\ddagger_a$ of the actual fields, the antifields $c^\ddagger_\alpha$ of the [[ghost fields]] as well as those of possibly further [[auxiliary fields]] )
we have
$$
\begin{aligned}
(\phi')^A
& \coloneqq
e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}}(\phi^A)
\\
& = \phi^A
\\
\phantom{A}
\\
(\phi')^\ddagger_A
& \coloneqq
e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}}
\left( \phi^\ddagger_A \right)
\\
& =
\phi^\ddagger_A - \frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}_{gf}}{\delta \phi^a}
\end{aligned}
$$
(and similarly for the higher jets); and the corresponding transformed [[Lagrangian density]] (eq:GaugeFixedLagrangianDensity)
may be written as
$$
\begin{aligned}
-\mathbf{L}' + \mathbf{L}'_{BRST}
& \coloneqq
e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}}\left( -\mathbf{L} + \mathbf{L}_{BRST} \right)
\\
& =
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi', (\phi')^\ddagger \right)
\end{aligned}
\,,
$$
where the notation on the right denotes that $\phi'$ is [[substitution|substituted]] for $\phi$ and $\phi'_\ddagger$ for $\phi_\ddagger$.
This means that the defining condition that $\mathbf{L}'$ be the antifield-independent summand (eq:GaugeFixedLagrangianDensity), which we may write as
$$
\mathbf{L}'
\coloneqq
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi'(\phi), \phi_\ddagger = 0 \right)
$$
translates into
$$
\mathbf{L}'
\coloneqq
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi', (\phi')^\ddagger_A = -\frac{\overset{\leftarrow}{\delta}_{EL} L_{gf}}{\delta \phi^A} \right)
\,.
$$
In this form BV-gauge fixing is considered traditionally
(e.g. [Hennaux 90, section 8.3, page 83, equation (76b) and item (iii)](gauge+fixing#Henneaux90)).
=--
$\,$
**adjoining [[contractible chain complexes|contractible cochain complexes]] of [[auxiliary fields]]**
{#AdjoiningAuiliaryFields}
Typically a [[Lagrangian field theory]] $(E,\mathbf{L})$ for given choice of [[field bundle]], even after finding appropriate [[gauge parameter bundles]] $\mathcal{G}$, does not yet admit a [[gauge fixing Lagrangian density]] (def. \ref{GaugeFixingLagrangianDensity}).
But if the [[gauge parameter bundle]] has been
chosen suitably, then the remaining [[obstruction]] vanishes "up to [[homotopy]]" in that a [[gauge fixing Lagrangian density]]
does exist if only one adjoins sufficiently many [[auxiliary fields]] forming a [[contractible chain complex|contractible complex]], hence without changing the [[cochain cohomology]] of the [[BV-BRST complex]]:
+-- {: .num_defn #AuxiliaryFields}
###### Definition
**([[auxiliary fields]] and [[antighost fields]])**
Over [[Minkowski spacetime]] $\Sigma$, let
$$
A \overset{aux}{\longrightarrow} \Sigma
$$
be any [[graded manifold|graded]] [[vector bundle]] (remark \ref{dgManifolds}), to be regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) for _[[auxiliary fields]]_. If this is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) we denote its field [[coordinates]] by $(b^i)$. On the corresponding graded bundle with degrees shifted down by one
$$
A[-1] \overset{aux[-1]}{\longrightarrow} \Sigma
$$
we write $(\overline{c}^i)$ for the induced field coordinates.
Accordingly, the shifted infinitesimal [[vertical cotangent bundle]] (def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle})
of the [[fiber product]] of these bundles
$$
T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A^\ast[-1] \right)
$$
has the following coordinates:
$$
\array{
\text{name:}
&
\array{
\text{antifield of}
\\
\text{antighost field}
}
&
\array{
\text{antifield of}
\\
\text{auxiliary field}
}
&
\text{antighost field}
&
\text{auxiliary field}
\\
\text{symbol:} & \overline{c}^\ddagger_i & b^\ddagger_i & \overline c^i & b^i
\\
deg = & -(deg(b^i)-1)-1 & -deg(b^i)-1 & deg(b^i)-1 & deg(b^i)
\\
& = -deg(b^i)
}
$$
On this [[fiber bundle]] consider the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
$$
\label{LagrangianDensityForAuxiliaryFields}
\mathbf{L}_{aux}
\;\in\;
\Omega^{p+1,0}_\Sigma( T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A[-1] \right) )
$$
given in [[local coordinates]] by
$$
\mathbf{L}_{aux}
\;\coloneqq\;
\overline{c}^\ddagger_i b^i \, dvol_\Sigma
\,.
$$
This is such that the [[local antibracket]] (def. \ref{LocalAntibracket}) with this Lagrangian
acts on generators as follows:
$$
\label{BVDifferentialOnauxiliaryFields}
\array{
&& \left\{ \mathbf{L}_{aux},- \right\}
\\
\text{auxiliary field}
&
b^i &\mapsto& 0
\\
\text{antighost field}
&
\overline{c}^i &\mapsto& b^i
\\
\text{antifield of auxiliary field}
&
b^\ddagger_i &\mapsto& - \overline{c}^\ddagger_i
\\
\text{antifield of antighost field}
&
\overline{c}^\ddagger_i &\mapsto& 0
}
$$
=--
+-- {: .num_remark}
###### Remark
**(warning on terminology)**
Beware that when adjoining [[antifields]] as in def. \ref{AuxiliaryFields} to a [[Lagrangian field theory]] which also has [[ghost fields]] $(c^\alpha)$ adjoined (example \ref{LocalOffShellBRSTComplex}) then there is _no_ relation, a priori, between
* the "antighost field" $\overline{c}^i$
and
* the "antifield of the ghost field" $c^\ddagger_\alpha$
In particular there is also the
* "antifield of the antighost field" $\overline{c}^\ddagger_i$
The terminology and notation is maybe unfortunate but entirely established.
=--
The following is immediate from def. \ref{AuxiliaryFields}, in fact this is the purpose of the definition:
+-- {: .num_prop #QuasiIsomorphismAdjoiningAuxiliaryFields}
###### Proposition
**(adjoining [[auxiliary fields]] is [[quasi-isomorphism]] of [[BV-BRST complexes]])**
Let
$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
-
\right\}
}
}
\;+\;
d
\right)
$$
be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}).
Let moreover $A \overset{aux}{\longrightarrow} \Sigma$ be any [[auxiliary field bundle]] (def. \ref{AuxiliaryFields}).
Then on the [[fiber product]] of the original [[field bundle]] $E$ and the shifted [[gauge parameter bundle]] $\mathcal{G}[1]$ with the [[auxiliary field bundle]] $A$
the sum of the original [[BV-Lagrangian density]] $-\mathbf{L} + \mathbf{L}_{BRST}$ with the
auxiliary Lagrangian density $\mathbf{L}_{aux}$ (eq:LagrangianDensityForAuxiliaryFields)
induce a new [[differential graded-commutative superalgebra]]:
$$
\begin{aligned}
& CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right)
\\
& \coloneqq\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]
\left(
E \times_\Sigma \mathcal{G}[1] \times_\Sigma \left( A \times_\Sigma A[-1]\right) \right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
\left( - L + L_{BRST} + \mathbf{L}_{aux} \right) dvol_\Sigma
\,,\,
-
\right\}
}
}
\;+\;
d
\right)
\end{aligned}
$$
with generators
$$
\array{
\text{fields} & \phi^a & E & \phi^\ddagger_a & \text{antifields}
\\
\\
\text{ghost fields} & c^\alpha & \mathcal{G}[1] & c^\ddagger_\alpha & \array{ \text{antifields of} \\ \text{ghost fields} }
\\
\\
\text{ auxiliary fields } & b^i & A & b^\ddagger_i & \array{ \text{antifields of} \\ \text{auxiliary fields} }
\\
\\
\text{ antighost fields } & \overline{c}^i & A[-1] & \overline{c}^{\ddagger}_i & \array{ \text{antifields of} \\ \text{antighost fields} }
}
$$
Moreover, the [[differential graded-commutative superalgebra]] of [[auxiliary fields]] and their [[antighost fields]] is a [[contractible chain complex]]
$$
\left(
\Omega^{0,0}_\Sigma( A \times_{\Sigma} A[-1] )
\,,\,
d_{CE} = \left\{ \overline{c}^\ddagger_i b^i \, dvol_\Sigma \,,\, - \right\}
\right)
\overset{\simeq_{qi}}{\longrightarrow}
0
$$
and thus the canonical inclusion map
$$
CE\left( E/(\mathcal{G} \times_\Sigma \times_\Sigma T \Sigma)_{\delta_{EL} L \simeq 0} \right)
\overset{\phantom{AA} \simeq_{qi} \phantom{aa}}{\hookrightarrow}
CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right)
$$
(of the original [[BV-BRST complex]] into its [[tensor product]] with that for the [[auxiliary fields]] and their [[antighost fields]])
is a [[quasi-isomorphism]].
=--
+-- {: .proof}
###### Proof
From (eq:BVDifferentialOnauxiliaryFields) we read off that
1. the map $s_{aux} \coloneqq \left\{ \mathbf{L}_{aux},- \right\}$ is a [[differential]] (squares to zero), and
the auxiliary [[Lagrangian density]] satisfies its [[classical master equation]] (remark \ref{ClassicalMasterEquationLocal}) strictly
$$
\{\mathbf{L}_{aux}, \mathbf{L}_{aux}\} = 0
$$
1. the [[cochain cohomology]] of this differential is trivial:
$$
H^\bullet( s_{aux} )\;=\;0
$$
1. The [[local antibracket]] of the [[BV-Lagrangian density]] with the auxiliary Lagrangian density vanishes:
$$
\left\{
- \mathbf{L} + \mathbf{L}_{BRST}
\,,\,
\mathbf{L}_{aux}
\right\}
\;=\;
0
$$
Together this implies that the sum $-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$ satisfies the [[classical master equation]] (remark \ref{ClassicalMasterEquationLocal})
$$
\left\{
\left(
- \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\right)
\,,\,
\left(
- \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\right)
\right\}
\;=\;
0
$$
and hence that
$$
s + s_{aux}
\;\coloneqq\;
\left\{
- \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\,,\,
-
\right\}
$$
is indeed a [[differential]]; such that its [[cochain cohomology]] is identified with that of $s = \left\{-\mathbf{L} + \mathbf{L}_{BRST},-\right\}$ under the canonical inclusion map.
=--
+-- {: .num_remark #FieldBundleBVBRST}
###### Remark
**([[gauge fixing|gauge fixed]] [[BV-BRST formalism|BV-BRST]] [[field bundle]])**
In conclusion, we have that, given
1. $(E,\mathbf{L})$ a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), with [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles});
1. $\mathcal{G}$ a choice of [[gauge parameters]] (def. \ref{GaugeParameters}),
hence
$\mathcal{G}[1]$ a choice of [[ghost fields]] (example \ref{LocalOffShellBRSTComplex});
1. $A$ a choice of [[auxiliary fields]] (def. \ref{AuxiliaryFields}),
hence
$A[-1]$ a choice of [[antighost fields]] (def. \ref{AuxiliaryFields})
1. $T^\ast_{\Sigma,inf}[-1](\cdots)$ the corresponding [[antifields]] (def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle})
1. a [[gauge fixing Lagrangian density]] $\mathbf{L}_{gf}$ (def. \ref{GaugeFixingLagrangianDensity})
then the result is a new [[Lagrangian field theory]]
$$
\left( E_{\text{BV-BRST}}, \mathbf{L}' \right)
$$
now with [[graded manifold|graded]] [[field bundle]] (remark \ref{dgManifolds}) the [[fiber product]]
$$
E_{\text{BV-BRST}}
\;\coloneqq\;
\underset{
\array{ \text{anti-} \\ \text{fields} }
}{
\underbrace{
T^\ast_{\Sigma,inf}
}
}
\left(
\underset{\text{fields}}{\underbrace{E}}
\times_\Sigma
\underset{ \array{ \text{ghost} \\ \text{fields} }}{\underbrace{\mathcal{G}[1]}}
\times_\Sigma
\underset{\array{ \text{auxiliary} \\ \text{fields} }}{\underbrace{A}}
\times_{\Sigma}
\underset{ \array{ \text{antighost} \\ \text{fields} } }{\underbrace{A[-1]}}
\right)
$$
and with [[Lagrangian density]] $\mathbf{L'}$ independent of the [[antifields]], but complemented by an auxiliary Lagrangian density $\mathbf{L}'_{BRST}$.
The key point being that $\mathbf{L}'$ admits a [[covariant phase space]] (while $\mathbf{L}$ may not), while in [[BV-BRST cohomology]] both theories still have the same gauge-invariant on-shell observables.
=--
$\,$
**Gauge fixed electromagnetic field**
{#GaugeFixingExamples}
As an example of the general theory of BV-BRST [[gauge fixing]] above we now discuss the gauge fixing of the [[electromagnetic field]].
+-- {: .num_example #NLGaugeFixingOfElectromagnetism}
###### Example
**([[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] of [[vacuum]] [[electromagnetism]])**
Consider the [[local BV-BRST complex]] for the [[free field theory|free]] [[electromagnetic field]] on [[Minkowski spacetime]] from example \ref{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}:
The [[field bundle]] is $E \coloneqq T^\ast \Sigma$ and the [[gauge parameter bundle]] is $\mathcal{G} \coloneqq \Sigma \times \mathbb{R}$.
The 0-jet field coordinates are
$$
\array{
& c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
\\
deg =
&
-2 & -1 & 0 & 1
}
$$
the [[Lagrangian density]] is (eq:ElectromagnetismLagrangian)
$$
\label{VacuumEMLagrangianDensityRecalledForNLFields}
\mathbf{L}_{EM} \coloneqq \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
$$
and the [[BV-BRST differential]] acts as:
$$
\array{
& &\array{ \text{BV-BRST} \\ \text{differential} }&
\\
\array{
\text{ electromagnetic field }
\\
\text{ ("vector potential") }
}
& a_\mu &\mapsto& c_{,\mu} & \text{gauge transformation}
\\
\phantom{A}
\\
\text{ ghost field }
& c &\mapsto& 0 & \text{abelian Lie algebra}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{electromagnetic field}
}
& (a^\ddagger)^\mu &\mapsto& f^{\nu \mu}_{,\nu} & \text{equations of motion}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{ghostfield}
}
& c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{Noether identity}
\\
\phantom{A}
\\
\text{Nakanishi-Lautrup field}
&
b &\mapsto& 0 & \text{vanishing of auxiliary fields...}
\\
\phantom{A}
\\
\text{antighost field}
&
\overline{c} &\mapsto& b & \text{... in cohomology}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{ Nakanishi-Lautrup field }
}
&
b^\ddagger &\mapsto& -\overline{c}^\ddagger
&
\text{vanishing of antifields of auxiliary fields...}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{antighost field}
}
&
\overline{c}^\ddagger &\mapsto& 0
&
\text{... in cohomology}
}
$$
Introduce a [[trivial vector bundle|trivial]] [[real line bundle]] for [[auxiliary fields]] $b$
in degree 0 and their [[antighost fields]] $\overline{c}$ (def. \ref{AuxiliaryFields}) in degree -1:
$$
\array{
&
\Sigma \times \langle \overline{c}\rangle
&\overset{ \overline{c} \mapsto b}{\longrightarrow}&
\Sigma \times\langle b\rangle
\\
deg = & -1 && 0
}
\,.
$$
In the present context the [[auxiliary field]] $b$ is called the _[[abelian Lie algebra|abelian]] [[Nakanishi-Lautrup field]]_.
The corresponding [[BV-BRST complex]] with [[auxiliary fields]] adjoined,
which, by prop. \ref{QuasiIsomorphismAdjoiningAuxiliaryFields}, is [[quasi-isomorphism|quasi-isomorphic]] to the original one above,
has coordinate generators
$$
\array{
& c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
\\
& & \overline{c} & b
\\
& & b^{\ddagger} & \overline{c}^\ddagger
\\
deg =
&
-2 & -1 & 0 & 1
}
\,.
$$
and [[BV-BRST differential]] given by the [[local antibracket]] (def. \ref{LocalAntibracket}) with $-\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$:
$$
s
\;=\;
\left\{
\left(
- \underset{ = L_{EM}}{\underbrace{\tfrac{1}{2}f_{\mu \nu} f^{\mu \nu}}}
+
\underset{ = L_{BRST} }{\underbrace{ c_{,\mu} (a^\ddagger)^\mu }}
+
\underset{ = L_{aux} }{\underbrace{ b \overline{c}^{\ddagger} }}
\right)
dvol_\Sigma
\,,\,
(-)
\right\}
$$
We say that the [[gauge fixing Lagrangian]] (def. \ref{GaugeFixingLagrangianDensity}) for [[Gaussian-averaged Lorenz gauge]]_ for the [[electromagnetic field]]
$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1}_\Sigma\left(
E \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1]
\right)
\,.
$$
is given by ([Henneaux 90 (103a)](Nakanishi-Lautrup+field#Henneaux90))
$$
\label{GaugeFixingLagrangianForGaussianAveragedLorentzGauge}
\mathbf{L}_{gf}
\;\coloneqq\;
\underset{deg = -1}{
\underbrace{
\phantom{A}\overline{c}\phantom{A}
}}
\underset{deg = 0}{\underbrace{( b - a^{\mu}_{,\mu} )}} \, dvol_\Sigma
\,.
$$
We check that this really is a [[gauge fixing Lagrangian density]] according to def. \ref{GaugeFixingLagrangianDensity}:
From (eq:VacuumEMLagrangianDensityRecalledForNLFields) and (eq:GaugeFixingLagrangianForGaussianAveragedLorentzGauge) we find the [[local antibracket|local antibrackets]] (def. \ref{LocalAntibracket}) with this [[gauge fixing Lagrangian density]] to be
$$
\begin{aligned}
\left\{\mathbf{L}_{gf}\,,\,\left( - \mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \right\}
& =
\left\{
\overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
\left(
-\tfrac{1}{2}f_{\mu \nu}f^{\mu \nu}
+
c_{,\mu} (a^\ddagger)^\mu
+
b \overline{c}^\ddagger
\right)
dvol_\Sigma
\right\}
\\
& =
\left\{
\overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
b \overline{c}^{\ddagger} \, dvol_\Sigma
\right\}
+
\left\{
\overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
c_{,\mu} (a^{\ddagger})^\mu \, dvol_\Sigma
\right\}
\\
& =
-
\left(
b ( b - a^{\mu}_{,\mu} ) + \overline{c}_{,\mu} c^{,\mu}
\right)
\, dvol_\Sigma
\\
\phantom{A}
\\
\{ \mathbf{L}_{gf}, \{ \mathbf{L}_{gf} , (-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} )\}\}
& = 0
\end{aligned}
$$
(So we are in the traditional situation of example \ref{GaugeFixingViaAntiLagrangianSubspaces}.)
Therefore the corresponding [[gauge fixing|gauge fixed]] [[Lagrangian density]] (eq:GaugeFixedLagrangianDensity) is (see also [Henneaux 90 (103b)](Nakanishi-Lautrup+field#Henneaux90)):
$$
\label{GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime}
\begin{aligned}
-\mathbf{L}' + \mathbf{L}'_{BRST}
& \coloneqq
e^{\left\{ \mathbf{L}_{gf} ,-\right\}}\left( -\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right)
\\
& =
-
\underset{
= \mathbf{L}'
}{
\underbrace{
\left(
\underset{ = L_{EM} }{
\underbrace{
\tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
}
}
+
\underset{ = -\left\{ L_{gf}, L_{BRST} + L_{aux} \right\} }{
\underbrace{
b ( b - a^{\mu}_{,\mu} )
+
\overline{c}_{,\mu} c^{,\mu}
}
}
\right) dvol_\Sigma
}
}
\;+\;
\underset{ = \mathbf{L}'_{BRST} }{
\underbrace{
\left(
\underset{ = L_{BRST} }{
\underbrace{
c_{,\mu} (a^\ddagger)^\mu
}
}
+
\underset{ = L_{aux} }{
\underbrace{
b \overline{c}^\ddagger
}
}
\right)
dvol_\Sigma
}
}
\end{aligned}
\,.
$$
The [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (def.
\ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime})
induced by the gauge fixed Lagrangian density $\mathbf{L}'$ at antifield degree 0 are (using (eq:ElectromagneticFieldEulerLagrangeForm)):
$$
\label{LorenzGaugeFixedEOMForVacuumElectromagnetism}
\delta_{EL} \mathbf{L}' \;=\; 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\left\{
\begin{aligned}
-\frac{d}{d x^\mu} f^{\mu \nu} & = b^{,\nu}
\\
b & = \tfrac{1}{2} a^\mu_{,\mu}
\\
c_{,\mu}{}^{,\mu} & = 0
\\
\overline{c}_{,\mu}{}^{,\mu} & = 0
\end{aligned}
\right.
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\left\{
\begin{aligned}
\Box a^\mu & = 0
\\
b & = \tfrac{1}{2} div a
\\
\Box c & = 0
\\
\Box \overline{c} & = 0
\end{aligned}
\right.
$$
(e.g. [Rejzner 16 (7.15) and (7.16)](Nakanishi-Lautrup+field#Rejzner16)).
(Here in the middle we show the equations as the appear directly from the [[Euler-Lagrange variational derivative]]
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}).
The [[differential operator]] $\Box = \eta^{\mu \nu} \frac{d}{d x^\mu} \frac{d}{d x^\nu} $ on the right is the [[wave operator]] (example \ref{EquationOfMotionOfFreeRealScalarField})
and $div$ denotes the [[divergence]].
The equivalence to the equations on the right follows from using in the first equation the derivative of the second equation
on the left, which is $b^{,\nu} = \tfrac{1}{2} a^{\mu,\nu}{}_{,\mu}$
and recalling the definition of the universal [[Faraday tensor]] (eq:FaradayTensorJet):
$\frac{d}{d x^\mu} f^{\mu \nu} = \tfrac{1}{2} \left( a^{\nu,\mu}{}_{,\mu} - a^{\mu,\nu}{}_{,\mu} \right)$.)
Now the [[differential equations]] for [[gauge fixing|gauge-fixed]] [[electromagnetism]] on the right in (eq:LorenzGaugeFixedEOMForVacuumElectromagnetism) are
nothing but the [[wave equations]] [[equations of motion|of motion]] of $(p+1) + 1 + 1$ [[free field theory|free]] [[mass|massless]] [[scalar fields]] (example \ref{EquationOfMotionOfFreeRealScalarField}).
As such, by example \ref{GreenHyperbolicKleinGordonEquation} they are a system of [[Green hyperbolic differential equations]] (def. \ref{GreenHyperbolicDifferentialOperator}),
hence admit [[Cauchy surfaces]] (def. \ref{CauchySurface}).
Therefore (eq:GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime) indeed is a [[gauge fixing]] of
the [[Lagrangian density]] of the [[electromagnetic field]] on [[Minkowski spacetime]] according to def. \ref{GaugeFixingLagrangianDensity}.
The gauge-fixed [[BRST operator]] induced from the gauge fixed Lagrangian density (eq:GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime) acts as
$$
\label{GaussianAveragedLorentzGaugeFixedBRSTOperator}
\array{
& \array{ s'_{BRST} = \\ \left\{ \left( c_{,\mu} (a^\ddagger)^\mu + b \overline{c}^{\ddagger}\right) dvol_\Sigma, (-) \right\} }
\\
a_\mu &\mapsto& c_{,\mu}
\\
b &\mapsto& 0
\\
\overline{c} &\mapsto & b
}
$$
=--
From this we immediately obtain the [[propagators]] for the gauge-fixed [[electromagnetic field]]:
+-- {: .num_prop #PhotonPropagatorInGaussianAveragedLorenzGauge}
###### Proposition
**([[photon propagator]] in [[Gaussian-averaged Lorenz gauge]])**
After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the [[causal propagator]] (prop. \ref{GreenFunctionsAreContinuous}) of the combined [[electromagnetic field]] and [[Nakanishi-Lautrup field]] is of the form
$$
\Delta^{EM, EL}
\;=\;
\left(
\array{
\Delta^{photon} & \ast
\\
\ast & \ast
}
\right)
$$
with
$$
\Delta^{photon}_{\mu \nu}(x,y)
\;=\;
\eta_{\mu \nu} \Delta(x,y)
\,,
$$
where
1. $\eta_{\mu \nu}$ is the [[Minkowski metric]] [[tensor]] (def. \ref{MinkowskiSpacetime});
1. $\Delta(x,y)$ is the [[causal propagator]] of the [[free field theory]] [[mass|massless]] [[real scalar field]] (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}).
Accordingly the [[Feynman propagator]] of the [[electromagnetic field]] in [[Gaussian-averaged Lorenz gauge]] is
$$
(\Delta^{photon}_F)_{\mu \nu}(x,y)
\;=\;
\eta_{\mu \nu} \Delta_F(x,y)
\,,
$$
where on the right $\Delta_F(x,y)$ is the [[Feynman propagator]] of the [[free field theory|free]] [[mass|massless]] [[real scalar field]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}).
This is also called the _[[photon propagator]]_.
Hence by prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the [[Fourier transform of distributions|distributional Fourier transform]] of the photon propagator is
$$
\widehat{\Delta^{photon}_F}_{\mu \nu}(k)
\;=\;
\frac{1}{- k^\mu k_\mu + i 0^+}
\,.
$$
=--
(this is a special case of [Khavkine 14 (99)](gauge+fixing#Khavkine14), see also [Rejzner 16, (7.20)](perturbative+algebraic+quantum+field+theory#Rejzner16))
+-- {: .proof}
###### Proof
The Gaussian-averaged Lorenz gauge-fixed equations of motion (eq:LorenzGaugeFixedEOMForVacuumElectromagnetism) of the electromagnetic field are just $(p+1)$ uncoupled [[mass|massless]] [[Klein-Gordon equations]], hence [[wave equations]] (example \ref{EquationOfMotionOfFreeRealScalarField}) for the $(p+1)$ real components of the [[electromagnetic field]] ("[[vector potential]]")
$$
\Box A_\mu = 0\phantom{AAAA} \mu \in \{0,1,\cdots, p\}
\,.
$$
This shows that the propoagator is proportional to that of the [[real scalar field]].
To see that the index structure is as claimed, recall that the [[domain]] and [[codomain]] of the [[advanced and retarded propagators]] in def. \ref{AdvancedAndRetardedGreenFunctions} is
$$
\array{
\Gamma_\Sigma(T\Sigma)
&\overset{\left( (\mathrm{G}_{\pm})_{\mu \nu} \right)}{\longrightarrow}&
\Gamma_\Sigma(T^\ast \Sigma)
}
$$
corresponding to a [[differential operator]] for the [[equations of motion]] which by (eq:ElectromagneticFieldEulerLagrangeForm) and (eq:LorenzGaugeFixedEOMForVacuumElectromagnetism) is given by
$$
\array{
\Gamma_\Sigma(T^\ast \Sigma)
&\overset{ \eta^{-1} \circ \Box }{\longrightarrow}&
\Gamma_\Sigma(T \Sigma)
\\
A_\mu &\mapsto& \eta^{\mu \nu} \Box A_\nu
}
$$
Then the defining equation (eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator) for the [[advanced and retarded Green functions]] is, in terms of their [[integral kernels]], the [[advanced and retarded propagators]] $\Delta_{\pm}$
$$
\eta^{\mu' \mu} \Box \underset{y \in X}{\int} (\Delta_{\pm})_{\mu \nu}((-),y) A^{\nu}(y) \, dvol_\Sigma(x)
=
A^\nu(x)
\,.
$$
This shows that
$$
(\Delta_{\pm})_{\mu \nu}
\;=\;
\eta_{\mu\nu} \Delta_{\pm}
$$
with $\Delta_{\pm}$ the [[advanced and retarded propagator]] of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}), and hence
$$
\begin{aligned}
\Delta_{\mu \nu}
&=
(\Delta_+)_{\mu \nu}
-
(\Delta_-)_{\mu \nu}
\\
& =
\eta_{\mu \nu} (\Delta_+ - \Delta_-)
\\
& =
\eta_{\mu \nu} \Delta
\end{aligned}
$$
=--
Next we compute the gauge-invariant on-shell polynomial observables of the electromagnetic field. The result will involve the following concept:
+-- {: .num_defn #LinearObservablesOfElectromagenticFieldWavePolarization}
###### Definition
**([[wave polarization]] of linear [[observables]] of the [[electromagnetic field]])**
Consider the [[electromagnetic field]] on [[Minkowski spacetime]] $\Sigma$, with [[field bundle]] the [[cotangent bundle]]
The space of off-shell linear observables is spanned by the point evaluation observables
$$
e^\mu \mathbf{A}_\mu(x)
\;\in\;
LinObs(T^\ast \Sigma)
$$
where
1. $e = (e^\mu) \in \mathbb{R}^{p,1}$ is some vector;
1. $x \in \mathbb{R}^{p,1}$ is some point in Minkowski spacetime
1. $\mathbf{A}_\mu(x) \;\colon\; A \mapsto A_\mu(x)$
is the functional which sends a section $A \in \Gamma_\Sigma(E) = \Omega^1(\Sigma)$ to its $\mu$-component at $x$.
After [[Fourier transform of distributions]] this is
$$
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\in\;
LinObs(T^\ast \Sigma)
$$
for $k = (k_\mu) \in (\mathbb{R}^{p,1})^\ast$ the _[[wave vector]]_
for $e = (e^\mu) \in \mathbb{R}^{p,1}$ the _[[wave polarization]]_
The linear [[on-shell]] observables are spanned by the same expressions, but subject to the condition that
$$
{\vert k\vert}_\eta^2 = k^\mu k_\mu = 0
$$
hence
$$
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})
\;=\;
\left\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0
\right\rangle
$$
We say that the space of _[[transversal polarization|transversally polarized]]_ linear on-shell observables
is the [[quotient vector space]]
$$
\label{ElectromagneticFieldLinearObservablesTransversallyPolarized}
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\;\coloneqq\;
\frac{
\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0
\rangle
}{
\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0 \,\, \text{and} \,\, e_\mu \propto k_\mu
\rangle
}
$$
of those observables whose [[Fourier transform|Fourier modes]] involve [[wave polarization]] vectors $e$ that vanish when contracted with the
[[wave vector]] $k$, modulo those whose [[wave polarization]] vector $e$ is proportional to the [[wave vector]].
For example if $k = (\kappa, 0, \cdots, \kappa)$, then the corresponding space of transversal polarization vectors may be identified
with $\left\{e \,\vert\, e = (0,e_1, e_2, \cdots, e_{p-1}, 0) \right\}$.
=--
+-- {: .num_prop #GausianAveragedLorenzGaugeFixedLinearObservablesOfTheElectromagneticField}
###### Proposition
**([[BRST cohomology]] on linear [[on-shell]] [[observables]] of the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] [[electromagnetic field]])**
After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the global [[BRST cohomology]] (def. \ref{ComplexBVBRSTGlobal}) on the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] (def. \ref{NLGaugeFixingOfElectromagnetism}) [[on-shell]] [[linear observables]] (def. \ref{LinearObservables}) at $deg_{gh} = 0$ (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) is [[isomorphism|isomorphic]] to the space of transversally polarized linear observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}:
$$
H^0( LinObs( T^\ast \Sigma \times_\Sigma A \times_\Sigma A[-1] \times_\Sigma \mathcal{G}[1], \mathbf{L}' ), s'_{BRST} )
\;\simeq\;
LinObs( T^\ast \Sigma, \mathbf{L}_{EM})_{trans}
\,.
$$
=--
(e.g. [Dermisek 09 II-5, p. 325](wave+polarization#Dermisek09))
+-- {: .proof}
###### Proof
The gauge fixed BRST differential (eq:GaussianAveragedLorentzGaugeFixedBRSTOperator) acts on the [[Fourier transform|Fourier modes]] of the linear observables (def. \ref{LinearObservables}) as follows
$$
\array{
& & s'_{BRST}
\\
\array{ \text{antighost} \\ \text{field} }
&
\widehat{\overline{\mathbf{C}}}(k)
&\mapsto&
\widehat{\mathbf{B}}(k) & \array{ \text{Nakanishi-Lautrup} \\ \text{field} }
\\
\phantom{a}
\\
&&& \underset{\text{on-shell}}{=}
\tfrac{i}{2} k^\mu \widehat{\mathbf{A}}_\mu(k)
&
\array{ \text{Lorenz gauge} \\ \text{condition} }
\\
\phantom{A}
\\
\array{ \text{electromagnetic} \\ \text{field} }
&
e^\mu \widehat{\mathbf{A}}_\mu(k)
&\mapsto&
i \left(e^\mu k_\mu\right) \widehat{\mathbf{C}}(k)
&
\array{
\text{polarization contracted}
\\
\text{with wave vector}
\\
\text{times ghost field}
}
\\
\phantom{A}
\\
\array{ \text{Nakanishi-Lautrup} \\ \text{field} }
& \widehat{\mathbf{B}}
&\mapsto&
0
}
$$
This impies that the gauge fixed [[BRST cohomology]] on linear on-shell observables at $deg_{gh} = 0$ is the space of transversally polarized linear observables (def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}):
$$
\label{LinearOnShellObservablesGaugeFixedBRSTCohomologyForEMField}
\begin{aligned}
H^0(LinObs(E,\mathbf{L}_{EM}), s'_{BRST})
& =
\left\langle
\frac{
\left\{
e^\mu \widehat{\mathbf{A}}_{\mu}(k)
\,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\,0 = d_{BRST}\left( e^\mu \widehat{\mathbf{A}}_\mu(k)
\right) = i (e^\mu k_\mu) \widehat{\mathbf{C}}(k)
\right\}
}{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k)
\,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\, e^\mu \widehat{\mathbf{A}}_\mu(k) \propto s'_{BRST}( \widehat{\overline{\mathbf{C}}}(k) ) = \tfrac{i}{2} k^\mu \widehat{ \mathbf{A} }_\mu(k)
\right\}
}
\right\rangle
\\
& =
\left\langle
\frac{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0
\right\}
}
{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu \propto k^\mu
\right\}
}
\right\rangle
\\
& =
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\end{aligned}
$$
Here the first line is the definition of [[cochain cohomology]] (using that both $\widehat{\mathbf{B}}$ and $\widehat{\overline{\mathbf{C}}}$ are immediately seen to vanish in cohomology), the second line is spelling out the action of the BRST operator and using the on-shell relations (eq:LorenzGaugeFixedEOMForVacuumElectromagnetism) for $\widehat{\mathbf{B}}$ and the last line is by def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}.
=--
As a corollary we obtain:
+-- {: .num_prop #BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}
###### Proposition
**([[BRST cohomology]] on polynomial [[on-shell]] [[observables]] of the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] [[electromagnetic field]])**
After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the global [[BRST cohomology]] (def. \ref{ComplexBVBRSTGlobal}) on the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] (def. \ref{NLGaugeFixingOfElectromagnetism}) polynomial on-shell observables (def. \ref{PolynomialObservables}) at $deg_{gh} = 0$ (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) is [[isomorphism|isomorphic]] to the distributional polynomial algebra on transversally polarized linear observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}:
$$
\label{EMBRSTCohomologyOnPolynomialOnShellObservables}
H^0(PolyObs( T^\ast \Sigma \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1] ,\mathbf{L}), s'_{BRST})
\;\simeq\;
Sym\left(
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\right)
$$
=--
+-- {: .proof}
###### Proof
Generally, if $(V^\bullet,d)$ is a cochain complex over a [[ground field]] of [[characteristic zero]]
(such as the [[real numbers]] in the present case) and $Sym(V^\bullet,d)$ the differential graded-[[symmetric algebra]] that it induces ([this example](symmetric+algebra#SymmetricAlgebraInCoChainComplexes)), then
$$
H^\bullet(Sym(V,d)) = Sym(H^\bullet(V,d))
\,.
$$
(by [this prop.](CochainCohomologyOfSymmetricAlgebraOnCochainComplex)).
=--
In conclusion we finally obtain:
+-- {: .num_prop #GaugeInvariantPolynomialOnShellObservablesOfFreeElectromagneticField}
###### Proposition
**(gauge-invariant polynomial [[on-shell]] [[observables]] of the [[free field theory]] [[electromagnetic field]])**
The [[BV-BRST cohomology]] on infinitesimal observables (def. \ref{LocalObservablesOnInfinitesimalNeighbourhood})
of the [[free field theory|free]] [[electromagnetic field]] on [[Minkowski spacetime]] (example \ref{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}) at $deg_{gh} = 0$
is the distributional polynomial algebra in the transversally polarized linear on-shell observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}, as in prop. \ref{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}.
=--
+-- {: .proof}
###### Proof
By the classes of [[quasi-isomorphisms]] of prop. \ref{ExponentialOfLocalAntibracket} and prop. \ref{QuasiIsomorphismAdjoiningAuxiliaryFields} we may equivalently compute the cohomology if the [[BV-BRST complex]] with differential $s'$, obtained after [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] from example \ref{NLGaugeFixingOfElectromagnetism}. Since the [[equations of motion]] (eq:LorenzGaugeFixedEOMForVacuumElectromagnetism) are manifestly [[Green hyperbolic differential equations]] after this gauge fixing [[Cauchy surfaces]] for the [[equations of motion]] exist and hence prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} together with prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres} implies that the gauge fixed BV-complex $s'_{BV}$ has its cohomology concentrated in degree zero on the [[on-shell]] observables. Therefore prop. \ref{CochainCohomologyOfBVBRSTComplexInDegreeZero} (i.e. the collapsing of the [[spectral sequence]] for the BV/BRST [[bicomplex]]) implies that the gauge fixed BV-BRST cohomology at ghost number
zero is given by the on-shell BRST-cohomology. This is characterized by prop. \ref{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}.
=--
$\,$
This concludes our discussion of [[gauge fixing]]. With the [[covariant phase space]] for [[gauge theories]] obtained thereby,
we may finally pass to the [[quantization]] of field theory to [[quantum field theory]] proper, in the [next chapter](#Quantization).
|
A first idea of quantum field theory -- Gauge symmetries | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Gauge+symmetries |
## Gauge symmetries
{#GaugeSymmetries}
In this chapter we discuss these topics:
* [compactly supported infinitesimal symmetries obstruct covariant phase space](#CompactlySupportedInfinitesimalSymmetriesObstructTheCovariantPhaseSpace)
* [Infinitesimal gauge symmetries](#InfinitesimalGauge)
* [Lie algebra action and Lie algebroids](#LieAlgebraActionAndLieAlgebroids)
* [BRST complex](#BRSTComplex)
* [Examples of local BRST complexes](#ExamplesForLocalBRSTComplexes)
$\,$
An [[infinitesimal gauge symmetry]] of a [[Lagrangian field theory]] (def. \ref{GaugeParameters} below) is a [[infinitesimal symmetry of the Lagrangian]] which may be freely parameterized, hence "gauged", by a _[[gauge parameter]]_. A [[Lagrangian field theory]]
exhibiting these is also called a _[[gauge theory]]_.
By choosing the [[gauge parameter]] to have [[compact support]], [[infinitesimal gauge symmetries]] in particular yield
[[infinitesimal symmetries of the Lagrangian]] with compact spacetime support.
One finds (prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} below) that the existence of [[on-shell]] non-trivial symmetries of this form is an [[obstruction]] to
the existence of the [[covariant phase space]] of the theory (prop. \ref{CovariantPhaseSpace}).
$\,$
**[[gauge symmetries]]**
| name | meaning | def. |
|------|----------|------|
| [[infinitesimal symmetry of the Lagrangian]] | [[evolutionary vector field]] which leaves [[invariant]] the [[Lagrangian density]] up to a [[total spacetime derivative]] | def. \ref{SymmetriesAndConservedCurrents} |
| spacetime-compactly supported [[infinitesimal symmetry of the Lagrangian]] | [[obstruction|obstructs]] existence of the [[covariant phase space]] (if non-trivial [[on-shell]]) | prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} |
| [[infinitesimal gauge symmetry]] | [[gauge parameter|gauge parameterized]] collection of [[infinitesimal symmetries of the Lagrangian]]; <br/> for [[compact support|compactly supported]] [[gauge parameter]] this yields spacetime-compactly supported infinitesimal symmetries | def. \ref{GaugeParameters} |
| [[rigid infinitesimal symmetry of the Lagrangian]] | infinitesimal symmetry modulo gauge symmetry | def. \ref{RigidInfinitesimalSymmetriesOfTheLagrangian} |
| generating set of [[gauge parameters]] | reflects all the [[Noether identities]] | remark \ref{GeneratingSetOfGaugeTransformations} |
| closed [[gauge parameters]] | [[Lie bracket]] of [[infinitesimal gauge symmetries]] closes on [[gauge parameters]] | def. \ref{GaugeParametersClosed} |
$\,$
But we may hard-wire these [[gauge equivalences]] into the very [[geometry]] of the [[types]] of [[field (physics)|fields]]
by forming the _[[homotopy quotient]]_ of the _[[action]]_ of the [[infinitesimal gauge symmetries]] on the [[jet bundle]].
This [[homotopy quotient]] is modeled by the _[[action Lie algebroid]]_ (def. \ref{ActionLieAlgebroid} below).
Its [[algebra of functions]] is the _[[local BRST complex]]_ of the theory (def. \ref{LocalOffShellBRSTComplex}) below.
In this construction the [[gauge parameters]] appear as [[auxiliary fields]] whose [[field bundle]]
is a _[[graded manifold|graded]]_ version of the [[gauge parameter]]-bundle. As such they are called
_[[ghost fields]]_. The ghost fields may have [[infinitesimal gauge symmetries]] themselves
which leads to [[ghost-of-ghost fields]], etc. (example \ref{LocalBRSTComplexBFieldMinkowskiSpacetime}) below.
It is these [[auxiliary fields|auxiliary]] [[ghost fields]] and [[ghost-of-ghost fields]] which will serve to remove the [[obstruction]] to the existence of the [[covariant phase space]] for [[gauge theories]], this we arrive at in _[Gauge fixing](#GaugeFixing)_, further below.
**[[gauge parameters]] and [[ghost fields]]**
| symbol | meaning | def. |
|--------|---------|--------|
| $\mathcal{G} \overset{gb}{\to} \Sigma$ | [[gauge parameter]] bundle | def. \ref{GaugeParameters} |
| $c^\alpha \in C^\infty(\mathcal{G})$ | [[coordinate function]] on [[gauge parameter]] bundle | |
| $\epsilon \in \Gamma_\Sigma(\mathcal{G})$ | [[gauge parameter]] | |
| $\mathcal{G}[1]$ | [[gauge parameter]] [[bundle]] regarded as [[graded manifold]] in degree 1 | expl. \ref{LocalOffShellBRSTComplex} |
| $C \in \Gamma_\Sigma(\mathcal{G}[1])$ | [[ghost field|gost]] [[field history]] | |
| $\underset{deg = 1}{\underbrace{c^\alpha}} \in C^\infty(\mathcal{G}[1])$ | [[ghost field]] component function | |
| $\underset{deg = 1}{\underbrace{c^\alpha_{,\mu_1 \cdots \mu_k}}} \in C^\infty(J^\infty_\Sigma(\mathcal{G}[1]))$ | [[ghost field]] [[jet]] component function | |
| $\phantom{A}$ | | |
| $\overset{(2)}{\mathcal{G}} \overset{gb}{\to} \Sigma$ | [[gauge-of-gauge transformation|gauge-of-gauge parameter]] bundle | expl. \ref{LocalBRSTComplexBFieldMinkowskiSpacetime} |
| $\overset{(2)}{c}^\alpha \in C^\infty(\overset{(2)}{\mathcal{G}})$ | [[coordinate function]] on [[gauge-of-gauge transformation|gauge-of-gauge parameter]] bundle | |
| $\overset{(2)}{\epsilon} \in \Gamma_\Sigma(\mathcal{G})$ | gauge-of-gauge parameter | |
| $\overset{(2)}{\mathcal{G}}[2]$ | [[gauge-of-gauge transformation|gauge-of-gauge parameter]] [[bundle]] regarded as [[graded manifold]] in degree 1 | |
| $\overset{(2)}{C} \in \Gamma_\Sigma(\mathcal{G}[1])$ | [[ghost-of-ghost field|gost-of-ghost]] [[field history]] | |
| $\underset{deg = 2}{\underbrace{\overset{(2)}{c}{}^\alpha}} \in C^\infty(\overset{(2)}{\mathcal{G}}[2])$ | [[ghost-of-ghost field]] component function | |
| $\underset{deg = 2}{\underbrace{{\overset{(2)}{c}}{}^\alpha_{,\mu_1 \cdots \mu_k}}} \in C^\infty(J^\infty_\Sigma(\overset{(2)}{\mathcal{G}}[2]))$ | [[ghost-of-ghost field]] [[jet]] component function | |
The mathematical theory capturing these phenomena is the [[higher Lie theory]] of [[Lie-∞ algebroids]] (def. \ref{LInfinityAlgebroid} below).
$\,$
**[[compact support|compactly supported]] [[infinitesimal symmetries of the Lagrangian|infinitesimal symmetries]] [[obstruction|obstruc]] the [[covariant phase space]]**
{#CompactlySupportedInfinitesimalSymmetriesObstructTheCovariantPhaseSpace}
As an immediate corollary of prop. \ref{FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories} we have the following important observation:
+-- {: .num_prop #NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces}
###### Proposition
**(spacetime-compactly supported and [[on-shell]] non-trivial [[infinitesimal symmetries of the Lagrangian]] [[obstruction|obstruct]] the [[covariant phase space]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]].
If there exists a single [[infinitesimal symmetry of the Lagrangian]] $v$ (def. \ref{SymmetriesAndConservedCurrents}) such that
1. it has compact spacetime support (def. \ref{SpacetimeSupport})
1. it does not vanish [[on-shell]] (eq:ProlongedShellInJetBundle) (so not a trivial one, example \ref{TrivialImplicialInfinitesimalGaugeTransformations})
then there does not exist any [[Cauchy surface]] (def. \ref{CauchySurface}) for the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) outside the spacetime support of $v$.
=--
+-- {: .proof}
###### Proof
By prop. \ref{FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories}
the flow along $\hat v$ preserves the [[on-shell]] [[space of field histories]]. Now by the assumption that $\hat v$ does not vanish [[on-shell]] implies that this flow is non-trivial, hence that it does continuously change the [[field histories]] over some points of spacetime, while the assumption that it has compact spacetime support means that these changes are confined to a [[compact subset]] of [[spacetime]].
This means that there is a continuum of solutions to the [[equations of motion]] whose restriction to the [[infinitesimal neighbourhood]] of any [[codimension]]-1 suface $\Sigma_p \hookrightarrow \Sigma$ outside of this compact support coincides. Therefore this restriction map is not an [[isomorphism]] and $\Sigma_p$ not a [[Cauchy surface]] for the [[equations of motion]].
=--
Notice that there always exist spacetime-compactly supported infinitesimal symmetries that however do vanish [[on-shell]]:
+-- {: .num_example #TrivialImplicialInfinitesimalGaugeTransformations}
###### Example
**(trivial compactly-supported [[infinitesimal symmetries of the Lagrangian]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}), so that the [[Lagrangian density]]
is canonically of the form
$$
\mathbf{L} = L \, dvol_\Sigma
$$
with [[Lagrangian function]] $L \in \Omega^{0,0}_\Sigma(E) = C^\infty(J^\infty_\Sigma(E))$ a [[smooth function]] of the [[jet bundle]] (characterized by prop. \ref{JetBundleIsLocallyProManifold}).
Then every [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) whose [[coefficients]]
which is proportional to the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral) of the [[Lagrangian function]] $L$
$$
v
\;
\coloneqq
\;
\frac{\delta_{EL} L }{\delta \phi^a} \kappa^{[a b]} \, \partial_{\phi^a}
\;\in\;
\Gamma_E^{ev}( T_\Sigma E )
$$
by smooth coefficient functions $\kappa^{a b}$
$$
\kappa^{[a b]} \;\in\; \Omega^{0,0}_\Sigma(E)
$$
such that
1. each $\kappa^{a b}$ has compact spacetime support (def. \ref{SpacetimeSupport})
1. $\kappa$ is skew-symmetric in its indices: $\kappa^{[a b]} = - \kappa^{[b a]}$
is an implicit infinitesimal gauge symmetry (def. \ref{ImplicitInfinitesimalGaugeSymmetry}).
This is so for a "trivial reason" namely due to that that skew symmetry:
$$
\begin{aligned}
\mathcal{L}_{\hat v} \mathbf{L}
& =
\iota_{\hat v} \delta \mathbf{L}
\\
&=
\iota_{\hat v} ( \delta_{EL}\mathbf{L} - d \Theta_{BFV} )
\\
& =
\iota_\epsilon \frac{\delta_{EL}L}{\delta \phi^a} \delta \phi^a + d \iota_{\hat v}\Theta_{BFV}
\\
& =
\underset{= 0}{
\underbrace{
\left( \frac{\delta_{EL} L }{\delta \phi^a} \right)
\left( \frac{\delta_{EL} L }{\delta \phi^b} \right)
\kappa^{[a b]}
}
}
\,
dvol_\Sigma
\;+\; d \iota_{\hat v} \Theta_{BFV}
\\
& =
d \iota_{\hat v} \Theta_{BFV}
\end{aligned}
$$
Here the first steps are just recalling those in the proof of [[Noether's theorem|Noether's theorem I]]
(prop. \ref{NoethersFirstTheorem}) while the last step follows with the skew-symmetry of $\kappa$.
Notice that this means that
1. the [[Noether current]] (eq:NoetherCurrent) vanishes: $J_{\hat v} = 0$;
1. the infinitesimal symmetry vanishes [[on-shell]] (eq:InclusionOfOnShellSpaceOfFieldHistories): $\hat v \vert_{\mathcal{E}} = 0$.
Therefore these implicit infinitesimal gauge symmetries
are called the _trivial infinitesimal gauge transformations_.
=--
(e.g. [Henneaux 90, section 2.5](BRST+complex#Henneaux90))
$\,$
Proposition \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} implies
that we need a good handle on determining whether the space of non-trivial
compactly supported [[infinitesimal symmetries of the Lagrangian]] modulo trivial ones is non-zero. This
[[obstruction]] turns out to be neatly captured by methods of [[homological algebra]]
applied to the [[local BV-complex]] (def. \ref{BVComplexOfOrdinaryLagrangianDensity}):
+-- {: .num_example #InterpretationCohomologyOfBVComplex}
###### Example
**([[cochain cohomology]] of [[local BV-complex]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}),
and let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle).
By inspection we find that the [[cochain cohomology]] of the local [[BV-complex]]
$\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}$ (def. \ref{BVComplexOfOrdinaryLagrangianDensity}) has the following interpretation:
In degree 0 the [[image]] of the [[BV-differential]] coming from degree -1 and modulo $d$-exact terms
$$
im\left( \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma(E,\varphi)) \overset{s_{BV}}{\to} \Omega^{0,0}_\Sigma(E,\varphi)/im(d) \right)
$$
is the ideal of functions modulo $im(d)$ that vanish [[on-shell]]. Since the differential going _from_ degree 0 to degree 1 vanishes,
the [[cochain cohomology]] in this degree is the [[quotient ring]]
$$
\begin{aligned}
H^0\left(\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}\vert d\right)
& \simeq
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}/im(d)
\end{aligned}
$$
of functions on the [[shell]] $\mathcal{E}$ (eq:ObservablesOnInfinitesimalNeighbourhoodOfZeroInShellInFieldFiber).
In degree -1 the [[kernel]] of the [[BV-differential]] going to degree 0
$$
ker\left( \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma(E,\varphi)) \overset{s_{BV}}{\to} \Omega^{0,0}_\Sigma(E,\varphi)\right)
$$
is the space of implicit [[infinitesimal gauge symmetries]] (def. \ref{ImplicitInfinitesimalGaugeSymmetry})
and the [[image]] of the differential coming from degree -2
$$
im\left(
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)
\wedge_{\Omega^{0,0}_{\Sigma,cp}(E,\varphi)}
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)
\overset{s_{BV}}{\longrightarrow}
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)
\right)
$$
is the trivial implicit infinitesimal gauge transformations (example \ref{TrivialImplicialInfinitesimalGaugeTransformations}).
Therefore the [[cochain cohomology]] in degree -1 is the [[quotient space]] of implicit infinitesimal gauge transformations
modulo the trivial ones:
$$
\label{NegativeOneCohomologyBV}
H^{-1}\left( \Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}} \right)
\simeq
\frac{
\left\{
\text{implicit infinitesimal gauge transformations}
\right\}
}
{
\left\{
\text{ trivial implicit infinitesimal gauge transformations}
\right\}
}
$$
=--
+-- {: .num_prop #BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres}
###### Proposition
**(local [[BV-complex]] is [[homological resolution]] of the [[shell]] iff there are no non-trivial compactly supported [[infinitesimal symmetries of the Lagrangian|infinitesimal symmetries]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians})
and let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle).
Furthermore assume that $\mathbf{L}$ is at least quadratic in the vertical coordinates around $\varphi$.
Then the local [[BV-complex]] $\Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}$ of local observables (def. \ref{BVComplexOfOrdinaryLagrangianDensity}) is a [[homological resolution]] of the algebra of functions on the [[infinitesimal neighbourhood]] of $\varphi$ in the [[shell]] (example \ref{ShellForSpacetimeIndependentLagrangians}),
hence the canonical comparison morphisms (eq:ComparisonMorphismFromOrdinaryBVComplexToLocalObservables)
is a [[quasi-isomorphism]]
precisely if there is no non-trivial (example \ref{TrivialImplicialInfinitesimalGaugeTransformations}) implicit [[infinitesimal gauge symmetry]] (def. \ref{ImplicitInfinitesimalGaugeSymmetry}):
$$
\left(
\Omega^{0,0}_{\Sigma}(E,\varphi)\vert_{\mathcal{E}_{BV}}
\overset{\simeq}{\longrightarrow}
\Omega^{0,0}_{\Sigma}(E,\varphi)\vert_{\mathcal{E}}
\right)
\;\Leftrightarrow\;
\left(
\array{
\text{there are no non-trivial}
\\
\text{compactly supported infinitesimal symmetries}
}
\right)
\,.
$$
=--
+-- {: .proof}
###### Proof
By example \ref{InterpretationCohomologyOfBVComplex} the vanishing of compactly supported infinitesimal symmetries
is equivalent to the vanishing of the cochain cohomology of the local BV-complex in degree -1 (eq:NegativeOneCohomologyBV).
Therefore the statement to be proven is
equivalently that the [[Koszul complex]] of the sequence of elements
$$
\left(
\frac{\delta_{EL} L}{\delta \phi^a} \in \Omega^{0,0}_{\Sigma,\varphi}(E)
\right)_{a = 1}^s
$$
is a [[homological resolution]] of $\Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}}$, hence has vanishing cohomology in all negative degrees,
already if it has vanishing cohomology in degree -1.
By a standard fact about [[Koszul complexes]] ([this prop.](Koszul+complex#KoszulResolutionForNoetherianRngAndElementsInJacobson))
a sufficient condition for this to be the case is that
1. the ring $\Omega^{0,0}_{\Sigma}(E,\varphi)$ is the tensor product of $C^\infty(\Sigma)$ with a [[Noetherian ring]];
1. the elements $\frac{\delta_{EL} L }{\delta \phi^a}$ are contained in its [[Jacobson radical]].
The first condition is the case since $\Omega^{0,0}_{\Sigma}(E,\varphi)$ is by definition a [[formal power series ring]] over a [[field]]
tensored with $C^\infty(\Sigma)$
(by [this example](noetherian+ring#PolynomialAlgebraOverNoetherianRingIsNoetherian)).
Since the Jacobson radical of a power series algebra consists of those elements whose constant term
vanishes (see [this example](Koszul+complex#KoszulComplexForFormalPowerSeriesAlgebras)),
the assumption that $\mathbf{L}$ is at least quadratic, hence that $\delta_{EL}\mathbf{L}$
is at least linear in the fields, guarantees that all $\frac{\delta_{EL}L}{\delta \phi^a}$ are contained
in the Jacobson radical.
=--
Prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres}
says what [[gauge fixing]] has to accomplish: given a [[local BV-BRST complex]] we need
to find a [[quasi-isomorphism]] to another complex which is such that it comes from a
_graded_ Lagrangian density whose BV-cohomology vanishes in degree -1
and hence induces a graded covariant phase space, and such that
the remaining BRST differential respects the Poisson bracket on this graded covariant phase space.
$\,$
**[[infinitesimal gauge symmetries]]**
{#InfinitesimalGauge}
Prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} says that the problem is to identify the presence of spacetime-compactly supported infinitesimal symmetries that are on-shell non-trivial.
One way they may be identified is if [[infinitesimal symmetries]] appear in _linearly [[dependent type|parameterized collections]]_, where the parameter itself is an _arbitrary_ [[spacetime]]-dependent [[section]] of some [[fiber bundle]] (hence is itself like a [[field history]]), because then choosing the parameter to have [[compact support]] yields an [[infinitesimal symmetry of the Lagrangian]] with compact spacetime support (remark \ref{GaugeParametrizedInfinitesimalGaugeTransformation} below).
In this case we speak of a _[[gauge parameter]]_ (def. \ref{GaugeParameters} below). It turns out that in most examples of [[Lagrangian field theories]] of interest, the compactly supported infinitesimal symmetries all come from [[gauge parameters]] this way. Therefore we now consider this case in detail.
+-- {: .num_defn #GaugeParameters}
###### Definition
**([[infinitesimal gauge symmetries]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Then a collection of _[[infinitesimal gauge symmetries]]_ of $(E,\mathbf{L})$ is
1. a [[vector bundle]] $\mathcal{G} \overset{gb}{\longrightarrow} \Sigma$ over [[spacetime]] $\Sigma$ of [[positive number|positive]] [[rank of a vector bundle|rank]], to be called a _[[gauge parameter bundle]] _;
1. a [[bundle morphism]] (def. \ref{BundlesAndFibers}) $R$ from the
[[jet bundle]] of the [[fiber product]] $\mathcal{G} \times_\Sigma E$ with the [[field bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) to the [[vertical tangent bundle]] of $E$ (def. \ref{VerticalTangentBundle}):
$$
\array{
J^\infty_\Sigma( \mathcal{G} \times_\Sigma E )
&& \overset{R}{\longrightarrow} &&
T_\Sigma E
& \overset{i}{\hookrightarrow} &
T_\Sigma (\mathcal{G} \times_\Sigma E)
\\
& \searrow && \swarrow
\\
&& E
}
$$
such that
1. $R$ is [[linear map|linear]] in the first argument (in the [[gauge parameter]]);
1. $i \circ R$ is an [[evolutionary vector field]] on $\mathcal{G} \times_\Sigma E$ (def. \ref{EvolutionaryVectorField});
1. $R$ is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) in the second argument.
We may express this equivalently in components
in the case that the [[field bundle]] $E$ is a [[trivial vector bundle]] with [[field (physics)|field]] [[fiber]] coordinates
$(\phi^a)$ (example \ref{TrivialVectorBundleAsAFieldBundle})
and also $\mathcal{G}$ happens to be a [[trivial vector bundle]]
$$
\mathcal{G} = \Sigma \times \mathfrak{g}
$$
where $\mathfrak{g}$ is a [[vector space]] with [[coordinate functions]] $\{c^\alpha\}$.
Then $R$ may be expanded in the form
$$
\label{CoordinateExpressionForGaugeParameterized}
R
\;=\;
\left(
c^\alpha R^a_\alpha
+
c^\alpha_{,\mu} R^{a \mu}_\alpha
+
c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha
+
\cdots
\right)
\partial_{\phi^a}
\,,
$$
where the components
$$
R^{a \mu_1 \cdots \mu_k}_\alpha
=
R^{a \mu_1 \cdots \mu_k}_\alpha\left( (\phi^b), (\phi^b_{,\mu}), \cdots \right)
\;\in\; \Omega^{0,0}_\Sigma(E) = C^\infty(J^\infty_\Sigma(E))
$$
are [[smooth functions]] on the jet bundle of $E$,
locally of finite order (prop. \ref{JetBundleIsLocallyProManifold}), and such that the [[Lie derivative]]
of the [[Lagrangian density]] along $R(e)$ is a [[total spacetime derivative]], which by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) means in components that
$$
\left(
c^\alpha R^a_\alpha
+
c^\alpha_{,\mu} R^{a \mu}_\alpha
+
c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha
+
\cdots
\right)
\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}
\;=\;
\frac{d}{d x^\mu} J^\mu_{R}
\,.
$$
=--
(e.g. [Henneaux 90 (3)](BRST+complex#Henneaux90))
The point is that [[infinitesimal gauge symmetries]] in particular yield spacetime-compactly supported infinitesimal gauge symmetries as in prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces}:
+-- {: .num_remark #GaugeParametrizedInfinitesimalGaugeTransformation}
###### Remark
**([[infinitesimal gauge symmetries]] yield spacetime-compactly supported [[infinitesimal symmetries of the Lagrangian]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) and $\mathcal{G} \overset{gb}{\to} \Sigma$ a bundle of [[gauge parameters]] for it (def. \ref{GaugeParameters}) with gauge parametrization
$$
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E)
\overset{R}{\longrightarrow}
T_\Sigma E
\,.
$$
Then for _every_ smooth [[section]] $\epsilon \in \Gamma_\Sigma(\mathcal{G})$ of the [[gauge parameter]] bundle (def. \ref{Sections})
there is an induced [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) given by the [[composition]] of $R$ with the [[jet prolongation]] of $\epsilon$ (def. \ref{JetProlongation})
$$
R(\epsilon)
\;\colon\;
J^\infty_\Sigma(E)
=
\Sigma \times_\Sigma J^\infty_\Sigma(E)
\overset{(j^\infty_\Sigma(\epsilon),id)}{\longrightarrow}
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E)
\overset{R}{\longrightarrow}
T_\Sigma E
\,.
$$
In terms of the components (eq:CoordinateExpressionForGaugeParameterized) this means that
$$
R(\epsilon)
\;=\;
\left(
\epsilon^\alpha R^a_\alpha
+
\frac{\partial^2 \epsilon^\alpha}{\partial x^\mu} R^{a \mu}_\alpha
+
\frac{\partial \epsilon^\alpha}{\partial x^\mu \partial x^\nu} R^{a \mu_1 \mu_2}_\alpha
+
\cdots
\right)
\,,
$$
where now
$$
\frac{\partial^k \epsilon^\alpha}{\partial x^{\mu_1} \cdots \partial x^{\mu_k}}
\;=\;
\frac{\partial^k \epsilon^\alpha}{\partial x^{\mu_1} \cdots \partial x^{\mu_k}}((x^\mu))
$$
are the actual [[spacetime]] [[partial derivatives]] of the [[gauge parameter]] [[section]] (which are functions of spacetime).
In particular, since $\mathcal{G} \overset{gb}{\to} \Sigma$ is assumed to be a [[vector bundle]], there always exists [[gauge parameter]] [[sections]] $\epsilon$ that have [[compact support]] ([[bump functions]]). For such compactly supported $\epsilon$ the infinitesimal symmetry $R(\epsilon)$ is
spacetime-compactly supported as in prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces}.
=--
$\,$
The following remark \ref{GeneratingSetOfGaugeTransformations} and def. \ref{RigidInfinitesimalSymmetriesOfTheLagrangian} introduce some useful terminology:
+-- {: .num_remark #GeneratingSetOfGaugeTransformations}
###### Remark
**(generating set of [[gauge transformations]])**
Given a [[Lagrangian field theory]], then a choice of [[gauge parameter]] bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ with gauge parameterized [[infinitesimal gauge symmetries]] $J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E$ (def. \ref{GaugeParameters}) is indeed a _choice_ and not uniquely fixed.
For example given any such bundle one may form the [[direct sum of vector bundles]] $\mathcal{G} \oplus_\Sigma \mathcal{G}'$ with any other [[smooth vector bundle]] $\mathcal{G}'$ over $\Sigma$, extend $R$ by zero to $\mathbb{G}'$, and thereby obtain another [[gauge parameter|gauge parameterized]] of [[infinitesimal gauge symmetries]]
$$
J^\infty_\Sigma((\mathcal{G}' \oplus_\Sigma \mathcal{G}) \times_\Sigma E) \overset{(0,R)}{\longrightarrow} T_\Sigma E
\,.
$$
Conversely, given any [[subbundle]] $\mathcal{G}' \hookrightarrow \mathcal{G}$, then the [[restriction]] of $R$ to $\mathcal{G}'$ is still a [[gauge parameter|gauge parameterized]] collection of [[infinitesimal gauge symmetries]].
We will see that for the purpose of removing the [[obstruction]] to the existence of the [[covariant phase space]], the gauge parameters have to capture all [[Noether identities]] (prop. \ref{NoetherIdentities}). In this case one says that the gauge parameter bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ is a _generating set_.
=--
(e.g. [Henneaux 90, section (2.8)](BRST+complex#Henneaux90))
+-- {: .num_defn #RigidInfinitesimalSymmetriesOfTheLagrangian}
###### Definition
**([[rigid infinitesimal symmetries of the Lagrangian]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
and let $J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E$
be [[infinitesimal gauge symmetries]] (def. \ref{GaugeParameters}) whose [[gauge parameters]]
form a generating set (remark \ref{GeneratingSetOfGaugeTransformations}).
Then the [[vector space]] of _[[rigid infinitesimal symmetries of the Lagrangian]]_ is the [[quotient space]]
of the [[infinitesimal symmetries of the Lagrangian]] by the [[image]] of the [[infinitesimal gauge symmetries]]:
$$
\left\{ \text{rigid infinitesimal symmetries} \right\}
\;=\;
\left\{
\text{infinitesimal symmetries}
\right\} \,/\,
\left\{ \text{infinitesimal gauge symmetries} \right\}
\,.
$$
=--
The following is a way to identify [[infinitesimal gauge symmetries]]:
+-- {: .num_prop #NoetherIdentities}
###### Proposition
**([[Noether's theorem|Noether's theorem II]] -- [[Noether identities]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[vector bundle]].
Then a [[bundle morphism]] of the form
$$
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E
$$
is a collection of [[infinitesimal gauge symmetries]] (def. \ref{GaugeParameters}) with local components (eq:CoordinateExpressionForGaugeParameterized)
$$
R
\;=\;
\left(
c^\alpha R^a_\alpha
+
c^\alpha_{,\mu} R^{a \mu}_\alpha
+
c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha
+
\cdots
\right)
\partial_{\phi^a}
$$
precisely if the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
satisfies the following conditions:
$$
\left(
R^{a}_\alpha \frac{\delta_{EL}\mathbf{L}}{\delta \phi^a}
-
\frac{d}{d x^\mu}
\left(
R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{\delta \phi^a}
\right)
+
\frac{d^2}{d x^{\mu_1} d x^{\mu_2}}
\left(
R^{a \mu_1 \mu_2}_\alpha
\frac{\delta_{EL}\mathbf{L}}{\delta \phi^a}
\right)
-
\cdots
\right)
\;=\;
0
\,.
$$
These relations are called the _[[Noether identities]]_ of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] (def \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}).
=--
+-- {: .proof}
###### Proof
By [[Noether's theorem|Noether's theorem I]],
$R$ is an [[infinitesimal symmetry of the Lagrangian]] precisely if the contraction (def. \ref{ContractionOfFormsWithVectorFields})
of $R$ with the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
is horizontally exact:
$$
\iota_{R} \delta_{EL}\mathbf{L} = d J_{\hat R}
\,.
$$
From (eq:CoordinateExpressionForGaugeParameterized) this means that
$$
\label{NoetherIExpressionForInfinitesimalGaugeSymmetry}
\begin{aligned}
d J_{\hat R}
& =
\iota_{R} \delta_{EL} \mathbf{L}
\\
& =
\underset{k \in \mathbb{N}}{\sum}
c^\alpha_{,\mu_1 \cdots \mu_k} R^{a \mu_1 \cdots \mu_k}_\alpha
\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}
\\
& =
\underset{A}{
\underbrace{
c^\alpha
\underset{k \in \mathbb{N}}{\sum}
(-1)^k
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R^{a \mu_1 \cdots \mu_k}_\alpha
\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}
\right)
}
}
+
d K
\,,
\end{aligned}
$$
where in the last step we used jet-level [[integration by parts]] (example \ref{IntegrationByPartsOnJetBundle}) to move the [[total spacetime derivatives]] off of $c^\alpha$, thereby picking up some horizontally exact correction term, as shown.
This means that the term $A$ over the brace is horizontally exact:
$$
\label{NoetherIdentityTermIsHorizontallyExact}
c^\alpha
\underset{k \in \mathbb{N}}{\sum}
(-1)^k
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R^{a \mu_1 \cdots \mu_k}_\alpha
\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}
\right)
\;=\;
d(...)
$$
But now the term on the left is independent of the jet coordinates $\epsilon^\alpha_{,\mu_1 \cdots \mu_k}$
of positive order $k \geq 1$, while the horizontal derivative increases the dependency on the jet order by one.
Therefore the term on the left is horizontally exact precisely if it vanishes, which is the case precisely if the coefficients of $c^\alpha$ vanish, which is the statement of the Noether identities.
Alternatively we may reach this conclusion from (eq:NoetherIdentityTermIsHorizontallyExact) by applying to both sides of (eq:NoetherIdentityTermIsHorizontallyExact) the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral) _with respect to $c^\alpha$_. On the left this yields again the coefficients of $c^\alpha$,
while by the argument from example \ref{TrivialLagrangianDensities} it makes the right hand side vanish.
=--
As a corollary we obtain:
+-- {: .num_prop #ConservedChargeOfInfinitesimalGaugeSymmetryVanishes}
###### Proposition
**([[conserved charge]] of [[infinitesimal gauge symmetry]] vanishes)**
The [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents})
$$
J_R
\;\in\;
\Omega^{p,0}_\Sigma(E \times_\Sigma \mathcal{G})
$$
which corresponds to an [[infinitesimal gauge symmetry]] $R$ (def. \ref{GaugeParameters}) by [[Noether's theorem|Noether's first theorem]] (prop. \ref{NoethersFirstTheorem}), is up to a term which vanishes [[on-shell]] (eq:ProlongedShellInJetBundle)
$$
K \;\in\; \Omega^p_\Sigma(E \times_\Sigma \mathcal{G})
\phantom{AA}\,,
\phantom{AA}
K\vert_{\mathcal{E}^\infty} = 0
\,,
$$
not just [[on-shell]]-conserved, but [[off-shell]]-conserved, in that its [[total spacetime derivative]] vanishes identically:
$$
d( J_R - K ) \;=\; 0
\,.
$$
Moreover, if the [[field bundle]] as well as the [[gauge parameter]]-bundles are [[trivial vector bundles]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) then $J_R$ is horizontally exact [[on-shell]] (eq:ProlongedShellInJetBundle)
$$
J_R \vert_{\mathcal{E}^\infty} = d(...)
\,.
$$
In particular the [[conserved charge]] (prop. \ref{ConservedCharge})
$$
Q_R \;\coloneqq\; \tau_{\Sigma_p}(J_R)
\;\in\;
C^\infty\left( \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0} \right)
$$
corresponding to an [[infinitesimal gauge symmetry]] vanishes on every [[codimension]] one [[submanifold]] $\Sigma_p \hookrightarrow \Sigma$ of [[spacetime]] (without [[manifold with boundary|boundary]], $\partial \Sigma_p = \emptyset$):
$$
Q_R = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
Take $K$ to be as in equation (eq:NoetherIExpressionForInfinitesimalGaugeSymmetry):
$$
d J_R = A + d K
\,.
$$
By the construction there, $K$ manifestly vanishes on the [[prolonged shell]] $\mathcal{E}^\infty$ (eq:ProlongedShellInJetBundle), being a sum of [[total spacetime derivatives]] of terms proportional to the components of the [[Euler-Lagrange form]].
By [[Noether's theorem|Noether's second theorem]] (prop. \ref{NoetherIdentities}) we have $A = 0$ and hence
$$
d(J_R - K) = 0
\,.
$$
Now if the [[field bundle]] and [[gauge parameter]] bundle are trivial, then prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} implies that
$$
\label{DecompositionOfGaugSymmetryConservedCurrent}
J_R - K = d(...)
\,.
$$
By restricting this equation to the [[prolonged shell]] and using that $K\vert_{\mathcal{E}^\infty} = 0$, it follows that $ J_R \vert_{\mathcal{E}^\infty} = d(...)$.
This implies $Q_R = 0$ by prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} and [[Stokes' theorem]] (prop. \ref{StokesTheorem}).
=--
This situation has a concise [[homological algebra|cohomological]] incarnation:
+-- {: .num_example #dWCohomology}
###### Example
**([[Noether's theorem|Noether's theorems I and II]] in terms of [[local BV-cohomology]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] $\Sigma$ of [[dimension]] $ p + 1$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Assume that both are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with field coordinates as in prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}.
Then in the [[local BV-complex]] (def. \ref{BVComplexOfOrdinaryLagrangianDensity}) we have:
The $(s_{BV} + d)$-closure of an element in total degree $p$ is characterizes as the [[direct sum]] of an [[evolutionary vector field]] which is an [[infinitesimal symmetry of the Lagrangian]] and the[[conserved current]] that corresponds to it under [[Noether's theorem|Noether's first theorem]] (prop. \ref{NoethersFirstTheorem}).
Moreover, such a pair is $(s_{BV} + d)$-exact precisely if the [[infinitesimal symmetry of the Lagrangian]] is in fact an [[infinitesimal gauge symmetry]] as witnessed by [[Noether's theorem|Noether's second theorem]] (prop. \ref{NoetherIdentities}).
=--
([Barnich-Brandt-Henneaux 94, top of p. 20](local+BRST+cohomology#BarnichBrandtHenneaux94))
+-- {: .proof}
###### Proof
An element of the [[local BV-complex]] in degee $p$ is the [[direct sum]] of a [[horizontal differential form]] of degree $p$ with the product of a horizontal form of degree $(p+1)$ times a function proportional to the [[antifields]]:
$$
\array{
\{J_v\} &&
\\
&&
\\
&& \{ v^a \phi^\ddagger_a dvol_\Sigma\}
}
$$
Its closure means that
$$
\array{
\{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \}
\\
&& \uparrow\mathrlap{s_{BV}}
\\
&& \{ v^a \phi^\ddagger_a dvol_\Sigma\}
}
$$
where the equality in the top right corner is euqation
It being exact means that
$$
\array{
\left\{ ... \right\}
&
\overset{d}{\longrightarrow}
&
\left\{
J_R
=
K + d(...)
\right\}
&\overset{d}{\longrightarrow}&
\left\{
d J_R
\right\}
\\
&&
\uparrow
\\
&&
\left\{
K^{a \mu} \phi^\ddagger_a \iota_{\partial_\mu} dvol_\Sigma
\right\}
}
$$
where now the equality in the second term from the left is equation (eq:DecompositionOfGaugSymmetryConservedCurrent) for [[conserved currents]] corresponding to [[infinitesimal gauge symmetries]] (prop. \ref{ConservedChargeOfInfinitesimalGaugeSymmetryVanishes}).
=--
We will need some further technical results on [[Noether identities]]:
+-- {: .num_defn #NoetherOperator}
###### Definition
**([[Noether operator]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] $\Sigma$ of [[dimension]] $ p + 1$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Assume that both are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with field coordinates as in prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}.
A _Noether operator_ $N$ is a [[differential operator]] (def. \ref{DifferentialOperator}) from the [[vertical cotangent bundle]] of $E$ (example \ref{VerticalTangentBundle}) to the [[trivial vector bundle|trivial]] [[real line bundle]]
$$
N(\omega)
\;=\;
\underset{k \in \mathbb{N}}{\sum}
N^{a \mu_1 \cdots \mu_k}
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \omega_a
$$
such that it annihilates the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):
$$
\underset{k \in \mathbb{N}}{\sum}
N^{a \mu_1 \cdots \mu_k}
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \frac{\delta_{EL} L}{\delta \phi^a}
\;=\; 0
\,.
$$
Given For $v$ an [[evolutionary vector field]] which is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}), we define a new differentia opeator $v \cdot N$ by
$$
\label{MultiplyingANoetherOperatorWithAnInfinitesimalSymmetry}
(v \cdot N)^{a \mu_1 \cdots \mu_k}
\;\coloneqq\;
\widehat{v}\left(
N^{a \mu_1 \cdots \mu_k}
\right)
\;-\;
N^a \circ (\mathrm{D}_v)^\ast_a
\,,
$$
where $\widehat{v}$ denotes the prolongation of the [[evolutionary vector field]] $v$ (prop. \ref{EvolutionaryVectorFieldProlongation})
and where $(\mathrm{D}_v)^\ast$ denotes the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators})
of the [[evolutionary derivative]] of $v$ (def. \ref{FieldDependentDifferentialOperatorDerivative}).
=--
([Barnich 10 (3.1) and (3.5)](BRST+complex#Barnich10))
+-- {: .num_prop #LieAlgebraActionOfInfinitesimalSymmetriesOfTheLagrangianOnNoetherOperators}
###### Proposition
**([[Lie algebra action]] of [[infinitesimal symmetries of the Lagrangian]] on [[Noether operators]])**
The operation (eq:MultiplyingANoetherOperatorWithAnInfinitesimalSymmetry) exhibits a [[Lie algebra action]]
of the [[Lie algebra]] of [[infinitesimal symmetries of the Lagrangian]] (prop. \ref{EvolutionaryVectorFieldLieAlgebra})
on Noether operators (def. \ref{NoetherOperator}), in that
1. $v \cdot N$ is again a Noether operator;
1. $v_1 \cdot (v_2 \cdot N) - v_2 \cdot (v_1 \cdot N) = [v_1, v_2] \cdot N$.
Moreover, if $\rho$ denotes the map which identifies a [[Noether identity]] with an [[infinitesimal gauge symmetry]]
by [[Noether's theorem|Noether's second theorem]] (def. \ref{NoetherIdentities}) then
$$
\label{LieActionOnNoetherOperatorGivesLieBracketUnderNoetherTheorem}
\rho \left(
v \cdot N
\right)
\;=\;
\left[ v, \rho(N)\right]
\,,
$$
where on the right we have again the [[Lie bracket]] of [[evolutionary vector fields]] from (prop. \ref{EvolutionaryVectorFieldLieAlgebra}).
=--
([Barnich 10, prop. 3.1 and (3.8)](BRST+complex#Barnich10))
+-- {: .proof}
###### Proof
For the first statement observe that by the [[product law]] for [[differentiation]] we have
$$
\begin{aligned}
0
& =
\widehat{v}\left(
N(\delta_{EL} L)
\right)
\\
& =
\widehat{v}
\left(
\underset{k \in \mathbb{N}}{\sum} N^{a \mu_1 \cdots \mu_k}
\right)
-
\left(
N^a \circ (\mathrm{D}_v)_a^b\left( \frac{\delta_{EL} L}{\delta \phi^a} \right)
\right)
\,,
\end{aligned}
$$
where on the right we used (eq:TowardsProofThatSymmetriesPreserveTheShell).
=--
$\,$
Here are examples of [[infinitesimal gauge symmetries]] in [[Lagrangian field theory]]:
+-- {: .num_example #InfinitesimalGaugeSymmetryElectromagnetism}
###### Example
**([[infinitesimal gauge symmetry]] of [[electromagnetic field]])**
Consider the [[Lagrangian field theory]] $(E,\mathbf{L})$ of [[free field|free]] [[electromagnetism]]
on [[Minkowski spacetime]] $\Sigma$ from example \ref{ElectromagnetismLagrangianDensity}. With field coordinates denoted $((x^\mu), (a_\mu))$ the [[Lagrangian density]] is
$$
\mathbf{L}
\;=\;
\tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
\, dvol_\Sigma
\,,
$$
where $f_{\mu \nu} \coloneqq a_{\nu,\mu}$ is the universal [[Faraday tensor]] from example \ref{JetFaraday}.
Let $\mathcal{G} \coloneqq \Sigma \times \mathbb{R}$ be the [[trivial line bundle]], regarded as a
[[gauge parameter]] bundle (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) with coordinate functions $((x^\mu), c)$.
Then a [[gauge parameter|gauge parametrized]] [[evolutionary vector field]] (eq:CoordinateExpressionForGaugeParameterized) is given by
$$
R
\;=\;
c_{,\mu} \partial_{a_\mu}
$$
with prolongation (prop. \ref{EvolutionaryVectorFieldProlongation})
$$
\label{EMProlongedSymmetryVectorField}
\widehat R
\;=\;
c_{,\mu} \partial_{a_\mu}
+
c_{,\mu \nu} \partial_{a_{\mu,\nu}}
+
\cdots
\,.
$$
This is because already the universal [[Faraday tensor]] is [[invariant]] under this flow:
$$
\begin{aligned}
\widehat {R} f_{\mu \nu}
&=
\tfrac{1}{2}
c_{,\mu' \nu'} \partial_{a_{\mu',\nu'}}
\left(
a_{\nu, \mu} - a_{\mu,\nu}
\right)
\\
& =
\tfrac{1}{2}
\left(
c_{,\nu\mu} - c_{,\mu \nu}
\right)
\\
& = 0
\,,
\end{aligned}
$$
because [[partial derivatives]] commute with each other: $c_{,\mu \nu} = c_{,\nu \mu}$ (eq:JetCoodinatesSymmetry).
Equivalently, the [[Euler-Lagrange form]]
$$
\delta_{EL}\mathbf{L}
\;=\;
\frac{d}{d x^\mu }f^{\mu \nu} \delta a_\nu \, dvol_\Sigma
$$
of the theory (example \ref{ElectromagnetismEl}),
corresponding to the [[vacuum]] [[Maxwell equations]] (example \ref{MaxwellVacuumEquation}),
satisfies the following [[Noether identity]] (prop. \ref{NoetherIdentities}):
$$
\frac{d}{d x^\mu} \frac{d}{d x^\nu} f^{\mu \nu} = 0
\,,
$$
again due to the fact that partial derivatives commute with each other.
This is the archetypical _[[infinitesimal gauge symmetry]]_ that gives [[gauge theory]] its name.
=--
More generally:
+-- {: .num_example #InfinitesimalGaugeSymmetryOfYangMillsTheory}
###### Example
**([[infinitesimal gauge symmetry]] of [[Yang-Mills theory]])**
For $\mathfrak{g}$ a [[semisimple Lie algebra]],
consider the [[Lagrangian field theory]] of [[Yang-Mills theory]] on [[Minkowski spacetime]] from example \ref{YangMillsLagrangian}, with [[Lagrangian density]]
$$
\mathbf{L}
\;=\;
\tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu}
$$
given by the universal [[field strength]] (eq:YangMillsJetFieldStrengthMinkowski)
$$
f^\alpha_{\mu \nu}
\;\coloneqq\;
\tfrac{1}{2}
\left(
a^\alpha_{[\nu,\mu]}
+
\tfrac{1}{2} \gamma^\alpha_{\beta \gamma} a^\beta_{[\mu} a^\gamma_{\nu]}
\right)
\,.
$$
Let $\mathcal{G} \coloneqq \Sigma \times \mathfrak{g}$ be the [[trivial vector bundle]] with [[fiber]] $\mathfrak{g}$, regarded as a
[[gauge parameter]] bundle (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) with coordinate functions $((x^\mu), c^\alpha)$.
Then a [[gauge parameter|gauge parametrized]] [[evolutionary vector field]] (eq:CoordinateExpressionForGaugeParameterized) is given by
$$
R
\;=\;
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
$$
with prolongation (prop. \ref{EvolutionaryVectorFieldProlongation})
$$
\label{OnMinkowskiInfinitesimalGaugeSymmetryForYangMills}
\widehat{R}
\;=\;
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
\;+\;
\left(
c^\alpha_{,\mu \nu}
-
\gamma^\alpha_{\beta \gamma}
\left(
c^\beta_{,\nu} a^\gamma_\mu
+
c^\beta a^\gamma_{\mu,\nu}
\right)
\right)
\partial_{a^\alpha_{\mu,\nu}}
\;+\;
\cdots
\,.
$$
We compute the [[derivative]] of the [[Lagrangian function]] along this vector field:
$$
\begin{aligned}
\widehat{R}
\left(
\tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu}
\right)
& =
\left(
R f^\alpha_{\mu \nu}
\right)
f_\alpha^{\mu \nu}
\\
& =
\left(
R
\left(
a_{\nu,\mu}
+
\tfrac{1}{2}\gamma^\alpha_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu}
\right)
\right)
f_\alpha^{\mu \nu}
\\
& =
\left(
c^\alpha_{,\nu \mu}
-
\gamma^\alpha_{\beta \gamma}
\left(
c^\beta_{,\mu} a^\gamma_\nu
+
c^\beta a^\gamma_{\nu,\mu}
\right)
+
\gamma^\alpha_{\beta \gamma}
\left(
c^\beta_{,\mu}
-
\gamma^\beta_{\beta' \gamma'}
c^{\beta'} a^{\gamma'}_\mu
\right)
a^\gamma_{\nu}
\right)
f_\alpha^{\mu \nu}
\\
& =
- \gamma^{\alpha}_{\beta \gamma} c^\beta
\underset{
= 2 f^\gamma_{\mu \nu}
}{
\underbrace{
\left(
a^\gamma_{\nu,\mu}
+
\gamma^\gamma_{\beta' \gamma'}
a^{\beta'}_\mu a^{\gamma'}_\nu
\right)
}
}
f_\alpha{}^{\mu \nu}
\\
&=
2
\gamma_{\alpha \beta \gamma}
c^\alpha f^\beta_{\mu \nu} f^{\gamma \mu \nu}
\\
& = 0
\,.
\end{aligned}
$$
Here in the third step we used that $c^\alpha_{,\nu \mu} = + c^\alpha_{,\mu \nu}$ (eq:JetCoodinatesSymmetry), so that its contraction with the skew-symmetric $f_\alpha^{\mu\nu}$ vanishes, and in the last step we used that for a [[semisimple Lie algebra]] $\gamma_{\alpha \beta \gamma} \coloneqq k_{\alpha \alpha'} \gamma^{\alpha'}{}_{\beta \gamma}$ is totally skew symmetric.
So the [[Lagrangian density]] of [[Yang-Mills theory]] is strictly invariant under these [[infinitesimal gauge symmetries]].
=--
+-- {: .num_example #InfinitesimalGaugeSymmetryOfTheBField}
###### Example
**([[infinitesimal gauge symmetry]] of the [[B-field]])**
Consider the [[Lagrangian field theory]] of the [[B-field]] on [[Minkowski spacetime]] from example \ref{BFieldLagrangianDensity}, with [[field bundle]] the [[differential 2-form]]-bundle $E = \wedge^2_\Sigma T^\ast \Sigma$ with coordinates $((x^\mu), (b_{\mu \nu}))$ subject to $b_{\mu \nu} = - b_{\nu \mu}$; and with [[Lagrangian density]]
$$
\mathbf{L}
\;=\;
\tfrac{1}{2}
h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma
$$
for
$$
h_{\mu_1 \mu_2 \mu_3} = b_{[\mu_1 \mu_2, \mu_3]}
$$
the universal [[B-field|B-]][[field strength]] (example \ref{BFieldJetFaraday}).
Let $\mathcal{G} \coloneqq T^\ast \Sigma$ be the [[cotangent bundle]] (def. \ref{Differential1FormsOnCartesianSpaces}), regarded as a [[gauge parameter]] bundle (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) with coordinate functions $((x^\mu), (c_\mu))$ as in example \ref{Electromagnetism}.
Then a [[gauge parameter|gauge parametrized]] [[evolutionary vector field]] (eq:CoordinateExpressionForGaugeParameterized) is given by
$$
R
\;=\;
c_{\mu,\nu} \partial_{b_{\mu \nu}}
$$
with prolongation (prop. \ref{EvolutionaryVectorFieldProlongation})
$$
\label{InfinitesimalGaugeSymmetryForBFieldOnMinkowskiSpacetime}
\widehat R
\;=\;
c_{\mu,\nu} \partial_{b_{\mu \nu}}
+
c_{\mu,\nu \rho} \partial_{b_{\mu \nu, \rho}}
+
\cdots
$$
In fact this leaves the [[Lagrangian function]] [[invariant]], in direct higher analogy to example \ref{InfinitesimalGaugeSymmetryElectromagnetism}:
$$
\begin{aligned}
\widehat{R} \tfrac{1}{2} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3}
& =
\left(
\widehat{R} b_{\mu_1 \mu_2, \mu_3}
\right)
h^{\mu_1 \mu_2 \mu_3}
\\
& =
c_{\mu_1, \mu_2 \mu_3}
h^{\mu_1 \mu_2 \mu_3}
\\
& = 0
\end{aligned}
$$
due to the symmetry of [[partial derivatives]] (eq:JetCoodinatesSymmetry).
$$
h_{,\mu}\partial_{c_{\mu}}
+
h_{,\mu \nu}\partial_{c_{\mu,\nu}}
$$
$$
R_\alpha^{a, \mu}
=
c_{\mu,\nu}
R^{\mu, \nu}_{\mu' \nu'}
\partial_{b_{\mu' \nu'}}
\,.
$$
While so far all this is in direct analogy to the case of the [[electromagnetic field]] (example \ref{InfinitesimalGaugeSymmetryElectromagnetism}), just with [[field histories]] being [[differential 1-forms]]
now replaced by [[differential 2-forms]], a key difference is that now the [[gauge parameter|gauge parameterization]] $R$
itself has [[infinitesimal gauge symmetries]]:
Let
$$
\label{SecondOrderGaugeParameterBundleForBFieldOnMinkowskiSpacetime}
\array{
\overset{(2)}{\mathcal{G}} &\coloneqq& \Sigma \times \mathbb{R}
\\
{}^{\overset{(2)}{gb}}\downarrow && \downarrow^{\mathrlap{pr_1}}
\\
\Sigma &=& \Sigma
}
$$
be the [[trivial vector bundle|trivial]] [[real line bundle]]
with coordinates $((x^\mu), \overset{(2)}{c})$, to be regarded as a second order [[infinitesimal gauge-of-gauge symmetry]], then
$$
\overset{(2)}{R}
\;\coloneqq\;
\overset{(2)}{c}_{,\mu} \partial_{c_\mu}
$$
with prolongation
$$
\label{SecondOderGaugeSymmetryOfBFieldOnMinkowski}
\widehat{\overset{(2)}{R}}
\;\coloneqq\;
\overset{(2)}{c}_{,\mu} \partial_{c_\mu}
+
\overset{(2)}{c}_{,\mu \nu} \partial_{c_{\mu,\nu}}
+
\cdots
$$
has the property that
$$
\label{NoetherIdentitySecondOrderForBFieldOnMinkowskiSpacetime}
\begin{aligned}
\widehat{\overset{(2)}{R}} (R)
&=
\overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}}
\left(
c_{\mu',\nu'} \partial_{b_{\mu' \nu'}}
\right)
\\
& = \overset{(2)}{c}_{,\mu \nu} \partial_{b_{\mu \nu}}
\\
& 0
\,.
\end{aligned}
$$
We further discuss these _[[higher gauge transformations]]_ below.
=--
$\,$
**[[Lie algebra actions]] and [[Lie algebroids]]**
{#LieAlgebraActionAndLieAlgebroids}
We have seen above [[infinitesimal gauge symmetries]] _implied_ by a [[Lagrangian field theory]],
exhibited by [[infinitesimal symmetries of the Lagrangian]]. In order to remove the [[obstructions]] that these
[[infinitesimal gauge symmetries]] cause for the existence of the [[covariant phase space]] (via prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} and remark \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) we will need (discussed below in _[Gauge fixing](#GaugeFixing)_) to make these
symmetries manifest by hard-wiring them into the geometry of the [[type]] of [[field (physics)|fields]].
Mathematically this means that we need
to take the _[[homotopy quotient]]_ of the [[jet bundle]] of the [[field bundle]] by the [[action]] of the
[[infinitesimal gauge symmetries]], which is modeled by their _[[action Lie algebroid]]_.
Here we introduce the required _[[higher Lie theory]]_ of [[Lie ∞-algebroids]] (def. \ref{LInfinityAlgebroid} below).
Further [below](#BRSTComplex) we specify this to actions by [[infinitesimal gauge symmetries]] to obtain the
_[[local BRST complex|local]] [[BRST complex]]_ of a [[Lagrangian field theory]] (def. \ref{LocalOffShellBRSTComplex}) below.
$\,$
The following discussion introduces and uses the tremendously useful fact that ([[higher Lie theory|higher]]) [[Lie theory]] may usefully
be dually expressed in terms of [[differential graded-commutative algebra]] (def. \ref{differentialgradedcommutativeSuperalgebra} below), namely in terms of "[[Chevalley-Eilenberg algebras]]".
In the [[physics]] literature, besides the [[BRST-BV formalism]], this fact underlies the [[D'Auria-Fré formulation of supergravity]] ("[[FDAs]]", see the convoluted [history of the concept](L-infinity-algebra#History)).
Mathematically the deep underlying phenomenon is called the "[[Koszul duality]] between the [[Lie operad]]
and the [[commutative algebra operad]]", but this need not concern us here. The phenomenon is readily seen in direct
application:
Before we proceed, we make explicit a [[structure]] wich we already encountered in example \ref{DifferentialFormOnSuperCartesianSpaces}.
+-- {: .num_defn #differentialgradedcommutativeSuperalgebra}
###### Definition
**([[differential graded-commutative superalgebra]])**
A _[[differential graded-commutative superalgebra]]_ is
1. a [[cochain complex]] $A_\bullet$ of [[super vector spaces]], hence for each $n \in \mathbb{Z}$
1 a [[super vector space]] $A_n = (A_n)_{even} \oplus (A_n)_{odd}$;
1. a super-degree preserving [[linear map]]
$$
d \;\colon\; A_{n} \longrightarrow A_{n+1}
$$
such that
$$
d \circ d = 0
$$
1, an [[associative algebra]]-[[structure]] on $A \coloneqq \underset{n \in \mathbb{Z}}{\oplus} A_n$
such that for all $a_1, a_2 \in A$ with homogenous bidegree $a_i \in (A_{n_a})_{\sigma_a}$ we have
the [[signs in supergeometry|super sign rule]]
1. $a b = (-1)^{n_a n_b} (-1)^{\sigma_a \sigma_b} \, b a$
1. $d(a b) = (d a) b + (-1)^{n_1} a (d b)$.
A [[homomorphism]] between two [[differential graded-commutative superalgebras]] is a [[linear map]] between the underlying [[super vector spaces]] which preserves both degrees, and respects the product as well as the [[differential]] $d$.
We write $dgcSAlg$ for the [[category]] of [[differential graded-commutative superalgebra]].
=--
For the [[signs in supergeometry|super sigsn rule]] appearing here see also e.g. [Castellani-D'Auria-Fré 91 (II.2.106) and (II.2.109)](signs+in+supergeometry#CastellaniDAuriaFre91), [Deligne-Freed 99, section 6](signs+in+supergeometry#DeligneFreed99).
+-- {: .num_example #deRhamAlgebraOfSuperDifferentialFormsIsDifferentialGradedCommutativeSuperalgebra}
###### Example
**([[de Rham algebra]] of [[super differential forms]] is [[differential graded-commutative superalgebra]])**
For $X$ a [[super Cartesian space]], def. \ref{SuperCartesianSpace} (or more generally a [[supermanifold]], def. \ref{SuperSmoothManifolds}) the [[de Rham algebra]] of [[super differential forms]] from def. \ref{DifferentialFormOnSuperCartesianSpaces}
$$
(\Omega^\bullet(X), d)
$$
is a [[differential graded-commutative superalgebra]] (def. \ref{differentialgradedcommutativeSuperalgebra}) with
product the [[wedge product]] of differential forms and differential the [[de Rham differential]].
We will recognize the [[formal duality|dual]] incarnation of this in [[higher Lie theory]] below in example \ref{HorizontalTangentLieAlgebroid}.
=--
+-- {: .num_prop #LieAlgebraInTermsOfChevalleyEilenbergAlgebra}
###### Proposition
**([[Lie algebra]] in terms of [[Chevalley-Eilenberg algebra]])**
Let $\mathfrak{g}$ be a [[finite dimensional vector space|finite dimensional]] [[super vector space]]
equipped with a [[super Lie algebra|super]] [[Lie bracket]] $[-,-] \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}$.
Write $\mathfrak{g}^\ast$ for the [[dual vector space]] and $[-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast$ for the [[linear dual]] map of the [[Lie bracket]]. Then on the [[Grassmann algebra]] $\wedge^\bullet \mathfrak{g}^\ast$
(which is $\mathbb{Z} \times \mathbb{Z}/2$-bigraded as in def. \ref{DifferentialFormOnSuperCartesianSpaces})
the graded [[derivation]] $d_{CE}$ of degree $(1,even)$, which on $\mathfrak{g}^\ast$ is given by $[-,-]^\ast$ constitutes a [[differential]]
in that $(d_{CE})^2 = 0$. The resulting [[differential graded-commutative algebra]] is called the [[Chevalley-Eilenberg algebra]]
$$
CE(\mathfrak{g})
\;\coloneqq\;
\left(
\wedge^\bullet \mathfrak{g}^\ast
\,,
d_{CE} = [-,-]^\ast
\right)
\,.
$$
In components:
If $\{c_\alpha\}$ is a [[linear basis]] of $\mathfrak{g}$, so that the [[Lie bracket]] is given by the structure constants $(\gamma^\alpha{}_{\beta \gamma})$ as
$$
[c_\beta, c_\gamma] = \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c_\gamma
$$
and if $\{c^\alpha\}$ denotes the corresponding dual basis, then $\wedge^\bullet \mathfrak{g}^\ast$ is equivalently the [[differential graded-commutative superalgebra]] (def. \ref{differentialgradedcommutativeSuperalgebra}) generated from the $c^\alpha$ in bi-degree $(1,\sigma)$, where $\sigma \in \mathbb{Z}/2$ is the super-degree of $c_\alpha$ as in def. \ref{DifferentialFormOnSuperCartesianSpaces} subject to the relation
$$
c^\alpha \wedge c^\beta = (-1) (-1)^{\sigma_\alpha \sigma_\beta} c^\beta \wedge c^\alpha
$$
and the differential is given by
$$
d_{CE} c^\alpha = \gamma^\alpha{}_{\beta \gamma} c^\beta \wedge c^\gamma
\,.
$$
Notice that by degree-reasons _every_ degree +1 [[derivation]] on $\wedge^\bullet \mathfrak{g}^\ast$ is of this form,
$$
\left\{
\array{
\text{derivations}
\\
\text{of degree}\, (1,even)
\\
\text{on} \, \wedge^\bullet \mathfrak{g}^\ast
}
\right\}
\;\;\simeq\;\,
\left\{
\array{
\text{super-skew}
\\
\text{bilinear maps}
\\
\mathfrak{g} \otimes \mathfrak{g} \overset{[-,-]}{\longrightarrow} \mathfrak{g}
}
\right\}
$$
The condition that $(d_{CE})^2 = 0$ is equivalently the (super-)[[Jacobi identity]] on the bracket $[-,-]$, making
it an actual (super-)[[Lie bracket]]:
$$
\label{JacobiIdentity}
(d_{CE})^2 = 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\gamma^\alpha{}_{\beta [\gamma} \gamma^{\beta}{}_{\beta' \gamma']}
= 0
$$
(where the square brackets on the right denote super-skew-symmetrization).
Hence not only is $CE(\mathfrak{g})$ a [[differential graded-commutative superalgebra]] (def. \ref{differentialgradedcommutativeSuperalgebra}) whenever $\mathfrak{g}$ is a [[super Lie algebra]],
but conversely [[super Lie algebra]]-[[structure]] on a [[super vector space]] $\mathfrak{g}$ is the same as
a differential of degree $(1,even)$ on the [[Grassmann algebra]] $\wedge^\bullet \mathfrak{g}^\ast$.
We may state this equivalence in a more refined form: A [[homomorphism]] $\phi \;\colon\; \mathfrak{g} \longrightarrow \mathfrak{h}$
between [[super vector space]] is, by degree-reasons, the same as a graded algebra homomorphism
$\phi^\ast \;\colon\; \wedge^\bullet \mathfrak{h}^\ast \longrightarrow \wedge^\bullet \mathfrak{g}^\ast$
and it is immediate to check that $\phi$ is a [[homomorphism]] of [[super Lie algebras]] precisely if
$\phi^\ast$ is a homomorpism of differential algebras:
$$
d_{CE(\mathfrak{g})} \circ \phi^\ast
=
\phi^\ast \circ d_{CE(\mathfrak{h})}
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\phi^{\alpha_1}{}_{\beta_1} \gamma^{\beta_1}_{\mathfrak{g}}{}_{\beta_2 \beta_3}
=
\gamma^{\alpha_1}_{\mathfrak{h}}{}_{\alpha_2 \alpha_3} \phi^{\alpha_2}{}_{\beta_2} \phi^{\alpha_3}{}_{\beta_3}
\,.
$$
This is a [[natural bijection]] between homomrophism of [[super Lie algebras]] and of [[differential graded-commutative superalgebras]] (def. \ref{differentialgradedcommutativeSuperalgebra})
$$
Hom_{SuperLieAlg}( \mathfrak{g}, \mathfrak{h} )
\;\simeq\;
Hom_{dgcSAlg}\left( CE(\mathfrak{h}), CE(\mathfrak{g}) \right)
\,.
$$
Stated more [[category theory|abstractly]] this means that forming [[Chevalley-Eilenberg algebras]] is a
[[fully faithful functor]]
$$
CE
\;\colon\;
SuperLieAlg^{fin}
\overset{\phantom{AAA}}{\hookrightarrow}
dgcSAlg^{op}
\,.
$$
=--
Notice that prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra} establishes a [[formal dual|dual]] algebraic incarnation of ([[super Lie algebras|super-]])[[Lie algebras]] which is of analogous form as the dual algebraic characterization of ([[super Cartesian space|super-]])[[Cartesian spaces]] from prop. \ref{AlgebraicFactsOfDifferentialGeometry} and def. \ref{SuperCartesianSpace}.
In fact both these concepts unify into the concept of an [[action Lie algebroid]]:
+-- {: .num_defn #InfinitesimalActionByLieAlgebra}
###### Definition
**([[action]] of [[Lie algebra]] by [[infinitesimal]] [[diffeomorphism]])**
Let $X$ be a [[supermanifold]] (def. \ref{SuperSmoothManifolds}), for instance a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}), and let $\mathfrak{g}$ be a [[finite dimensional vector space|finite dimensional]] [[super Lie algebra]] as in prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra}.
An _[[action]]_ of $\mathfrak{g}$ on $X$ by
[[infinitesimal]] [[diffeomorphisms]], is a [[homomorphism]]
of [[super Lie algebras]]
$$
\rho \;\colon \mathfrak{g} \longrightarrow ( Vect(X), [-,-] )
$$
to the [[tangent vector fields]] on $X$ (example \ref{TangentVectorFields})
Equivalently -- to bring out the relation to
the [[gauge parameter|gauge parameterized]] [[infinitesimal gauge transformations]] in def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation} -- this is a $\mathfrak{g}$-parameterized [[section]]
$$
\array{
\mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X
\\
& {\mathllap{pr_2}}\searrow && \swarrow_{\mathrlap{p}}
\\
&& X
}
$$
of the [[tangent bundle]], such that for all pairs of points $e_1, e_2$ in $\mathfrak{g}$
we have
$$
\left[R(e_1,-), R(e_2,-)\right] = R([e_1,e_2],-)
$$
(with the [[Lie bracket]] of [[tangent vector fields]] on the left).
In components:
If $\{c^\alpha\}$ is a linear basis of $\mathfrak{g}^\ast$
with corresponding structure constants $(\gamma^\alpha{}_{\beta \gamma})$ (as in prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra})
and if $\{\phi^a\}$ is a [[coordinate chart]] of $X$, then $R$ is given by
$$
R \;=\; c^\alpha R_\alpha^a \frac{\partial}{\partial \phi^a}
\,.
$$
=--
Now the construction of the [[Chevalley-Eilenberg algebra]] of a [[super Lie algebra]] (prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra})
extends to the case where this super Lie algebra [[action|acts]] on a [[supermanifold]] (def. \ref{InfinitesimalActionByLieAlgebra}):
+-- {: .num_defn #ActionLieAlgebroid}
###### Definition
**([[action Lie algebroid]])**
Given a [[Lie algebra action]]
$$
\mathfrak{g} \times X
\overset{R}{\longrightarrow}
T X
$$
of a [[finite-dimensional vector space|finite-dimensional]] [[super Lie algebra]] $\mathfrak{g}$ on a [[supermanifold]] $X$ (def. \ref{InfinitesimalActionByLieAlgebra}) we obtain a [[differential graded-commutative superalgebra]] to be denoted $CE(X/\mathfrak{g})$
1. whose underlying graded-commutative superalgebra is the [[Grassmann algebra]] of the $C^\infty(X)$-[[free module]] on $\mathfrak{g}^\ast$ over $C^\infty(X)$
$$
\wedge^\bullet_{C^\infty(X)} (\mathfrak{g}^\ast \otimes C^\infty(X))
\;=\;
\underset{
deg = 0
}{
\underbrace{
C^\infty(X)
}}
\oplus
\underset{
deg = 1
}{
\underbrace{
C^\infty(X) \otimes \mathfrak{g}^\ast
}}
\oplus
\underset{
def = 2
}{
\underbrace{
C^\infty(X)
\otimes
\mathfrak{g}^\ast \wedge \mathfrak{g}^\ast
}}
\oplus
\cdots
$$
which means that the [[graded manifold]] underlying the action Lie algebroid according to remark \ref{dgManifolds} is
$$
\label{ActionLieAlgebroidGradedManifold}
X/\mathfrak{g}
\;=_{grmfd}\;
\mathfrak{g}[1] \times X
\,,
$$
1. whose [[differential]] $d_{CE}$ is given
1. on functions $f \in C^\infty(X)$ by the [[linear dual]] of the Lie algebra action
$$
d_{CE} f \coloneqq \rho(-)(f) \in C^\infty(X) \otimes \mathfrak{g}^\ast
$$
1. on dual Lie algebra elements $\omega \in \mathfrak{g}^\ast$ by the [[linear dual]] of the [[Lie bracket]]
$$
d_{CE} \omega \coloneqq \omega([-,-]) \;\in \; \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast
\,.
$$
In components:
Assume that $X = \mathbb{R}^n$ is a [[super Cartesian space]]
with [[coordinate functions]] $(\phi^a)$ and let $\{c_\alpha\}$ be a [[linear basis]] for $\mathfrak{g}$
with dual basis $(c^\alpha)$ for $\mathfrak{g}^\ast$ and structure constants $(\gamma^\alpha){}_{\beta \gamma}$ as in prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra} and
with the Lie action given in components $(R_\alpha^a)$ as in def. \ref{InfinitesimalActionByLieAlgebra}.
Then the [[differential]] is given by
$$
\begin{aligned}
d_{CE(X/\mathfrak{g})} c^\alpha & = \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} \, c^\beta \wedge c^\gamma
\\
d_{CE(X/\mathfrak{g})} \phi^a & = R^a_\alpha c^\alpha
\end{aligned}
$$
We may summarize this by writing the [[derivation]] $d_{CE(X/\mathfrak{g})}$ as follows:
$$
\label{DifferentialOnActionLieAlgebroid}
d_{CE(X/\mathfrak{g})}
\;=\;
c^\alpha R_\alpha^a \frac{\partial}{\partial \phi^a}
\;+\;
\tfrac{1}{2} \gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}
\,.
$$
That this squares to zero is equivalently
* in degree 0 the [[Lie algebra action|action property]]: $\rho([t, t']) = [\rho(t), \rho(t')]$
* in degree 1 the [[Jacobi identity]] (eq:JacobiIdentity).
$$
(d_{CE(X/\mathfrak{g})})^2 = 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\array{
\phantom{and} \, \text{Jacobi identity}
\\
\text{and} \, \text{action property}
}
$$
Hence as before in prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra} the [[Lie theory|Lie theoretic]] [[structure]] is faithfully captured dually by [[differential graded-commutative superalgebra]].
We call the [[formal duality|formal dual]] of this dgc-superalgebra the _[[action Lie algebroid]]_ $X/\mathfrak{g}$ of
$\mathfrak{g}$ acting on $X$.
=--
The concept emerging by this example we may consider generally:
+-- {: .num_defn #LInfinityAlgebroid}
###### Definition
**([[super L-∞ algebra|super]]-[[Lie ∞-algebroid]])**
Let $X$ be a [[supermanifold]] (def. \ref{SuperSmoothManifolds}) (for instance a [[super Cartesian space]], def. \ref{SuperCartesianSpace}) and write $C^\infty(X)$ for its [[algebra of functions]].
Then a _connected [[super L-∞ algebra|super]] [[Lie ∞-algebroid]]_ $\mathfrak{a}$ over $X$ of [[finite type]] is a
1. a sequence $(\mathfrak{a}_k)_{ k = 1 }^\infty$ of [[free modules]] of [[finite number|finite]] [[rank]] over $C^\infty(X)$, hence a
[[graded module]] $\mathfrak{a}_\bullet$ in degrees $k \in \mathbb{N}$; $k \geq 1$
1. a [[differential]] $d_{CE}$ that makes the [[graded-commutative algebra]] $Sym_{C^\infty(X)}(\mathfrak{a}^\ast_\bullet)$
into a cochain [[differential graded-commutative algebra]] (hence with $d_{CE}$ of degree +1) over $\mathbb{R}$
(not necessarily over $C^\infty(X)$), to be called the _[[Chevalley-Eilenberg algebra]]_ of $\mathfrak{a}$:
$$
\label{CEAlgebra}
CE(\mathfrak{a})
\;\coloneqq\;
\left(
Sym_{C^\infty(X)}(\mathfrak{a}^\ast_\bullet)
\,,\,
d_{CE}
\right)
\,.
$$
If we allow $\mathfrak{a}_\bullet$ to also have terms in non-positive degree, then we speak of a _[[derived Lie algebroid]]_.
If $\mathfrak{a}_\bullet$ is _only_ concentrated in negative degrees, we also speak of a _[[derived manifold]]_.
With $C^\infty(X)$ canonically itself regarded as a [[differential graded-commutative superalgebra]], there is a canonical
dg-algebra homomorphism
$$
CE(\mathfrak{a}) \longrightarrow C^\infty(X)
$$
which is the identity on $C^\infty(X)$ and zero on $\mathfrak{a}^\ast_{\neq 0}$.
=--
(We discuss [[homomorphism]] between [[Lie ∞-algebroid]] below in def. \ref{HomomorphismBetweenLieAlgebroids}.)
+-- {: .num_remark #dgManifolds}
###### Remark
**([[Lie algebroids]] as [[differential graded manifolds]])**
Definition \ref{LInfinityAlgebroid} of _[[derived Lie algebroids]]_ is an encoding in [[higher algebra]] ([[homological algebra]], in this case) of a situation that is usefully thought of in terms of
[[higher differential geometry]].
To see this, recall the magic algebraic properties of ordinary [[differential geometry]] (prop. \ref{AlgebraicFactsOfDifferentialGeometry})
1. [[embedding of smooth manifolds into formal duals of R-algebras]];
1. [[smooth Serre-Swan theorem|embedding of smooth vector bundles into formal duals of modules]]
Together these imply that we may think of the [[graded algebra]] underlying a [[Chevalley-Eilenberg algebra]]
as being the [[algebra of functions]] on a [[graded manifold]]
$$
\cdots \times \mathfrak{a}_2 \times \mathfrak{a}_1 \times X \times \mathfrak{a}_{-1} \times \cdots
$$
which is [[infinitesimal]] in non-vanishing degree.
The "higher" in [[higher differential geometry]] refers to the degrees higher than zero. See at
_[[schreiber:Higher Structures]]_ for exposition.
Specifically if $\mathfrak{a}_\bullet$ has components in negative degrees,
these are also called _[[derived manifolds]]_.
=--
+-- {: .num_example #BasicExamplesOfLieAlgebroids}
###### Example
**(basic examples of [[Lie algebroids]])**
Two basic examples of [[Lie algebroids]] are:
1. For $X$ any [[supermanifold]] (def. \ref{SuperSmoothManifolds}), for instance a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) then setting $\mathfrak{a}_{\neq 0 } \coloneqq 0$ and $d_{CE} \coloneqq 0$ makes it a Lie algebroid in the sense of def. \ref{LInfinityAlgebroid}.
1. For $\mathfrak{g}$ a [[finite-dimensional vector space|finite-dimensional]] [[super Lie algebra]], its [[Chevalley-Eilenberg algebra]] (prop. \ref{LieAlgebraInTermsOfChevalleyEilenbergAlgebra}) $CE(\mathfrak{g})$ exhibits $\mathfrak{g}$ as a [[Lie algebroid]] in the sense of def. \ref{LInfinityAlgebroid}. We write $B\mathfrak{g}$ or $\ast/\mathfrak{g}$ for $\mathfrak{g}$ regarded as a [[Lie algebroid]] this way.
1. For $X$ and $\mathfrak{g}$ as in the previous items, and for $R \colon \mathfrak{g} \times X \to T X$ a [[Lie algebra action]] (def. \ref{InfinitesimalActionByLieAlgebra}) of $\mathfrak{g}$ on $X$, then the dgs-superalegbra $CE(X/\mathfrak{g})$ from def. \ref{ActionLieAlgebroid} defines a [[Lie algebroid]] in the sense of def. \ref{LInfinityAlgebroid}, the _[[action Lie algebroid]]_.
In the special case that $\mathfrak{g} = 0$ this reduces to the first example, while for $X = \ast$ this reduces to the second example.
=--
Here is another basic examples of [[Lie algebroids]] that will plays a role:
+-- {: .num_example #HorizontalTangentLieAlgebroid}
###### Example
**(horizontal [[tangent Lie algebroid]])**
Let $\Sigma$ be a [[smooth manifold]] or more generally a [[supermanifold]] or more generally a [[locally pro-manifold]] (prop. \ref{JetBundleIsLocallyProManifold}).
Then we write $\Sigma/T\Sigma$ for the [[Lie algebroid]] over $X$ and whose [[Chevalley-Eilenberg algebra]] is generated
over $C^\infty(X)$ in degree 1 from the [[module]]
$$
\mathfrak{a}_1^\ast \coloneqq (\Gamma(T \Sigma))^\ast \simeq\Gamma(T^\ast \Sigma) = \Omega^1(\Sigma)
$$
of [[differential 1-forms]] and whose [[Chevalley-Eilenberg differential]] is the [[de Rham differential]],
so that the [[Chevalley-Eilenberg algebra]] is the [[de Rham dg-algebra]] of [[super differential forms]] (example \ref{deRhamAlgebraOfSuperDifferentialFormsIsDifferentialGradedCommutativeSuperalgebra})
$$
CE( \Sigma/T\Sigma ) \coloneqq (\Omega^\bullet(\Sigma), d_{dR})
\,.
$$
This is called the _[[tangent Lie algebroid]]_ of $\Sigma$.
As a [[graded manifold]] (via remark \ref{dgManifolds}) this is called the "[[shifted tangent bundle]]" $T[1] \Sigma$ of $X$.
More generally, let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]] over $\Sigma$. Then there is a [[Lie algebroid]]
$J^\infty_\Sigma(E)/T\Sigma$ over the [[jet bundle]] of $E$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
defined by its [[Chevalley-Eilenberg algebra]] being the [[horizontal derivative|horizontal]] part of the
[[variational bicomplex]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):
$$
CE\left(
J^\infty_\Sigma(E)/T\Sigma
\right)
\;\coloneqq\;
\left(\Omega^{\bullet,0}_\Sigma(E), d\right)
\,.
$$
The underlying [[graded manifold]] of $J^\infty_\Sigma(E)/T\Sigma$ is the [[fiber product]]
$J^\infty_\Sigma(E)\times_\Sigma T[1]\Sigma$ of the [[jet bundle]] of $E$ with the [[shifted tangent bundle]]
of $\Sigma$.
There is then a canonical homomorphism of Lie algebroids (def. \ref{HomomorphismBetweenLieAlgebroids})
$$
\array{
J^\infty_\Sigma(E)/T\Sigma
\\
\downarrow
\\
\Sigma/T\Sigma
}
$$
=--
$\,$
**[[local BRST complex|local]] [[off-shell]] [[BRST complex]]**
{#BRSTComplex}
With the general concept of [[Lie algebra action]] (def. \ref{InfinitesimalActionByLieAlgebra}) and the corresponding [[action Lie algebroids]] (def. \ref{ActionLieAlgebroid}) and more general [[Lie ∞-algebroids]] in hand (def. \ref{LInfinityAlgebroid})
we now apply this to the [[action]] of [[infinitesimal gauge symmetries]] (def. \ref{GaugeParameters}) on field histories of a [[Lagrangian field theory]], but we consider this _[[local field theory|locally]]_, namely on the [[jet bundle]]. The [[Chevalley-Eilenberg algebra]] of the resulting [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) is known as the _[[local BRST complex|local]] [[BRST complex]]_, example \ref{LocalOffShellBRSTComplex} below.
The [[Lie algebroid]]-perspective on [[BV-BRST formalism]] has been made explicit in ([Barnich 10](BRST+complex#Barnich10)).
+-- {: .num_defn #GaugeParametersClosed}
###### Definition
**(closed [[gauge parameters]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Then a [[gauge parameter]] bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ parameterizing [[infinitesimal gauge symmetries]] (def. \ref{GaugeParameters})
$$
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E
$$
is called _closed_ if it is closed under the [[Lie bracket]] of [[evolutionary vector fields]] (prop. \ref{EvolutionaryVectorFieldLieAlgebra}) in that there exists a morphism (not necessarily uniquely)
$$
\label{ClosedGaugeParametersBracket}
[-,-]_{\mathcal{G}}
\;\colon\;
J^\infty_\Sigma( \mathcal{G} \times_\Sigma \mathcal{G} \times_\Sigma E )
\longrightarrow
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E)
$$
such that
$$
\left[ R(-) , R(-)\right]
\;=\;
R([-,-]_{\mathcal{G}})
\,,
$$
where on the left we have the Lie bracket of [[evolutionary vector fields]] from prop. \ref{EvolutionaryVectorFieldLieAlgebra}.
Beware that $[-,-]_{\mathcal{G}}$ may be a function of the fields, namely of the [[jet bundle]] of the [[field bundle]] $E$. Hence for closed [[gauge parameters]] $[-,-]_{\mathcal{G}}$ in general defines a [[Lie algebroid]]-structure (def. \ref{LInfinityAlgebroid}).
Notice that the collection of all [[infinitesimal symmetries of the Lagrangian]] by necessity always forms a (very large) [[Lie algebra]]. The condition of closed [[gauge parameters]] is a condition on the _choice_ of parameterization of the [[infinitesimal gauge symmetries]], see remark \ref{GeneratingSetOfGaugeTransformations}.
=--
([Henneaux 90, section 2.9](BRST+complex#Henneaux90))
Recall the general concept of a [[Lie algebra action]] from def. \ref{InfinitesimalActionByLieAlgebra}.
The following realizes this for the action of closed [[infinitesimal gauge symmetries]] on the [[jet bundle]]
of a [[Lagrangian field theory]].
+-- {: .num_example #ActionOfGaugeParameterizedInfinitesimalGaugeSymmetriesOnJetBundle}
###### Example
**([[action]] of closed [[infinitesimal gauge symmetries]] on [[field (physics)|fields]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), and let
$\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of [[gauge parameters]] (def. \ref{GaugeParameters}) paramaterizing [[infinitesimal gauge symmetries]]
$$
J^\infty_\Sigma(\mathcal{G} \times_\Sigma E)
\overset{R}{\longrightarrow}
T_\Sigma E
$$
which are closed (def. \ref{GaugeParametersClosed}), via a bracket $[-,-]_{\mathcal{G}}$.
By passing from these [[evolutionary vector fields]] $R$ (def. \ref{EvolutionaryVectorField})
to their prolongations $\widehat{R}$, being actual vector fields on the jet bundle (prop. \ref{EvolutionaryVectorFieldProlongation}),
we obtain a bundle morphism of the form
$$
\array{
J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma (E)
&& \overset{\widehat{R(e)}}{\longrightarrow} &&
T_\Sigma J^\infty_\Sigma(E)
\\
& \searrow && \swarrow
\\
&& J^\infty_\Sigma(E)
}
$$
and via the assumed bracket $[-,-]_{\mathcal{G}}$ on [[gauge parameters]] this exhibits [[Lie algebroid]] structure on $J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E) \overset{pr_2}{\to} J^\infty_\Sigma(E)$.
In the case that $\mathcal{G} = \mathfrak{g} \times \Sigma$ is a [[trivial vector bundle]], with [[fiber]] $\mathfrak{g}$, then so is its [[jet bundle]]
$$
J^\infty_\Sigma(\mathfrak{g} \times \Sigma) = \mathfrak{g}^\infty \times \Sigma
\,.
$$
If moreover the bracket (eq:ClosedGaugeParametersBracket) on the [[infinitesimal gauge symmetries]] is independent of the fields, then this induces a [[Lie algebra]] structure on $\mathfrak{g}^\infty$ and exhibits an [[Lie algebra action]]
$$
\array{
\mathfrak{g}^\infty \times J^\infty_\Sigma E
&& \overset{\widehat{R(e)}}{\longrightarrow} &&
T_\Sigma J^\infty_\Sigma(E)
\\
& \searrow && \swarrow
\\
&& J^\infty_\Sigma(E)
}
\,.
$$
of the [[gauge parameter|gauge parameterized]] [[infinitesimal gauge symmetries]] on the [[jet bundle]] of the [[field bundle]] by [[infinitesimal]] [[diffeomorphisms]].
=--
+-- {: .num_example #LocalOffShellBRSTComplex}
###### Example
**([[local BRST complex|local]] [[BRST complex]] and [[ghost fields]] for closed [[infinitesimal gauge symmetries]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), and let
$\mathcal{G} \overset{gb}{\longrightarrow} \Sigma$ be a bundle of irreducible closed [[gauge parameters]]
for the theory (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) with
bundle morphism
$$
\array{
J^\infty_\Sigma( \mathcal{G} \times_\Sigma E )
&& \overset{R}{\longrightarrow} &&
T_\Sigma E
\\
& \searrow && \swarrow
\\
&& E
}
\,.
$$
Assuming that the gauge parameter bundle is [[trivial vector bundle|trivial]], $\mathcal{G} = \mathfrak{g} \times \Sigma$,
then by example \ref{ActionOfGaugeParameterizedInfinitesimalGaugeSymmetriesOnJetBundle} this
induces an [[action]] $\hat R$
of a Lie algebra $\mathfrak{g}^\infty$ on $J^\infty_\Sigma E$ by [[infinitesimal]] [[diffeomorphisms]].
The corresponding [[action Lie algebroid]] $J^\infty_\Sigma(E)/\mathfrak{g}^\infty$
(def. \ref{ActionLieAlgebroid}) has as underlying [[graded manifold]] (remark \ref{dgManifolds})
$$
\mathfrak{g}^\infty[1] \times J^\infty_\Sigma(E)
\;\simeq\;
J^\infty_\Sigma( \mathcal{G}[1] \times_\Sigma E )
$$
the [[jet bundle]] of the _[[graded manifold|graded]] [[field bundle]]_
$$
E_{BRST} \;\coloneqq\; E \times_\Sigma \mathcal{G}[1]
$$
which regards the [[gauge parameters]] as [[field (physics)|fields]] in degree 1.
As such these are called _[[ghost fields]]_:
$$
\left\{
\text{ghost field histories}
\right\}
\;\coloneqq\;
\Gamma_\Sigma( \mathcal{G}[1] )
\,.
$$
Therefore we write suggestively
$$
E/\mathcal{G}
\;\coloneqq\;
J^\infty_\Sigma(E)/\mathfrak{g}^\infty
$$
for the [[action Lie algebroid]] of the [[gauge parameter|gauge parameterized]] implicit [[infinitesimal gauge symmetries]]
on the [[jet bundle]] of the [[field bundle]].
The Chevalley-Eilenberg [[differential]] of the [[BRST complex]] is traditionally denoted
$$
s_{BRST} \coloneqq d_{CE}
\,.
$$
To express this in [[coordinates]], assume that the [[field bundle]] $E$ as well as the [[gauge parameter]]
bundle are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle})
with $(\phi^a)$ the [[field (physics)|field]] coordinates on the [[fiber]] of $E$
with induced jet coordinates $((x^\mu), (\phi^a), (\phi^a_{\mu}), \cdots)$ and
$(c^\alpha)$ are [[ghost field]] coordinates on the fiber of $\mathcal{G}[1]$ with induced
jet coordinates $((x^\mu), (c^\alpha), (c^\alpha_\mu), \cdots)$.
Then in terms of the
corresponding coordinate expression for the gauge symmetries $R$ (eq:CoordinateExpressionForGaugeParameterized)
the [[BRST differential]] is given on the [[field (physics)|fields]] by
$$
s_{BRST} \, \phi^a
\;=\;
c^\alpha_{,\mu_1 \cdots \mu_k}
\underset{k \in \mathbb{N}}{\sum}
R^{a \mu_1 \cdots \mu_k}_{\alpha}
$$
and on the [[ghost fields]] by
$$
s_{BRST} \, c^\alpha = \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
\,,
$$
and it extends from there, via prop. \ref{EvolutionaryVectorFieldProlongation},
to jets of fields and ghost fields by (anti-)commutativity with the [[total derivative|total spacetime derivative]].
Moreover, since the action of the [[infinitesimal gauge symmetries]] is by definition via
prolongations (prop. \ref{EvolutionaryVectorFieldProlongation}) of [[evolutionary vector fields]] (def. \ref{EvolutionaryVectorField}) and hence compatible with the [[total derivative|total spacetime derivative]] (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative) this construction
descends to the horizontal tangent Lie algebroid $J^\infty_\Sigma(E)/T\Sigma$ (example \ref{HorizontalTangentLieAlgebroid})
to yield
$$
E/(\mathcal{G}\times_\Sigma T \Sigma)
\;\coloneqq\;
\left(J^\infty_\Sigma(E)/T\Sigma\right)/\mathfrak{g}^\infty
$$
The [[Chevalley-Eilenberg differential]] on $E/(\mathcal{G}\times_\Sigma T \Sigma)$ is
$$
d - s_{BRST}
$$
The [[Chevalley-Eilenberg algebra]] of functions on this [[differential graded manifold]] (eq:CEAlgebra)
is called the [[off-shell]] _[[local BRST complex]]_.
=--
([Barnich-Brandt-Henneaux 94](local+BRST+cohomology#BarnichBrandtHenneaux94), [Barnich 10 (35)](BRST+complex#Barnich10)).
+-- {: .num_defn}
###### Definition
**(global [[BRST complex]])**
We may pass from the [[off-shell]] [[local BRST complex]] (def. \ref{LocalOffShellBRSTComplex}) on the [[jet bundle]] to the "global" BRST
complex by [[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}):
Write $Obs(E \times_\Sigma \mathcal{G}[1])$ for the induced graded [[off-shell]] [[algebra of observables]] (def. \ref{LocalObservables}).
For $A \in \Omega^{p+1,\bullet}_\Sigma(E \times_\Sigma \mathcal{G}[1])$ with corresponding [[local observable]]
$\tau_\Sigma(A) \in LocObs_\Sigma(E \times_\Sigma \mathcal{G}[1])$ its BRST differential is defined by
$$
s_{BRST} \tau_{\Sigma}(A) \;\coloneqq\; \tau_{\Sigma}(s_{BRST} A)
$$
and extended from there to $Obs(E \times_\Sigma \mathcal{G}[1])$ as a graded derivation.
=--
$\,$
**Examples of [[local BRST complexes]] of [[Lagrangian field theory|Lagrangian]] [[gauge theories]]**
{#ExamplesForLocalBRSTComplexes}
+-- {: .num_prop #LocalBRSTComplexForFreeElectromagneticFieldOnMinkowskiSpacetim}
###### Example
**([[local BRST complex]] for [[free field|free]] [[electromagnetic field]] on [[Minkowski spacetime]])**
Consider the [[Lagrangian field theory]] of [[free field|free]] [[electromagnetism]] on [[Minkowski spacetime]] (example \ref{ElectromagnetismLagrangianDensity}) with its [[gauge parameter]] bundle as in
example \ref{InfinitesimalGaugeSymmetryElectromagnetism}.
By (eq:EMProlongedSymmetryVectorField) the action of the [[BRST differential]] is the derivation
$$
s_{BRST}
\;=\;
c_{,\mu} \frac{\partial}{\partial a_\mu}
+
c_{, \mu \nu} \frac{\partial}{\partial a_{\mu, \nu}}
+
\cdots
\,.
$$
In particular the [[Lagrangian density]] is BRST-closed
$$
\begin{aligned}
s_{BRST} \mathbf{L}
& =
s_{BRST} f_{\mu \nu} f^{\mu \nu} dvol_\Sigma
\\
& =
c_{,\mu \nu} f^{\mu \nu} dvol_\Sigma
\\
& =
0
\end{aligned}
$$
as is the [[Euler-Lagrange form]] (due to the symmetry $c_{,\mu \nu} = c_{,\nu \mu}$ (eq:JetCoodinatesSymmetry) and in contrast to the
skew-symmetry $f_{\mu \nu} = - f_{\nu \mu}$).
=--
+-- {: .num_example #YangMillsLocalBRSTComplex}
###### Example
**([[local BRST complex]] for the [[Yang-Mills field]] on [[Minkowski spacetime]])**
For $\mathfrak{g}$ a [[semisimple Lie algebra]], consider the [[Lagrangian field theory]] of [[Yang-Mills theory]] on [[Minkowski spacetime]] from example \ref{YangMillsLagrangian}, with [[Lagrangian density]]
$$
\mathbf{L}
\;=\;
\tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu}
$$
given by the universal [[field strength]] (eq:YangMillsJetFieldStrengthMinkowski)
$$
f^\alpha_{\mu \nu}
\;\coloneqq\;
\tfrac{1}{2}
\left(
a^\alpha_{[\nu,\mu]}
+
\tfrac{1}{2} \gamma^\alpha_{\beta \gamma} a^\beta_{[\mu} a^\gamma_{\nu]}
\right)
\,.
$$
Let $\mathcal{G} \coloneqq \Sigma \times \mathfrak{g}$ be the [[trivial vector bundle]] with [[fiber]] $\mathfrak{g}$, regarded as a
[[gauge parameter]] bundle (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) with coordinate functions $((x^\mu), c^\alpha)$ and consider the [[gauge parameter|gauge parametrized]] [[evolutionary vector field]] (eq:CoordinateExpressionForGaugeParameterized)
$$
R
\;=\;
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
$$
from example \ref{InfinitesimalGaugeSymmetryOfYangMillsTheory}.
We claim that these are _closed [[gauge parameters]]_ in the sense of def. \ref{GaugeParametersClosed}, hence that the [[local BRST complex]] in the form of example \ref{LocalOffShellBRSTComplex} exists.
To see this, observe that, by def. \ref{ActionLieAlgebroid} the candidate BRST differential needs to be of the form (eq:OnMinkowskiInfinitesimalGaugeSymmetryForYangMills) plus the [[linear dual]] of the [[Lie bracket]] $[-,-]_{\mathcal{G}}^\ast$
$$
s_{BRST}
\;=\;
\left(
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
\;+\;
\text{prolongation}
\right)
+
([-,-]_{\mathcal{G}})^\ast
\,.
$$
Moreover, by def. \ref{ActionLieAlgebroid} we may equivalently make an Ansatz for $([-,-]_{\mathcal{G}})^\ast$ and if the resulting differential $s_{BRST}$ squares to zero, as this dually defines the required closure bracket $[-,-]_\mathcal{G}$.
We claim that
$$
\label{OffShellYangMillsOnMinkowskiBRSTOperator}
s_{BRST}
\;\coloneqq\;
\widehat{
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\frac{\partial}{\partial a^\alpha_\mu}
}
+
\widehat{
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} \, c^\beta c^\gamma
\frac{\partial}{\partial c^\alpha}
}
\,,
$$
where the hat denotes prolongation (prop. \ref{EvolutionaryVectorFieldProlongation}). This is the [[local BRST complex|local]] ([[jet bundle]]) [[BRST differential]] for [[Yang-Mills theory]] on [[Minkowski spacetime]].
=--
(e.g. [Barnich-Brandt-Henneaux 00 (7.2)](local+BRST+cohomology#BarnichBrandtHenneaux00))
+-- {: .proof}
###### Proof
We need to show that (eq:OffShellYangMillsOnMinkowskiBRSTOperator) squares to zero. Consider the two terms that appear:
$$
(s_{BRST})^2
=
\left[
\widehat{
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
}
\;,\;
\widehat{
\left(
c^{\alpha'}_{,\mu}
-
\gamma^{\alpha'}_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^{\alpha'}_\mu}
}
\right]
\;+\;
2
\left[
\widehat{
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
}
\;,\;
\widehat{
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma}
\,
c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}
}
\right]
\,.
$$
The first term is
$$
\begin{aligned}
\left[
\widehat{
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
}
\;,\;
\widehat{
\left(
c^{\alpha'}_{,\mu}
-
\gamma^{\alpha'}_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^{\alpha'}_\mu}
}
\right]
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
\left(
c^\gamma_{,\mu}
-
\gamma^\gamma_{\beta' \gamma'} c^{\beta'} a^{\gamma'}_\mu
\right)
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\\
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
c^\gamma_{,\mu}
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
+
2
\gamma^{\alpha'}_{\beta \gamma}
\gamma^\gamma_{\beta' \gamma'}
\widehat{
c^\beta c^{\beta'} a^{\gamma'}_\mu
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\\
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
c^\gamma_{,\mu}
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
+
\gamma^{\alpha'}_{\beta \gamma}
\gamma^\gamma_{\beta' \gamma'}
\widehat{
\left(
c^\beta c^{\beta'} a^{\gamma'}_\mu
-
c^{\beta'} c^{\beta} a^{\gamma'}_\mu
\right)
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\\
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
c^\gamma_{,\mu}
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
+
\gamma^{\alpha'}_{\beta \gamma}
\gamma^\gamma_{\beta' \gamma'}
\widehat{
\left(
-
c^\beta c^{\gamma'} a^{\beta'}_\mu
-
c^{\beta'} c^{\beta} a^{\gamma'}_\mu
\right)
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\\
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
c^\gamma_{,\mu}
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
+
\gamma^{\alpha'}_{\beta \gamma}
\gamma^\gamma_{\beta' \gamma'}
\widehat{
c^{\gamma'} c^{\beta'} a^{\beta}_\mu
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\\
& =
-
2
\gamma^{\alpha'}_{\beta \gamma}
\widehat{
c^\beta
c^\gamma_{,\mu}
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
+
\gamma^{\alpha'}_{\gamma \beta}
\gamma^\beta_{\beta' \gamma'}
\widehat{
c^{\beta'} c^{\gamma'} a^{\gamma}_\mu
\frac{\partial}{\partial a^{\alpha'}_\mu}
}
\end{aligned}
$$
Here first we expanded out, then in the second-but-last line we used the [[Jacobi identity]] (eq:JacobiIdentity) and in the last line we adjusted indices, just for convenience of comparison with the next term. That next term is
$$
\left[
\widehat{
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu
\right)
\partial_{a^\alpha_\mu}
}
\;,\;
\gamma^\alpha{}_{\beta \gamma}
\,
\widehat{c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}}
\right]
=
2
\gamma^\alpha_{\beta \gamma}
\widehat{
c^\beta_{,\mu} c^\gamma
\frac{\partial}{\partial a^\alpha_\mu}
}
-
\gamma^\alpha_{\beta \gamma}
\gamma^\beta_{\beta' \gamma'}
\widehat{
c^{\beta'} c^{\gamma'}
a^\gamma_\mu
\frac{\partial}{\partial a^\alpha_\mu}
}
\,,
$$
where the first summand on the right comes from the prolongation.
This shows that the two terms cancel.
=--
+-- {: .num_example #LocalBRSTComplexBFieldMinkowskiSpacetime}
###### Example
**([[local BRST complex]] for [[B-field]] on [[Minkowski spacetime]])**
Consider the [[Lagrangian field theory]] of the [[B-field]] on [[Minkowski spacetime]] from example \ref{BFieldLagrangianDensity}, with [[field bundle]] the [[differential 2-form]]-bundle $E = \wedge^2_\Sigma T^\ast \Sigma$ with coordinates $((x^\mu), (b_{\mu \nu}))$ subject to $b_{\mu \nu} = - b_{\nu \mu}$; and with [[Lagrangian density]].
By example \ref{InfinitesimalGaugeSymmetryOfTheBField} the [[local BRST complex]] (example \ref{example}) has BRST differential of the form
$$
c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}}
+
c_{\mu,\nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}}
+
\cdots
\,.
$$
In this case this enhanced to an [[L-infinity algebroid|Lie 2-algebroid]] by regarding the second-order [[gauge parameters]] (eq:SecondOrderGaugeParameterBundleForBFieldOnMinkowskiSpacetime) in degree 2 to form a [[graded manifold|graded]] [[field bundle]]
$$
\underset{ \{\overset{(2)}{c}\} }{
\underbrace{
\overset{(2)}{\mathcal{G}}[2]
}}
\times_\Sigma
\underset{\{c_\mu\}}{
\underbrace{
\mathcal{G}[1]
}
}
\times_\Sigma
\underset{
(b_{\mu \nu})
}{
\underbrace{
E
}}
\;=\;
\mathbb{R}[2] \times T^\ast \Sigma [1] \times_\Sigma E
$$
by adding the [[ghost-of-ghost field]] $(\overset{(2)}{c})$ (eq:SecondOderGaugeSymmetryOfBFieldOnMinkowski) and taking the local BRST differential to be the sum of the first order [[infinitesimal gauge symmetries]] (eq:InfinitesimalGaugeSymmetryForBFieldOnMinkowskiSpacetime) and the second order [[infinitesimal gauge-of-gauge symmetry]] (eq:SecondOderGaugeSymmetryOfBFieldOnMinkowski):
$$
s_{BRST}
\;=\;
\left(
c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}}
+
c_{\mu,\nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}}
+
\cdots
\right)
+
\left(
\overset{(2)}{c}_{,\mu} \frac{\partial}{\partial c_\mu}
+
\overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}}
+
\cdots
\right)
\,.
$$
Notice that this indeed still squares to zero, due to the second-order [[Noether identity]] (eq:NoetherIdentitySecondOrderForBFieldOnMinkowskiSpacetime):
$$
\begin{aligned}
\left( s_{BSRT} \right)^2
& =
\left[
\overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}},
c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}}
\right]
\;+\;
\left[
\overset{(2)}{c}_{,\mu \nu_1 \nu_2} \frac{\partial}{\partial c_{\mu,\nu_1 \nu_2}},
c_{\mu, \nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}}
\right]
\\
& =
\underset{ = 0 }{
\underbrace{
\overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial b_{\mu \nu}}
}}
\;+\;
\underset{ = 0 }{
\underbrace{
\overset{(2)}{c}_{,\mu \nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}}
}}
\;+\;
\cdots
\\
& = 0
\,.
\end{aligned}
$$
=--
$\,$
This concludes our discussion of [[infinitesimal gauge symmetries]], their [[off-shell]] [[action]] on the [[jet bundle]] of the [[field bundle]] and the corresponding [[homotopy quotient]] exhibited by the [[local BRST complex]]. In the [next chapter](#ReducedPhaseSpace) we discuss the [[homotopy intersection]] of this construction with the [[shell]]: the _[[reduced phase space]]_.
|
A first idea of quantum field theory -- Geometry | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Geometry |
## Geometry
{#Geometry}
The [[geometry of physics]] is _[[differential geometry]]_. This is the flavor of [[geometry]] which is modeled on
[[Cartesian spaces]] $\mathbb{R}^n$ with [[smooth functions]] between them. Here we briefly review the basics of
[[differential geometry]] on [[Cartesian spaces]].
In principle the only **background** assumed of the reader here is
1. usual _[[structural set theory|naive set theory]]_ (e.g. [[Sets for Mathematics|Lawvere-Rosebrugh 03]]);
1. the concept of the _[[continuum]]_: the [[real line]] $\mathbb{R}$, the [[plane]] $\mathbb{R}^2$, etc.
1. the concepts of _[[differentiation]]_ and _[[integration]]_ of functions on such [[Cartesian spaces]];
hence essentially the content of multi-variable [[differential calculus]].
We now discuss:
* _[Abstract coordinate systems](#SmoothFunctions)_
* _[Fiber bundles](#BundlesAndSections)_
* _[Synthetic differential geometry](#SyntheticDifferentialGeometry)_
* _[Differential forms](#DifferentialFormsAndCartanCalculus)_
As we uncover [[Lagrangian field theory]] further below, we discover ever more general concepts of
"[[space]]" in differential geometry, such as
_[[smooth manifolds]]_, _[[diffeological spaces]]_, _[[infinitesimal neighbourhoods]]_, _[[supermanifolds]]_, _[[Lie algebroids]]_
and _[[super L-∞ algebra|super]] [[Lie ∞-algebroids]]_.
We introduce these incrementally as we go along:
**more general [[spaces]] in [[differential geometry]] introduced further below**
{#NotionsOfGeometry}
| | | | | | | | | | | | [[higher differential geometry]] |
|--|---|---|---|---|---|---|---|---|---|---|---|
| **[[differential geometry]]** | [[smooth manifolds]] <br/> (def. \ref{SmoothManifoldInsideDiffeologicalSpaces}) | $\hookrightarrow$ | [[diffeological spaces]] <br/> (def. \ref{DiffeologicalSpace}) | $\hookrightarrow$ | [[smooth sets]] <br/> (def. \ref{SmoothSet}) | $\hookrightarrow$ | [[formal smooth sets]] <br/> (def. \ref{FormalSmoothSet}) | $\hookrightarrow$ | [[super formal smooth sets]] <br/> (def. \ref{SuperFormalSmoothSet}) | $\hookrightarrow$ | [[super formal smooth ∞-groupoids]] <br/> (not needed in fully [[perturbative QFT]]) |
| **[[infinitesimal]] [[formal geometry|geometry]], <br/> [[Lie theory]]** | | | | | | | [[infinitesimally thickened points]] <br/> (def. \ref{InfinitesimallyThickendSmoothManifold}) | | [[superpoints]] <br/> (def. \ref{SuperCartesianSpace}) | | [[Lie ∞-algebroids]] <br/> (def. \ref{LInfinityAlgebroid}) |
| | | | | | | | | | | | **[[higher Lie theory]]** |
| **needed in [[QFT]] for:** | [[spacetime]] (def. \ref{MinkowskiSpacetime}) | | [[space of field histories]] <br/> (def. \ref{DiffeologicalSpaceOfFieldHistories}) | | | | [[Cauchy surface]] (def. \ref{CauchySurface}), <br/> [[perturbation theory]] (def. \ref{LocalObservablesOnInfinitesimalNeighbourhood}) | | [[Dirac field]] (expl. \ref{DiracFieldBundle}), [[Pauli exclusion principle]] | | [[infinitesimal gauge symmetry]]/[[BRST complex]] <br/> (expl. \ref{LocalOffShellBRSTComplex}) |
$\,$
**Abstract coordinate systems**
{#SmoothFunctions}
What characterizes [[differential geometry]] is that it models [[geometry]] on _the [[continuum]]_,
namely the [[real line]] $\mathbb{R}$, together with its [[Cartesian products]] $\mathbb{R}^n$,
regarded with its canonical [[smooth structure]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem} below).
We may think of these _[[Cartesian spaces]]_ $\mathbb{R}^n$ as the "abstract [[coordinate systems]]"
and of the [[smooth functions]] between them as the "abstract [[coordinate transformations]]".
We will eventually consider [below](#FieldBundles) much more general "[[smooth spaces]]" $X$ than just the [[Cartesian spaces]] $\mathbb{R}^n$;
but all of them are going to be understood by "laying out abstract coordinate systems" inside them,
in the general sense of having smooth functions $f \colon \mathbb{R}^n \to X$ mapping a Cartesian space smoothly into them.
All structure on [[generalized smooth spaces]] $X$ is thereby reduced to _compatible systems_ of structures on just [[Cartesian spaces]],
one for each smooth "probe" $f\colon \mathbb{R}^n \to X$. This is called "[[functorial geometry]]".
Notice that the popular concept of a _[[smooth manifold]]_ (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below)
is essentially that of a [[smooth space]] which _locally looks just like_ a [[Cartesian space]],
in that there exist sufficiently many $f \colon \mathbb{R}^n \to X$ which are ([[open map|open]]) _[[isomorphisms]]_
onto their [[images]]. Historically it was a long process to arrive at the insight that it is wrong to _fix_
such local coordinate identifications $f$, or to have any structure depend on such a choice. But it is useful
to go one step further:
In [[functorial geometry]] we do not even focus attention on those $f \colon \mathbb{R}^n \to X$ that are
isomorphisms onto their image, but consider _all_ "probes" of $X$ by "abstract coordinate systems".
This makes [[differential geometry]] both simpler as well as more powerful. The analogous insight
for [[algebraic geometry]] is due to [Grothendieck 65](functorial+geometry#Grothendieck65); it was transported
to [[differential geometry]] by [Lawvere 67](synthetic+differential+geometry#Lawvere67).
This allows to combine the best of two superficially disjoint worlds: On the one hand we may reduce
all constructions and computations to [[coordinates]], the way traditionally done in the [[physics]] literature;
on the other hand we have full conceptual control over the coordinate-free generalized spaces analyzed thereby.
What makes this work is that all [[coordinate]]-constructions are _[[functorial geometry|functorially]]_ considered
over all abstract coordinate systems.
$\,$
+-- {: .num_defn #CartesianSpacesAndSmoothFunctionsBetweenThem}
###### Definition
**([[Cartesian spaces]] and [[smooth functions]] between them)**
For $n \in \mathbb{N}$ we say that the set $\mathbb{R}^n$ of [[n-tuples]] of [[real numbers]]
is a _[[Cartesian space]]_. This comes with the canonical [[coordinate functions]]
$$
x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
$$
which send an [[n-tuple]] of real numbers to the $k$th element in the tuple, for $k \in \{1, \cdots, n\}$.
For
$$
f \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'}
$$
any [[function]] between [[Cartesian spaces]], we may ask whether its [[partial derivative]]
along the $k$th coordinate exists, denoted
$$
\frac{\partial f}{\partial x^k}
\;\colon\;
\mathbb{R}^{n}
\longrightarrow
\mathbb{R}^{n'}
\,.
$$
If this exists, we may in turn ask that the [[partial derivative]] of the partial derivative exists
$$
\frac{\partial^2 f}{\partial x^{k_1} \partial x^{k_2}}
\coloneqq
\frac{\partial}{\partial x^{k_2}} \frac{\partial f}{\partial x^{k_1}}
$$
and so on.
A general higher [[partial derivative]] obtained this way is, if it exists, indexed by an [[n-tuple]]
of [[natural numbers]] $\alpha \in \mathbb{N}^n$ and denoted
$$
\label{PartialDerivativeWithManyIndices}
\partial_\alpha
\;\coloneqq\;
\frac{
\partial^{\vert \alpha \vert} f
}{
\partial^{\alpha_1} x^1
\partial^{\alpha_2} x^2
\cdots
\partial^{\alpha_n} x^n
}
\,,
$$
where ${\vert \alpha\vert} \coloneqq \underoverset{n}{i = 1}{\sum} \alpha_i$ is the total _order_ of the partial derivative.
If all partial derivative to all orders $\alpha \in \mathbb{N}^n$ of a [[function]] $f \colon \mathbb{R}^n \to \mathbb{R}^{n'}$
exist, then $f$ is called a _[[smooth function]]_.
=--
Of course the [[composition]] $g \circ f$ of two smooth functions is again a [[smooth function]].
$$
\array{
&& \mathbb{R}^{n_2}
\\
& {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
\\
\mathbb{R}^{n_1}
&& \underset{g \circ f}{\longrightarrow} &&
\mathbb{R}^{n_3}
}
\,.
$$
The inclined reader may notice that this means that [[Cartesian spaces]] with [[smooth functions]]
between them constitute a _[[category]]_ ("[[CartSp]]"); but the reader not so inclined may ignore this.
For the following it is useful to think of each [[Cartesian space]] as an _abstract [[coordinate system]]_.
We will be dealing with various [[generalized smooth spaces]] (see the table [below](#NotionsOfGeometry)), but they will all be characterized by
a prescription for how to smoothly map abstract coordinate systems into them.
+-- {: .num_example #CoordinateFunctionsAreSmoothFunctions}
###### Example
**([[coordinate functions]] are [[smooth functions]])**
Given a [[Cartesian space]] $\mathbb{R}^n$, then all its [[coordinate functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
$$
x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
$$
are [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).
For
$$
f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
$$
any [[smooth function]] and $a \in \{1, 2, \cdots, n_2\}$ write
$$
f^a \coloneqq x^a \circ f
\;\colon\;
\mathbb{R}^{n_1}
\overset{f}{\longrightarrow}
\mathbb{R}^{n_2}
\overset{x^a}{\longrightarrow}
\mathbb{R}
$$
.
for its [[composition]] with this [[coordinate function]].
=--
+-- {: .num_example #AlgebraOfSmoothFunctionsOnCartesianSpaces}
###### Example
**([[algebra of functions|algebra of]] [[smooth functions]] on [[Cartesian spaces]])**
For each $n \in \mathbb{N}$, the set
$$
C^\infty(\mathbb{R}^n)
\;\coloneqq\;
Hom_{CartSp}(\mathbb{R}^n, \mathbb{R})
$$
of [[real number]]-valued [[smooth functions]] $f \colon \mathbb{R}^n \to \mathbb{R}$
on the $n$-dimensional [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
becomes a [[commutative algebra|commutative]] [[associative algebra]] over the [[ring]] of [[real numbers]]
by pointwise addition and multiplication in $\mathbb{R}$: for $f,g \in C^\infty(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$
1. $(f + g)(x) \coloneqq f(x) + g(x)$
1. $(f \cdot g)(x) \coloneqq f(x) \cdot g(x)$.
The inclusion
$$
\mathbb{R} \overset{const}{\hookrightarrow} C^\infty(\mathbb{R}^n)
$$
is given by the [[constant functions]].
We call this the _[[real numbers|real]] [[algebra of functions|algebra of]] [[smooth functions]]_ on $\mathbb{R}^n$:
$$
C^\infty(\mathbb{R}^n)
\;\in\;
\mathbb{R} Alg
\,.
$$
If
$$
f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
$$
is any [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) then
[[composition|pre-composition]] with $f$ ("[[pullback of functions]]")
$$
\array{
C^\infty(\mathbb{R}^{n_2})
&\overset{f^\ast}{\longrightarrow}&
C^\infty(\mathbb{R}^{n_1})
\\
g &\mapsto& f^\ast g \coloneqq g \circ f
}
$$
is an [[associative algebra|algebra]] [[homomorphism]]. Moreover, this is clearly compatible with [[composition]] in that
$$
f_1^\ast(f_2^\ast g) = (f_2 \circ f_1)^\ast g
\,.
$$
Stated more [[category theory|abstractly]], this means that assigning [[algebra of functions|algebras]] of [[smooth functions]]
is a [[functor]]
$$
C^\infty(-)
\;\colon\;
CartSp \longrightarrow \mathbb{R} Alg^{op}
$$
from the [[category]] [[CartSp]] of [[Cartesian spaces]] and [[smooth functions]] between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), to the
[[opposite category|opposite]] of the category $\mathbb{R}$[[Alg]] of $\mathbb{R}$-[[associative algebra|algebras]].
=--
+-- {: .num_defn #LocalDiffeomorphismBetweenCartesianSpaces}
###### Definition
**([[local diffeomorphisms]] and [[open embeddings]] of [[Cartesian spaces]])**
A [[smooth function]] $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ from one [[Cartesian space]] to itself
(def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is called a _[[local diffeomorphism]]_, denoted
$$
f \;\colon\; \mathbb{R}^{n} \overset{et}{\longrightarrow} \mathbb{R}^n
$$
if the [[determinant]] of the [[matrix]] of [[partial derivatives]] (the "[[Jacobian]]" of $f$) is
everywhere non-vanishing
$$
det
\left(
\array{
\frac{\partial f^1}{\partial x^1}(x)
&\cdots&
\frac{\partial f^n}{\partial x^1}(x)
\\
\vdots && \vdots
\\
\frac{\partial f^1}{\partial x^n}(x)
&\cdots&
\frac{\partial f^n}{\partial x^n}(x)
}
\right)
\;\neq\;
0
\phantom{AAAA}
\text{for all} \, x \in \mathbb{R}^n
\,.
$$
If the function $f$ is both a [[local diffeomorphism]], as above, as well as an [[injective function]] then
we call it an _[[open embedding]]_, denoted
$$
f \;\colon\;
\mathbb{R}^n \overset{\phantom{A}et\phantom{A}}{\hookrightarrow} \mathbb{R}^n
\,.
$$
=--
+-- {: .num_defn #DifferentiablyGoodOpenCover}
###### Definition
**([[good open cover]] of [[Cartesian spaces]])**
For $\mathbb{R}^n$ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}),
a _[[differentiably good open cover]]_ is
* an [[indexed set]]
$$
\left\{
\mathbb{R}^n
\underoverset{et}{\phantom{AA}f_i\phantom{AA}}{\hookrightarrow}
\mathbb{R}^n
\right\}_{i \in I}
$$
of [[open embeddings]]
(def. \ref{LocalDiffeomorphismBetweenCartesianSpaces})
such that the [[images]]
$$
U_i \coloneqq im(f_i) \subset \mathbb{R}^n
$$
satisfy:
1. ([[open cover]]) every point of $\mathbb{R}^n$ is contained in at least one of the $U_i$;
1. ([[good open cover|good]]) all [[finite set|finite]] [[intersections]] $U_{i_1} \cap \cdots \cap U_{i_k} \subset \mathbb{R}^n$ are either [[empty set]] or themselves images of [[open embeddings]] according to def. \ref{LocalDiffeomorphismBetweenCartesianSpaces}.
The inclined reader may notice that the concept of [[differentiably good open covers]] from def. \ref{DifferentiablyGoodOpenCover}
is a _[[coverage]]_ on the [[category]] _[[CartSp]]_ of [[Cartesian spaces]] with [[smooth functions]] between them, making it a
_[[site]]_, but the reader not so inclined may ignore this.
=--
([[schreiber:Cech Cocycles for Differential characteristic Classes|Fiorenza-Schreiber-Stasheff 12, def. 6.3.9]])
$\,$
$\,$
**[[fiber bundles]]**
{#BundlesAndSections}
Given any context of [[objects]] and [[morphisms]] between them, such as the [[Cartesian spaces]]
and [[smooth functions]] from def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}
it is of interest to fix one [[object]] $X$ and consider other objects _[[dependent type|parameterized over]]_
it. These are called _[[bundles]]_ (def. \ref{BundlesAndFibers}) below.
For reference, we briefly discuss here the basic concepts related to [[bundles]] in the context of
[[Cartesian spaces]].
Of course the theory of bundles is mostly trivial over Cartesian spaces;
it gains its main interest from its generalization to more general [[smooth manifolds]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below).
It is still worthwhile for our development to first consider the relevant concepts in this simple case first.
For more exposition see at _[[fiber bundles in physics]]_.
$\,$
+-- {: .num_defn #BundlesAndFibers}
###### Definition
**([[bundles]])**
We say that a [[smooth function]] $E \overset{fb}{\to} X$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
is a _[[bundle]]_ just to amplify that we think of it as exhibiting $E$ as being a "space over $X$":
$$
\array{
E
\\
\downarrow\mathrlap{fb}
\\
X
}
\,.
$$
For $x \in X$ a point, we say that the _[[fiber]]_ of this [[bundle]] over $x$ is the [[pre-image]]
$$
\label{FiberOfAFiberBundle}
E_x \coloneqq fb^{-1}(\{x\}) \subset E
$$
of the point $x$ under the smooth function. We think of $fb$ as exhibiting a "smoothly varying" set of [[fiber]] spaces over $X$.
Given two [[bundles]] $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ over $X$,
a _[[homomorphism]] of bundles_ between them is a [[smooth function]]
$f \colon E_1 \to E_2$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) between their
total spaces which respects the bundle projections, in that
$$
fb_2 \circ f = fb_1
\phantom{AAAA}
\text{i.e.}
\phantom{AAA}
\array{
E_1 && \overset{f}{\longrightarrow} && E_2
\\
& {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
\\
&& X
}
\,.
$$
Hence a bundle homomorphism is a smooth function that sends [[fibers]] to [[fibers]] over the same point:
$$
f\left(
(E_1)_x
\right)
\;\subset\;
(E_2)_x
\,.
$$
The inclined reader may notice that this defines a [[category]] of [[bundles]] over $X$,
which is in fact just the _[[slice category]]_ $CartSp_{/X}$; the reader not so inclined may ignore this.
=--
+-- {: .num_defn #Sections}
###### Definition
**([[sections]])**
Given a [[bundle]] $E \overset{fb}{\to} X$ (def. \ref{BundlesAndFibers})
a _[[section]]_ is a [[smooth function]] $s \colon X \to E$ such that
$$
fb \circ s = id_X
\phantom{AAAAA}
\array{
&& E
\\
&
{}^{\mathllap{s}}\nearrow
& \downarrow\mathrlap{fb}
\\
X &=& X
}
\,.
$$
This means that $s$ sends every point $x \in X$ to an element in the [[fiber]] over that point
$$
s(x) \in E_x
\,.
$$
We write
$$
\Gamma_X(E) \coloneqq
\left\{
\array{
&& E
\\
& {}^{\mathllap{s}}\nearrow & \downarrow^\mathrlap{fb}
\\
X &=& X
}
\phantom{fb}
\right\}
$$
for the [[space of sections|set of sections]] of a bundle.
For $E_1 \overset{f_1}{\to} X$ and $E_2 \overset{f_2}{\to} X$ two [[bundles]] and for
$$
\array{
E_1
&& \overset{f}{\longrightarrow} &&
E_2
\\
& {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
\\
&& X
}
$$
a bundle [[homomorphism]] between them (def. \ref{BundlesAndFibers}), then [[composition]] with $f$ sends [[sections]] to [[sections]]
and hence yields a [[function]] denoted
$$
\array{
\Gamma_X(E_1)
&\overset{f_\ast}{\longrightarrow}&
\Gamma_X(E_2)
\\
s &\mapsto& f \circ s
}
\,.
$$
=--
+-- {: .num_example #TrivialBundleOnCartesianSpace}
###### Example
**([[trivial bundle]])**
For $X$ and $F$ [[Cartesian spaces]], then the [[Cartesian product]] $X \times F$ equipped with the [[projection]]
$$
\array{
X \times F
\\
\downarrow^\mathrlap{pr_1}
\\
X
}
$$
to $X$ is a [[bundle]] (def. \ref{BundlesAndFibers}), called the _[[trivial bundle]]_
with [[fiber]] $F$. This represents the _constant_ smoothly varying set of [[fibers]], constant on $F$
If $F = \ast$ is the point, then this is the identity bundle
$$
\array{
X
\\
\downarrow\mathrlap{id}
\\
X
}
\,.
$$
Given any [[bundle]] $E \overset{fb}{\to} X$, then a bundle homomorphism (def. \ref{BundlesAndFibers})
from the identity bundle to $E \overset{fb}{\to} X$ is equivalently a [[section]] of $E \overset{fb}{\to} X$ (def. \ref{Sections})
$$
\array{
X && \overset{s}{\longrightarrow} && E
\\
& {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{fb}}
\\
&& X
}
$$
=--
+-- {: .num_defn #FiberBundle}
###### Definition
**([[fiber bundle]])**
A [[bundle]] $E \overset{fb}{\to} X$ (def. \ref{BundlesAndFibers}) is called a _[[fiber bundle]]_ with _typical fiber_ $F$ if
there exists a [[differentiably good open cover]] $\{U_i \hookrightarrow X\}_{i \in I}$ (def. \ref{DifferentiablyGoodOpenCover})
such that the restriction of $fb$ to each $U_i$ is [[isomorphism|isomorphic]] to the [[trivial fiber bundle]] with
fiber $F$ over $U_i$. Such [[diffeomorphisms]] $f_i \colon U_i \times F \overset{\simeq}{\to} E\vert_{U_i}$
are called _[[local trivializations]]_ of the fiber bundle:
$$
\array{
U_i \times F &\underoverset{\simeq}{f_i}{\longrightarrow}& E\vert_{U_i}
\\
& {}_{\mathllap{pr_1}}\searrow & \downarrow\mathrlap{fb\vert_{U_i}}
\\
&& U_i
}
\,.
$$
=--
+-- {: .num_defn #VectorBundle}
###### Definition
**([[vector bundle]])**
A _[[vector bundle]]_ is a [[fiber bundle]] $E \overset{vb}{\to} X$ (def. \ref{FiberBundle}) with typical fiber a [[vector space]] $V$ such that
there exists a [[local trivialization]] $\{U_i \times V \underoverset{\simeq}{f_i}{\to} E\vert_{U_i}\}_{i \in I}$
whose _gluing functions_
$$
U_i \cap U_j \times V
\overset{f_i\vert_{U_i \cap U_j}}{\longrightarrow}
E\vert_{U_i \cap U_j}
\overset{f_j^{-1}\vert_{U_i \cap U_j}}{\longrightarrow}
U_i \cap U_j \times V
$$
for all $i,j \in I$ are [[linear functions]] over each point $x \in U_i \cap U_j$.
A [[homomorphism]] of [[vector bundle]] is a bundle morphism $f$ (def. \ref{BundlesAndFibers})
such that there exist [[local trivializations]] on both sides with respect to which $g$ is
[[fiber]]-wise a [[linear map]].
The inclined reader may notice that this makes vector bundles over $X$ a [[category]] (denoted $Vect_{/X}$);
the reader not so inclined may ignore this.
=--
+-- {: .num_example #ModuleOfSectionsOfAVectorBundle}
###### Example
**([[module]] of [[sections]] of a [[vector bundle]])**
Given a [[vector bundle]] $E \overset{vb}{\to} X$ (def. \ref{VectorBundle}),
then its [[space of sections|set of sections]] $\Gamma_X(E)$ (def. \ref{BundlesAndFibers})
becomes a [[real vector space]] by [[fiber]]-wise multiplication with [[real numbers]].
Moreover, it becomes a [[module]] over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(X)$
(example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) by the same [[fiber]]-wise multiplication:
$$
\array{
C^\infty(X) \otimes_{\mathbb{R}} \Gamma_X(E) &\longrightarrow& \Gamma_X(E)
\\
(f,s) &\mapsto& (x \mapsto f(x) \cdot s(x))
}
\,.
$$
For $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ two [[vector bundles]] and
$$
\array{
E_1 && \overset{f}{\longrightarrow} && E_2
\\
& {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
\\
&& X
}
$$
a vector bundle homomorphism (def. \ref{VectorBundle}) then the induced function on sections (def. \ref{Sections})
$$
f_\ast \;\colon\; \Gamma_X(E_1) \longrightarrow \Gamma_X(E_2)
$$
is compatible with this [[action]] by smooth functions and hence constitutes a [[homomorphism]] of $C^\infty(X)$-[[modules]].
The inclined reader may notice that this means that taking [[spaces of sections]] yields a [[functor]]
$$
\Gamma_X(-)
\;\colon\;
Vect_{/X}
\longrightarrow
C^\infty(X) Mod
$$
from the [[category of vector bundles]] over $X$ to that over [[modules]] over $C^\infty(X)$.
=--
+-- {: .num_example #TangentVectorFields}
###### Example
**([[tangent vector fields]] and [[tangent bundle]])**
For $\mathbb{R}^n $ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
the [[trivial vector bundle]] (example \ref{TrivialBundleOnCartesianSpace}, def. \ref{VectorBundle})
$$
\array{
T \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times \mathbb{R}^n
\\
\mathllap{tb}\downarrow && \downarrow\mathrlap{pr_1}
\\
\mathbb{R}^n &=& \mathbb{R}^n
}
$$
is called the _[[tangent bundle]]_ of $\mathbb{R}^n$. With $(x^a)_{a = 1}^n$ the [[coordinate functions]]
on $\mathbb{R}^n$ (def. \ref{CoordinateFunctionsAreSmoothFunctions}) we write $(\partial_a)_{a = 1}^n$
for the corresponding [[linear basis]] of $\mathbb{R}^n$ regarded as a [[vector space]]. Then
a general [[section]] (def. \ref{Sections})
$$
\array{
&& T \mathbb{R}^n
\\
& {}^{\mathllap{v}}\nearrow& \downarrow\mathrlap{tb}
\\
\mathbb{R}^n &=& \mathbb{R}^n
}
$$
of the [[tangent bundle]] has a unique expansion of the form
$$
v = v^a \partial_a
$$
where a sum over indices is understood ([[Einstein summation convention]]) and where
the components $(v^a \in C^\infty(\mathbb{R}^n))_{a = 1}^n$
are [[smooth functions]] on $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).
Such a $v$ is also called a smooth _[[tangent vector field]]_ on $\mathbb{R}^n$.
Each tangent vector field $v$ on $\mathbb{R}^n$ determines a [[partial derivative]]
on [[smooth functions]]
$$
\array{
C^\infty(\mathbb{R}^n) &\overset{D_v}{\longrightarrow}& C^\infty(\mathbb{R}^n)
\\
f &\mapsto& \mathrlap{ D_v f \coloneqq v^a \partial_a (f) \coloneqq \sum_a v^a \frac{\partial f}{\partial x^a} }
}
\,.
$$
By the [[product law]] of [[differentiation]], this is a [[derivation]] on the [[algebra of functions|algebra of]] [[smooth functions]]
(example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) in that
1. it is an $\mathbb{R}$-[[linear map]] in that
$$
D_v( c_1 f_1 + c_2 f_2 ) = c_1 D_v f_1 + c_2 D_v f_2
$$
1. it satisfies the [[Leibniz rule]]
$$
D_v(f_1 \cdot f_2) = (D_v f_1) \cdot f_2 + f_1 \cdot (D_v f_2)
$$
for all $c_1, c_2 \in \mathbb{R}$ and all $f_1, f_2 \in C^\infty(\mathbb{R}^n)$.
Hence regarding [[tangent vector fields]] as [[partial derivatives]] constitutes a [[linear function]]
$$
D \;\colon\; \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \longrightarrow Der(C^\infty(\mathbb{R}^n))
$$
from the [[space of sections]] of the [[tangent bundle]]. In fact this is
a [[homomorphism]] of $C^\infty(\mathbb{R}^n)$-[[modules]] (example \ref{ModuleOfSectionsOfAVectorBundle}),
in that for $f \in C^\infty(\mathbb{R}^n)$ and $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ we have
$$
D_{f v}(-) = f \cdot D_v(-)
\,.
$$
=--
+-- {: .num_example #VerticalTangentBundle}
###### Example
**([[vertical tangent bundle]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. Then its _[[vertical tangent bundle]]_ $T_\Sigma E \overset{T fb}{\to} \Sigma$ is the [[fiber bundle]] (def. \ref{FiberBundle}) over $\Sigma$ whose [[fiber]] over a point is the [[tangent bundle]] (def. \ref{TangentVectorFields}) of the fiber of $E \overset{fb}{\to}\Sigma$ over that point:
$$
(T_\Sigma E)_x \coloneqq T(E_x)
\,.
$$
If $E \simeq \Sigma \times F$ is a [[trivial fiber bundle]] with [[fiber]] $F$, then its vertical vector bundle is the trivial fiber bundle with fiber $T F$.
=--
+-- {: .num_defn #DualVectorBundle}
###### Definition
**([[dual vector bundle]])**
For $E \overset{vb}{\to} \Sigma$ a [[vector bundle]] (def. \ref{VectorBundle}), its
_[[dual vector bundle]]_ is the vector bundle whose [[fiber]] (eq:FiberOfAFiberBundle) over $x \in \Sigma$ is the
[[dual vector space]] of the corresponding fiber of $E \to \Sigma$:
$$
(E^\ast)_x \;\coloneqq\; (E_x)^\ast
\,.
$$
The defining pairing of [[dual vector spaces]] $(E_x)^\ast \otimes E_x \to \mathbb{R}$ applied
pointwise induces a pairing on the [[modules]] of [[sections]] (def. \ref{ModuleOfSectionsOfAVectorBundle})
of the original vector bundle and its dual with values in the [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}):
$$
\label{PairingOfDualSections}
\array{
\Gamma_\Sigma(E) \otimes_{C^\infty(X)} \Gamma_\Sigma(E^\ast)
&\longrightarrow&
C^\infty(\Sigma)
\\
(v,\alpha) &\mapsto& (v \cdot \alpha \colon x \mapsto \alpha_x(v_x) )
}
$$
=--
$\,$
**[[synthetic differential geometry]]**
{#SyntheticDifferentialGeometry}
Below we encounter generalizations of ordinary [[differential geometry]] that include
explicit "[[infinitesimals]]" in the guise of _[[infinitesimally thickened points]]_,
as well as "super-graded infinitesimals", in the guise of _[[superpoints]]_ (necessary for the
description of [[fermion fields]] such as the [[Dirac field]]). As we discuss [below](#FieldBundles), these structures are
naturally incorporated into [[differential geometry]] in just the same way as [[Grothendieck]]
introduced them into [[algebraic geometry]] (in the guise of "[[formal schemes]]"),
namely in terms of [[formal dual|formally dual]] [[rings of functions]] with [[nilpotent ideals]].
That this also works well for [[differential geometry]] rests on the following three
basic but important properties, which say that [[smooth functions]] behave "more algebraically"
than their definition might superficially suggest:
+-- {: .num_prop #AlgebraicFactsOfDifferentialGeometry}
###### Proposition
**(the three magic algebraic properties of [[differential geometry]])**
1. **[[embedding of smooth manifolds into formal duals of R-algebras|embedding of Cartesian spaces into formal duals of R-algebras]]**
For $X$ and $Y$ two [[Cartesian spaces]], the [[smooth functions]] $f \colon X \longrightarrow Y$
between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) are in [[natural bijection]] with their
induced algebra [[homomorphisms]] $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), so that
one may equivalently handle [[Cartesian spaces]] entirely via their $\mathbb{R}$-algebras of smooth functions.
Stated more [[category theory|abstractly]], this means equivalently that the [[functor]] $C^\infty(-)$ that
sends a [[smooth manifold]] $X$ to its $\mathbb{R}$-[[associative algebra|algebra]]
$C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is a _[[fully faithful functor]]_:
$$
C^\infty(-)
\;\colon\;
SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op}
\,.
$$
([Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10](embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93))
1. **[[smooth Serre-Swan theorem|embedding of smooth vector bundles into formal duals of R-algebra modules]]**
For $E_1 \overset{vb_1}{\to} X$ and $E_2 \overset{vb_2}{\to} X$ two [[vector bundle]] (def. \ref{VectorBundle})
there is then a [[natural bijection]] between vector bundle [[homomorphisms]] $f \colon E_1 \to E_2$
and the [[homomorphisms]] of [[modules]] $f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)$
that these induces between the [[spaces of sections]] (example \ref{ModuleOfSectionsOfAVectorBundle}).
More [[category theory|abstractly]] this means that the [[functor]] $\Gamma_X(-)$ is a [[fully faithful functor]]
$$
\Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod
$$
([Nestruev 03, theorem 11.29](smooth+Serre-Swan+theorem#Nestruev03))
Moreover, the [[modules]] over the $\mathbb{R}$-algebra $C^\infty(X)$
of [[smooth functions]] on $X$ which arise this way as [[sections]] of [[smooth vector bundles]] over
a [[Cartesian space]] $X$
are precisely the [[finitely generated module|finitely generated]] [[free modules]] over $C^\infty(X)$.
([Nestruev 03, theorem 11.32](smooth+Serre-Swan+theorem#Nestruev03))
1. **[[derivations of smooth functions are vector fields|vector fields are derivations of smooth functions]]**.
For $X$ a [[Cartesian space]] (example \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}),
then any [[derivation]] $D \colon C^\infty(X) \to C^\infty(X)$ on
the $\mathbb{R}$-[[associative algebra|algebra]]
$C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is given by [[differentiation]] with respect to a uniquely defined smooth [[tangent vector field]]: The function that regards [[tangent vector fields]]
with [[derivations]] from example \ref{TangentVectorFields}
$$
\array{
\Gamma_X(T X)
&\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}&
Der(C^\infty(X))
\\
v &\mapsto& D_v
}
$$
is in fact an [[isomorphism]].
(This follows directly from the _[[Hadamard lemma]]_.)
=--
Actually all three statements in prop. \ref{AlgebraicFactsOfDifferentialGeometry}
hold not just for [[Cartesian spaces]], but generally for [[smooth manifolds]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below; if only we generalize in the second statement from [[free modules]] to [[projective modules]].
However for our development here it is useful to first focus on just [[Cartesian spaces]] and then bootstrap
the theory of [[smooth manifolds]] and much more from that, which we do [below](#FieldBundles).
$\,$
$\,$
**[[differential forms]]**
{#DifferentialFormsAndCartanCalculus}
We introduce and discuss [[differential forms]] on [[Cartesian spaces]].
+-- {: .num_defn #Differential1FormsOnCartesianSpaces}
###### Definition
**([[differential 1-forms]] on [[Cartesian spaces]] and the [[cotangent bundle]])**
For $n \in \mathbb{N}$ a _[[smooth differential 1-form]]_ $\omega$ on a [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is an [[n-tuple]]
$$
\left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n
$$
of [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), which we think of equivalently as the [[coefficients]] of a [[formal linear combination]]
$$
\omega = \omega_i d x^i
$$
on a [[set]] $\{d x^1, d x^2, \cdots, d x^n\}$ of [[cardinality]] $n$.
Here a sum over repeated indices is tacitly understood ([[Einstein summation convention]]).
Write
$$
\Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set
$$
for the set of smooth [[differential 1-forms]] on $\mathbb{R}^k$.
We may think of the expressions $(d x^a)_{a = 1}^n$ as a [[linear basis]] for the [[dual vector space]]
$\mathbb{R}^n$. With this the [[differential 1-forms]] are equivalently the [[sections]] (def. \ref{Sections})
of the [[trivial vector bundle]] (example \ref{TrivialBundleOnCartesianSpace}, def. \ref{VectorBundle})
$$
\array{
T^\ast \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times (\mathbb{R}^n)^\ast
\\
\mathllap{cb}\downarrow && \downarrow\mathrlap{pr_1}
\\
\mathbb{R}^n &=& \mathbb{R}^n
}
$$
called the _[[cotangent bundle]]_ of $\mathbb{R}^n$ (def. \ref{Differential1FormsOnCartesianSpaces}):
$$
\Omega^1(\mathbb{R}^n) = \Gamma_{\mathbb{R}^n}(T^\ast \mathbb{R}^n)
\,.
$$
This amplifies via example \ref{ModuleOfSectionsOfAVectorBundle} that $\Omega^1(\mathbb{R}^n)$
has the [[structure]] of a [[module]] over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^n)$,
by the evident multiplication of [[differential 1-forms]] with [[smooth functions]]:
1. The set $\Omega^1(\mathbb{R}^k)$ of [[differential 1-forms]] in a [[Cartesian space]] (def. \ref{Differential1FormsOnCartesianSpaces})
is naturally an [[abelian group]] with addition given by componentwise addition
$$
\begin{aligned}
\omega + \lambda & =
\omega_i d x^i + \lambda_i d x^i
\\
& = (\omega_i + \lambda_i) d x^i
\end{aligned}
\,,
$$
1. The abelian group $\Omega^1(\mathbb{R}^k)$ is naturally equipped with the structure of a [[module]] over the
[[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), where the [[action]] $C^\infty(\mathbb{R}^k) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k)$ is given by componentwise multiplication
$$
f \cdot \omega = ( f \cdot \omega_i) d x^i
\,.
$$
Accordingly there is a canonical pairing between [[differential 1-forms]] and [[tangent vector fields]] (example \ref{TangentVectorFields})
$$
\label{PairingVectorFieldsWithDifferential1Forms}
\array{
\Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)
\otimes_{\mathbb{R}}
\Gamma_{\mathbb{R}^n}(T \ast \mathbb{R}^n)
&\overset{\iota_{(-)}(-) }{\longrightarrow}&
C^\infty(\mathbb{R}^n)
\\
(v,\omega)
&\mapsto&
\mathrlap{
\iota_v \omega \coloneqq v^a \omega_a
}
}
$$
With [[differential 1-forms]] in hand, we may collect all the first-order [[partial derivatives]] of a [[smooth function]]
into a single object: the _[[exterior derivative]]_ or _[[de Rham differential]]_ is the $\mathbb{R}$-[[linear function]]
$$
\label{deRhamDifferentialOnFunctionsOnCartesianSpace}
\array{
C^\infty(\mathbb{R}^n) &\overset{d}{\longrightarrow}& \Omega^1(\mathbb{R}^n)
\\
f &\mapsto& \mathrlap{ d f \coloneqq \frac{\partial f}{ \partial x^a} d x^a }
}
\,.
$$
Under the above pairing with [[tangent vector fields]] $v$ this yields the particular [[partial derivative]] along $v$:
$$
\iota_v d f = D_v f = v^a \frac{\partial f}{\partial x^a}
\,.
$$
=--
We think of $d x^i$ as a measure for [[infinitesimal space|infinitesimal]] displacements along the $x^i$-[[coordinate]] of a [[Cartesian space]]. If we have a measure of infintesimal displacement on some $\mathbb{R}^n$ and a smooth function $f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n$, then this induces a measure for infinitesimal displacement on $\mathbb{R}^{\tilde n}$ by sending whatever happens there first with $f$ to $\mathbb{R}^n$ and then applying the given measure there. This is captured by the following definition:
+-- {: .num_defn #PullbackOfDifferential1FormsOnCartesianSpaces}
###### Definition
**([[pullback of differential forms|pullback of differential 1-forms]])**
For $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ a [[smooth function]], the **[[pullback of differential forms|pullback of differential 1-forms]]** along $\phi$ is the [[function]]
$$
\phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})
$$
between sets of differential 1-forms, def. \ref{Differential1FormsOnCartesianSpaces}, which is defined on [[basis]]-elements by
$$
\phi^* d x^i
\;\coloneqq\;
\frac{\partial \phi^i}{\partial \tilde x^j} d \tilde x^j
$$
and then extended linearly by
$$
\begin{aligned}
\phi^* \omega & = \phi^* \left( \omega_i d x^i \right)
\\
& \coloneqq
\left(\phi^* \omega\right)_i \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j
\\
& =
(\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j
\end{aligned}
\,.
$$
This is compatible with [[identity morphisms]] and [[composition]] in that
$$
\label{PullbackOfDifferentialFormsCompatibleWithComposition}
(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)}
\phantom{AAAA}
(g \circ f)^\ast = f^\ast \circ g^\ast
\,.
$$
Stated more [[category theory|abstractly]], this just means that [[pullback of differential forms|pullback of differential 1-forms]]
makes the assignment of sets of differential 1-forms to [[Cartesian spaces]] a [[contravariant functor|contravariant]] [[functor]]
$$
\Omega^1(-) \;\colon\; CartSp^{op} \longrightarrow Set
\,.
$$
=--
The following definition captures the idea that if $d x^i$ is a measure for displacement along the $x^i$-[[coordinate]], and $d x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way to get a measure, to be called $d x^i \wedge d x^j$, for [[infinitesimal]] _[[surfaces]]_ (squares) in the $x^i$-$x^j$-plane. And this should keep track of the [[orientation]] of these squares, with
$$
d x^j \wedge d x^i = - d x^i \wedge d x^j
$$
being the same infinitesimal measure with orientation reversed.
+-- {: .num_defn #DifferentialnForms}
###### Definition
**([[exterior algebra]] of [[differential n-forms]])**
For $k,n \in \mathbb{N}$, the **smooth [[differential forms]]** on a [[Cartesian space]] $\mathbb{R}^k$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is the [[exterior algebra]]
$$
\Omega^\bullet(\mathbb{R}^k)
\coloneqq
\wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)
$$
over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) of the [[module]] $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms.
We write $\Omega^n(\mathbb{R}^k)$ for the sub-module of degree $n$ and call its elements the _[[differential n-forms]]_.
Explicitly this means that a [[differential n-form]] $\omega \in \Omega^n(\mathbb{R}^k)$ on $\mathbb{R}^k$ is a [[formal linear combination]] over $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) of [[basis]] elements of the form $d x^{i_1} \wedge \cdots \wedge d x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:
$$
\omega = \omega_{i_1, \cdots, i_n} d x^{i_1} \wedge \cdots \wedge d x^{i_n}
\,.
$$
=--
Now all the constructions for [[differential 1-forms]] above extent naturally to [[differential n-forms]]:
+-- {: .num_defn #deRhamDifferential}
###### Definition
**([[exterior derivative]] or [[de Rham differential]])**
For $\mathbb{R}^n$ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
the [[de Rham differential]] $d \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$
(eq:deRhamDifferentialOnFunctionsOnCartesianSpace) uniquely extended as a [[derivation]]
of degree +1 to the [[exterior algebra]] of [[differential forms]] (def. \ref{DifferentialnForms})
$$
d
\;\colon\;
\Omega^\bullet(\mathbb{R}^n)
\longrightarrow
\Omega^\bullet(\mathbb{R}^n)
$$
meaning that for $\omega_i \in \Omega^{k_i}(\mathbb{R})$ then
$$
d(\omega_1 \wedge \omega_2)
\;=\;
(d \omega_1) \wedge \omega_2
+
\omega_1 \wedge d \omega_2
\,.
$$
In components this simply means that
$$
\begin{aligned}
d \omega
& =
d \left(\omega_{i_1 \cdots i_k} d x^{i_1} \wedge \cdots \wedge d x^{i_k}\right)
\\
& =
\frac{\partial \omega_{i_1 \cdots i_k}}{\partial x^{a}}
d x^a \wedge d x^{i_1} \wedge \cdots \wedge d x^{i_k}
\end{aligned}
\,.
$$
Since [[partial derivatives]] commute with each other, while differential 1-form anti-commute, this implies
that $d$ is nilpotent
$$
d^2 = d \circ d = 0
\,.
$$
We say hence that [[differential forms]] form a _[[cochain complex]]_, the _[[de Rham complex]]_
$(\Omega^\bullet(\mathbb{R}^n), d)$.
=--
+-- {: .num_defn #ContractionOfFormsWithVectorFields}
###### Definition
**(contraction of [[differential n-forms]] with [[tangent vector fields]])**
The pairing $\iota_v \omega = \omega(v)$ of [[tangent vector fields]] $v$ with [[differential 1-forms]] $\omega$ (eq:PairingVectorFieldsWithDifferential1Forms) uniquely [[extension|extends]] to the [[exterior algebra]] $\Omega^\bullet(\mathbb{R}^n)$ of [[differential forms]] (def. \ref{DifferentialnForms}) as a [[derivation]] of degree -1
$$
\iota_v \;\colon\; \Omega^{\bullet+1}(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n)
\,.
$$
In particular for $\omega_1, \omega_2 \in \Omega^1(\mathbb{R}^n)$ two [[differential 1-forms]], then
$$
\iota_{v} (\omega_1 \wedge \omega_2)
\;=\;
\omega_1(v) \omega_2 - \omega_2(v) \omega_1
\;\in\;
\Omega^1(\mathbb{R}^n)
\,.
$$
=--
+-- {: .num_prop #PullbackOfDifferentialForms}
###### Proposition
**([[pullback of differential forms|pullback of differential n-forms]])**
For $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]] between [[Cartesian spaces]]
(def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
the operationf of [[pullback of differential forms|pullback of differential 1-forms]] of def. \ref{Differential1FormsOnCartesianSpaces} extends as an $C^\infty(\mathbb{R}^k)$-[[associative algebra|algebra]] [[homomorphism]] to the [[exterior algebra]]
of [[differential forms]] (def. \ref{DifferentialnForms}),
$$
f^\ast
\;\colon\;
\Omega^\bullet(\mathbb{R}^{n_2})
\longrightarrow
\Omega^\bullet(\mathbb{R}^{n_1})
$$
given on basis elements by
$$
f^* \left( dx^{i_1} \wedge \cdots \wedge dx^{i_n} \right)
=
\left(f^* dx^{i_1} \wedge \cdots \wedge f^* dx^{i_n} \right)
\,.
$$
This commutes with the [[de Rham differential]] $d$ on both sides (def. \ref{deRhamDifferential}) in that
$$
d \circ f^\ast
=
f^\ast \circ d
\phantom{AAAAA}
\array{
\Omega^\bullet(X) &\overset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y)
\\
\mathllap{d}\downarrow && \downarrow\mathrlap{d}
\\
\Omega^\bullet(X) &\underset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y)
}
$$
hence that [[pullback of differential forms]] is a _[[chain map]]_ of [[de Rham complexes]].
This is still compatible with [[identity morphisms]] and [[composition]] in that
$$
\label{PullbackOfDiffereentialFormsCompatibleWithComposition}
(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)}
\phantom{AAAA}
(g \circ f)^\ast = f^\ast \circ g^\ast
\,.
$$
Stated more [[category theory|abstractly]], this just means that [[pullback of differential forms|pullback of differential n-forms]]
makes the assignment of sets of [[differential n-forms]] to [[Cartesian spaces]] a [[contravariant functor|contravariant]] [[functor]]
$$
\Omega^n(-) \;\colon\; CartSp^{op} \longrightarrow Set
\,.
$$
=--
+-- {: .num_prop #CartanHomotopyFormula}
###### Proposition
**([[Cartan's homotopy formula]])**
Let $X$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), and let $v \in \Gamma(T X)$ be
a smooth [[tangent vector field]] (example \ref{TangentVectorFields}).
For $t \in \mathbb{R}$ write $\exp(t v) \colon X \overset{\simeq}{\to} X$ for the [[flow]] by [[diffeomorphisms]]
along $v$ of parameter length $t$.
Then the [[derivative]] with respect to $t$ of the [[pullback of differential forms]] along $\exp(t v)$, hence the [[Lie derivative]]
$\mathcal{L}_v \colon \Omega^\bullet(X) \to \Omega^\bullet(X)$,
is given by the [[anticommutator]] of the contraction derivation $\iota_v$ (def. \ref{ContractionOfFormsWithVectorFields}) with the [[de Rham differential]] $d$ (def. \ref{deRhamDifferential}):
$$
\begin{aligned}
\mathcal{L}_v
&\coloneqq
\frac{d}{d t } \exp(t v)^\ast \omega \vert_{t = 0}
\\
& =
\iota_v d \omega + d \iota_v \omega
\,.
\end{aligned}
$$
=--
Finally we turn to the concept of [[integration of differential forms]] (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces} below).
First we need to say what it is that differential forms may be integrated over:
+-- {: .num_defn #SingularSimplicesInCartesianSpaces}
###### Definition
**(smooth [[singular simplicial chains]] in [[Cartesian spaces]])**
For $n \in \mathbb{N}$, the _standard [[n-simplex]]_ in the [[Cartesian space]]
$\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is the
[[subset]]
$$
\Delta^n
\;\coloneqq\;
\left\{
(x^i)_{i = 1}^n
\;\vert\;
0 \leq x^1 \leq \cdots \leq x^n
\right\}
\;\subset\;
\mathbb{R}^n
\,.
$$
More generally, a _smooth [[singular simplicial complex|singular n-simplex]]_ in a [[Cartesian space]]
$\mathbb{R}^k$ is a [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})
$$
\sigma
\;\colon\;
\mathbb{R}^n
\longrightarrow
\mathbb{R}^k
\,,
$$
to be thought of as a smooth extension of its restriction
$$
\sigma\vert_{\Delta^n}
\;\colon\;
\Delta^n
\longrightarrow
\mathbb{R}^k
\,.
$$
(This is called a _[[singular simplicial complex|singular]]_ simplex because there is no condition that
$\Sigma$ be an [[embedding]] in any way, in particular $\sigma$ may be a [[constant function]].)
A [[singular chain]] in $\mathbb{R}^k$ of [[dimension]] $n$ is a [[formal linear combination]] of singular $n$-simplices in $\mathbb{R}^k$.
In particular, given a singular $n+1$-simplex $\sigma$, then its _[[boundary of a simplex|boundary]]_
is a [[singular chain]] of singular $n$-simplices $\partial \sigma$.
=--
+-- {: .num_defn #IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces}
###### Definition
**([[fiber integration|fiber]]-[[integration of differential forms]]) over smooth [[singular chains]] in [[Cartesian spaces]])**
For $n \in \mathbb{N}$ and $\omega \in \Omega^n(\mathbb{R}^n)$ a [[differential n-form]] (def. \ref{DifferentialnForms}),
which may be written as
$$
\omega = f d x^1 \wedge \cdots d x^n
\,,
$$
then its [[integration of differential forms|integration]] over the standard [[n-simplex]] $\Delta^n \subset \mathbb{R}^n$ (def. \ref{SingularSimplicesInCartesianSpaces}) is the ordinary [[integral]] (e.g. [[Riemann integral]])
$$
\int_{\Delta^n} \omega
\;\coloneqq\;
\underset{0 \leq x^1 \leq \cdots \leq x^n \leq 1}{\int} f(x^1, \cdots, x^n) \, d x^1 \cdots d x^n
\,.
$$
More generally, for
1. $\omega \in \Omega^n(\mathbb{R}^k)$ a [[differential n-forms]];
1. $C = \underset{i}{\sum} c_i \sigma_i $ a singular $n$-chain (def. \ref{SingularSimplicesInCartesianSpaces})
in any [[Cartesian space]] $\mathbb{R}^k$. Then the _[[integration of differential forms|integration]]_
of $\omega$ over $x$ is the [[sum]] of the integrations, as above, of the [[pullback of differential forms]] (def. \ref{PullbackOfDifferentialForms}) along all the singular [[n-simplices]] in the chain:
$$
\int_C \omega
\;\coloneqq\;
\underset{i}{\sum}
c_i
\int_{\Delta^n} (\sigma_i)^\ast \omega
\,.
$$
Finally, for $U$ another Cartesian space, then _[[fiber integration]] of differential forms along $U \times C \to U$_
is the linear map
$$
\int_C \;\colon\; \Omega^{\bullet + dim(C)}(U \times C) \longrightarrow \Omega^\bullet(U)
$$
which on differential forms of the form $\omega_U \wedge \omega$ is given by
$$
\int_C \omega_U \wedge \omega
\;\coloneqq\;
(-1)^{\vert \omega_U\vert} \int_C \omega
\,.
$$
=--
+-- {: .num_prop #StokesTheorem}
###### Proposition
**([[Stokes theorem]] for [[fiber integration|fiber]]-[[integration of differential forms]])**
For $\Sigma$ a smooth [[singular simplicial chain]] (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces})
the operation of [[fiber integration|fiber]]-[[integration of differential forms]] along
$U \times \Sigma \overset{pr_1}{\longrightarrow} U$ (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces})
is compatible with the [[exterior derivative]] $d_U$ on $U$ (def. \ref{deRhamDifferential}) in that
$$
\begin{aligned}
d \int_\Sigma \omega
& =
(-1)^{dim(\Sigma)} \int_\Sigma d_U \omega
\\
& =
(-1)^{dim(\Sigma)}
\left(
\int_\Sigma d \omega
-
\int_{\partial \Sigma} \omega
\right)
\end{aligned}
\,,
$$
where $d = d_U + d_\Sigma$ is the [[de Rham differential]] on $U \times \Sigma$ (def. \ref{deRhamDifferential}) and where the second
equality is the _[[Stokes theorem]]_ along $\Sigma$:
$$
\int_\Sigma d_\Sigma \omega = \int_{\partial \Sigma} \omega
\,.
$$
=--
$\,$
This concludes our review of the basics of ([[synthetic differential geometry|synthetic]]) [[differential geometry]] on which the following development of quantum field theory is based.
In the [next chapter](#Spacetime) we consider _[[spacetime]]_ and _[[spin]]_.
|
A first idea of quantum field theory -- Interacting quantum fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Interacting+quantum+fields |
## Interacting quantum fields
{#QuantumObservables}
In this chapter we discuss the following topics:
* _[Free field vacua](#FreeFieldVacua)_
* _[Perturbative S-matrices](#PerturbativeSMatrixAndTimeOrderedProducts)_
* _[Conceptual remarks](#RemarksOnCausalPerturbationTheoryAxioms)_
* _[Interacting field observables](#LocalNetsOfInteractingFieldObservables)_
* _[Time-ordered products](#TimeOrderedProducts)_
* _[("Re"-)Normalization](#ExistenceAndRenormalization)
* _[Feynman perturbation series](#FeynmanDiagrams)_
* _[Effective action](#EffectiveAction)_
* _[Vacuum diagrams](#VacuumDiagrams)_
* _[Interacting quantum BV-differential](#InteractingQantumBVDifferential)_
* _[Ward identities](#WardIdentities)_
$\,$
In the [previous chapter](#FreeQuantumFields) we have found the [[quantization]] of _[[free field theories|free]]_ [[Lagrangian field theories]] by first choosing a [[gauge fixing|gauge fixed]] [[BV-BRST complex|BV-BRST]]-[[homological resolution|resolution]] of the [[algebra of observables|algebra of]] [[gauge invariance|gauge invariant]] [[on-shell]] observabes, then applying [[algebraic deformation quantization]] induced by the resulting [[Peierls-Poisson bracket]] on the graded [[covariant phase space]] to pass to a [[non-commutative algebra]] of quantum observables, such that, finally, the [[BV-BRST differential]] is respected.
Of course most [[quantum field theories]] of interest are non-[[free field theories|free]]; they are _[[interacting field theories]]_ whose [[equations of motion]] is a _non-linear_ differential equation. The archetypical example is the coupling of the [[Dirac field]] to the [[electromagnetic field]] via the [[electron-photon interaction]], corresponding to the [[interacting field theory]] called _[[quantum electrodynamics]]_ (discussed [below](#QuantumElectrodynamics)).
In principle the [[perturbative quantum field theory|perturbative]] [[quantization]] of such non-[[free field theory]] [[interacting field theories]] proceeds the same way: One picks a [[BV-BRST complex|BV-BRST]]-[[gauge fixing]], computes the [[Peierls-Poisson bracket]] on the resulting [[covariant phase space]] ([Khavkine 14](Peierls+bracket#Khavkine14)) and then finds a [[formal deformation quantization]] of this [[Poisson structure]] to obtain the quantized [[non-commutative algebra]] of [[quantum observables]], as [[formal power series]] in [[Planck's constant]] $\hbar$.
It turns out ([Collini 16](perturbative+algebraic+quantum+field+theory#Collini16), [Hawkins-Rejzner 16](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16), prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below)
that the resulting [[interacting field theory|interacting]] [[formal deformation quantization]] may equivalently be expressed in terms of _[[scattering amplitudes]]_ (example \ref{ScatteringAmplitudeFromInteractingFieldObservables} below): These are the [[probability amplitudes]] for [[plane waves]] of [[free fields]] to come in from the far [[past]], then [[interaction|interact]] in a compact region of [[spacetime]] via the given [[interaction]] ([[adiabatic switching|adiabatically switched-off]] outside that region) and to emerge again as [[free fields]] into the far [[future]].
The collection of all these [[scattering amplitudes]], as the [[types]] and [[wave vectors]] of the incoming and outgoing [[free fields]] varies, is called the _[[perturbative S-matrix|perturbative scattering matrix]]_ of the [[interacting field theory]], or just _[[S-matrix]]_ for short. It may equivalently be expressed as the [[exponential]] of [[time-ordered products]] of the [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]] with itself (def. \ref{LagrangianFieldTheoryPerturbativeScattering} below). The [[combinatorics]] of the terms in this exponential is captured by _[[Feynman diagrams]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below), which, with some care (remark \ref{WorldlineFormalism} below), may be thought of as [[finite multigraphs]] (def. \ref{Graphs} below) whose [[edges]] are [[worldlines]] of [[virtual particles]] and whose [[vertices]] are the [[interactions]] that these particles undergo (def. \ref{FeynmanDiagram} below).
The [[axiom|axiomatic]] definition of [[S-matrices]] for [[relativistic field theory|relativistic]] [[Lagrangian field theories]] and their rigorous construction via [[renormalization|("re"-)normalization]] of [[time-ordered products]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below) is called _[[causal perturbation theory]]_, due to ([Epstein-Glaser 73](causal+perturbation+theory#EpsteinGlaser73)). This makes precise and well-defined the would-be [[path integral quantization]] of [[interacting field theories]] (remark \ref{InterpretationOfPerturbativeSMatrix} below) and removes the errors (remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} below) and ensuing puzzlements (expressed in [Feynman 85](Schwinger-Tomonaga-Feynman-Dyson#Feynman85SuchABunchOfWords)) that plagued the original informal conception of [[perturbative quantum field theory]] due to [[Schwinger-Tomonaga-Feynman-Dyson]] (remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences} below).
The equivalent re-formulation of the [[formal deformation quantization]] of [[interacting field theories]] in terms of [[scattering amplitudes]] (prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below) has the advantage that it gives a direct handle on those [[observables]] that are measured in [[scattering]] [[experiments]], such as the [[LHC]]-experiment. The bulk of mankind's knowledge about realistic [[perturbative quantum field theory]] -- such as notably the [[standard model of particle physics]] -- is reflected in such [[scattering amplitudes]] given via their [[Feynman perturbation series]] in [[formal power series|formal powers]] of [[Planck's constant]] and the [[coupling constant]].
Moreover, the mathematical passage from [[scattering amplitudes]] to the actual [[interacting field algebra]] [[algebra of quantum observables|of quantum observables]] (def. \ref{QuntumMollerOperator} below) corresponding to the [[formal deformation quantization]] is well understood, given via "[[Bogoliubov's formula]]" by the _[[quantum Møller operators]]_ (def. \ref{InteractingFieldObservables} below).
Via [[Bogoliubov's formula]] every perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) induces for every choice of [[adiabatic switching|adiabatically switched]]
[[interaction]] [[action functional]] a notion of [[perturbative QFT|perturbative]] [[interacting field observables]] (def. \ref{InteractingFieldObservables}). These generate an algebra (def. \ref{QuntumMollerOperator} below).
By [[Bogoliubov's formula]], in general this algebra depends on the choice of [[adiabatic switching]];
which however is not meant to be part of the [[physics]], but just a mathematical device for grasping global field structures locally.
But this spurious dependence goes away (prop. \ref{IsomorphismFromChangeOfAdiabaticSwitching} below) when restricting attention to
observables whose spacetime support is inside a compact [[causally closed subsets]] $\mathcal{O}$ of spacetime (def. \ref{PerturbativeGeneratingLocalNetOfObservables} below). This is a sensible condition for an [[observable]] in [[physics]],
where any realistic [[experiment]] nessecarily probes only a compact subset of spacetime, see also remark \ref{AdiabaticLimit}.
The resulting system (a "[[co-presheaf]]") of well-defined perturbative [[interacting field algebras of observables]] (def. \ref{SystemOfAlgebrasOfQuantumObservables} below)
$$
\mathcal{O} \mapsto IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
$$
is in fact [[causal locality|causally local]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below).
This fact was presupposed without proof already in [Il'in-Slavnov 78](perturbative+algebraic+quantum+field+theory#IlinSlavnov78);
because this is one of two key properties that the [[Haag-Kastler axioms]] ([Haag-Kastler 64](Haag-Kastler+axioms#HaagKastler64)) demand of an intrinsically defined [[quantum field theory]]
(i.e. defined without necessarily making recourse to the geometric backdrop of [[Lagrangian field theory]]). The only other key property demanded by the [[Haag-Kastler axioms]]
is that the [[algebras of observables]] be [[C*-algebras]]; this however must be regarded as the axiom encoding
[[non-perturbative quantum field theory]] and hence is necessarily violated in the present context of [[perturbative QFT]].
Since quantum field theory following the full [[Haag-Kastler axioms]] is commonly known as _[[AQFT]]_, this perturbative version,
with [[causally local nets of observables]] but without the [[C*-algebra]]-condition on them, has come to be called
_[[perturbative AQFT]]_ ([Dütsch-Fredenhagen 01](perturbative+algebraic+quantum+field+theory#DuetschFredenhagen01),
[Fredenhagen-Rejzner 12](perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12)).
In this terminology the content of prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below is that
_while the input of [[causal perturbation theory]] is a [[gauge fixing|gauge fixed]] [[Lagrangian field theory]],
the output is a [[perturbative algebraic quantum field theory]]_:
$$
\array{
\array{
\text{gauge-fixed}
\\
\text{Lagrangian}
\\
\text{field theory}
}
&
\overset{
\array{
\text{causal}
\\
\text{perturbation theory}
\\
}
}{\longrightarrow}&
\array{
\text{perturbative}
\\
\text{algebraic}
\\
\text{quantum}
\\
\text{field theory}
}
\\
\underset{
\array{
\text{(Becchi-Rouet-Stora 76,}
\\
\text{Batalin-Vilkovisky 80s)}
}
}{\,}
&
\underset{
\array{
\text{(Bogoliubov-Shirkov 59,}
\\
\text{Epstein-Glaser 73)}
}
}{\,}
&
\underset{
\array{
\text{ (Il'in-Slavnov 78, }
\\
\text{Brunetti-Fredenhagen 99,}
\\
\text{Dütsch-Fredenhagen 01)}
}
}{\,}
}
$$
The independence of the [[causally local net]] of localized [[interacting field algebras of observables]]
$IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int} )(\mathcal{O})$ from the choice of [[adiabatic switching]]
implies a well-defined spacetime-global [[algebra of observables]] by forming the [[inductive limit]]
$$
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})
\;\coloneqq\;
\underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim}
\left(
{\, \atop \,}
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
{\, \atop \,}
\right)
\,.
$$
This is also called the _[[algebraic adiabatic limit]]_, defining the [[algebras of observables]] of [[perturbative QFT]]
"in the infrared". The only remaining step in the construction of a [[perturbative QFT]] that remains is then
to find an [[interacting vacuum state]]
$$
\left\langle
-
\right\rangle_{int}
\;\colon\;
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})
\longrightarrow
\mathbb{C}[ [ \hbar, g] ]
$$
on the global [[interacting field algebra]] $Obs_{\mathbf{L}_{int}}$.
This is related to the actual _[[adiabatic limit]]_, and it is by and large an open problem, see remark \ref{AdiabaticLimit} below.
In conclusion so far, the [[algebraic adiabatic limit]] yields, starting with a [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]],
the perturbative construction of [[interacting field algebras of observables]] (def. \ref{QuntumMollerOperator}) and their organization
in increasing powers of $\hbar$ and $g$ ([[loop order]], prop. \ref{FeynmanDiagramLoopOrder}) via the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}, example \ref{SMatrixVacuumContribution}).
But this [[interacting field algebra of observables]] still involves all the [[auxiliary fields]] of the [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]] (as in example \ref{FieldSpeciesQED} for QED), while the
actual physical [[gauge invariance|gauge invariant]] [[on-shell]] observables should be (just) the [[cochain cohomology]] of the
[[BV-BRST differential]] on this enlarged space of observables. Hence for the construction of [[perturbative QFT]] to conclude,
it remains to pass the [[BV-BRST differential]] of the [[free field]] [[Wick algebra]] of observables to
a [[differential]] on the [[interacting field algebra]], such that its [[cochain cohomology]] is well defined.
Since the [[time-ordered products]] away from coinciding interaction points are uniquely fixed (prop. \ref{TimeOrderedProductAwayFromDiagonal} below), one finds that also this _interacting quantum BV-differential_
is uniquely fixed, on [[regular polynomial observables]], by [[conjugation]] with the [[quantum Møller operators]] (def. \ref{BVDifferentialInteractingQuantum}).
The formula that characterizes it there is called the _[[quantum master equation]]_ or equivalently the _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation} below).
In its incarnation as the [[master Ward identity]], this expresses the difference between the [[shell]] of the free classical field theory
and that of the interacting quantum field theory, thus generalizing the [[Schwinger-Dyson equation]] to [[interacting field theory]] (example \ref{SchwingerDysonReductionOfQuantumMasterWardIdentity} below). Applied to [[Noether's theorem]] it expresses the possible
failure of [[conserved currents]] associated with [[infinitesimal symmetries of the Lagrangian]] to still be conserved
in the [[interacting field theory|interacting]] [[perturbative QFT]] (example \ref{NoetherCurrentConservationQuantumCorrection} below).
As one [[extension of distributions|extends]] the [[time-ordered products]] to coinciding interaction points in [[renormalization|("re"-)normalization]]
of the [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below), the [[quantum master equation]]/[[master Ward identity]] becomes a _[[renormalization condition]]_ (prop. \ref{BasicConditionsRenormalization} below). If this condition fails
one says that the [[interacting field theory|interacting]] [[perturbative QFT]] has a _[[quantum anomaly]]_, specifically a _[[gauge anomaly]]_
if the [[Ward identity]] of an [[infinitesimal gauge symmetry]] is violated.
These issues of [[renormalization|"(re)-"normalization]] we discuss in detail in the [next chapter](#Renormalization).
$\,$
**Free field vacua**
{#FreeFieldVacua}
In considering [[perturbative QFT]], we are considering [[perturbation theory]] in formal [[deformation]] parameters around a fixed [[free field theory|free]]
[[Lagrangian field theory|Lagrangian]] [[quantum field theory]] in a chosen [[Hadamard vacuum state]].
For convenient referencing we collect all the structure and notation that goes into this in the following definitions:
+-- {: .num_defn #VacuumFree}
###### Definition
**([[free field theory|free]] [[relativistic field theory|relativistic]] [[Lagrangian field theory|Lagrangian]] [[quantum field theory|quantum field]] [[vacuum]])**
Let
1. $\Sigma$ be a [[spacetime]] (e.g. [[Minkowski spacetime]]);
1. $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}), with [[field bundle]] $E \overset{fb}{\to} \Sigma$;
1. $\mathcal{G} \overset{fb}{\to} \Sigma$ a [[gauge parameter bundle]] for $(E,\mathbf{L})$ (def. \ref{GaugeParameters}), with induced [[BRST-complex|BRST]]-[[reduced phase space|reduced]] [[Lagrangian field theory]] $\left( E \times_\Sigma \mathcal{G}[1], \mathbf{L} - \mathbf{L}_{BRST}\right)$ (example \ref{LocalOffShellBRSTComplex});
1. $(E_{\text{BV-BRST}}, \mathbf{L}' - \mathbf{L}'_{BRST})$ a [[gauge fixing]] (def. \ref{GaugeFixingLagrangianDensity}) with [[graded manifold|graded]] [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}} = T^\ast_{\Sigma}[-1]\left( E\times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1]\right)$ (remark \ref{FieldBundleBVBRST});
1. $\Delta_H \in \Gamma'( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )$ a [[Wightman propagator]] $\Delta_H = \tfrac{i}{2} \Delta + H$ compatible with the [[causal propagator]] $\Delta$ which corresponds to the [[Green hyperbolic partial differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] induced by the [[gauge fixing|gauge-fixed]] [[Lagrangian density]] $\mathbf{L}'$.
Given this, we write
$$
\left(
{\, \atop \,}
PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ] \;,\;
\star_H
{\, \atop \,}
\right)
$$
for the corresponding [[Wick algebra]]-[[structure]] on [[formal power series]] in $\hbar$ ([[Planck's constant]]) of [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable}). This is a [[star algebra]] with respect to ([[coefficient]]-wise) [[complex conjugation]] (prop. \ref{MoyalStarProductOnMicrocausal}).
Write
$$
\label{HadamardVacuumStateForFreeFieldTheory}
\array{
PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
&\overset{\langle - \rangle}{\longrightarrow}&
\mathbb{C}[ [\hbar] ]
\\
A &\mapsto& A(\Phi = 0)
}
$$
for the induced [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}), hence the [[state on a star-algebra|state]] whose [[distribution|distributional]] [[2-point function]] is the chosen [[Wightman propagator]]:
$$
\left\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)\right\rangle
\;=\;
\hbar \, \Delta_H^{a b}(x,y)
\,.
$$
Given any [[microcausal polynomial observable]] $A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ then its value in this state is called its _free [[vacuum expectation value]]_
$$
\left\langle
A
\right\rangle
\;\in\;
\mathbb{C}[ [ \hbar, g, j] ]
\,.
$$
Write
$$
\label{NormalOrderingLocalObservables}
\array{
LocObs(E_{\text{BV-BRST}})
&\overset{\phantom{A}:(-):\phantom{A}}{\hookrightarrow}&
PolyObs(E_{\text{BV-BRST}})_{mc}
\\
A &\mapsto& :A:
}
$$
for the inclusion of [[local observables]] (def. \ref{LocalObservables}) into [[microcausal polynomial observables]] (example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}), thought of as forming [[normal-ordered products]] in the [[Wick algebra]] (by def. \ref{NormalOrderedProductNotation}).
We denote the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] $\Delta_H$ according to prop. \ref{PropagatorStarProduct}) by juxtaposition (def. \ref{NormalOrderedProductNotation})
$$
A_1 A_2 \;\coloneqq\; A_1 \star_H A_2
\,.
$$
If an element $A \in PolyObs(E_{\text{BV-BRST}})$ has an [[inverse]] with respect to this product, we denote that by
$A^{-1}$:
$$
A^{-1} A = 1
\,.
$$
Finally, for $A \in LocObs(E_{\text{BV-BRST}})$ we write $supp(A) \subset \Sigma$ for its spacetime support (def. \ref{SpacetimeSupport}).
For $S_1, S_2 \subset \Sigma$ two [[subsets]] of [[spacetime]] we write
$$
S_1 {\vee\!\!\!\wedge} S_2
\phantom{AAA}
\left\{
\array{
\text{"}S_1 \, \text{does not intersect the past of} \, S_2\text{"}
\\
\Updownarrow
\\
\text{"}S_2 \, \text{does not intersect the future of} \, S_1\text{"}
}
\right.
$$
for the [[causal order]]-[[relation]] (def. \ref{CausalOrdering}) and
$$
S_1 {\gt\!\!\!\!\lt} S_2
\phantom{AAA}
\text{for}
\phantom{AAA}
\array{
S_1 {\vee\!\!\!\wedge} S_2
\\
\text{and}
\\
S_2 {\vee\!\!\!\wedge} A_1
}
$$
for _[[spacelike]] separation_.
=--
Being concerned with [[perturbative QFT|perturbation theory]] means mathematically that we consider _[[formal power series]]_
in [[deformation]] parameters $\hbar$ ("[[Planck's constant]]") and $g$ ("[[coupling constant]]"), also in $j$ ("[[source field]]"),
see also remark \ref{AsymptoticSeriesObservables}. The following collects our notational conventions for these matters:
+-- {: .num_defn #FormalParameters}
###### Definition
**([[formal power series]] of [[observables]] for [[perturbative QFT]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Write
$$
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
\;\coloneqq\;
\underset{ k_1, k_2, k_3 \in \mathbb{N}}{\prod}
LocObs(E_{\text{BV-BRST}})\langle \hbar^{k_1} g^{k_2} j^{k^3}\rangle
$$
for the space of [[formal power series]] in three formal [[variables]]
1. $\hbar$ ("[[Planck's constant]]"),
1. $g$ ("[[coupling constant]]")
1. $j$ ("[[source field]]")
with [[coefficients]] in the [[topological vector spaces]] of the [[off-shell]] polynomial [[local observables]] of the [[free field]] theory (def. \ref{LocalObservables}); similarly for the [[off-shell]] [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable}):
$$
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j ] ]
\;\coloneqq\;
\underset{ k_1, k_2, k_3 \in \mathbb{N}}{\prod}
PolyObs(E_{\text{BV-BRST}})_{mc}\langle \hbar^{k_1} g^{k_2} j^{k^3}\rangle
\,.
$$
Similary
$$
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
\,,
\phantom{AAA}
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
$$
denotes the subspace for which no powers of $j$ appear, etc.
Accordingly
$$
C^\infty_{cp}(\Sigma) \langle g \rangle
$$
denotes the vector space of [[bump functions]] on [[spacetime]] tensored with the vector space spanned by a single copy of $g$.
The elements
$$
g_{sw} \in C^\infty_{cp}(\Sigma)\langle g \rangle
$$
may be regarded as [[spacetime]]-dependent "[[coupling constants]]" with compact support,
called _[[adiabatic switching|adiabatically switched]] couplings_.
Similarly then
$$
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g , j \rangle
$$
is the subspace of those formal power series that are at least linear in $g$ or $j$ (hence those that vanish if one sets $g,j = 0$ ).
Hence every element of this space may be written in the form
$$
O
=
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g , j \rangle
\,,
$$
where the notation is to suggest that we will think of the coefficient of $g$ as an ([[adiabatic switching|adiabatically switched]])
[[interaction]] [[action functional]] and of the coefficient of $j$ as an external [[source field]] (reflected by internal and external
vertices, respectively, in [[Feynman diagrams]], see def. \ref{VerticesAndFieldSpecies} below).
In particular for
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g] ]
$$
a [[formal power series]] in $\hbar$ and $g$ of [[local Lagrangian densities]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), thought of as a local [[interaction]] Lagrangians, and if
$$
g_{sw}
\;\in\;
C^\infty_{cp}(\Sigma) \langle g \rangle
$$
is an [[adiabatic switching|adiabatically switched]] coupling as before, then the [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the product
$$
g_{sw} \mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})[ [ \hbar ,g ] ]\langle g \rangle
$$
is such an [[adiabatic switching|adiabatically switched]] [[interaction]]
$$
g S_{int}
\;=\;
\tau_\Sigma( g_{sw} \mathbf{L}_{int} )
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle
\,.
$$
We also consider the space of [[off-shell]] [[microcausal polynomial observables]] of the [[free field theory]] with formal parameters adjoined
$$
PolyObs(E_{\text{BV-BRST}})_{mc} ((\hbar)) [ [ g , j] ]
\,,
$$
which, in its $\hbar$-dependent, is the space of _[[Laurent series]]_ in $\hbar$, hence the space exhibiting also [[negative number|negative]] formal powers of $\hbar$.
=--
$\,$
**Perturbative S-Matrices**
{#PerturbativeSMatrixAndTimeOrderedProducts}
We introduce now the [[axioms]] for perturbative [[scattering matrices]] relative to a fixed [[relativistic field theory|relativistic]] [[free field theory|free]] [[Lagrangian field theory|Lagrangian]] [[quantum field theory|quantum field]] [[vacuum]] (def. \ref{VacuumFree} below) according to _[[causal perturbation theory]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering} below).
Since the first of these axioms requires the S-matrix to be a formal sum of [[multilinear map|multi-]][[linear continuous functionals]], it is convenient to impose axioms on these directly: this is the axiomatics for _[[time-ordered products]]_
in def. \ref{TimeOrderedProduct} below. That these latter axioms already imply the former
is the statement of prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below . Its proof requires a close look at the "[[reverse-time ordered products]]" for the inverse S-matrix (def. \ref{ReverseTimeOrderedProduct} below)
and their induced reverse-causal factorization (prop. \ref{ReverseCausalFactorizationOfReverseTimeOrderedProducts} below).
+-- {: .num_defn #LagrangianFieldTheoryPerturbativeScattering}
###### Definition
**([[S-matrix]] [[axioms]] -- [[causal perturbation theory]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Then a _perturbative [[S-matrix]] [[renormalization scheme|scheme]]_ for [[perturbative QFT]] around this [[free field|free]] [[vacuum]] is a [[function]]
$$
\mathcal{S}
\;\;\colon\;\;
LocObs(E_{\text{BV-BRST}})[ [\hbar , g, j] ]\langle g, j \rangle
\overset{\phantom{AAA}}{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
$$
from [[local observables]] to [[microcausal polynomial observables]] of the free vacuum theory,
with formal parameters adjoined as indicated (def. \ref{FormalParameters}),
such that the following two conditions "perturbation" and "causal additivity (jointly: "[[causal perturbation theory]]") hold:
1. ([[perturbative quantum field theory|perturbation]])
There exist [[multilinear map|multi-]][[linear continuous functionals]] (over $\mathbb{C}[ [\hbar, g, j] ]$) of the form
$$
\label{TimeOrderedProductsInSMatrix}
T_k
\;\colon\;
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\langle g, j \rangle
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
$$
for all $k \in \mathbb{N}$, such that:
1. The nullary map is [[constant function|constant]] on the [[neutral element|unit]] of the [[Wick algebra]]
$$
T_0( g S_{int} + j A) = 1
$$
1. The unary map is the inclusion of [[local observables]] as [[normal-ordered products]] (eq:NormalOrderingLocalObservables)
$$
T_1(g S_{int} + j A) = g :S_{int}: + j :A:
$$
1. The perturbative S-matrix is the [[exponential series]] of these maps in that for
all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [\hbar , g, j] ]\langle g,j\rangle $
$$
\label{ExponentialSeriesScatteringMatrix}
\begin{aligned}
\mathcal{S}( g S_{int} + j A)
& =
T
\left(
\exp_{\otimes}
\left(
\tfrac{ 1 }{i \hbar}
\left(
g S_{int} + j A
\right)
\right)
\right)
\\
& \coloneqq
\underoverset{k = 0}{\infty}{\sum}
\frac{1}{k!}
\left( \frac{1}{i \hbar} \right)^k
T_k
\left(
{\, \atop \,}
\underset{k\,\text{arguments}}{\underbrace{ (g S_{int} + jA) , \cdots, (g S_{int} + j A) }}
{\, \atop \,}
\right)
\end{aligned}
$$
1. ([[causal additivity]])
For all perturbative [[local observables]] $ O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$ we have
$$
\label{CausalAdditivity}
\left(
{\, \atop \,}
supp( O_1 ) {\vee\!\!\!\wedge} supp( O_2 )
{\, \atop \,}
\right)
\;\; \Rightarrow \;\;
\left(
{\, \atop \,}
\mathcal{S}( O_0 + O_1 + O_2 )
\;\,
\mathcal{S}( O_0 + O_1 )
\,
\mathcal{S}( O_0 )^{-1}
\,
\mathcal{S}(O_0 + O_2)
{\, \atop \,}
\right)
\,.
$$
(The [[inverse]] $\mathcal{S}(O)^{-1}$ of $\mathcal{S}(O)$ with respect to the [[Wick algebra]]-[[structure]]
is implied to exist by the axiom "perturbation", see remark \ref{PerturbativeSMatrixInverse} below.)
=--
Def. \ref{LagrangianFieldTheoryPerturbativeScattering} is due to ([Epstein-Glaser 73 (1)](causal+perturbation+theory#EpsteinGlaser73)),
following ([Stückelberg 49-53](causal+perturbation+theory#Stueckelberg49), [Bogoliubov-Shirkov 59](causal+perturbation+theory#BogoliubovShirkov59)).
That the [[domain]] of an S-matrix scheme is indeed the space of [[local observables]] was made explicit (in terms of axioms for the [[time-ordered products]], see def. \ref{TimeOrderedProduct} below), in ([Brunetti-Fredenhagen 99, section 3](S-matrix#BrunettiFredenhagen99), [Dütsch-Fredenhagen 04, appendix E](S-matrix#DuetschFredenhagen04), [Hollands-Wald 04,around (20)](S-matrix#HollandsWald04)). Review includes ([Rejzner 16, around def. 6.7](S-matrix#Rejzner16), [Dütsch 18, section 3.3](S-matrix#Duetsch18)).
+-- {: .num_remark #PerturbativeSMatrixInverse}
###### Remark
**([[inverse|invertibility]] of the [[S-matrix]])**
The mutliplicative inverse $S(-)^{-1}$ of the perturbative [[S-matrix]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering}
with respect to the [[Wick algebra]]-product indeed exists, so that the list of axioms is indeed well defined: By the axiom "perturbation" this follows with the usual formula for the multiplicative inverse of [[formal power series]] that are non-vanishing in degree 0:
If we write
$$
\mathcal{S}(g S_{int} + j A) = 1 + \mathcal{D}(g S_{int} + j A)
$$
then
$$
\label{InfverseOfPerturbativeSMatrix}
\begin{aligned}
\left(
{\, \atop \,}
\mathcal{S}(g S_{int} + j A)
{\, \atop \,}
\right)^{-1}
&=
\left(
{\, \atop \,}
1 + \mathcal{D}(g S_{int} + j A)
{\, \atop \,}
\right)^{-1}
\\
& =
\underoverset{r = 0}{\infty}{\sum}
\left(
{\, \atop \,}
-\mathcal{D}(g S_{int} + j A)
{\, \atop \,}
\right)^r
\end{aligned}
$$
where the sum does exist in $PolyObs(E_{\text{BV-BRST}})((\hbar))[ [[ g,j ] ]$, because (by
the axiom "perturbation")
$\mathcal{D}(g S_{int} + j A)$ has vanishing coefficient in zeroth order in the formal parameters $g$ and $j$, so that
only a finite sub-sum of the formal infinite sum contributes in each order in $g$ and $j$.
This expression for the inverse of S-matrix may usefully be re-organized in terms of "rever-time ordered products" (def. \ref{ReverseTimeOrderedProduct} below), see prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix} below.
Notice that $\mathcal{S}(-g S_{int} - j A )$ is instead the inverse with respect to the [[time-ordered products]] (eq:TimeOrderedProductsInSMatrix) in that
$$
T( \mathcal{S}(-g S_{int} - j A ) \,,\, \mathcal{S}(g S_{int} + j A) )
\;=\;
1
\;=\;
T( \mathcal{S}(g S_{int} + j A ) \,,\, \mathcal{S}(-g S_{in} - j A ) )
\,.
$$
(Since the time-ordered product is, by definition, symmetric in its arguments, the usual formula for the multiplicative inverse of
an [[exponential series]] applies).
=--
+-- {: .num_remark #MoreDeformationParameters}
###### Remark
**(adjoining further [[deformation]] parameters)
The definition of [[S-matrix]] schemes in def. \ref{LagrangianFieldTheoryPerturbativeScattering} has immediate variants where arbitrary countable sets $\{g_n\}$ and $\{j_m\}$ of formal [[deformation]] parameters are considered, instead of just a single [[coupling constant]] $g$ and a single [[source field]] $j$. The more such constants are considered, the "more perturbative" the theory becomes and the stronger the implications.
=--
Given a perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) it immediately
induces a corresponding concept of [[observables]]:
+-- {: .num_defn #SchemeGeneratingFunction}
###### Definition
**([[generating function]] scheme for [[interacting field observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.
The corresponding _[[generating function]] [[renormalization scheme|scheme]]_
(for [[interacting field observables]], def. \ref{InteractingFieldObservables} below) is the functional
$$
\mathcal{Z}_{(-)}(-)
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [\hbar, g] ]\langle g \rangle
\;\times\;
LocObs(E_{\text{BV-BRST}})[ [\hbar, j] ]\langle j \rangle
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [g , j] ]
$$
given by
$$
\label{GeneratingFunctionInducedFromSMatrix}
\mathcal{Z}_{g S_{int}}(j A)
\;\coloneqq\;
\mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )
\,.
$$
=--
+-- {: .num_prop #ZCausalAdditivity}
###### Proposition
**([[causal additivity]] in terms of [[generating functions]])**
In terms of the [[generating functions]] $\mathcal{Z}$ (def. \ref{SchemeGeneratingFunction}) the axiom "[[causal additivity]]"
on the [[S-matrix]] [[renormalization scheme|scheme]] $\mathcal{S}$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) is equivalent to:
* ([[causal additivity]] in terms of $\mathcal{Z}$)
For all [[local observables]] $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\otimes\mathbb{C}\langle g,j\rangle$ we have
$$
\label{GeneratingFunctionCausalAdditivity}
\begin{aligned}
\left(
{\, \atop \,}
supp(O_1) {\vee\!\!\!\wedge} supp(O_2)
{\, \atop \,}
\right)
& \;\; \Rightarrow \;\;
\left(
{\, \atop \,}
\mathcal{Z}_{O_0}( O_1 ) \, \mathcal{Z}_{O_0}( O_2)
=
\mathcal{Z}_{ O_0 }( O_1 + O_2 )
{\, \atop \,}
\right)
\\
& \;\; \Leftrightarrow \;\;
\left(
{\, \atop \,}
\mathcal{Z}_{ O_0 + O_1 }( O_2 )
=
\mathcal{Z}_{ O_0 }( O_2 )
{\, \atop \,}
\right)
\end{aligned}
\,.
$$
(Whence "additivity".)
=--
+-- {: .proof}
###### Proof
This follows by elementary manipulations:
Multiplying both sides of (eq:CausalAdditivity) by $\mathcal{S}(O_0)^{-1}$ yields
$$
\underset{
\mathcal{Z}_{ O_0 }( O_1 + O_2 )
}{
\underbrace{
\mathcal{S}( O_0 )^{-1}
\mathcal{S}( O_0 + O_1 + O_2 )
}
}
\;=\;
\underset{
\mathcal{Z}_{ O_0 }( O_1 )
}{
\underbrace{
\mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_1 )
}
}
\underset{
\mathcal{Z}_{ O_0 }( O_2 )
}{
\underbrace{
\mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_2 )
}
}
$$
This is the first line of (eq:GeneratingFunctionCausalAdditivity).
Multiplying both sides of (eq:CausalAdditivity) by $\mathcal{S}( O_0 + O_1 )^{-1}$ yields
$$
\underset{
= \mathcal{Z}_{ O_0 + O_1 }( O_2 )
}{
\underbrace{
\mathcal{S}( O_0 + O_1 )^{-1}
\mathcal{S}( O_0 + O_1 + O_2 )
}
}
\;=\;
\underset{
= \mathcal{Z}_{ O_0 }( O_2 )
}{
\underbrace{
\mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_2 )
}
}
\,.
$$
This is the second line of (eq:GeneratingFunctionCausalAdditivity).
=--
+-- {: .num_defn #InteractingFieldObservables}
###### Definition
**([[interacting field observables]] -- [[Bogoliubov's formula]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].
Then for $A \in LocObs(E_{\text{BV-BRST}})[ [\hbar , g] ]$ a [[local observable]] of the [[free field theory]],
we say that the corresponding [[local interacting field observable]]
$$
A_{int} \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar, g] ]
$$
is the [[coefficient]] of $j^1$ in the [[generating function]] (eq:GeneratingFunctionInducedFromSMatrix):
$$
\label{BogoliubovsFormula}
\begin{aligned}
A_{int}
&\coloneqq
i \hbar \frac{d}{d j}
\left(
{\, \atop \,}
\mathcal{Z}_{ g S_{int} }( j A )
{\, \atop \,}
\right)_{\vert_{j = 0}}
\\
& \coloneqq
i \hbar \frac{d}{d j}
\left(
{\, \atop \,}
\mathcal{S}(g S_{int})^{-1} \, \mathcal{S}( g S_{int} + j A )
{\, \atop \,}
\right)_{\vert_{j = 0}}
\\
& =
\mathcal{S}(g S_{int})^{-1}
T\left(
\mathcal{S}(g S_{int}), A
\right)
\,.
\end{aligned}
$$
This expression is called _[[Bogoliubov's formula]]_, due to ([Bogoliubov-Shirkov 59](S-matrix#BogoliubovShirkov59)).
One thinks of $A_{int}$ as the [[deformation]] of the [[local observable]] $A$ as the [[interaction]] $S_{int}$ is turned on; and speaks of an element of the _[[interacting field algebra of observables]]_. Their value ("[[expectation value]]") in the given free [[Hadamard vacuum state]]
$\langle -\rangle$ (def. \ref{VacuumFree}) is a [[formal power series]] in [[Planck's constant]] $\hbar$ and in the [[coupling constant]] $g$, with [[coefficients]] in the [[complex numbers]]
$$
\left\langle
A_{int}
\right\rangle
\;\in\;
\mathbb{C}[ [\hbar, g] ]
$$
which express the [[probability amplitudes]] that reflect the predictions of the [[perturbative QFT]], which may be compared to [[experiment]].
=--
([Epstein-Glaser 73, around (74)](S-matrix#EpsteinGlaser73)); review includes ([Dütsch-Fredenhagen 00, around (17)](S-matrix#DuetschFredenhagen00), [Dütsch 18, around (3.212)](pAQFT#Duetsch18)).
+-- {: .num_example #FormalPowerSeriesInteractingFieldObservables}
###### Remark
**([[interacting field observables]] are [[formal deformation quantization]])**
The [[interacting field observables]] in def. \ref{InteractingFieldObservables} are indeed [[formal power series]] in the formal parameter $\hbar$ ([[Planck's constant]]), as opposed to being more general [[Laurent series]], hence they involve no [[negative number|negative]] powers of $\hbar$ ([Dütsch-Fredenhagen 00, prop. 2 (ii)](interacting+field+observable#DuetschFredenhagen00), [Hawkins-Rejzner 16, cor. 5.2](interacting+field+observable#HawkinsRejzner16)). This is not immediate, since by def. \ref{LagrangianFieldTheoryPerturbativeScattering} the [[S-matrix]] that they are defined from does involve negative powers of $\hbar$.
It follows in particular that the [[interacting field observables]] have a [[classical limit]] $\hbar \to 0$, which
is not the case for the [[S-matrix]] itself (due to it involving negative powers of $\hbar$).
Indeed the [[interacting field observables]] constitute a _[[formal deformation quantization]]_ of the [[covariant phase space]]
of the [[interacting field theory]] (prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below)
and are thus the more fundamental concept.
=--
As the name suggests, the [[S-matrices]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering} serve to express [[scattering amplitudes]] (example \ref{ScatteringAmplitudeFromInteractingFieldObservables} below). But by remark \ref{FormalPowerSeriesInteractingFieldObservables}
the more fundamental concept is that of the [[interacting field observables]]. Their perspective reveals that
consistent interpretation of [[scattering amplitudes]] requires the following condition on the relation between the
[[vacuum state]] and the [[interaction]] term:
+-- {: .num_defn #VacuumStability}
###### Definition
**([[vacuum stability]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle$ be a [[local observable]], regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
We say that the given [[Hadamard vacuum state|Hadamard]] [[vacuum state]] (prop. \ref{WickAlgebraCanonicalState})
$$
\langle - \rangle
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ]
\longrightarrow
\mathbb{C}[ [ \hbar, g, j ] ]
$$
is _[[vacuum stability|stable]]_ with respect to the [[interaction]] $S_{int}$, if for all elements of the [[Wick algebra]]
$$
A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]
$$
we have
$$
\left\langle
A \mathcal{S}(g S_{int})
\right\rangle
\;=\;
\left\langle
\mathcal{S}(g S_{int})
\right\rangle
\,
\left\langle
A
\right\rangle
\phantom{AA}
\text{and}
\phantom{AA}
\left\langle
\mathcal{S}(g S_{int})^{-1}
A
\right\rangle
\;=\;
\frac{1}
{
\left\langle
\mathcal{S}(g S_{int})
\right\rangle
}
\left\langle
A
\right\rangle
$$
=--
+-- {: .num_example #InteractinFieldTimeOrderedProduct}
###### Example
**([[time-ordered product]] of [[interacting field observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].
Consider two [[local observables]]
$$
A_1, A_2
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
$$
with [[causal ordering|causally ordered]] spacetime support
$$
supp(A_1) {\vee\!\!\!\!\wedge} supp(A_2)
$$
Then [[causal additivity]] according to prop. \ref{ZCausalAdditivity} implies that the
[[Wick algebra]]-product of the corresponding [[interacting field observables]]
$(A_1)_{int}, (A_2)_{int} \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ] $ (def. \ref{InteractingFieldObservables})
is
$$
\begin{aligned}
(A_1)_{int}
(A_2)_{int}
& =
\left(
\frac{\partial}{\partial j} \mathcal{Z}(j A_1 )
\right)_{\vert j = 0}
\left(
\frac{\partial}{\partial j} \mathcal{Z}( j A_2 )
\right)_{\vert j = 0}
\\
& =
\frac{\partial^2}{\partial j_1 \partial j_2}
\left(
{\, \atop \,}
\mathcal{Z}( j_1 A_1 )
\mathcal{Z}( j_2 A_2 )
{\, \atop \,}
\right)_{
\left\vert { {j_1 = 0}, \atop {j_2 = 0} } \right.
}
\\
& =
\frac{\partial^2}{\partial j_1 \partial j_2}
\left(
{\, \atop \,}
\mathcal{Z}( j_1 A_1 + j_2 A_2 )
{\, \atop \,}
\right)_{
\left\vert { {j_1 = 0}, \atop {j_2 = 0} } \right.
}
\end{aligned}
$$
Here the last line makes sense if one extends the axioms on the [[S-matrix]] in prop. \ref{LagrangianFieldTheoryPerturbativeScattering}
from formal power series in $\hbar, g, j$ to formal power series in $\hbar, g, j_1, j_2, \cdots$ (remark \ref{MoreDeformationParameters}).
Hence in this generalization, the [[generating functions]] $\mathcal{Z}$ are not just generating functions
for [[interacting field observables]] themselves, but in fact for _[[time-ordered products]]_ of interacting field observables.
=--
An important special case of [[time-ordered products]] of [[interacting field observables]] as in example \ref{InteractinFieldTimeOrderedProduct}
is the following special case of _[[scattering amplitudes]]_, which is the example that gives the _[[scattering matrix]]_ in def. \ref{LagrangianFieldTheoryPerturbativeScattering} its name:
+-- {: .num_example #ScatteringAmplitudeFromInteractingFieldObservables}
###### Example
**([[scattering amplitudes]] as [[vacuum expectation values]] of [[interacting field observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]],
such that the [[vacuum state]] is [[vacuum stability|stable]] with respect to $g S_{int}$ (def. \ref{VacuumStability}).
Consider [[local observables]]
$$
\array{
A_{in,1}, \cdots, A_{in , n_{in}},
\\
A_{out,1}, \cdots, A_{out, n_{out}}
}
\;\;\in\;\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$
whose spacetime support satisfies the following [[causal ordering]]:
$$
A_{out, i_{out} }
{\gt\!\!\!\!\lt}
A_{out, j_{out}}
\phantom{AAA}
A_{out, i_{out} }
{\vee\!\!\!\wedge}
S_{int}
{\vee\!\!\!\wedge}
A_{in, i_{in}}
\phantom{AAA}
A_{in, i_{in} }
{\gt\!\!\!\!\lt}
A_{in, j_{in}}
$$
for all $1 \leq i_{out} \lt j_{out} \leq n_{out}$ and $1 \leq i_{in} \lt j_{in} \leq n_{in}$.
Then the [[vacuum expectation value]] of the [[Wick algebra]]-product of the corresponding [[interacting field observables]] (def. \ref{InteractingFieldObservables})
is
$$
\begin{aligned}
&
\left\langle
{\, \atop \,}
(A_{out, 1})_{int}
\cdots
(A_{out,n_{out}})_{int}
\,
(A_{in, 1})_{int}
\cdots
(A_{in,n_{in}})_{int}
{\, \atop \,}
\right\rangle
\\
& =
\left\langle
{\, \atop \,}
A_{out,1}
\cdots
A_{out,n_{out}}
\right|
\;
\mathcal{S}(g S_{int})
\;
\left|
A_{in,1}
\cdots
A_{in, n_{in}}
{\, \atop \,}
\right\rangle
\\
& \coloneqq
\frac{1}{
\left\langle \mathcal{S}(g S_{int}) \right\rangle
}
\left\langle
{\, \atop \,}
A_{out,1}
\cdots
A_{out,n_{out}}
\;
\mathcal{S}(g S_{int})
\;
A_{in,1}
\cdots
A_{in, n_{in}}
{\, \atop \,}
\right\rangle
\,.
\end{aligned}
$$
These [[vacuum expectation values]] are interpreted, in the [[adiabatic limit]] where $g_{sw} \to 1$, as _[[scattering amplitudes]]_ (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes} below).
=--
+-- {: .proof}
###### Proof
For notational convenience, we spell out the argument for $n_{in} = 1 = n_{out}$. The general case
is directly analogous.
So assuming the [[causal order]] (def. \ref{CausalOrdering})
$$
supp(A_{out})
{\vee\!\!\!\wedge}
supp(S_{int})
{\vee\!\!\!\wedge}
supp(A_{in})
$$
we compute with [[causal additivity]] via prop. \ref{ZCausalAdditivity} as follows:
$$
\begin{aligned}
(A_{out})_{int} (A_{in})_{int}
& =
\frac{d^2 }{\partial j_{out} \partial j_{in}}
\left(
\mathcal{Z}( j_{out} A_{out} )
\mathcal{Z}( j_{in} A_{in} )
\right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } } \right.}
\\
& =
\frac{\partial^2 }{\partial j_{out} \partial j_{in}}
\left(
\mathcal{S}(g S_{int})^{-1}
\underset{
=
\mathcal{S}(j_{out} A_{out})
\mathcal{S}(g S_{int})
}{
\underbrace{
\mathcal{S}(g S_{int} + j_{out} A_{out})
}
}
\mathcal{S}(g S_{int})^{-1}
\underset{
= \mathcal{S}(g S_{int}) \mathcal{S}(j_{in} A_{in})
}{
\underbrace{
\mathcal{S}(g S_{int} + j_{in}A_{in})
}
}
\right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } } \right.}
\\
& =
\frac{\partial^2 }{\partial j_{out} \partial j_{in}}
\left(
\mathcal{S}(g S_{int})^{-1}
\mathcal{S}(j_{out} A_{out})
\underset{
= \mathcal{S}(g S_{int})
}{
\underbrace{
\mathcal{S}(g S_{int})
\mathcal{S}(g S_{int})^{-1}
\mathcal{S}(g S_{int})
}
}
\mathcal{S}(j_{in} A_{in})
\right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } } \right.}
\\
& =
\mathcal{S}(g S_{int})^{-1}
\,
\left(
{\, \atop \,}
A_{out}
\mathcal{S}(g S_{int})
A_{in}
{\, \atop \,}
\right)
\,.
\end{aligned}
$$
With this the statement follows by the definition of [[vacuum stability]] (def. \ref{VacuumStability}).
=--
+-- {: .num_remark}
###### Remark
**(computing [[S-matrices]] via [[Feynman perturbation series]])**
For practical computation of [[vacuum expectation values]] of [[interacting field observables]] (example \ref{InteractinFieldTimeOrderedProduct})
and hence in particular, via example \ref{ScatteringAmplitudeFromInteractingFieldObservables}, of [[scattering amplitudes]],
one needs some method for collecting all the contributions to the [[formal power series]] in increasing order in $\hbar$ and $g$.
Such a method is provided by the _[[Feynman perturbation series]]_ (example \ref{FeynmanPerturbationSeries} below)
and the _[[effective action]]_ (def. \ref{InPerturbationTheoryActionEffective}), see example \ref{SMatrixVacuumContribution} below.
=--
$\,$
**Conceptual remarks**
{#RemarksOnCausalPerturbationTheoryAxioms}
The simple axioms for [[S-matrix]] [[renormalization scheme|schemes]] in [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and hence for [[interacting field observables]] (def. \ref{InteractingFieldObservables}) have a wealth of implications and consequences. Before discussing these formally below, we here make a few informal remarks meant to put various relevant concepts into perspective:
+-- {: .num_remark #AsymptoticSeriesObservables}
###### Remark
**([[perturbative quantum field theory|perturbative QFT]] and [[asymptotic expansion]] of [[probability amplitudes]])**
Given a [[perturbative quantum field theory|perturbative]] [[S-matrix]] [[renormalization scheme|scheme]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), then by remark \ref{FormalPowerSeriesInteractingFieldObservables} the
[[expectation values]] of [[interacting field observables]] (def. \ref{InteractingFieldObservables}) are [[formal power series]]
in the formal parameters $\hbar$ and $g$ (which are interpreted as [[Planck's constant]], and as the [[coupling constant]], respectively):
$$
\left\langle A_{int} \right\rangle
\;\in\;
\mathbb{C}[ [\hbar, g] ]
\,.
$$
This means that there is _no_ guarantee that these series _[[convergence|converge]]_ for any [[positive real number|positive]]
value of $\hbar$ and/or $g$.
In terms of [[synthetic differential geometry]] this means that in [[perturbative QFT]] the [[deformation]] of the [[classical field theory|classical]] [[free field theory]] by quantum effects (measured by $\hbar$) and [[interactions]] (meaured by $g$) is so very tiny as to actually be [[infinitesimal]]: formal power series may be read as functions on the [[infinitesimal neighbourhood]] in a space of [[Lagrangian field theories]] at the point $\hbar = 0$, $g = 0$.
In fact, a simple argument (due to [Dyson 52](perturbative+quantum+field+theory#Dyson52))
suggests that in realistic field theories these series _never_ converge for _any_ [[positive real number|positive]] value of $\hbar$ and/or $g$. Namely convergence for $g$ would imply a [[positive real number|positive]] _[[radius of convergence]]_ around $g = 0$, which would imply convergence also for $-g$ and even for [[imaginary number|imaginary]] values of $g$, which would however correspond to unstable [[interactions]] for which no converging field theory is to be expected. (See [Helling, p. 4](perturbative+quantum+field+theory#Helling) for the example of [[phi^4 theory]].)
In physical practice one tries to interpret these non-converging [[formal power series]] as _[[asymptotic expansions]]_ of
actual but hypothetical functions in $\hbar, g$, which reflect the actual but hypothetical _[[non-perturbative quantum field theory]]_
that one imagines is being approximated by [[perturbative QFT]] methods. An _[[asymptotic expansion]]_ of a function is a [[power series]] which may not converge, but which has for every $n \in \mathbb{N}$ an estimate for how far the [[sum]] of the first $n$ terms in the series may differ from the function being approximated.
For examples such as [[quantum electrodynamics]] and [[quantum chromodynamics]], as in the [[standard model of particle physics]], the truncation of these [[formal power series]] [[scattering amplitudes]] to the first handful of [[loop orders]] in $\hbar$ happens to agree with [[experiment]] (such as at the [[LHC]] collider) to high precision (for [[QED]]) or at least decent precision (for [[QCD]]), at least away from infrared phenomena (see remark \ref{AdiabaticLimit}).
In summary this says that [[perturbative QFT]] is an extremely _coarse_ and restrictive approximation to what should be genuine [[non-perturbative quantum field theory]], while at the same time it happens to match certain experimental observations to remarkable degree, albeit only if some ad-hoc truncation of the resulting power series is considered.
This is strong motivation for going beyond [[perturbative QFT]] to understand and construct genuine [[non-perturbative quantum field theory]]. Unfortunately, this is a wide-open problem, away from toy examples. Not a single [[interacting field theory]] in [[spacetime]] [[dimension]] $\geq 4$ has been non-perturbatively quantized. Already a single aspect of the [[non-perturbative quantum field theory|non-perturbative]] [[quantization of Yang-Mills theory]] (as in [[QCD]]) has famously been advertized as one of the _[Millennium Problems](http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap)_ of our age; and speculation about [[non-perturbative quantum field theory|non-perturbative]] [[quantum gravity]] is the subject of much activity.
Now, as the name indicates, the [[axioms]] of _[[causal perturbation theory]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) do
_not_ address [[non-perturbative effect|non-perturbative aspects]] of [[non-perturbative field theory]]; the convergence or non-convergence of the [[formal power series]] that are axiomatized by [[Bogoliubov's formula]] (def. \ref{InteractingFieldObservables}) is _not_ addressed by the theory.
The point of the axioms of [[causal perturbation theory]] is to give rigorous mathematical meaning to _everything else_ in [[perturbative QFT]].
=--
+-- {: .num_remark #DysonCausalFactorization}
###### Remark
**([[Dyson series]] and [[Schrödinger equation]] in [[interaction picture]])**
The axiom "[[causal additivity]]" (eq:CausalAdditivity) on an [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
implies immediately this seemingly weaker condition (which turns out to be equivalent, this is prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below):
* ([[causal factorization]])
For all [[local observables]] $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, h, j] ]\langle g , j\rangle $ we have
$$
\left(
{\, \atop \,}
supp(O_1) {\vee\!\!\!\wedge} supp(O_2)
{\, \atop \,}
\right)
\;\; \Rightarrow \;\;
\left(
{\, \atop \,}
\mathcal{S}( O_1 + O_2 )
=
\mathcal{S}( O_1 ) \, \mathcal{S}( O_2 )
{\, \atop \,}
\right)
$$
(This is the special case of "causal additivity" for $O_0 = 0$, using that by the axiom "perturbation" (eq:ExponentialSeriesScatteringMatrix) we have $\mathcal{S}(0) = 1$.)
If we now think of $O_1 = g S_{1}$ and $O_2 = g S_2$ themselves as [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functionals]], then this becomes
$$
\left(
{\, \atop \,}
supp(S_1) {\vee\!\!\!\wedge} supp(S_2)
{\, \atop \,}
\right)
\;\; \Rightarrow \;\;
\left(
{\, \atop \,}
\mathcal{S}( g S_1 + g S_2 )
=
\mathcal{S}( g S_1) \, \mathcal{S}( g sS_2)
{\, \atop \,}
\right)
$$
This exhibits the [[S-matrix]]-scheme as a "[[causal ordering|causally ordered]] [[exponential]]"
or "[[Dyson series]]" of the [[interaction]], hence as a refinement to [[relativistic field theory]] of what in
[[quantum mechanics]] is the "integral version of the [[Schrödinger equation]] in the [[interaction picture]]" (see [this equation](S-matrix#IntegralVersionSchroedingerEquationInInteractionPicture) at _[[S-matrix]]_; see also [Scharf 95, second half of 0.3](S-matrix#Scharf95)).
The relevance of manifest [[causal additivity]] of the [[S-matrix]], over just [[causal factorization]] (even though both conditions happen to be equivalent, see prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below), is that it directly implies that
the induced [[interacting field algebra of observables]] (def. \ref{InteractingFieldObservables}) forms a [[causally local net]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below).
=--
+-- {: .num_remark #InterpretationOfPerturbativeSMatrix}
###### Remark
**([[path integral]]-intuition)**
In informal discussion of [[perturbative QFT]] going back to informal ideas of [[Schwinger-Tomonaga-Feynman-Dyson]], the
perturbative [[S-matrix]] is thought of in terms of a would-be _[[path integral]]_, symbolically written
$$
\mathcal{S}\left(
g S_{int} + j A
\right)
\;\overset{\text{not really!}}{=}\;
\!\!\!\!\!
\underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
\!\!\!\!\!\!
\exp\left(
\tfrac{1}{i \hbar}
\int_\Sigma
\left(
g L_{int}(\Phi) + j A(\Phi)
\right)
\right)
\,
\exp\left(
\tfrac{1}{i \hbar}\int_\Sigma L_{free}(\Phi)
\right) D[\Phi]
\,.
$$
Here the would-be [[integration]] is thought to be over the [[space of field histories]] $\Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}$ (the [[space of sections]] of the given [[field bundle]], remark \ref{PossibleFieldHistories}) for [[field histories]] which satisfy given asymptotic conditions at $x^0 \to \pm \infty$; and as these boundary conditions vary the above is regarded as a would-be [[integral kernel]] that defines the required operator in the [[Wick algebra]] (e.g. [Weinberg 95, around (9.3.10) and (9.4.1)](S-matrix#Weinberg95)). This is related to the intuitive picture of the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries} below) expressing a sum over all possible interactions of [[virtual particles]] (remark \ref{WorldlineFormalism}).
Beyond toy examples, it is not known how to define the would-be [[measure]] $D[\Phi]$ and it is not known how to make sense of this expression as an actual [[integral]].
The analogous path-integral intuition for [[Bogoliubov's formula]] for [[interacting field observables]] (def. \ref{InteractingFieldObservables}) symbolically reads
$$
\begin{aligned}
A_{int}
& \overset{\text{not really!}}{=}
\frac{d}{d j}
\ln
\left(
\underset{\Phi \in \Gamma_\Sigma(E)_{asm}}{\int}
\!\!\!\!
\exp\left(
\underset{\Sigma}{\int}
g L_{int}(\Phi) + j A(\Phi)
\right)
\,
\exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)
D[\Phi]
\right)
\vert_{j = 0}
\end{aligned}
$$
If here we were to regard the expression
$$
\mu(\Phi)
\;\overset{\text{not really!}}{\coloneqq}\;
\frac{
\exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)\, D[\Phi]
}
{
\underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
\!\!\!\!
\exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)\, D[\Phi]
}
$$
as a would-be [[Gaussian measure]] on the [[space of field histories]], normalized to a would-be [[probability measure]], then this formula
would express interacting field observables as ordinary [[expectation values]]
$$
A_{int}
\overset{\text{not really!}}{=}
\!\!\!
\underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
\!\!\!\!\!\!
A(\Phi)
\,\mu(\Phi)
\,.
$$
As before, beyond toy examples it is not known how to make sense of this as an actual [[integration]].
But we may think of the axioms for the [[S-matrix]] in [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) as rigorously _defining_ the [[path integral]], not analytically as an actual [[integration]], but _[[synthetic mathematics|synthetically]]_ by axiomatizing the properties of the desired _outcome_ of the would-be integration:
The analogy with a well-defined [[integral]] and the usual properties of an [[exponential]] vividly _suggest_
that the would-be [[path integral]] should obey [[causal factorization]]. Instead of trying to make sense of
[[path integral|path integration]] so that this factorization property could then be appealed to as a _consequence_ of general properties
of [[integration]] and [[exponentials]], the axioms of [[causal perturbation theory]] directly prescribe the
desired factorization property, without insisting that it derives from an actual integration.
The great success of [[path integral]]-intuition in the development of [[quantum field theory]], despite the dearth of
actual constructions, indicates that it is not the would-be integration process as such that actually matters in field
theory, but only the resulting properties that this _suggests_ the S-matrix should have; which is what
[[causal perturbation theory]] axiomatizes. Indeed, the simple [[axioms]] of [[causal perturbation theory]]
rigorously _imply_ finite (i.e. [[renormalization|("re"-)normalized]]) [[perturbative quantum field theory]] (see remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}).
$$
\array{
\array{
\text{would-be}
\\
\text{path integral}
\\
\text{intuition}
}
&
\overset{
\array{
\text{informally}
\\
\text{suggests}
}
}{\longrightarrow}
&
\array{
\text{causally additive}
\\
\text{scattering matrix}
}
&
\overset{
\array{
\text{rigorously}
\\
\text{implies}
}
}{\longrightarrow}
&
\array{
\text{UV-finite}
\\
\text{(i.e. (re-)normalized)}
\\
\text{perturbative QFT}
}
}
$$
=--
+-- {: .num_remark #FromAxiomaticSMatrixScatteringAmplitudes}
###### Remark
**([[scattering amplitudes]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
$$
be a [[local observable]], regarded
as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Then for
$$
A_{in}, A_{out} \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
$$
two [[microcausal polynomial observables]], with [[causal ordering]]
$$
supp(A_{out}) {\vee\!\!\!\wedge} supp(A_{int})
$$
the corresponding _[[scattering amplitude]]_ (as in example \ref{ScatteringAmplitudeFromInteractingFieldObservables}) is the value (called "[[expectation value]]" when referring to $A^\ast_{out} \, \mathcal{S}(S_{int}) \, A_{in}$, or "matrix element" when referring to $\mathcal{S}(S_{int})$, or "transition amplitude" when referring to $\left\langle A_{out} \right\vert$ and $\left\vert A_{in} \right\rangle$)
$$
\left\langle
A_{out}
\,\vert\,
\mathcal{S}(S_{int})
\,\vert\,
A_{in}
\right\rangle
\;\coloneqq\;
\left\langle
A^\ast_{out} \, \mathcal{S}(S_{int}) \, A_{in}
\right\rangle
\;\in\;
\mathbb{C}[ [ \hbar, g ] ]
\,.
$$
for the [[Wick algebra]]-product $A^\ast_{out} \, \mathcal{S}(S_{int})\, A_{in} \in PolyObs(E_{\text{BV-BRST}})[ [\hbar, g ] ]$
in the given [[Hadamard vacuum state]] $\langle -\rangle \colon PolyObs(E_{\text{BV-BRST}})[ [\hbar, g] ] \to \mathbb{C}[ [\hbar,g] ]$.
If here $A_{in}$ and $A_{out}$ are monomials in [[Wick algebra]]-products of the [[field observables]] $\mathbf{\Phi}^a(x) \in Obs(E_{\text{BV-BRST}})[ [\hbar] ]$,
then this [[scattering amplitude]] comes from the [[integral kernel]]
$$
\begin{aligned}
&
\left\langle
\mathbf{\Phi}^{a_{out,1}}(x_{out,1}) \cdots \mathbf{\Phi}^{a_{out,s}}(x_{out,s})
\vert
\,
\mathcal{S}(S_{int})
\,
\vert
\mathbf{\Phi}^{a_{in,1}}(x_{in,1})
\cdots
\mathbf{\Phi}^{a_{in,r}}(x_{in,r})
\right\rangle
\\
& \coloneqq
\left\langle
\left(\mathbf{\Phi}^{a_{out,1}}(x_{out,1})\right)^\ast
\cdots
\left(\mathbf{\Phi}^{a_{out,s}}(x_{out,s})\right)^\ast
\;\mathcal{S}(S_{int})\;
\mathbf{\Phi}^{a_{in,1}}(x_{in,1})
\cdots
\mathbf{\Phi}^{a_{in,r}}(x_{in,r})
\right\rangle
\end{aligned}
$$
or similarly, under [[Fourier transform of distributions]],
$$
\label{ScatteringPlaneWaves}
\begin{aligned}
&
\left\langle
\widehat{\mathbf{\Phi}}^{a_{out,1}}(k_{out,1}) \cdots \widehat{\mathbf{\Phi}}^{a_{out,s}}(k_{out,s})
\vert
\,
\mathcal{S}(S_{int})
\,
\vert
\widehat{\mathbf{\Phi}}^{a_{in,1}}(k_{in,1})
\cdots
\widehat{\mathbf{\Phi}}^{a_{in,r}}(k_{in,r})
\right\rangle
\\
& \coloneqq
\left\langle
\left(\widehat{\mathbf{\Phi}}^{a_{out,1}}(k_{out,1})\right)^\ast
\cdots
\left(\widehat{\mathbf{\Phi}}^{a_{out,s}}(k_{out,s})\right)^\ast
\;\mathcal{S}(S_{int})\;
\widehat{\mathbf{\Phi}}^{a_{in,1}}(k_{in,1})
\cdots
\widehat{\mathbf{\Phi}}^{a_{in,r}}(k_{in,r})
\right\rangle
\end{aligned}
\,.
$$
These are interpreted as the (distributional) _[[probability amplitudes]]_ for [[plane waves]] of field species $a_{in,\cdot}$
with [[wave vector]] $k_{in,\cdot}$ to come in from the far past, ineract with each other via $S_{int}$,
and emerge in the far future as [[plane waves]] of field species $a_{out,\cdot}$ with [[wave vectors]] $k_{out,\cdot}$.
=--
Or rather:
+-- {: .num_remark #AdiabaticLimit}
###### Remark
**([[adiabatic limit]], [[infrared divergences]] and [[interacting vacuum]])**
Since a [[local observable]] $S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$ by definition
has compact spacetime support, the [[scattering amplitudes]] in remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}
describe [[scattering]] processes for [[interactions]] that vanish (are "[[adiabatic switching|adiabatically switched off]]") outside a compact subset of [[spacetime]]. This constraint is crucial for [[causal perturbation theory]] to work.
There are several aspects to this:
* ([[adiabatic limit]]) On the one hand, real physical interactions $\mathbf{L}_{int}$ (say the [[electron-photon interaction]])
are not _really_ supposed to vanish outside a compact region of spacetime. In order to reflect this mathematically, one may consider a [[sequence]] of [[adiabatic switchings]] $g_{sw} \in C^\infty_{cp}(\Sigma)\langle g \rangle$ (each of [[compact support]]) whose [[limit of a sequence|limit]] is the [[constant function]] $g \in C^\infty(\Sigma)\langle g\rangle$ (the actual [[coupling constant]]), then consider the corresponding [[sequence]] of [[interaction]] [[action functionals]] $S_{int,sw} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$ and finally consider:
1. as the true [[scattering amplitude]] the corresponding [[limit of a sequence|limit]]
$$
\left\langle
A_{out} \vert \mathcal{S}(S_{int}) \vert A_{int}
\right\rangle
\;\coloneqq\;
\underset{g_{sw} \to 1}{\lim}
\left\langle
A_{out} \vert \mathcal{S}(S_{int,sw}) \vert A_{int}
\right\rangle
$$
of adiabatically switched [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) -- if it exists. This is called the _[[strong adiabatic limit]]_.
1. as the true [[n-point functions]] the corresponding [[limit of a sequence|limit]]
$$
\begin{aligned}
&
\left\langle
\mathbf{\Phi}^{a_1}_{int}(x_1)
\mathbf{\Phi}^{a_2}_{int}(x_2)
\cdots
\mathbf{\Phi}^{a_{n-1}}_{int}(x_{n-1})
\mathbf{\Phi}^{a_n}_{int,sw}(x_n)
\right\rangle
\\
& =
\underset{\underset{g_{sw} \to 1}{\longrightarrow}}{\lim}
\left\langle
\mathbf{\Phi}^{a_1}_{int,sw}(x_1)
\mathbf{\Phi}^{a_2}_{int,sw}(x_2)
\cdots
\mathbf{\Phi}^{a_{n-1}}_{int,sw}(x_{n-1})
\mathbf{\Phi}^{a_n}_{int,sw}(x_n)
\right\rangle
\end{aligned}
$$
of [[tempered distribution|tempered distributional]] [[expectation values]] of products of [[interacting field algebra|interacting]] [[field observables]] (def. \ref{InteractingFieldObservables}) -- if it exists. (Similarly for [[time-ordered products]].) This is called the _[[weak adiabatic limit]]_.
Beware that the left hand sides here are symbolic: Even if the limit exists in [[expectation values]], in general there is no actual observable whose expectation value is that limit.
The strong and weak adiabatic limits have been shown to exist if all [[field (physics)|fields]] are [[mass|massive]] ([Epstein-Glaser 73](S-matrix#EpsteinGlaser73)). The weak adiabatic limit has been shown to exists for [[quantum electrodynamics]] and for [[mass]]-less [[phi^4 theory]] ([Blanchard-Seneor 75](adiabatic+switching#BlanchardSeneor75)) and for larger classes of field theories in ([Duch 17, p. 113, 114](adiabatic+switching#Duch17)).
If these limits do not exist, one says that the [[perturbative QFT]] has an _[[infrared divergence]]_.
* ([[algebraic adiabatic limit]]) On the other hand, it is equally unrealistic that an actual [[experiment]] _detects_ phenomena outside a given compact subset of spacetime. Realistic scattering [[experiments]] (such as the [[LHC]]) do not really prepare or measure [[plane waves]] filling all of [[spacetime]] as described by the [[scattering amplitudes]] (eq:ScatteringPlaneWaves). Any [[observable]] that is realistically measurable must have compact spacetime support. We see below in prop. \ref{IsomorphismFromChangeOfAdiabaticSwitching} that such [[interacting field observables]] with compact spacetime support may be computed without taking the [[adiabatic limit]]: It is sufficient to use any [[adiabatic switching]] which is constant on the support of the observable.
This way one obtains for each [[causally closed subset]] $\mathcal{O}$ of spacetime an algebra of observables $\mathcal{A}_{int}(\mathcal{O})$ whose support is in $\mathcal{O}$, and for each inclusion of subsets a corresponding inclusion of algebras of observables (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below). Of this system of observables one may form the [[category theory|category-theoretic]] [[inductive limit]] to obtain a single global algebra of observables.
$$
\mathcal{A}_{int}
\;\coloneqq\;
\underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim}
\mathcal{A}_{int}(\mathcal{O})
$$
This always exists. It is called the _[[algebraic adiabatic limit]]_ (going back to [Brunetti-Fredenhagen 00, section 8](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)).
For [[quantum electrodynamics]] the [[algebraic adiabatic limit]] was worked out in ([Dütsch-Fredenhagen 98](quantum+electrodynamics#DuetschFredenhagen98), reviewed in [Dütsch 18, 5,3](QED+Duetsch18)).
* ([[interacting vacuum]]) While, via the above [[algebraic adiabatic limit]], [[causal perturbation theory]] yields the correct [[interacting field algebra of quantum observables]] independent of choices of [[adiabatic switching]], a theory of _[[quantum probability]]_ requires, on top of the [[algebra of observables]], also a _[[state on a star-algebra|state]]_
$$
\langle - \rangle_{int} \;\colon\; \mathcal{A}_{int} \longrightarrow \mathbb{C}[ [\hbar] ]
$$
Just as the [[interacting field algebra of observables]] $\mathcal{A}_{int}$ is a [[deformation]] of the free field algebra of observables ([[Wick algebra]]), there ought to be a corresponding deformation of the free [[Hadamard vacuum state]] $\langle- \rangle$ into an "[[interacting vacuum state]]" $\langle - \rangle_{int}$.
Sometimes the [[weak adiabatic limit]] serves to define the [[interacting vacuum]] (see [Duch 17, p. 113-114](adiabatic+switching#Duch17)).
A stark example of these infrared issues is the phenomenon of _[[confinement]]_ of [[quarks]] to [[hadron]] [[bound states]] (notably to [[protons]] and [[neutrons]]) at large [[wavelengths]]. This is paramount in [[experiment|observation]] and reproduced in numerical [[lattice gauge theory]] simulation, but is invisible to [[perturbative QFT|perturbative]] [[quantum chromodynamics]] in its [[free field]] [[vacuum state]], due to [[infrared divergences]].
It is expected that this should be rectified by the proper [[interacting vacuum]] of [[QCD]] ([Rafelski 90, pages 12-16](confinement#Rafelski90)), which is possibly a "[[theta-vacuum]]" exhibiting [[superposition]] of [[instanton in QCD|QCD instantons]] ([Schäfer-Shuryak 98, section III.D](instanton+in+QCD#SchaeferShuryak98)). This remains open, closely related to the _[Millennium Problem](http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap)_ of [[quantization of Yang-Mills theory]].
=--
In contrast to the above subtleties about the [[infrared divergences]], any would-be [[UV-divergences]] in [[perturbative QFT]] are
dealt with by [[causal perturbation theory]]:
+-- {: .num_remark #TheTraditionalErrorThatLeadsToTheNotoriouDivergencies}
###### Remark
**(the traditional error leading to [[UV-divergences]])**
Naively it might seem that (say over [[Minkowski spacetime]], for simplicity)
examples of [[time-ordered products]] according to def. \ref{TimeOrderedProduct}
might simply be obtained by multiplying [[Wick algebra]]-products with [[step functions]] $\Theta$ of the time coordinates, hence to write, in the notation as [[generalized functions]] (remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}):
$$
T(x_1, x_2)
\overset{\text{no!}}{=}
\Theta(x_1^0 - x_2^0) \, T(x_1) \, T(x_2)
+
\Theta(x_2^0 - x_1^0) \, T(x_2) \, T(x_1)
$$
and analogously for time-ordered products of more arguments (for instance [Weinberg 95, p. 143, between (3.5.9) and (3.5.10)](S-matrix#Weinberg95)).
This however is simply a mathematical error (as amplified in [Scharf 95, below (3.2.4), below (3.2.44) and in fig. 3](causal+perturbation+theory#Scharf95)):
Both $T$ as well as $\Theta$ are [[distributions]] and their [[product of distributions]] is in general not defined, as [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}), which is exactly what guarantees absence of [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}), may be violated. The notorious [[ultraviolet divergences]] which plagued ([Feynman 85](Schwinger-Tomonaga-Feynman-Dyson#Feynman85SuchABunchOfWords)) the original conception of [[perturbative QFT]] due to [[Schwinger-Tomonaga-Feynman-Dyson]] are the signature of this ill-defined product (see remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}).
On the other hand, when both distributions are [[restriction of distributions|restricted]] to the [[complement]] of the [[diagonal]]
(i.e. restricted away from coinciding points $x_1 = x_2$), then the [[step function]] becomes a [[non-singular distribution]]
so that the above expression happens to be well defined and does solve the axioms for time-ordered products.
Hence what needs to be done to properly define the [[time-ordered product]] is to choose an [[extension of distributions]] of the above product expression back from the complement of the diagonal to the whole space of [[tuples]] of points. Any such extension will produce time-ordered products.
There are in general several different such [[extension of distributions|extensions]]. This freedom of choice is the freedom
of _[[renormalization|"re-"normalization]]_; or equivalently, by the [[main theorem of perturbative renormalization theory]] (theorem \ref{PerturbativeRenormalizationMainTheorem} below),
this is the freedom of choosing "[[counterterms]]" (remark \ref{TermCounter} below) for the [[local observable|local]] [[interactions]]. This we discuss [below](#ExistenceAndRenormalization) and in more detail in the [next chapter](#Renormalization).
=--
+-- {: .num_remark #CausalPerturbationTheoryAbsenceOfUVDivergences}
###### Remark
**(absence of [[ultraviolet divergences]] and [[renormalization|re-normalization]])**
The simple axioms of [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
do fully capture [[perturbative quantum field theory]] "in the ultraviolet": A solution to these axioms
induces, by definition, well-defined [[perturbative QFT|perturbative]] [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes})
and well-defined [[perturbative QFT|perturbative]] [[probability amplitudes]] of [[interacting field observables]] (def. \ref{InteractingFieldObservables}) induced by _[[local observables|local]]_ [[action functionals]] (describing point-interactions such as the [[electron-photon interaction]]). By the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) such solutions exist. This means that, while these are necessarily [[formal power series]] in $\hbar$ and $g$ (remark \ref{AsymptoticSeriesObservables}), all the [[coefficients]] of these formal power series ("[[loop order]] contributions") are well defined.
This is in contrast to the original informal conception of [[perturbative QFT]] due to [[Schwinger-Tomonaga-Feynman-Dyson]],
which in a first stage produced ill-defined [[divergence|diverging]] expressions for the [[coefficients]] (due to the mathematical error discussed in remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} below), which were then "[[renormalization|re-normalized]]" to finite values, by further informal arguments.
Here in [[causal perturbation theory]] no [[divergences]] in the [[coefficients]] of the [[formal power series]] are considered in the first place, all coefficients are well-defined, hence "finite". In this sense [[causal perturbation theory]] is about "finite" perturbative QFT, where instead of "re-normalization" of ill-defined expressions one just encounters "normalization" (prominently highlighted in [Scharf 95, see title, introduction, and section 4.3](causal+perturbation+theoryscatt#Scharf95)), namely compatible choices of these finite values. The actual "re-normalization" in the sense of "change of normalization" is expressed by the [[Stückelberg-Petermann renormalization group]].
This refers to those [[divergences]] that are known as _[[UV-divergences]]_, namely short-distance effects,
which are mathematically reflected in the fact that the perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
is defined on _[[local observables]]_, which, by their very locality, encode point-[[interactions]].
See also remark \ref{AdiabaticLimit} on _[[infrared divergences]]_.
=--
+-- {: .num_remark #WorldlineFormalism}
###### Remark
**([[virtual particles]], [[worldline formalism]] and [[perturbative string theory]])**
It is suggestive to think of the [[edges]] in the [[Feynman diagrams]] (def. \ref{FeynmanDiagram}) as [[worldlines]] of "[[virtual particles]]" and of the [[vertices]] as the points where they collide and transmute. (Care must be exercised not to confuse this with concepts of real [[particles]].) With this interpretation prop. \ref{FeynmanDiagramAmplitude} may be read as saying that the [[scattering amplitude]] for given external [[source fields]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) is the [[superposition]] of the [[Feynman amplitudes]] of all possible ways that these may interact; which is closely related to the intuition for the [[path integral]] (remark \ref{InterpretationOfPerturbativeSMatrix}).
This intuition is made precise by the _[[worldline formalism]]_ of [[perturbative quantum field theory]] ([Strassler 92](worldline+formalism#Strassler92)). This is the perspective on [[perturbative QFT]] which directly relates [[perturbative QFT]] to [[perturbative string theory]] ([Schmidt-Schubert 94](worldline+formalism#SchmidtSchubert94)). In fact the [[worldline formalism]] for [[perturbative QFT]] was originally found by taking thre point-particle limit of [[string scattering amplitudes]] ([Bern-Kosower 91](worldline+formalism#BernKosower91), [Bern-Kosower 92](worldline+formalism#BernKosower92)).
=--
+-- {: .num_remark #calSFunctionIsRenormalizationScheme}
###### Remark
**([[renormalization scheme]])**
Beware the terminology in def. \ref{LagrangianFieldTheoryPerturbativeScattering}: A _single_ S-matrix is one single observable
$$
\mathcal{S}(S_{int})
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [g,j] ]
$$
for a fixed ([[adiabatic switching|adiabatically switched]] [[local observable|local]]) [[interaction]] $S_{int}$, reflecting the [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) with respect to that particular interaction. Hence the function
$$
\mathcal{S}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [\hbar, g,j] ]\langle g, j
\rangle
\longrightarrow
PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g,j] ]
$$
axiomatized in def. \ref{LagrangianFieldTheoryPerturbativeScattering} is really a whole _scheme_ for constructing compatible S-matrices for _all_ possible (adiabatically switched, local) interactions at once.
Since the usual proof of the construction of such schemes of S-matrices involves _[[renormalization|("re"-)normalization]]_, the function $\mathcal{S}$ axiomatized by def. \ref{LagrangianFieldTheoryPerturbativeScattering} may also be referred to as a _[[renormalization scheme|("re"-)normalization scheme]]_.
This perspective on $\mathcal{S}$ as a [[renormalization scheme]] is amplified by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem})
wich states that the space of choices for $\mathcal{S}$ is a [[torsor]] over the [[Stückelberg-Petermann renormalization group]].
=--
+-- {: .num_remark}
###### Remark
**([[quantum anomalies]])**
The [[axioms]] for the [[S-matrix]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering}
(and similarly that for the [[time-ordered products]] below in def. \ref{TimeOrderedProduct})
are sufficient to imply a [[causally local net]] of perturbative [[interacting field algebras of quantum observables]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below), and thus its [[algebraic adiabatic limit]] (remark \ref{AdiabaticLimit}).
It does not guarantee, however, that the [[BV-BRST differential]] passes to those
[[algebras of quantum observables]], hence it does not guarantee that the [[infinitesimal symmetries of the Lagrangian]]
are respected by the [[quantization]] process (there may be "[[quantum anomalies]]").
The extra condition that does ensure this is the _[[quantum master Ward identity]]_ or _[[quantum master equation]]_.
This we discuss elsewhere.
Apart from [[gauge symmetries]] one also wants to require that rigid symmetries are preserved by the S-matrix, notably [[Poincare group]]-symmetry for scattering on [[Minkowski spacetime]].
=--
$\,$
**Interacting field observables**
{#LocalNetsOfInteractingFieldObservables}
We now discuss how the perturbative [[interacting field observables]] which are induced from an [[S-matrix]]
enjoy good properties expected of any abstractly defined [[perturbative algebraic quantum field theory]].
$\,$
+-- {: .num_defn #QuntumMollerOperator}
###### Definition
**([[interacting field algebra of observables]] -- [[quantum Møller operator]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].
We write
$$
LocIntObs_{\mathcal{S}}(E_{\text{BV-BRST}}, g S_{int})
\;\coloneqq\;
\left\{
{\, \atop \,}
A_{int} \;\vert\; A \in LocObs(E_{BV-BRST})[ [ \hbar, g ] ]
{\, \atop \,}
\right\}
\hookrightarrow
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$
for the subspace of [[interacting field observables]] $A_{int}$ (def. \ref{InteractingFieldObservables}) corresponding to [[local observables]] $A$,
the _[[local interacting field observables]]_.
Furthermore we write
$$
\array{
LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
&
\underoverset{\simeq}{\phantom{A}\mathcal{R}^{-1}\phantom{A}}{\longrightarrow}
&
IntLocObs(E_{\text{BV-BRST}}, g S_{int})[ [ \hbar , g ] ]
\\
A
&\mapsto&
A_{int}
\coloneqq
\mathcal{S}(g S_{int})^{-1} T( \mathcal{S}(g S_{int}), A )
}
$$
for the factorization of the function $A \mapsto A_{int}$ through its image, which, by remark \ref{PerturbativeSMatrixInverse},
is a [[linear isomorphism]] with [[inverse]]
$$
\array{
IntLocObs(E_{\text{BV-BRST}}, g S_{int})[ [ \hbar , g ] ]
&
\underoverset{\simeq}{\phantom{A}\mathcal{R}\phantom{A}}{\longrightarrow}
&
LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
\\
A_{int}
&\mapsto&
A
\coloneqq
T\left(
\mathcal{S}(-g S_{int})
,
\left( \mathcal{S}(g S_{int}) A_{int} \right)
\right)
}
$$
This may be called the _[[quantum Møller operator]]_ ([Hawkins-Rejzner 16, (33)](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16)).
Finally we write
$$
\begin{aligned}
IntObs(E_{\text{BV-BRST}}, S_{int})
& \coloneqq
\left\langle
{\, \atop \,}
IntLocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
{\, \atop \,}
\right\rangle
\\
&
\phantom{\coloneqq}
\hookrightarrow
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
\end{aligned}
$$
for the smallest subalgebra of the [[Wick algebra]] containing the [[interacting local observables]].
This is the _perturbative [[interacting field algebra of observables]]_.
=--
The definition of the [[interacting field algebra of observables]] from the data of a [[scattering matrix]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) via [[Bogoliubov's formula]] (def. \ref{InteractingFieldObservables}) is physically well-motivated, but is not immediately recognizable as the result of applying a systematic concept of [[quantization]] (such as [[formal deformation quantization]]) to the given [[Lagrangian field theory]]. The following
proposition \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} says that this is nevertheless the case.
(The special case of this statement for [[free field theory]] is discussed at _[[Wick algebra]]_, see remark \ref{WickAlgebraIsFormalDeformationQuantization}).
+-- {: .num_prop #InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}
###### Proposition
**([[interacting field algebra of observables]] is [[formal deformation quantization]] of [[interacting field theory|interacting]] [[Lagrangian field theory]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g_{sw} \mathbf{L}_{int} \in \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})[ [\hbar, g ] ]\langle g\rangle$ be an [[adiabatic switching|adiabatically switched]] [[interaction]] [[Lagrangian density]] with corresponding
[[action functional]] $g S_{int} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$.
Then, at least on [[regular polynomial observables]], the construction of perturbative [[interacting field algebras of observables]] in def. \ref{QuntumMollerOperator} is a [[formal deformation quantization]] of the [[interacting field theory|interacting]] [[Lagrangian field theory]]
$(E_{\text{BV-BRST}}, \mathbf{L}' + g_{sw} \mathbf{L}_{int})$.
=--
([Hawkins-Rejzner 16, prop. 5.4](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16), [Collini 16](pAQFT#Collini16))
The following definition collects the system (a [[co-presheaf]]) of [[generating functions]] for [[interacting field observables]]
which are localized in spacetime as the spacetime localization region varies:
+-- {: .num_defn #PerturbativeGeneratingLocalNetOfObservables}
###### Definition
**(system of spacetime-localized [[generating functions]] for [[interacting field observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
$$
be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]
$$
S_{int,sw}
\;\coloneqq\;
\tau_\Sigma(g_{sw} \mathbf{L}_{int})
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$
is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
For $\mathcal{O} \subset \Sigma$ a [[causally closed subset]] of [[spacetime]]
(def. \ref{CausalComplementOfSubsetOfLorentzianManifold})
and for $g_{sw} \in Cutoffs(\mathcal{O})$ an [[adiabatic switching]] function (def. \ref{CutoffFunctions})
which is constant on a [[neighbourhood]] of $\mathcal{O}$, write
$$
Gen(E_{\text{BV-BRST}}, S_{int,sw} )(\mathcal{O})
\;\coloneqq\;
\left\langle
\mathcal{Z}_{S_{int,sw}}(j A)
\;\vert\;
A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
\,\text{with}\,
supp(A) \subset \mathcal{O}
\right\rangle
\;\subset\;
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$
for the smallest subalgebra of the [[Wick algebra]] which contains the
[[generating functions]] (def. \ref{SchemeGeneratingFunction})
with respect to $S_{int,sw}$ for all those [[local observables]] $A$ whose spacetime support is in $\mathcal{O}$.
Moreover, write
$$
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
\;\subset\;
\underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod}
Gen(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
$$
be the subalgebra of the [[Cartesian product]] of all these algebras as $g_{sw}$ ranges over cutoffs, which is generated by the
[[tuples]]
$$
\mathcal{Z}_{\mathbf{L}_{int}}(A)
\;\coloneqq\;
\left(
\mathcal{Z}_{S_{int,sw}}(j A)
\right)_{g_{sw} \in Cutoffs(\mathcal{O})}
$$
for $A$ with $supp(A) \subset \mathcal{O}$.
We call $Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int} )(\mathcal{O})$ the _algebra of [[generating functions]] for [[interacting field observables]] localized in $\mathcal{O}$_.
Finally, for $\mathcal{O}_1 \subset \mathcal{O}_2$ an inclusion of two [[causally closed subsets]], let
$$
i_{\mathcal{O}_1, \mathcal{O}_2}
\;\colon\;
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1)
\longrightarrow
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
$$
be the algebra [[homomorphism]] which is given simply by restricting the index set of [[tuples]].
This construction defines a [[functor]]
$$
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})
\;\colon\;
CausClsdSubsets(\Sigma)
\longrightarrow
Algebras
$$
from the [[poset]] of [[causally closed subsets]] of [[spacetime]] to the [[category]] of [[algebras]].
> (extends to [[star algebras]] if scattering matrices are chosen unitary...)
=--
([Brunetti-Fredenhagen 99, (65)-(67)](S-matrix#BrunettiFredenhagen99))
The key technical fact is the following:
+-- {: .num_prop #IsomorphismFromChangeOfAdiabaticSwitching}
###### Proposition
**(localized [[interacting field observables]] independent of [[adiabatic switching]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
$$
be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]
$$
g S_{int,sw} \;\coloneqq\; \tau_\Sigma(g_{sw} \mathbf{L}_{int})
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$
is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
If two such [[adiabatic switchings]] $g_{sw,1}, g_{sw,2} \in C^\infty_{cp}(\Sigma)$ agree on a
[[causally closed subset]]
$$
\mathcal{O} \;\subset\; \Sigma
$$
in that
$$
g_{sw,1}\vert_{\mathcal{O}} = g_{sw,2}\vert_{\mathcal{O}}
$$
then there exists a [[microcausal polynomial observable]]
$$
K \;\in\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j ] ]
$$
such that for every [[local observable]]
$$
A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
$$
with spacetime support in $\mathcal{O}$
$$
supp(A) \;\subset\; \mathcal{O}
$$
the corresponding two [[generating functions]] (eq:GeneratingFunctionInducedFromSMatrix)
are related via [[conjugation]] by $K$:
$$
\label{AdiabaticSwitchingRelationGeneratingFunctions}
\mathcal{Z}_{S_{int,sw_2}}
\left(
j A
\right)
\;=\;
K^{-1}
\,
\left(
\mathcal{Z}_{S_{int,sw_1}}
\left(
j A
\right)
\right)
\,
K
\,.
$$
In particular this means that for every choice of [[adiabatic switching]] $g_{sw} \in Cutoffs(\mathcal{O})$
the algebra $Gen_{S_{int,sw}}(\mathcal{O})$ of [[generating functions]] for [[interacting field observables]]
computed with $g_{sw}$ is canonically [[isomorphism|isomorphic]] to the
abstract algebra $Gen_{\mathbf{L}_{int}}(\mathcal{O})$ (def. \ref{PerturbativeGeneratingLocalNetOfObservables}),
by the evident map on generators:
$$
\label{AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching}
\array{
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{o})
&\overset{\simeq}{\longrightarrow}&
Gen(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
\\
\left(
\mathcal{Z}_{S_{int,sw'}}
\right)_{g_{sw'} \in Cutoffs(\mathcal{O})}
&\mapsto&
\mathcal{Z}_{S_{int,sw}}
}
\,.
$$
=--
([Brunetti-Fredenhagen 99, prop. 8.1](S-matrix#BrunettiFredenhagen99))
+-- {: .proof}
###### Proof
By causal closure of $\mathcal{O}$, lemma \ref{CausalPartition}
says that there are [[bump functions]]
$$
a, r \in C^\infty_{cp}(\Sigma)\langle g \rangle
$$
which decompose the difference of [[adiabatic switchings]]
$$
g_{sw,2} - g_{sw,1} = a + r
$$
subject to the [[causal ordering]]
$$
supp(a) \,{\vee\!\!\!\wedge}\, \mathcal{O} \,{\vee\!\!\!\wedge}\, supp(r)
\,.
$$
With this the result follows from repeated use of [[causal additivity]] in its various equivalent
incarnations from prop. \ref{ZCausalAdditivity}:
$$
\begin{aligned}
& \mathcal{Z}_{g S_{int,sw_2}}(j A)
\\
& =
\mathcal{Z}_{
\left(
\tau_\Sigma
\left(
g_{sw,2}
\mathbf{L}_{int}
\right)
\right)
}
\left(
j A
\right)
\\
& =
\mathcal{Z}_{
\left(
\tau_\Sigma
\left(
(g_{sw,1} + a + r)\mathbf{L}_{int}
\right)
\right)
}
\left(
j A
\right)
\\
& =
\mathcal{Z}_{
\left(
g S_{int,sw_1}
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
+
\tau_\Sigma
\left(
a \mathbf{L}_{int}
\right)
\right)
}
\left(
j A
\right)
\\
& =
\mathcal{Z}_{
\left(
g S_{int,sw_1}
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
}
\left(
j A
\right)
\\
& =
\mathcal{S}
\left(
g S_{int,sw_1}
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)^{-1}
\,
\mathcal{S}
\left(
g S_{int,sw_1}
+
j A
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
\\
& =
\mathcal{S}
\left(
g S_{int,sw_1}
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)^{-1}
\,
\mathcal{S}
\left(
g S_{int,sw_1}
+
j A
\right)
\,
\mathcal{S}
\left(
g S_{int,sw_1}
\right)^{-1}
\,
\mathcal{S}
\left(
j A
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
\\
& =
\mathcal{S}
\left(
g S_{int,sw_1}
+
\tau_\Sigma
\left(
r\mathbf{L}_{int}
\right)
\right)^{-1}
\,
\underset{
= id
}{
\underbrace{
\mathcal{S}
\left(
g S_{int,sw_1}
\right)
\,
\mathcal{S}
\left(
g S_{int,sw_1}
\right)^{-1}
}
}
\,
\mathcal{S}
\left(
g S_{int,sw_1}
+
j A
\right)
\,
\mathcal{S}
\left(
g S_{int , sw_1}
\right)^{-1}
\,
\mathcal{S}
\left(
j A
+
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
\\
& =
\underset{
K^{-1}
}{
\underbrace{
\left(
\mathcal{Z}_{
g S_{int,sw_1}
}
\left(
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
\right)^{-1}
}
}
\,
\mathcal{Z}_{
g S_{int,sw_1}
}
\left(
j A
\right)
\,\,
\underset{
K
}{
\underbrace{
\mathcal{Z}_{
g S_{int,sw_1}
}
\left(
\tau_\Sigma
\left(
r \mathbf{L}_{int}
\right)
\right)
}}
\end{aligned}
$$
This proves the existence of elements $K$ as claimed.
It is clear that conjugation induces an algebra homomorphism,
and since the map is a linear isomorphism on the space of generators,
it is an algebra isomorphism on the algebras being generated (eq:AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching).
(While the elements $K$ in (eq:AdiabaticSwitchingRelationGeneratingFunctions) are far from being unique themselves,
equation (eq:AdiabaticSwitchingRelationGeneratingFunctions) says that the map on generators induced
by conjugation with $K$ is independent of this choice.)
=--
+-- {: .num_prop #GeneratingAlgebrasIsLocalNet}
###### Proposition
**(system of generating algebras is [[causally local net]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
$$
be a [[Lagrangian density]], to be thought of as an [[interaction]].
Then the system
$$
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})
\;\colon\;
CausCldSubsets(\Sigma)
\longrightarrow
Algebra
$$
of localized [[generating functions]] for [[interacting field observables]] (def. \ref{PerturbativeGeneratingLocalNetOfObservables})
is a _[[causally local net]]_ in that it satisfies the following conditions:
1. (isotony) For every inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$ of [[causally closed subsets]] of [[spacetime]]
the corresponding algebra homomorphism is a [[monomorphism]]
$$
i_{\mathcal{O}_1, \mathcal{O}_2}
\;\colon\;
Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1) \hookrightarrow Gen(E_{\text{BV-BRST}},\mathbf{L}_{int})(\mathcal{O}_2)
$$
1. ([[causal locality]]) For $\mathcal{O}_1, \mathcal{O}_2 \subset X$ two [[causally closed subsets]]
which are [[spacelike]] separated, in that their [[causal ordering]] (def. \ref{CausalOrdering}) satisfies
$$
\mathcal{O}_1 {\vee\!\!\!\wedge} \mathcal{O}_2
\;\text{and}\;
\mathcal{O}_2 {\vee\!\!\!\wedge} \mathcal{O}_1
$$
and for $\mathcal{O} \subset \Sigma$ any further [[causally closed subset]] which contains both
$$
\mathcal{O}_1 , \mathcal{O}_2 \subset \mathcal{O}
$$
then the corresponding images of the generating function algebras of interacting field observables
localized in $\mathcal{O}_1$ and in $\mathcal{O}_2$,
respectively, commute with each other as subalgebras of the generating function algebras of interacting
field observables localized in $\mathcal{O}$:
$$
\left[
i_{\mathcal{O}_1,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_1))
\;,\;
i_{\mathcal{O}_2,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_2))
\right]
\;=\;
0
\;\;\;
\in Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
\,.
$$
=--
([Dütsch-Fredenhagen 00, section 3](S-matrix#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](S-matrix#BrunettiFredenhagen99),
[Il'in-Slavnov 78](S-matrix#IlinSlavnov78))
+-- {: .proof}
###### Proof
Isotony is immediate from the definition of the algebra homomorphisms in def. \ref{PerturbativeGeneratingLocalNetOfObservables}.
By the isomorphism (eq:AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching) we may check causal localizy with respect to any
choice of [[adiabatic switching]] $g_{sw} \in Cautoff(\mathcal{O})$ constant over $\mathcal{O}$. For this
the statement follows, with the assumption of spacelike separation, by [[causal additivity]] (prop. \ref{ZCausalAdditivity}):
For $supp(A_1) \subset \mathcal{O}_1$ and $supp(A_2) \subset \mathcal{O}_2$ we have:
$$
\begin{aligned}
\mathcal{Z}_{g S_{int,sw}}( j A_1 )
\mathcal{Z}_{g S_{int,sw}}( j A_2 )
& =
\mathcal{S}_{g S_{int,sw}}( j A_1 + j A_2)
\\
& =
\mathcal{S}_{g S_{int,sw}}( j A_2 + j A_1)
\\
& =
\mathcal{Z}_{g S_{int,sw}}( j A_2 )
\mathcal{Z}_{g S_{int,sw}}( j A_1 )
\end{aligned}
$$
=--
With the [[causally local net]] of localized [[generating functions]] for
[[interacting field observables]] in hand, it is now immediate to get the
+-- {: .num_defn #SystemOfAlgebrasOfQuantumObservables}
###### Definition
**(system of [[interacting field algebras of observables]])
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
$$
be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]
$$
g S_{int,sw} \;\coloneqq\; g \tau_\Sigma(g_{sw} \mathbf{L}_{int})
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle
$$
is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
For $\mathcal{O} \subset \Sigma$ a [[causally closed subset]]
of [[spacetime]] (def. \ref{CausalComplementOfSubsetOfLorentzianManifold})
and for $g_{sw} \in Cutoffs(\mathcal{O})$ an compatible [[adiabatic switching]] function (def. \ref{CutoffFunctions}) write
$$
IntObs(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
\coloneqq
\left\langle
i \hbar \frac{d}{d j}
\mathcal{Z}_{S_{int}}(j A)\vert_{j = 0} \;\vert\; supp(A) \subset \mathcal{O}
\right\rangle
\;\subset\;
PolyObs((\hbar))[ [ g ] ]
$$
for the [[interacting field algebra of observables]] (def. \ref{QuntumMollerOperator}) with spacetime support in $\mathcal{O}$.
Let then
$$
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
\subset
\underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod} IntObs(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
$$
be the subalgebra of the [[Cartesian product]] of all these algebras as $g_{sw}$ ranges, which is generated by the
[[tuples]]
$$
i \hbar
\frac{d}{d j } \mathcal{Z}_{\mathbf{L}_{int}}\vert_{j = 0}
\;\coloneqq\;
\left(
i \hbar
\frac{d}{d j } \mathcal{Z}_{S_{int,sw}} (j A)\vert_{j = 0}
\right)_{g_{sw} \in Cutoffs(\mathcal{O})}
$$
for $supp(A) \subset \mathcal{O}$.
Finally, for $\mathcal{O}_1 \subset \mathcal{O}_2$ an inclusion of two [[causally closed subsets]], let
$$
i_{\mathcal{O}_1, \mathcal{O}_2}
\;\colon\;
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1)
\longrightarrow
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
$$
be the algebra [[homomorphism]] which is given simply by restricting the index set of [[tuples]].
This construction defines a [[functor]]
$$
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int}) \;\colon\; CausClsdSubsets(\Sigma) \longrightarrow Algebras
$$
from the [[poset]] of [[causally closed subsets]] in the [[spacetime]] $\Sigma$ to the [[category]] of [[star algebras]].
=--
Finally, as a direct corollary of prop. \ref{GeneratingAlgebrasIsLocalNet}, we obtain the key result:
+-- {: .num_prop #PerturbativeQuantumObservablesIsLocalnet}
###### Proposition
**(system of [[interacting field algebras of observables]] is [[causally local net|causally local]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let
$$
\mathbf{L}_{int}
\;\in\;
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
\,.
$$
be a [[Lagrangian density]], to be thought of as an [[interaction]],
then the system of [[algebras of observables]] $Obs_{L_{int}}$ (def. \ref{SystemOfAlgebrasOfQuantumObservables})
is a [[local net of observables]] in that
1. (isotony) For every inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$ of [[causally closed subsets]]
the corresponding algebra homomorphism is a [[monomorphism]]
$$
i_{\mathcal{O}_1, \mathcal{O}_2}
\;\colon\;
IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1) \hookrightarrow IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
$$
1. ([[causal locality]]) For $\mathcal{O}_1, \mathcal{O}_2 \subset X$ two [[causally closed subsets]]
which are [[spacelike]] separated, in that their [[causal ordering]] (def. \ref{CausalOrdering}) satisfies
$$
\mathcal{O}_1 {\vee\!\!\!\wedge} \mathcal{O}_2
\;\text{and}\;
\mathcal{O}_2 {\vee\!\!\!\wedge} \mathcal{O}_1
$$
and for $\mathcal{O} \subset \Sigma$ any further causally closed subset which contains both
$$
\mathcal{O}_1 , \mathcal{O}_2 \subset \mathcal{O}
$$
then the corresponding images of the generating algebras of $\mathcal{O}_1$ and $\mathcal{O}_2$,
respectively, commute with each other as subalgebras of the generating algebra of $\mathcal{O}$:
$$
\left[
i_{\mathcal{O}_1,\mathcal{O}}(Obs_{\mathbf{L}_{int}}(\mathcal{O}_1))
\;,\;
i_{\mathcal{O}_2,\mathcal{O}}(Obs_{\mathbf{L}_{int}}(\mathcal{O}_2))
\right]
\;=\;
0
\;\;\;
\in IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
\,.
$$
=--
([Dütsch-Fredenhagen 00, below (17)](S-matrix#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](S-matrix#BrunettiFredenhagen99),
[Il'in-Slavnov 78](S-matrix#IlinSlavnov78))
+-- {: .proof}
###### Proof
The first point is again immediate from the definition (def. \ref{SystemOfAlgebrasOfQuantumObservables}).
For the second point it is sufficient to check the commutativity relation on generators. For these
the statement follows with prop. \ref{GeneratingAlgebrasIsLocalNet}:
$$
\begin{aligned}
& \left[
i \hbar \frac{d}{d j} \mathcal{Z}_{S_{int,sw}}(j A_1)\vert_{j = 0}
\;,\;
i \hbar \frac{d}{d j} \mathcal{Z}_{S_{int,sw}}(j J_2)\vert_{j = 0}
\right]
\\
& =
(i \hbar)^2
\frac{
\partial^2
}{
\partial j_1 \partial j_2
}
\underset{ = 0}{
\underbrace{
\left[
\mathcal{Z}_{S_{int,sw}}(j_1 A_1)
\;,\;
\mathcal{Z}_{S_{int,sw}}(j_1 A_2)
\right]}}_{ \left\vert { {j_1 = 0} \atop {j_2 = 0} } \right. }
\\
& = 0
\end{aligned}
$$
=--
$\,$
**[[time-ordered products]]**
{#TimeOrderedProducts}
Definition \ref{LagrangianFieldTheoryPerturbativeScattering} suggests to focus on the multilinear operations $T(...)$ which define the perturbative [[S-matrix]] order-by-order in $\hbar$. We impose [[axioms]] on these _[[time-ordered products]]_ directly (def. \ref{TimeOrderedProduct}) and then prove that these axioms imply the axioms for the corresponding [[S-matrix]] (prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below).
+-- {: .num_defn #TimeOrderedProduct}
###### Definition
**([[time-ordered products]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
A _[[time-ordered product]]_ is a sequence of [[multilinear map|multi-]][[linear continuous functionals]] for all $k \in \mathbb{N}$ of the form
$$
T_k
\;\colon\;
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\langle g,j \rangle
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$
(from [[tensor products]] of [[local observables]] to [[microcausal polynomial observables]], with formal parameters adjoined
according to def. \ref{FormalParameters})
such that the following conditions hold for all possible arguments:
1. (normalization)
$$
T_0(O) = 1
$$
1. (perturbation)
$$
T_1(O) = :O:
$$
1. (symmetry) each $T_k$ is symmetric in its arguments, in that for every [[permutation]]
$\sigma \in \Sigma(k)$ of $k$ elements
$$
T_k(O_{\sigma(1)}, O_{\sigma(2)}, \cdots, O_{\sigma(k)})
\;=\;
T_k(O_1, O_2, \cdots, O_k)
$$
1. ([[causal factorization]]) If the spacetime support (def. \ref{SpacetimeSupport}) of [[local observables]] satisfies the [[causal ordering]] (def. \ref{CausalOrdering})
$$
\left(
{\, \atop \,}
supp(O_1)
\cup
\cdots
\cup
supp(O_r)
{\, \atop \,}
\right)
\;{\vee\!\!\!\wedge}\;
\left(
{\, \atop \,}
supp(O_{r+1})
\cup
\cdots
\cup
supp(O_k)
{\, \atop \,}
\right)
$$
then the time-ordered product of these $k$ arguments factors as
the [[Wick algebra]]-product of the time-ordered product of the first $r$ and that of the second $k-r$ arguments:
$$
T(O_1, \cdots, O_k)
\; = \;
T( O_1, \cdots , O_r )
\,
T( O_{r+1}, \cdots , O_k )
\,.
$$
=--
+-- {: .num_example #TimeOrderedProductsFromSMatrixScheme}
###### Example
**([[S-matrix]] scheme implies [[time-ordered products]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and let
$$
\mathcal{S}
\;=\;
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}\frac{1}{(i \hbar)^k}
T_k
$$
be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.
Then the $\{T_k\}_{k \in \mathbb{N}}$ are [[time-ordered products]] in the sense of def. \ref{TimeOrderedProduct}.
=--
+-- {: .proof}
###### Proof
We need to show that the $\{T_k\}_{k \in \mathbb{N}}$ satisfy [[causal factorization]].
For
$$
O_j\;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
$$
a local observable, consider the continuous linear function that muliplies this by any [[real number]]
$$
\array{
\mathbb{R}
&\longrightarrow&
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
\\
\kappa_j &\mapsto& \kappa_j O_j
}
\,.
$$
Since the $T_k$ by definition are [[continuous linear functionals]], they are in particular
[[differentiable maps]], and hence so is the S-matrix $\mathcal{S}$. We may extract $T_k$ from
$\mathcal{S}$ by [[differentiation]] with respect to the parameters $\kappa_j$ at $\kappa_j = 0$:
$$
T_k(O_1, \cdots, O_k)
\;=\;
\frac{\partial^k}{
\partial \kappa_1
\cdots
\partial \kappa_k
}
\mathcal{S}\left( \kappa_1 O_1 + \cdots + \kappa_k O_k \right)\vert_{\kappa_1, \cdots, \kappa_k = 0}
$$
for all $k \in \mathbb{N}$.
Now the [[causal additivity]] of the S-matrix $\mathcal{S}$ implies its [[causal factorization]] (remark \ref{DysonCausalFactorization})
and this implies the causal factorization of the $\{T_k\}$ by the [[product law]] of [[differentiation]]:
$$
\begin{aligned}
T_k(O_1, \cdots, O_k)
& =
(i \hbar)^k
\frac{\partial^k}{
\partial \kappa_1
\cdots
\partial \kappa_k
}
\mathcal{S}\left( \kappa_1 O_1 + \cdots + \kappa_k O_k \right)\vert_{\kappa_1, \cdots, \kappa_k = 0}
\\
& =
(i \hbar)^k
\frac{\partial^k}{
\partial \kappa_1
\cdots
\partial \kappa_k
}
\left(
{\, \atop \,}
\mathcal{S}(\kappa_1 O_1 + \cdots + \kappa_r O_r)
\,
\mathcal{S}(\kappa_{r+1} O_{r+1} + \cdots + \kappa_k O_k)
{\, \atop \,}
\right)
\vert_{\kappa_1, \cdots, \kappa_k = 0}
\\
& =
(i \hbar)^r
\frac{\partial^r}{
\partial \kappa_1
\cdots
\partial \kappa_r
}
\mathcal{S}(\kappa_1 O_1 + \cdots + \kappa_r O_r)
\vert_{\kappa_1, \cdots, \kappa_r = 0}
\;
(i \hbar)^{k-r}
\frac{\partial^{k-r}}{
\partial \kappa_{r+1}
\cdots
\partial \kappa_k
}
\mathcal{S}(\kappa_{r+1} O_{r+1} + \cdots + \kappa_k O_k)
\vert_{\kappa_{r+1}, \cdots, \kappa_k = 0}
\\
& =
T_{r}( O_1, \cdots, O_{r} )
\,
T_{k-r}( O_{r+1}, \cdots, O_{k} )
\end{aligned}
\,.
$$
=--
The converse implication, that [[time-ordered products]] induce an [[S-matrix]] scheme involves
more work (prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below).
+-- {: .num_remark #NotationForTimeOrderedProductsAsGeneralizedFunctions}
###### Remark
**([[time-ordered products]] as [[generalized functions]])**
It is convenient (as in [Epstein-Glaser 73](S-matrix#EpsteinGlaser73)) to think of [[time-ordered products]] (def. \ref{TimeOrderedProduct}), being
[[Wick algebra]]-valued [[distributions]] (hence [[operator-valued distributions]] if we were to choose a [[representation]] of the
[[Wick algebra]] by [[linear operators]] on a [[Hilbert space]]), as [[generalized functions]] depending on spacetime points:
If
$$
\left\{
\alpha_i \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})\langle g \rangle
\right\}
\cup
\left\{
\beta_j \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})\langle j \rangle
\right\}
$$
is a [[finite set]] of [[horizontal differential forms]], and
$$
\left\{
g_i, j_{j} \in C^\infty_{cp}(\Sigma)
\right\}
$$
is a corresponding set of [[bump functions]] on [[spacetime]] ([[adiabatic switchings]]), so that
$$
\left\{
S_j
\colon
\Phi
\mapsto
\underset{\Sigma}{\int} g_i(x) \, \left(j^\infty_\Sigma(\Phi)^\ast \alpha_i\right)(x)\, dvol_\Sigma(x)
\right\}
\;\cup\;
\left\{
A_j
\colon
\Phi
\mapsto
\underset{\Sigma}{\int} j_i(x) \, \left(j^\infty_\Sigma(\Phi)^\ast \beta_i\right)(x)\, dvol_\Sigma(x)
\right\}
$$
is the corresponding set of [[local observables]], then we may write the [[time-ordered product]]
of these observables as the [[integration]] of these [[bump functions]] against a [[generalized function]]
$T_{(\alpha_i)}$ with values in the [[Wick algebra]]:
$$
\begin{aligned}
&
\underset{\Sigma^n}{\int}
T_{(\alpha_i), (\beta_j)}(x_1, \cdots, x_{r}, x_{r+1}, \cdots x_{n})
g_1(x_1) \cdots g_r(x_r)
\,
j_1(x_{r+1}) \cdots j_n(x_n)
\, dvol_{\Sigma^n}(x_1, \cdots x_n)
\\
& \coloneqq
T( S_1, \cdots, S_r, A_{r+1}, \cdots, A_n )
\end{aligned}
\,.
$$
Moreover, the subscripts on these [[generalized functions]] will always be
clear from the context, so that in computations we may notationally suppress these.
Finally, due to the "symmetry" axiom in def. \ref{TimeOrderedProduct}, a time-ordered product
depends, up to signs, only on its [[set]] of arguments, not on the order of the arguments. We will write
$\mathbf{X} \coloneqq \{x_1, \cdots, x_r\}$ and $\mathbf{Y} \coloneqq \{y_1, \cdots y_r\}$
for sets of spacetime points, and hence abbreviate the expression for the "value" of the
generalized function in the above as $T(\mathbf{X}, \mathbf{Y})$ etc.
In this condensed notation the above reads
$$
\underset{\Sigma^{r+s}}{\int}
T(\mathbf{X}, \mathbf{Y})
\,
g_1(x_1) \cdots g_r(x_r)
j_{r+1}(x_{r+1}) \cdots j_n(x_n)
\,
dvol_{\Sigma^{r+s}}(\mathbf{X})
\,.
$$
=--
This condensed notation turns out to be greatly simplify computations, as it absorbs all the "relative"
combinatorial prefactors:
+-- {: .num_example #ProductOfPerturbationSeriesInGenealizedFunctionNotation}
###### Example
**(product of perturbation series in [[generalized function]]-notation)**
Let
$$
U(g)
\coloneqq
\underoverset{n = 0}{\infty}{\sum}
\frac{1}{n!}
\int U(x_1, \cdots, x_n)
\,
g(x_1) \cdots g(x_n) \, dvol
$$
and
$$
V(g)
\coloneqq
\underoverset{n = 0}{\infty}{\sum}
\frac{1}{n!}
\int V(x_1, \cdots, x_n)
\,
g(x_1) \cdots g(x_n) \, dvol
$$
be power series of [[Wick algebra]]-valued [[distributions]] in the [[generalized function]]-notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}.
Then their product $W(g) \coloneqq U(g) V(g)$ with [[generalized function]]-representation
$$
W(g)
\coloneqq
\underoverset{n = 0}{\infty}{\sum}
\frac{1}{n!}
\int W(x_1, \cdots, x_n)
\,
g(x_1) \cdots g(x_n) \, dvol
$$
is given simply by
$$
W(\mathbf{X})
\;=\;
\underset{\mathbf{I} \subset \mathbf{X}}{\sum} U(\mathbf{I}) V(\mathbf{X} \setminus \mathbf{I})
\,.
$$
=--
([Epstein-Glaser 73 (5)](S-matrix#EpsteinGlaser73))
+-- {: .proof}
###### Proof
For fixed [[cardinality]] ${\vert \mathbf{I} \vert} = n_1$
the sum over all subsets $\mathbf{I} \subset \mathbf{X}$ overcounts the sum over
[[partitions]] of the coordinates as $(x_1, \cdots x_{n_1}, x_{n_1 + 1}, \cdots x_n)$ precisely by the
[[binomial coefficient]] $\frac{n!}{n_1! (n - n_1) !}$. Here the factor of $n!$ cancels
against the "global" combinatorial prefactor in the above expansion of $W(g)$, while the remaining
factor $\frac{1}{n_1! (n - n_1) !}$ is just the "relative" combinatorial prefactor seen at total order $n$
when expanding the product $U(g)V(g)$.
=--
In order to prove that the axioms for [[time-ordered products]] do imply those for a perturbative [[S-matrix]]
(prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below) we need to consider
the corresponding reverse-time ordered products:
+-- {: .num_defn #ReverseTimeOrderedProduct}
###### Definition
**([[reverse-time ordered products]])**
Given a [[time-ordered product]] $T = \{T_k\}_{k \in \mathbb{N}}$ (def. \ref{TimeOrderedProduct}),
its _[[reverse-time ordered product]]_
$$
\overline{T}_k
\;\colon\;
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
{\, \atop \,}
\right)
\longrightarrow
PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g, j] ]
$$
for $k \in \mathbb{N}$ is defined by
$$
\overline{T}( A_1 \cdots A_n )
\;\coloneqq\;
\left\{
\array{
\underoverset{r = 1}{n}{\sum}
(-1)^r
\underset{\sigma \in Unshuffl(n,r)}{\sum}
T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
\,
T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
\cdots
T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} )
&\vert& k \geq 1
\\
1 &\vert& k = 0
}
\right.
\,,
$$
where the sum is over all [[unshuffles]] $\sigma$ of $(1 \leq \cdots \leq n)$ into $r$ non-empty
ordered subsequences. Alternatively, in the [[generalized function]]-notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}, this reads
$$
\overline{T}( \mathbf{X} )
=
\underoverset{r = 1}{{\vert \mathbf{X} \vert}}{\sum}
(-1)^r
\underset{ \array{
\mathbf{I}_1, \cdots, \mathbf{I}_r \neq \emptyset
\\
\underset{j \neq k}{\forall}\left( \mathbf{I}_j \cap \mathbf{I}_k = \emptyset \right)
\\
\mathbf{I}_1 \cup \cdots \cup \mathbf{I}_r = \mathbf{X}
} }{\sum} T( \mathbf{I}_1 ) \cdots T(\mathbf{I}_r)
$$
=--
([Epstein-Glaser 73, (11)](S-matrix#EpsteinGlaser73))
+-- {: .num_prop #ReverseTimOrderedProductsGiveReverseSMatrix}
###### Proposition
**([[reverse-time ordered products]] express [[inverse]] [[S-matrix]])
Given [[time-ordered products]] $T(-)$ (def. \ref{TimeOrderedProduct}), then the
corresponding reverse time-ordered product $\overline{T}(-)$ (def. \ref{ReverseTimeOrderedProduct})
expresses the [[inverse]] $S(-)^{-1}$ (according to remark \ref{PerturbativeSMatrixInverse}) of the corresponding
perturbative [[S-matrix]] scheme $\mathcal{S}(S_{int}) \coloneqq \underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} T(\underset{k\,\text{args}}{\underbrace{S_{int}, \cdots , S_{int}}})$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}):
$$
\left(
{\, \atop \,}
\mathcal{S}(g S_{int} + j A )
{\, \atop \,}
\right)^{-1}
\;=\;
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}
\left(
\frac{1}{i \hbar}
\right)^k
\overline{T}( \underset{k \, \text{arguments}}{\underbrace{ (g S_{int} + j A), \cdots, (g S_{int} + j A)}} )
\,.
$$
=--
+-- {: .proof}
###### Proof
For brevity we write just "$A$" for $\tfrac{1}{i \hbar}(g S_{int} + j A)$.
(Hence we assume without restriction that $A$ is not independent of
powers of $g$ and $j$; this is just for making all sums in the following be order-wise finite sums.)
By definition we have
$$
\begin{aligned}
&
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}
\overline{T}( \underset{k \, \text{args}}{\underbrace{A, \cdots , A}} )
\\
& =
\underset{ k \in \mathbb{N}}{\sum}
\frac{1}{k!}
\underoverset{r = 1}{k}{\sum}
(-1)^r
\!\!\!\underset{\sigma \in Unshuffl(k,r)}{\sum}\!\!\!
T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
\cdots
T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} )
\end{aligned}
$$
where all the $A_k$ happen to coincide: $A_k = A$.
If instead of [[unshuffles]] (i.e. [[partitions]] into non-empty subsequences preserving the original order) we took
partitions into arbitrarily ordered subsequences,
we would be overcounting by the [[factorial]] of the length of the subsequences, and hence the
above may be equivalently written as:
$$
\cdots
=
\underset{k \in \mathbb{N}}{\sum}
\tfrac{1}{k!}
\underoverset{r = 1}{k}{\sum}
(-1)^r
\!\!\!
\underset{ {\sigma \in \Sigma(k)} \atop { { k_1 + \cdots + k_r = k } \atop { \underset{i}{\forall} (k_i \geq 1) } } }{\sum}
\!\!\!
\tfrac{1}{k_1!} \cdots \tfrac{1}{k_r !}
\,
T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
\,
T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
\cdots
T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} )
\,,
$$
where $\Sigma(k)$ denotes the [[symmetric group]] (the set of all [[permutations]] of $k$ elements).
Moreover, since all the $A_k$ are equal, the sum is in fact independent of $\sigma$, it only depends
on the length of the subsequences. Since there are $k!$ permutations of $k$ elements
the above reduces to
$$
\begin{aligned}
\cdots
& =
\underset{k \in \mathbb{N}}{\sum}
\underoverset{r = 1}{k}{\sum}
(-1)^r
\!\!\!
\underset{ k_1 + \cdots + k_r = k }{\sum}
\tfrac{1}{k_1!} \cdots \tfrac{1}{k_r !}
T( \underset{k_1 \, \text{factors}}{\underbrace{ A, \cdots , A }} )
T( \underset{k_2 \, \text{factors}}{\underbrace{ A, \cdots , A }} )
\cdots
T( \underset{k_r \, \text{factors}}{\underbrace{ A, \cdots , A }} )
\\
& =
\underoverset{r = 0}{\infty}{\sum}
\left(
- \underoverset{k = 0}{\infty}{\sum} T ( \underset{k\,\text{factors}}{\underbrace{A, \cdots , A}} )
\right)^r
\\
& =
\mathcal{S}(A)^{-1}
\,,
\end{aligned}
$$
where in the last line we used (eq:InfverseOfPerturbativeSMatrix).
=--
In fact prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix} is a special case of the
following more general statement:
+-- {: .num_prop #InversionFormulaForTimeOrderedProducts}
###### Proposition
**(inversion relation for [[reverse-time ordered products]])
Let $\{T_k\}_{k \in \mathbb{N}}$ be [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then the [[reverse-time ordered products]] according to def. \ref{ReverseTimeOrderedProduct}
satisfies the following inversion relation for all $\mathbf{X} \neq \emptyset$
(in the condensed notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}):
$$
\underset{\mathbf{J} \subset \mathbf{X}}{\sum}
T(\mathbf{J}) \overline{T}(\mathbf{X} \setminus \mathbf{J})
\;=\;
0
$$
and
$$
\underset{\mathbf{J} \subset \mathbf{X}}{\sum}
\overline{T}(\mathbf{X} \setminus \mathbf{J}) T(\mathbf{J})
\;=\;
0
$$
=--
+-- {: .proof}
###### Proof
This is immediate from unwinding the definitions.
=--
+-- {: .num_prop #ReverseCausalFactorizationOfReverseTimeOrderedProducts}
###### Proposition
**(reverse [[causal factorization]] of [[reverse-time ordered products]])**
Let $\{T_k\}_{k \in \mathbb{N}}$ be [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then the reverse-time ordered products according to def. \ref{ReverseTimeOrderedProduct} satisfies
reverse-[[causal factorization]].
=--
([Epstein-Glaser 73, around (15)](S-matrix#EpsteinGlaser73))
+-- {: .proof}
###### Proof
In the condensed notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions},
we need to show that for $\mathbf{X} = \mathbf{P} \cup \mathbf{Q}$ with $\mathbf{P} \cap \mathbf{Q} = \emptyset$
then
$$
\left(
\mathbf{P} {\vee\!\!\!\wedge} \mathbf{Q}
\right)
\;\Rightarrow\;
\left(
\overline{T}(\mathbf{X})
=
\overline{T}(\mathbf{Q}) \overline{T}(\mathbf{P})
\right)
\,.
$$
We proceed by [[induction]]. If ${\vert \mathbf{X}\vert} = 1$ the statement is immediate.
So assume that the statement is true for sets of [[cardinality]] $n \geq 1$ and consider
$\mathbf{X}$ with ${\vert \mathbf{X}\vert} = n+1$.
We make free use of the condensed notation as in example \ref{ProductOfPerturbationSeriesInGenealizedFunctionNotation}.
From the formal inversion
$$
\underset{\mathbf{J} \subset \mathbf{X}}{\sum} \overline{T}(\mathbf{J}) T(\mathbf{X}\setminus \mathbf{J})
= 0
$$
(which uses the induction assumption that ${\vert \mathbf{X}\vert} \geq 1$) it follows that
$$
\begin{aligned}
\overline{T}(\mathbf{X})
& =
- \underset{ { \mathbf{J} \subset \mathbf{X} } \atop { \mathbf{J} \neq \mathbf{X} } }{\sum}
\overline{T}(\mathbf{J}) T( \mathbf{X} \setminus \mathbf{J} )
\\
& =
- \underset{
{ \mathbf{J} \cup \mathbf{J}' = \mathbf{X} }
\atop
{
{ \mathbf{J} \cap \mathbf{J}' = \emptyset }
\atop
{ \mathbf{J}' \neq \emptyset }
}
}{\sum}
\overline{T}( \mathbf{Q} \cap \mathbf{J} )
\overline{T}( \mathbf{P} \cap \mathbf{J} )
T ( \mathbf{P} \cap ( \mathbf{J}' ) )
T ( \mathbf{Q} \cap ( \mathbf{J}' ) )
\\
& =
- \underset{
{ \mathbf{L} \cup \mathbf{L}' = \mathbf{Q} \,,\, \mathbf{L} \cap \mathbf{L}' = \emptyset }
\atop
{ \mathbf{L}' \neq \emptyset }
}{\sum}
\!\!\!
\overline{T}( \mathbf{L} )
\underset{ = 0}{
\underbrace{
\left(
\underset{
\mathbf{K} \subset \mathbf{P}
}{\sum}
\overline{T}( \mathbf{K} ) T( \mathbf{P} \setminus \mathbf{K})
\right)
}
}
T(\mathbf{L'})
-
\overline{T}(\mathbf{Q})
\underset{
= - \overline{T}(\mathbf{P})
}{
\underbrace{
\underset{
{\mathbf{K} \subset \mathbf{P}}
\atop
{ \mathbf{K} \neq \emptyset }
}{\sum}
\overline{T}(\mathbf{K})
T (\mathbf{P} \setminus \mathbf{K} )
}}
\\
& =
\overline{T}(\mathbf{Q}) \overline{T}(\mathbf{P})
\end{aligned}
\,.
$$
Here
1. in the second line we used that $\mathbf{X} = \mathbf{Q} \sqcup \mathbf{P}$, together with the
[[causal factorization]] property of $T(-)$ (which holds by def. \ref{TimeOrderedProduct}) and that of $\overline{T}(-)$
(which holds by the induction assumption, using that $\mathbf{J} \neq \mathbf{X}$ hence that
${\vert \mathbf{J}\vert} \lt {\vert \mathbf{X}\vert}$).
1. in the third line we decomposed the sum over $\mathbf{J}, \mathbf{J}' \subset \mathbf{X}$
into two sums over subsets of $\mathbf{Q}$ and $\mathbf{P}$:
1. The first summand in the third line is the contribution where $\mathbf{J}'$ has a non-empty intersection with $\mathbf{Q}$. This makes $\mathbf{K}$ range without constraint, and therefore the sum in the middle vanishes, as indicated, as it is the contribution at order ${\vert \mathbf{Q}\vert}$ of the inversion formula from prop. \ref{InversionFormulaForTimeOrderedProducts}.
1. The second summand in the third line is the contribution where $\mathbf{J}'$ does not intersect $\mathbf{Q}$.
Now the sum over $\mathbf{K}$ is the inversion formula from prop. \ref{InversionFormulaForTimeOrderedProducts} except for one term, and so it equals that term.
=--
Using these facts about the reverse-time ordered products,
we may finally prove that [[time-ordered products]] indeed do induced a perturbative S-matrix:
+-- {: .num_prop #TimeOrderedProductInducesPerturbativeSMatrix}
###### Proposition
**([[time-ordered products]] induce [[S-matrix]])**
Let $\{T_k\}_{k \in \mathbb{N}}$ be a system of [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then
$$
\begin{aligned}
\mathcal{S}(-)
& \coloneqq
T
\left(
\exp_\otimes
\left(
\tfrac{1}{i \hbar}(-)
\right)
\right)
\\
&
\coloneqq
\underset{k \in \mathbb{N}}{\sum}
\tfrac{1}{k!}
\tfrac{1}{(i \hbar)^k} T( \underset{k \, \text{factors}}{\underbrace{-, \cdots , -}} )
\end{aligned}
$$
is indeed a perturbative S-matrix according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.
=--
+-- {: .proof}
###### Proof
The axiom "perturbation" of the S-matrix is
immediate from the axioms "perturbation" and "normalization" of the time-ordered products.
What requires proof is that [[causal additivity]] of the S-matrix follows from
the [[causal factorization]] property of the time-ordered products.
Notice that also the weaker [[causal factorization]] property of the S-matrix (remark \ref{DysonCausalFactorization})
is immediate from the causal factorization condition on the time-ordered products.
But [[causal additivity]] is stronger. It is remarkable that this, too,
follows from just the time-ordering ([Epstein-Glaser 73, around (73)](S-matrix#EpsteinGlaser73)):
To see this, first expand the
generating function $\mathcal{Z}$ (eq:GeneratingFunctionInducedFromSMatrix) into
powers of $g$ and $j$
$$
\mathcal{Z}_{g S_{int}}(j A)
\;=\;
\underoverset{n,m = 0}{\infty}{\sum}
\frac{1}{n! m!}
R\left(
{\, \atop \,}
\underset{n\, \text{factors}}{\underbrace{g S_{int}, \cdots ,g S_{int}}},
( \underset{m \, \text{factors}}{ \underbrace{ j A , \cdots , j A } } )
{\, \atop \,}
\right)
$$
and then compare order-by-order with the given time-ordered product $T$ and
its induced reverse-time ordered product (def. \ref{ReverseTimeOrderedProduct}) via prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix}.
(These $R(-,-)$ are also called the "generating [[retarded products]], discussed in their own right around def. \ref{RetardedProductFromPerturbativeSMatrix} below.)
In the condensed notation of
remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}
and its way of absorbing combinatorial prefactors as in example \ref{ProductOfPerturbationSeriesInGenealizedFunctionNotation}
this
yields at order $(g/\hbar)^{\vert \mathbf{Y}\vert} (j/\hbar)^{\vert \mathbf{X}\vert}$ the coefficient
$$
\label{CoefficientOfgeneratingRetardedProduct}
R(\mathbf{Y}, \mathbf{X})
\;=\;
\underset{\mathbf{I} \subset \mathbf{Y}}{\sum}
\overline{T}(\mathbf{I})
T( (\mathbf{Y} \setminus \mathbf{I}) , \mathbf{X} )
\,.
$$
We claim now that the [[support of a distribution|support]] of $R$ is inside the subset for which
$\mathbf{Y}$ is in the [[causal past]] of $\mathbf{X}$. This will imply the claim, because by
multi-linearity of $R(-,-)$ it then follows that
$$
\left(supp(A_1) {\vee\!\!\!\wedge} supp(A_2)\right)
\Rightarrow
\left( Z_{(g S_{int} + j A_1)}(j A_2) = Z_{S_{int}}(A_2) \right)
$$
and by prop. \ref{ZCausalAdditivity} this is equivalent to [[causal additivity]] of the S-matrix.
It remains to prove the claim:
Consider $\mathbf{X}, \mathbf{Y} \subset \Sigma$ such that the subset $\mathbf{P} \subset \mathbf{Y}$
of points not in the past of $\mathbf{X}$, hence the maximal subset with [[causal ordering]]
$$
\mathbf{P} {\vee\!\!\!\wedge} \mathbf{X}
\,,
$$
is non-empty. We need to show that in this case $R(\mathbf{Y}, \mathbf{X}) = 0$ (in the sense of generalized functions).
Write $\mathbf{Q} \coloneqq \mathbf{Y} \setminus \mathbf{P}$
for the complementary set of points, so that all points of $\mathbf{Q}$ are in the past of $\mathbf{X}$.
Notice that this implies that $\mathbf{P}$ is also not in the past of $\mathbf{Q}$:
$$
\mathbf{P} {\vee\!\!\!\wedge} \mathbf{Q}
\,.
$$
With this decomposition of $\mathbf{Y}$, the sum in (eq:CoefficientOfgeneratingRetardedProduct)
over subsets $\mathbf{I}$ of $\mathbf{Y}$ may be decomposed into a sum over
subsets $\mathbf{J}$ of $\mathbf{P}$ and $\mathbf{K}$ of $\mathbf{Q}$, respectively.
These subsets inherit the above causal ordering, so that by the causal factorization property of
$T(-)$ (def. \ref{TimeOrderedProduct}) and $\overline{T}(-)$ (prop. \ref{ReverseCausalFactorizationOfReverseTimeOrderedProducts})
the time-ordered and reverse time-ordered products
factor on these arguments:
$$
\begin{aligned}
R(\mathbf{Y}, \mathbf{X})
& =
\underset{ {\mathbf{J} \subset \mathbf{P}} \atop { \mathbf{K} \subset \mathbf{Q} } }{\sum}
\,
\overline{T}( \mathbf{J} \cup \mathbf{K} )
T( (\mathbf{P} \setminus \mathbf{J}) \cup (\mathbf{Q} \setminus \mathbf{K}), \mathbf{X} )
\\
& =
\underset{ {\mathbf{J} \subset \mathbf{P}} \atop { \mathbf{K} \subset \mathbf{Q} } }{\sum}
\,
\overline{T}( \mathbf{K} )
\overline{T}( \mathbf{J} )
T( \mathbf{P} \setminus \mathbf{J} )
T( \mathbf{Q} \setminus \mathbf{K}, \mathbf{X} )
\\
& =
\underset{ \mathbf{K} \subset \mathbf{Q} }{\sum}
\overline{T}(\mathbf{K})
\underset{= 0}{
\underbrace{
\left(
\underset{\mathbf{J} \subset \mathbf{P}}{\sum}
\overline{T}(\mathbf{J})
T( \mathbf{P} \setminus \mathbf{J} )
\right)
}}
T(\mathbf{Q} \setminus \mathbf{K}, \mathbf{X})
\end{aligned}
\,.
$$
Here the sub-sum in brackets vanishes by the inversion formula, prop. \ref{InversionFormulaForTimeOrderedProducts}.
=--
In conclusion:
+-- {: .num_prop #CausalFactorizationAlreadyImpliesSMatrix}
###### Proposition
**([[S-matrix]] scheme via [[causal factorization]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and consider a function
$$
\mathcal{S}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j \rangle
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j] ]
$$
from [[local observables]] to [[microcausal polynomial observables]] which satisfies the condition "perturbation"
from def. \ref{LagrangianFieldTheoryPerturbativeScattering}. Then the following two conditions
on $\mathcal{S}$ are equivalent
1. [[causal additivity]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
1. [[causal factorization]] (remark \ref{DysonCausalFactorization})
and hence either of them is necessary and sufficient for $\mathcal{S}$ to be a perturbative [[S-matrix]] scheme
according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.
=--
+-- {: .proof}
###### Proof
That causal factorization follows from causal additivity is immediate (remark \ref{DysonCausalFactorization}).
Conversely, causal factorization of $\mathcal{S}$ implies that its expansion coefficients $\{T_k\}_{k \in \mathbb{N}}$
are [[time-ordered products]] (def. \ref{TimeOrderedProduct}), via the proof of example \ref{TimeOrderedProductsFromSMatrixScheme},
and this implies causal additivity by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}.
=--
$\,$
**("Re"-)Normalization**
{#ExistenceAndRenormalization}
We discuss now that [[time-ordered products]] as in def. \ref{TimeOrderedProduct}, hence, by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, perturbative [[S-matrix]] schemes (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) exist in fact uniquely away from coinciding interaction points (prop. \ref{TimeOrderedProductAwayFromDiagonal} below).
This means that the construction of full [[time-ordered products]]/[[S-matrix]] schemes may be phrased as an [[extension of distributions]] of time-ordered products to the [[diagonal]] locus of coinciding spacetime arguments (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} below).
This choice in their definition is called the choice of _[[renormalization|("re"-)normalization]]_ of the [[time-ordered products]] (remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}), and hence of the [[interacting field theory|interacting]] [[pQFT]] that these define (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below).
The space of these choices may be accurately characterized, it is a [[torsor]] over a [[group]] of re-definitions of the
[[interaction]]-terms, called the "[[Stückelberg-Petermann renormalization group]]". This is called the
_[[main theorem of perturbative renormalization]]_, theorem \ref{PerturbativeRenormalizationMainTheorem} below.
Here we discuss just enough of the ingredients needed to _state_ this theorem. We give the proof in the [next chapter](#Renormalization).
$\,$
+-- {: .num_defn #TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport}
###### Definition
**([[tuples]] of [[local observables]] with pairwise disjoint spacetime support)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
For $k \in \mathbb{N}$, write
$$
\left(
{\, \atop \,}
LocPoly(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j ] ]}}_{pds}
\hookrightarrow
\left(
{\, \atop \,}
LocPoly(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j ] ]}}
$$
for the linear subspace of the $k$-fold [[tensor product]] of [[local observables]] (as in def. \ref{LagrangianFieldTheoryPerturbativeScattering}, def. \ref{TimeOrderedProduct}) on those tensor products $A_1 \otimes \cdots A_k$ of [[tuples]] with disjoint spacetime [[support]]:
$$
supp(A_j) \cap supp(A_k) = \emptyset
\phantom{AAA}
\text{for} \, i \neq j \in \{1, \cdots, k\}
\,.
$$
=--
+-- {: .num_prop #TimeOrderedProductAwayFromDiagonal}
###### Proposition
**([[time-ordered product]] unique away from coinciding spacetime arguments)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $T = \{T_k\}_{k \in \mathbb{N}}$ be a sequence of [[time-ordered products]] (def. \ref{TimeOrderedProduct})
$$
\array{
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
& \longrightarrow &
PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
\\
\uparrow & \nearrow_{(-) \star_F \cdots \star_F (-)}
\\
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}_{pds}
}
$$
Then their [[restriction]] to the subspace of [[tuples]] of [[local observables]] of pairwise disjoint spacetime support (def. \ref{TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport}) is unique (independent of the [[renormalization|"re-"normalization]] freedom in choosing $T$) and is given by the [[star product]]
$$
A_1 \star_{F} A_2
\;\coloneqq\;
((-)\cdot (-)) \circ
\exp\left(
\hbar
\left(
\underset{\Sigma \times \Sigma}{\int}
\Delta_F^{a b}(x,y)
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}^b(y)}
\,
dvol_\Sigma(x)\, dvol_\Sigma(y)
\right)
\right)
(A_1 \otimes A_2)
$$
that is induced (def. \ref{PropagatorStarProduct}) by the [[Feynman propagator]] $\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_- + H)$ (corresponding to the [[Wightman propagator]] $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ which is given by the choice of [[free field|free]] [[vacuum]]), in that
$$
T
\left(
{\, \atop \,}
A_1 , \cdots, A_k
{\, \atop \,}
\right)
\;=\;
A_1
\star_F
\cdots
\star_F
A_k
\,.
$$
In particular the time-ordered product extends from the restricted domain of tensor products of local observables to a restricted domain of [[microcausal polynomial observables]], where it becomes an [[associativity|associative]] product:
$$
\label{RestrictedTimeOrderedProductAssociative}
\begin{aligned}
T(A_1, \cdots, A_{k_n})
& =
T(A_1, \cdots, A_{k_1})
\star_F
T(A_{k_1 + 1}, \cdots, A_{k_2})
\star_F
\cdots
\star_F
T(A_{k_{n-1} + 1}, \cdots, A_{k_n})
\\
& =
A_1 \star_F \cdots \star_F A_{k_n}
\end{aligned}
$$
for all tuples of local observables $A_1, \cdots, A_{k_1}, A_{k_1+1}, \cdots, A_{k_2}, \cdots, \cdots A_{k_n}$ with pairwise disjoint spacetime support.
=--
The idea of this statement goes back at least to [Epstein-Glaser 73](S-matrix#EpsteinGlaser73), as in remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies}. One formulation appears as ([Brunetti-Fredenhagen 00, theorem 4.3](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)). The above formulation in terms of the [[star product]] is stated in ([Fredenhagen-Rejzner 12, p. 27](perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12), [Dütsch 18, lemma 3.63 (b)](perturbative+algebraic+quantum+field+theory#Duetsch18)).
+-- {: .proof}
###### Proof
By [[induction]] over the number of arguments, it is sufficient to see that, more generally, for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]] with disjoint spacetime support the star product $A_1 \star_F A_2$ is well-defined and satisfies causal factorization.
Consider two [[partitions of unity]]
$$
(\chi_{1,i} \in C^\infty_{cp}(\Sigma))_{i}
\phantom{AAA}
(\chi_{1,j} \in C^\infty_{cp}(\Sigma))_{j}
$$
and write $(A_{1,i})_i$ and $(A_{2,j})_{j}$ for the collection of [[microcausal polynomial observables]] obtained by multiplying all the [[distribution|distributional]] [[coefficients]] of $A_1$ and of $A_2$ with $\chi_{1,i}$ and with $\chi_{2,j}$, respectively, for all $i$ and $j$, hence such that
$$
A_1 \;=\; \underset{i}{\sum} A_{1,i}
\phantom{AAA}
A_2 \;=\; \underset{j}{\sum} A_{1,j}
\,.
$$
By linearity, it is sufficient to prove that $A_{1,i} \star_F A_{2,j}$ is well defined for all $i,j$ and satisfies causal factorization.
Since the spacetime supports of $A_1$ and $A_2$ are assumed to be disjoint
$$
supp(A_1) \cap supp(A_2) \;=\; \emptyset
$$
we may find partitions such that each resulting pair of smaller supports is in fact in [[causal order]]-relation:
$$
\array{
\left(
supp(A_1) \cap supp(\chi_{1,i})
\right)
{\vee\!\!\!\wedge}
\left(
supp(A_2) \cap supp(\chi_{2,j})
\right)
\\
\text{or}
\\
\left(
supp(A_2) \cap supp(\chi_{2,j})
\right)
{\vee\!\!\!\wedge}
\left(
supp(A_1) \cap supp(\chi_{1,u})
\right)
}
\phantom{AAAAA}
\text{for all}\,\, i,j
\,.
$$
But now it follows as in the proof of prop. \ref{CausalOrderingTimeOrderedProductOnRegular}) via (eq:CausallyOrderedWickProductViaFeynmanPropagator) that
$$
A_{1,i}
\star_F
A_{2,j}
\;=\;
\left\{
\array{
A_{1,i} \star_H A_{2,j} &\vert& supp(A_{1,i}) {\vee\!\!\!\wedge} supp(A_{2,j})
\\
A_{2,j} \star_H A_{1,i} &\vert& supp(A_{2,j}) {\vee\!\!\!\wedge} supp(A_{1,i})
}
\right.
$$
Finally the [[associativity]]-statement follows as in prop. \ref{AssociativeAndUnitalStarProduct}.
=--
Before using the unqueness of the [[time-ordered products]] away from coinciding spacetime arguments (prop. \ref{TimeOrderedProductAwayFromDiagonal}) to characterize the freedom in [[renormalization|("re"-)normalizing]] [[time-ordered products]], we pause to observe that in the same vein the [[time-ordered products]] have a unique extension of their domain also to [[regular polynomial observables]]. This is in itself a trivial statement (since all [[star products]] are defined on [[regular polynomial observables]], def. \ref{PropagatorStarProduct}) but for understanding the behaviour under [[renormalization|("re"-)normalization]] of other structures, such as the interacting [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum} below) it is useful to understand renormalization as a process that starts extending awa from [[regular polynomial observables]].
By prop. \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}, on [[regular polynomial observables]] the
[[S-matrix]] is given as follows:
+-- {: .num_defn #OnRegularObservablesPerturbativeSMatrix}
###### Definition
**([[perturbative S-matrix]] on [[regular polynomial observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Recall that the _[[time-ordered product]] on [[regular polynomial observables]]_
is the [[star product]] $\star_F$ induced by the [[Feynman propagator]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) and that, due to the [[non-singular distribution|non-singular]]
nature of [[regular polynomial observables]], this is given by [[conjugation]] of the pointwise product (eq:ObservablesPointwiseProduct) with $\mathcal{T}$ (eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism) as
$$
T(A_1, A_2)
\;=\;
A_1 \star_F A_2
\;=\;
\mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))
$$
(prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}).
We say that the _[[perturbative S-matrix]] scheme_ on [[regular polynomial observables]]
is the [[exponential]] with respect to $\star_F$:
$$
\mathcal{S}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g , j] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [ g, j] ]
$$
given by
$$
\mathcal{S}(S_{int})
=
\exp_{\star_F}
\left(
\tfrac{1}{i \hbar} S_{int})
\right)
\coloneqq
1
+
\tfrac{1}{\i \hbar} S_{int}
+
\tfrac{1}{2}
\tfrac{1}{(i \hbar)^2}
S_{int} \star_F S_{int}
+
\cdots
\,.
$$
We think of $S_{int}$ here as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].
We write $\mathcal{S}(S_{int})^{-1}$ for the [[inverse]] with respect to the [[Wick algebra|Wick product]] (which exists by remark \ref{PerturbativeSMatrixInverse})
$$
\mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int})
=
1
\,.
$$
Notice that this is in general different form the inverse with respect to the [[time-ordered product]] $\star_F$, which is $\mathcal{S}(-S_{int})$:
$$
\mathcal{S}(-S_{int})
\star_F
\mathcal{S}(S_{int})
=
1
\,.
$$
=--
Similarly, by def. \ref{QuntumMollerOperator}, on [[regular polynomial observables]] the [[quantum Møller operator]] is given as follows:
+-- {: .num_defn #MollerOperatorOnRegularPolynomialObservables}
###### Definition
**([[quantum Møller operator]] on [[regular polynomial observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] of degree 0
$$
S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar , g, j] ]
$$
then the corresponding _[[quantum Møller operator]]_ on [[regular polynomial observables]]
$$
\mathcal{R}^{-1}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar , g, j] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
$$
is given by the [[derivative]] of [[Bogoliubov's formula]]
$$
\mathcal{R}^{-1}
\;\coloneqq\;
\mathcal{S}(S_{int})^{-1}
\star_H
(\mathcal{S}(S_{int}) \star_F (-))
\,,
$$
where $\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$ is the [[perturbative S-matrix]] from def. \ref{OnRegularObservablesPerturbativeSMatrix}.
This indeed lands in [[formal power series]] in [[Planck's constant]] $\hbar$ (by remark \ref{PowersInPlancksConstant}), instead of in more general [[Laurent series]] as the [[perturbative S-matrix]] does (def. \ref{OnRegularObservablesPerturbativeSMatrix}).
Hence the inverse map is
$$
\mathcal{R}
\;=\;
\mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) )
\,.
$$
=--
([Bogoliubov-Shirkov 59](Bogoliubov's+formula#BogoliubovShirkov59); the above terminology follows [Hawkins-Rejzner 16, below def. 5.1](Møller+operator#HawkinsRejzner16))
(Beware that compared to Fredenhagen, Rejzner et. al. we change notation conventions $\mathcal{R} \leftrightarrow \mathcal{R}^{-1}$ in order to bring out the analogy to (the conventions for the) [[time-ordered product]] $A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))$ on regular polynomial observables.)
Still by def. \ref{QuntumMollerOperator}, on [[regular polynomial observables]] the [[interacting field algebra of observables]] is given as follows:
+-- {: .num_defn #FieldAlgebraObservablesInteracting}
###### Definition
**([[interacting field algebra]] [[structure]] on [[regular polynomial observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}. Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0
$$
S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
\,,
$$
then the _[[interacting field algebra]]_ [[structure]] on [[regular polynomial observables]]
$$
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, h] ]
\overset{ \star_{int} }{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
$$
is the [[conjugation]] of the [[Wick algebra]]-[[structure]] by the [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}):
$$
A_1 \star_{int} A_2
\;\coloneqq\;
\mathcal{R}
\left(
\mathcal{R}^{-1}(A_1)
\star_H
\mathcal{R}^{-1}(A_2)
\right)
$$
=--
(e.g. [Fredenhagen-Rejzner 11b, (19)](quantum+master+equation#FredenhagenRejzner11b))
Notice the following dependencies of these defnitions, which we leave notationally implicit:
| [[endomorphism]] of <br/> [[regular polynomial observables]] | meaning | depends on choice of |
|--------|---------|----------------------|
| $\phantom{AA}\mathcal{T}$ | [[time-ordered product|time-ordering]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{S}$ | [[S-matrix]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{R}$ | [[quantum Møller operator]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] and [[interaction]] |
$\,$
After having discussed the uniqueness of the [[time-ordered products]] away from coinciding spacetime arguments (prop. \ref{TimeOrderedProductAwayFromDiagonal}) we now phrase and then discuss the freedom in defining these products at coinciding arguments, thus [[renormalization|("re"-)normalizing]] them.
+-- {: .num_defn #ExtensionOfTimeOrderedProoductsRenormalization}
###### Definition
**([[Epstein-Glaser renormalization|Epstein-Glaser ("re"-)normalization]] of [[perturbative QFT]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Prop. \ref{TimeOrderedProductAwayFromDiagonal} implies that the
problem of constructing a sequence of [[time-ordered products]] (def. \ref{TimeOrderedProduct}), hence, by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, an [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) for [[perturbative quantum field theory]] around the given [[free field]] [[vacuum]], is equivalently a problem of a sequence of compatible _[[extensions of distributions]]_ of the [[star products]] $\underset{k \; \text{arguments}}{\underbrace{(-)\star_F \cdots \star_F (-)}}$ of the [[Feynman propagator]] on $k$ arguments from the [[complement]] of coinciding [[events]] inside the [[Cartesian products]] $\Sigma^k$ of [[spacetime]] $\Sigma$, along the canonical inclusion
$$
\Sigma^k \setminus \left\{ (x_i) \,\vert\, \underset{i \neq j}{\exists} (x_i = x_j) \right\}
\overset{\phantom{AAA}}{\hookrightarrow}
\Sigma^k
\,.
$$
Via the [[associativity]] (eq:RestrictedTimeOrderedProductAssociative) of the restricted [[time-ordered product]] thesese choices are naturally made by [[induction]] over $k$, choosing the $(k+1)$-ary [[time-ordered product]] $T_{k+1}$ as an [[extension of distributions]] of $T_k(\underset{k \, \text{args}}{\underbrace{-, \cdots, -}}) \star_F (-)$.
This [[induction|inductive]] choice of [[extension of distributions]] of the [[time-ordered product]] to coinciding interaction points deserves to be called a choice of _normalization_ of the [[time-ordered product]] (e.g. [Scharf 94, section 4.3](#Scharf95)), but for historical reasons (see remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} and remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}) it is known as _[[renormalization|re-normalization]]_. Specifically the inductive construction by extension to coinciding interaction points is known as _[[Epstein-Glaser renormalization]]_.
=--
In ([Epstein-Glaser 73](S-matrix#EpsteinGlaser73)) this is phrased in terms of splitting of distributions. In ([Brunetti-Fredenhagen 00, sections 4 and 7](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)) the perspective via [[extension of distributions]] is introduced, following ([Stora 93](perturbative+algebraic+quantum+field+theory#Stora93)). Review is in ([Dütsch 18, section 3.3.2](perturbative+algebraic+quantum+field+theory#Duetsch18)).
Proposition \ref{TimeOrderedProductAwayFromDiagonal} already shows that the freedom in choosing the [[renormalization|("re"-)normalization]] of [[time-ordered products]] is at most that of [[extensions of distributions|extending]] them to the "[[fat diagonal]]", where at least one pair of interaction points coincides. The following proposition \ref{RenormalizationIsInductivelyExtensionToDiagonal} says that when making these choices [[induction|inductively]] in the arity of the [[time-ordered products]] as in def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} then the available choice of [[renormalization|("re"-)normalization)]] at each stage is in fact only that of extension to the actual [[diagonal]], where _all_ interaction points coincide:
+-- {: .num_prop #RenormalizationIsInductivelyExtensionToDiagonal}
###### Proposition
**([[renormalization|("re"-)normalization]] is [[induction|inductive]] [[extension of distributions|extension]] of [[time-ordered products]] to [[diagonal]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Assume that for $n \in \mathbb{N}$, [[time-ordered products]]
$\{T_{k}\}_{k \leq n}$ of arity $k \leq n$ have been constructed
in the sense of def. \ref{TimeOrderedProduct}.
Then the time-ordered product $T_{n+1}$ of arity $n+1$ is uniquely fixed on the [[complement]]
$$
\Sigma^{n+1} \setminus diag(n)
\;=\;
\left\{
(x_i \in \Sigma)_{i = 1}^n
\;\vert\;
\underset{i,j}{\exists} (x_i \neq x_j)
\right\}
$$
of the [[image]] of the [[diagonal]] inclusion $\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}$
(where we regarded $T_{n+1}$ as a [[generalized function]] on $\Sigma^{n+1}$ according to remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}).
=--
This statement appears in ([Popineau-Stora 82](renormalization#PopineauStora82)), with (unpublished) details in ([Stora 93](renormalization#Stora93)), following personal communication by [[Henri Epstein]] (according to [Dütsch 18, footnote 57](S-matrix#Duetsch18)). Following this, statement and detailed proof appeared in ([Brunetti-Fredenhagen 99](S-matrix#BrunettiFredenhagen99)).
+-- {: .proof}
###### Proof
We will construct an [[open cover]] of $\Sigma^{n+1} \setminus \Sigma$ by subsets $\mathcal{C}_I \subset \Sigma^{n+1}$ which are [[disjoint unions]]
of [[inhabited set|non-empty]] sets that are in [[causal order]], so that by [[causal factorization]] the
time-ordered products $T_{n+1}$ on these subsets are uniquely given by $T_{k}(-) \star_H T_{n-k}(-)$.
Then we show that these unique products on these special subsets do coincide on [[intersections]].
This yields the claim by a [[partition of unity]].
We now say this in detail:
For $I \subset \{1, \cdots, n+1\}$ write $\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I$.
For $I, \overline{I} \neq \emptyset$, define the subset
$$
\mathcal{C}_I
\;\coloneqq\;
\left\{
(x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1}
\;\vert\;
\{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I}
\right\}
\;\subset\;
\Sigma^{n+1}
\,.
$$
Since the [[causal order]]-relation involves the [[closed future cones]]/[[closed past cones]], respectively, it is clear that these are [[open subsets]]. Moreover it is immediate that they form an [[open cover]] of the [[complement]] of the [[diagonal]]:
$$
\underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } } }{\cup}
\mathcal{C}_I
\;=\;
\Sigma^{n+1} \setminus diag(\Sigma)
\,.
$$
(Because any two distinct points in the [[globally hyperbolic spacetime]] $\Sigma$ may be causally separated by a [[Cauchy surface]],
and any such may be deformed a little such as not to intersect any of a given finite set of points. )
Hence the condition of [[causal factorization]] on $T_{n+1}$ implies that
[[restriction of distributions|restricted]] to any $\mathcal{C}_{I}$ these have to be given (in the condensed [[generalized function]]-notation from remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions} on
any unordered tuple $\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I$ with corresponding induced tuples
$\mathbf{I} \coloneqq \{x_i\}_{i \in I}$ and $\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}$ by
$$
\label{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}
T_{n+1}( \mathbf{X} )
\;=\;
T(\mathbf{I}) T(\overline{\mathbf{I}})
\phantom{AA}
\text{for}
\phantom{A}
\mathcal{X} \in \mathcal{C}_I
\,.
$$
This shows that $T_{n+1}$ is unique on $\Sigma^{n+1} \setminus diag(\Sigma)$ if it exists at all,
hence if these local identifications glue to a global definition of $T_{n+1}$. To see that this is the case,
we have to consider any two such subsets
$$
I_1, I_2
\subset
\{1, \cdots, n+1\}
\,,
\phantom{AA}
I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset
\,.
$$
By definition this implies that for
$$
\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}
$$
a tuple of spacetime points which decomposes into causal order with respect to both these subsets,
the corresponding mixed intersections of tuples are spacelike separated:
$$
\mathbf{I}_1 \cap \overline{\mathbf{I}_2}
\;
{\gt\!\!\!\!\lt}
\;
\overline{\mathbf{I}_1} \cap \mathbf{I}_2
\,.
$$
By the assumption that the $\{T_k\}_{k \neq n}$ satisfy causal factorization, this implies that the corresponding
time-ordered products commute:
$$
\label{TimeOrderedProductsOfMixedIntersectionsCommute}
T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
\,
T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
\;=\;
T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
\,
T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
\,.
$$
Using this we find that the identifications of $T_{n+1}$
on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal),
agree on the intersection: in that for $ \mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have
$$
\begin{aligned}
T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} )
& =
T( \mathbf{I}_1 \cap \mathbf{I}_2 )
T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
\,
T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
\\
& =
T( \mathbf{I}_1 \cap \mathbf{I}_2 )
\underbrace{
T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
}
T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
\\
& =
T( \mathbf{I}_2 )
T( \overline{\mathbf{I}_2} )
\end{aligned}
$$
Here in the first step we expanded out the two factors using (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal) for $I_2$,
then under the brace we used (eq:TimeOrderedProductsOfMixedIntersectionsCommute) and in the last step we used again
(eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal), but now for $I_1$.
To conclude, let
$$
\left(
\chi_I
\in
C^\infty_{cp}(\Sigma^{n+1})
\right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }
$$
be a [[partition of unity]] subordinate to the [[open cover]] formed by the $\mathcal{C}_I$. Then the above implies that
setting for any $\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)$
$$
T_{n+1}(\mathbf{X})
\;\coloneqq\;
\underset{
{ I \in \{1, \cdots, n+1\} }
\atop
{ I, \overline{I} \neq \emptyset }
}{\sum}
\chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} )
$$
is well defined and satisfies causal factorization.
=--
Since [[renormalization|("re"-)normalization]] involves making choices, there is the freedom to impose
further conditions that one may want to have satisfied. These are called _[[renormalization conditions]]_.
+-- {: .num_defn #RenormalizationConditions}
###### Definition
**([[renormalization conditions]], [[protection from quantum corrections]] and [[quantum anomalies]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Then a condition $P$ on $k$-ary functions of the form
$$
T_k
\;\colon\;
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$
is called a _renormalization condition_ if
1. it holds for the unique [[time-ordered products]] away from coinciding spacetime arguments (according to prop. \ref{TimeOrderedProductAwayFromDiagonal});
1. whenever it holds for all unrestricted $T_{k \leq n}$ for some $n \in \mathbb{N}$, then it also holds
for $T_{n+1}$ restricted away from the diagonal:
$$
P(T_k)_{k \leq n}
\;\Rightarrow\;
P\left(
T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}
\right)
\,.
$$
This means that a renormalization condition is a condition that may consistently be imposed degreewise
in an [[induction|inductive]] construction of [[time-ordered products]] by degreewise [[extension of distributions|extension]]
to the [[diagonal]], according to prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}.
If specified renormalization conditions $\{P_i\}$ completely remove any freedom in the choice of time-ordered products for a given [[quantum observable]], one says that the renormalization conditions _[[protection from quantum corrections|protects the observable against quantum corrections]]_.
If for specified renormalization conditions $\{P_i\}$ there is _no_ choice of [[time-ordered products]] $\{T_k\}_{k \in \mathbb{N}}$
(def. \ref{TimeOrderedProduct}) that satisfies all these conditions, then one says that an
[[interacting field theory|interacting]] [[perturbative QFT]] satisfying $\{P_i\}$ fails to exist due to a _[[quantum anomaly]]_.
=--
+-- {: .num_prop #BasicConditionsRenormalization}
###### Proposition
**(basic [[renormalization conditions]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Then the following conditions are [[renormalization conditions]] (def. \ref{RenormalizationConditions}):
1. (field independence) The [[functional derivative]] of a [[polynomial observable]] arising as a [[time-ordered product]] takes contributions only from the arguments, not from the product operation itself; in [[generalized function]]-notation:
$$
\label{FieldIndependenceRenormalizationCondition}
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
T(A_1, \cdots, A_n)
\;=\;
\underset{1 \leq k \leq n}{\sum}
T\left(
A_1, \cdots, A_{k-1}, \frac{\delta}{\delta \mathbf{\Phi}^a(x)}A_k, A_{k+1}, \cdots, A_n
\right)
$$
1. ([[translation group|translation]] equivariance) If the underlying [[spacetime]] is [[Minkowski spacetime]], $\Sigma = \mathbb{R}^{p,1}$,
with the induced [[action]] of the [[translation group]] on [[polynomial observables]]
$$
\rho
\;\colon\;
\mathbb{R}^{p,1} \times PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
$$
then
$$
\rho_v \left( {\, \atop \,} T(A_1, \cdots, A_n) {\, \atop \,}\right)
\;=\;
T(\rho_{v}(A_1), \cdots, \rho_v(A_n))
$$
1. ([[quantum master equation]], [[master Ward identity]]) see prop. \ref{QuantumMasterEquation}
(if this condition fails, the corresponding [[quantum anomaly]] (def. \ref{RenormalizationConditions}) is called a _[[gauge anomaly]]_)
=--
([Dütsch 18, p. 150 and section 4.2](S-matrix#Duetsch18))
+-- {: .proof}
###### Proof
For the first two statements this is obvious from prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} and prop. \ref{TimeOrderedProductAwayFromDiagonal}, which imply that $T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}$ is
uniquely specified from $\{T_k\}_{k \leq n}$ via the [[star product]] induced by the [[Feynman propagator]],
and the fact that, on [[Minkowski spacetime]], this is manifestly translation invariant and independent of the fields
(e.q. prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}).
The third statement requires work. That the [[quantum master equation]]/([[master Ward identity]]
always holds on [[regular polynomial observables]] is prop. \ref{QuantumMasterEquation} below.
That it holds for $T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}$ if it holds for $\{T_k\}_{k \leq n}$
is shown in ([Duetsch 18, section 4.2.2](S-matrix#Duetsch18)).
=--
$\,$
We discuss methods for [[renormalization|normalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal})
and [[renormalization|re-normalization]] in detail in the [next chapter](#Renormalization).
$\,$
**Feynman perturbation series**
{#FeynmanDiagrams}
By def \ref{ExtensionOfTimeOrderedProoductsRenormalization} and the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}), the construction of perturbative [[S-matrix]] schemes/[[time-ordered products]] may be phrased as [[renormalization|("re-")normalization]] of the [[star product]] induced by the [[Feynman propagator]], namely as a choice of [[extension of distributions]] of the this star-product to the locus of coinciding interaction points.
Since the [[star product]] is the [[exponential]] of the binary contraction with the [[Feynman propagator]], it is naturally expanded as a [[sum]] of [[products of distributions]] labeled by [[finite multigraphs]] (def. \ref{Graphs} below),
where each [[vertex]] corresponds to an [[interaction]] or [[source field]] insertion, and where each [[edge]]
corresponds to one contractions of two of these with the [[Feynman propagator]]. The [[products of distributions]]
arising this way are the _[[Feynman amplitudes]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below).
If the [[free field]] [[vacuum]] is decomposed as a [[direct sum]] of distinct [[free field]] [[types]]/species (def. \ref{VerticesAndFieldSpecies} below), then
in addition to the [[vertices]] also the edges in these [[graphs]] receive labels, now by the field species whose particular [[Feynman propagator]] is being used in the contraction at that edges.
These labeled graphs are now called _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below) and the [[products of distributions]] which they encode are their _[[Feynman amplitudes]]_ built by the _[[Feynman rules]]_ (prop. \ref{FeynmanDiagramAmplitude} below).
The choice of [[renormalization|("re"-)normalization]] of the [[time-ordered products]]/[[S-matrix]] is thus equivalently a choice of [[renormalization|("re"-)normalization]] of the [[Feynman amplitudes]] for all possible [[Feynman diagrams]].
These are usefully organized in powers of $\hbar$ by their _[[loop order]]_ (prop. \ref{FeynmanDiagramLoopOrder} below).
In conclusion, the [[Feynman rules]] make the perturbative [[S-matrix]] be equal to a [[formal power series]]
of [[Feynman amplitudes]] labeled by [[Feynman graphs]]. As such it is known as the _[[Feynman perturbation series]]_ (example \ref{FeynmanPerturbationSeries} below).
Notice how it is therefore the [[combinatorics]] of [[star products]] that governs both [[Wick's lemma]] in [[free field theory]] as well as [[Feynman diagram|Feynman diagrammatics]] in [[interacting field theory]]:
[[!include Wick algebra -- table]]
$\,$
{#FeynmanDiagramInTwoStages} We now discuss [[Feynman diagrams]] and their [[Feynman amplitudes]] in two stages:
First we consider plain [[finite multigraphs]] with [[linear order|linearly ordered]] vertices
but no other labels (def. \ref{Graphs} below) and discuss how these generally organize an expansion
of the [[time-ordered products]] as a sum of [[products of distributions|distributional products]]
of the given [[Feynman propagator]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below).
These summands (or their [[vacuum expectation values]]) are called the _[[Feynman amplitudes]]_ if
one thinks of the underlying [[free field]] [[vacuum]] as having a single "field species"
and of the chosen [[interaction]] to be a single "interaction vertex".
But often it is possible and useful to identify different field species and different interaction vertices.
In fact in applications this choice is typically evident and not highlighted as a choice. We
make it explicit below as def. \ref{VerticesAndFieldSpecies}. Such a choice makes
both the [[interaction]] term as well as the [[Feynman propagator]] decompose as sums (remark \ref{FeynmanPropagatorFieldSpecies} below).
Accordingly then, after "multiplying out" the products of these sums that appear in the Feynman amplitudes, these, too, decompose
further as as sums indexed by multigraphs whose edges are labeled by field species, and whose vertices are labeled
by interactions. These labeled multigraphs are the _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below) and the
corresponding summands are the [[Feynman amplitudes]] proper (prop. \ref{FeynmanDiagramAmplitude} below).
+-- {: .num_defn #Graphs}
###### Definition
**([[finite graph|finite]] [[multigraphs]])**
A _[[finite graph|finite]] [[multigraph]]_ is
1. a [[finite set]] $V$ ("of [[vertices]]");
1. a [[finite set]] $E$ ("of [[edges]]");
1. a [[function]] $E \overset{p}{\to} \left\{ {\,\atop \,} \{v_1, v_2\} = \{v_2, v_1\} \;\vert\; v_1, v_2 \in V \,,\; v_1 \neq v_2 {\, \atop \,} \right\}$
(sending any [[edge]] to the unordered pair of distinct [[vertices]] that it goes between).
A choice of [[linear order]] on the set of vertices of a finite multigraph is a choice of [[bijection]] of the form
$$
V \simeq \{1, 2, \cdots, \nu\}
\,.
$$
Hence the [[isomorphism classes]] of a [[finite graph|finite]] [[multigraphs]] with [[linear order|linearly ordered]]
[[vertices]] are characterized by
1. a [[natural number]]
$$
\nu \coloneqq {\vert V\vert} \in \mathbb{N}
$$
(the number of [[vertices]]);
1. for each $i \lt j \in \{1, \cdots, \nu\}$ a natural number
$$
e_{i,j} \coloneqq {\vert p^{-1}(\{v_i,v_j\})\vert} \in \mathbb{N}
$$
(the number of [[edges]] between the $i$th and the $j$th vertex).
We write $\mathcal{G}_\nu$ for the set of such [[isomorphism classes]] of finite multigraphs with linearly ordered vertices
identified with $\{1, 2, \cdots, \nu\}$; and we write
$$
\mathcal{G} \;\coloneqq\; \underset{\nu \in \mathbb{N}}{\sqcup} \mathcal{G}_\nu
$$
for the set of [[isomorphism classes]] of finite multigraphs with linearly ordered vertices of any number.
=--
+-- {: .num_prop #FeynmanPerturbationSeriesAwayFromCoincidingPoints}
###### Proposition
**([[Feynman amplitudes]] of [[finite multigraphs]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
For $\nu \in \mathbb{N}$, the $\nu$-fold [[time-ordered product]] away from coinciding interaction points, given by prop. \ref{TimeOrderedProductAwayFromDiagonal}
$$
T_\nu
\;\colon\;
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
{\, \atop \,}
\right)^{\otimes^\nu_{\mathbb{C}[ [\hbar, g, j] ]}}_{pds}
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$
is equal to the following [[formal power series]] labeled by [[isomorphism classes]] of [[finite multigraphs]] with $\nu$ [[linear order|linearly ordered]] [[vertices]], $\Gamma \in \mathcal{G}_\nu$ (def. \ref{Graphs}):
$$
\label{FeynmanAmplitudeExpansionOfTimeOrderedProductAwayFromDiagonal}
\begin{aligned}
& T_\nu(O_1, \cdots , O_\nu)
\\
& =
\underset{\Gamma \in \mathcal{G}_\nu}{\sum}
\Gamma\left(O_i)_{i = 1}^\nu\right)
\\
& \coloneqq
\underset{ \Gamma \in \mathcal{G}_\nu }{\sum}
prod
\circ
\underset{ r \lt s \in \{1, \cdots, \nu\} }{\prod}
\frac{\hbar^{e_{r,s}}}{e_{r,s}!}
\left\langle
(\Delta_{F})^{e_{r,s}}
,
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
\right\rangle
\left(
O_1 \otimes \cdots \otimes O_{\nu}
\right)
\\
& \coloneqq
\underset{ \Gamma \in \mathcal{G}_\nu }{\sum}
((-) \cdot \cdots \cdot (-))
\circ
\underset{ r \lt s \in \{1, \cdots,\nu\} }{\prod}
\frac{\hbar^{e_{r,s}}}{e_{r,s}!}
\\
&
\phantom{AAA}
\underset{i = 1, \cdots e_{r,s}}{\prod}
\underset{\Sigma \times \Sigma}{\int}
dvol_\Sigma(x_i)
dvol_\Sigma(y_i)
\,
\Delta_F^{a_i b_i}(x_i,y_i)
\\
& \phantom{AAAAAA}
\left(
O_1
\otimes
\cdots
\otimes
O_{r-1}
\otimes
\frac{
\delta^{e_{r,s}} O_r
}{
\delta \mathbf{\Phi}^{a_1}(x_1)
\cdots
\delta \mathbf{\Phi}^{a_{e_{r,s}}}(x_{e_{r,s}})
}
\otimes
O_{r+1}
\otimes
\cdots
\otimes
O_{s-1}
\otimes
\frac{
\delta^{e_{r,s}} O_s
}{
\delta \mathbf{\Phi}^{b_1}(y_1)
\cdots
\delta \mathbf{\Phi}^{b_{e_{r,s}}}(y_{e_{r,s}})
}
\otimes
O_{s+1}
\otimes
\cdots
\otimes
O_\nu
\right)
\,,
\end{aligned}
$$
where $e_{r,s} \coloneqq e_{r,s}(\Gamma)$ is, for short, the number of [[edges]] between vertex $r$ and vertex $s$ in the
[[finite multigraph]] $\Gamma$ of the outer sum, according to def. \ref{Graphs}.
Here the summands of the expansion (eq:FeynmanAmplitudeExpansionOfTimeOrderedProductAwayFromDiagonal)
$$
\label{FeynmanAmplitude}
\Gamma\left( (O_i)_{i = 1}^\nu\right)
\;\coloneqq\;
prod
\circ
\underset{ r \lt s \in \{1, \cdots,\nu\} }{\prod}
\frac{\hbar^{e_{r,s}}}{e_{r,s}!}
\left\langle
(\Delta_{F})^{e_{r,s}}
,
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
\right\rangle
\left(
O_1 \otimes \cdots \otimes O_{\nu}
\right)
\;\in\;
PolyObs(E_{\text{BV-BRT}})((\hbar))[ [g,j ] ]
$$
and/or their [[vacuum expectation values]]
$$
\left\langle
\Gamma\left((V_i)_{i = 1}^v\right)
\right\rangle
\;\in\;
\mathbb{C}((\hbar))[ [ h, j] ]
$$
are called the _[[Feynman amplitudes]]_ for scattering processes in the given [[free field]] [[vacuum]] of shape $\Gamma$ with [[interaction]] [[vertices]] $O_i$. Their expression as [[products of distributions]] via algebraic expression on the right hand side of (eq:FeynmanAmplitude) is also called the _[[Feynman rules]]_.
=--
([Keller 10, IV.1](S-matrix#Keller10))
+-- {: .proof}
###### Proof
We proceed by [[induction]] over the number $v$ of [[vertices]].
The statement is trivially true for a single vertex.
So assume that it is true for $v \geq 1$ vertices. It follows that
$$
\begin{aligned}
& T(O_1, \cdots, O_\nu, O_{\nu+1})
\\
& =
T( T(O_1, \cdots ,O_\nu), O_{\nu+1} )
\\
&=
prod
\circ
\exp\left(
\left\langle
\hbar \Delta_F,
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right\rangle
\right)
\left(
\left(
prod
\circ
\!\!\!\!
\underset{\Gamma \in \mathcal{G}_\nu }{\sum}
\underset{ { r \lt s } \atop { \in \{1, \cdots, \nu\} } }{\prod}
\frac{1}{e_{r,s}!}
\left\langle
(\hbar \Delta_F)^{e_{r,s}}
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \mathbf{\Phi}_s^{e_{r,s}} }
\right\rangle
(O_1 \otimes \cdots \otimes O_\nu)
\right)
\,\otimes\,
O_{\nu+1}
\right)
\\
& =
prod
\circ
\underset{\Gamma \in \mathcal{G}_\nu }{\sum}
\\
& \phantom{=}
\underset{ { r \lt s } \atop { \in \{1,\cdots, \nu\}} }{\prod}
\frac{1}{e_{r,s}!}
\left\langle
(\hbar \Delta_F)^{e_{r,s}}
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{ \delta \mathbf{\Phi}_s^{e_{r,s}} }
\right\rangle
\\
& \phantom{=}
\underset{
{ e_{\nu+1} =}
\atop
{ e_{1,{\nu+1}} + \cdots + e_{\nu,\nu + 1} }
}{\sum}
\underset{
= (e_{1,\nu + 1}) \cdots (e_{\nu,\nu+1}))
}{
\underbrace{
\frac{
\left(
{ e_{\nu + 1} }
\atop
{ (e_{1, \nu + 1}), \cdots, (e_{\nu , \nu+1}) }
\right)
}{
( e_{\nu+1} ) !
}
}
}
\left\langle
(\hbar \Delta_F)^{e_{\nu+1}}
\left(
\frac{\delta^{e_{1,\nu+1}} O_1 }{\delta \mathbf{\Phi}^{e_{1,\nu+1}}}
\otimes
\cdots
\otimes
\frac{
\delta^{e_{\nu,\nu+1}} O_\nu
}{
\delta \mathbf{\Phi}^{e_{\nu,\nu+1}}
}
\;\otimes\;
\frac{
\delta^{ e_{\nu + 1} } O_{\nu+1}
}{
\delta \mathbf{\Phi}^{e_{1,\nu+1} + \cdots + e_{\nu,\nu+1}}
}
\right\rangle
\right)
\\
&=
prod
\circ
\underset{\Gamma \in \mathcal{G}_{\nu+1} }{\sum}
\underset{ { r \lt s } \atop { \in \{1, \cdots, \nu+1\} } }{\prod}
\tfrac{1}{e_{r,s}!}
\left\langle
(\hbar \Delta_F)^{e_{r,s}}
\,,\,
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
\frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
\right\rangle
(O_1 \otimes \cdots \otimes O_{\nu+1})
\end{aligned}
$$
The combinatorial factor over the brace is the [[multinomial coefficient]] expressing the number of ways of distributing $e_{\nu+1}$-many functional derivatives to $v$ factors, via the [[product rule]], and quotiented by the [[factorial]] that comes from the [[exponential]] in the definition of the [[star product]].
Here in the first step we used the [[associativity]] (eq:RestrictedTimeOrderedProductAssociative) of the restricted time-ordered product, in the second step we used the induction assumption,
in the third we passed the outer functional derivatives through the pointwise product using the [[product rule]], and in the fourth step we recognized that this amounts to summing
in addition over all possible choices of sets of edges from the first $v$ vertices to the new $\nu+1$st vertex,
which yield in total the sum over all diagrams with $\nu+1$ vertices.
=--
If the [[free field theory]] is decomposed as a [[direct sum]] of free field theories (def. \ref{VerticesAndFieldSpecies} below), we obtain a more fine-grained concept of [[Feynman amplitudes]], associated not just with a [[finite multigraph]], but also with a
labelling of this graph by field species and interaction types. These labeled multigraphs are the genuine _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below):
+-- {: .num_defn #VerticesAndFieldSpecies}
###### Definition
**(field species and interaction vertices)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Then
1. a choice of _field species_ is a choice of decomposition of the [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}}$
as a [[fiber product]] over [[finite set]] $Spec = \{sp_1, sp_2, \cdots, sp_n\}$ of ([[graded manifold|graded]] [[supermanifold|super-]]) [[field bundles]]
$$
E_{\text{BV-BRST}}
\;\simeq\;
E_{sp_1} \times_{\Sigma} \cdots \times_\Sigma E_{sp_n}
\,,
$$
such that the [[gauge fixing|gauge fixed]] [[free field|free]] [[Lagrangian density]] $\mathbf{L}'$ is the [[sum]]
$$
\mathbf{L}' \;=\; \mathbf{L}'_{sp_1} + \cdots + \mathbf{L}'_{sp_n}
$$
of [[free field theory|free]] [[Lagrangian densities]]
$$
\mathbf{L}'_{sp_i} \in \Omega^{p+1,0}_\Sigma(E_i)
$$
on these separate field bundles.
1. a choice of _interaction vertices and external vertices_ is a choice of sum decomposition
$$
g S_{int} + j A
\;=\;
\underset{i \in Ext}{\sum} g S_{int,i}
+
\underset{j \in Int}{\sum} j A_j
$$
parameterized by [[finite sets]] $Int$ and $Ext$, to be called the sets of _internal vertex labels_ and _external vertex labels_,
respectively.
=--
+-- {: .num_remark #FeynmanPropagatorFieldSpecies}
###### Remark
**(Feynman propagator for separate field species)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Then a choice of field species as in def. \ref{VerticesAndFieldSpecies}
induces a corresponding decomposition of the [[Feynman propagator]] of the gauge fixed free field theory
$$
\Delta_F
\;\in\;
\Gamma'_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
$$
as the sum of Feynman propagators for each of the chosen field species:
$$
\Delta_F
\;=\;
\Delta_{F,1} + \cdots + \Delta_{F,n}
\;\in\;
\underoverset{i = 1}{n}{\oplus}
\Gamma'_{\Sigma \times \Sigma}( E_{sp_i} \boxtimes E_{sp_i} )
\;\subset\;
\Gamma'_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
$$
hence in components, with $(\phi^A$ the collective field coordinates on $E_{\text{BV-BRST}}$,
this decomposition is of the form
$$
\left(
\Delta_F^{A, B}
\right)
\;=\;
\left(
\array{
(\Delta_{F,1}^{a b}) & 0 & 0 & \cdots & 0
\\
0 & (\Delta_{F,2}^{\alpha \beta}) & 0 & \cdots & 0
\\
\vdots & & & & \vdots
\\
0 & \cdots & \cdots & 0 & (\Delta_{F,n}^{i j})
}
\right)
$$
=--
+-- {: .num_example #FieldSpeciesQED}
###### Example
**(field species in [[quantum electrodynamics]])**
The [[field bundle]] for [[Gaussian-averaged Lorenz gauge|Lorenz]] [[gauge fixing|gauge fixed]] [[quantum electrodynamics]]
on [[Minkowski spacetime]] $\Sigma$ admits a decomposition into field species, according to def. \ref{VerticesAndFieldSpecies}, as
$$
E_{\text{BV-BRST}}
\;=\;
\underset{
\text{Dirac}
\atop
\text{field}
}{
\underbrace{
(S_{odd} \times \Sigma)
}}
\times_\Sigma
\underset{
{\text{electromagnetic field &}}
\atop
{\text{Nakanishi-Lautrup field}}
}{
\underbrace{
T^\ast\Sigma
\times_\Sigma
(\mathbb{R} \times \Sigma)
}}
\times_\Sigma
\underset{
\text{ghost field}
}{
\underbrace{
(\mathbb{R}[1] \times \Sigma)
}
}
\times_\Sigma
\underset{
\text{antighost field}
}{
\underbrace{
(\mathbb{R}[-1] \times \Sigma)
}
}
$$
(by example \ref{LagrangianQED}) and example \ref{NLGaugeFixingOfElectromagnetism})).
The corresponding sum decomposition of the Feynman propagator, according to remark \ref{FeynmanPropagatorFieldSpecies},
is
$$
\Delta_F
\;=\;
\underset{
\text{Dirac}
\atop
\text{field}
}{
\underbrace{
\Delta_F^{\text{electron}}
}
}
+
\underset{
\text{electromagnetic field &}
\atop
\text{Nakanishi-Lautrup field}
}{
\underbrace{
\left(
\array{
\Delta_F^{photon} & *
\\
* & *
}
\right)
}
}
+
\Delta_F^{ghost}
+
\Delta_F^{\text{antighost}}
\,,
$$
where
1. $\Delta_F^{\text{electron}}$ is the [[electron propagator]] (def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetim}));
1. $\Delta_F^{photon}$ is the [[photon propagator]] in [[Gaussian-averaged Lorenz gauge]] (prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge});
1. the [[ghost field]] and [[antighost field]] [[Feynman propagators]] $\Delta_F^{ghost}$, and $\Delta_F^{antighost}$ are each one copy of the [[Feynman propagator]] of the [[real scalar field]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}), while the [[Nakanishi-Lautrup field]] contributes a mixing with the [[photon propagator]], notationally suppressed behind the star-symbols above.
=--
+-- {: .num_defn #FeynmanDiagram}
###### Definition
**([[Feynman diagrams]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Let moreover
$$
E_{\text{BV-BRST}}
\;\simeq\;
\underset{sp \in Spec}{\times} E_{sp}
\,,
$$
be a choice of field species, according to def \ref{VerticesAndFieldSpecies},
$$
g S_{int} + j A
\;=\;
\underset{i \in Ext}{\sum} g S_{int,i}
+
\underset{j \in Int}{\sum} j A_j
$$
a choice of internal and external interaction vertices according to def. \ref{VerticesAndFieldSpecies}.
With these choices, we say that a _[[Feynman diagram]]_ $(\Gamma, vertlab, edgelab)$ is
1. a [[finite multigraph]] with [[linear order|linearly ordered]] vertices (def. \ref{Graphs})
$$
\Gamma \in \mathcal{G}
\,,
$$
1. a [[function]] from its [[vertices]]
$$
vertlab \;\colon\; V_{\Gamma} \longrightarrow Int \sqcup Ext
$$
to the [[disjoint union]] of the chosen sets of internal and external vertex labels;
1. a [[function]] from its [[edges]]
$$
edgelab \;\colon\; E_{\Gamma} \to Spec
$$
to the chosen set of field species.
We write
$$
\array{
\mathcal{G}^{Feyn}
&\overset{\text{forget} \atop \text{labels}}{\longrightarrow}&
\mathcal{G}
\\
(\Gamma,vertlab, edgelab) &\mapsto& \Gamma
}
$$
for the set of [[isomorphism classes]] of Feynman diagrams with labels in $Sp$, refining the set of
isomorphisms of plain [[finite multigraphs]] with [[linear order|linearly ordered]] [[vertices]] from def. \ref{Graphs}.
=--
+-- {: .num_prop #FeynmanDiagramAmplitude}
###### Proposition
**([[Feynman amplitudes]] for [[Feynman diagrams]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Let moreover
$$
E_{\text{BV-BRST}}
\;\simeq\;
\underset{sp \in Spec}{\times} E_{sp}
\,,
$$
be a choice of field species, according to def \ref{VerticesAndFieldSpecies}, hence inducing, by remark \ref{FeynmanPropagatorFieldSpecies},
a sum decomposition of the [[Feynman propagator]]
$$
\label{FeynmanPropagatorSumOverFieldSpecies}
\Delta_F
\;=\;
\underset{sp \in Spec}{\sum}\Delta_{F,sp}
\,,
$$
and let
$$
\label{VertexDecompositionFeynmanAmplitude}
g S_{int} + j A
\;=\;
\underset{i \in Ext}{\sum} g S_{int,i}
+
\underset{j \in Int}{\Sum} j A_j
$$
be a choice of internal and external interaction vertices according to def. \ref{VerticesAndFieldSpecies}.
Then by "multiplying out" the products of the sums (eq:FeynmanPropagatorSumOverFieldSpecies) and (eq:VertexDecompositionFeynmanAmplitude) in the
formula (eq:FeynmanAmplitude) for the [[Feynman amplitude]] $\Gamma\left( (g S_{int} + j A))_{i = 1}^\nu \right)$ (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) this decomposes as a sum of the form
$$
\Gamma\left(
(g S_{int} + j A)_{i = 1}^\nu
\right)
\;=\;
\underset{
{ V_\Gamma \overset{vertlab}{\longrightarrow} Int \sqcup Ext}
\atop
{ E_\Gamma \overset{edgelab}{\longrightarrow} Spec }
}{\sum}
\left(
\Gamma,
edgelab, vertlab
\right)
(g S_{int} + j A)
$$
over all ways of labeling the [[vertices]] $v$ of $\Gamma$ by the internal or external vertex labels,
and the [[edges]] $e$ of $\Gamma$ by field species. The corresponding summands
$$
\left(
\Gamma,
edgelab,
vertlab
\right)
(g S_{int} + j A)
\;\in\;
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$
or rather their [[vacuum expectation value]]
$$
\left\langle
\left(
\Gamma,
edgelab, vertlab
\right)
(g S_{int} + j A)
\right\rangle
\;\in\;
\mathbb{C}[ [ \hbar, g, j ] ]
$$
are called the _[[Feynman amplitude]] associated with these [[Feynman diagrams]].
=--
+-- {: .num_example #FeynmanAmplitudesInCausalPerturbationTheoryExampleOfQED}
###### Example
**([[Feynman amplitudes]] in [[causal perturbation theory]] -- example of [[QED]])**
To recall, in [[perturbative quantum field theory]], [[Feynman diagrams]] (def. \ref{FeynmanDiagram}) are labeled [[finite multigraphs]] (def. \ref{Graphs}) that encode [[product of distributions|products of]] [[Feynman propagators]], called _[[Feynman amplitudes]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) which in turn contribute to [[probability amplitudes]] for physical [[scattering]] processes -- _[[scattering amplitudes]]_ (example \ref{ScatteringAmplitudeFromInteractingFieldObservables}):
The [[Feynman amplitudes]] are the summands in the [[Feynman perturbation series]]-expansion (example \ref{FeynmanPerturbationSeries}) of the _[[scattering matrix]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
$$
\mathcal{S}
\left(
S_{int}
\right)
=
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}
\frac{1}{(i \hbar)^k}
T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} )
$$
of a given [[interaction]] [[Lagrangian density]] $L_{int}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
The [[Feynman amplitudes]] are the summands in an expansion of the _[[time-ordered products]]_ $T(\cdots)$ (def. \ref{TimeOrderedProduct}) of the [[interaction]] with itself, which, away from coincident vertices, is given by the [[star product]] of the [[Feynman propagator]] $\Delta_F$ (prop. \ref{TimeOrderedProductAwayFromDiagonal}), via the [[exponential]] contraction
$$
T(S_{int}, S_{int})
\;=\;
prod \circ \exp
\left(
\hbar
\int \Delta_{F}^{a b}(x,y)
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}(y)}
\right)
( S_{int} \otimes S_{int} )
\,.
$$
Each [[edge]] in a [[Feynman diagram]] corresponds to a factor of a [[Feynman propagator]] in $T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )$, being a [[distribution of two variables]]; and each [[vertex]] corresponds to a factor of the [[interaction]] [[Lagrangian density]] at $x_i$.
For example [[quantum electrodynamics]] (example \ref{LagrangianQED}) in [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) involves (via example \ref{FieldSpeciesQED}):
1. the [[Dirac field]] modelling the [[electron]], with [[Feynman propagator]] called the _[[electron propagator]]_ (def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}), here to be denoted
$$
\Delta \phantom{AAAA} \text{electron propagator}
$$
1. the [[electromagnetic field]] modelling the [[photon]], with [[Feynman propagator]] called the _[[photon propagator]]_ (prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}), here to be denoted
$$
G \phantom{AAAA} \text{photon propagator}
$$
1. the [[electron-photon interaction]] (eq:ElectronPhotonInteractionLocalLagrangian)
$$
L_{int}
\;=\;
\underset{
\text{interaction}
}{
\underbrace{
i g
(\gamma^\mu)^\alpha{}_\beta
}
}
\,
\underset{
{ \text{incoming} \atop \text{electron} }
\atop
\text{field}
}{\underbrace{\overline{\psi_\alpha}}}
\;
\underset{
{
\,
\atop
\text{photon}
}
\atop
\text{field}
}{\underbrace{a_\mu}}
\;
\underset{
{\text{outgoing} \atop \text{electron} }
\atop
\text{field}
}{\underbrace{\psi^\beta}}
$$
The [[Feynman diagram]] for the [[electron-photon interaction]] alone is
<center>
<img src="https://ncatlab.org/nlab/files/InteractionVertexOfQED.jpg" width="150">
</center>
where the solid lines correspond to the [[electron]], and the wiggly line to the [[photon]]. The corresponding [[product of distributions]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) is (written in [[generalized function]]-notation, example \ref{SomeNonSingularTemperedDistributions})
$$
\underset{
\text{loop order}
}{
\underbrace{
\hbar^{3/2-1}
}
}
\underset{
\text{electron-photon}
\atop
\text{interaction}
}{
\underbrace{
i g (\gamma^\mu)^\alpha{}_\beta
}
}
\,.
\,
\underset{
{\text{incoming} \atop \text{electron}}
\atop
\text{propagator}
}{
\underbrace{
\overline{\Delta(-,x)}_{-, \alpha}
}
}
\underset{
{
\,
\atop
\text{photon}
}
\atop
\text{propagator}
}{
\underbrace{
G(x,-)_{\mu,-}
}
}
\underset{
{ \text{outgoing} \atop \text{electron} }
\atop
\text{propagator}
}{
\underbrace{
\Delta(x,-)^{\beta, -}
}
}
$$
Hence a typical [[Feynman diagram]] in the [[QED]] [[Feynman perturbation series]] induced by this [[electron-photon interaction]] looks as follows:
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="560"/>
</center>
where on the bottom the corresponding [[Feynman amplitude]] [[product of distributions]] is shown;
now notationally suppressing the contraction of the internal indices and all prefactors.
For instance the two solid [[edges]] between the [[vertices]] $x_2$ and $x_3$ correspond to the two factors of $\Delta(x_2,x_2)$:
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponent1.jpg" width="560"/>
</center>
This way each sub-graph encodes its corresponding subset of factors in the [[Feynman amplitude]]:
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentTwo.jpg" width="560"/>
</center>
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentThree.jpg" width="560"/>
</center>
> graphics grabbed from [Brouder 10](Feynman+diagram#Brouder10)
A priori this [[product of distributions]] is defined away from coincident vertices: $x_i \neq x_j$ (prop. \ref{TimeOrderedProductAwayFromDiagonal} below). The definition at coincident vertices $x_i = x_j$ requires a choice of _[[extension of distributions]]_ (def. \ref{ExtensionOfDistributions} below) to the [[diagonal]] locus of coincident interaction points. This choice is the _[[renormalization|("re-")normalization]]_ (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below) of the [[Feynman amplitude]].
=--
+-- {: .num_example #FeynmanPerturbationSeries}
###### Example
**([[Feynman perturbation series]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]\langle g , j\rangle
$$
be a [[local observable]], regarded as a [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
By prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} every choice of perturbative [[S-matrix]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
$$
\mathcal{S}(g S_{int} + j A)
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ] +
$$
has an expansion as a [[formal power series]] of the form
$$
\mathcal{S}(g S_{int} + j A)
\;=\;
\underset{\Gamma \in \mathcal{G}}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right)
\,,
$$
where the series is over all [[finite multigraphs]] with [[linear order|linearly ordered]] [[vertices]] $\Gamma$ (def. \ref{Graphs}), and the summands are the corresponding [[renormalization|("re"-)normalized]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).
If moreover a choice of field species and of internal and external interaction vertices is made, according to def. \ref{VerticesAndFieldSpecies},
then this series expansion refines to an expansion over all [[Feynman diagrams]] $(\Gamma,edgelab, vertlab)$ (def. \ref{FeynmanDiagram})
of [[Feynman amplitudes]] $(\Gamma, edgelab,vertlab)(g S_{int} + j A)$ (def. \ref{FeynmanDiagramAmplitude}):
$$
\mathcal{S}(g S_{int} + j A)
\;=\;
\underset{(\Gamma,edgelab, vertlab) \in \mathcal{G}^{Feyn}}{\sum}
(\Gamma, edgelab,vertlab)(g S_{int} + j A)
\,,
$$
Expressed in this form the [[S-matrix]] is known as the _[[Feynman perturbation series]]_.
=--
+-- {: .num_remark #Tadpoles}
###### Remark
**(no [[tadpole]] [[Feynman diagrams]])**
In the definition of [[finite multigraphs]] in def. \ref{Graphs} there are _no_ edges considered that go from any [[vertex]] to _itself_. Accordingly, there are _no_ such labeled edges in [[Feynman diagrams]] (def. \ref{FeynmanDiagram}):
<center>
<img src="https://ncatlab.org/nlab/files/tadpole.png" width="70">
</center>
In [[pQFT]] these diagrams are called _[[tadpoles]]_, and their non-appearance is considered part of the _[[Feynman rules]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).
Via prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} this condition reflects the nature of the [[star product]] (def. \ref{PropagatorStarProduct}) which always contracts _different_ [[tensor product]] factors with the [[Feynman propagator]] before taking their pointwise product.
Beware that in [[graph theory]] these [[tadpoles]] are called "[[loops]]", while here in [[pQFT]] a "loop" in a [[planar graph]] refers instead to what in [[graph theory]] is called a _[[face]]_ of the graph, see the discussion of _[[loop order]]_ in prop. \ref{FeynmanDiagramLoopOrder} below.
=--
([Keller 10, remark II.8 and proof of prop. II.7](S-matrix#Keller10))
$\,$
**Effective action**
{#EffectiveAction}
We have seen that the [[Feynman perturbation series]] expresses the [[S-matrix]] as a [[formal power series]] of _[[Feynman amplitudes]]_ labeled by _[[Feynman diagrams]]_. Now the [[Feynman amplitude]] associated with a [[disjoint union]] of [[connected graph|connected]] [[Feynman diagrams]] (def. \ref{ConnectedGraphs} below) is just the product of the amplitudes of the [[connected components]] (prop. \ref{LogarithmEffectiveAction} below). This allows to re-organize the [[Feynman perturbation series]] as the ordinary [[exponential]] of the Feynman perturbation series restricted to just [[connected graph|connected]] Feynman diagrams. The latter is called the _[[effective action]]_ (def. \ref{InPerturbationTheoryActionEffective} below) because it allows to express [[vacuum expectation values]] of the [[S-matrix]] as an ordinary exponential (equation (eq:ExponentialSeffVEVOfSMatrix) below).
+-- {: .num_defn #ConnectedGraphs}
###### Definition
**([[connected graphs]])**
Given two [[finite multigraphs]] $\Gamma_1, \Gamma_2 \in \mathcal{G}$ (def. \ref{Graphs}), their [[disjoint union]]
$$
\Gamma_1 \sqcup \Gamma_2
\;\in\;
\mathcal{G}
$$
is the finite multigraph whose set of [[vertices]] and set of [[edges]] are the [[disjoint unions]] of the corresponding sets of $\Gamma_1$ and $\Gamma_2$
$$
V_{\Gamma_1 \sqcup \Gamma_2}
\;\coloneqq\;
V_{\Gamma_1} \sqcup V_{\Gamma_2}
$$
$$
E_{\Gamma_1 \sqcup \Gamma_2}
\;\coloneqq\;
E_{\Gamma_1} \sqcup E_{\Gamma_2}
$$
and whose vertex-assigning function $p$ is the corresponding function on disjoint unions
$$
p_{\Gamma_1 \sqcup \Gamma_2}
\;\coloneqq\;
p_{\Gamma_1} \sqcup p_{\Gamma_2}
\,.
$$
The operation induces a pairing on the set $\mathcal{G}$ of [[isomorphism classes]] of [[finite multigraphs]]
$$
(-) \sqcup (-)
\;\colon\;
\mathcal{G} \times \mathcal{G}
\longrightarrow
\mathcal{G}
\,.
$$
A [[finite multigraph]] $\Gamma \in \mathcal{G}$ (def. \ref{Graphs}) is called _[[connected graph|connected]]_ if it is not the [[disjoint union]] of two [[inhabited|non-empty]] finite multigraphs.
We write
$$
\mathcal{G}_{conn}
\subset
\mathcal{G}
$$
for the subset of [[isomorphism classes]] of [[connected graph|connected]] [[finite multigraphs]].
=--
+-- {: .num_lemma #MultiplicativeFeynmanAmplitudes}
###### Lemma
**([[Feynman amplitudes]] multiply under [[disjoint union]] of [[graphs]])**
Let
$$
\Gamma
\;=\;
\Gamma_1
\sqcup
\Gamma_2
\sqcup
\cdots
\sqcup
\Gamma_n
\;\in\; \mathcal{G}
$$
be [[disjoint union]] of graphs (def. \ref{ConnectedGraphs}). then then corresponding [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints})
multiply by the pointwise product (def. \ref{Observable}):
$$
\Gamma\left( g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
\;=\;
\Gamma_1\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_1)}\right)
\cdot
\Gamma_2\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_2)} \right)
\cdot
\cdots
\cdot
\Gamma_n\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_n)} \right)
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{TimeOrderedProductAwayFromDiagonal} the contributions to the S-matrix away from coinciding interaction points are given by the [[star product]] induced by the [[Feynman propagator]], and specifically, by prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}, the [[Feynman amplitudes]] are given this way.
Moreover the [[star product]] (def. \ref{PropagatorStarProduct}) is given by first contracting with powers of the [[Feynman propagator]] and then multiplying all resulting terms with the pointwise product of observables.
This implies the claim by the nature of the combinatorial factor in the definition of the [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).
=--
+-- {: .num_defn #InPerturbationTheoryActionEffective}
###### Definition
**([[effective action]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be an [[S-matrix]] scheme for [[perturbative QFT]] around this vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and let
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]
$$
be a [[local observable]].
Recall that for each [[finite multigraph]] $\Gamma \in \mathcal{G}$ (def. \ref{Graphs}) the [[Feynman perturbation series]] for $\mathcal{S}(g S_{int} + j A)$ (example \ref{FeynmanPerturbationSeries})
$$
\mathcal{S}(g S_{int} + j A)
\;=\;
\underset{\Gamma \in \mathcal{G}}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
$$
contributes with a [[renormalization|("re"-)nromalized]] [[Feynman amplitude]] $\Gamma\left( (g S_{int} + j A)_{i = 1}^v\right) \in PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j ] ]$.
We say that the corresponding _[[effective action]]_ is $i \hbar$ times the sub-series
$$
\label{ExpansionEffectiveAction}
S_{eff}(g,j)
\;\coloneqq\;
i \hbar
\underset{\Gamma \in \mathcal{G}_{conn}}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
\;\in\;
PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j ] ]
$$
of [[Feynman amplitudes]] that are labeled only by the _[[connected graphs]]_ $\Gamma \in \mathcal{G}_{conn} \subset \mathcal{G}$ (def. \ref{ConnectedGraphs}).
(A priori $S_{eff}(g,j)$ could contain negative powers of $\hbar$, but it turns out that it does not; this is prop. \ref{FeynmanDiagramLoopOrder} below.)
=--
+-- {: .num_remark #TerminologyForEffectiveAction}
###### Remark
**(terminology for "effective action")**
Beware differing conventions of terminology:
1. In the perspective of [[effective quantum field theory]] (remark \ref{pQFTEffective} below), the [[effective action]] in def. \ref{InPerturbationTheoryActionEffective} is sometimes called the _effective potential_ at scale $\Lambda = 0$ (see prop. \ref{InPerturbationTheoryActionEffective} below).
This terminology originates in restriction to the special example of the [[scalar field]] (example \ref{RealScalarFieldBundle}), where the non-derivative [[Phi^n interactions]] $g S_{int} = \underset{n}{\sum} \underset{\Sigma}{\int} g_{sw}^{(n)}(x) (\mathbf{\Phi}(x))^n \, dvol_\Sigma(x)$ (example \ref{phintheoryLagrangian}) are naturally thought of as [[potential energy]]-terms.
From this perspective the [[effective action]] in def. \ref{InPerturbationTheoryActionEffective} is a special case of _[[relative effective actions]]_ $S_{eff,\Lambda}$ ("relative effective potentials", in the case of [[Phi^n interactions]]) relative to an arbitrary [[UV cutoff]]-scales $\Lambda$ (def. \ref{EffectiveActionRelative} below).
1. For the special case that
$$
j A
\coloneqq
\underset{\Sigma}{\int} j_{sw,a}(x) \mathbf{\Phi}^a(x)\, dvol_{\Sigma}(x)
$$
is a [[regular polynomial observable|regular]] [[linear observable]] (def. \ref{RegularLinearFieldObservables}) the [[effective action]] according to def. \ref{InPerturbationTheoryActionEffective} is often denoted $W(j)$ or $E(j)$, and then its _functional [[Legendre transform]]_ (if that makes sense) is instead called the effective action, instead.
This is because the latter encodes the [[equations of motion]] for the [[vacuum expectation values]] $\langle \mathbf{\Phi}(x)_int\rangle$ of the [[interacting field observables|interacting]] [[field observables]]; see example \ref{EquationsOfMotionForVacuumExpectationValues} below.
Notice the different meaning of "effective" in both cases: In the first case it refers to what is effectively seen of the full [[pQFT]] _at some [[UV-cutoff scale]]_, while in the second case it refers to what is effectively seen when restricting attention only to the [[vacuum expectation values]] of [[regular polynomial observable|regular]] [[linear observables]].
=--
+-- {: .num_prop #LogarithmEffectiveAction}
###### Proposition
**([[effective action]] is [[logarithm]] of [[S-matrix]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be an [[S-matrix]] scheme for [[perturbative QFT]] around this vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and let
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]
$$
be a [[local observable]] and let
$$
S_{eff}(g,j) \;\in\; PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j] ]
$$
be the corresponding [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}).
Then then [[S-matrix]] for $g S_{int} + j A$ is the [[exponential]] of the [[effective action]] with respect to the pointwise product $(-)\cdot (-)$ of observables (def. \ref{Observable}):
$$
\begin{aligned}
\mathcal{S}(g S_{int} + j A)
& =
\exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff}(g,j) \right)
\\
& \coloneqq
1
+
\frac{1}{i \hbar} S_{eff}(g,j)
+
\frac{1}{(i \hbar)^2} S_{eff}(g,j) \cdot S_{eff}(g,j)
+
\frac{1}{(i \hbar)^3} S_{eff}(g,j) \cdot S_{eff}(g,j) \cdot S_{eff}(g,j)
+
\cdots
\end{aligned}
$$
Moreover, this relation passes to the [[vacuum expectation values]]:
$$
\label{ExponentialSeffVEVOfSMatrix}
\begin{aligned}
\left\langle
{\, \atop \,}
\mathcal{S}(g S_{int} + j A)
{\, \atop \,}
\right\rangle
& =
\left\langle
{\, \atop \,}
\exp\left(
\tfrac{1}{i \hbar} S_{eff}(g,j)
\right)
{\, \atop \,}
\right\rangle
\\
& =
e^{\tfrac{1}{i \hbar} \langle S_{eff}(g,j) \rangle}
\end{aligned}
\,.
$$
Conversely the [[vacuum expectation value]] of the [[effective action]] is to the [[logarithm]] of that of the S-matrix:
$$
\left\langle
S_{eff}(g,j)
\right\rangle
\;=\;
i \hbar
\,
\ln
\left\langle
\mathcal{S}(g S_{int} + j A)
\right\rangle
\,.
$$
=--
+-- {: .proof}
###### Proof
By lemma \ref{MultiplicativeFeynmanAmplitudes} the summands in the $n$th pointwise power of $\frac{1}{i \hbar}$ times the effective action are precisely the Feynman amplitudes $\Gamma\left((g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right)$ of [[finite multigraphs]] $\Gamma$ with $n$ [[connected components]], where each such appears with multiplicity given by the [[factorial]] of $n$:
$$
\frac{1}{n!}
\left(
\frac{1}{i \hbar}
S_{eff}(g,j)
\right)^n
\;=\;
\underset{
{ \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j }
\atop
{ \Gamma_j \in \mathcal{G}_{conn} }
}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
\,.
$$
It follows that
$$
\begin{aligned}
\exp_\cdot\left(
\frac{1}{i \hbar} S_{int}
\right)
& =
\underset{n \in \mathbb{N}}{\sum}
\underset{
{ \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j }
\atop
{ \Gamma_j \in \mathcal{G}_{conn} }
}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
\\
& =
\underset{\Gamma \in \mathcal{G}}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
\end{aligned}
$$
yields the [[Feynman perturbation series]] by expressing it as a series (re-)organized by number of [[connected components]] of the [[Feynman diagrams]].
To conclude the proof it is now sufficient to observe that taking [[vacuum expectation values]] of [[polynomial observables]] respects the pointwise product of observables
$$
\left\langle A_1 \cdot A_2 \right\rangle
\;=\;
\left\langle
A_1
\right\rangle
\,
\left\langle
A_2
\right\rangle
\,.
$$
This is because the [[Hadamard vacuum state]] $\langle -\rangle \colon PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \to \mathbb{C}[ [\hbar, g, j ] ]$ simply picks the zero-order monomial term, by prop. \ref{WickAlgebraCanonicalState}),
and under multiplication of polynomials the zero-order terms are multiplied.
=--
This immediately implies the following important fact:
+-- {: .num_prop #EffectiveActionIsGeneratingFunction}
###### Proposition
**(in [[vacuum stability|stable vacuum]] the [[effective action]] is [[generating function]] for [[vacuum expectation values]] of [[interacting field observables]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
If the given [[vacuum state]] is [[vacuum stability|stable]] (def. \ref{VacuumStability}) then the [[vacuum expectation value]] $\langle S_{eff}(g,j)\rangle$ of the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) is the generating function for the [[vacuum expectation value]] of the [[interacting field observable]] $A_{int}$ (def. \ref{InteractingFieldObservables}) in that
$$
\left\langle
A_{int}
\right\rangle
\;=\;
\frac{d}{d j}
S_{eff}(g,j)\vert_{j = 0}
\,.
$$
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
\frac{d}{d j}
S_{eff}(g,j)
& =
i \hbar
\frac{d}{d j}
\ln
\left\langle
\mathcal{S}(g S_{int} + j A)
\right\rangle
\vert_{j = 0}
\\
& =
i \hbar
\left\langle
\mathcal{S}(g S_{int})
\right\rangle^{-1}
\frac{d}{d j}
\left\langle
\mathcal{S}(g S_{int} + j A)
\right\rangle
\vert_{j = 0}
\\
& =
\left\langle
\frac{d}{d j}
\underset{ \mathcal{Z}(j A) }{
\underbrace{\mathcal{S}(g S_{int})^{-1}
\mathcal{S}(g S_{int} + j A)
}}
\vert_{j = 0}
\right\rangle
\\
& =
\left\langle
A_{int}
\right\rangle
\,.
\end{aligned}
$$
Here in the first step we used prop \ref{LogarithmEffectiveAction}, in the second step we applied the [[chain rule]] of [[differentiation]], in the third step we used the definition of [[vacuum stability]] (def. \ref{VacuumStability}) and in the fourth step we recognized the definition of the [[interacting field observables]] (def. \ref{InteractingFieldObservables}).
=--
+-- {: .num_example #EquationsOfMotionForVacuumExpectationValues}
###### Example
**([[equations of motion]] for [[vacuum expectation values]] of [[interacting field observables]])**
Consider the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) for the case that
$$
\begin{aligned}
j A
& =
\tau{\Sigma}( j_{sw} \phi)
\\
& =
\underset{\Sigma}{\int}
j_{sw}(x) \mathbf{\Phi}(x)
\, dvol_\Sigma(x)
\end{aligned}
$$
is a [[regular polynomial observable|regular]] [[linear observable]] ([this def.](A+first+idea+of+quantum+field+theory#RegularLinearFieldObservables)), hence the smearing of a [[field observable]] ([this def.](A+first+idea+of+quantum+field+theory#PointEvaluationObservables)) by an [[adiabatic switching]] of the [[source field]]
$$
j_{sw} \;\in\; C^\infty_{cp}(\Sigma) \langle j\rangle
\,.
$$
(Here we are notationally suppressing internal field indices, for convenience.)
In this case the [[vacuum expectation value]] of the corresponding [[effective action]] is often denoted
$$
W(j_{sw})
$$
and regarded as a functional of the [[adiabatic switching]] $j_{sw}$ of the [[source field]].
In this case prop. \ref{EffectiveActionIsGeneratingFunction} says that if the [[vacuum state]] is [[vacuum stability|stable]], then $W$ is the [[generating functional]] for [[interacting field observables|interacting]] (def. \ref{InteractingFieldObservables}) [[field observables]] (def. \ref{PointEvaluationObservables}) in that
$$
\label{WFunctionalDerivative}
\left\langle
\mathbf{\Phi}(x)_{int}
\right\rangle
\;=\;
\frac{\delta}{\delta j_{sw}(x)}
W(j_{sw} = 0)
\,.
$$
Assume then that there exists a corresponding functional $\Gamma(\Phi)$ of the [[field histories]] $\Phi \in \Gamma_{\Sigma}(E_{\text{BV-BRST}})$ (def. \ref{FieldsAndFieldBundles}), which behaves like a functional [[Legendre transform]] of $W$ in that it satisfies the functional version of the defining equation of Legendre transforms (first derivatives are [[inverse functions]] of each other, see [this equation](Legendre+transformation#eq:DerivativesOfLegendreTransformsAreInverseFunctions)):
$$
\frac{\delta }{\delta \Phi(x)}
\Gamma
\left(
\frac{\delta}{\delta j_{sw}(y)} W
\right)
\;=\;
\delta(x,y) j_{sw}(x)
\,.
$$
By (eq:WFunctionalDerivative) this implies that
$$
\frac{\delta }{\delta \Phi(x)}
\Gamma
\left(
\left\langle \mathbf{\Phi}(x)_{int} \right\rangle \right)
\;=\;
0
\,.
$$
This may be read as a quantum version of the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction})
formulated now not for the [[field histories]] $\Phi(x)$, but for the [[vacuum expectation values]] $\langle \mathbf{\Phi}(x)_{int}\rangle$ of their corresponding [[interacting quantum field observables]].
Beware, (as in remark \ref{TerminologyForEffectiveAction}) that many texts refer to $\Gamma(\Phi)$ as the _effective action_, instead of its [[Legendre transform]], the generating functional $W(j_{sw})$.
=--
The perspective of the [[effective action]] gives a transparent picture of the order of quantum effects involved in the
[[S-matrix]], this is prop. \ref{FeynmanDiagramLoopOrder} below. In order to state this conveniently, we invoke
two basic concepts from [[graph theory]]:
+-- {: .num_defn #GraphPlanar}
###### Definition
**([[planar graphs]] and [[trees]])**
A [[finite multigraph]] (def. \ref{Graphs}) is called a _[[planar graph]]_ if it
admits an [[embedding]] into the [[plane]], hence if it may be "drawn into the plane" without intersections, in the evident way.
A [[finite multigraph]] is called a _[[tree]]_ if for any two of its [[vertices]]
there is at most one [[path]] of [[edges]] connecting them, these are examples of planar graphs. We write
$$
\mathcal{G}_{tree}
\subset
\mathcal{G}
$$
for the [[subset]] of [[isomorphism classes]] of [[finite multigraphs]] with [[linear order|linearly orrdered]] [[vertices]] (def. \ref{Graphs})
on those which are [[trees]].
=--
+-- {: .num_prop #FeynmanDiagramLoopOrder}
###### Proposition
**([[loop order]] and [[tree level]] of [[Feynman perturbation series]])**
The [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) contains no negative powers of $\hbar$, hence is indeed
a [[formal power series]] also in $\hbar$:
$$
S_{eff}(g,j)
\;\in\;
PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
\,.
$$
and in particular
$$
\left\langle
S_{eff}(g,j)
\right\rangle
\;\in\;
\mathbb{C}[ [ \hbar, g, j] ]
\,.
$$
Moreover, the contribution to the effective action in the [[classical limit]] $\hbar \to 0$
is precisely that of [[Feynman amplitudes]] of those [[finite multigraphs]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) which are [[trees]] (def. \ref{GraphPlanar}); thus called the _[[tree level]]_-contribution:
$$
S_{eff}(g,j)\vert_{\hbar = 0}
\;=\;
i \hbar
\underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum}
\Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
\,.
$$
Finally, a [[finite multigraph]] $\Gamma$ (def. \ref{Graphs}) which is [[planar graph|planar]] (def. \ref{GraphPlanar}) and [[connected graph|connected]] (def. \ref{ConnectedGraphs}) contributes to the effective action
precisely at order
$$
\hbar^{L(\Gamma)}
\,,
$$
where $L(\Gamma) \in \mathbb{N}$ is the number of _[[faces]]_ of $\Gamma$, here called the _number of loops_ of the diagram;
here usually called the _[[loop order]]_ of $\Gamma$.
(Beware the terminology clash with [[graph theory]], see the discussion of [[tadpoles]] in remark \ref{Tadpoles}.)
=--
+-- {: .proof}
###### Proof
By def. \ref{LagrangianFieldTheoryPerturbativeScattering} the explicit $\hbar$-dependence of the [[S-matrix]] is
$$
\mathcal{S}
\left(
S_{int}
\right)
\;=\;
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}
\frac{1}{(i \hbar)^k}
T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots, S_{int}}} )
$$
and by prop. \ref{TimeOrderedProductAwayFromDiagonal} the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is
$$
T(S_{int}, S_{int})
\;=\;
prod \circ
\exp\left(
\hbar
\left\langle
\Delta_F,
\frac{\delta}{\delta \mathbf{\Phi}}
\otimes
\frac{\delta}{\delta \mathbf{\Phi}}
\right\rangle
\right)
( S_{int} \otimes S_{int} )
\,,
$$
By the [[Feynman rules]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) this means that
1. each [[vertex]] of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;
1. each [[edge]] of a Feynman diagram contributes a power $\hbar^{+1}$ to its Feynman amplitude.
If we write
$$
E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}
$$
for the total number of [[vertices]] and [[edges]], respectively, in $\Gamma$, this means that a Feynman amplitude
corresponding to some $\Gamma \in \mathcal{G}$ contributes precisely at order
$$
\label{GeneralFeynmanDiagramhbarContribution}
\hbar^{E(\Gamma) - V(\Gamma)}
\,.
$$
So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is [[connected graph|connected]] (def. \ref{ConnectedGraphs}) and [[planar graph|planar]] (def. \ref{GraphPlanar}), then _[[Euler's formula]]_ asserts that
$$
\label{ConnectedPlanarGraphEulerCharacteristic}
E(\Gamma) - V(\Gamma)
\;=\;
L(\Gamma) - 1
\,.
$$
Hence $\hbar^{L(\Gamma)- 1}$ is the order of $\hbar$ at which $\Gamma$ contributes to the [[scattering matrix]] expressed as the [[Feynman perturbation series]].
But the [[effective action]], by definition (eq:ExpansionEffectiveAction), has the same contributions
of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order
$$
\hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)}
\,.
$$
This proves the second claim on [[loop order]].
The first claim, due to the extra factor of $\hbar$ in the definition of the effective action, is equivalent to saying
that the Feynman amplitude of every [[connected graph|connected]] [[finite multigraph]]
contributes powers in $\hbar$ of order $\geq -1$ and contributes at order $\hbar^{-1}$ precisely if the graph is a tree.
Observe that a [[connected graph|connected]] [[finite multigraph]] $\Gamma$ with $\nu \in \mathbb{N}$ vertices (necessarily $\nu \geq 1$) has at least $\nu-1$ edges and precisely $\nu - 1$ edges if it is a tree.
To see this, consecutively remove edges from $\Gamma$ as long as possible while retaining connectivity. When this process stops, the result must be a connected tree $\Gamma'$, hence a [[connected graph|connected]] [[planar graph]] with $L(\Gamma') = 0$. Therefore [[Euler's formula]] (eq:ConnectedPlanarGraphEulerCharacteristic) implies that that $E(\Gamma') = V(\Gamma') -1$.
This means that the connected multigraph $\Gamma$ in general has a Feynman amplitude of order
$$
\hbar^{E(\Gamma) - V(\Gamma)}
=
\hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}} }
$$
and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.
=--
$\,$
**Vacuum diagrams**
{#VacuumDiagrams}
With the [[Feynman perturbation series]] and the [[effective action]] in hand, it is now immediate to see that there
is a general contribution by [[vacuum diagrams]] (def. \ref{VacuumDiagram} below) in the [[scattering matrix]] which,
in a [[vacuum stability|stable vacuum state]],
cancels out against the prefactor $\mathcal{S}(g S_{int})$ in [[Bogoliubov's formula]] for [[interacting field observables]].
+-- {: .num_defn #VacuumDiagram}
###### Definition
**([[vacuum diagrams]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]],
and consider a choice of decomposition for field species and interaction vertices according to def. \ref{VerticesAndFieldSpecies}.
Then a [[Feynman diagram]] all whose vertices are internal vertices (def. \ref{FeynmanDiagram})
is called a _[[vacuum diagram]]_.
Write
$$
\mathcal{G}^{Feyn}_{vac}
\subset
\mathcal{G}^{Feyn}
$$
for the subset of [[isomorphism classes]] of vacuum diagrams among the set of isomorphism classes of all Feynman diagrams, def. \ref{FeynmanDiagram}. Similarly write
$$
\mathcal{G}^{Feyn}_{conn,vac}
\;\coloneqq\;
\mathcal{G}^{Feyn}_{conn}
\cap
\mathcal{G}^{Feyn}_{vac}
\;\subset\;
\mathcal{G}^{Feyn}
$$
for the subset of [[isomorphism classes]] of Feynman diagrams which are both vacuum diagrams as well as [[connected graphs]] (def. \ref{ConnectedGraphs}).
Finally write
$$
S_{eff,vac}(g)
\;\coloneqq\;
\underset{ { (\Gamma,vertlab,edgelab) } \atop { \in \mathcal{G}_{conn,vac} } }{\sum}
(\Gamma,vertlab, edgelab)(g S_{int})
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g ] ]
$$
for the sub-series of that for the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) given
only by those connected diagrams which are also vacuum diagrams.
=--
+-- {: .num_example}
###### Example
**(2-vertex [[vacuum diagram]] in [[QED]])**
The [[vacuum diagram]] (def. \ref{VacuumDiagram}) with two [[electron-photon interaction]]-vertices in [[quantum electrodynamics]] (example \ref{LagrangianQED}) is:
<center>
<img src="https://ncatlab.org/nlab/files/QEDVacuumDiagram.png" width="200">
</center>
=--
+-- {: .num_example #SMatrixVacuumContribution}
###### Example
**([[vacuum diagram]]-contribution to [[S-matrices]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]],
and consider a choice of decomposition for field species and interaction vertices according to def. \ref{VerticesAndFieldSpecies}.
Then the [[Feynman perturbation series]]-expansion of the [[S-matrix]] (example \ref{FeynmanPerturbationSeries})
of the [[interaction]]-term $g S_{int}$ alone (no [[source field]]-contribution) is the series of [[Feynman amplitudes]] that are labeled by [[vacuum diagrams]] (def. \ref{VacuumDiagram}), hence (by prop. \ref{LogarithmEffectiveAction}) the exponential of the
vacuum [[effective action]] $S_{eff,vac}$ (def. \ref{VacuumDiagram}):
$$
\begin{aligned}
\mathcal{S}(g S_{int})
& =
\exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff,vac}(g,j) \right)
\\
& =
\underset{\Gamma \in \mathcal{G}_{vac}}{\sum}
\Gamma\left(g S_{int}\right)
\end{aligned}
\,.
$$
More generally, the S-matrix with [[source field]]-contribution $j A$ included always splits as a
_pointwise_ product of the vacuum S_matrix with the [[Feynman perturbation series]] over all [[Feynman graphs]] with at least
one external vertex:
$$
\begin{aligned}
\mathcal{S}(g S_{int} + j A)
\;=\;
\mathcal{S}(g S_{int})
\cdot
\underset{
\text{Feynman perturbation series}
\atop
\text{over diagrams with at least one external vertex}
}{
\underbrace{
\exp_\cdot
\left(
\tfrac{1}{i \hbar} \left( S_{eff}(g,j) - S_{eff,vac}(g) \right)
\right)
}
}
\,,
\end{aligned}
$$
Hence if the [[free field]] [[vacuum state]] is stable with respect to the interaction $g S_{int}$, according to def. \ref{VacuumStability}, then the [[vacuum expectation value]] of a [[time-ordered product]] of [[interacting field observables]] $j (A_i)_{int}$ (example \ref{InteractinFieldTimeOrderedProduct}) and hence in particular of [[scattering amplitudes]] (example \ref{ScatteringAmplitudeFromInteractingFieldObservables}) is
given by the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}) over
just the non-vacuum [[Feynman diagrams]], hence over all those diagram that have at least one one external vertex
$$
\begin{aligned}
&
\left(
{\, \atop \,}
supp(A_1)
{\vee\!\!\!\wedge}
supp(A_2)
{\vee\!\!\!\wedge}
\cdots
{\vee\!\!\!\wedge}
supp(A_n)
{\, \atop \,}
\right)
\\
& \Rightarrow
\left\langle
{\, \atop \,}
(A_1)_int
(A_2)_{int}
\cdots
(A_n)_{int}
{\, \atop \,}
\right\rangle
\;=\;
\frac{d^n}{ d j_1 \cdots d j_n}
\left(
\underset{\Gamma \in \mathcal{G} \setminus \mathcal{G}_{vac} }{\sum}
\Gamma(g S_{int} + \sum_i j_i A_i)
\right)_{
\vert j_1, \cdots, j_n = 0
}
\,.
\end{aligned}
$$
This is the way in which the [[Feynman perturbation series]] is used in practice for computing [[scattering amplitudes]].
=--
$\,$
**Interacting quantum BV-Differential**
{#InteractingQantumBVDifferential}
So far we have discussed, starting with a [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]],
the perturbative construction of [[interacting field algebras of observables]] (def. \ref{QuntumMollerOperator}) and their organization
in increasing powers of $\hbar$ and $g$ ([[loop order]], prop. \ref{FeynmanDiagramLoopOrder}) via the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}, example \ref{SMatrixVacuumContribution}).
But this [[interacting field algebra of observables]] still involves all the [[auxiliary fields]] of the [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]] (example \ref{FieldSpeciesQED}), while the
actual physical [[gauge invariance|gauge invariant]] [[on-shell]] observables should be (just) the [[cochain cohomology]] of the
[[BV-BRST differential]] on this enlarged space of observables. Hence for the construction of [[perturbative QFT]] to conclude,
it remains to pass the [[BV-BRST differential]] of the [[free field]] [[Wick algebra]] of observables to
a [[differential]] on the [[interacting field algebra]], such that its [[cochain cohomology]] is well defined.
Since the [[time-ordered products]] away from coinciding interaction points and as well as on [[regular polynomial observables]] are uniquely fixed
(prop. \ref{TimeOrderedProductAwayFromDiagonal}), one finds that also this _interacting quantum BV-differential_
is uniquely fixed, on [[regular polynomial observables]], by [[conjugation]] with the [[quantum Møller operators]] (def. \ref{BVDifferentialInteractingQuantum}).
The formula that characterizes it there is called the _[[quantum master equation]]_ or equivalently the _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation} below).
When [[extension of distributions|extending]] to coinciding interaction points via [[renormalization|("re"-)normalization]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) these identities are not guaranteed to hold anymore, but may
be imposed as [[renormalization conditions]] (def. \ref{RenormalizationConditions}, prop. \ref{BasicConditionsRenormalization}).
Quantum correction to the [[master Ward identity]] then imply corrections to [[Noether's theorem|Noether current]] [[conserved current|conservation laws]]; this we discuss [below](#WardIdentities).
$\,$
For the following discussion, recall from the [previous chapter](#FreeQuantumFields) how the global BV-differential
$$
\{S',-\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
on [[regular polynomial observables]] (def. \ref{BVDifferentialGlobal}) as well as the global [[antibracket]] $\{-,-\}$ (def. \ref{ComplexBVBRSTGlobal}) are [[conjugation|conjugated]] into the [[time-ordered product]] via the time ordering operator $\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ (def. \ref{AntibracketTimeOrdered}, prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}), which makes
In the same way we may use the [[quantum Møller operators]] to conjugate the BV-differential into the regular part of the [[interacting field algebra of observables]]:
+-- {: .num_defn #BVDifferentialInteractingQuantum}
###### Definition
**(interacting quantum [[BV-differential]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and let
$$
S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g, j] ]
$$
be a [[regular polynomial observables]], regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].
Then the _interacting quantum [[BV-differential]]_ on the [[interacting field algebra]] on [[regular polynomial observables]]
(def. \ref{FieldAlgebraObservablesInteracting}) is the [[conjugation]] of the plain global [[BV-differential]] $\{-S',-\}$ (def. \ref{ComplexBVBRSTGlobal}) by the [[quantum Møller operator]] induced by $S_{int}$ (def. \ref{MollerOperatorOnRegularPolynomialObservables}):
$$
\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\,.
$$
=--
([Rejzner 11, (5.38)](quantum+master+equation#Rejzner11))
+-- {: .num_prop #QuantumMasterEquation}
###### Proposition
**([[quantum master equation]] and [[quantum master Ward identity]] on [[regular polynomial observables]])**
Consider an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0
$$
S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
\,,
$$
Then the following are equivalent:
1. The _[[quantum master equation]]_ (QME)
$$
\label{OnRegularObservablesQuantumMasterEquation}
\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
+
i \hbar \Delta_{BV}( S' + S_{int} )
\;=\;
0
\,.
$$
1. The [[perturbative S-matrix]] (def. \ref{OnRegularObservablesPerturbativeSMatrix}) is $BV$-closed
$$
\{-S', \mathcal{S}(S_{int})\} = 0
\,.
$$
1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[retarded products]]_:
$$
\label{OnRegularObservablesQuantumMasterWardIdentity}
\mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1}
\;=\;
-
\left(
\left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}}
+ i \hbar \Delta_{BV}
\right)
$$
([Dütsch 18, (4.2)](Ward+identity#Duetsch18))
expressing the interacting quantum [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum}) as the sum of the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the _total_ [[action functional]] $S' + S_{int}$ and $i \hbar$ times the [[BV-operator]] ([BV-operator](BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator)).
1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[time-ordered products]]_:
$$
\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
\mathcal{S}(-S_{int})
\star_F
\{-S', \mathcal{S}(S_{int}) \star_F (-)\}
\;=\;
-
\left(
\left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}}
+ i \hbar \Delta_{BV}
\right)
$$
([Dütsch 18, (4.8)](Ward+identity#Duetsch18))
=--
([Rejzner 11, (5.35) - (5.38)](quantum+master+equation#Rejzner11), following [Hollands 07, (342)-(345)](Ward+identity#Hollands07))
+-- {: .proof}
###### Proof
To see that the first two conditions are equivalent, we compute as follows
$$
\label{QuantumMasterOnRegularObservablesBVDifferentialOfSMatrixInTerms}
\begin{aligned}
\left\{
-S', \mathcal{S}(S_{int})
\right\}
& =
\left\{
-S'
,
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar} S_{int}
\right)
\right\}
\\
& =
\underset{
{
\tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}}
}
\atop
{
\star_F
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar} S_{int}
\right)
}
}{
\underbrace{
\left\{
-S'
,
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar} S_{int}
\right)
\right\}_{\mathcal{T}}
}
}
-
i \hbar
\underset{
{
\left(
\tfrac{1}{i \hbar}
\Delta_{BV}(S_{int})
+
\tfrac{1}{2 (i \hbar)^2}
\left\{
S_{int}, S_{int}
\right\}_{\mathcal{T}}
\right)
}
\atop
{
\star_{F}
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar}
S_{int}
\right)
}
}{
\underbrace{
\Delta_{BV}
\left(
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar} S_{int}
\right)
\right)
}
}
\\
& =
\tfrac{-1}{i \hbar}
\underset{ \text{QME} }{
\underbrace{
\left(
\{S',S_{int}\}
+
\tfrac{1}{2}\{S_{int}, S_{int}\}
+
i \hbar \Delta_{BV}(S_{int})
\right)
}
}
\star_F
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar}
S_{int}
\right)
\end{aligned}
$$
Here in the first step we used the definition of the [[BV-operator]] (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to rewrite the plain antibracket in terms of the time-ordered antibracket (def. \ref{AntibracketTimeOrdered}), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (prop. \ref{AntibracketBVOperatorRelation}) and under the first brace the consequence of this statement for application to exponentials (example \ref{TimeOrderedExponentialBVOperator}). Finally we collected terms, and to "complete the square" we added the terms on the left of
$$
\frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}}
-
i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0
$$
which vanish because, by definition of [[gauge fixing]] (def. \ref{GaugeFixingLagrangianDensity}), the free gauge-fixed action functional $S'$ is independent of [[antifields]].
But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the [[inverse]] $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$, this implies the claim.
Next we show that the [[quantum master equation]] implies the [[quantum master Ward identities]].
We use that the BV-differential $\{-S',-\}$ is a [[derivation]] of the [[Wick algebra]] product $\star_H$ (lemma \ref{DerivationBVDifferentialForWickAlgebra}).
First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also
$\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.
Thus we compute as follows:
$$
\begin{aligned}
\{-S', -\} \circ \mathcal{R}^{-1}(A)
& =
\{-S', \mathcal{R}^{-1}(A)\}
\\
& =
\left\{
{ \, \atop \, }
-S',
\mathcal{S}(S_{int})^{-1}
\star_H
\left( \mathcal{S}(S_{int}) \star_F a \right)
{\, \atop \,}
\right\}
\\
& = \phantom{+}
\underset{
= 0
}{
\underbrace{
\left\{
-S', \mathcal{S}(S_{int})^{-1}
\right\}
}
}
\star_H
\left(
\mathcal{S}(S_{int}) \star_F A
\right)
\\
& \phantom{=}
+
\mathcal{S}(S_{int})^{-1}
\star_H
\left\{
-S',
\mathcal{S}(S_{int}) \star_F A
\right\}
\\
& =
\mathcal{S}(S_{int})^{-1}
\star_H
\left(
\underset{
= 1
}{
\underbrace{
\mathcal{S}(+ S_{int})
\star_F
\mathcal{S}(- S_{int})
}
}
\star_F
\left\{
-S',
\mathcal{S}(S_{int}) \star_F A
\right\}
\right)
\\
& =
\mathcal{S}(S_{int})^{-1}
\star_H
\left(
\mathcal{S}(+ S_{int})
\star_F
\underset{ (\ast) }{
\underbrace{
\mathcal{S}(- S_{int})
\star_F
\left\{
-S',
\mathcal{S}(S_{int}) \star_F A
\right\}
}
}
\right)
\\
& =
\mathcal{R}^{-1}
\left(
\underset{ (\ast) }{
\underbrace{
\phantom{\, \atop \,}
\mathcal{S}(-S_{int})
\star_F
\left\{
-S',
\mathcal{S}(S_{int}) \star_F A
\right\}
}
}
\right)
\end{aligned}
$$
By applying $\mathcal{R}$ to both sides of this equation, this means
first of all that the interacting quantum BV-differential
is equivalently given by
$$
\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
\;=\;
\mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\}
\,,
$$
hence that if either version (eq:OnRegularObservablesQuantumMasterWardIdentity) or (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) of the [[master Ward identity]] holds, it implies the other.
Now expanding out the definition of $\mathcal{S}$ (def. \ref{OnRegularObservablesPerturbativeSMatrix}) and expressing $\{-S',-\}$ via the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) and the [[BV-operator]] $\Delta_{BV}$ (prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) as
$$
\{-S',-\}
\;=\;
\{-S',-\}_{\mathcal{T}} - i \hbar \Delta_{BV}
$$
(on [[regular polynomial observables]]), we continue computing as follows:
$$
\label{QMESecondStep}
\begin{aligned}
& \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A )
\\
& =
\exp_{\mathcal{T}}
\left(
\tfrac{-1}{i \hbar} S_{int}
\right)
\star_F
\left\{
-S',
\exp_{\mathcal{T}}
\left(
\tfrac{1}{i \hbar} S_{int}
\right)
\star_F
A
\right\}
\\
& =
\exp_{\mathcal{T}}
\left(
\tfrac{-1}{i \hbar}
S_{int}
\right)
\star_F
\left(
\left\{
-S',
\exp_{\mathcal{T}}
\left(
\tfrac{ 1 }{i \hbar} S_{int}
\right)
\star_F
A
\right\}_{\mathcal{T}}
-
i \hbar
\Delta_{BV}
\left(
\exp_{\mathcal{T}}
\left(
\tfrac{ 1 }{i \hbar} S_{int}
\right)
\star_F
A
\right)
\right)
\\
&
\phantom{+}
=
\tfrac{1}{i \hbar} \{ -S', S_{int} \}_{\mathcal{T}} \star_F A
+
\{-S', A\}_{\mathcal{T}}
\\
&
\phantom{=}
-
i \hbar
\exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right)
\star_F
\left(
\underset{
{
\left(
\tfrac{1}{i \hbar}\Delta_{BV}(S_{int})
+
\tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}
\right)
}
\atop
{
\star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right)
}
}{
\underbrace{
\Delta_{BV}
\left(
\exp_{\mathcal{T}}
\left(
\tfrac{ 1}{i \hbar} S_{int}
\right)
\right)
}
}
\star_F A
\,+\,
\exp_{\mathcal{T}}
\left(
\tfrac{ 1}{i \hbar} S_{int}
\right)
\star_F
\Delta_{BV}(A)
\,+\,
\underset{
{
\exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)
}
\atop
{
\star_F
\tfrac{ 1}{i \hbar}
\{S_{int}, A\}
}
}{
\underbrace{
\left\{
\exp_{\mathcal{T}}
\left(
\tfrac{ 1}{i \hbar} S_{int}
\right)
\,,\,
A
\right\}_{\mathcal{T}}
}
}
\right)
\\
& =
-
\left(
\{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
+
i \hbar \Delta_{BV}(A)
\right)
\\
& \phantom{=}
-
\tfrac{1}{i \hbar}
\underset{ \text{QME} }{
\underbrace{
\left(
\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
+
i \hbar \Delta_{BV}( S' + S_{int} )
\right)
}}
\star_F A
\\
& =
-
\left(
\{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
+
i \hbar \Delta_{BV}(A)
\right)
\end{aligned}
$$
Here in the line with the braces we used that the [[BV-operator]] is a [[derivation]] of the [[time-ordered product]] up to correction by the time-ordered [[antibracket]] (prop. \ref{AntibracketBVOperatorRelation}), and under the first brace we used the effect of that property on time-ordered exponentials (example \ref{TimeOrderedExponentialBVOperator}), while under the second brace we used that $\{(-),A\}_{\mathcal{T}}$ is a derivation of the time-ordered product. Finally we have collected terms, added $0 = \{S',S'\} + i \hbar \Delta_{BV}(S')$ as before, and then used the QME.
This shows that the quantum [[master Ward identities]] follow from the [[quantum master equation]]. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.
To see this, observe that with the BV-differential being nilpotent, also its conjugation by $\mathcal{R}$ is, so that with the above we have:
$$
\begin{aligned}
& \left( \{-S',-\}\right)^2 = 0
\\
\Leftrightarrow
\;
&
\left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0
\\
\Leftrightarrow
\;
&
\underset{
\left\{
{\, \atop \,}
\tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}}
+
i \hbar \Delta_{BV}(S' + S_{int})
\,,\,
(-)
\right\}
}{
\underbrace{
\left(
\{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
\right)^2
}
} = 0
\end{aligned}
$$
Here under the brace we computed as follows:
$$
\begin{aligned}
\left(
\{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
\right)^2
& =
\phantom{+}
\underset{
\tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
}{
\underbrace{
\{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
}}
\\
&
\phantom{=}
+
i \hbar
\underset{
\{ \Delta_{BV}(S'+ S)\,,\, (-) \}_{\mathcal{T}}
}{
\underbrace{
\left(
\{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV}
+
\Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}}
\right)
}}
\\
&
\phantom{=}
+
(i \hbar)^2
\underset{= 0}
{
\underbrace{
\Delta_{BV} \circ \Delta_{BV}
}
}
\end{aligned}
\,.
$$
where, in turn, the term under the first brace follows by the graded [[Jacobi identity]], the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).
=--
$\,$
**[[Ward identities]]**
{#WardIdentities}
The _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation}) expresses the relation between the [[quantum field theory|quantum]] (measured by [[Planck's constant]] $\hbar$) [[interacting field theory|interacting]] (measured by the [[coupling constant]] $g$) [[equations of motion]] to the [[classical field theory|classical]] [[free field]] [[equations of motion]] at $\hbar, g\to 0$ (remark \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below). As such it generalizes the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}), to which it reduces for $g = 0$ (example \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below) as well as the _classical master Ward identity_, which is the case for $\hbar = 0$ (example \ref{MasterWardIdentityClassical} below).
Applied to products of the [[equations of motion]] with any given [[observable]], the master Ward identity becomes a particular _Ward identity_.
This is of interest notably in view of [[Noether's theorem]] (prop. \ref{NoethersFirstTheorem}), which says that every [[infinitesimal symmetry of the Lagrangian]] of, in particular, the given [[free field theory]], corresponds to a [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), hence a [[horizontal differential form]] whose [[total spacetime derivative]] vanishes up to a term proportional to the [[equations of motion]]. Under [[transgression of variational differential forms|transgression]] to [[local observables]] this is a relation of the form
$$
div \mathbf{J} = 0 \phantom{AAA} \text{on-shell}
\,,
$$
where "on shell" means up to the ideal generated by the [[classical field theory|classical]] [[free field theory|free]] [[equations of motion]]. Hence for the case of [[local observables]] of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original [[conserved current]] to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example \ref{NoetherCurrentConservationQuantumCorrection} below).
As one [[extension of distributions|extends]] the [[time-ordered products]] to coinciding interaction points in [[renormalization|("re"-)normalization]]
of the [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}), the [[quantum master equation]]/[[master Ward identity]] becomes a _[[renormalization condition]]_ (def. \ref{RenormalizationConditions}, prop. \ref{BasicConditionsRenormalization}). If this condition fails, one speaks of a _[[quantum anomaly]]_. Specifically if the Ward identity for an [[infinitesimal gauge symmetry]] is violated, one speaks of a _[[gauge anomaly]]_.
$\,$
+-- {: .num_defn #OnRegularPolynomialObservablesMasterWardIdentity}
###### Definition
Consider a [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] $(E_{\text{BV-BRST}}, \mathbf{L}')$ (def. \ref{GaugeFixingLagrangianDensity}) with global [[BV-differential]] on [[regular polynomial observables]]
$$
\{-S',(-)\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
(def. \ref{ComplexBVBRSTGlobal}).
Let moreover
$$
g S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g ] ]
$$
be a [[regular polynomial observable]] (regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]]) such that the total action $S' + g S_{int}$ satisfies the [[quantum master equation]] (prop. \ref{QuantumMasterEquation}); and write
$$
\mathcal{R}^{-1}(-)
\;\coloneqq\;
\mathcal{S}(g S_{int})^{-1}
\star_H
(\mathcal{S}(g S_{int}) \star_F (-))
$$
for the corresponding [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}).
Then by prop. \ref{QuantumMasterEquation} we have
$$
\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
\{-S',(-)\} \circ \mathcal{R}^{-1} \;=\;
\mathcal{R}^{-1}
\left(\left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}} -i \hbar \Delta_{BV}\right)
$$
This is the _quantum master Ward identity_ on [[regular polynomial observables]], i.e. before [[renormalization]].
=--
([Rejzner 13, (37)](Ward+identity#Rejzner13))
+-- {: .num_remark #QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}
###### Remark
**([[quantum master Ward identity]] relates [[quantum field theory|quantum]] [[interacting field theory|interacting field]] [[equation of motion|EOMs]] to [[classical field theory|classical]] [[free field]] [[equation of motion|EOMs]])**
For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ] $ the [[quantum master Ward identity]] on [[regular polynomial observables]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reads
$$
\label{RearrangedMasterQuantumWard}
\mathcal{R}^{-1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\right)
\;=\;
\{-S', \mathcal{R}^{-1}(A) \}
$$
The term on the right is manifestly in the [[image]] of the global [[BV-differential]] $\{-S',-\}$ of the [[free field theory]] (def. \ref{ComplexBVBRSTGlobal}) and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] (eq:OnShellPolynomialObservablesAsBVCohomology)
$$
\underset{
\text{on-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
}}
\;\simeq\;
\underset{
\text{off-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
}}/im(\{-S',-\})
$$
(by example \ref{BVDifferentialGlobal}).
Hence
$$
\mathcal{R}^{-1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\right)
\;=\;
0
\phantom{AAA}
\text{on-shell}
$$
In contrast, the left hand side is the [[interacting field observable]] (via def. \ref{MollerOperatorOnRegularPolynomialObservables}) of the sum of the [[time-ordered product|time-ordered]] [[antibracket]] with the [[action functional]] of the [[interacting field theory]] and a quantum correction given by the [[BV-operator]]. If we use the definition of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{RearrangedMasterWardWithOnShell}) we may equivalently re-write this as
$$
\label{RearrangedMasterWardWithOnShell}
\mathcal{R}^{-1}
\left(
\left\{ -S' \,,\, A \right\}
+
\left\{ -g S_{int} \,,\, A \right\}_{\mathcal{T}}
\right)
\;=\;
0
\phantom{AAA}
\text{on-shell}
$$
Hence the [[quantum master Ward identity]] expresses a relation between the ideal spanned by the [[classical field theory|classical]] [[free field theory|free field]] [[equations of motion]] and the [[quantum field theory|quantum]] [[interacting field theory|interacting field]] equations of motion.
=--
+-- {: .num_example #SchwingerDysonReductionOfQuantumMasterWardIdentity}
###### Example
**([[free field]]-limit of [[master Ward identity]] is [[Schwinger-Dyson equation]])**
In the [[free field]]-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to
$$
\left\{ -S' \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\;=\;
\{-S', A \}
$$
which is the defining equation for the [[BV-operator]] (eq:BVOperatorDefiningRelation), hence is isomorphic (under $\mathcal{T}$) to the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger})
=--
+-- {: .num_example #MasterWardIdentityClassical}
###### Example
**([[classical limit]] of [[quantum master Ward identity]])**
In the [[classical limit]] $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to
$$
\mathcal{R}^{1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}
\right)
\;=\;
\{-S', \mathcal{R}^{-1}(A) \}
$$
This says that the [[interacting field observable]] corresponding to the global [[antibracket]] with the action functional of the [[interacting field theory]] vanishes on-shell, classically.
Applied to an observable which is [[linear map|linear]] in the [[antifields]]
$$
A
\;=\;
\underset{\Sigma}{\int}
A^a(x)
\mathbf{\Phi}^\ddagger_a(x)
\,
dvol_\Sigma(x)
$$
this yields
$$
\begin{aligned}
0
& =
\{-S', \mathcal{R}^{-1}(A)\}
+
\mathcal{R}^{-1}
\left(
\left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}}
\right)
\\
& =
\underset{\Sigma}{\int}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
\, dvol_\Sigma(x)
+
\mathcal{R}^{-1}
\left(
\underset{\Sigma}{\int}
A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
\, dvol_\Sigma(x)
\right)
\end{aligned}
$$
This is the _classical master Ward identity_ according to ([Dütsch-Fredenhagen 02](Ward+identity#DuetschFredenhagen02), [Brennecke-Dütsch 07, (5.5)](Ward+identity#BrennecketDuetsch07)), following ([Dütsch-Boas 02](Ward+identity#DuetschBoas02)).
=--
+-- {: .num_example #NoetherCurrentConservationQuantumCorrection}
###### Example
**(quantum correction to [[Noether's theorem|Noether current]] [[conserved current|conservation]])**
Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an [[evolutionary vector field]], which is an [[infinitesimal symmetry of the Lagrangian]] $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding [[conserved current]], by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}), so that
$$
\begin{aligned}
d J_{\hat v}
& =
\iota_{\hat v} \delta \mathbf{L}'
\\
& =
(v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a}
\phantom{AAA}
\in
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
\end{aligned}
$$
by (eq:CurrentNoetherConservation), where in the second line we just rewrote the expression in components (eq:EulerLagrangeEquationGeneral)
$$
v^a
\,,
\frac{\delta_{EL} L'}{\delta \phi^a}
\;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})
$$
and re-arranged suggestively.
Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of [[bump function]], we obtain the [[local observables]]
$$
\begin{aligned}
A_{sw}
& \coloneqq
\underset{\Sigma}{\int}
\underset{
A^a(x)
}{
\underbrace{
a_{sw}(x)
v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
}
}
\mathbf{\Phi}^\ddagger_a(x)
\,
dvol_\Sigma(x)
\\
& \coloneqq
\tau_\Sigma( a_{sw} v^a \phi^{\ddagger}_a \, dvol_\Sigma)
\end{aligned}
$$
and
$$
\begin{aligned}
(div \mathbf{J})_{sw}
& \coloneqq
\underset{\Sigma}{\int}
\underset{
A^a(x)
}{
\underbrace{
a_{sw}(x)
v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
}
}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
\,
dvol_\Sigma(x)
\\
& \coloneqq
\tau_\Sigma
\left(
a_{sw} v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a}
\, dvol_\Sigma
\right)
\end{aligned}
$$
by [[transgression of variational differential forms]].
This is such that
$$
\left\{
-S' , A_{sw}
\right\}
=
(div \mathbf{J})_{sw}
\,.
$$
Hence applied to this choice of local observable $A$, the quantum master Ward identity (eq:RearrangedMasterWardWithOnShell) now says that
$$
\mathcal{R}^{-1}
\left(
{\, \atop \,}
(div \mathbf{J})_{sw}
\right)
\;=\;
\mathcal{R}^{-1}
\left(
{\, \atop \,}
\{g S_{int}, A_{sw} \}_{\mathcal{T}}
{\, \atop \,}
\right)
\phantom{AAA}
\text{on-shell}
$$
Hence the [[interacting field observable]]-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.
=--
$\,$
This concludes our discussion of perturbative [[quantum observables]] of [[interacting field theories]]. In the _[next chapter](#Renormalization)_ wé discuss explicitly the [[induction|inductive]] construction via _[[renormalization|("re"-)normalization]]_
of [[time-ordered products]]/[[Feynman amplitudes]] as well as the various incarnations of the [[renormalization group|re-normalization group]]
passing between different choices of such [[renormalization|("re"-)normalizations]].
|
A first idea of quantum field theory -- Lagrangians | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Lagrangians | ## Lagrangians
{#Lagrangians}
In this chapter we discuss the following topics:
* _[Lagrangian densities](#LagrangianDensities)_
* _[Euler-Lagrange forms and Presymplectic currents](#ELFormsAndPresymplecticCurrents)_
* _[Euler-Lagrange equations of motion](#ELEquationsOfMotion)_
$\,$
Given any [[type]] of [[field (physics)|fields]] (def. \ref{FieldsAndFieldBundles}), those [[field histories]] that
are to be regarded as "physically realizable" (if we think of the field theory as a description of the [[observable universe]])
should satisfy some [[differential equation]] -- the _[[equation of motion]]_ -- meaning that realizability of any field histories may
be checked upon restricting the configuration to the [[infinitesimal neighbourhoods]] (example \ref{InfinitesimalNeighbourhood}) of each spacetime point. This expresses the physical absence of "action at a distance" and is one aspect of what it means to have
a _[[local field theory]]_. By remark \ref{JetBundleInTermsOfSyntheticDifferentialGeometry} this means that
[[equations of motion]] of a field theory are [[equations]] among the [[coordinates]] of the [[jet bundle]] of the [[field bundle]].
For many field theories of interest, their [[differential equation|differential]] [[equation of motion]]
is not a random [[partial differential equations]], but is of the special kind that exhibits the "[[principle of extremal action]]" (prop. \ref{PrincipleOfExtremalAction} below)
determined by a _[[local Lagrangian density]]_
(def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below). These are called _[[Lagrangian field theories]]_, and this is what we consider here.
Namely among all the [[variational differential forms]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
two kinds stand out, namley the 0-forms in $\Omega^{0,0}_\Sigma(E)$ -- the smooth functions -- and the horizontal $p+1$-forms
$\Omega^{p+1,0}_\Sigma(E)$ -- to be called the _[[Lagrangian densities]] $\mathbf{L}$_ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below) -- since these occupy the two "corners" of the [[variational bicomplex]] (eq:VariationalBicomplexDiagram).
There is not much to say about the 0-forms, but the [[Lagrangian densities]] $\mathbf{L}$ do inherit special structure
from their special position in the [[variational bicomplex]]:
Their [[variational derivative]] $\delta \mathbf{L}$ uniquely decomposes as
1. the _[[Euler-Lagrange derivative]]_ $\delta_{EL} \mathbf{L}$ which is proportional to the variation of the fields
(instead of their derivatives)
1. the [[total derivative|total spacetime derivative]] $d \Theta_{BFV}$
of a potential $\Theta_{BFV}$ for a _[[presymplectic current]]_ $\Omega_{BFV} \coloneqq \delta \Theta_{BFV}$.
This is prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} below:
$$
\delta \mathbf{L}
\;=\;
\underset{ \text{Euler-Lagrange variation}
}{\underbrace{\delta_{EL}\mathbf{L}}} - d \underset{\text{presymplectic current}}{\underbrace{\Theta_{BFV}}}
\,.
$$
These two terms play a pivotal role in the theory: The condition that the first term vanishes
on [[field histories]] is a [[differential equation]] on field histories,
called the _[[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]_ (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} below).
The space of solutions to this [[differential equation]], called the _[[on-shell]] [[space of field histories]]_
$$
\label{InclusionOfOnShellSpaceOfFieldHistories}
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
\overset{\phantom{AAA}}{\hookrightarrow}
\Gamma_\Sigma(E)
$$
has the interpretation of the space of "physically realizable field histories".
This is the key object of study in the following chapters. Often this is
referred to as the space of _[[classical field theory|classical]] field histories_,
indicating that this does not yet reflect the full [[quantum field theory]].
Indeed, there is also the second term in the variational derivative of the Lagrangian density,
the [[presymplectic current]] $\Theta_{BFV}$, and this implies a [[presymplectic structure]] on
the on-shell space of field histories (def. \ref{PhaseSpaceAssociatedWithCauchySurface} below)
which encodes [[deformations]] of the algebra of
smooth functions on $\Gamma_\Sigma(E)$. This deformation is the _[[quantization]]_ of the field
theory to an actual [[quantum field theory]], which we discuss [below](#Quantization).
$$
\array{
&&& \delta \mathbf{L}
\\
&&& =
\\
& & \delta_{EL}\mathbf{L} &- & d \Theta_{BFV} &
\\
& \swarrow && && \searrow
\\
\array{
\text{classical}
\\
\text{field theory}
}
&& && &&
\array{
\text{deformation to}
\\
\text{quantum}
\\
\text{field theory}
}
}
$$
$\,$
**[[Lagrangian densities]]**
{#LagrangianDensities}
+-- {: .num_defn #LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}
###### Definition
**([[local Lagrangian density]])**
Given a [[field bundle]] $E$ over a $(p+1)$-dimensional [[Minkowski spacetime]] $\Sigma$ as in example \ref{TrivialVectorBundleAsAFieldBundle}, then a _[[local Lagrangian density]]_ $\mathbf{L}$ (for the type of field thus defined) is a [[horizontal differential form]] of degree $(p+1)$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) on the corresponding [[jet bundle]]
(def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}):
$$
\mathbf{L} \;\in \; \Omega^{p+1,0}_{\Sigma}(E)
\,.
$$
By example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}
in terms of the given [[volume form]] on spacetimes, any such Lagrangian density may uniquely be written as
$$
\label{LagrangianFunctionViaVolumeForm}
\mathbf{L} = L \, dvol_\Sigma
$$
where the [[coefficient]] function (the _Lagrangian function_)
is a smooth function on the spacetime and field coordinates:
$$
L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots )
\,.
$$
where by prop. \ref{JetBundleIsLocallyProManifold} $L((x^\mu), \cdots)$ depends locally on an arbitrary but finite
order of derivatives $\phi^a_{,\mu_1 \cdots \mu_k}$.
We say that a [[field bundle]] $E \overset{fb}{\to} \Sigma$ (def. \ref{FieldsAndFieldBundles}) equipped with a [[local Lagrangian density]] $\mathbf{L}$
is (or defines) a _[[prequantum field theory|prequantum]] [[Lagrangian field theory]]_ on the [[spacetime]] $\Sigma$.
=--
+-- {: .num_remark #ParameterizedLagrangianDensities}
###### Remark
**(parameterized and [[physical unit]]-less [[Lagrangian densities]])**
More generally we may consider parameterized collections of [[Lagrangian densities]], i.e. functions
$$
\mathbf{L}_{(-)}
\;\colon\;
U \longrightarrow \Omega^{p+1,0}_\Sigma(E)
$$
for $U$ some [[Cartesian space]] or generally some [[super Cartesian space]].
For example all [[Lagrangian densities]] considered in [[relativistic field theory]] are naturally [[smooth functions]] of the scale of the [[metric]] $\eta$ (def. \ref{SpacetimeAsMatrices})
$$
\array{
\mathbb{R}_{\gt 0}
&\overset{}{\longrightarrow}&
\Omega^{p+1,0}_\Sigma(E)
\\
r &\mapsto& \mathbf{L}_{r^2\eta}
}
$$
But by the discussion in remark \ref{MinkowskiMetricAndPhysicalUnitOfLength}, in [[physics]] a rescaling of the [[metric]] is interpreted as reflecting but a change of [[physical units]] of [[length]]/[[distance]]. Hence if a [[Lagrangian density]] is supposed to express intrinsic content of a [[theory (physics)|physical theory]], it should remain unchanged under such a change of [[physical units]].
This is achieved by having the Lagrangian be parameterized by _further_ parameters, whose corresponding [[physical units]] compensate that of the metric such as to make the Lagrangian density "[[physical unit]]-less".
This means to consider parameter spaces $U$ equipped with an [[action]] of the multiplicative [[group]] $\mathbb{R}_{\gt 0}$ of [[positive real numbers]], and parameterized Lagrangians
$$
\mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E)
$$
which are [[invariant]] under this [[action]].
=--
+-- {: .num_remark #LocallyVariationalFieldTheory}
###### Remark
**([[locally variational field theory]] and Lagrangian [[circle n-bundle with connection|p-gerbe connection]])**
If the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) is not just a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) then a Lagrangian density for a given [[equation of motion]] may not exist
as a globally defined differential $(p+1)$-form, but only as a [[circle n-bundle with connection|p-gerbe connection]].
This is the case for _[[locally variational field theories]]_ such as the _[[charged particle]]_, the _[[WZW model]]_
and generally theories involving _[[higher WZW terms]]_. For more on this see the exposition at _[[schreiber:Higher Structures|Higher Structures in Physics]]_.
=--
+-- {: .num_example #LagrangianForFreeScalarFieldOnMinkowskiSpacetime}
###### Example
**([[local Lagrangian density]] for [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]])**
Consider the [[field bundle]] for the [[real scalar field]] from example \ref{RealScalarFieldBundle},
i.e. the [[trivial line bundle]] over [[Minkowski spacetime]].
According to def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}
its [[jet bundle]] $J^\infty_\Sigma(E)$ has canonical coordinates
$$
\left\{
\{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu_1 \mu_2}\}, \cdots
\right\}
\,.
$$
In these coordinates, the [[local Lagrangian density]] $L \in \Omega^{p+1,0}(\Sigma)$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
defining the [[free field|free]] [[real scalar field]] of [[mass]] $m \in \mathbb{R}$ on $\Sigma$ is
$$
L
\coloneqq
\tfrac{1}{2}
\left(
\eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
-
m^2 \phi^2
\right)
\mathrm{dvol}_\Sigma
\,.
$$
This is naturally thought of as a collection of Lagrangians smoothly parameterized by the
[[metric]] $\eta$ and the [[mass]] $m$. For this to be [[physical unit]]-free in the sense of
remark \ref{ParameterizedLagrangianDensities} the [[physical unit]] of the parameter $m$ must be
that of the inverse metric, hence must be an inverse [[length]] according to remark \ref{MinkowskiMetricAndPhysicalUnitOfLength}
This is the _inverse [[Compton wavelength]]_ $\ell_m = \hbar / m c$ (eq:ComptonWavelength) and hence
the [[physical unit]]-free version of the Lagrangian density for the free scalar particle is
$$
\mathbf{L}_{\eta,\ell_m}
\:\coloneqq\;
\tfrac{\ell_m^2}{2}
\left(
\eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
-
\left( \tfrac{m c}{\hbar} \right)^2 \phi^2
\right)
\mathrm{dvol}_\Sigma
\,.
$$
=--
+-- {: .num_example #phintheoryLagrangian}
###### Example
**([[phi^n theory]])**
Consider the [[field bundle]] for the [[real scalar field]] from example \ref{RealScalarFieldBundle},
i.e. the [[trivial line bundle]] over [[Minkowski spacetime]]. More generally we may
consider adding to the [[free field]] [[Lagrangian density]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}
some power of the field coordinate
$$
\mathbf{L}_{int}
\;\coloneqq\;
g \phi^n \, dvol_\Sigma
\,,
$$
for $g \in \mathbb{R}$ some number, here called the _[[coupling constant]]_.
The [[interacting field theory|interacting]] [[Lagrangian field theory]] defined by the resulting [[Lagrangian density]]
$$
\mathbf{L}
+
\mathbf{L}_{int}
\;=\;
\tfrac{1}{2}
\left(
\eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
-
m^2 \phi^2
+
g \phi^n
\right)
\mathrm{dvol}_\Sigma
$$
is usually called just _[[phi^n theory]]_.
=--
+-- {: .num_example #ElectromagnetismLagrangianDensity}
###### Example
**([[local Lagrangian density]] for [[free field|free]] [[electromagnetic field]])**
Consider the [[field bundle]] $T^\ast \Sigma \to \Sigma$ for the [[electromagnetic field]] on [[Minkowski spacetime]] from example \ref{Electromagnetism},
i.e. the [[cotangent bundle]], which over Minkowski spacetime happens to be a [[trivial vector bundle]] of [[rank of a vector bundle|rank]]
$p+1$. With [[fiber]] coordinates taken to be $(a_\mu)_{\mu = 0}^p$, the induced fiber coordinates on the
corresponding [[jet bundle]] $J^\infty_\Sigma(T^\ast \Sigma)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) are
$( (x^\mu), (a_\mu), (a_{\mu,\nu}), (a_{\mu,\nu_1 \nu_2}), \cdots )$.
Consider then the [[local Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) given by
$$
\label{ElectromagnetismLagrangian}
\mathbf{L}
\;\coloneqq\;
\tfrac{1}{2}
f_{\mu \nu} f^{\mu \nu} dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_\Sigma(T^\ast \Sigma)
\,,
$$
where $f_{\mu \nu} \coloneqq \tfrac{1}{2}(a_{\nu,\mu} - a_{\mu,\nu})$ are the components of the universal [[Faraday tensor]] on the [[jet bundle]] from example \ref{JetFaraday}.
This is the [[Lagrangian density]] that defines the Lagrangian field theory of _[[free field|free]] [[electromagnetism]]_.
Here for $A \in \Gamma_\Sigma(T^\ast \Sigma)$ an [[electromagnetic field]] history ([[vector potential]]),
then the [[pullback of differential forms|pullback]] of $f_{\mu \nu}$ along its [[jet prolongation]] (def. \ref{JetProlongation})
is the corresponding component of the [[Faraday tensor]] (eq:TensorFaraday):
$$
\begin{aligned}
\left(
j^\infty_\Sigma(A)
\right)^\ast(f_{\mu \nu})
& =
(d A)_{\mu \nu}
\\
& = F_{\mu \nu}
\end{aligned}
$$
It follows that the pullback of the Lagrangian (eq:ElectromagnetismLagrangian) along the
jet prologation of the electromagnetic field is
$$
\begin{aligned}
\left(
j^\infty_\Sigma(A)
\right)^\ast \mathbf{L}
& =
\tfrac{1}{2}
F_{\mu \nu} F^{\mu \nu} dvol_\Sigma
\\
& =
\tfrac{1}{2} F \wedge \star_\eta F
\end{aligned}
$$
Here $\star_\eta$ denotes the [[Hodge star operator]] of [[Minkowski spacetime]].
=--
More generally:
+-- {: .num_example #YangMillsLagrangian}
###### Example
**([[Lagrangian density]] for [[Yang-Mills theory]] on [[Minkowski spacetime]])**
Let $\mathfrak{g}$ be a [[finite number|finite]] [[dimension|dimensional]] [[Lie algebra]] which is [[semisimple Lie algebra|semisimple]]. This means that the [[Killing form]] [[invariant polynomial]]
$$
k \colon \mathfrak{g} \otimes \mathfrak{g} \longrightarrow \mathbb{R}
$$
is a non-degenerate [[bilinear form]]. Examples include the [[special unitary Lie algebras]] $\mathfrak{so}(n)$.
Then for $E = T^\ast \Sigma \otimes \mathfrak{g}$
the [[field bundle]] for [[Yang-Mills theory]] as in example \ref{YangMillsFieldOverMinkowski},
the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) $\mathfrak{g}$-[[Yang-Mills theory]] on [[Minkowski spacetime]] is
$$
\mathbf{L}
\;\coloneqq\;
\tfrac{1}{2}
k_{\alpha \beta} f^\alpha_{\mu \nu} f^{\beta \mu \nu}
\,
dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_\Sigma(T^\ast \Sigma)
\,,
$$
where
$$
f^\alpha_{\mu \nu}
\;\coloneqq\;
\tfrac{1}{2}
\left(
a^\alpha_{\nu,\mu}
-
a^\alpha_{\mu,\nu}
+
\gamma^{\alpha}{}_{\beta \gamma}
a^\beta_{\mu} a^\gamma_{\nu}
\right)
\;\in\;
\Omega^{0,0}_\Sigma(E)
$$
is the universal [[Yang-Mills theory|Yang-Mills]] [[field strength]] (eq:YangMillsJetFieldStrengthMinkowski).
For the purposes of [[perturbative quantum field theory]] (to be discussed below in chapter _[15. Interacting quantum fields](#InteractingQuantumFields)_) we may allow for a rescaling of the structure constants by (at this point) a [[real number]] $g$, to be called the _[[coupling constant]]_, and decompose the Lagrangian into a sum of a [[free field theory|free field theory Lagrangian]] (def. \ref{FreeFieldTheory}) and an [[interaction]] term:
$$
\begin{aligned}
\mathbf{L}
& =
\tfrac{1}{2}
k_{\alpha \beta}
\tfrac{1}{2}
\left(
a^\alpha_{\nu,\mu}
-
a^\alpha_{\mu,\nu}
+
g \gamma^{\alpha}{}_{\beta' \gamma'}
a^{\beta'}_{\mu} a^{\gamma'}_{\nu}
\right)
\tfrac{1}{2}
\left(
a^{\beta\nu,\mu}
-
a^{\beta \mu,\nu}
+
g \gamma^{\beta}{}_{\beta'' \gamma''}
a^{\beta''}_{\mu} a^{\gamma''}_{\nu}
\right)
\,dvol_\Sigma
\\
& =
\underset{
\mathbf{L}_{\mathrm{free}}
}{
\underbrace{
\tfrac{1}{2}
k_{\alpha \beta}
\tfrac{1}{2}
\left(
a^\alpha_{\nu,\mu}
-
a^\alpha_{\mu,\nu}
\right)
\tfrac{1}{2}
\left(
a^{\beta\nu,\mu}
-
a^{\beta \mu,\nu}
\right)
\,dvol_\Sigma
}
}
\\
& \phantom{=}
+
\underset{
\mathbf{L}_{int}
}{
\underbrace{
g
\,
k_{\alpha \beta}
\tfrac{1}{2}
\left(
a^\alpha_{\nu,\mu}
-
a^\alpha_{\mu,\nu}
\right)
\tfrac{1}{2}
\left(
\gamma^{\beta}{}_{\beta'' \gamma''}
a^{\beta''}_{\mu} a^{\gamma''}_{\nu}
\right)
\,dvol_\Sigma
\;
+
\;
g^2
\,
\tfrac{1}{2}
k_{\alpha \beta}
\tfrac{1}{2}
\left(
\gamma^{\alpha}{}_{\beta' \gamma'}
a^{\beta'}_{\mu} a^{\gamma'}_{\nu}
\right)
\tfrac{1}{2}
\left(
\gamma^{\beta}{}_{\beta'' \gamma''}
a^{\beta''}_{\mu} a^{\gamma''}_{\nu}
\right)
\,dvol_\Sigma
}
}
\\
\end{aligned}
\,,
$$
Notice that $\mathbf{L}_{free}$ is equivalently a sum of $dim(\mathfrak{g})$-copies of the Lagrangian for the [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity}).
On the other hand, for the purpose of exhibiting "[[non-perturbative effects]] due to [[instantons]]" in [[Yang-Mills theory]], one consider the rescaled Yang-Mills field coordinate
$$
\tilde a^\alpha_\mu
\;\coloneqq\;
\frac{1}{g} a^\alpha_\mu
$$
with corresponding [[field strength]]
$$
\tilde f^\alpha_{\mu \nu}
\;\coloneqq\;
\tfrac{1}{2}
\left(
\tilde a^\alpha_{\nu,\mu}
-
\tilde a^\alpha_{\mu,\nu}
+
\gamma^{\alpha}{}_{\beta \gamma}
\tilde a^\beta_{\mu} \tilde a^\gamma_{\nu}
\right)
\;\in\;
\Omega^{0,0}_\Sigma(E)
\,.
$$
In terms of this the expression for the Lagrangian is brought back to the abstract form it had before rescaling the structure constants by the [[coupling constant]], up to a _global_ rescaling of all terms by the _inverse square_ of the coupling constant:
$$
\label{MinkowskiYangMillsLagrangianWithCouplingConstantPulledOut}
\mathbf{L}
\;=\;
\frac{1}{g^2}
\tfrac{1}{2}
k_{\alpha \beta} \tilde f^\alpha_{\mu \nu} \tilde f^{\beta \mu \nu}
\,
dvol_\Sigma
\,.
$$
=--
+-- {: .num_example #BFieldLagrangianDensity}
###### Example
**([[local Lagrangian density]] for [[free field|free]] [[B-field]])**
Consider the [[field bundle]] $\wedge^2_\Sigma T^\ast \Sigma \to \Sigma$ for the [[B-field]] on [[Minkowski spacetime]] from example \ref{BField}. With [[fiber]] coordinates taken to be $(b_{\mu \nu})$ with
$$
b_{\mu \nu} = - b_{\nu \mu}
\,,
$$
the induced fiber coordinates on the
corresponding [[jet bundle]] $J^\infty_\Sigma(T^\ast \Sigma)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) are
$( (x^\mu), (b_{\mu \nu}), (b_{\mu \nu, \mu_1}), (b_{\mu \nu, \mu_1 \mu_2}), \cdots )$.
Consider then the [[local Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) given by
$$
\label{LagrangianForBField}
\mathbf{L}
\;\coloneqq\;
\tfrac{1}{2}
h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_\Sigma(\wedge^2_\Sigma T^\ast \Sigma)
\,,
$$
where $h_{\mu_1 \mu_2 \mu_3}$ are the components of the universal [[B-field|B-]][[field strength]] on the [[jet bundle]] from example \ref{BFieldJetFaraday}.
=--
+-- {: .num_example #LagrangianDensityForDiracField}
###### Example
**([[Lagrangian density]] for [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]])**
For $\Sigma$ [[Minkowski spacetime]] of [[dimension]] $p + 1 \in \{3,4,6,10\}$ (def. \ref{MinkowskiSpacetime}),
consider the [[field bundle]] $\Sigma \times S_{odd} \to \Sigma$ for the [[Dirac field]] from example \ref{DiracFieldBundle}.
With the two-component [[spinor]] [[field fiber]] coordinates from remark \ref{TwoComponentSpinorNotation}, the [[jet bundle]] has induced fiber coordinates as follows:
$$
\left(
\left(\psi^\alpha\right)
,
\left(
\psi^\alpha_{,\mu}
\right)
,
\cdots
\right)
\;=\;
\left(
\left(
(\chi_a), (\chi_{a,\mu}), \cdots
\right),
\left(
( \xi^{\dagger \dot a}), (\xi^{\dagger \dot a}_{,\mu}), \cdots
\right)
\right)
$$
All of these are odd-graded elements (def. \ref{SupercommutativeSuperalgebra}) in a [[Grassmann algebra]] (example \ref{GrassmannAlgebra}), hence anti-commute with each other, in generalization of (eq:DiracFieldCoordinatesAnticommute):
$$
\label{DiracFieldJetCoordinatesAnticommute}
\psi^\alpha_{,\mu_1 \cdots \mu_r}
\psi^\beta_{,\mu_1 \cdots \mu_s}
\;=\;
-
\psi^\beta_{,\mu_1 \cdots \mu_s}
\psi^\alpha_{,\mu_1 \cdots \mu_r}
\,.
$$
The [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
of the _massless [[free field theory|free]] [[Dirac field]]_ on [[Minkowski spacetime]] is
$$
\label{DiracFieldLagrangianMassless}
\mathbf{L}
\;\coloneqq\;
\overline{\psi} \, \gamma^\mu \psi_{,\mu}\, dvol_\Sigma
\,,
$$
given by the bilinear pairing $\overline{(-)}\Gamma(-)$ from prop. \ref{RealSpinorPairingsViaDivisionAlg}
of the field coordinate with its first spacetime derivative and
expressed here in two-component spinor field coordinates as in (eq:TwoComponentNotationForSpinorToVectorPairing), hence with the [[Dirac conjugate]] $\overline{\psi}$ (eq:DiracConjugate) on the left.
Specifically in [[spacetime]] [[dimension]] $p + 1 = 4$, the [[Lagrangian function]] for the _massive [[Dirac field]]_
of [[mass]] $m \in \mathbb{R}$ is
$$
\begin{aligned}
L
& \coloneqq
\underset{
\text{kinetic term}
}{
\underbrace{
i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu}
}
}
+
\underset{
\text{mass term}
}{
\underbrace{
m \overline{\psi} \psi
}}
\end{aligned}
$$
This is naturally thought of as a collection of Lagrangians smoothly parameterized by the
[[metric]] $\eta$ and the [[mass]] $m$. For this to be [[physical unit]]-free in the sense of
remark \ref{ParameterizedLagrangianDensities} the [[physical unit]] of the parameter $m$ must be
that of the inverse metric, hence must be an inverse [[length]] according to remark \ref{MinkowskiMetricAndPhysicalUnitOfLength}
This is the _inverse [[Compton wavelength]]_ $\ell_m = \hbar / m c$ (eq:ComptonWavelength) and hence
the [[physical unit]]-free version of the Lagrangian density for the free Dirac field is
$$
\mathbf{L}_{\eta,\ell_m}
\;\coloneqq\;
\ell_m
\left(
i \overline{\psi} \gamma^\mu \psi_{,\mu} + \left( \tfrac{m c}{\hbar} \right) \overline{\psi} \psi
\right) dvol_\Sigma
\,.
$$
=--
+-- {: .num_remark #RealityOfLagrangianDensityOfTheDiracField}
###### Remark
**([[real part|reality]] of the [[Lagrangian density]] of the [[Dirac field]])**
The kinetic term of the [[Lagrangian density]] for the [[Dirac field]] form def. \ref{LagrangianDensityForDiracField}
is a sum of two contributions, one for each [[chiral spinor]] component in the full [[Dirac spinor]] (remark \ref{TwoComponentSpinorNotation}):
$$
\begin{aligned}
i \overline{\psi} \gamma^\mu \psi_{,\mu}
& =
i
\underset{
-(\partial_\mu \xi^a ) \sigma^\mu_{a \dot c} \xi^{\dagger \dot c}
+ \partial_\mu(\chi^a \sigma^\mu_{a \dot c} \chi^{\dagger \dot c})
}{
\underbrace{
\xi^a \sigma^\mu_{a \dot c} \partial_\mu \xi^{\dagger \dot c}
}
}
+
\xi^\dagger_{\dot a} \tilde \sigma^{\mu \dot a c} \partial_\mu \xi_c
\\
& =
\xi^\dagger \tilde \sigma^\mu \partial_\mu \xi
+
\chi^\dagger \tilde \sigma^\mu \partial_\mu \chi
+
\partial_\mu(\xi \sigma^\mu \xi^\dagger)
\end{aligned}
$$
Here the computation shown under the brace crucially uses that all these jet coordinates
for the Dirac field are anti-commuting, due to their [[supergeometry|supergeometric]] nature (eq:DiracFieldJetCoordinatesAnticommute).
Notice that a priori this is a function on the jet bundle with values in $\mathbb{K}$.
But in fact for $\mathbb{K} = \mathbb{C}$ it is real up to a [[total spacetime derivative]]:, because
$$
\begin{aligned}
\left(
i \chi^\dagger \tilde \sigma^\mu \partial_\mu\chi
\right)^\dagger
& =
-i \left( \partial_\mu \chi\right)^\dagger \sigma^\mu \chi
\\
& =
i \chi^\dagger \sigma^\mu \partial_\mu \chi + i \partial_\mu\left( \chi^\dagger \sigma^\mu \chi \right)
\end{aligned}
$$
and similarly for $i \xi^\dagger \tilde \sigma^\mu \partial_\mu\xi$
=--
(e.g. [Dermisek I-9](Dirac+field#DermisekI9))
+-- {: .num_example #LagrangianQED}
###### Example
**([[Lagrangian density]] for [[quantum electrodynamics]])**
Consider the [[fiber product]] of the [[field bundles]] for the [[electromagnetic field]] (example \ref{Electromagnetism})
and the [[Dirac field]] (example \ref{DiracFieldBundle}) over 4-dimensional [[Minkowski spacetime]] $\Sigma \coloneqq \mathbb{R}^{3,1}$ (def. \ref{MinkowskiSpacetime}):
$$
E
\;\coloneqq\;
\underset{ \array{ \text{electromagnetic} \\ \text{field} } }{\underbrace{T^\ast \Sigma}}
\times
\underset{
\array{
\text{Dirac} \\ \text{field}
}
}{
\underbrace{
S_{odd}
}
}
\,.
$$
This means that now a [[field history]] is a [[pair]] $(A,\Psi)$, with $A$ a field history of the [[electromagnetic field]]
and $\Psi$ a field history of the [[Dirac field]].
On the resulting [[jet bundle]] consider the [[Lagrangian density]]
$$
\label{ElectronPhotonInteractionLocalLagrangian}
L_{int}
\;\coloneqq\;
i g \, \overline{\psi} \gamma^\mu \psi a_\mu
$$
for $g \in \mathbb{R}$ some number, called the _[[coupling constant]]_. This is called the _[[electron-photon interaction]]_.
Then the sum of the [[Lagrangian densities]] for
1. the [[free field|free]] [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity});
1. the [[free field|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField})
1. the above [[electron-photon interaction]]
$$
\mathbf{L}_{EM} + \mathbf{L}_{Dir} + \mathbf{L}_{int}
\;=\;
\left(
\tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
\;+\;
i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} + m \overline{\psi} \psi
\;+\;
i g \, \overline{\psi} \gamma^\mu \psi a_\mu
\right)
\, dvol_\Sigma
$$
defines the [[interacting field theory]] [[Lagrangian field theory]] whose [[perturbative quantum field theory|perturbative quantization]] is called _[[quantum electrodynamics]]_.
In this context the square of the [[coupling constant]]
$$
\alpha \coloneqq \frac{g^2}{4 \pi}
$$
is called the _[[fine structure constant]]_.
=--
$\,$
**[[Euler-Lagrange forms]] and [[presymplectic currents]]**
{#ELFormsAndPresymplecticCurrents}
The beauty of [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) is that a choice of [[Lagrangian density]] determines
both the [[equations of motion]] of the fields as well as a [[presymplectic manifold|presymplectic structure]]
on the space of solutions to this equation (the "[[shell]]"), making it the "[[covariant phase space]]" of the theory.
All this we discuss [below](#PhaseSpace). But in fact all this key structure of the field theory is
nothing but the shadow (under "[[transgression of variational differential forms]]", def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces} below) of the following
simple relation in the [[variational bicomplex]]:
+-- {: .num_prop #EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}
###### Proposition
**([[Euler-Lagrange form]] and [[presymplectic current]])**
Given a [[Lagrangian density]] $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ as in def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}, then its de Rham differential
$\mathbf{d}\mathbf{L}$, which by degree reasons equals $\delta \mathbf{L}$, has a _unique_ decomposition as a sum of two terms
$$
\label{dLDecomposition}
\mathbf{d} \mathbf{L}
=
\delta_{EL} \mathbf{L}
-
d \Theta_{BFV}
$$
such that $\delta_{EL}\mathbf{L}$ is proportional to the [[variational derivative]] of the fields (but not their derivatives, called a "[[source form]]"):
$$
\delta_{EL} \mathbf{L}
\;\in\;
\Omega^{p+1,0}_{\Sigma}(E) \wedge \delta C^\infty(E)
\;\subset\;
\Omega^{p+1,1}_{\Sigma}(E)
\,.
$$
The map
$$
\delta_{EL} \;\colon\; \Omega^{p+1,0}_{\Sigma}(E) \longrightarrow \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta \Omega^{0,0}_{\Sigma}(E)
$$
thus defined is called the _[[Euler-Lagrange operator]]_ and is explicitly given by the _[[Euler-Lagrange derivative]]_:
$$
\label{EulerLagrangeEquationGeneral}
\begin{aligned}
\delta_{EL} L \, dvol_\Sigma
& \coloneqq
\frac{\delta_{EL} L}{\delta \phi^a}
\delta \phi^a \wedge dvol_\Sigma
\\
& \coloneqq
\left(
\frac{\partial L}{\partial \phi^a}
-
\frac{d}{d x^\mu}
\frac{\partial L}{\partial \phi^a_{,\mu}}
+
\frac{d^2}{d x^{\mu_1} d x^{\mu_2}}
\frac{\partial L}{\partial \phi^a_{\mu_1, \mu_2}}
-
\cdots
\right)
\delta \phi^a
\wedge
dvol_\Sigma
\,.
\end{aligned}
$$
The [[smooth space|smooth subspace]] of the [[jet bundle]] on which the [[Euler-Lagrange form]]
vanishes
$$
\label{ShellInJetBundle}
\mathcal{E}
\;\coloneqq\;
\left\{
x \in J^\infty_\Sigma(E)
\;\vert\;
\delta_{EL}\mathbf{L}(x) = 0
\right\}
\;\overset{i_{\mathcal{E}}}{\hookrightarrow}\;
J^\infty_\Sigma(E)
\,.
$$
is called the _[[shell]]_. The smaller subspace on which also all [[total spacetime derivatives]]
vanish (the "[[formally integrable PDE|formally integrable prolongation]]")
is the _prolonged [[shell]]_
$$
\label{ProlongedShellInJetBundle}
\mathcal{E}^\infty
\;\coloneqq\;
\left\{
x \in J^\infty_\Sigma(E)
\;\vert\;
\left(
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\delta_{EL}\mathbf{L}
\right)(x) = 0
\right\}
\overset{i_{\mathcal{E}^\infty}}{\hookrightarrow}
J^\infty_\Sigma(E)
\,.
$$
Saying something holds "[[on-shell]]" is to mean that it holds after restriction to this
subspace. For example a [[variational differential form]] $\alpha \in \Omega^{\bullet,\bullet}_\Sigma(E)$
is said to _vanish on shell_ if $\alpha\vert_{\mathcal{E}^\infty} = 0$.
The remaining term $d \Theta_{BFV}$ in (eq:dLDecomposition) is unique, while the _presymplectic potential_
$$
\label{PresymplecticPotential}
\Theta_{BFV} \in \Omega^{p,1}_{\Sigma}(E)
$$
is not unique.
(For a [[field bundle]] which is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}), prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} says
that $\Theta_{BFV}$ is unique up to addition of total spacetime derivatives $d \kappa$, for $\kappa \in \Omega^{p-1,1}_\Sigma(E)$.)
One possible choice for the presymplectic current $\Theta_{BFV}$ is
$$
\label{StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime}
\begin{aligned}
\Theta_{BFV}
& \coloneqq \phantom{+}
\frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a
\; \wedge \iota_{\partial_\mu} dvol_\Sigma
\\
& \phantom{=}
+
\left(
\frac{\partial L}{\partial \phi^a_{,\nu \mu}}
\delta \phi^a_{,\nu}
-
\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}}
\delta \phi^a_{,\mu}
\right)
\wedge \iota_{\partial_\mu} dvol_\Sigma
\\
& \phantom{=} + \cdots
\,,
\end{aligned}
$$
where
$$
\iota_{\partial_{\mu}} dvol_\Sigma
\;\coloneqq\;
(-1)^{\mu} d x^0 \wedge \cdots d x^{\mu-1} \wedge d x^{\mu+1} \wedge \cdots \wedge d x^p
$$
denotes the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[volume form]] with the [[vector field]] $\partial_\mu$.
The [[vertical derivative]] of a chosen presymplectic potential $\Theta_{BFV}$ is called a _[[pre-symplectic current]]_ for $\mathbf{L}$:
$$
\label{PresymplecticCurrent}
\Omega_{BFV}
\;\coloneqq\;
\delta \Theta_{BFV} \;\;\; \in \Omega^{p,2}_{\Sigma}(E)
\,.
$$
Given a choice of $\Theta_{BFV}$ then the sum
$$
\label{TheLepage}
\mathbf{L} + \Theta_{BFV} \;\in\; \Omega^{p+1,0}_\Sigma(E) \oplus \Omega^{p,1}_\Sigma(E)
$$
is called the corresponding _[[Lepage form]]_. Its de Rham derivative is the sum of
the Euler-Lagrange variation and the presymplectic current:
$$
\label{DerivativeOfLepageForm}
\mathbf{d}( \mathbf{L} + \Theta_{BFV} )
\;=\;
\delta_{EL} \mathbf{L} + \Omega_{BFV}
\,.
$$
(Its conceptual nature will be elucidated after the introduction of the [[local BV-complex]]
in example \ref{DerivedPresymplecticCurrentOfRealScalarField} below.)
=--
+-- {: .proof}
###### Proof
Using $\mathbf{L} = L dvol_\Sigma$ and that $d \mathbf{L} = 0$ by degree reasons (example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}), we find
$$
\begin{aligned}
\mathbf{d}\mathbf{L}
& =
\left(
\frac{\partial L}{\partial \phi^a} \delta \phi^a
+
\frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu}
+
\frac{\partial L}{\partial \phi^a_{,\mu_1 \mu_2}} \delta \phi^a_{,\mu_1 \mu_2}
+
\cdots
\right)
\wedge dvol_{\Sigma}
\end{aligned}
\,.
$$
The idea now is to have $d \Theta_{BFV}$ pick up those terms that would appear as [[boundary]] terms under the [[integral]]
$\int_\Sigma j^\infty_\Sigma(\Phi)^\ast \mathbf{d}L$ if we were to consider [[integration by parts]] to remove
spacetime derivatives of $\delta \phi^a$.
We compute, using example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle},
the total horizontal derivative of $\Theta_{BFV}$ from (eq:StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime) as follows:
$$
\begin{aligned}
d \Theta_{BFV}
& =
\left(
d
\left(
\frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a
\right)
+
d
\left(
\frac{\partial L}{\partial \phi^a_{,\nu \mu}}
\delta \phi^a_{,\nu}
-
\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{\mu \nu}}
\delta \phi^a
\right)
+
\cdots
\right)
\wedge \iota_{\partial_\mu} dvol_\Sigma
\\
& =
\left(
\left(
\left(
d \frac{\partial L}{\partial \phi^a_{,\mu}}
\right) \wedge \delta \phi^a
-
\frac{\partial L}{\partial \phi^a_{,\mu}} \delta d \phi^a
\right)
+
\left(
\left(d \frac{\partial L}{\partial \phi^a_{,\nu \mu}}\right) \wedge \delta \phi^a_{,\nu}
-
\frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta d \phi^a_{,\nu}
-
\left( d \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \right) \wedge
\delta \phi^a
+
\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}}
\delta d \phi^a
\right)
+
\cdots
\right)
\wedge \iota_{\partial_\mu} dvol_\Sigma
\\
& =
-
\left(
\left(
\frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a
+
\frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a_{,\mu}
\right)
+
\left(
\frac{d}{d x^\mu}
\frac{\partial L}{\partial \phi^a_{,\nu \mu}}
\delta \phi^a_{,\nu}
+
\frac{\partial L}{\partial \phi^a_{,\nu \mu}}
\delta \phi^a_{,\nu \mu}
-
\frac{d^2}{ d x^\mu d x^\nu}
\frac{\partial L}{\partial \phi^a_{,\mu \nu}}
\delta \phi^a
-
\frac{d}{d x^\nu}
\frac{\partial L}{\partial \phi^a_{,\mu \nu}}
\delta \phi^a_{,\mu}
\right)
+ \cdots
\right)
\wedge dvol_\Sigma
\,,
\end{aligned}
$$
where in the last line we used that
$$
d x^{\mu_1} \wedge \iota_{\partial_{\mu_2}} dvol_\Sigma
=
\left\{
\array{
dvol_\Sigma &\vert& \text{if}\, \mu_1 = \mu_2
\\
0 &\vert& \text{otherwise}
}
\right.
$$
Here the two terms proportional to $\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}$
cancel out, and we are left with
$$
d \Theta_{BFV}
\;=\;
-
\left(
\frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}}
-
\frac{d^2}{ d x^\mu d x^\nu}
\frac{\partial L}{\partial \phi^a_{,\mu \nu}}
+
\cdots
\right)
\delta \phi^a \wedge dvol_\Sigma
-
\left(
\frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a_{,\mu}
+
\frac{\partial L}{\partial \phi^a_{,\nu \mu}}
\delta \phi^a_{,\nu \mu}
+
\cdots
\right)
\wedge dvol_\Sigma
$$
Hence $-d \Theta_{BFV}$ shares with $\mathbf{d} \mathbf{L}$ the terms that are proportional to
$\delta \phi^a_{,\mu_1 \cdots \mu_k}$ for $k \geq 1$,
and so the remaining terms are proportional to $\delta \phi^a$, as claimed:
$$
\mathbf{d}\mathbf{L} + d \Theta_{BFV}
=
\underset{
= \delta_{EL}\mathbf{L}
}{
\underbrace{
\left(
\frac{\partial L}{\partial \phi^a}
-
\frac{d}{d x^\mu}\frac{\partial L}{\partial \phi^a_{,\mu}}
+
\frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}}
+
\cdots
\right)
\delta \phi^a \wedge dvol_\Sigma
}}
\,.
$$
=--
The following fact is immediate from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime},
but of central importance, we futher amplify this in remark \ref{PresymplecticCurrentInterpretation} below:
+-- {: .num_prop #HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}
###### Proposition
**([[total derivative|total spacetime derivative]] of [[presymplectic current]] vanishes [[on-shell]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Then the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$ and the [[presymplectic current]]
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) are related by
$$
d \Omega_{BFV} = - \delta(\delta_{EL}\mathbf{L})
\,.
$$
In particular this means that restricted to the prolonged shell $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ (eq:ProlongedShellInJetBundle) the total spacetime derivative of the [[presymplectic current]] vanishes:
$$
\label{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}
d \Omega_{BFV} \vert_{\mathcal{E}^\infty} \;=\; 0
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} we have
$$
\delta \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta_{BFV}
\,.
$$
The claim follows from applying the [[variational derivative]] $\delta$ to both sides,
using (eq:HorizontalAndVerticalDerivativeAnticommute): $\delta^2 = 0$ and $\delta \circ d = - d \circ \delta$.
=--
Many examples of interest fall into the following two special cases of prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}:
+-- {: .num_prop #ShellForSpacetimeIndependentLagrangians}
###### Example
**([[Euler-Lagrange form]] for [[spacetime]]-independent [[Lagrangian densities]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] $E \simeq \Sigma \times F$ over [[Minkowski spacetime]] $\Sigma$
(example \ref{TrivialVectorBundleAsAFieldBundle}).
In general the [[Lagrangian density]] $\mathbf{L}$ is a function of all the spacetime and field coordinates
$$
\mathbf{L} = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots) dvol_\Sigma
\,.
$$
Consider the special case that $\mathbf{L}$ is _[[spacetime]]-independent_ in that the Lagrangian function $L$
is independent of the spacetime coordinate $(x^\mu)$.
Then the same evidently holds for the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}).
Therefore in this case the [[shell]] (eq:ProlongedShellInJetBundle) is itself a [[trivial bundle]] over spacetime.
In this situation every point $\varphi$ in the jet fiber defines a constant section of the shell:
$$
\label{ConstantSectionOfTrivialShellBundle}
\Sigma \times \{\varphi\} \subset \mathcal{E}^\infty
\,.
$$
=--
+-- {: .num_example #CanonicalMomentum}
###### Example
**([[canonical momentum]])**
Consider a [[Lagrangian field theory]] $(E, \mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[Lagrangian density]] $\mathbf{L}$
1. does not depend on the [[spacetime]]-[[coordinates]] (example \ref{ShellForSpacetimeIndependentLagrangians});
1. depends on spacetime derivatives of [[field (physics)|field]] coordinates (hence on [[jet bundle]] coordinates) at most to first order.
Hence if the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) this means to consider the case that
$$
\mathbf{L}
\;=\;
L\left(
(\phi^a), (\phi^a_{,\mu})
\right) \wedge dvol_\Sigma
\,.
$$
Then the [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is (up to possibly a horizontally exact part) of the form
$$
\label{CanonicalMomentumPresymplecticCurrent}
\Omega_{BFV}
\;=\;
\delta p_a^\mu
\wedge
\delta \phi^a
\wedge
\iota_{\partial_\mu} dvol_\Sigma
$$
where
$$
\label{CanonicalMomentumInCoordinates}
p_a^\mu
\;\coloneqq\;
\frac{\partial L}{ \partial \phi^a_{,\mu}}
$$
denotes the [[partial derivative]] of the [[Lagrangian density|Lagrangian function]] with respect to the spacetime-[[derivatives]] of the [[field (physics)|field]] [[coordinates]].
Here
$$
\begin{aligned}
p_a
& \coloneqq
p_a^0
\\
& =
\frac{\partial L}{\partial \phi^a_{,0}}
\end{aligned}
$$
is called the _[[canonical momentum]]_ corresponding to the "[[canonical coordinate|canonical field coordinate]]" $\phi^a$.
In the language of [[multisymplectic geometry]] the full expression
$$
p_a^\mu \wedge \iota_{\partial_\mu} dvol_\Sigma
\;\in\;
\Omega^{p,1}_\Sigma(E)
$$
is also called the "canonical multi-momentum", or similar.
=--
+-- {: .proof}
###### Proof
We compute:
$$
\begin{aligned}
\mathbf{d} \mathbf{L}
& =
\left(
\frac{\partial L}{\partial \phi^a} \delta \phi^a
+
\frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu}
\right)
\delta \phi^a
\wedge
dvol_\Sigma
\\
& =
\left(
\frac{\partial L}{\partial \phi^a}
-
\frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}}
\right)
\wedge
dvol_\Sigma
-
d
\underset{
\Theta_{BFV}
}{
\underbrace{
\left(
\frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a
\right)
\wedge
\iota_{\partial_\mu} dvol_\Sigma
}
}
\end{aligned}
\,.
$$
Hence
$$
\begin{aligned}
\Omega_{BFV}
& \coloneqq
\delta \Theta_{BFV}
\\
& =
\delta
\left(
\frac{\partial L}{\partial \phi^a_{,\mu}}
\delta \phi^a_{,\mu}
\wedge \iota_{\partial_\mu} dvol_\Sigma
\right)
\\
& =
\delta
\frac{\partial L}{\partial \phi^a_{,\mu}}
\wedge
\delta \phi^a_{,\mu}
\wedge
\iota_{\partial_\mu} dvol_\Sigma
\\
& =
\delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma
\end{aligned}
$$
=--
+-- {: .num_remark #PresymplecticCurrentInterpretation}
###### Remark
**([[presymplectic current]] is local version of ([[presymplectic form|pre-]])[[symplectic form]] of [[Hamiltonian mechanics]])**
In the simple but very common situation of example \ref{CanonicalMomentum} the [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) takes the form (eq:CanonicalMomentumInCoordinates)
$$
\Omega_{BFV}
\;=\;
\delta p_a^\mu
\wedge
\delta \phi^a
\wedge
\iota_{\partial_\mu} dvol_\Sigma
$$
with $\phi^a$ the [[field (physics)|field]] [[coordinates]] ("[[canonical coordinates]]")
and $p_a^\mu$ the "[[canonical momentum]]" (eq:CanonicalMomentumInCoordinates).
Notice that this is of the schematic form "$(\delta p_a \wedge \delta q^a) \wedge dvol_{\Sigma_p}$",
which is reminiscent of the wedge product of a [[symplectic form]] expressed in [[Darboux coordinates]] with
a [[volume form]] for a $p$-dimensional [[manifold]]. Indeed, below in _[Phase space](#PhaseSpace)_
we discuss that this [[presymplectic current]] "[[transgression of variational differential forms|transgresses]]" (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces} below)
to a [[presymplectic form]] of the schematic form "$d P_a \wedge d Q^a$" on the [[on-shell]] [[space of field histories]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime})
by [[integration of differential forms|integrating]] it over a [[Cauchy surface]] of [[dimension]] $p$. In good situations
this [[presymplectic form]] is in fact a [[symplectic form]] on the [[on-shell]] [[space of field histories]] (theorem \ref{PPeierlsBracket} below).
This shows that the [[presymplectic current]] $\Omega_{BFV}$ is the [[local field theory|local]] (i.e. [[jet bundle|jet level]]) avatar of the
[[symplectic form]] that governs the formulation of [[Hamiltonian mechanics]] in terms of [[symplectic geometry]].
In fact prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell} may be read as saying that
the [[presymplectic current]] is a _[[conserved current]]_ (def. \ref{SymmetriesAndConservedCurrents} below),
only that it takes values not in [[smooth functions]] of the field coordinates and jets, but in
[[variational differential form|variational 2-forms]] on fields. There is a [[conserved charge]] associated with
every [[conserved current]] (prop. \ref{ConservedCharge} below) and the conserved charge associated with the
[[presymplectic current]] is the ([[presymplectic form|pre-]])[[symplectic form]] on the [[phase space]]
of the field theory (def. \ref{PhaseSpaceAssociatedWithCauchySurface} below).
=--
+-- {: .num_example #FreeScalarFieldEOM}
###### Example
**([[Euler-Lagrange form]] and [[presymplectic current]] for [[free field|free]] [[real scalar field]])**
Consider the [[Lagrangian field theory]] of the [[free field|free]] [[real scalar field]]
from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.
Then the [[Euler-Lagrange operator|Euler-Lagrange form]] and [[presymplectic current]]
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) are
$$
\label{RealScalarFieldLEForm}
\delta_{EL}\mathbf{L}
\;=\;
\left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma
\;\in\;
\Omega^{p+1,1}_{\Sigma}(E)
\,.
$$
and
$$
\Omega_{BFV}
\;=\;
\left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma}
\;\in\;
\Omega^{p,2}_{\Sigma}(E)
\,,
$$
respectively.
=--
+-- {: .proof}
###### Proof
This is a special case of example \ref{CanonicalMomentum}, but we spell it out in detail again:
We need to show that [[Euler-Lagrange operator]] $\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma)$ takes the [[local Lagrangian density]] for the [[free field|free]] [[scalar field]] to
$$
\delta_EL L
\;=\;
\left(
\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi
\right)
\delta \phi \wedge \mathrm{dvol}_\Sigma
\,.
$$
First of all, using just the [[variational derivative]] ([[vertical derivative]]) $\delta$ is a graded [[derivation]],
the result of applying it to the local Lagrangian density is
$$
\delta L
\;=\;
\left(
\eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu}
-
m^2 \phi \delta \phi
\right)
\wedge \mathrm{dvol}_\Sigma
\,.
$$
By definition of the [[Euler-Lagrange operator]], in order to find $\delta_{EL}\mathbf{L}$ and $\Theta_{BFV}$, we need to exhibit this as the
sum of the form $(-) \wedge \delta \phi - d \Theta_{BFV}$.
The key to find $\Theta_{BFV}$ is to realize $\delta \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma$ as a
[[total derivative|total spacetime derivative]] ([[horizontal derivative]]). Since $d \phi = \phi_{,\mu} d x^\mu$ this is accomplished by
$$
\delta \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma
=
\delta d \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma
\,,
$$
where on the right we have the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[tangent vector field]]
along $x^\nu$ into the [[volume form]].
Hence we may take the presymplectic potential (eq:PresymplecticPotential) of the free scalar field to be
$$
\label{PresymplecticPotentialOfFreeScalarField}
\Theta_{BFV}
\coloneqq
\eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu}
\mathrm{dvol}_\Sigma
\,,
$$
because with this we have
$$
d \Theta_{BFV}
=
\eta^{\mu \nu}
\left(
\phi_{,\mu \nu} \delta \phi
-
\eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu}
\right) \wedge \mathrm{dvol}_\Sigma
\,.
$$
In conclusion this yields the decomposition of the vertical differential of the Lagrangian density
$$
\delta L
=
\underset{
= \delta_{EL} \mathcal{L}
}{
\underbrace{
\left(
\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi
\right)
\delta \phi \wedge \mathrm{dvol}_\Sigma
}
}
-
d \Theta_{BFV}
\,,
$$
which shows that $\delta_{EL} L$ is as claimed, and that $\Theta_{BFV}$ is a presymplectic potential current (eq:PresymplecticPotential).
Hence the presymplectic current itself is
$$
\begin{aligned}
\Omega_{BFV} &\coloneqq \delta \Theta_{BFV}
\\
& =
\delta \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right)
\\
& =
\left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma}
\end{aligned}
\,.
$$
=--
+-- {: .num_example #ElectromagnetismEl}
###### Example
**([[Euler-Lagrange form]] for [[free field|free]] [[electromagnetic field]])**
Consider the [[Lagrangian field theory]] of [[free field|free]] [[electromagnetism]] from example \ref{ElectromagnetismLagrangianDensity}.
The [[Euler-Lagrange variational derivative]] is
$$
\label{ElectromagneticFieldEulerLagrangeForm}
\delta_{EL} \mathbf{L}
\;=\;
- \frac{d}{d x^\mu} f^{\mu \nu} \delta a_\nu
\,.
$$
Hence the [[shell]] (eq:ShellInJetBundle) in this case is
$$
\mathcal{E}
=
\Sigma \times
\left\{
\left(
(a_\mu) , (a_{\mu,\mu_1}), (a_{\mu,\mu_1 \mu_2}), \cdots
\right)
\;\vert\; f^{\mu \nu}{}_{,\mu} = 0 \right\}
\;\subset\;
J^\infty_\Sigma(T^\ast \Sigma)
\,.
$$
=--
+-- {: .proof}
###### Proof
By (eq:EulerLagrangeEquationGeneral) we have
$$
\begin{aligned}
\frac{\delta_{EL} L}{\delta a_\mu} \delta a_\mu
& =
\left(
\underset{
= 0
}{
\underbrace{
\frac{\partial}{\partial a_\mu}
\tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]}
}
}
-
\frac{d}{d x^\rho}
\frac{\partial}{\partial a_{\alpha,\rho}}
\tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]}
\right)
\delta a_\alpha
\\
& =
-
\tfrac{1}{2}
\left(
\frac{d}{d x^\rho}
\frac{\partial}{\partial a_{\alpha,\rho}}
a_{\mu,\nu} a^{[\mu,\nu]}
\right)
\delta a_\alpha
\\
& =
-
\left(
\frac{d}{d x^\rho}
a^{[\alpha,\rho]}
\right)
\delta a_{\alpha}
\\
& =
- f^{\mu \nu}{}_{,\mu} \delta a_{\nu}
\,.
\end{aligned}
$$
=--
More generally:
+-- {: .num_example #YangMillsOnMinkowskiEl}
###### Example
**([[Euler-Lagrange form]] for [[Yang-Mills theory]] on [[Minkowski spacetime]])**
Let $\mathfrak{g}$ be a [[semisimple Lie algebra]]
and consider the [[Lagrangian field theory]] $(E,\mathbf{L})$ of $\mathfrak{g}$-[[Yang-Mills theory]] from example \ref{YangMillsLagrangian}.
Its [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is
$$
\begin{aligned}
\delta_{EL}\mathbf{L}
& =
-
\left(
f^{\mu \nu \alpha}_{,\mu}
+
\gamma^\alpha{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma}
\right)
k_{\alpha \beta}
\,\delta a_\mu^\beta
\, dvol_\Sigma
\,,
\end{aligned}
$$
where
$$
f^\alpha_{\mu \nu}
\;\in\;
\Omega^{0,0}_\Sigma(E)
$$
is the universal [[Yang-Mills theory|Yang-Mills]] [[field strength]] (eq:YangMillsJetFieldStrengthMinkowski).
=--
+-- {: .proof}
###### Proof
With the explicit form (eq:EulerLagrangeEquationGeneral) for the [[Euler-Lagrange derivative]] we compute as follows:
$$
\begin{aligned}
\delta_{EL}
\left(
\tfrac{1}{2}
k_{\alpha \beta} f^\alpha_{\mu\nu} f^{\beta \mu \nu}
\right)
& =
\left(
\left(
\frac{\partial}{\partial a_{\mu'}^{\alpha'}}
\left(
a_{\nu,\mu}^\alpha
+
\tfrac{1}{2}
\gamma^{\alpha}{}_{\alpha_2 \alpha_3}
a_{\mu}^{\alpha_2} a_\nu^{\alpha_3}
\right)
\right)
k_{\alpha \beta}
f^{\beta \mu \nu}
-
\left(
\frac{d}{d x^{\nu'}}
\frac{\partial}{\partial a_{\mu',\nu'}^{\alpha'}}
\left(
a_{\nu,\mu}^\alpha
+
\tfrac{1}{2}
\gamma^{\alpha}{}_{\alpha_2 \alpha_3}
a_{\mu}^{\alpha_2} a_\nu^{\alpha_3}
\right)
\right)
k_{\alpha \beta}
f^{\beta \mu \nu}
\right)
\delta a_{\mu'}^{\alpha'}
\\
& =
\gamma^{\alpha}{}_{\alpha' \alpha_3} a_\nu^{\alpha_3}
f^{\beta \mu \nu}
k_{\alpha \beta}
\delta a_{\mu}^{\alpha'}
-
\left(
\frac{d}{d x^{\mu}} f^{\beta \mu \nu}
\right)
k_{\alpha \beta}
\delta a_{\nu}^{\alpha}
\\
&=
-
\left(
f^{\alpha \mu \nu}_{,\mu}
+
\gamma^\alpha{}_{\beta \gamma} a_\mu^\beta f^{\gamma \mu \nu}
\right)
k_{\alpha \beta}
\delta a_\nu^\beta
\end{aligned}
$$
In the last step we used that for a [[semisimple Lie algebra]] $\gamma_{\alpha \beta \gamma} \coloneqq k_{\alpha \alpha'} \gamma^{\alpha'}{}_{\beta \gamma}$ is totally skew-symmetric in its indices (this being the coefficients of the [[Lie algebra cocycle]])
which is in transgression with the [[Killing form]] [[invariant polynomial]] $k$.
=--
+-- {: .num_example #EulerLagrangeFormBField}
###### Example
**([[Euler-Lagrange form]] of [[free field|free]] [[B-field]])**
Consider the [[Lagrangian field theory]] of the [[free field|free]] [[B-field]] from example \ref{BField}.
The [[Euler-Lagrange variational derivative]] is
$$
\delta_{EL} \mathbf{L}
\;=\;
h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu}
\,,
$$
where $h_{\mu_1 \mu_2 \mu_3}$ is the universal [[B-field|B-]][[field strength]] from example \ref{BFieldJetFaraday}.
=--
+-- {: .proof}
###### Proof
By (eq:EulerLagrangeEquationGeneral) we have
$$
\begin{aligned}
\frac{\delta_{EL} L}{\delta b_{\mu \nu}} \delta b_{\mu \nu}
& =
\left(
\underset{
= 0
}{
\underbrace{
\frac{\partial}{\partial b_{\mu \nu}}
\tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]}
}
}
-
\frac{d}{d x^\rho}
\frac{\partial}{\partial b_{\mu \nu, \rho}}
\tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]}
\right)
\delta b_{\mu \nu}
\\
& =
-
\left(
\frac{d}{d x^\rho}
\frac{\partial}{\partial b_{\mu \nu, \rho}}
\tfrac{1}{2} b_{\mu_1 \mu_2, \mu_3} b^{[\mu_1 \mu_2, \mu_3]}
\right)
\delta b_{\mu \nu}
\\
& =
-
\left(
\frac{d}{d x^\rho}
b^{[\mu \nu, \rho]}
\right)
\delta b_{\mu \nu}
\\
& =
-
h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu}
\,.
\end{aligned}
$$
=--
+-- {: .num_example #PresymplecticCurrentDiracField}
###### Example
**([[Euler-Lagrange form]] and [[presymplectic current]] of [[Dirac field]])**
Consider the [[Lagrangian field theory]] of the [[Dirac field]] on [[Minkowski spacetime]] of [[dimension]] $p + 1 \in \{3,4,6,10\}$ (example \ref{LagrangianDensityForDiracField}).
Then
* the [[Euler-Lagrange variational derivative]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) in the case of vanishing [[mass]] $m$ is
$$
\delta_{EL} \mathbf{L}
\;=\;
2 i\,
\overline{\delta \psi}
\,\gamma^\mu\, \psi_{,\mu}
\,
\wedge dvol_\Sigma
$$
and in the case that [[spacetime]] [[dimension]] is $p +1 = 4$ and arbitrary [[mass]] $m\in \mathbb{R}$, it is
$$
\delta_{EL} \mathbf{L}
\;=\;
\left(
\overline{\delta \psi}
\left(
i \gamma^\mu \psi_{,\mu} + m \psi
\right)
+
\left(
- i \gamma^\mu\overline{\psi_{,\mu}} + m \overline{\psi}
\right)
(\delta \psi)
\right)
\,
dvol_\Sigma
$$
* its [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is
$$
\Omega_{BFV}
\;=\;
\overline{\delta \psi}\,\gamma^\mu \,\delta \psi \, \iota_{\partial_\mu} dvol_\Sigma
$$
=--
+-- {: .proof}
###### Proof
In any case the [[canonical momentum]] of the [[Dirac field]] according to example \ref{CanonicalMomentum} is
$$
\begin{aligned}
p^\alpha_\mu
& \coloneqq
\frac{\partial }{\partial \psi^\alpha_{,\mu}}
\left(
i \overline {\psi} \, \gamma^\nu \, \psi_{,\nu}
+
m \overline{\psi} \psi
\right)
\\
& =
\overline{\psi}^\beta (\gamma^\mu)_\beta{}^\alpha
\end{aligned}
$$
This yields the [[presymplectic current]] as claimed, by example \ref{CanonicalMomentum}.
Now regarding the [[Euler-Lagrange form]], first consider the massless case in spacetime dimension $p+1 \in \{3,4,6,10\}$, where
$$
L
\;=\;
i \overline{\psi} \, \gamma^\mu \, \psi_{,\mu}
\,.
$$
Then we compute as follows:
$$
\begin{aligned}
\delta_{EL} L
& =
i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu}
\underset{
= +
i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu}
}{
\underbrace{
-
i \overline{\psi_{,\mu}} \, \gamma^\mu \, \delta \psi
}
}
\\
& =
2 i \, \overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu}
\end{aligned}
$$
Here the first equation is the general formula (eq:EulerLagrangeEquationGeneral) for the Euler-Lagrange variation, while the identity under the braces combines two facts (as in remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature} above):
1. the symmetry (eq:SpinorToVectorPairingIsSymmetric) of the spinor pairing $\overline{(-)}\gamma^\mu(-)$ (prop. \ref{RealSpinorPairingsViaDivisionAlg});
1. the anti-commutativity (eq:DiracFieldJetCoordinatesAnticommute) of the Dirac field and jet coordinates, due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}).
Finally in the special case of the massive Dirac field in spacetime dimension $p+1 = 4$ the Lagrangian function is
$$
L
\;=\;
i \, \overline{\psi} \gamma^\mu \psi_{,\mu} + m \overline{\psi}\psi
$$
where now $\psi_\alpha$ takes values in the [[complex numbers]] $\mathbb{C}$ (as opposed to in $\mathbb{R}$, $\mathbb{H}$ or $\mathbb{O}$). Therefore we may now form the [[derivative]] equivalently by treeating $\psi$ and $\overline{\psi}$ as independent components of the field. This immediately yields the claim.
=--
+-- {: .num_example #TrivialLagrangianDensities}
###### Example
**(trivial [[Lagrangian densities]] and the [[Euler-Lagrange complex]])**
If a [[Lagrangian density]] $\mathbf{L}$ (def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime})
is in the image of the [[total spacetime derivative]], hence horizontally exact (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
$$
\mathbf{L} \;=\; d \mathbf{\ell}
$$
for any $\mathbf{\ell} \in \Omega^{p,0}_\Sigma(E)$, then both its [[Euler-Lagrange form]] as well as its [[presymplectic current]]
(def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) vanish:
$$
\delta_{EL}\mathbf{L} = 0
\phantom{AA}
\,,
\phantom{AA}
\Omega_{BFV} = 0
\,.
$$
This is because with $\delta \circ d = - d \circ \delta$ (eq:HorizontalAndVerticalDerivativeAnticommute) the defining unique decomposition (eq:dLDecomposition) of $\delta \mathbf{L}$ is given by
$$
\begin{aligned}
\delta \mathbf{L}
& =
\delta d \mathbf{\ell}
\\
& =
\underset{= \delta_{EL}\mathbf{L}}{\underbrace{0}}
-
d \underset{\Theta_{BFV}}{\underbrace{\delta \mathbf{l}}}
\end{aligned}
$$
which then implies with (eq:PresymplecticCurrent) that
$$
\begin{aligned}
\Omega_{BFV}
& \coloneqq
\delta \Theta_{BFV}
\\
& =
\delta \delta \mathbf{\ell}
\\
& =
0
\end{aligned}
$$
Therefore the [[Lagrangian densities]] which are [[total spacetime derivatives]]
are also called _trivial Lagrangian densities_.
If the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle})
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) then also the converse is true:
Every Lagrangian density whose [[Euler-Lagrange form]] vanishes is a total spacetime derivative.
Stated more [[category theory|abstractly]], this means that the [[exact sequence]] of the total spacetime from prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} extends to the right via the [[Euler-Lagrange variational derivative]]
$\delta_{EL}$ to an [[exact sequence]] of the form
$$
\mathbb{R}
\overset{}{\hookrightarrow}
\Omega^{0,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{1,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{2,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\cdots
\overset{d}{\longrightarrow}
\Omega^{p,0}_\Sigma(E)
\overset{d}{\longrightarrow}
\Omega^{p+1,0}_\Sigma(E)
\overset{\delta_{EL}}{\longrightarrow}
\Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E))
\overset{\delta_{H}}{\longrightarrow}
\cdots
\,.
$$
In fact, as shown, this [[exact sequence]] keeps going to the right;
this is also called the _[[Euler-Lagrange complex]]_.
([Anderson 89, theorem 5.1](Euler-Lagrange+complex#Anderson89))
The next [[differential]] $\delta_{H}$ after the [[Euler-Lagrange variational derivative]] $\delta_{EL}$
is known as the _[[Helmholtz operator]]_. By definition of [[exact sequence]], the
[[Helmholtz operator]] detects whether a [[partial differential equation]] on [[field histories]],
induced by a [[variational differential form]] $P \in \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E))$
as in (eq:EquationOfMotionEL) comes from varying a [[Lagrangian density]], hence whether it is the
[[equation of motion]] of a [[Lagrangian field theory]] via def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}.
This way [[homological algebra]] is brought to bear on core questions of [[field theory]].
For more on this see the exposition at _[[schreiber:Higher Structures|Higher Structures in Physics]]_.
=--
+-- {: .num_remark #LagrangianDensityOfDiracFieldSupergeometricNature}
###### Remark
**([[supergeometry|supergeometric]] nature of [[Lagrangian density]] of the [[Dirac field]])**
Observe that the [[Lagrangian density]] for the [[Dirac field]] (def. \ref{LagrangianDensityForDiracField}) makes sense (only) due to the [[supergeometry|supergeometric]] nature of the [[Dirac field]] (remark \ref{DiracFieldSupergeometric}): If the field jet coordinates $\psi_{,\mu_1 \cdots \mu_k}$ were not anti-commuting (eq:DiracFieldJetCoordinatesAnticommute) then the
Dirac's field Lagrangian density (def. \ref{LagrangianDensityForDiracField}) would be a [[total spacetime derivative]]
and hence be trivial according to example \ref{TrivialLagrangianDensities}.
This is because
$$
d \left(
\tfrac{1}{2}
\overline{\psi} \,\gamma^\mu\, \psi
\,
\iota_{\partial_\mu} dvol_\Sigma
\right)
=
\tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma
+
\underset{ = (-1) \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma }{
\underbrace{
\tfrac{1}{2}\overline{\psi} \,\gamma^\mu\, \psi_{,\mu} \, dvol_\Sigma
}}
\,.
$$
Here the identification under the brace uses two facts:
1. the symmetry (eq:SpinorToVectorPairingIsSymmetric) of the spinor bilinear pairing $\overline{(-)}\Gamma (-)$;
1. the anti-commutativity (eq:DiracFieldJetCoordinatesAnticommute) of the Dirac field and jet coordinates, due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}).
The second fact gives the minus sign under the brace, which makes the total expression vanish, if the
Dirac field and jet coordinates indeed are anti-commuting (which, incidentally, means that we found
an "[[off-shell]] [[conserved current]]" for the Dirac field, see example \ref{DiracCurrent} below).
If however the Dirac field and jet coordinates did commute with each other, we would instead
have a plus sign under the brace, in which case the total horizontal derivative expression
above would equal the massless Dirac field Lagrangian (eq:DiracFieldLagrangianMassless),
thus rendering it trivial in the sense of example \ref{TrivialLagrangianDensities}.
The same [[supergeometry|supergeometric]] nature of the [[Dirac field]] will be necessary
for its intended [[equation of motion]], the _[[Dirac equation]]_ (example \ref{EquationOfMotionOfDiracFieldIsDiracEquation}) to derive from a
[[Lagrangian density]]; see the proof of example \ref{PresymplecticCurrentDiracField} below, and see remark \ref{SupergeometricNatureOfDiracEquation} below.
=--
$\,$
**[[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]**
{#ELEquationsOfMotion}
The key implication of the [[Euler-Lagrange form]] on the [[jet bundle]] is that it induces
the _[[equation of motion]]_ on the [[space of field histories]]:
+-- {: .num_defn #EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}
###### Definition
**([[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]])**
Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} then the corresponding _[[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]_ is the condition on [[field histories]] (def. \ref{SupergeometricSpaceOfFieldHistories})
$$
\Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E)
$$
to have a [[jet prolongation]] (def. \ref{JetProlongation})
$$
j^\infty_\Sigma(\Phi_{(-)}(-) ) \;\colon\; U \times \Sigma \longrightarrow J^\infty_\Sigma(E)
$$
that factors through the [[shell]] inclusion $\mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E)$
(eq:ShellInJetBundle) defined by vanishing of the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
$$
\label{EquationOfMotionEL}
j^\infty_\Sigma(\Phi_{(-)}(-))
\;\colon\;
U \times \Sigma
\longrightarrow
\mathcal{E}
\overset{i_{\mathcal{E}}}{\hookrightarrow}
J^\infty_\Sigma(E)
\,.
$$
(This implies that $j^\infty_\Sigma(\Phi_{(-)})$ factors even through the prolonged shell $\mathcal{E}^\infty \overset{i_{\mathcal{E}^\infty}}{\hookrightarrow} J^\infty_\Sigma(E)$ (eq:ProlongedShellInJetBundle).)
In the case that the field bundle is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle} this is the condition that
$\Phi_{(-)}$ satisfies the following [[differential equation]] (again using prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):
$$
\frac{\delta_{EL} L}{\delta \phi^a}
\;\coloneqq\;
\left(
\frac{\partial L}{\partial \phi^a}
-
\frac{d}{d x^\mu}
\frac{\partial L}{\partial \phi^a_{,\mu}}
+
\frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}}
-
\cdots
\right)
\left(
(x^\mu),
(\Phi^a),
\left( \frac{\partial \Phi^a_{(-)}}{\partial x^\mu}\right),
\left( \frac{\partial^2 \Phi^a_{(-)}}{\partial x^\mu \partial x^\nu} \right),
\cdots
\right)
\;=\;
0
\,,
$$
where the [[differential operator]] (def. \ref{DifferentialOperator})
$$
\label{DifferentialOperatorEulerLagrangeDerivative}
j^\infty_\Sigma(-)^\ast
\left(
\frac{\delta_{EL}L}{\delta \phi^{(-)}}
\right)
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\Gamma_\Sigma(T^\ast_\Sigma E)
$$
from the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) to its [[vertical cotangent bundle]] (def. \ref{VerticalTangentBundle}) is given by the _[[Euler-Lagrange derivative]]_ (eq:EulerLagrangeEquationGeneral).
The _[[on-shell]] [[space of field histories]]_ is the space of solutions to this condition, namely the
the sub-[[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet})
of the full [[space of field histories]] (eq:SpaceOfFieldHistories) (def. \ref{SupergeometricSpaceOfFieldHistories})
$$
\label{OnShellFieldHistories}
\Gamma_\Sigma(E)_{\delta_{EL} L = 0}
\overset{\phantom{AAA}}{\hookrightarrow}
\Gamma_\Sigma(E)
$$
whose plots are those $\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)$ that factor through the shell (eq:EquationOfMotionEL).
More generally for $\Sigma_r \hookrightarrow \Sigma$ a [[submanifold]] of [[spacetime]], we write
$$
\label{OnShellFieldHistoriesInHigherCodimension}
\Gamma_{\Sigma_r}(E)_{\delta_{EL} L = 0}
\overset{\phantom{AAA}}{\hookrightarrow}
\Gamma_{\Sigma_r}(E)
$$
for the sub-[[super formal smooth set|super smooth ste]] of on-shell field histories restricted to the [[infinitesimal neighbourhood]]
of $\Sigma_r$ in $\Sigma$ (eq:SpaceOfFieldHistoriesInHigherCodimension).
=--
+-- {: .num_defn #FreeFieldTheory}
###### Definition
**([[free field theory]])**
A [[Lagrangian field theory]] $(E, \mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
with [[field bundle]] $E \overset{fb}{\to} \Sigma$ a [[vector bundle]]
(e.g. a [[trivial vector bundle]] as in example \ref{TrivialVectorBundleAsAFieldBundle})
is called a _[[free field theory]]_ if its [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]
(def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is a
[[differential equation]] that is _[[linear differential equation]]_, in that with
$$
\Phi_1, \Phi_2 \;\in\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$
any two [[on-shell]] [[field histories]] (eq:OnShellFieldHistories) and $c_1, c_2 \in \mathbb{R}$
any two [[real numbers]], also the [[linear combination]]
$$
c_1 \Phi_1 + c_2 \Phi_2 \;\in\; \Gamma_\Sigma(E)
\,,
$$
which a priori exists only as an element in the off-shell [[space of field histories]],
is again a solution to the [[equations of motion]] and hence an element of $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$.
A [[Lagrangian field theory]] which is not a [[free field theory]] is called an _[[interaction|interacting]]_ [[field theory]].
=--
+-- {: .num_remark #FreeFieldTheoryRelevance}
###### Remark
**(relevance of [[free field theory]])**
In [[perturbative quantum field theory]] one considers [[interaction|interacting]] [[field theories]]
in the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of [[free field theories]] (def. \ref{FreeFieldTheory})
inside some [[super formal smooth set|super smooth set]] of general [[Lagrangian field theories]].
While [[free field theories]] are typically of limited interest in themselves,
this [[perturbative quantum field theory|perturbation theory]] around them exhausts much of what is
known about [[quantum field theory]] in general, and therefore [[free field theories]] are of paramount importance
for the general theory.
We discuss the [[covariant phase space]] of [[free field theories]] below in _[Propagators](#Propagators)_
and their [[quantization]] below in _[Free quantum fields](#FreeQuantumFields) _.
=--
+-- {: .num_prop #EquationOfMotionOfFreeRealScalarField}
###### Example
**([[equation of motion]] of [[free field|free]] [[real scalar field]] is [[Klein-Gordon equation]])**
Consider the [[Lagrangian field theory]] of the [[free field|free]] [[real scalar field]]
from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.
By example \ref{FreeScalarFieldEOM} its [[Euler-Lagrange form]] is
$$
\delta_{EL}\mathbf{L}
\;=\;
\left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma
$$
Hence for $\Phi \in \Gamma_\Sigma(E) = C^\infty(X)$ a [[field history]],
its [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]
according to def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} is
$$
\eta^{\mu \nu} \frac{\partial^2 }{\partial x^\mu \partial x^\nu} \Phi - m^2 \Phi \;=\; 0
$$
often abbreviated as
$$
\label{KleinGordonEquation}
(\Box - m^2) \Phi \;=\; 0
\,.
$$
This [[PDE]] is called the _[[Klein-Gordon equation]]_ on Minowski spacetime. If the [[mass]] $m$
vanishes, $m = 0$, then this is the _relativistic [[wave equation]]_.
Hence this is indeed a [[free field theory]] according to def. \ref{FreeFieldTheory}.
The corresponding [[linear differential operator]] (def. \ref{DifferentialOperator})
$$
\label{KleinGordonOperator}
(\Box - m^2)
\;\colon\;
\Gamma_\Sigma(\Sigma \times \mathbb{R})
\longrightarrow
\Gamma_\Sigma(\Sigma \times \mathbb{R})
$$
is called the _[[Klein-Gordon operator]]_.
=--
For later use we record the following basic fact about the [[Klein-Gordon equation]]:
+-- {: .num_example #FormallySelfAdjointKleinGordonOperator}
###### Example
**([[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] )**
The [[Klein-Gordon operator]] (eq:KleinGordonOperator)
is its own [[formal adjoint differential operator|formal adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) witnessed by the bilinear differential operator (eq:FormallyAdjointDifferentialOperatorWitness) given by
$$
\label{WitnessForFormalSelfadjointnessOfKleinGordonEquation}
K(\Phi_1, \Phi_2)
\;\coloneqq\;
\left(
\frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
-
\Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
\right)
\eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
\,.
$$
=--
+-- {: .proof}
###### Proof
$$
\begin{aligned}
d K(\Phi_1, \Phi_2)
& =
d
\left(
\frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
-
\Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
\right)
\eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
\\
&=
\left(
\left(
\eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
+
\eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu}
\right)
-
\left(
\eta^{\mu \nu}
\frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu}
+
\Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
\right)
\right)
dvol_\Sigma
\\
& =
\left(
\eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
-
\Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
\right)
dvol_\Sigma
\\
& =
\Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2)
\end{aligned}
$$
=--
+-- {: .num_prop #MaxwellVacuumEquation}
###### Example
**([[equations of motion]] of [[vacuum]] [[electromagnetism]] are [[vacuum]] [[Maxwell's equations]])**
Consider the [[Lagrangian field theory]] of [[free field|free]] [[electromagnetism]]
on [[Minkowski spacetime]] from example \ref{ElectromagnetismLagrangianDensity}.
By example \ref{ElectromagnetismEl} its [[Euler-Lagrange form]] is
$$
\delta_{EL}\mathbf{L}
\;=\;
\frac{d}{d x^\mu}f^{\mu \nu} \delta a_\nu
\,.
$$
Hence for $A \in \Gamma_{\Sigma}(T^\ast \Sigma) = \Omega^1(\Sigma)$ a [[field history]] ("[[vector potential]]"),
its [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]
according to def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} is
$$
\begin{aligned}
& \frac{\partial}{\partial x^\mu} F^{\mu \nu} = 0
\\
\Leftrightarrow\;\; &
d \star_\eta F = 0
\end{aligned}
\,,
$$
where $F = d A$ is the [[Faraday tensor]] (eq:TensorFaraday).
(In the coordinate-free formulation in the second line "$\star_\eta$" denotes
the [[Hodge star operator]] induced by the [[pseudo-Riemannian metric]] $\eta$ on [[Minkowski spacetime]].)
These [[PDEs]] are called the _[[vacuum]] [[Maxwell's equations]]_.
This, too, is a [[free field theory]] according to def. \ref{FreeFieldTheory}.
=--
+-- {: .num_example #EquationOfMotionOfDiracFieldIsDiracEquation}
###### Example
**([[equation of motion]] of [[Dirac field]] is [[Dirac equation]])**
Consider the [[Lagrangian field theory]] of the [[Dirac field]]
on [[Minkowski spacetime]] from example \ref{LagrangianDensityForDiracField},
with [[field fiber]] the [[spin representation]] $S$ regarded as a [[superpoint]] $S_{odd}$ and
[[Lagrangian density]] given by the spinor bilinear pairing
$$
L
\;=\;
i \overline{\psi} \gamma^\mu \partial_\mu \psi + m \overline{\psi}\psi
$$
(in spacetime dimension $p+1 \in \{3,4,6,10\}$ with $m = 0$ unless $p+1 = 4$).
By example \ref{PresymplecticCurrentDiracField} the [[Euler-Lagrange derivative|Euler-Lagrange]] [[differential operator]] (eq:DifferentialOperatorEulerLagrangeDerivative) for the [[Dirac field]] is of the form
$$
\label{DiracOperatorAsELOperator}
\array{
\Gamma_\Sigma(\Sigma \times S)
&\overset{ }{\longrightarrow}&
\Gamma_\Sigma(\Sigma \times S^\ast)
\\
\Psi &\mapsto& \overline{(-)} D \psi
}
$$
so that the corresponding [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is equivalently
$$
\label{DiracEquation}
\underset{D}{
\underbrace{
\left(-i \gamma^\mu \partial_\mu + m\right)
}}
\psi \;=\; 0
\,.
$$
This is the _[[Dirac equation]]_ and $D$ is called a _[[Dirac operator]]_. In terms of the _[[Feynman slash notation]]_ from (eq:FeynmanSlashNotationForMasslessDiracOperator)
the corresponding [[differential operator]], the _[[Dirac operator]]_ reads
$$
D
\;=\;
\left(
- i \partial\!\!\!/\, + m
\right)
\,.
$$
Hence this is a [[free field theory]] according to def. \ref{FreeFieldTheory}.
Observe that the "square" of the [[Dirac operator]] is the [[Klein-Gordon operator]] $\Box - m^2$ (eq:KleinGordonEquation)
$$
\begin{aligned}
\left(
+i \gamma^\mu \partial_\mu + m
\right)
\left(-i \gamma^\mu \partial_\mu + m\right)\psi
&
=
\left(\partial_\mu \partial^\mu - m^2\right) \psi
\\
& =
\left(\Box - m^2\right) \psi
\end{aligned}
\,.
$$
This means that a [[Dirac field]] which solves the [[Dirac equations]] is
in particular (on [[Minkowski spacetime]]) componentwise a [[solution]] to the [[Klein-Gordon equation]].
=--
+-- {: .num_prop #SupergeometricNatureOfDiracEquation}
###### Remark
**([[supergeometry|supergeometric]] nature of the [[Dirac equation]] as an [[Euler-Lagrange equation]])**
While the [[Dirac equation]] (eq:DiracEquation) of example \ref{EquationOfMotionOfDiracFieldIsDiracEquation}
would make sense in itself also if the field coordinates $\psi$ and jet coordinates $\psi_{,\mu}$ of the [[Dirac field]]
were not anti-commuting (eq:DiracFieldJetCoordinatesAnticommute), due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}), it would,
by remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature}, then no longer be the [[Euler-Lagrange equation]] of a [[Lagrangian density]],
hence then Dirac field theory would not be a [[Lagrangian field theory]].
=--
+-- {: .num_example #DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}
###### Example
**([[Dirac operator]] on [[Dirac spinors]] is [[formally self-adjoint differential operator]])**
The _[[Dirac operator]], hence the [[differential operator]] corresponding to the [[Dirac equation]] of example \ref{EquationOfMotionOfDiracFieldIsDiracEquation} via def. \ref{DifferentialOperator}
is a [[formally self-adjoint differential operator|formally anti-self adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):
$$
D^\ast = - D
\,.
$$
=--
+-- {: .proof}
###### Proof
By (eq:DiracOperatorAsELOperator) we are to regard the Dirac operator as taking values in the [[dual vector bundle|dual]] [[spin bundle]] by using the [[Dirac conjugate]] $\overline{(-)}$ (eq:DiracConjugate):
$$
\array{
\Gamma_\Sigma(\Sigma \times S)
&\overset{}{\longrightarrow}&
\Gamma_\Sigma(\Sigma \times S^\ast)
\\
\Psi &\mapsto& \overline{(-)} D \Psi
}
$$
Then we need to show that there is $K(-,-)$ such that for all [[pairs]] of [[spinor]] [[sections]] $\Psi_1, \Psi_2$ we have
$$
\overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1)
-
\overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2)
\;=\;
d K(\psi_1, \psi_2)
\,.
$$
But the spinor-to-vector pairing is symmetric (eq:SpinorToVectorPairingIsSymmetric), hence this is equivalent to
$$
\overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2
+
\overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2)
\;=\;
d K(\psi_1, \psi_2)
\,.
$$
By the [[product law]] of [[differentiation]], this is solved, for all $\Psi_1, \Psi_2$, by
$$
K(\Psi_1, \Psi_2)
\;\coloneqq\;
\left( \overline{\Psi_1} \gamma^\mu \Psi_2\right)
\,
\iota_{\partial_\mu} dvol
\,.
$$
=--
$\,$
This concludes our discussion of [[Lagrangian densities]] and their [[variational calculus]].
In the [next chapter](#Symmetries) we consider the [[infinitesimal symmetries of Lagrangians]]
and the [[conserved currents]] that these induce via [[Noether's theorem]].
|
A first idea of quantum field theory -- Observables | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Observables |
## Observables
{#Observables}
In this chapter we discuss these topics:
* _[General observables](#GeneralObservables)_
* _[Polynomial off-shell observables and Distributions](#LinearOffShellObservablesAreDistributions)_
* _[Polynomial on-shell observables and Distributional solutions to PDEs](#PolynomialOnShellObservablesAreDistributionalSolutionsToTheEquationsOfMotion)_
* _[Local observables and Transgression](#LocalObservablesByTransgression)_
* _[Infinitesimal observables](#InfinitesimalObservables)_
* _[States](#StatesArePositiveLinearFunctionals)_
$\,$
Given a [[Lagrangian field theory]] (def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}), then a general _[[observable]] quantity_ or just _[[observable]]_ for short (def. \ref{Observable} below), is a [[smooth function]]
$$
A \;\colon\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C}
$$
on the [[on-shell]] [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}, example \ref{SupergeometricSpaceOfFieldHistories})
hence a [[smooth function|smooth]] "[[functional]]" of field histories. We think
of this as assigning to each physically realizable field history $\Phi$ the value $A(\Phi)$ of the given quantity as exhibited by
that field history. For instance concepts like "average [[field strength]] in the compact spacetime region $\mathcal{O}$" should be
observables. In particular the _field [[amplitude]] at spacetime point $x$_ should be an observable, the "[[field observable]]" denoted $\mathbf{\Phi}^a(x)$.
Beware that in much of the literature on [[field theory]], these point-evaluation [[field observables]] $\mathbf{\Phi}^a(x)$
(example below \ref{PointEvaluationObservables}) are eventually referred to as "fields" themselves, blurring the distinction between
1. [[types]] of fields/[[field bundles]] $E$,
1. [[field histories]]/[[sections]] $\Phi$,
1. [[functions]] on the [[space of field histories]] $\mathbf{\Phi}^a(x)$.
In particular, the process of _[[quantization]]_ (discussed in _[Quantization](#Quantization)_ below)
affects the third of these concepts only, in that it [[deformation theory|deforms]] the [[associative algebra|algebra]] [[structure]]
on observables to a [[non-commutative algebra|non-commutative]] [[algebra of quantum observables]].
For this reason the [[field observables]] $\mathbf{\Phi}^a(x)$ are often referred to as _quantum fields_.
But to understand the conceptual nature of [[quantum field theory]] it is important that the
$\mathbf{\Phi}^a(x)$ are really the _[[observables]]_ or _[[quantum observables]]_ on the [[space of field histories]].
**[[field (physics)|fields]]**
| aspect | [[term]] | [[type]] | description | def. |
|--|------------|-|---------|----|
| [[field bundle|field component]] | $\phi^a$, $\phi^a_{,\mu}$ | $J^\infty_\Sigma(E) \to \mathbb{R}$ |[[coordinate function]] on [[jet bundle]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} |
| [[field history]] | $\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$ | $\Sigma \to J^\infty_\Sigma(E)$ | [[jet prolongation]] of [[section]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetProlongation} |
| [[field observable]] | $\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x), $ | $\Gamma_{\Sigma}(E) \to \mathbb{R}$ | [[derivative of a distribution|derivatives]] of [[delta-distribution|delta]]-[[functional]] on [[space of sections]] | def. \ref{Observable}, example \ref{PointEvaluationObservables} |
| averaging of [[field observable]] | $\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$ | $\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$ | [[operator-valued distribution|observable-valued distribution]] | def. \ref{RegularLinearFieldObservables} |
| [[algebra of quantum observables]] | $\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$ | $\mathbb{C}Alg$ | [[non-commutative algebra]] [[structure]] on [[field observables]] | def. \ref{WickAlgebraOfFreeQuantumField}, def. \ref{GeneratingFunctionsForCorrelationFunctions} |
$\,$
There are various further conditions on [[observables]] which we will eventually consider, forming subspaces of
_[[gauge invariant]] observables_ (def. \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}),
_[[local observables]]_ (def. \ref{LocalObservables} below), _Hamiltonian local observables_ (def. \ref{HamiltonianLocalObservables} below) and _[[microcausal functional|microcausal observables]]_ (def. \ref{MicrocausalFunctionals}). While in the end it is
only these special kinds of observables that matter, it is useful to first consider the unconstrained concept
and then consecutively characterize smaller subspaces of well-behaved observables.
In fact it is useful to consider yet more generally the observables on the full [[space of field histories]]
(not just the [[on-shell]] subspace), called the _off-shell observables_.
In the case that the [[field bundle]] is a [[vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}),
the [[off-shell]] [[space of field histories]]
is canonically a [[vector space]] and hence it makes sense to consider _[[linear map|linear]]_ off-shell observables, i.e.
those observables $A$ with $A(c \Phi) = c A(\Phi)$ and $A(\Phi_1 + \Phi_2) = A(\Phi_1) + A(\Phi_2)$.
It turns out that these are precisely the _[[compactly supported distributions]]_ in the sense of [[Laurent Schwartz]]
(prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} below).
This fact makes powerful tools from [[functional analysis]] and [[microlocal analysis]] available for the analysis of [[field theory]]
(discussed [below](#MicrolocalAnalysisAndUltravioletDivergence)).
More generally there are the [[multilinear map|multilinear]] off-shell observables, and these are analogously given by
_[[distributions of several variables]]_ (def. \ref{PolynomialObservables} below). In fully [[perturbative quantum field theory]]
one considers only the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of a single [[on-shell]] [[field history]]
and in this case all [[observables]] are in fact given by such multilinear observables (def. \ref{LocalObservablesOnInfinitesimalNeighbourhood} below).
For a [[free field theory]] (def. \ref{FreeFieldTheory}) whose [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] are given by a [[linear differential operator]] which behaves well in that it is "[[Green hyperbolic differential operator|Green hyperbolic]]" (def. \ref{GreenHyperbolicDifferentialOperator} below) it follows that the actual [[on-shell]] [[linear observables]] are equivalently those off-shell observables which are
_spatially [[compactly supported distribution|compactly supported]] [[distributional solution of a PDE|distributional solutions]]_ to the [[formally adjoint differential operator|formally adjoint]]
[[equation of motion]] (prop. \ref{DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions} below);
and this equivalence is exhibited by [[composition]] with the _[[causal Green function]]_
(def. \ref{AdvancedAndRetardedGreenFunctions} below):
This is theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} below,
which is pivotal for passing from [[classical field theory]] to [[quantum field theory]]:
$$
\left\{
\,\,
\array{
\text{polynomial}
\\
\text{on-shell}
\\
\text{observables}
}
\,\,
\right\}
\underoverset{\simeq}{\text{restriction}}{\longleftarrow}
\left\{
\array{
\text{polynomial}
\\
\text{off-shell}
\\
\text{observables}
\\
\text{modulo equations of motion}
}
\right\}
\underoverset{\simeq}{\text{causal propagator}}{\longleftarrow}
\left\{
\array{
\text{spatially compactly supported}
\\
\text{distributions in several variables}
\\
\text{which are distributional solutions}
\\
\text{to the adjoint equations of motion}
}
\right\}
$$
This fact makes, in addition, the distributional analysis of [[linear differential equations]]
available for the analysis of [[free field theory]], notably the theory of _[[propagators]]_, such as _[[Feynman propagators]]_ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below), which we turn to
in _[Propagators](#Propagators)_ below.
The [[functional analysis]] and [[microlocal analysis]] ([below](#MicrolocalAnalysisAndUltravioletDivergence)) of linear [[observables]] re-expressed in [[distribution|distribution theory]]
via theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} solves the
issues that the original formulation of [[perturbative quantum field theory]] by [[Schwinger-Tomonaga-Feynman-Dyson]] in the 1940s
was notorious for suffering from ([Feynman 85](Schwinger-Tomonaga-Feynman-Dyson#Feynman85SuchABunchOfWords)): The [[normal ordered product]] of
quantum observables in a [[Wick algebra]] of observables follows from [[Hörmander's criterion]] for the [[product of distributions]]
to be well-defined (this we discuss in _[Free quantum fields](#FreeQuantumFields)_ below) and the
_[[renormalization]]_ freedom in the construction of the [[S-matrix]] is governed by the mechanism of
_[[extensions of distributions]]_ (this we discuss in _[Renormalization](#Renormalization)_ below).
Among the polynomial on-shell observables characterized this way, the focus is furthermore on the _[[local observables]]_:
In _[[local field theory]]_ the idea is that both the [[equations of motion]] as well as the observations
are fully determined by their restriction to [[infinitesimal neighbourhoods]] of spacetime points ([[events]]).
For the equations of motion this means that they are [[partial differential equations]] as we have seen
[above](#EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime). For the observables
it should mean that they must be averages over regions of spacetime of functions of the value of the field histories and their derivatives
_at any point_ of spacetime.
Now a "smooth function of the value of the field histories and their derivatives at any point"
is precisely a smooth function on the [[jet bundle]] of the [[field bundle]] (example \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
pulled back via [[jet prolongation]] (def. \ref{JetProlongation}).
If this is to be averaged over spacetime it needs to be the coefficient of a horizontal $p+1$-form (prop. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}).
In mathematical terminology these desiderata say that the [[local observables]] in a local field theory should be precisely the
"[[transgression of variational differential forms|transgressions]]" (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces} below) of horizontal variational $p+1$-forms
(with [[compact topological space|compact]] [[spacetime support]], def. \ref{SpacetimeSupport} below) to the [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}). This is
def. \ref{LocalObservables} below.
A key example of a [[local observable]] in [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) is the _[[action functional]]_ (example \ref{ActionFunctional} below). This is the [[transgression of variational differential forms|transgression]]
of the [[Lagrangian density]] itself, or rather of its product with an "[[adiabatic switching]] function" that
localizes its [[support]] in a compact spacetime region. In typical cases the physical quantity whose observation is
represented by the action functional is the difference of the [[kinetic energy|kinetic]] energy-momentum minus the [[potential energy]]
of a field history averaged over the given region of spacetime.
The [[equations of motion]] of a [[Lagrangian field theory]] say that those field histories
are physically realized which are [[critical points]] of this [[action functional]] observable. This is
the _[[principle of extremal action]]_ (prop. \ref{PrincipleOfExtremalAction} below).
In summary we find the following system of types of observables:
[[!include perturbative observables -- table]]
In the chapter _[Free quantum fields](#FreeQuantumFields)_ we will see that the space of all [[polynomial observables]]
is too large to admit [[quantization]], while the space of [[regular polynomial observables|regular]] [[local observables]]
is too small to contain the usual [[interaction]] terms for [[perturbative quantum field theory]] (example \ref{RegularPolynomialLocalObservablesAreNecessarilyLinear}) below. The space of [[microcausal polynomial observables]]
(def. \ref{MicrocausalObservable} below) is in between these two extremes, and evades both of these obstacles.
$\,$
Given the concept of [[observables]], it remains to formalize what it means for the [[physical system]] to be in some definite _[[state]]_
so that the [[observable]] quantities take some definite value, reflecting the properties of that state.
Whatever formalization for _[[states]]_ of a [[field theory]] one considers, at the very least
the [[space of states]] $States$ should come with a pairing [[linear map]]
$$
\array{
Obs \otimes States & \longrightarrow& \mathcal{C}
\\
\left( A , \langle - \rangle \right) &\mapsto& \langle A \rangle
}
$$
which reads in an [[observable]] quantity $A$ and a state, to be denoted $\langle - \rangle$,
and produces the [[complex number]] $\langle A \rangle$
which is the "value of the observable quantity $A$ in the case that the physical system is in the state $\langle -\rangle$".
One might imagine that
it is fundamentally possible to pinpoint the exact [[field history]] that the
[[physical system]] is found in. From this perspective, fixing a [[state]] should simply mean to
pick such a [[field history]], namely an element $\Phi \in \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$
in the [[on-shell]] [[space of field histories]].
If we write $\langle -\rangle_{\Phi}$ for this state, its pairing map with the [[observables]] would simply be
[[evaluation]] of the observable, being a function on the field history space, on that particular element in this space:
$$
\langle A \rangle_{\Phi} \coloneqq A(\Phi)
\,.
$$
However, in the practice of [[experiment]] a field history can never be known precisely, without remaining uncertainty.
Moreover, [[quantum physics]] (to which we finally come [below](#Quantization)), suggests that this is
true not just in practice, but even in principle. Therefore we should allow [[states]] to be a kind of [[probability distributions]]
on the [[space of field histories]], and regard the pairing $\langle A \rangle$ of a state $\langle - \rangle$ with an observable
$A$ as a kind of _[[expectation value]]_ of the function $A$ averaged with respect to this probability distribution.
Specifically, if the observable quantity $A$ is (a smooth approximation to) a [[characteristic function]]
of a [[subset]] $S \subset \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ of the [[space of field histories]], then its value in a given state should be the [[probability]] to find the [[physical system]] in that subset of field histories.
But, moreover, the [[superposition principle]] of [[quantum physics]] says that the actually observable observables
are only those of the form $A^\ast A$ (for $A^\ast$ the image under the star-operation on the [[star algebra]]
of observables.
This finally leads to the definition of _[[states]]_ in def. \ref{StateOnAStarAlgebra} below.
$\,$
**General observables**
{#GeneralObservables}
+-- {: .num_defn #Observable}
###### Definition
**([[observables]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
with $\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0}$ its [[on-shell]] [[space of field histories]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}).
Then the _space of [[observables]]_ is the [[super formal smooth set]] (def. \ref{SuperFormalSmoothSet})
which is the [[mapping space]]
$$
Obs(E,\mathbf{L})
\;\coloneqq\;
\left[
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
\,,\,
\mathbb{C}
\right]
$$
from the [[on-shell]] [[space of field histories]] to the [[complex numbers]].
Similarly there is the space of _off-shell observables_
$$
\label{GlobalObservables}
Obs(E)
\;\coloneqq\;
\left[
\Gamma_\Sigma(E)
\,,\,
\mathbb{C}
\right]
\,.
$$
Every off-shell observables induces an on-shell observable by [[restriction]], this yields a smooth function
$$
\label{OffShellObservablesRestrictToOnShellObservables}
Obs(E)
\overset{(-)_{\delta_{EL}\mathbf{L} = 0}}{\longrightarrow}
Obs(E,\mathbf{L})
$$
similarly we may consider the observables on the sup-spaces of field histories with restricted causal support
according to def. \ref{CompactlySourceCausalSupport}. We write
$$
Obs(E_{scp})
\;\coloneqq\;
\left[ \Gamma_{\Sigma,scp}(E), \mathbb{C} \right]
$$
and
$$
\label{SpaceOfObservablesOnFieldHistoriesOfSpatiallyCompactSupport}
Obs(E_{scp}, \mathbf{L})
\;\coloneqq\;
\left[ \Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}, \mathbb{C} \right]
$$
for the spaces of (off-shell) observables on [[field histories]] with spatially compact support (def. \ref{CompactlySourceCausalSupport}).
$\,$
Observables form a [[commutative algebra]] under pointwise product:
$$
\label{ObservablesPointwiseProduct}
\array{
Obs(E) \otimes Obs(E) &\overset{(-)\cdot (-)}{\longrightarrow}& Obs(E)
\\
(A_1, A_2) &\mapsto& A_1 \cdot A_2
}
$$
given by
$$
(A_1 \cdot A_2)(\Phi_{(-)}) \coloneqq A_1(\Phi_{(-)}) \cdot A_2(\Phi_{(-)})
\,,
$$
where on the right we have the product in $\mathbb{C}$.
(Suitable subspaces of observables will in addition carry other products, notably [[non-commutative algebra]] [[structures]], this is the topic of the chapters _[Free quantum fields](#FreeQuantumFields)_ and _[Quantum observables](#QuantumObservables)_ below.)
$\,$
**Observables on bosonic fields**
In the case that $E$ is a purely [[bosonic]] [[field bundle]] in [[smooth manifolds]] so that $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$
is a [[diffeological space]] (def. \ref{DiffeologicalSpaceOfFieldHistories}, def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime})
this means that a single [[observable]] $A \in Obs_{E,\mathbf{L}}$ is equivalently a [[smooth function]] (def. \ref{DiffeologicalSpace})
$$
A
\;\colon\;
\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0}
\longrightarrow
\mathbb{C}
\,.
$$
Explicitly, by def. \ref{SmoothSet} (and similarly by def. \ref{SuperFormalSmoothSet}) this means that
$A$ is for each [[Cartesian space]] $U$ (generally: [[super Cartesian space]], def. \ref{SuperCartesianSpace})
a [[natural transformation|natural]] function of plots
$$
A_U
\;\colon\;
\left\{
\array{
U \times \Sigma && \overset{\Phi_{(-)}}{\longrightarrow} && E
\\
& {}_{\mathllap{pr_2}}\searrow && \swarrow_{\mathrlap{fb}}
\\
&& \Sigma
}
\right\}_{\delta_{EL}\mathbf{L} = 0}
\;\overset{}{\longrightarrow}\;
\left\{
U \to \mathbb{C}
\right\}
\,.
$$
**Observables on fermionic fields**
In the case that $E$ has purely [[fermionic]] [[fibers]] (def. \ref{FermionicBosonicFields}), such as for the [[Dirac field]] (example \ref{DiracFieldBundle})
with $E = \Sigma\times S_{odd}$ then the only [[global element|points]] in $Obs_{E}$, namely
morphisms $\mathbb{R}^0 \to Obs_E$ are observables depending on an even power of [[field histories]];
while general observables appear as possibly odd-parameterized families
$$
(\theta \mapsto \theta \Psi)
\;\colon\;
\mathbb{R}^{0\vert 1}
\longrightarrow
Obs_{E,\mathbf{L}}
$$
whose component $\mathbf{\Psi}$ is a section of the even-graded field bundle, regarded in odd degree, via prop. \ref{DiracSpaceOfFieldHistories}. See example \ref{DiracFieldPolynomialObservables} below.
=--
The most basic kind of observables are the following:
+-- {: .num_example #PointEvaluationObservables}
###### Example
**(point evaluation observables -- [[field observables]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] (def. \ref{FieldsAndFieldBundles}) over some [[spacetime]] $\Sigma$
happens to be a [[trivial vector bundle]] in even degree (i.e. bosonic) with [[field fiber]] [[coordinates]] $(\phi^a)$ (example \ref{TrivialVectorBundleAsAFieldBundle}).
With respect to these coordinates a [[field history]], hence a [[section]] of the [[field bundle]]
$$
\Phi \;\in \; \Gamma_\Sigma(E)
$$
has components $(\Phi^a)$ which are [[smooth functions]] on [[spacetime]].
Then for every index $a$ and every point $x \in \Sigma$ in [[spacetime]] (every [[event]]) there is an [[observable]] (def. \ref{Observable})
denoted $\mathbf{\Phi}^a(x)$ which is given by
$$
\mathbf{\Phi}^a(x)
\;\colon\;
\Phi_{(-)}
\mapsto
\Phi_{(-)}^a(x)
\,,
$$
hence which on a test space $U$ (a [[Cartesian space]] or more generally [[super Cartesian space]], def. \ref{SuperCartesianSpace}) sends a $U$-parameterized collection of fields
$$
\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$
to their $U$-parameterized collection of values at $x$ of their $a$-th component.
Notice how the various aspects of the concept of "field" are involved here, all closely related
but crucially different:
$$
\array{
\mathbf{\Phi}^a(x)
&\colon&
\Phi
&\overset{\phantom{AA}}{\mapsto}&
\Phi^a(x)
&=&
\phi^a & \circ \Phi(x)
\\
\array{
\text{field}
\\
\text{observable}
}
&&
\array{
\text{field}
\\
\text{history}
}
&&
\array{
\text{field}
\\
\text{value}
}
&&
\array{
\text{field}
\\
\text{component}
}
}
$$
=--
$\,$
**Polynomial off-shell Observables and Distributions**
{#LinearOffShellObservablesAreDistributions}
We consider here _[[linear observables]]_ (def. \ref{LinearObservables} below) and more generally
_quadratic observables_ (def. \ref{QuadraticObservables}) and generally _[[polynomial observables]]_ (def. \ref{PolynomialObservables} below) for [[free field theories]] and discuss how these are equivalently given by
[[integration]] against [[generalized functions]] called _[[distributions]]_ (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} and prop. \ref{DistributionsAreGeneralizedFunctions} below).
This is the basis for the discussion of [[quantum observables]] for [[free field theories]] [further below](#FreeQuantumFields).
$\,$
+-- {: .num_defn #LinearObservables}
###### Definition
**([[linear map|linear]] [[off-shell]] [[observables]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is a [[super vector bundle|super]] [[vector bundle]]
(as in example \ref{TrivialVectorBundleAsAFieldBundle} and as opposed to more general non-linear [[fiber bundles]]).
This means that the
[[off-shell]] [[space of field histories]] $\Gamma_\Sigma(E)$ (example \ref{SupergeometricSpaceOfFieldHistories})
inherits the structure of a [[super vector space|super]] [[vector space]] by spacetime-pointwise (i.e. [[event]]-wise) scaling and addition of
[[field histories]].
Then an [[off-shell]] [[observable]] (def. \ref{Observable})
$$
A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}
$$
is a _[[linear observable]]_ if it is a [[linear function]] with respect to this vector space structure, hence if
$$
A\left( c \Phi_{(-)}) = c A(\Phi_{(-)} \right)
\phantom{AAAA}
\text{and}
\phantom{AAAA}
A\left(\Phi_{(-)} + \Phi'_{(-)} \right)
=
A\left( \Phi_{(-)}) + A(\Phi'_{(-)} \right)
$$
for all plots of [[field histories]] $\Phi_{(-)}, \Phi'_{(-)}$.
If moreover $(E,\mathbf{L})$ is a [[free field theory]] (def. \ref{FreeFieldTheory}) then the [[on-shell]] [[space of field histories]] inherits this linear structure and we may similarly speak of linear on-shell observables.
We write
$$
LinObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$
for the subspace of [[linear observables]] inside all [[observables]] (def. \ref{Observable}) and similarly
$$
LinObs(E) \hookrightarrow Obs(E)
$$
for the linear off-shell observables inside all off-shell observables, and similarly
for the subspaces of [[linear observables]] on [[field histories]] of spatially compact supprt (eq:SpaceOfObservablesOnFieldHistoriesOfSpatiallyCompactSupport):
$$
\label{LinearObservablesOnSpatiallyCompactlySupportedOnShellFieldHistories}
LinObs(E_{scp}, \mathbf{L})
\hookrightarrow
Obs(E_{scp}, \mathbf{L})
$$
and
$$
LinObs(E_{scp})
\hookrightarrow
Obs(E_{scp})
\,.
$$
=--
+-- {: .num_example #LinearPointEvaluationObservables}
###### Example
**(point evaluation observables are linear)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}),
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is the [[trivial vector bundle]]
with field [[coordinates]] $(\phi^a)$ (example \ref{TrivialVectorBundleAsAFieldBundle}).
Then for each field component index $a$ and point $x \in \Sigma$ of [[spacetime]] (each [[event]])
the point evaluation observable (example \ref{PointEvaluationObservables})
$$
\array{
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
&\overset{\mathbf{\Phi}^a(x)}{\longrightarrow}&
\mathbb{C}
\\
\phi &\mapsto& \phi^a(x)
}
$$
is a [[linear observable]] according to def. \ref{LinearObservables}. The [[distribution]]
that it corresponds to under prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}
is the _[[Dirac delta-distribution]]_ at the point $x$ combined with the [[Kronecker delta]] on the index $a$:
In the [[generalized function]]-notation of remark \ref{LinearObservablesAsGeneralizedFunctions} this reads:
$$
\Phi^a(x) \;\colon\; \Phi \mapsto \int_\Sigma \Phi^b(y) \delta_b^a \delta(x,y) \, dvol_\Sigma(y)
\,.
$$
=--
+-- {: .num_prop #LinearObservablesAreTheCompactlySupportedDistributions}
###### Proposition
**([[linear map|linear]] [[off-shell]] [[observables]] of [[scalar field]] are the [[compactly supported distributions]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}),
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is the [[trivial vector bundle|trivial]] [[real line bundle]]
(as for the [[real scalar field]], example \ref{RealScalarFieldBundle}).
This means that the [[off-shell]] [[space of field histories]] $\Gamma_\Sigma(E) \simeq C^\infty(\Sigma)$ (eq:SpaceOfFieldHistoriesOfRealScalarField) is the
[[real vector space]] of [[smooth functions]] on [[Minkowski spacetime]] and that every [[linear observable]] $A$ (def. \ref{LinearObservables})
gives a [[linear function]]
$$
A_\ast \;\colon\; C^\infty(\Sigma)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C}
\,.
$$
This [[linear function]] $A_\ast$ is in fact
a _[[compactly supported distribution]]_, in the sense of [[functional analysis]], in that it satisfies
the following [[Fréchet vector space]] [[continuous function|continuity condition]]:
* **[[Fréchet space|Fréchet]] [[continuous linear functional]]**
A [[linear function]] $A_\ast \;\colon\; C^\infty(\mathbb{R}^{p,1}) \to \mathbb{R}$ is called _[[continuous function|continuous]]_ if
there exists
1. a [[compact subset]] $K \subset \mathbb{R}^{p,1}$ of [[Minkowski spacetime]];
1. a [[natural number]] $k \in \mathbb{N}$;
1. a [[positive number|positive]] [[real number]] $C \in \mathbb{R}_+$
such that for all [[on-shell]] [[field histories]]
$$
\Phi \in C^\infty(\Sigma)_{\delta_{EL}\mathbf{L} = 0}
$$
the following [[inequality]] of [[absolute values]] ${\vert -\vert}$ of [[partial derivatives]] holds
$$
{\vert A_\ast(\Phi)\vert}
\;\leq\;
C \underset{{\vert \alpha \vert} \leq k}{\sum} \, \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x)\vert}
\,,
$$
where the sum is over all multi-indices $\alpha \in \mathbb{N}^{p+1}$ (eq:PartialDerivativeWithManyIndices)
whose total degree ${\vert \alpha\vert} \coloneqq \alpha_0 + \cdots + \alpha_{p}$ is bounded by $k$,
and where
$$
\partial^\alpha \Phi
\;\coloneqq\;
\frac{\partial^{{\vert \alpha\vert}} \Phi }{ \partial^{\alpha_0} x^0 \partial^{\alpha_1} x^1 \cdots \partial^{\alpha^p} x^p }
$$
denotes the corresponding [[partial derivative]] (eq:PartialDerivativeWithManyIndices).
This identification constitutes a [[linear isomorphism]]
$$
\array{
LinObs(\Sigma \times \mathbb{R}) &\overset{\simeq}{\longrightarrow}& \mathcal{E}'(\Sigma)
\\
\array{
\text{linear off-shell}
\\
\text{observables}
\\
\text{of the scalar field}
}
&&
\array{
\text{compactly supported}
\\
\text{distributions}
\\
\text{on spacetime}
}
}
\,,
$$
saying that all [[compactly supported distributions]] arise from linear off-shell observables of the [[scalar field]] this way, and uniquely so.
=--
For **proof** see at _[[distributions are the smooth linear functionals]]_, [this prop.](distributions+are+the+smooth+linear+functionals#CompactlySupportedDistributionsAreTheSmoothLinearFunctionals)
The identification from prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} of linear off-shell observables
with [[compactly supported distributions]] makes available powerful tools from [[functional analysis]]. The key fact is the
following:
+-- {: .num_prop #DistributionsAreGeneralizedFunctions}
###### Proposition
**([[distributions]] are [[generalized functions]])**
For $n \in \mathbb{N}$, every [[compactly supported function|compactly supported]] [[smooth function]]
$b \in C^\infty_{cp}(\mathbb{R}^n)$ on the [[Cartesian space]] $\mathbb{R}^n$ induces a [[distribution]] (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}), hence a [[continuous linear functional]],
by [[integration]] against $b$ times the [[volume form]].
$$
\array{
C^\infty(\mathbb{R}^n) &\longrightarrow& \mathbb{R}
\\
f &\mapsto& \int_{\mathbb{R}^n} f(x) b(x) \, dvol(x)
}
$$
The distributions arising this way are called the _[[non-singular distributions]]_.
This construction is clearly a [[linear function|linear]] [[monomorphism|inclusion]]
$$
C^\infty_{cp}(\mathbb{R}^n) \overset{\phantom{AAA}}{\hookrightarrow} \mathcal{E}'(\mathbb{R}^n)
$$
and in fact this is a [[dense subspace]] inclusion for the space of compactly supported distributions
$\mathcal{E}'(\mathbb{R}^n)$ equipped with the [[dual space]] topology ([this def.](dual+vector+space#LinearDualOfATopologicalVectorSpace)) to the [[Fréchet space]] structure on $C^\infty(\mathbb{R}^n)$ from prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}.
Hence every [[compactly supported distribution]] $u$ is the [[limit of a sequence|limit]] of a [[sequence]] $\{b_n\}_{n \in \mathbb{N}}$
of [[compactly supported function|compactly supported]] [[smooth functions]] in that for every [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ we have that the value $u(f) \in \mathbb{R}$ is the [[limit of a sequence|limit]] of [[integrals]]
against $b_n dvol$:
$$
u(f)
\;=\;
\underset{n \to \infty}{\lim}\, \int_{\mathbb{R}^n} f(x) b_n(x) dvol(x)
\,.
$$
=--
(e. g. [Hörmander 90, theorem 4.1.5](non-singular+distributionFourier+transform#Hoermander90Fourier+transform#Hoermander90))
Proposition \ref{DistributionsAreGeneralizedFunctions} with prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} implies that with due care we may think of _all_ linear off-shell observables as arising from [[integration]] of [[field histories]]
against some "[[generalized function|generalized smooth functions]]" (namely a [[limit of a sequence|limit]] of actual smooth functions):
+-- {: .num_remark #LinearObservablesAsGeneralizedFunctions}
###### Remark
**([[linear map|linear]] [[off-shell]] [[observables]] of [[real scalar field]] as [[integration]] against [[generalized functions]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}),
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is a [[trivial vector bundle]] with field
coordinates $(\phi^a)$.
Prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} implies immediately that in this
situation linear off-shell observables $A$ (def. \ref{LinearObservables}) correspond to [[tuples]] $(A_a)$
of [[compactly supported distributions]] via
$$
A(\Phi) = \underset{a}{\sum} A_a(\Phi^a)
\,.
$$
With prop. \ref{DistributionsAreGeneralizedFunctions} it follows furthermore that
there is a [[sequence]] of [[tuples]] of [[smooth functions]] $\{(\alpha_n)_{a}\}_{n \in \mathbb{N}}$ such that $A_a$
is the [[limit of a sequence|limit]] of the [[integrations]] against these:
$$
A(\Phi)
\;=\;
\underset{n \to \infty}{\lim}
\,
\int_\Sigma \Phi^a(x) (\alpha_n)_a(x) \, dvol(x)
\,,
$$
where now the [[sum]] over the index $a$ is again left notationally implicit.
For handling distributions/linear off-shell observables it is therefore useful to adopt, with due care,
shorthand notation as if the [[limit of a sequence|limits]] of the [[sequences]] of [[smooth functions]] $(\alpha_n)_a$ actually existed,
as "[[generalized functions]]" $\alpha_a$, and to set
$$
\int_\Sigma \Phi^a(x) \alpha_a(x) \, dvol(x)
\;\coloneqq\;
A(\Phi)
\,,
$$
This suggests that basic operations on functions, such as their pointwise product,
should be [[extension|extended]] to [[distributions]], e.g. to a _[[product of distributions]]_.
This turns out to exist, as long as the high-frequency modes in the [[Fourier transform of distributions|Fourier transform of]]
the distributions being multiplied cancel out -- the mathematical reflection of "[[UV-divergences]]" in [[quantum field theory]].
This we turn to in _[Free quantum fields](#FreeQuantumFields)_ below.
=--
These considerations generalize from the [[field bundle]] of the [[real scalar field]] to general [[field bundles]] (def. \ref{FieldsAndFieldBundles})
as long as they are [[smooth vector bundles]] (def. \ref{VectorBundle}):
+-- {: .num_defn #TVSStructureOnSpacesOfSmoothSections}
###### Definition
**([[Fréchet space|Fréchet]] [[topological vector space]] on [[spaces of smooth sections]] of a [[smooth vector bundle]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) which is a [[smooth vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime});
hence, up to [[isomorphism]], a [[trivial vector bundle]] as in example \ref{TrivialVectorBundleAsAFieldBundle}.
On its [[real vector space]] $\Gamma_\Sigma(E)$ [[space of sections|of smooth sections]] consider the [[seminorms]] indexed by a [[compact subset]] $K \subset \Sigma$ and a [[natural number]] $k \in \mathbb{N}$ and given by
$$
\array{
\Gamma_\Sigma(E) &\overset{p_K^k}{\longrightarrow}&
[0,\infty)
\\
\Phi &\mapsto& \underset{ {\vert \alpha\vert} \leq k}{max} \left( \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x)\vert}\right) \,,
}
$$
where on the right we have the [[absolute values]] of the [[partial derivatives]] of $\Phi$ index by $\alpha$ (eq:PartialDerivativeWithManyIndices) with respect to any choice of [[norm]] on the [[fibers]].
This makes $\Gamma_\Sigma(E)$ a [[Fréchet space|Fréchet]] [[topological vector space]].
For $K \subset \Sigma$ any [[closed subset]] then the sub-space of sections
$$
\Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)
$$
of sections whose [[support]] is inside $K$ becomes a [[Fréchet space|Fréchet]] [[topological vector spaces]] with the induced [[subspace topology]], which makes these be [[closed subspaces]].
Finally, the [[vector spaces]] of smooth sections with prescribed causal support (def. \ref{CompactlySourceCausalSupport})
are [[inductive limits]] of vector spaces $\Gamma_{\Sigma,K}(E)$ as above, and hence they inherit
[[topological vector space]] [[structure]] by forming the corresponding [[inductive limit]] in the [[category]]
of [[topological vector spaces]]. For instance
$$
\Gamma_{\Sigma,cp}(E)
\;\coloneqq\;
\underset{\underset{ {K \subset \Sigma} \atop {K\, \text{compact}} }{\longrightarrow}}{\lim}
\Gamma_{\Sigma,K}(E)
$$
etc.
=--
([Bär 14, 2.1](space+of+sections#Baer14))
+-- {: .num_defn #DistributionalSections}
###### Definition
**([[distribution|distributional]] [[sections]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}).
The [[vector space|vector]] [[spaces of smooth sections]] with restricted support from def. \ref{CompactlySourceCausalSupport} structures of [[topological vector spaces]] via def. \ref{TVSStructureOnSpacesOfSmoothSections}. We denote the [[dual topological vector spaces]] by
$$
\Gamma'_{\Sigma}( E ^*)
\;\coloneqq\;
(\Gamma_{\Sigma,cp}(E))^*
\,.
$$
This is called the space of _distributional sections_ of the [[dual vector bundle]] ${E}^*$.
The _[[support of a distribution|support of a distributional section]]_ $supp(u)$ is the set of points in $\Sigma$ such that
for every neighbourhood of that point $u$ does not vanish on all sections with support in that neighbourhood.
Imposing the same restrictions to the [[supports of distributions|supports of distributional sections]] as in def. \ref{CompactlySourceCausalSupport}, we have the following subspaces of distributional sections:
$$
\Gamma'_{\Sigma,cp}(E^\ast) ,
\Gamma'_{\Sigma,\pm cp}(E^\ast) ,
\Gamma'_{\Sigma,scp}(E^\ast) ,
\Gamma'_{\Sigma,fcp}(E^\ast) ,
\Gamma'_{\Sigma,pcp}(E^\ast) ,
\Gamma'_{\Sigma,tcp}(E^\ast)
\;\subset\;
\Gamma'_{\Sigma}(E^\ast) .
$$
=--
([Sanders 13](Green+hyperbolic+differential+operator#Sanders12), [Bär 14](Green+hyperbolic+differential+operator#Baer14))
As before in prop. \ref{DistributionsAreGeneralizedFunctions} the actual [[smooth sections]] yield
examples of distributional sections, and all distributional sections arise as [[limit of a sequence|limits]]
of integrations against smooth sections:
+-- {: .num_prop #NonSingularDistributionalSections}
###### Proposition
**([[non-singular distribution|non-singular distributional sections]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over [[Minkowski spacetime]] and let
$s \in \{cp, \pm cp, scp, tcp\}$ be any of the [[support]] conditions from def. \ref{CompactlySourceCausalSupport}.
Then the operation of regarding a [[compact support|compactly supported]] [[smooth section]] of the [[dual vector bundle]]
as a [[functional]] on sections with this support property is a [[dense subspace]] inclusion into the
[[topological vector space]] of distributional sections from def. \ref{DistributionalSections}:
$$
\array{
\Gamma_{\Sigma,cp}(E^\ast)
&\overset{\phantom{A}u_{(-)}\phantom{A} }{\hookrightarrow}&
\Gamma'_{\Sigma,s}(E)
\\
b &\mapsto& \left( \Phi \mapsto \underset{\Sigma}{\int} b(x) \cdot \Phi(x) \, dvol_\Sigma(x) \right)
}
$$
=--
([Bär 14, lemma 2.15](Green+hyperbolic+differential+equation#Baer14}))
+-- {: .num_prop #DistributionsWithCausalSupports}
###### Proposition
**(distribution dualities with causally restricted supports)**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}).
Then there are the following [[isomorphisms]] of [[topological vector spaces]] between a) [[dual spaces]] of [[spaces of sections]]
with restricted causal support (def. \ref{CompactlySourceCausalSupport}) and equipped with the topology from def. \ref{TVSStructureOnSpacesOfSmoothSections} and b) spaces of distributional sections with restricted supports, according to def. \ref{DistributionalSections}:
$$
\begin{aligned}
\Gamma_{\Sigma,cp}(E)^* &\simeq \Gamma'_{\Sigma}(E^\ast) ,
\\
\Gamma_{\Sigma,+cp}(E)^* &\simeq \Gamma'_{\Sigma,fcp}(E^\ast) ,
\\
\Gamma_{\Sigma,-cp}(E)^* &\simeq \Gamma'_{\Sigma,pcp}(E^\ast) ,
\\
\Gamma_{\Sigma,scp}(E)^* &\simeq \Gamma'_{\Sigma,tcp}(E^\ast) ,
\\
\Gamma_{\Sigma,fcp}(E)^* &\simeq \Gamma'_{\Sigma,+cp}(E^\ast) ,
\\
\Gamma_{\Sigma,pcp}(E)^* &\simeq \Gamma'_{\Sigma,-cp}(E^\ast) ,
\\
\Gamma_{\Sigma,tcp}(E)^* &\simeq \Gamma'_{\Sigma,scp}(E^\ast) ,
\\
\Gamma_{\Sigma}(E)^* &\simeq \Gamma'_{\Sigma,cp}(E^\ast) .
\end{aligned}
$$
=--
([Sanders 13, thm. 4.3](Green+hyperbolic+differential+operator#Sanders12), [Bär 14, lem. 2.14](Green+hyperbolic+differential+operator#Baer14))
The concept of [[linear observables]] naturally generalizes to that of [[multilinear map|multilinear observables]]:
+-- {: .num_defn #QuadraticObservables}
###### Definition
**([[quadratic form|quadratic]] [[off-shell]] [[observables]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over a [[spacetime]] $\Sigma$
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is a [[super vector bundle|super]] [[vector bundle]].
The _[[external tensor product|external]] [[tensor product of vector bundles]]_ of the [[field bundle]]
$E \overset{fb}{\to} \Sigma$ with itself, denoted
$$
E \boxtimes E \overset{}{\to} \Sigma \times \Sigma
$$
is the [[vector bundle]] over the [[Cartesian product]] $\Sigma \times \Sigma$, of [[spacetime]] with itself,
whose [[fiber]] over a pair of points $(x_1,x_2)$ is the [[tensor product]] $E_{x_1} \otimes E_{x_2}$ of the corresponding field fibers.
Given a [[field history]], hence a [[section]] $\phi \in \Gamma_\Sigma(E)$ of the [[field bundle]], there is then
the induced section $\phi \boxtimes \phi \in \Gamma_{\Sigma \times \Sigma}(E \boxtimes E)$.
We say that an [[off-shell]] [[observable]]
$$
A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}
$$
is _[[quadratic form|quadratic]]_ if it comes from a "graded-symmetric [[bilinear map|bilinear]] observable", namely a smooth function on the [[space of sections]]
of the [[external tensor product|external]] [[tensor product of vector bundles|tensor product of]] the [[field bundle]]
with itself
$$
B
\;\colon\;
\Gamma_{\Sigma \times \Sigma}(E \boxtimes E)_{\delta_{EL}\mathbf{L} = 0}
\longrightarrow
\mathbb{C}
\,,
$$
as
$$
A(\Phi) = B(\Phi,\Phi)
\,.
$$
More explicitly: By prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions} the quadratic observable $A$
is given by a [[compactly supported distribution]] [[distribution of two variables|of two variables]]
which in the notation of remark \ref{LinearObservablesAsGeneralizedFunctions} comes from a graded-symmetric [[matrix]] of
[[generalized functions]]
$\beta_{a_1 a_2} \in \mathcal{E}'(\Sigma \times \Sigma, E \boxtimes E)$ as
$$
A(\Phi)
\;=\;
\int_{\Sigma \times \Sigma}
\beta_{a_1 a_2}(x_1,x_2)
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2)\, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
\,.
$$
This notation makes manifest how the concept of quadratic observables is a generalization of that of [[quadratic forms]] coming from [[bilinear forms]].
=--
+-- {: .num_defn #PolynomialObservables}
###### Definition
**([[off-shell]] [[polynomial observables]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over a [[spacetime]] $\Sigma$
whose [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}) is a [[super vector bundle|super]] [[vector bundle]].
An [[off-shell]] [[observable]] (def. \ref{Observable})
$$
A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}
$$
is a _[[polynomial observable]]_ if it is the [[sum]] of a constant, and a [[linear observable]] (def. \ref{LinearObservables}),
and a quadratic observable (def. \ref{QuadraticObservables}) and so on:
$$
\label{ExpansionOfPolynomialObservables}
\begin{aligned}
A(\Phi)
& = \phantom{+}
\alpha^{(0)}
\\
&
\phantom{=}
+
\int_{\Sigma} \Phi^a(x) \alpha^{(1)}_a(x) \, dvol_\Sigma(x)
\\
&
\phantom{=}
+
\int_{\Sigma^2}
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2)
\alpha^{(2)}_{a_1 a_2}(x_1, x_2) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
\\
&
\phantom{=}
+
\int_{\Sigma^3}
\Phi^{a_1}(x_1) \cdot \Phi^{a_2}(x_2) \cdot \Phi^{a_3}(x_3)
\alpha^{(3)}_{a_1 a_2 a_3}(x_1,x_2,x_3)
\, dvol_\Sigma(x_1) dvol_\Sigma(x_2) dvol_\Sigma(x^3)
\\
&
\phantom{=}
+
\cdots
\,.
\end{aligned}
$$
If all the [[coefficient]] [[distributions]] $\alpha^{(k)}$ are [[non-singular distributions]], then we say that $A$ is a _[[regular polynomial observable]]_.
We write
$$
PolyObs(E)_{reg} \hookrightarrow PolyObs(E) \hookrightarrow Obs(E)
$$
for the subspace of (regular) polynomial off-shell observables.
=--
+-- {: .num_example #DiracFieldPolynomialObservables}
###### Example
**([[polynomial observables]] of the [[Dirac field]])**
Let $E = \Sigma \times S_{odd}$ be the [[field bundle]] of the [[Dirac field]] (example \ref{DiracFieldBundle}).
Then, by prop. \ref{DiracSpaceOfFieldHistories},
an $\mathbb{R}^{0\vert 1}$-parameterized plot of the space of [[off-shell]] [[polynomial observables]] (def. \ref{PolynomialObservables})
$$
A_{(-)}
\;\colon\;
\mathbb{R}^{0 \vert 1}
\longrightarrow
PolyObs(\Sigma \times S_{odd})
$$
is of the form
$$
\begin{aligned}
A_{(-)}
& =
a^{(0)}
\\
& \phantom{=}
+
\theta
\underset{\Sigma}{\int}
a^{(1)}_{\alpha}(x)
\mathbf{\Psi}^\alpha(x)
dvol_\Sigma(x)
\\
&
\phantom{=}
+
\underset{\Sigma^2}{\int}
a^{(2)}_{\alpha_1 \alpha_2}(x,y)
\mathbf{\Psi}^{\alpha_1}(x_1)
\cdot
\mathbf{\Psi}^{\alpha_2}(x_2)
\,
dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
\\
& \phantom{=}
+
\theta
\underset{\Sigma}{\int}
a^{(3)}_{\alpha_1 \alpha_2 \alpha_3}(x_1, x_2, x_3)
\mathbf{\Psi}^{\alpha_1}(x_1)
\cdot
\mathbf{\Psi}^{\alpha_2}(x_2)
\cdot
\mathbf{\Psi}^{\alpha_3}(x_3)
\, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \, dvol_\Sigma(x_3)
\\
& \phantom{=}
+ \cdots
\end{aligned}
$$
for any [[distributions of several variables]] $a^{(k)}_{\alpha_1, \cdots , \alpha_k}$. Here
$$
\mathbf{\Psi}^\alpha(x)
\;\colon\;
\Gamma_\Sigma(\Sigma \times S_{even})
\longrightarrow
\mathbb{C}
$$
are the point-evaluation [[field observables]] (example \ref{PointEvaluationObservables}) on the [[spinor bundle]],
and
$$
\theta \in C^\infty(\mathbb{R}^{0\vert 1})_{odd}
$$
is the canonical odd-graded coordinate function
on the [[superpoint]] $\mathbb{R}^{0 \vert 1}$ (def. \ref{SuperCartesianSpace}).
Hence all the _odd_ powers of the [[Dirac field|Dirac]]-[[field observables]] are proportional to $\theta$.
In particular if one considers just a point in the space of polynomial observables
$$
A \;\colon\; \mathbb{R}^{0} \longrightarrow PolyObs(E \times S_{odd})
$$
then all the odd monomials in the [[field observables]] of the [[Dirac field]] disappear.
=--
+-- {: .proof}
###### Proof
By definition of supergeometric [[mapping spaces]] (def. \ref{MappingSpaceOutOfASuperCartesianSpace}),
there is a [[natural bijection]] between $\mathbb{R}^{0 \vert 1}$-plots $A_{(-)}$ of the space of observables
and smooth functionss out of the [[Cartesian product]] of $\mathbb{R}^{0 \vert 1}$ with the [[space of field histories]]
to the [[complex numbers]]:
$$
\frac{
\mathbb{R}^{0\vert 1}
\overset{ A_{(-)} }{\longrightarrow}
[ \Gamma_\Sigma(\Sigma \times S_{odd}), \mathbb{C} ]
}
{
\mathbb{R}^{0 \vert 1} \times \Gamma_\Sigma(\Sigma \times S_{odd})
\longrightarrow
\mathbb{C}
}
$$
Moreover, by prop. \ref{DiracSpaceOfFieldHistories} we have that the coordinate functions
on the space of field histories of the Dirac bundle are given by the field observables $\mathbf{\Psi}^\alpha(x)$
regarded in odd degree. Now a homomorphism as above has to pull back the even coordinate function
on $\mathbb{C}$ to even coordinate functions on this Cartesian product, hence to joint even powers of
$\theta$ and $\mathbf{\Psi}^\alpha(x)$.
=--
$\,$
Next we discuss the restriction of these off-shell polynomial observables to the [[shell]] to yield [[on-shell]] polynomial observables, characterized by theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} below.
$\,$
**Polynomial on-shell Observables and Distributional solutions to PDEs**
{#PolynomialOnShellObservablesAreDistributionalSolutionsToTheEquationsOfMotion}
The evident [[on-shell]] version of def. \ref{PolynomialObservables} is this:
+-- {: .num_defn #PolynomialObservablesOnShell}
###### Definition
**([[on-shell]] [[polynomial observables]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[on-shell]] [[space of field histories]] $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Then an [[on-shell]] [[observable]] (def. \ref{Observable})
$$
A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}
$$
is an _[[on-shell]] [[polynomial observable]]_ if it is the [[restriction]] of an [[off-shell]] [[polynomial observable]] $A_{off}$ according to def. \ref{PolynomialObservables}:
$$
\array{
\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} &\overset{\phantom{A}A\phantom{A}}{\longrightarrow}& \mathbb{C}
\\
\downarrow & \nearrow_{\mathrlap{A_{off}}}
\\
\Gamma_\Sigma(E)
}
\,.
$$
Similarly $A$ is an [[on-shell]] [[linear observable]] or [[on-shell]] [[regular polynomial observable]] etc. if it is the [[restriction]] of a [[linear observable]] or [[regular polynomial observable]], respectively, according to def. \ref{PolynomialObservables}. We write
$$
PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$
for the subspace of polynomial on-shell observables inside all on-shell observables, and similarly
$$
LinObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$
and
$$
PolyObs(E,\mathbf{L})_{reg} \hookrightarrow Obs(E,\mathbf{L})
$$
etc.
=--
While by def. \ref{PolynomialObservablesOnShell} every [[off-shell]] [[observable]] induces an [[on-shell]] [[observable]]
simply by [[restriction]] (eq:OffShellObservablesRestrictToOnShellObservables),
different off-shell observables may restrict to the _same_ on-shell observable. It is therefore useful to find a condition on off-shell observables that makes them equivalent to on-shell observables under restriction.
We now discuss such precise characterizations of the off-shell polynomial observables for the case of sufficiently well behaved [[free field theory|free field]] [[equations of motion]] -- namely
[[Green hyperbolic differential equations]], def. \ref{GreenHyperbolicDifferentialOperator} below.
The main result is theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} below.
While in general the [[equations of motion]] are not [[Green hyperbolic differential equations|Green hyperbolic]] -- namely not in the presence of implicit [[infinitesimal gauge symmetries]] discussed in _[Gauge symmetries](#GaugeSymmetries)_ below -- it turns out that up to a suitable notion of [[equivalence]] they are equivalent to those that are; this we discuss in the chapter _[Gauge fixing](#GaugeFixing)_ below.
$\,$
+-- {: .num_defn #DistributionalDerivatives}
###### Definition
**([[derivatives of distributions]] and [[distributional solutions of PDEs]])**
Given a [[pair]] of [[formally adjoint differential operators]] $P, P^\ast \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$
(def. \ref{FormallyAdjointDifferentialOperators})
then the _[[derivative of distributions|distributional derivative]]_
of a [[distribution|distributional section]] $u \in \Gamma'_\Sigma(E)$ (def. \ref{DistributionalSections})
by $P$ is the distributional section $P u \in \Gamma'_\Sigma(E^\ast)$
$$
P u
\;\coloneqq\;
u(P^\ast(-)) \;\colon\; \Gamma_{\Sigma,cp}(E^\ast)
\,.
$$
If
$$
P u = 0 \;\in\; \Gamma'_\Sigma(E^\ast)
$$
then we say that $u$ is a [[distributional solution]] (or [[generalized solution]]) of the homogeneous [[differential equation]]
defined by $P$.
=--
+-- {: .num_example #DistributionalPDESolutionsFromOrdinaryPDESolutions}
###### Example
**(ordinary [[PDE]] solutions are [[generalized solutions of a PDE|generalized solutions]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over [[Minkowski spacetime]]
and let $P, P^\ast \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$ be a [[pair]] of [[formally adjoint differential operators]].
Then for every [[non-singular distribution|non-singular distributional section]] $u_{\Phi} \in \Gamma'_{\Sigma}(E^\ast)$
coming from an actual smooth section $\Phi \in \Gamma_\Sigma(E)$ via prop. \ref{NonSingularDistributionalSections}
the [[derivative of distributions]] (def. \ref{DistributionalDerivatives}) is the distributional section
induced from the ordinary derivative of smooth functions:
$$
P u_\Phi \;=\; u_{P \Phi}
\,.
$$
In particular $u_\Phi$ is a [[generalized solution of a PDE|distributional solution]] to the [[PDE]] precisely
if $\Phi$ is an ordinary solution:
$$
P u_\Phi \;=\; 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
P \Phi = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
For all $b \in \Gamma_{\Sigma,cp}(E)$ we have
$$
\begin{aligned}
(P u_\Phi)(b)
& =
u_\Phi(P^\ast b)
\\
& =
\int u \cdot P^\ast b \, dvol
\\
& =
\int (P u) \cdot b \, dvol
\\
& =
u_{P \Phi}(b)
\end{aligned}
$$
where all steps are by the definitions except the third, which is by the definition of
[[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}),
using that by the [[compact support]] of $b$ and the [[Stokes theorem]] (prop. \ref{StokesTheorem})
the term $K(\Phi,b)$ in def. \ref{FormallyAdjointDifferentialOperators} does not contribute to the [[integral]].
=--
+-- {: .num_defn #AdvancedAndRetardedGreenFunctions}
###### Definition
**([[advanced and retarded Green functions]] and [[causal Green function]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) which is a [[vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}).
Let $P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(E^\ast)$ be a [[differential operator]] (def. \ref{DifferentialOperator}) on its [[space of smooth sections]].
Then a [[linear map]]
$$
\mathrm{G}_{P,\pm}
\;\colon\;
\Gamma_{\Sigma, cp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, \pm cp}(E)
$$
from [[spaces of smooth sections]] of [[compact support]] to spaces of sections of causally sourced future/past support (def. \ref{CompactlySourceCausalSupport}) is called an _[[advanced or retarded Green function]]_ for $P$, respectively, if
1. for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have
$$
\label{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator}
G_{P,\pm} \circ P(\Phi) = \Phi
$$
and
$$
\label{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator}
P \circ G_{P,\pm}(\Phi) = \Phi
$$
1. the [[support]] of $G_{P,\pm}(\Phi)$ is in the [[closed future cone]] or [[closed past cone]] of the support of $\Phi$, respectively.
If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference
$$
\label{CausalGreenFunction}
\mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$
is called the _[[causal Green function]]_.
=--
(e.g. [Bär 14, def. 3.2, cor. 3.10](Green+hyperbolic+differential+equation#Baer14}))
+-- {: .num_defn #GreenHyperbolicDifferentialOperator}
###### Definition
**([[Green hyperbolic differential equation]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) which is a [[vector bundle]] (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}).
A [[differential operator]] (def. \ref{NormallyHyperbolicDifferentialOperator})
$$
P
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\Gamma_{\Sigma}(E^\ast)
$$
is called a _[[Green hyperbolic differential operator]]_ if $P$ as well as its [[formal adjoint differential operator]] $P^\ast$ (def. \ref{FormallyAdjointDifferentialOperators}) admit [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctions}).
=--
([Bär 14, def. 3.2](Green+hyperbolic+partial+differential+equation##Baer14}), [Khavkine 14, def. 2.2](Green+hyperbolic+partial+differential+equation#Khavkine14))
The two archtypical examples of [[Green hyperbolic differential equations]] are the [[Klein-Gordon equation]]
and the [[Dirac equation]] on [[Minkowski spacetime]]. For the moment we just cite the existence of the
[[advanced and retarded Green functions]] for these, we will work these out in detail below in _[Propagators](#Propagators)_.
+-- {: .num_example #GreenHyperbolicKleinGordonEquation}
###### Example
**([[Klein-Gordon equation]] is a [[Green hyperbolic differential equation]])**
The [[Klein-Gordon equation]], hence the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]
of the [[free field theory|free]] [[scalar field]] (example \ref{EquationOfMotionOfFreeRealScalarField})
is a [[Green hyperbolic differential equation]] (def. \ref{GreenHyperbolicDifferentialOperator})
and [[formal adjoint differential operator|formally self-adjoint]] (example \ref{FormallySelfAdjointKleinGordonOperator}).
=--
(e. g. [Bär-Ginoux-Pfaeffle 07](Klein-Gordon+equation#BaerGinouxPfaeffle07), [Bär 14, example 3.3](#Baer14))
+-- {: .num_example #GreenHyperbolicDiracOperator}
###### Example
**([[Dirac operator]] is [[Green hyperbolic differential operator|Green hyperbolic]])**
The [[Dirac equation]], hence the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]
of the [[mass|massive]] [[free field theory|free]] [[Dirac field]] (example \ref{EquationOfMotionOfDiracFieldIsDiracEquation}) is a [[Green hyperbolic differential equation]] (def. \ref{GreenHyperbolicDifferentialOperator})
and [[formally adjoint differential operator|formally anti self-adjoint]] (example \ref{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}).
=--
([Bär 14, corollary 3.15, example 3.16](#Baer14))
+-- {: .num_example #CausalGreenFunctionOfFormallyAdjointDifferentialOperatorAreFormallyAdjoint}
###### Example
**([[causal Green functions]] of [[formally adjoint differential operator|formally adjoint]] [[Green hyperbolic differential operators]] are [[formally adjoint differential operator|formally adjoint]])**
Let
$$
P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(E^\ast)
$$
be a pair of [[Green hyperbolic differential operators]] (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}). Then also their [[causal Green functions]] $\mathrm{G}_P$ and $G_{P^\ast}$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[formally adjoint differential operators]], up to a sign:
$$
\left(
\mathrm{G}_P
\right)^\ast
\;=\;
- \mathrm{G}_{P^\ast}
\,.
$$
=--
([Khavkine 14, (24), (25)](Green+hyperbolic+partial+differential+equation#Khavkine14))
We did not require that the [[advanced and retarded Green functions]] of a [[Green hyperbolic differential operator]] are unique;
in fact this is automatic:
+-- {: .num_prop #AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}
###### Proposition
**([[advanced and retarded Green functions]] of [[Green hyperbolic differential operator]] are unique)**
The [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctions}) of a [[Green hyperbolic differential operator]] (def. \ref{GreenHyperbolicDifferentialOperator}) are unique.
=--
([Bär 14, cor. 3.12](Green+hyperbolic+differential+operator#Baer14})
Moreover we did not require that the [[advanced and retarded Green functions]] of a [[Green hyperbolic differential operator]]
come from [[integral kernels]] ("[[propagators]]"). This, too, is automatic:
+-- {: .num_prop #GreenFunctionsAreContinuous}
###### Proposition
**([[causal Green functions]] of [[Green hyperbolic differential operators]] are [[continuous linear maps]])**
Given a [[Green hyperbolic differential operator]] $P$ (def. \ref{GreenHyperbolicDifferentialOperator}), the advanced, retarded and causal Green functions of $P$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[continuous linear maps]] with respect to the [[topological vector space]] structure from def. \ref{TVSStructureOnSpacesOfSmoothSections} and also have a unique continuous [[extension]] to the spaces of sections with larger support (def. \ref{CompactlySourceCausalSupport}) as follows:
$$
\begin{aligned}
\mathrm{G}_{P,+}
&\;\colon\;
\Gamma_{\Sigma, pcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, pcp}(E) ,
\\
\mathrm{G}_{P,-}
&\;\colon\;
\Gamma_{\Sigma, fcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma, fcp}(E) ,
\\
\mathrm{G}_{P}
&\;\colon\;
\Gamma_{\Sigma, tcp}(E^\ast)
\longrightarrow
\Gamma_{\Sigma}(E) ,
\end{aligned}
$$
such that we still have the relation
$$
\mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$
and
$$
P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id
$$
and
$$
supp \mathrm{G}_{P,\pm}({\alpha}^*) \subseteq J^\pm(supp {\alpha}^*)
\,.
$$
By the _[[Schwartz kernel theorem]]_ the continuity of $\mathrm{G}_{\pm}, \mathrm{G}$
implies that there are [[integral kernels]]
$$
\Delta_{\pm} \;\in\; \Gamma'_{\Sigma \times \Sigma}( E \boxtimes_\Sigma E )
$$
such that, in the notation of [[generalized functions]],
$$
(G_{\pm} \alpha^\ast)(x)
\;=\;
\underset{\Sigma}{\int}
\Delta_\pm(x,y) \cdot \alpha^\ast(y) \, dvol_\Sigma(y)
\,.
$$
These [[integral kernels]] are called the _[[advanced and retarded propagators]]_. Similarly the combination
$$
\label{CausalPropagator}
\Delta \;\coloneqq\; \Delta_+ - \Delta_-
$$
is called the _[[causal propagator]]_.
=--
([Bär 14, thm. 3.8, cor. 3.11](Green+hyperbolic+differential+operator#Baer14))
We now come to the main theorem on [[polynomial observables]]:
+-- {: .num_lemma #ExactSequenceOfGreenHyperbolicSystem}
###### Lemma
**([[exact sequence]] of [[Green hyperbolic differential operator]])**
Let $\Gamma_\Sigma(E) \overset{P}{\longrightarrow} \Gamma_\Sigma(E^\ast)$ be a [[Green hyperbolic differential operator]] (def. \ref{GreenHyperbolicDifferentialOperator}) with [[causal Green function]] $\mathrm{G}$ (def. \ref{GreenHyperbolicDifferentialOperator}). Then the sequences
$$
\label{GreenOperatorExactSequenceFirst}
\array{
0
&\to&
\Gamma_{\Sigma,cp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,cp}(E^\ast)
&\overset{\mathrm{G}_P}{\longrightarrow}&
\Gamma_{\Sigma,scp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,scp}(E^\ast)
&\to&
0
\\
\\
0
&\to&
\Gamma_{\Sigma,tcp}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma,tcp}(E^\ast)
&\overset{\mathrm{G}_P}{\longrightarrow}&
\Gamma_{\Sigma}(E)
&\overset{P}{\longrightarrow}&
\Gamma_{\Sigma}(E^\ast)
&\to&
0
}
$$
of these operators restricted to functions with causally restricted supports as indicated (def. \ref{CompactlySourceCausalSupport})
are [[exact sequence]]s of [[topological vector spaces]] and continuous [[linear map]]s between them.
Under passing to [[dual spaces]] and using the isomorphisms of spaces of distributional sections (def. \ref{DistributionalSections}) from prop. \ref{DistributionsWithCausalSupports} this yields the following dual [[exact sequence]] of [[topological vector spaces]] and continuous [[linear maps]] between them:
$$
\label{GreenHyperbolicOperatorDualExactSequence}
\array{
0
&\to&
\Gamma'_{\Sigma,tcp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma'_{\Sigma,tcp}(E^\ast)
&\overset{-\mathrm{G}_{P^*}}{\longrightarrow}&
\Gamma'_{\Sigma}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma'_{\Sigma}(E^\ast)
&\to&
0
\\
\\
0
&\to&
\Gamma'_{\Sigma,cp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma'_{\Sigma,cp}(E^\ast)
&\overset{-\mathrm{G}_{P^*}}{\longrightarrow}&
\Gamma'_{\Sigma,scp}(E)
&\overset{P^*}{\longrightarrow}&
\Gamma'_{\Sigma,scp}(E^\ast)
&\to&
0
}
$$
=--
This is due to [[Igor Khavkine]], based on ([Khavkine 14, prop. 2.1](Green+hyperbolic+partial+differential+equation#Khavkine14));
for **proof** see at _[[Green hyperbolic differential operator]]_ [this lemma](Green+hyperbolic+partial+differential+equation#ExactSequenceOfGreenHyperbolicSystem).
+-- {: .num_cor #OnShellSpaceOfFieldHistoriesForFreeFieldTheoryGreenHyperbolic}
###### Corollary
**([[on-shell]] [[space of field histories]] for [[Green hyperbolic differential operator|Green hyperbolic]] [[free field theories]])**
Let $(E,\mathbf{L})$ be a [[free field theory]] [[Lagrangian field theory]] (def. \ref{LagrangianDensityForDiracField})
whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] $P \Phi = 0$ is [[Green hyperbolic differential operator|Green hyperbolic]] (def. \ref{GreenHyperbolicDifferentialOperator}).
Then the [[on-shell]] [[space of field histories]] (or of [[field histories]] with spatially compact support, def. \ref{CompactlySourceCausalSupport}) is, as a [[vector space]], [[linear isomorphism|linearly isomorphic]]
to the [[quotient space]] of [[compact support|compactly supported]] sections (or of temporally compactly supported sections, def. \ref{CompactlySourceCausalSupport}) by the [[image]] of the [[differential operator]] $P$, and this isomorphism
is given by the [[causal Green function]] $\mathrm{G}_P$ (eq:CausalGreenFunction)
$$
\label{SolutionSpaceIsomorphicToQuotientByImP}
\array{
\Gamma_{\Sigma,tcp}(E^\ast)/im(P)
&\underoverset{\simeq}{\phantom{A}\mathrm{G}_P \phantom{A}}{\longrightarrow}&
ker(P) \;=\; \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}
\\
\Gamma_{\Sigma,cp}(E^\ast)/im(P)
&\underoverset{\simeq}{\phantom{A}\mathrm{G}_P\phantom{A}}{\longrightarrow}&
ker_{scp}(P) \;=\; \Gamma_{\Sigma,scp}(E)_{\delta_{EL}\mathbf{L} = 0}
\,.
}
$$
=--
+-- {: .proof}
###### Proof
This is a direct consequence of the [[exact sequence|exactness]] of the sequence (eq:GreenOperatorExactSequenceFirst)
in lemma \ref{ExactSequenceOfGreenHyperbolicSystem}.
We spell this out for the statement for $\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}$,
which follows from the first line in (eq:GreenOperatorExactSequenceFirst), the first statement
similarly follows from the second line of (eq:GreenOperatorExactSequenceFirst):
First the [[on-shell]] [[space of field histories]] is the [[kernel]] of $P$, by definition
of [[free field theory]] (def. \ref{LagrangianDensityForDiracField})
$$
\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}
\;=\;
ker_{scp}(P)
\,.
$$
Second, exactness of the sequence (eq:GreenOperatorExactSequenceFirst) at $\Gamma_{\Sigma,scp}(E)$ means that the
[[kernel]] $ker_{scp}(P)$ of $P$ equals the [[image]] $im(\mathrm{G}_{P})$. But by exactness of the sequence
at $\Gamma_{\Sigma,cp}(E^\ast)$ it follows that $\mathrm{G}_P$ becomes [[injective]] on the [[quotient space]]
$\Gamma_{\Sigma,cp}(E)^\ast/im(P)$. Therefore on this quotient space it becomes an isomorphism onto its [[image]].
=--
+-- {: .num_remark #LinearOnShellObservablesAreTheGeneralizedPDESolutionsNaiveVersion}
###### Remark
Under passing to [[dual vector spaces]], the linear isomorphism in corollary \ref{OnShellSpaceOfFieldHistoriesForFreeFieldTheoryGreenHyperbolic}
in turn yields [[linear isomorphisms]] of the form
$$
\label{DualSolutionSpaceIsomorphicToQuotientByImP}
\array{
\left(\Gamma_{\Sigma,cp}(E^\ast)/im(P)\right)^\ast
&\underoverset{\simeq}{(-)\circ \mathrm{G}_P}{\longleftarrow}&
\left(ker_{scp}(P)\right)^\ast
\\
\left(\Gamma_\Sigma(E^\ast)/im(P)\right)^\ast
&\underoverset{\simeq}{(-)\circ \mathrm{G}_P }{\longleftarrow}&
\left(ker(P)\right)^\ast
}
\,.
$$
Except possibly for the issue of [[continuous map|continuity]] this says that
the linear on-shell [[observables]] (def. \ref{LinearObservables}) of a [[Green hyperbolic differential equation|Green hyperbolic]] [[free field theory]] are equivalently those linear off-shell observables which
are [[generalized solution of a PDE|generalized solutions]] of the [[formally adjoint differential operator|formally dual]]
[[equation of motion]] according to def. \ref{DistributionalDerivatives}.
That this remains true also for [[topological vector space]] [[structure]] follows with the dual exact sequence
(eq:GreenHyperbolicOperatorDualExactSequence). This is the statement of prop. \ref{DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions} below.
=--
+-- {: .num_prop #DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}
###### Proposition
**([[distributions|distributional sections]] on a [[Green hyperbolic differential equation|Green hyperbolic]] solution space are the [[generalized PDE solutions]])**
Let $P, P \ast \;\colon\; \Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(E^\ast)$ be a pair of [[Green hyperbolic differential operators]] (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}).
Then
1. the canonical pairing (from prop. \ref{DistributionsWithCausalSupports})
$$
\array{
\Gamma'_{\Sigma,cp}(E^\ast) &\otimes& \Gamma_\Sigma(E)
&\overset{}{\longrightarrow}&
\mathbb{C}
\\
\alpha^\ast &,& \Phi &\mapsto& \int \alpha^\ast_a(x) \Phi^a(x)\, dvol_\Sigma(x)
}
$$
induces a [[continuous linear map|continuous]] [[linear isomorphism]]
$$
\label{LinearDualOfSolutionSpaceIsLinearDualOfFullSpaceModuloImageOfDifferentialOperator}
(ker(P))^\ast
\;\simeq\;
\Gamma'_{\Sigma,cp}(E^\ast)/im_{cp}(P^\ast)
$$
1. a [[continuous linear functional]] on the solution space
$$
u_{sol} \in \left(ker(P)\right)^\ast
$$
is equivalently a [[distribution|distributional section]] (def. \ref{DistributionalSections})
whose [[support of a distribution|support]] is spacelike
compact (def. \ref{CompactlySourceCausalSupport}, prop. \ref{DistributionsWithCausalSupports})
$$
u \in \Gamma'_{\Sigma,scp}(E)
$$
and which is a [[distributional solution of a PDE|distributional solution]] (def. \ref{DistributionalDerivatives})
to the differential equation
$$
P^\ast u = 0
\,.
$$
Similarly, a [[continuous linear functional]] on the subspace of solutions that have spatially compact support (def. \ref{CompactlySourceCausalSupport})
$$
u_{sol} \in \left(ker(P)_{scp}\right)^\ast
$$
is equivalently a [[distribution|distributional section]] (def. \ref{DistributionalSections})
without constraint on its [[support of a distribution|distributional support]]
$$
u \in \Gamma'_{\Sigma}(E)
$$
and which is a [[distributional solution of a PDE|distributional solution]] (def. \ref{DistributionalDerivatives})
to the differential equation
$$
P^\ast u = 0
\,.
$$
Moreover, these [[linear isomorphisms]] are both given by composition with
the [[causal Green function]] $\mathrm{G}$ (def. \ref{AdvancedAndRetardedGreenFunctions}):
$$
\array{
\left(ker(P)\right)^\ast
&\underoverset{\simeq}{(-)\circ \mathrm{G}}{\longrightarrow}&
\left\{
u \in \Gamma'_{\Sigma,scp}(E)
\,\vert\,
P^\ast u = 0
\right\}
\\
\left(ker_{scp}(P)\right)^\ast
&\underoverset{\simeq}{(-)\circ \mathrm{G}}{\longrightarrow}&
\left\{
u \in \Gamma'_{\Sigma}(E)
\,\vert\,
P^\ast u = 0
\right\}
}
\,.
$$
=--
This follows from the [[exact sequence]] in lemma \ref{ExactSequenceOfGreenHyperbolicSystem}.
For details of the **proof** see at _[[Green hyperbolic differential operator]]_ [this prop.](Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions),
due to [[Igor Khavkine]].
In conclusion we have found the following:
+-- {: .num_theorem #LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}
###### Theorem
**(linear [[observables]] of [[Green hyperbolic differential operator|Green]] [[free field theory]] are the [[distributional solution of a PDE|distributional solutions]] to the [[formally adjoint differential operator|formally adjoint]] [[equations of motion]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory|Lagrangian]] [[free field theory]] (def. \ref{FreeFieldTheory})
which is a [[free field theory]] (def. \ref{FreeFieldTheory})
whose [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equation|differential]] [[equation of motion]] $P \Phi = 0$ (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is [[Green hyperbolic differential equation|Green hyperbolic]] (def. \ref{GreenHyperbolicDifferentialOperator}), such as the [[Klein-Gordon equation]] (example \ref{GreenHyperbolicKleinGordonEquation})
or the [[Dirac equation]] (example \ref{GreenHyperbolicDiracOperator}). Then:
1. The linear off-shell observables (def. \ref{LinearObservables}) are equivalently the
[[compactly supported distribution|compactly supported distributional sections]] (def. \ref{DistributionalSections}) of the [[field bundle]]:
$$
LinObs(E)
\;\simeq\;
\Gamma'_{\Sigma,cp}(E)
$$
1. The linear on-shell [[observables]] (def. \ref{LinearObservables}) are equivalently the linear off-shell observables
modulo the image of the [[differential operator]] $P$:
$$
\label{LinearOnShellObservablesAreLinearOffShellobservableModuloTheEquationsOfMotion}
LinObs(E,\mathbf{L})
\simeq
LinObs(E)/im(P)
\,.
$$
More generally the on-shell [[polynomial observables]] are identified with the off-shell polynomial observables (def. \ref{PolynomialObservables}) modulo the image of $P$:
$$
\label{PolynomialOnShellObservablesArePolynomialOffShellobservableModuloTheEquationsOfMotion}
PolyObs(E,\mathbf{L})
\simeq
PolyObs(E)/im(P)
\,.
$$
1. The linear on-shell [[observables]] (def. \ref{LinearObservables}) are also equivalently
those spacelike compactly supported [[distribution|compactly distributional sections]] (def. \ref{DistributionalSections})
which are [[distributional solution of a PDE|distributional solutions]] of the [[formally adjoint differential operator|formally adjoint]] [[equations of motion]] (def. \ref{FormallyAdjointDifferentialOperators}), and this isomorphism is exhibited by precomposition with the [[causal propagator]] $\mathrm{G}$:
$$
LinObs(E,\mathbf{L})
\;\underoverset{\simeq}{\phantom{A}(-)\circ\mathrm{G}_P \phantom{A}}{\longrightarrow}\;
\left\{
A \in \Gamma'_{\Sigma,scp}(E)
\;\vert\;
P^\ast A = 0
\right\}
$$
Similarly the linear on-shell observables on spacelike compactly supported on-shell field histories (eq:SpaceOfObservablesOnFieldHistoriesOfSpatiallyCompactSupport) are equivalently the [[distributional solution of a PDE|distributional solutions]] without constraint on their [[support of a distribution|support]]:
$$
LinObs(E_{scp},\mathbf{L})
\;\underoverset{\simeq}{\phantom{A}(-) \circ \mathrm{G}_P \phantom{A}}{\longrightarrow}\;
\left\{
A \in \Gamma'_{\Sigma}(E)
\;\vert\;
P^\ast A = 0
\right\}
$$
=--
+-- {: .proof}
###### Proof
The first statement follows with prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}
applied componentwise. The same proof applies verbatim to the subspace of solutions,
showing that $LinObs(E,\mathbf{L}) \simeq \left( ker(P)\right)^\ast$, with the [[dual topological vector space]] on the right.
With this the second and third statement follows by prop. \ref{DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}.
=--
We will be interested in those [[linear observables]] which under the identification from theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} correspond to the [[non-singular distributions]] (because on these the [[Poisson-Peierls bracket]] of the theory is defined, theorem \ref{PPeierlsBracket} below):
+-- {: .num_defn #RegularLinearFieldObservables}
###### Definition
**([[regular observables|regular]] [[linear observables]] and [[operator-valued distribution|observable-valued distributions]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}) is [[Green hyperbolic differential equation|Green hyperbolic]] (def. \ref{GreenHyperbolicDifferentialOperator}).
According to def. \ref{PolynomialObservables} the _[[regular linear observable|regular]]_ [[linear observables]] among the linear on-shell observables (def. \ref{LinearObservables}) are the [[non-singular distributions]] on the [[on-shell]] [[space of field histories]], hence the [[image]]
$$
LinObs(E_{scp},\mathbf{L})_{reg}
\hookrightarrow
LinObs(E_{scp},\mathbf{L})
$$
of the map
$$
\label{RegularLinearObservables}
\array{
\mathbf{\Phi}
&\colon&
\Gamma_{\Sigma,cp}(E^\ast)
&\longrightarrow&
LinObs(E_{scp},\mathbf{L})
&\hookrightarrow&
Obs(E_{scp},\mathbf{L})
\\
&& \alpha^\ast
&\mapsto&
\left(
\Phi
\mapsto
\underset{\Sigma}{\int} \alpha^\ast_a(x) \Phi^a(x)
\, dvol_\Sigma(x)
\right)
}
$$
By theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion} we have the identification (eq:LinearDualOfSolutionSpaceIsLinearDualOfFullSpaceModuloImageOfDifferentialOperator) (eq:LinearOnShellObservablesAreLinearOffShellobservableModuloTheEquationsOfMotion)
$$
\label{RegularLinearObservablesAreCompactlySupportedSectionsModuloImageOfP}
LinObs(E_{scp},\mathbf{L})_{reg}
\;\simeq\;
\Gamma_{\Sigma,scp}(E^\ast)/im(P)
\,.
$$
The point-evaluation [[field observables]] $\mathbf{\Phi}^a(x)$ (example \ref{PointEvaluationObservables}) are [[linear observables]] (example \ref{LinearPointEvaluationObservables}) but far from being regular (eq:RegularLinearObservables)
(except in [[spacetime]] [[dimension]] $p +1 = 0+1$). But the regular observables are precisely the
averages ("smearings") of these point evaluation observables against compactly supported weights.
Viewed this way, the defining inclusion of the [[regular observable|regular]] [[linear observables]] (eq:RegularLinearObservables) is itself
an _[[operator-valued distribution|observable valued distribution]]_
$$
\label{AverageOfFieldObservableIsRegularLinearObservables}
\array{
\mathbf{\Phi} &\colon& \Gamma_{\Sigma,cp}(E^\ast) &\hookrightarrow& LinObs(E,\mathbf{L})
\\
&& \alpha^\ast &\mapsto& \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)
}
$$
which to a "smearing function" $\alpha^\ast$ assigns the observable which is the [[field observable]] smeared by (i.e. averaged against)
that smearing function.
Below in _[Free quantum fields](#FreeQuantumFields)_ we discuss how the [[polynomial Poisson algebra]] of regular polynomial observables of a [[free field theory]] may be [[deformation quantization|deformed]] to a [[non-commutative algebra|non-commutative]] [[algebra of quantum observables]]. Often this may be [[representation|represented]] by [[linear operators]] acting on some [[Hilbert space]].
In this case then $\mathbf{\Phi}$ above becomes a [[continuous linear functional]] from $\Gamma_{\Sigma,cp}(E)$ to a space of
linear operators on some Hilbert space. As such it is then called an _[[operator-valued distribution]]_.
=--
$\,$
**Local observables**
{#LocalObservablesByTransgression}
We now discuss the sub-class of those [[observables]] which are "[[local field theory|local]]".
+-- {: .num_defn #SpacetimeSupport}
###### Definition
**([[spacetime support]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{FieldsAndFieldBundles}),
with induced [[jet bundle]] $J^\infty_\Sigma(E)$
For every [[subset]] $S \subset \Sigma$ let
$$
\array{
J^\infty_\Sigma(E)\vert_S
&\overset{\iota_S}{\hookrightarrow}&
J^\infty_\Sigma(E)
\\
\downarrow &(pb)& \downarrow
\\
S &\hookrightarrow& \Sigma
}
$$
be the corresponding restriction of the [[jet bundle]] of $E$.
The _spacetime support _ $supp_\Sigma(A)$ of a [[differential form]] $A \in \Omega^\bullet(J^\infty_\Sigma(E))$
on the [[jet bundle]] of $E$ is the [[topological closure]] of the maximal subset $S \subset \Sigma$
such that the restriction of $A$ to the jet bundle restrited to this subset does not vanishes:
$$
supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma | \iota_{\{x\}}^\ast A \neq 0 \} )
$$
We write
$$
\Omega^{r,s}_{\Sigma,cp}(E)
\coloneqq
\left\{
A \in \Omega^{r,s}_\Sigma(E)
\;\vert\;
supp_\Sigma(A) \, \text{is compact}
\right\}
\;\hookrightarrow\;
\Omega^{r,s}_\Sigma(E)
$$
for the subspace of differential forms on the jet bundle whose spacetime support is a [[compact subspace]].
=--
+-- {: .num_defn #TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}
###### Definition
**([[transgression of variational differential forms]] to [[space of field histories]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{FieldsAndFieldBundles}).
and let
$$
\Sigma_r \hookrightarrow \Sigma
$$
be a [[submanifold]] of [[spacetime]] of dimension $r \in \mathbb{N}$. Recall the [[space of field histories]]
restricted to its [[infinitesimal neighbourhood]], denoted $\Gamma_{\Sigma_r}(E)$ (def. \ref{FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}).
Then the operation of _[[transgression of variational differential forms]]_ to $\Sigma_r$ is the [[linear map]]
$$
\tau_{\Sigma_r}
\;\colon\;
\Omega^{\bullet,\bullet}_{\Sigma,cp}(E)
\overset{ }{\longrightarrow}
\Omega^\bullet\left(
\Gamma_{\Sigma_r}(E)
\right)
$$
that sends a variational differential form $A \in \Omega^{\bullet,\bullet}_{\Sigma,cp}(E)$
to the differential form $\tau_{\Sigma_r}A\in \Omega^\bullet(\Gamma_{\Sigma_r}(E))$
(def. \ref{DifferentialFormsOnDiffeologicalSpaces}, example \ref{ModuliOfSDifferentialForms})
which to a smooth family on field histories
$$
\Phi_{(-)}(-) \;\colon\; U \times N_\Sigma \Sigma_r \longrightarrow E
$$
assigns the differential form given by first forming the [[pullback of differential forms]] along the family of [[jet prolongation]] $j^\infty_\Sigma(\Phi_{(-)})$ followed by the [[integration of differential forms]] over $\Sigma_r$:
$$
(\tau_{\Sigma}A)_\Phi
\;\coloneqq\;
\int_{\Sigma_r} (j^\infty_\Sigma(\Phi_{(-)}))^\ast A
\;\in\;
\Omega^\bullet(U)
\,.
$$
=--
+-- {: .num_remark #TransgressionToDimensionrSupportedOnHorizontalrForms}
###### Remark
**([[transgression of variational differential forms|transgression]] to dimension $r$ picks out horizontal $r$-forms)**
In def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}
we regard [[integration of differential forms]] over $\Sigma_r$ as an operation defined on differential forms of all degrees, which vanishes except on forms of degree $r$, and hence transgression of variational differential forms
to $\Sigma_r$ vanishes except on the subspace
$$
\Omega^{r,\bullet}_\Sigma(E)
\;\subset\;
\Omega^{\bullet,\bullet}_\Sigma(E)
$$
of forms of horizontal degree $r$.
=--
+-- {: .num_example #ActionFunctional}
###### Example
**([[adiabatic switching|adiabatically switched]] [[action functional]])**
Given a [[field bundle]] $E \overset{fb}{\longrightarrow} \Sigma$,
consider a [[local Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
$$
\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)
\,.
$$
For any [[bump function]] $b \in C^\infty_{cp}(\Sigma)$, the [[transgression of variational differential forms|transgression]] of $b \mathbf{L}$
(def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) is called the _[[action functional]]_
$$
\mathcal{S}_b \mathbf{L}
\coloneqq
\tau_{\Sigma}
\left(
b \mathbf{L}
\right)
\;\colon\;
\Gamma_\Sigma(E)
\longrightarrow
\mathbb{R}
$$
induced by $\mathbf{L}$, "[[adiabatic switching|adiabatically switched]]" by $b$.
Specifically if the field bundle is a [[trivial vector bundle]] as in example \ref{TrivialVectorBundleAsAFieldBundle},
such that the Lagrangian density may be written in the form
$$
\mathbf{L}
\;=\;
L
\left(
(x^\mu), (\phi^a), (\phi^a_{,\mu}),
\cdots
\right)
\,
b dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_{\Sigma,cp}( E )
\,.
$$
then its action functional takes a field history $\Phi$ to the value
$$
\mathcal{S}_{b \mathbf{L}}(\Phi)
\:\colon\;
\int_\Sigma
L
\left(
x,
\left( \Phi^a(x) \right),
\left(\frac{\partial \Phi^a}{\partial x^\mu}(x)\right),
\cdots
\right)
\,
b(x)
dvol_\Sigma(x)
$$
=--
+-- {: .num_prop #TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative}
###### Proposition
**([[transgression of variational differential forms|transgression]] compatible with [[variational derivative]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{FieldsAndFieldBundles})
and let $\Sigma_r \hookrightarrow \Sigma$ be a [[submanifold]] possibly [[manifold with boundary|with boundary]]
$\partial \Sigma_r \hookrightarrow \Sigma_r$. Write
$$
\Gamma_{\Sigma_r}(E) \overset{(-)\vert_{\partial \Sigma_r}}{\longrightarrow} \Gamma_{\partial \Sigma_r}(E)
$$
for the boundary restriction map.
Then the operation of [[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
$$
\tau_{\Sigma} \;\colon\; \Omega^{\bullet,\bullet}_{\Sigma,cp}(E) \longrightarrow \Omega^\bullet\left(\Gamma_{\Sigma_r}(E)\right)
$$
is compatible with the [[variational derivative]] $\delta$ and with the [[total derivative|total spacetime derivative]] $d$
in the following way:
1. On variational forms that are in the image of the total spacetime derivative
a transgressive variant of the [[Stokes' theorem]] (prop. \ref{StokesTheorem}) holds:
$$
\tau_{\Sigma_r}(d \alpha) \;=\; ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma_r}( \alpha)
$$
1. Transgression intertwines, up to a sign, the [[variational derivative]] $\delta$ on variational differential forms with the
plain de Rham differential on the space of field histories:
$$
\tau_{\Sigma}\left(
\delta \alpha
\right)
\;=\;
(-1)^{p+1}\, d \,\tau_{\Sigma}(\alpha)
\,.
$$
=--
+-- {: .proof}
###### Proof
Regarding the first statement, consider a horizontally exact variational form
$$
d \alpha \in \Omega^{r,s}_{\Sigma,cp}(E)
\,.
$$
By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} the pullback of this form
along the jet prolongation of fields is exact in the $\Sigma$-direction:
$$
(j^\infty_\Sigma\Phi_{(-)})^\ast(d \alpha )
\;=\;
d_\Sigma (j^\infty_\Sigma\Phi_{(-)})^\ast \alpha
\,,
$$
(where we write $d = d_U + d_\Sigma$ for the de Rham differential on $U \times \Sigma$).
Hence by the ordinary [[Stokes' theorem]] (prop. \ref{StokesTheorem}) restricted to any $\Phi_{(-)} \colon U \to \Gamma_{\Sigma_r}(E)$
with restriction $(-)\vert_{\partial \Sigma_r} \circ \Phi_{(-)} \colon U \to \Gamma_{\Sigma_r}(E)$
the relation
$$
\begin{aligned}
(\Phi_{(-)})^\ast \tau_{\Sigma_r}(d \alpha)
& =
\int_{\Sigma_r}
d_{\Sigma_r} (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha
\\
& = \int_{\partial \Sigma_r} (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha
\\
& = \int_{\partial \Sigma_r} (j^\infty_\Sigma ( (-)\vert_{\Sigma_r} \circ \Phi_{(-)}) )^\ast\alpha
\\
& =
( (-)\vert_{\Sigma_r} \circ \Phi_{(-)} )^\ast \tau_{\partial \Sigma_r}(\alpha)
\\
& =
(\Phi_{(-)})^\ast ((-)\vert_{\Sigma_r})^\ast \tau_{\partial \Sigma_r}(\alpha)
\,.
\end{aligned}
\,.
$$
Regarding the second statement: by the [[Leibniz rule]] for de Rham differential
([[product law]] of [[differentiation]]) it is sufficient to check the claim on variational
derivatives of local coordinate functions
$$
\delta \phi^a_{\mu_1 \cdots \mu_k} b
\in
\Omega^{0,1}_\Sigma(E)
\,.
$$
The [[pullback of differential forms]] (prop. \ref{PullbackOfDifferentialForms})
along the [[jet prolongation]] $j^\infty_\Sigma(\Phi_{(-)}) \colon U \times \Sigma \to J^\infty_\Sigma(E)$ has two contributions:
one from the variation along $\Sigma$, the other from variation along $U$:
1. By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative},
for _fixed_ $u \in U$ the pullback of $\delta \phi^a_{\mu_1 \cdots \mu_k}$ along the jet prolongation vanishes.
1. For fixed $x \in \Sigma$, the pullback of the full de Rham differential $\mathbf{d} \phi^a_{\mu_1\cdots \mu_k}$ is
$$
\begin{aligned}
(\Phi_{(-)}(x))^\ast( \mathbf{d} \phi^a_{\mu_1\cdots \mu_k} )
& =
d_U (\Phi_{(-)}(x))^\ast(\phi^a_{\mu_1\cdots \mu_k})
\\
& =
d_U \frac{ \partial^k \Phi_{(-)}(x)}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}
\end{aligned}
$$
(since the full de Rham differentials always commute with pullback of differential forms by prop. \ref{PullbackOfDifferentialForms}),
while the pullback of the horizontal derivative
$d \phi^a_{\mu_1\cdots \mu_k} = \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \mathbf{d}x^{\mu_{k+1}}$
vanishes at fixed $x \in \Sigma$.
This implies over the given smooth family $\Phi_{(-)}$ that
$$
\begin{aligned}
\tau_\Sigma\left(
\delta \phi^a_{,\mu_1 \cdots \mu_k} b
\right)\vert_{\Phi_{(-)}}
& =
\tau_\Sigma\left(
\mathbf{d} ( \phi^a_{,\mu_1 \cdots \mu_k} b)
\right)
\vert_{\Phi_{(-)}}
-
\underset{ = 0 }{
\underbrace{
\tau_\Sigma
\left(
d (\phi^a_{,\mu_1 \cdots \mu_k} b)
\right)\vert_{\Phi_{(-)}}
}}
\\
& =
\int_\Sigma d_U (\Phi_{(-)})^\ast ( \phi^a_{\mu_1\cdots \mu_k} b )
\\
& = (-1)^{p+1} d_U \int_\Sigma (\Phi_{(-)})^\ast ( \phi^a_{\mu_1\cdots \mu_k} b )
\\
& = (-1)^{p+1} d_U \tau_{\Sigma}( \Phi_{(-)} )^\ast ( \phi^a_{\mu_1 \cdots \mu_k} )
\,.
\end{aligned}
$$
and since this holds covariantly for all smooth families $\Phi_{(-)}$, this implies the claim.
=--
+-- {: .num_example #IntegrationByPartsOnJetBundle}
###### Example
**([[cochain cohomology|cohomological]] [[integration by parts]] on the [[jet bundle]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}).
Prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative} says in particular that
the operation of _[[integration by parts]]_ in an [[integral]] is "localized" to a cohomological statement on [[horizontal differential forms]]: Let
$$
\alpha_1, \alpha_2 \;\in\; \Omega^{\bullet,\bullet}_\Sigma(E)
$$
be two [[variational differential forms]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}), of total horizontal degree $p$ (hence one less than the [[dimension]] of [[spacetime]] $\Sigma$).
Then the [[derivation]]-property of the [[total spacetime derivative]] says that
$$
\label{IntegrationByPartsCohomologicallyOnJetBundle}
(d \alpha_1) \wedge \alpha_2
\;=\;
- (-1)^{deg(\alpha_1)} \alpha_1 \wedge ( d \alpha_2 )
\;\;
d( \alpha_1 \wedge \alpha_2 )
\;\in\;
\Omega^{p+1,\bullet}_\Sigma(E)
\,,
$$
hence that we may "throw over" the spacetime derivative from the factor $\alpha_1$ to the factor $\alpha_2$, up to a sign,
and up to a total spacetime derivative $d (\alpha_1 \wedge \alpha_2)$. By prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative} this last term vanishes
under [[transgression of variational differential forms|transgression]] $\tau_\sigma$ to a [[spacetime]] without [[manifold with boundary]],
so that the above equation becomes
$$
\tau_\Sigma( d \alpha_1) \wedge \alpha_2 )
\;=\;
- (-1)^{deg(\alpha_1)}
\tau_\Sigma( \alpha_1 \wedge d \alpha_2 )
\,,
$$
hence
$$
\underset{\Sigma}{\int}
(d j^\infty_\sigma(\alpha_1)) \wedge j^\infty_\Sigma(\alpha_2)
\;=\;
- (-1)^{deg(\alpha_1)}
\underset{\Sigma}{\int}
j^\infty_\Sigma(\alpha_1) \wedge d j^\infty_\Sigma(\alpha_2)
\,.
$$
This last statement is the statement of _[[integration by parts]]_ under an integral.
Notice that these [[integrals]] (and hence the actual [[integration by parts]]-rule) only exist
if $\alpha_1 \wedge \alpha_2$ has compact spacetime support, while the "cohomological" avatar (eq:IntegrationByPartsCohomologicallyOnJetBundle) of this relation on the jet bundle holds without such a restriction.
=--
+-- {: .num_example #VariationOfTheActionFunctional}
###### Example
**([[variational derivative|variation]] of the [[action functional]])**
Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
then the derivative of its [[adiabatic switching|adiabatically switched]] [[action functional]]
(def. \ref{ActionFunctional}) equals the [[transgression of variational differential forms|transgression]]
of the [[Euler-Lagrange variational derivative]] $\delta_{EL} \mathbf{L}$ (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):
$$
d \mathcal{S}_{b \mathbf{L}}
\;=\;
\tau_\Sigma( b \delta_{EL}\mathbf{L} )
\,.
$$
=--
+-- {: .proof}
###### Proof
By the second statement of prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative} we have
$$
\begin{aligned}
d \mathcal{S}_{b \mathbf{L}}
& =
\tau_\Sigma( \delta ( b \mathbf{L} ) )
\end{aligned}
\,,
$$
Moreover, by prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} this is
$$
\begin{aligned}
\cdots
& =
\tau_\Sigma( \delta_{EL} b \mathbf{L} + d \Theta_{BFV,b} )
\\
& =
\tau_\Sigma( \delta_{EL} b \mathbf{L} ) + \underset{= 0}{\underbrace{\tau_\Sigma( d \Theta_{BFV,b} )}}
\end{aligned}
\,,
$$
where the second term vanishes by the first statement of prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative}.
=--
+-- {: .num_prop #PrincipleOfExtremalAction}
###### Proposition
**([[principle of extremal action]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
The [[de Rham differential]] $d \mathcal{S}_{b\mathbf{L}}$ of the [[action functional]] (example \ref{VariationOfTheActionFunctional})
vanishes at a field history
$$
\Phi \in \Gamma_\Sigma(E)
$$
for all [[adiabatic switchings]] $b \in C^\infty_{cp}(\Sigma)$ constant on some
subset $\mathcal{O} \subset \Sigma$ (def. \ref{CutoffFunctions}) on those smooth collections of field histories
$$
\Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E)
$$
around $\Phi$ which, as functions on $U$, are constant outside $\mathcal{O}$ (example \ref{DiffeologicalSpaceOfFieldHistories}, example \ref{SupergeometricSpaceOfFieldHistories})
precisely if $\Phi$ solves the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]
(def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}):
$$
\left(
\underset{ { {\mathcal{O} \subset \Sigma} \atop { b\vert_{\mathcal{O}} = const } } \atop { \Phi_{(-)}\vert_{\Sigma \setminus \mathcal{O}} = const } }{\forall}
\left(
(\Phi_{(-)})^\ast d \mathcal{S}_{b \mathbf{L}}(\Phi) = 0
\right)
\right)
\;\Leftrightarrow\;
\left(
j^\infty_\Sigma(\Phi)^\ast \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) = 0
\right)
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative} we have
$$
(\Phi_{(-)})^\ast d \mathcal{S}_{b \mathbf{L}}
\;=\;
\int_\Sigma
j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} b \mathbf{L} )
\,.
$$
By the assumption on $\Phi_{(-)}$ it follows that after pullback to $U$ the switching function $b$ is constant, so that
it commutes with the differentials:
$$
(\Phi_{(-)})^\ast
d \mathcal{S}_{b \mathbf{L}}
\;=\;
\int_\Sigma
b
j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} \mathbf{L} )
\,.
$$
This vanishes at $\Phi$ for all $\Phi_{(-)}$ precisely if all components of $j^\infty_\Sigma(\Phi_{(-)})^\ast ( \delta_{EL} \mathbf{L} )$
vanish, which is the statement of the Euler-Lagrange equations of motion.
=--
+-- {: .num_defn #LocalObservables}
###### Definition
**([[local observables]])**
Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) the _[[local observables]]_ are the [[horizontal differential form|horizontal p+1-forms]]
1. of compact spacetime support (def. \ref{SpacetimeSupport})
1. modulo [[total spacetime derivatives]]
$$
LocObs(E)
\;\coloneqq\;
\left(\Omega^{p+1,0}_{\Sigma,cp}(E)/(im(d))\right)\vert_{\mathcal{E}^\infty}
$$
which we may identify with the subspace of all observables (eq:GlobalObservables)
that arises as the [[image]] under
[[transgression of variational differential forms]] $\tau_\Sigma$ (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
of local observables to [[functionals]] on the [[on-shell]] [[space of field histories]] (eq:OnShellFieldHistories):
$$
\array{
LocObs(E)
&\overset{\tau_\Sigma}{\hookrightarrow}&
Obs(E)
\\
\alpha &\maptos& \tau_\Sigma \alpha
}
\,.
$$
This is a sub-vector space inside all observables which is however not closed under the pointwise product of observables (eq:ObservablesPointwiseProduct) (unless $ E=0 $). We write
$$
MultiLocObs(E)
\hookrightarrow
Obs(E)
$$
for the smallest subalgebra of observables, under the pointwise product (eq:ObservablesPointwiseProduct), that contains all the local observables. This is called the algebra of _[[multilocal observables]]_.
The [[intersection]] of the (multi-)[[local observables]] with the [[off-shell]] [[polynomial observables]] (def. \ref{PolynomialObservables}) are the (multi-)local polynomial observables
$$
\label{InclusionOfPolynomialLocalObservablesIntoPolynomialObservables}
PolyLocObs(E)
\hookrightarrow
PolyMultiLocObs(E)
\overset{\text{dense}}{\hookrightarrow}
PolyObs(E)
\hookrightarrow
Obs(E)
$$
=--
+-- {: .num_example }
###### Example
**([[local observables]] of the [[real scalar field]])**
Consider the [[field bundle]] of the [[real scalar field]] (example \ref{RealScalarFieldBundle}).
A typical example of [[local observables]] (def. \ref{LocalObservables}) in this case is
the "field amplitude averaged over a given spacetime region"
determined by a [[bump function]] $b \in C^\infty_{cp}(\Sigma)$. On an on-shell field history $\Phi$ this observable
takes as value the integral
$$
\tau_\Sigma(b \phi)(\Phi) \;=\; \int_\Sigma \Phi(x) b(x) dvol_\Sigma(x)
\,.
$$
=--
+-- {: .num_example }
###### Example
**([[local observables]] of the [[electromagnetic field]])**
Consider the [[field bundle]] for [[free field theory|free]] [[electromagnetism]] on [[Minkowski spacetime]] $\Sigma$.
Then for $b \in C^\infty(\Sigma)$ a [[bump function]] on [[spacetime]], the [[transgression of variational differential forms|transgression]]
of the universal [[Faraday tensor]] (def. \ref{JetFaraday}) against $b$ times the [[volume form]] is a [[local observable]]
(def. \ref{LocalObservables}), namely the _[[field strength]]_ (eq:TensorFaraday) of the [[electromagnetic field]]
averaged over spacetime.
=--
For the construction of the [[algebra of quantum observables]] it will be important to notice that the
[[intersection]] between [[local observables]] and [[regular polynomial observables]] is very small:
+-- {: .num_example #RegularPolynomialLocalObservablesAreNecessarilyLinear}
###### Example
**([[local observable|local]] [[regular polynomial observables]] are [[linear observables]])**
An [[observable]] (def. \ref{Observable}) which is
1. a [[regular polynomial observable]] (def. \ref{PolynomialObservables});
1. a [[local observable]] (def. \ref{LocalObservables})
is necessarily
* a [[linear observable]] (def. \ref{LinearObservables}).
This is because non-linear local expressions are polynomials in the sense of def. \ref{PolynomialObservables}
with [[delta distribution]]-[[coefficients]], for instance for the [[real scalar field]] the $\Phi^2$ [[interaction]]
term is
$$
\int (\Phi(x))^2 \, dvol_\Sigma(x)
\;=\;
\int \int \Phi(x) \Phi(y) \underset{ = \alpha^{(2)}(x,y) }{\underbrace{\delta(x-y)}} \, dvol_\Sigma(y)
$$
and so its [[coefficient]] $\alpha^{(2)}$ is manifestly not a [[non-singular distribution]].
=--
$\,$
**Infinitesimal observables**
{#InfinitesimalObservables}
The definition of [[observables]] in def. \ref{Observable} and specifically of [[local observables]] in def. \ref{LocalObservables} uses explicit restriction to the [[shell]],
hence, by the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction}) to the "[[critical locus]]" of the [[action functional]]. Such [[critical loci]] are often hard to
handle explicitly. It helps to consider a "[[homological resolution]]" that is given, in good circumstances, by the corresponding
"[[derived critical locus]]". These we consider in detail below in _[Reduced phase space](#ReducedPhaseSpace)_.
In order to have good control over these resolutions, we here consider the first _[[perturbative quantum field theory|perturbative]]_
aspect of [[field theory]], namely we consider the restriction of [[local observables]] to just an
[[infinitesimal neighbourhood]] of a background [[on-shell]] field history:
+-- {: .num_defn #LocalObservablesOnInfinitesimalNeighbourhood}
###### Definition
**([[local observables]] around [[infinitesimal neighbourhood]] of background [[on-shell]] field history)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}).
Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle)
as in example \ref{ShellForSpacetimeIndependentLagrangians}.
Then we write
$$
LocObs_\Sigma(E,\varphi)
$$
for the restriction of the [[local observables]] (def. \ref{LocalObservables}) to the fiberwise
[[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of $\Sigma \times \{\varphi\}$.
Explicitly, this means the following:
First of all, by prop. \ref{JetBundleIsLocallyProManifold} the dependence of the Lagrangian density $\mathbf{L}$
on the order of field derivatives is bounded by some $k \in \mathbb{N}$ on some [[neighbourhood]] of $\varphi$ and hence,
by the spacetime independence of $\mathbf{L}$, on some neighbourhood of $\Sigma \times \{\varphi\}$.
Therefore we may restrict without loss to the order-$k$ jets. By slight abuse of notation we still write
$$
\mathcal{E} \hookrightarrow J^k_\Sigma(E)
$$
for the corresponding shell. It follows then that the restriction of the ring $\Omega^{0,0}_{\Sigma,cp}(E)$ of smooth functions
on the jet bundle to the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood})
is equivalently the [[formal power series ring]] over $C^\infty_{cp}(\Sigma)$
in the variables
$$
((\phi^a- \varphi^a),
(\phi^a_{,\mu}- \varphi^a_{,\mu}),
\cdots,
(\phi^a_{,\mu_1 \cdots \mu_k} - \varphi^a_{,\mu_1 \cdots \mu_k})
)
$$
We denote this by
$$
\label{FunctionsOnInfNbh}
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\;\coloneqq\;
C^\infty_{cp}(\Sigma)\left[ \left[ (\phi^a - \varphi^a ), (\phi^a_{,\mu} -\varphi^a_{,\mu}), \cdots, (\phi^a_{,\mu_1 \cdots \mu_k}- \varphi^a_{,\mu_1 \cdots \mu_k}) \right] \right]
\,.
$$
A key consequence is that the further restriction of this ring to the [[shell]] $\mathcal{E}^\infty$ (eq:ProlongedShellInJetBundle)
is now simply the further [[quotient ring]] by the ideal generated by the [[total spacetime derivatives]]
of the components $\frac{\partial_{EL}L}{\delta \phi^a}$ of
the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}).
$$
\label{ObservablesOnInfinitesimalNeighbourhoodOfZeroInShellInFieldFiber}
\begin{aligned}
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}
& \coloneqq
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
/
\left(
\frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\right)_{ { a \in \{1, \cdots, s\} } \atop { { l \in \{1, \cdots, k\} } \atop { \mu_r \in \{0, \cdots, p\} } } }
\\
& =
C^\infty_{cp}(\Sigma)\left[ \left[ (\phi^a - \varphi^a ), (\phi^a_{,\mu} -\varphi^a_{,\mu}), \cdots, (\phi^a_{,\mu_1 \cdots \mu_k}- \varphi^a_{,\mu_1 \cdots \mu_k}) \right] \right]
/
\left(
\frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\right)_{ { a \in \{1, \cdots, s\} } \atop { { l \in \{1, \cdots, k\} } \atop { \mu_r \in \{0, \cdots, p\} } } }
\end{aligned}
\,.
$$
Finally the [[local observables]] restricted to the infinitesimal neighbourhood is the module
$$
\label{LocalObservablesRestrictedToInfinitesimalNeighbourhood}
LocObs_\Sigma(E,\varphi)
\;\simeq\;
\left(
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}} \langle dvol_\Sigma \rangle
\right)/(im(d))
\,.
$$
=--
The space of local observables in def. \ref{LocalObservablesOnInfinitesimalNeighbourhood} is the
[[quotient]] of a [[formal power series algebra]] by the components of the [[Euler-Lagrange form]]
and by the [[image]] of the horizontal spacetime [[de Rham differential]]. It is convenient
to also conceive of the components of the [[Euler-Lagrange form]] as the [[image]] of a [[differential]],
for then the algebra of local observables obtaines a [[cochain cohomology|cohomological]] interpretation,
which will lend itself to computation. This differential, whose image is the components of the
[[Euler-Lagrange form]], is called the _[[BV-differential]]. We introduce this
now first (def. \ref{BVComplexOfOrdinaryLagrangianDensity} below) in a direct ad-hoc way.
Further [below](#ReducedPhaseSpace) we discuss the conceptual nature of this differential
as part of the construction of the [[reduced phase space]] as a [[derived critical locus]] (example \ref{DerivedProlongedShellInAbsenceOfExplicitGaugeSymmetries} below).
+-- {: .num_defn #BVComplexOfOrdinaryLagrangianDensity}
###### Definition
**([[local BV-complex]] of ordinary [[Lagrangian density]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{LocalObservablesOnInfinitesimalNeighbourhood}).
Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}^\infty$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle).
In correspondence with def. \ref{LocalObservablesOnInfinitesimalNeighbourhood}, write
$$
\Gamma_{\Sigma,cp}(T_\Sigma J^\infty_\Sigma E,\varphi)
\simeq
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)
\;\in\;
\Omega^{0,0}_{\Sigma,cp}(E) Mod
$$
for the restriction of [[vertical vector fields]] on the [[jet bundle]] to the fiberwise [[infinitesimal neighbourhood]]
(example \ref{InfinitesimalNeighbourhood}) of $\Sigma \times {\varphi}$.
Now we regard this as a _[[graded module]]_ over $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ (eq:FunctionsOnInfNbh) concentrated in degree $-1$:
$$
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\;\in\;
\Omega^{0,0}_{\Sigma,cp}(E) Mod^{\mathbb{Z}}
\,.
$$
This is called the module of _[[antifields]]_ corresponding the given [[type]] of [[field (physics)|fields]] encoded by $E$.
If the [[field bundle]] is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with field coordinates
$(\phi^a)$, then we write
$$
\label{AntifieldCoordinates}
\phi^{\ddagger}_{a,\mu_1 \cdots \mu_l}
\;\coloneqq\;
\left(
\partial_{(\phi^a_{\mu_1 \cdots \mu_l})}
\right)[-1]
\;\in\;
\Gamma_{\Sigma,cp}(T_\Sigma J^\infty_\Sigma E,\varphi)[-1]
$$
for the vector field generator that takes derivatives along $\partial_{\phi^a_{,\mu_1 \cdots \mu_k}}$, but regarded now in degree -1.
Evaluation of vector fields in thelocal BV-complex
[[total spacetime derivatives]] $\frac{d^l}{d x^{\mu_1} \cdots d x^{\mu_l}} \delta_{EL}\mathbf{L} \in \Omega^{p,0}_\Sigma(E) \wedge \delta \Omega^{0,0}_\Sigma(E)$ of the [[variational derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) yields a [[linear map]]
over $\Omega^{\bullet,\bullet}_{\Sigma,cp}(E,\varphi)$ (eq:ObservablesOnInfinitesimalNeighbourhoodOfZeroInShellInFieldFiber)
$$
\iota_{(-)}\delta_{EL} \mathbf{L}
\;\colon\;
\Gamma_{\Sigma,cp}( J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\longrightarrow
\Omega^{p+1,0}_{\Sigma,cp}(E,\varphi)
\,.
$$
If we use the [[volume form]] $dvol_\Sigma$ on [[spacetime]] $\Sigma$ to induce an identification
$$
\Omega^{p+1,0}_\Sigma(E) \;\simeq\; C^\infty(J^\infty_\Sigma(E))\langle dvol_\sigma\rangle
$$
with respect to which the [[Lagrangian density]] decomposes as
$$
\mathbf{L} = L dvol_\Sigma
$$
then this is a $\Omega^{0,0}_\sigma(E,\varphi)$-[[linear map]] of the form
$$
\iota_{(-)}{\delta L_{EL}}
\;\colon\;
\Gamma_{\Sigma,cp}^{ev}(T_\Sigma E,\varphi)[-1]
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\,.
$$
In the special case that the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]]
(example \ref{TrivialVectorBundleAsAFieldBundle}) with [[field (physics)|field]] coordinates $(\phi^a)$
so that the [[Euler-Lagrange form]] has the coordinate expansion
$$
\delta_{EL} \mathbf{L}
\;=\;
\frac{\delta_{EL}\mathbf{L}}{\delta \phi^a} \delta \phi^a
$$
then this map is given on the [[antifield]] basis elements (eq:AntifieldCoordinates) by
$$
\iota_{(-)} {\delta L_{EL}}
\;\colon\;
\phi^{\ddagger}_{a,\mu_1 \cdots \mu_l}
\;\mapsto\;
\frac{d^l}{d x^{\mu_1} \cdots d x^{\mu_l}} \frac{\delta_{EL} L}{\delta \phi^a}
\,.
$$
Consider then the [[symmetric algebra|graded symmetric algebra]]
$$
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\;\coloneqq\;
Sym_{\Omega^{0,0}_{\Sigma,cp}(E,\varphi)}\left(
\Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi)[-1]
\right)
$$
which is generated over $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ from the module of vector fields in degree -1.
If we think of a single vector field as a fiber-wise [[linear function]] on the [[cotangent bundle]],
and of a [[multivector field]] similarly as a [[multilinear function]] on the cotangent bundle, then we may
think of this as the algebra of functions on the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of $\varphi$
inside the [[graded manifold]] $(T_\Sigma E)[-1] \times_\Sigma E$.
Let now
$$
\label{BVDifferentialForOrdinaryLagrangian}
s_{BV}
\;\colon\;
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\;\longrightarrow\;
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
$$
be the unique extension of the linear map $\iota_{(-)}{\delta_{EL} L}$ to an $\mathbb{R}$-linear [[derivation]] of degree +1 on this algebra.
The resulting [[differential graded-commutative algebra]] over $\mathbb{R}$
$$
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}
\;\coloneqq\;
\left(
C^\infty( J^\infty_\Sigma((T_\Sigma E)[-1] \times_\Sigma E, \varphi) )
\,,\,
s_{BV}
\right)
$$
is called the _[[local BRST cohomology|local]] [[BV-complex]]_
of the Lagrangian field theory at the background solution $\varphi$.
This is the CE-algebra of the infintiesimal neighbourhood of $\Sigma \times \{\varphi\}$
in the derived prolonged shell (def. \ref{DerivedProlongedShell}).
In this case, in the absence of any explicit infinitesimal gauge symmetries,
this is an example of a _[[Koszul complex]]_.
There are canonical homomorphisms of [[dgc-algebras]], one from the
algebra of functions $\Omega^{0,0}_{\Sigma,cp}(E,\varphi)$ on the [[infinitesimal neighbourhood]]
of the background solution $\varphi$ to the local BV-complex
and from there to the
local observables on the neighbourhood of the background solution $\varphi$ (eq:ObservablesOnInfinitesimalNeighbourhoodOfZeroInShellInFieldFiber), all considered
with compact spacetime support:
$$
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}
\longrightarrow
\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}
$$
such that the composite is the canonical [[quotient]] [[coprojection]].
Similarly we obtain a factorization for the entire [[variational bicomplex]]:
$$
\label{ComparisonMorphismFromOrdinaryBVComplexToLocalObservables}
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)
\longrightarrow
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}
\longrightarrow
\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}}
\,,
$$
where $\Omega^{\bullet,\bullet}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}$
is now triply graded, with three anti-commuting differentials $d$ $\delta$ and $s_{BV}$.
By construction this is now such that the local observables (def. \ref{LocalObservables})
are the [[cochain cohomology]] of this complex in horizontal form degree p+1, vertical degree 0
and BV-degree 0:
$$
LocObs_\Sigma(E) \simeq \Omega^{p+1,0}_{\Sigma,cp}(E)/(im(s_{BV} + d))
\,.
$$
=--
$\,$
**States**
{#StatesArePositiveLinearFunctionals}
We introduce the basics of _[[quantum probability]]_ in terms of [[state on a star-algebra|states]] defined as positive linear maps on [[star-algebras]] of observables.
+-- {: .num_defn #StarAlgebra}
###### Definition
**([[star algebra]])**
A _[[star ring]]_ is a [[ring]] $R$ equipped with
* a [[linear map]]
$(-)^\ast \;\colon\; R \longrightarrow R$
such that
* ([[involution]]) $((-)^\ast)^\ast = id$;
* ([[antihomomorphism]])
1. $(a b)^\ast = b^\ast a^\ast$ for all $a,b \in R$
1. $1^\ast = 1$.
A [[homomorphism]] of star-rings
$$
f \;\colon\; (R_1, (-)^\ast) \longrightarrow (R_2, (-)^\dagger)
$$
is a [[homomorphism]] of the underlying [[rings]]
$$
f \;\colon\; R_1 \longrightarrow R_2
$$
which respects the star-[[involutions]] in that
$$
f \circ (-)^\ast \;=\; (-)^\dagger \circ f
\,.
$$
A _[[star algebra]]_ $\mathcal{A}$ over a [[commutative ring|commutative]] star-ring $R$ in an [[associative algebra]] $\mathcal{A}$ over $R$ such that the inclusion
$$
R \hookrightarrow \mathcal{A}
$$
is a star-homomorphism.
=--
+-- {: .num_example #StarAlgebraOfObservables}
###### Examples
**([[complex number]]-valued [[observables]] are [[star-algebra]] under pointwise product and pointwise [[complex conjugation]])**
The [[complex numbers]] $\mathbb{C}$ carry the [[structure]] of a [[star-ring]] (def. \ref{StarAlgebra}) with star-operation given by [[complex conjugation]].
Given any space $X$, then the [[algebra of functions]] on $X$ with values in the [[complex numbers]] carries the [[structure]] of a [[star-algebra]] over the star-ring $\mathbb{C}$ (def. \ref{StarAlgebra}) with star-operation given by pointwise [[complex conjugation]] in the [[complex numbers]].
In particular for $(E,\mathbf{L})$ a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) then its [[on-shell]] [[observables]] $Obs(E,\mathbf{L})$ (def. \ref{Observable}) carry the structure of a [[star-algebra]] this way.
=--
+-- {: .num_defn #StateOnAStarAlgebra}
###### Definition
**([[state on a star-algebra]])**
Given a [[star algebra]] $(\mathcal{A}, (-)^\ast)$ (def. \ref{StarAlgebra})
over the star-ring of [[complex numbers]] (def. \ref{StarAlgebraOfObservables}) a _[[state on a star-algebra|state]]_ is a [[function]] to the [[complex numbers]]
$$
\langle -\rangle
\;\colon\;
Obs_\Sigma \longrightarrow \mathbb{C}
$$
such that
1. (linearity) this is a complex-[[linear map]]:
$$
\left\langle
c_1 A_1 + c_2 A_2
\right\rangle
\;=\;
c_1 \langle A_1 \rangle + c_2 \langle A_2 \rangle
$$
1. (positivity) for all $A \in Obs$ we have that
$$
\langle A^\ast A \rangle \geq 0 \;\in\; \mathbb{R}
$$
where on the left $A^\ast$ is the [[star-algebra|star-operation]] from
1. (normalization)
$$
\langle 1 \rangle \;=\; 1
\,.
$$
=--
(e.g. [Bordemann-Waldmann 96](state+on+a+star-algebra#BordemannWaldmann96), [Fredenhagen-Rejzner 12, def. 2.4](state+on+a+star-algebra#FredenhagenRejzner12), [Khavkine-Moretti 15, def. 6](state+on+a+star-algebra#KhavkineMoretti15))
+-- {: .num_remark #ProbabilityTheoreticInterpretationOfStateOnAStarAlgebra}
###### Remark
**([[probability theory|probability theoretic]] interpretation of [[state on a star-algebra]])**
A [[star algebra]] $\mathcal{A}$ (def. \ref{StarAlgebra}) equipped with a [[state on a star-algebra|state]] $\mathcal{A} \overset{\langle -\rangle}{\longrightarrow} \mathbb{C}$ (def. \ref{StateOnAStarAlgebra}) is also called a _[[quantum probability space]]_, at least when $\mathcal{A}$ is in fact a [[von Neumann algebra]].
For this interpretation we think of each element $A \in \mathcal{A}$ as an [[observable]] as in example \ref{StarAlgebraOfObservables} and of the state as assigning _[[expectation values]]_.
=--
+-- {: .num_remark #StatesFormAConvexSet}
###### Remark
**([[state on a star-algebra|states]] form a [[convex set]])**
For $\mathcal{A}$ a unital [[star-algebra]] (def. \ref{StarAlgebra}), the [[set]] of [[state on a star-algebra|states]] on $\mathcal{A}$ according to def. \ref{StateOnAStarAlgebra} is naturally a [[convex set]]: For $\langle (-)\rangle_1, \langle - \rangle_2 \colon \mathcal{A} \to \mathbb{C}$ two [[state on a star-algebra|states]] then for every $p \in [0,1] \subset \mathbb{R}$ also the [[linear combination]]
$$
\array{
\mathcal{A}
&\overset{p \langle (-)\rangle_1 + (1-p) \langle (-)\rangle_2}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& p \langle A \rangle_1 + (1-p) \langle A \rangle_2
}
$$
is a [[state on a star-algebra|state]].
=--
+-- {: .num_defn #PureStateOnAStarAlgebra}
###### Definition
**([[pure state]])**
A [[state on a star-algebra|state]] $\rho \colon \mathcal{A} \to \mathbb{C}$ on a unital [[star-algebra]] (def. \ref{StateOnAStarAlgebra}) is called a _[[pure state]]_ if it is extremal in the [[convex set]] of all states (remark \ref{StatesFormAConvexSet}) in that an identification
$$
\langle (- )\rangle = p \langle (-)\rangle_1 + (1-p) \langle (-)\rangle_2
$$
for $p \in (0,1)$ implies that $\langle (-) \rangle_1 = \langle (-)\rangle_2$ (hence $= \langle (-)\rangle$).
=--
+-- {: .num_prop #ClassicalProbabilityMeasureAsStateOnMeasurableFunctions}
###### Proposition
**(classical [[probability measure]] as state on [[measurable functions]])**
For $\Omega$ classical [[probability space]], hence a [[measure space]] which normalized total measure $\int_\Omega d\mu = 1$, let $\mathcal{A} \cloneqq L^1(\Omega)$ be the algebra of Lebesgue [[measurable functions]] with values in the [[complex numbers]], regarded as a [[star algebra]] (def. \ref{StarAlgebra}) by pointwise [[complex conjugation]] as in example \ref{StarAlgebraOfObservables}. Then forming the [[expectation value]] with respect to $\mu$ defines a [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}):
$$
\array{
L^1(\Omega)
&\overset{\langle (-)\rangle_\mu}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& \int_\Omega A d\mu
}
$$
=--
+-- {: .num_example #ElementsOfHilbertSpaceAsPureStates}
###### Example
**(elements of a [[Hilbert space]] as [[pure states]] on [[bounded operators]])**
Let $\mathcal{H}$ be a [[complex numbers|complex]] [[separable Hilbert space|separable]] [[Hilbert space]] with [[inner product]] $\langle -,-\rangle$ and let $\mathcal{A} \coloneqq \mathcal{B}(\mathcal{H})$ be the algebra of [[bounded operators]], regarded as a [[star algebra]] (def. \ref{StarAlgebra}) under forming [[adjoint operators]]. Then for every element $\psi \in \mathcal{H}$ of unit [[norm]] $\langle \psi,\psi\rangle = 1$ there is the [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}) given by
$$
\array{
\mathcal{B}(\mathcal{H})
&\overset{\langle (-)\rangle_\psi}{\longrightarrow}&
\mathbb{C}
\\
A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &\coloneqq& \langle \psi, A \psi \rangle
}
$$
These are [[pure states]] (def. \ref{PureStateOnAStarAlgebra}).
More general states in this case are given by [[density matrices]].
=--
+-- {: .num_theorem #GNSConstruction}
###### Theorem
**([[GNS construction]])**
Given
1. a [[star-algebra]], $\mathcal{A}$ (def. \ref{StarAlgebra});
1. a [[state on a star-algebra|state]], $\langle (-)\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$ (def. \ref{StateOnAStarAlgebra})
there exists
1. a [[star-representation]]
$$
\pi
\;\colon\;
\mathcal{A}
\longrightarrow
End(\mathcal{H})
$$
of $\mathcal{A}$ on some [[Hilbert space]] $\mathcal{H}$
1. a [[cyclic vector]] $\psi \in \mathcal{H}$
such that $\langle (-)\rangle$ is the state corresponding to $\psi$ via example \ref{ElementsOfHilbertSpaceAsPureStates}, in that
$$
\begin{aligned}
\langle A \rangle
& = \langle \psi \vert\, A \, \vert \psi \rangle
\\
& \coloneqq
\langle \psi , \pi(A) \psi \rangle
\end{aligned}
$$
for all $A \in \mathcal{A}$.
=--
([Khavkine-Moretti 15, theorem 1](GNS+construction#KhavkineMoretti15))
+-- {: .num_defn #ClassicalState}
###### Definition
**(classical state)**
Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) then a _classical state_ is a [[state on a star-algebra|state on the star algebra]] (def. \ref{StateOnAStarAlgebra}) of [[on-shell]] [[observables]] (example \ref{StarAlgebraOfObservables}):
$$
\langle -\rangle
\;\colon\;
Obs(E,\mathbf{L})
\longrightarrow
\mathbb{C}
\,.
$$
=--
Below we consider _[[quantum states]]_. These are defined just as in def. \ref{ClassicalState}, only that now the algebra of observables is equipped with another product, which changes the meaning of the product expression $A^\ast A$ and hence the positivity condition in def. \ref{StateOnAStarAlgebra}.
$\,$
This concludes our discussion of [[observables]]. In the [next chapter](#PhaseSpace) we consider the
construction of the [[covariant phase space]] and of the [[Poisson-Peierls bracket]] on [[observables]].
|
A first idea of quantum field theory -- Phase space | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Phase+space | ## Phase space
{#PhaseSpace}
In this chapter we discuss these topics:
* _[Covariant phase space](#PreymplecticPhaseSpace)_
* _[BV-Resolution of the covariant phase space](#BVResolutionOfTheCovariantPhaseSpace)_
* _[Hamiltonian local observables](#HamiltonianLocalObservablesOnACauchySurface)_
$\,$
It might seem that with the construction of the [[local observables]] (def. \ref{LocalObservables})
on the [[on-shell]] [[space of field histories]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
the [[field theory]] defined by a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) has been completely analyzed:
This data specifies, in principle, which [[field histories]] are realized, and which [[observable]] properties these have.
In particular, if the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime})
admit [[Cauchy surfaces]] (def. \ref{CauchySurface} below), i.e. spatial [[codimension]] 1 slices of [[spacetimes]]
such that a [[field history]] is uniquely specified already by its restriction to the [[infinitesimal neighbourhood]]
of that spatial slice, then a sufficiently complete collection of [[local observables]] whose spacetime support (def. \ref{SpacetimeSupport})
[[covering|covers]] that Cauchy surface allows to _predict_ the evolution of the field histories through time
from that Cauchy surface.
This is all what one might think a theory of physical fields should accomplish, and in fact this is essentially
all that was thought to be required of a theory of nature from about [[Isaac Newton]]'s time to about [[Max Planck]]'s time.
But we have seen that a remarkable aspect of [[Lagrangian field theory]] is that the [[de Rham differential]] of the [[local Lagrangian density]] $\mathbf{L}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) decomposes into
_two_ kinds of [[variational differential forms]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}),
one of which is the [[Euler-Lagrange form]] which determines the [[equations of motion]] (eq:EulerLagrangeEquationGeneral).
However, there is a second contribution: The _[[presymplectic current]]_ $\Omega_{BFV} \in \Omega^{p,2}_{\Sigma}(E)$ (eq:PresymplecticCurrent).
Since this is of horizontal degree $p$,
its [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) implies a further structure on the [[space of field histories]]
restricted to [[spacetime]] [[submanifolds]] of dimension $p$ (i.e. of spacetime "[[codimension]] 1").
There may be such submanifolds such that this restriction to their [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) does not actually change the [[on-shell]] [[space of field histories]],
these are called the _[[Cauchy surfaces]]_ (def. \ref{CauchySurface} below).
By the [[Hamiltonian Noether theorem]] (prop. \ref{HamiltonianDifferentialForms}) the [[presymplectic current]] induces [[infinitesimal symmetries]] acting on [[field histories]] and [[local observables]], given by the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (prop. \ref{LocalPoissonBracket}).
The [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the [[presymplectic current]]
to these [[Cauchy surfaces]] yields the corresponding [[infinitesimal symmetry]] group acting on the [[on-shell]] [[field histories]],
whose [[Lie bracket]] is the _[[Poisson bracket]]_ pairing on [[on-shell]] [[observables]] (example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} below). This data, the [[on-shell]] [[space of field histories]] on the [[infinitesimal neighbourhood]] of a [[Cauchy surface]] equipped with [[infinitesimal symmetry]] exhibited by the [[Poisson bracket]] is called the _[[phase space]]_ of the theory (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) below.
In fact if enough [[Cauchy surfaces]] exist, then the [[presymplectic forms]] associated with
any one choice turn out do agree after [[pullback of differential forms|pullback]] to the full [[on-shell]] [[space of field histories]], exhibiting this as the _[[covariant phase space]]_ of the theory (prop. \ref{CovariantPhaseSpace} below) which is hence manifestly independent of aa choice of space/time splitting. Accordingly, also the [[Poisson bracket]] on [[on-shell]] [[observables]] exists in a covariant form; for [[free field theories]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] (def. \ref{GreenHyperbolicDifferentialOperator}) this is called the _[[Peierls-Poisson bracket]]_ (theorem \ref{PPeierlsBracket} below). The [[integral kernel]] for this [[Peierls-Poisson bracket]] is called the _[[causal propagator]]_ (prop. \ref{GreenFunctionsAreContinuous}).
Its "[[normal ordered product|normal ordered]]" or "[[positive real number|positive]] [[frequency]] component", called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below) as well as the corresponding [[time-ordered product|time-ordered]] variant, called the _[[Feynman propagator]]_ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below), which we discuss in detail in _[Propagators](#Propagators)_ below,
control the [[causal perturbation theory]] for constructing [[perturbative quantum field theory]] by [[deformation quantization|deforming]]
the commutative pointwise product of [[on-shell]] [[observables]] to a [[non-commutative algebra|non-commutative product]] governed to first order by the [[Peierls-Poisson bracket]].
To see how such a [[deformation quantization]] comes about conceptually from the [[phase space]] strucure,
notice from the basic principles of [[homotopy theory]] that given any [[structure]] on a [[space]]
which is [[invariant]] with respect to a [[symmetry group]] [[action|acting]] on the space
(here: the [[presymplectic current]])
then the true structure at hand is the [[homotopy quotient]] of that [[space]] by that [[symmetry group]].
We will explain this further below. This here just to point out that the [[homotopy quotient]] of the
[[phase space]] by the [[Hamiltonian vector field|infinitesimal symmetries of the presymplectic current]]
is called the _[[symplectic groupoid]]_ and that the _true_ [[algebra of observables]] is hence
the ([[polarization|polarized]]) [[groupoid convolution algebra|convolution algebra of functions]]
on this groupoid. This turns out to the "[[algebra of quantum observables]]" and the passage from the
naive [[local observables]] on [[presymplectic manifold|presymplectic]] [[phase space]] to
this non-commutative algebra of functions on its [[homotopy quotient]] to the [[symplectic groupoid]]
is called _[[quantization]]_. This we discuss in much detail [below](#Quantization); for the
moment this is just to motivate why the [[covariant phase space]] is the crucial construction to be
extracted from a [[Lagrangian field theory]].
$\,$
$$
\array{
\left\{
\array{
\text{on-shell space}
\\
\text{ of field histories}
\\
\text{restricted to}
\\
\text{Cauchy surface}
}
\right\}
&\overset{\array{ \text{homotopy} \\ \text{quotient} \\ \text{by} \\ \text{infinitesimal} \\ \text{symmetries} }}{\longrightarrow}
&
\left\{
\array{
\text{covariant}
\\
\text{phase space}
}
\right\}
&\overset{ \array{\text{Lie algebra} \\ \text{of functions} } }{\longrightarrow}&
\left\{
\array{
\text{Poisson algebra}
\\
\text{of observables}
}
\right\}
\\
&
\searrow
&
\Big\downarrow{}^\mathrlap{{\text{Lie integration}}}
&&
{}^{\mathllap{quantization}}\Big\downarrow
\\
&& \left\{
\array{
\text{symplectic}
\\
\text{groupoid}
}
\right\}
&
\overset{
\array{
\text{polarized}
\\
\text{convolution}
\\
\text{algebra}
}
}{\longrightarrow}&
\left\{
\array{
\text{quantum algebra}
\\
\text{of observables}
}
\right\}
}
$$
$\,$
**Covariant phase space**
{#PreymplecticPhaseSpace}
+-- {: .num_defn #CauchySurface}
###### Definition
**([[Cauchy surface]])**
Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), then a
_[[Cauchy surface]]_ is a [[submanifold]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{SmoothManifoldInsideDiffeologicalSpaces}) such that the
restriction map from the [[on-shell]] [[space of field histories]] $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$
(eq:OnShellFieldHistories) to the space $\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ (eq:OnShellFieldHistoriesInHigherCodimension) of on-shell field histories
restricted to the [[infinitesimal neighbourhood]] of $\Sigma_p$ (example \ref{InfinitesimalNeighbourhood}) is an [[isomorphism]]:
$$
\label{CauchySurfaceIsomorphismOnHistorySpace}
\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0 }
\underoverset{\simeq}{(-)\vert_{N_\Sigma \Sigma_p}}{\longrightarrow}
\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}
\,.
$$
=--
+-- {: .num_example #NormallyHyperbolicOperatorsHaveCauchySurfaces}
###### Example
**([[normally hyperbolic differential operators]] have [[Cauchy surfaces]])**
Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) are given by a [[normally hyperbolic differential operator]] (def. \ref{NormallyHyperbolicDifferentialOperator}), then it admits [[Cauchy surfaces]] in the sense of Def. \ref{CauchySurface}.
=--
(e.g. [Bär-Ginoux-Pfäffle 07, section 3.2](hyperbolic+differential+operator#BaerGinouxPfaeffle07))
+-- {: .num_defn #PhaseSpaceAssociatedWithCauchySurface}
###### Definition
**([[phase space]] associated with a [[Cauchy surface]])**
Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
and given a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface})
then the corresponding _[[phase space]]_ is
1. the [[super smooth set]] $\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ (eq:OnShellFieldHistoriesInHigherCodimension)
of [[on-shell]] [[field histories]] restricted to the [[infinitesimal neighbourhood]] of $\Sigma_p$;
1. equipped with the [[differential 2-form]] (as in def. \ref{DifferentialFormsOnDiffeologicalSpaces})
$$
\label{TransgressionOfPresymplecticCurrentToCauchySurface}
\omega_{\Sigma_p}
\;\coloneqq\;
\tau_{\Sigma_p}\left(\Omega_{BFV}\right)
\;\in\;
\Omega^2\left(
\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}
\right)
$$
which is the distributional [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
of the [[presymplectic current]] $\Omega_{BFV}$ (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
to $\Sigma_p$.
This $\omega_{\Sigma_p}$ is a [[closed differential form]] in the sense of def. \ref{DifferentialFormsOnDiffeologicalSpaces},
due to prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative}
and using that $\Omega_{BFV} = \delta \Theta_{BFV}$ is closed by definition (eq:PresymplecticCurrent).
As such this is called the _[[presymplectic form]]_ on the phase space.
=--
+-- {: .num_example #EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory}
###### Example
**(evaluation of [[transgression of variational differential forms|transgressed variational form]] on [[tangent vectors]] for [[free field theory]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) which is [[free field theory|free]] (def. \ref{FreeFieldTheory}) hence
whose [[field bundle]] is a some [[smooth vector bundle|smooth]] [[super vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle})
and whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is [[linear differential equation|linear]].
Then the [[synthetic differential geometry|synthetic]] [[tangent bundle]] (def. \ref{TangentBundleSynthetic}) of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ (eq:OnShellFieldHistories)
with spacelike compact support (def \ref{CompactlySourceCausalSupport})
is canonically identified with the [[Cartesian product]] of this [[super smooth set]] with itself
$$
T\left(
\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}
\right)
\;\simeq\;
\left(\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}\right)
\times
\left(\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}\right)
\,.
$$
With field coordinates as in example \ref{TrivialVectorBundleAsAFieldBundle}, we may expand the
[[presymplectic current]] as
$$
\Omega_{BFV}
=
\left(\Omega_{BFV}\right)^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}_{a_1 a_2}
\delta \phi^{a_1}_{\mu_1 \cdots \mu_k} \wedge \delta \phi^{a_2}_{\nu_1 \cdots \nu_{k_2}}
\wedge
\iota_{\partial_\kappa} dvol_\Sigma
\,,
$$
where the components $(\Omega_{BFV})_{a_1 a_2}^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}$
are smooth functions on the [[jet bundle]].
Under these identifications the value of the [[presymplectic form]] $\omega_{\Sigma_p}$ (eq:TransgressionOfPresymplecticCurrentToCauchySurface)
on two [[tangent vectors]] $\vec \Phi_1, \vec \Phi_2 \in \Gamma_{\Sigma,scp}(E)$
at a point $\Phi \in \Gamma_{\Sigma,scp}(E)$ is
$$
\omega_{\Sigma_p}(\vec \Phi_1, \vec \Phi_2)
\;=\;
\underset{\Sigma_p}{\int}
\left(\Omega_{BFV}\right)^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}_{a_1 a_2}(\Phi(x))
\left(
\frac{\partial}{\partial x^{\mu_1}}
\cdots
\frac{\partial}{\partial x^{\mu_{k_1}}}
\vec \Phi_1(x)
\right)
\left(
\frac{\partial}{\partial x^{\nu_1}}
\cdots
\frac{\partial}{\partial x^{\nu_{k_2}}}
\vec \Phi_2(x)
\right)
\,
\iota_{\partial_\kappa} dvol_\Sigma(x)
\,.
$$
=--
+-- {: .num_example #PresymplecticFormForFreeRealScalarField}
###### Example
**([[presymplectic form]] for [[free field|free]] [[real scalar field]])**
Consider the [[Lagrangian field theory]] for the [[free field|free]] [[real scalar field]]
from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.
Under the identification of example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} the [[presymplectic form]] on the [[phase space]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface})
associated with a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$
is given by
$$
\begin{aligned}
\omega_{\Sigma_p}(\vec \Phi_1, \vec\Phi_2)
& =
\int_{\Sigma_{p}}
\left(
\frac{\partial \vec \Phi_1}{\partial x^\mu}(x) \vec \Phi_2(x)
-
\vec \Phi_1(x) \frac{\partial \vec \Phi_2}{\partial x^\mu}(x)
\right)
\eta^{\mu \nu}
\iota_{\partial_\mu} dvol_{\Sigma_{p}}(x)
\\
& =
\underset{\Sigma_p}{\int} K(\vec \Phi_1, \vec \Phi_2)
\,.
\end{aligned}
$$
Here the first equation follows via example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} from the form of $\Omega_{BFV}$ from example \ref{FreeScalarFieldEOM},
while the second equation
identifies the integrand as the witness $K$ for the [[formally adjoint differential operator|formally self-adjointness]] of the [[Klein-Gordon equation]] from example \ref{FormallySelfAdjointKleinGordonOperator}.
=--
+-- {: .num_example #PresymplecticFormForFreeDiracField}
###### Example
**([[presymplectic form]] for [[free field theory|free]] [[Dirac field]])**
Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField}).
Under the identification of example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} the [[presymplectic form]] on the [[phase space]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) associated with a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$
is given by
$$
\begin{aligned}
\omega_{\Sigma_p}(\theta_1 \vec \Psi_1, \theta_2 \vec\Psi_2)
& =
\int_{\Sigma_{p}}
\left(
\overline{\theta_1 \vec \psi_1}\gamma^\mu \left( \theta_2 \vec \Psi_2 \right)
\right)
\iota_{\partial_\mu} dvol_{\Sigma_{p}}(x)
\\
& =
\underset{\Sigma_p}{\int} K(\vec \Phi_1, \vec \Phi_2)
\,.
\end{aligned}
$$
Here the first equation follows via example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} from the form of $\Omega_{BFV}$ from example \ref{PresymplecticCurrentDiracField},
while the second equation
identifies the integrand as the witness $K$ for the [[formally adjoint differential operator|formally self-adjointness]] of the [[Dirac equation]] from example \ref{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}.
=--
+-- {: .num_prop #CovariantPhaseSpace}
###### Proposition
**([[covariant phase space]])**
Consider $(E, \mathbf{L})$ a [[Lagrangian field theory]] on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Let
$$
\Sigma_{tra} \overset{tra}{\hookrightarrow} \Sigma
$$
be a [[submanifold]] [[manifold with boundary|with two boundary components]]
$\partial \Sigma_{tra} = \Sigma_{in} \sqcup \Sigma_{out}$ ,
both of which are [[Cauchy surfaces]] (def. \ref{CauchySurface}).
Then the corresponding inclusion diagram
$$
\array{
&& \Sigma_{tra}
\\
& {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrm{out}}
\\
\Sigma_{in}
&& &&
\Sigma_{out}
}
$$
induces a [[Lagrangian correspondence]] between the associated [[phase spaces]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface})
$$
\array{
&& \Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0}
\\
& {}^{\mathllap{ (-)\vert_{in} }}\swarrow && \searrow^{\mathrlap{ (-)\vert_{out} }}
\\
\Gamma_{\Sigma^{(in)}}(E)_{\delta_{EL}\mathbf{L}= 0}
&& &&
\Gamma_{\Sigma^{(out)}}(E)_{\delta_{EL}\mathbf{L}= 0}
\\
& {}_{\mathllap{\omega_{in}}}\searrow && \swarrow_{\mathrlap{\omega_{out}}}
\\
&& \mathbf{\Omega}^{2}
}
$$
in that the [[pullback of differential forms|pullback]] of the
two [[presymplectic forms]] (eq:TransgressionOfPresymplecticCurrentToCauchySurface) coincides on the
space of field histories:
$$
\left( (-)\vert_{in}\right)^\ast\left( \omega_{in}\right)
\;=\;
\left( (-)\vert_{out} \right)^\ast \left( \omega_{out} \right)
\phantom{AAAA}
\in
\Omega^2
\left(
\Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0}
\right)
\,.
$$
Hence there is a well defined [[presymplectic form]]
$$
\omega \in \Omega^2\left( \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}} = 0 \right)
$$
on the genuine [[space of field histories]], given by $\omega \coloneqq i^\ast \omega_{\Sigma_p}$
for any Cauchy surface $\Sigma_p \overset{i}{\hookrightarrow} \Sigma$. This [[presymplectic smooth space]]
$$
\left(
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}}
\,,\,
\omega
\right)
$$
is therefore called the _[[covariant phase space]]_ of the [[Lagrangian field theory]] $(E,\mathbf{L})$.
=--
+-- {: .proof}
###### Proof
By prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell} the
total spacetime derivative $d \Omega_{BFV}$ of the [[presymplectic current]] vanishes [[on-shell]]:
$$
d \Omega_{BFV} = - \delta \delta_{EL} \mathbf{L}
$$
in that the [[pullback of differential forms|pullback]] (def. \ref{PullbackOfDifferential1FormsOnCartesianSpaces})
along the [[shell]] inclusion $\mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E)$ (eq:ShellInJetBundle)
vanishes:
$$
\begin{aligned}
(i_{\mathcal{E}})^\ast
\left(
d \Omega_{BFV}
\right)
& =
- (i_{\mathcal{E}})^\ast
\left(
\delta \delta_{EL} \mathcal{L}
\right)
\\
& =
- \delta
\underset{
= 0
}{
\underbrace{
(i_{\mathcal{E}})^\ast
\left(
\delta_{EL} \mathbf{L}
\right)
}
}
\\
& = 0
\end{aligned}
$$
This implies that the transgression of $d \Omega_{BFV}$ to
the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma_{tra}}(E)_{\delta_{EL}\mathbf{L} = 0}$ vanishes
(since by definition (eq:EquationOfMotionEL) that involves pulling back through the shell inclusion)
$$
\tau_{\Sigma_{tra}}(d \Omega_{BFV}) = 0
\,.
$$
But then the claim follows with prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative}:
$$
\begin{aligned}
0
& = \tau_{\Sigma_{tra}}(d \Omega_{BFV})
\\
& = ((-)\vert_{\Sigma_{tra}})^\ast \tau_{\partial \Sigma_{tra}} \Omega_{BFV}
\,.
\end{aligned}
$$
=--
+-- {: .num_theorem #PPeierlsBracket}
###### Theorem
**([[polynomial Poisson algebra|polynomial Poisson bracket]] on [[covariant phase space]] -- the [[Peierls bracket]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) such that
1. it is a [[free field theory]] (def. \ref{FreeFieldTheory})
1. whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] $P \Phi = 0$ (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is
1. [[formally adjoint differential operator|formally self-adjoint]] or [[formally adjoint differential operator|formally anti self-adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) such that
* the integral over the witness $K$ (eq:FormallyAdjointDifferentialOperatorWitness) is the
[[presymplectic form]] (eq:TransgressionOfPresymplecticCurrentToCauchySurface): $\omega_{\Sigma_p} = \underset{\Sigma_p}{\int} K$;
1. [[Green hyperbolic differential operator|Green hyperbolic]] (def. \ref{GreenHyperbolicDifferentialOperator}).
Write
$$
\mathrm{G}_P
\;\colon\;
LinObs(E_{scp},\mathbf{L})^{reg}
\overset{\mathrm{G}_P}{\longrightarrow}
\Gamma_{\Sigma,scp}(E)_{\delta_{EL}\mathbf{L} = 0}
$$
for the linear map
from regular linear field observables (def. \ref{RegularLinearFieldObservables}) to on-shell [[field histories]] with spatially compact support (def. \ref{CompactlySourceCausalSupport}) given under the identification (eq:RegularLinearObservablesAreCompactlySupportedSectionsModuloImageOfP)
by the [[causal Green function]] $\mathrm{G}_P$ (def. \ref{AdvancedAndRetardedGreenFunctions}).
Then for every [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) this map is an inverse to the
[[presymplectic form]] $\omega_{\Sigma_p}$ (def. \ref{PhaseSpaceAssociatedWithCauchySurface})
in that, under the identification of tangent vectors to field histories from example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory}, we have that the composite
$$
\label{ForGreenHyperbolicFreeFieldTheoryCausalGreenFunctionIsInverseToPresymplecticFormOnRegularLinearObservables}
\array{
\omega_{\Sigma_p}(\mathrm{G}_P(-),(-))
\;=\;
ev
&\colon&
LinObs(E_{scp},\mathbf{L})^{reg} &\otimes& \Gamma_{\Sigma,scp}(E)
&\longrightarrow&
\mathbb{C}
\\
&& (A &,& \Phi) &\mapsto& A(\Phi)
}
$$
equals the [[evaluation map]] of observables on field histories.
This means that for every [[Cauchy surface]] $\Sigma_p$ the [[presymplectic form]] $\omega_{\Sigma_p}$ restricts to a _[[symplectic form]]_
on regular linear observables. The corresponding _[[Poisson bracket]]_ is
$$
\left\{
-,-
\right\}_{\Sigma_p}
\;\coloneqq\;
\omega_{\Sigma_p}(\mathrm{G}_P(-), \mathrm{G}_P(-))
\;\;\colon\;\;
LinObs(E_{scp},\mathbf{L})^{reg} \otimes LinObs(E_{scp},\mathbf{L})^{reg}
\longrightarrow
\mathbb{R}
\,.
$$
Moreover, equation (eq:ForGreenHyperbolicFreeFieldTheoryCausalGreenFunctionIsInverseToPresymplecticFormOnRegularLinearObservables)
implies that this is the _covariant [[Poisson bracket]]_ in the sense of the [[covariant phase space]] (def. \ref{CovariantPhaseSpace})
in that it does not actually depend on the choice of [[Cauchy surface]].
An equivalent expression for the Poisson bracket that makes its independence from the choice of Cauchy surface
manifest is the _$P$-[[Peierls bracket]]_ given by
$$
\label{ThePPeierlsBracket}
\array{
LinObs(E_{scp},\mathbf{L})^{reg} \otimes LinObs(E_{scp},\mathbf{L})^{reg}
&\overset{\{-,-\}}{\longrightarrow}&
\mathbb{R}
\\
(\alpha^\ast, \beta^\ast)
&\mapsto&
\underset{\Sigma}{\int}
\mathrm{G}(\alpha^\ast) \cdot \beta^\ast
\,
dvol_\Sigma
}
$$
where on the left $\alpha^\ast, \beta^\ast \in \Gamma_{\Sigma,cp}(E^\ast) \simeq LinObs(E_{scp},\mathbf{L})^{reg}$
Hence under the given assumptions, for every Cauchy surface the [[Poisson bracket]]
associated with that Cauchy surface equals the invariantly ("covariantly") defined [[Peierls bracket]]
$$
\{-,-\}_{\Sigma_p} = \{-,-\}
\,.
$$
Finally this means that in terms of the [[causal propagator]] $\Delta$ (eq:CausalPropagator) the
covariant [[Peierls-Poisson bracket]] is given in [[generalized function]]-notation by
$$
\label{CausalPropagatorPPeierlsBracket}
\{\alpha^\ast, \beta^\ast\}
\;=\;
\underset{\Sigma}{\int} \underset{\Sigma}{\int}
\alpha^\ast(x) \cdot \Delta(x,y) \cdot \beta^\ast(y)
\, dvol_\Sigma(x)\, dvol_\Sigma(y)
$$
Therefore, while the point-evaluation field observables $\mathbf{\Phi}^a(x)$ (def. \ref{PointEvaluationObservables}) are not themselves regular observables (def. \ref{RegularLinearFieldObservables}),
the [[Peierls-Poisson bracket]] (eq:CausalPropagatorPPeierlsBracket) is induced from the
following distributional bracket between them
$$
\left\{
\mathbf{\Phi}^a(x)
,
\mathbf{\Phi}^b(y)
\right\}
\;=\;
\Delta^{a b}(x,y)
$$
with the [[causal propagator]] (eq:CausalPropagator) on the right, in that with the identification (eq:AverageOfFieldObservableIsRegularLinearObservables) the [[Peierls-Poisson bracket]] on
regular linear observables arises as follows:
$$
\begin{aligned}
\left\{
\underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
\,,\,
\underset{\Sigma}{\int} \beta^\ast_b(y) \mathbf{\Phi}^b(y) \, dvol_\Sigma(y)
\right\}
& =
\underset{\Sigma}{\int} \underset{\Sigma}{\int}
\alpha^\ast_a(x)
\underset{= \Delta^{a b}(x,y)}{
\underbrace{
\left\{
\mathbf{\Phi}^a(x),
\mathbf{\Phi}^b(y)
\right\}
}
}
\beta^\ast_b(y)
\, dvol_\Sigma(x)\, dvol_\Sigma(y)
\\
& =
\underset{\Sigma}{\int} \underset{\Sigma}{\int}
\alpha^\ast_a(x)
\Delta^{a b}(x,y)
\beta^\ast_b(y)
\, dvol_\Sigma(x)\, dvol_\Sigma(y)
\end{aligned}
$$
=--
([Khavkine 14, lemma 2.5](Green+hyperbolic+partial+differential+equation#Khavkine14))
+-- {: .proof}
###### Proof
Consider two more Cauchy surfaces $\Sigma_p^\pm \hookrightarrow I^\pm(\Sigma) \hookrightarrow \Sigma$, in the [[future]] $I^+$ and in the [[past]] $I^-$ of $\Sigma$, respectively. Choose a [[partition of unity]] on $\Sigma$ consisting of two elements $\chi^\pm \in C^\infty(\Sigma)$ with [[support]] bounded by these Cauchy surfaces: $supp(\chi_\pm) \subset I^\pm(\Sigma^{\mp})$.
Then define
$$
\label{SplittingOfGreenExactSequenceType}
P_\chi
\;\colon\;
\Gamma_{\Sigma,scp}(E)
\longrightarrow
\Gamma_{\Sigma,cp}(E^\ast)
$$
by
$$
\label{SplittingOfGreenExactSequence}
\begin{aligned}
P_\chi(\Phi)
& \coloneqq
\phantom{-} P(\chi_+ \Phi)
\\
& =
-
P(\chi_- \Phi)
\,.
\end{aligned}
$$
Notice that the [[support]] of the partitioned field history is in the compactly sourced future/past cone
$$
\label{ChipmPhiIsSupportedInPastFuture}
\chi_\pm \Phi
\;\in\;
\Gamma_{\Sigma,\pm cp}(E)
$$
since $\Phi$ is supported in the compactly sourced causal cone, but that
$P(\chi_\pm \Phi)$ indeed has [[compact support]] as required by (eq:SplittingOfGreenExactSequenceType): Since $P(\Phi) = 0$, by assumption, the support is the intersection of that of $\Phi$ with that of $d \chi_\pm$, and the first is spacelike compact by assumption, while the latter is timelike compact, by definition of partition of unity.
Similarly, the equality in (eq:SplittingOfGreenExactSequence) holds because by [[partition of unity]] $P(\chi_+ \Phi) + P(\chi_-\Phi) = P((\chi_+ + \chi_-)\Phi ) = P(\Phi) = 0$.
It follows that
$$
\label{PchiIsRightInverseToGP}
\begin{aligned}
\mathrm{G}_P \circ P_\chi (\Phi)
& =
\left(
\mathrm{G}_{P,+} - \mathrm{G}_{P,-}
\right) P_\chi (\Phi)
\\
& =
\underset{ = \chi_+ \Phi}{\underbrace{\mathrm{G}_{P,+} P(\chi_+ \Phi)}}
+
\underset{ = \chi_- \Phi }{\underbrace{\mathrm{G}_{P,-} P(\chi_- \Phi)}}
\\
& =
(\chi_+ + \chi_-)\Phi
\\
& =
\Phi
\,,
\end{aligned}
$$
where in the second line we chose from the two equivalent expressions (eq:SplittingOfGreenExactSequence) such that via (eq:ChipmPhiIsSupportedInPastFuture) the defining property of the [[advanced and retarded Green functions|advanced or retarded Green function]], respectively, may be applied, as shown under the braces.
([Khavkine 14, lemma 2.1](Green+hyperbolic+differential+equation#Khavkine14))
Now we apply this to the computation of $\omega_{\Sigma_p}(\mathrm{G}_P(-),-)$:
$$
\begin{aligned}
\omega_{\Sigma_P}(\mathrm{G}_P(\alpha^\ast),\vec \Phi)
& =
\underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \vec \Phi)
\\
& =
\underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \chi_+\vec \Phi)
+
\underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \chi_-\vec \Phi)
\\
& =
\underset{I^-(\Sigma_P)}{\int} d K(\mathrm{G}_P(\alpha^\ast), \chi_+\vec \Phi)
-
\underset{I^+(\Sigma_P)}{\int} d K(\mathrm{G}_P(\alpha^\ast), \chi_-\vec \Phi)
\\
& =
\underset{I^-(\Sigma_P)}{\int}
\left(
\underset{= 0}{
\underbrace{
P(\mathrm{G}_P(\alpha^\ast))}} \cdot \chi_+\vec \Phi
\mp
\mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi)
\right)
dvol_\Sigma
-
\underset{I^+(\Sigma_P)}{\int}
\left(
\underset{= 0}{
\underbrace{
P(\mathrm{G}_P(\alpha^\ast))}} \cdot \chi_-\vec \Phi
\mp
\mathrm{G}_P(\alpha^\ast) \cdot P(\chi_- \vec \Phi)
\right)
dvol_\Sigma
\\
& =
\mp
\left(
\underset{I^-(\Sigma_P)}{\int}
\mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi)
dvol_\Sigma
+
\underset{I^+(\Sigma_P)}{\int}
\mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi)
dvol_\Sigma
\right)
\\
& =
\underset{\Sigma}{\int} \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi) dvol_\Sigma
\\
& =
\underset{\Sigma}{\int} \alpha^\ast \cdot \mathrm{G}_{P} (P (\chi_+ \vec \Phi))
\\
& =
\underset{\Sigma}{\int} \alpha^\ast \cdot \vec \Phi
\end{aligned}
$$
Here we computed as follows:
1. applied the assumption that $\omega_{\Sigma_p}(-,-) = \underset{\Sigma_p}{\int} K(-,-)$;
1. applied the above partition of unity;
1. used the [[Stokes theorem]] (prop. \ref{StokesTheorem}) for the past and the future of $\Sigma_p$, respectively;
1. applied the definition of $d K$ as the witness of the formal (anti-) self-adjointness of $P$ (def. \ref{FormallyAdjointDifferentialOperators});
1. used $P\circ \mathrm{G}_p = 0$ on $\Gamma_{\Sigma,cp}(E^\ast)$ (def. \ref{AdvancedAndRetardedGreenFunctions}) and used (eq:SplittingOfGreenExactSequence);
1. unified the two integration domains, now that the integrands are the same;
1. used the formally (anti-)self adjointness of the Green functions (example \ref{CausalGreenFunctionOfFormallyAdjointDifferentialOperatorAreFormallyAdjoint});
1. used (eq:PchiIsRightInverseToGP).
=--
+-- {: .num_example #PeierlsBracketEistsForScalarFieldAndDiracField}
###### Example
**([[scalar field]] and [[Dirac field]] have [[covariant phase space|covariant]] [[Peierls-Poisson bracket]])**
Examples of [[free field theory|free]] [[Lagrangian field theories]] for which the assumptions of theorem
\ref{PPeierlsBracket} are satisfied, so that the covariant [[Poisson bracket]] exists in the form
of the [[Peierls bracket]] include
* the [[free field theory|free]] [[real scalar field]] (example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime});
* the [[free field theory|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField}).
For the [[free field theory|free]] [[scalar field]] this is the statement of example \ref{GreenHyperbolicKleinGordonEquation}
with example \ref{PresymplecticFormForFreeRealScalarField}, while for the [[Dirac field]] this is the
statement of example \ref{GreenHyperbolicDiracOperator} with example \ref{PresymplecticFormForFreeDiracField}.
=--
For the [[free field theory|free]] [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity})
the assumptions of theorem \ref{PPeierlsBracket} are violated, the [[covariant phase space]] does not exist.
But in the discussion of _[Gauge fixing](#GaugeFixing)_, below,
we will find that for an equivalent re-incarnation of the electromagnetic field, they are met after all.
$\,$
**BV-resolution of the covariant phase space**
{#BVResolutionOfTheCovariantPhaseSpace}
So far we have discussed the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) in terms of
explicit restriction to the [[shell]]. We now turn to the more flexible perspective where
a [[homological resolution]] of the [[shell]] in terms of "[[antifields]]" is used (def. \ref{BVComplexOfOrdinaryLagrangianDensity}).
+-- {: .num_example #BVPresymplecticCurrent}
###### Example
**(BV-presymplectic current)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}).
Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle).
Then in the BV-variational bicomplex (eq:ComparisonMorphismFromOrdinaryBVComplexToLocalObservables)
there exists the _BV-presymplectic potential_
$$
\label{BVPresymplecticPotential}
\Theta_{BV}
\;\coloneqq\;
\phi^{\ddagger}_a \delta \phi^a \, dvol_\Sigma
\;\in\;
\Omega^{p,1}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}
$$
and the corresponding _BV-presymplectic current_
$$
\Omega_{BV}
;\in\;
\Omega^{p,2}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}
$$
defined by
$$
\begin{aligned}
\Omega_{BV}
& \coloneqq
\delta \Theta_{BV}
\\
& =
\delta \phi^{\ddagger}_a \wedge \delta \phi^a \wedge dvol_{\Sigma}
\end{aligned}
\,,
$$
where $(\phi^a)$ are the given [[field (physics)|field]]
[[coordinates]], $\phi^{\ddagger}_a$ the corresponding [[antifield]] coordinates (eq:AntifieldCoordinates)
and $\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}$ the corresponding components of the [[Euler-Lagrange form]]
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}).
=--
+-- {: .num_prop #ResolutionOfCovariantPhaseSpaceCorrespondence}
###### Proposition
**(local BV-BFV relation)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}).
Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell (eq:ConstantSectionOfTrivialShellBundle).
Then the BV-presymplectic current $\Omega_{BV}$ (def. \ref{BVPresymplecticCurrent})
witnesses the [[on-shell]] vanishing (prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}) of the [[total derivative|total spacetime derivative]] of the genuine [[presymplectic current]] $\Omega_{BFV}$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) in that the [[total derivative|total spacetime derivative]]
of $\Omega_{BFV}$ equals the BV-differential $s_{BV}$ of $\Omega_{BV}$:
$$
d \Omega_{BFV} = s \Omega_{BV}
\,.
$$
Hence if $\Sigma_{tra} \hookrightarrow \Sigma$ is a [[submanifold]] of [[spacetime]] of full dimension $p+1$
[[manifold with boundary|with boundary]] $\partial \Sigma_{tra} = \Sigma_{in} \sqcup \Sigma_{out}$
$$
\array{
&& \Sigma_{tra}
\\
& {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrm{out}}
\\
\Sigma_{in}
&& &&
\Sigma_{out}
}
$$
then the [[pullback of differential forms|pullback]] of the
two [[presymplectic forms]] (eq:TransgressionOfPresymplecticCurrentToCauchySurface)
on the incoming and outgoing [[spaces of field histories]], respectively, differ by the
BV-differential of the transgression of the BV-presymplectic current:
$$
\left( (-)\vert_{in}\right)^\ast\left( \omega_{in}\right)
\;-\;
\left( (-)\vert_{out} \right)^\ast \left( \omega_{out} \right)
=
\tau_{\mathbb{D} \times \Sigma_{tra}} ( s \Omega_{BV} )
\phantom{AAAA}
\in
\Omega^2
\left(
\Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0}
\right)
\,.
$$
This [[homological resolution]] of the [[Lagrangian correspondence]] that exhibits the "covariance" of the
[[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) is known as the _BV-BFV relation_
([Cattaneo-Mnev-Reshetikhin 12 (9)](BV-BRST+formalism#CattaneoMnevReshetikhin12)).
=--
+-- {: .proof}
###### Proof
For the first statement we compute as follows:
$$
\begin{aligned}
s \Omega_{BV}
& =
- \delta (s \phi^{\ddagger}_a) \delta \phi^a \wedge dvol_{\Sigma}
\\
& =
- \delta \frac{\delta_{EL}L }{\delta \phi^a} \delta \phi^a dvol_{\Sigma}
\\
& =
- \delta \delta_{EL}\mathbf{L}
\\
& =
d \Omega_{BFV}
\,,
\end{aligned}
$$
where the first steps simply unwind the definitions, and where the last step is prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}.
With this the second statement follows by immediate generalization of
the proof of prop. \ref{CovariantPhaseSpace}.
=--
+-- {: .num_example #DerivedPresymplecticCurrentOfRealScalarField}
###### Example
**(derived [[presymplectic current]] of [[real scalar field]])**
Consider a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
without any non-trivial implicit [[infinitesimal gauge transformations]] (def. \ref{ImplicitInfinitesimalGaugeSymmetry});
for instance the [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.
Inside its [[local BV-complex]] (def. \ref{BVComplexOfOrdinaryLagrangianDensity}) we may form the linear combination of
1. the [[presymplectic current]] $\Omega_{BFV}$ (example \ref{FreeScalarFieldEOM})
1. the BF-presymplectic current $\Omega_{BV}$ (example \ref{BVPresymplecticCurrent}).
This yields a vertical 2-form
$$
\Omega \;\coloneqq\; \Omega_{BV} + \Omega_{BFV} \;\; \in \Omega^{p,2}_\Sigma(E)\vert_{\mathcal{E}_{BV}}
$$
which might be called the _derived presymplectic current_.
Similarly we may form the linear combination of
1. the presymplectic potential current $\Theta_{BFV}$ (eq:dLDecomposition)
1. the BF-presymplectic potential current $\Theta_{BV}$ (eq:BVPresymplecticPotential)
1. the [[Lagrangian density]] $\mathbf{L}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
hence
$$
\Theta \;\coloneqq\; \Theta_{BV} + \underset{Lepage}{\underbrace{ \Theta_{BFV} + \mathbf{L} }}
$$
(where the sum of the two terms on the right is the [[Lepage form]] (eq:TheLepage)).
This might be called the _derived presymplectic potental current_.
We then have that
$$
(\delta + (d-s))\Omega \;=\; 0
$$
and in fact
$$
(\delta + (d-s))\Theta \;=\; \Omega
\,.
$$
=--
+-- {: .proof}
###### Proof
Of course the first statement follows from the second, but in fact the two contributions of the first statement
even vanish separately:
$$
\delta \Omega = 0
\,,
\phantom{AAAA}
(d-s)\Omega = 0
\,.
$$
The statement on the left is immediate from the definitions, since $\Omega = \delta \Theta$.
For the statement on the right we compute
$$
\begin{aligned}
(d - s) (\Omega_{BV} + \Omega_{BFV})
& =
\underset{= 0}{\underbrace{d \Omega_{BFV}
-
\underset{ = 0 }{\underbrace{ s \Omega_{BV}}} }}
+
\underset{ = 0}{\underbrace{ d \Omega_{BV} - s \Omega_{BFV} }}
\\
& = 0
\end{aligned}
$$
Here the first term vanishes via the local BV-BFV relation (prop. \ref{ResolutionOfCovariantPhaseSpaceCorrespondence})
while the other two terms vanish simply by degree reasons.
Similarly for the second statement we compute as follows:
$$
\begin{aligned}
(\delta + (d - s) ) \Theta
& =
\underset{ = \Omega_{BV} + \Omega_{BFV}}{\underbrace{ \delta (\Theta_{BV} + \Theta_{BFV}) }}
+
\underset{ = \delta \mathbf{L}}{\underbrace{\mathbf{d} \mathbf{L}}}
+
\underset{ = 0 }{\underbrace{ (d-s) \mathbf{L} }}
+
(d-s)(\Theta_{BV} + \Theta_{BFV})
\\
& =
\Omega_{BV} + \Omega_{BFV}
+
\delta \mathbf{L}
+
\underset{ = 0}{\underbrace{d \Theta_{BV}}}
-
\underset{ = \delta_{EL} \mathbf{L} }{\underbrace{ s \Theta_{BV}}}
+
\underset{ = \delta_{EL}\mathbf{L} - \delta \mathbf{L} }{\underbrace{ d \Theta_{BFV} } }
-
\underset{ = 0 }{\underbrace{ s \Theta_{BFV} }}
\\
& =
\Omega_{BV} + \Omega_{BFV}
\end{aligned}
\,.
$$
Here the direct vanishing of various terms is again by simple degree reasons, and
otherwise we used the definition of $\Omega$ and, crucially, the variational identity
$\delta \mathbf{L} = \delta_{EL}\mathbf{L} - d \Theta_{BFV}$ (eq:dLDecomposition).
=--
$\,$
**Hamiltonian local observables**
{#HamiltonianLocalObservablesOnACauchySurface}
We have defined the _[[local observables]]_ (def. \ref{LocalObservables}) as
the [[transgression of variational differential forms|transgressions]] of horizontal $p+1$-forms
(with compact spacetime support) to the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ over
all of [[spacetime]] $\Sigma$. More
explicitly, these could be called the _spacetime local observables_.
But with every choice of [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) comes another notion
of local observables: those that are [[transgression of variational differential forms|transgressions]]
of horizontal $p$-forms (instead of $p+1$-forms) to the [[on-shell]]
[[space of field histories]] restricted to the [[infinitesimal neighbourhood]] of that Cauchy surface (def. \ref{FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}): $\Gamma_{\Sigma_p}(E)_{\delta_{EL} \mathbf{L} = 0}$.
These are _spatially local observables_, with respect to the given choice of [[Cauchy surface]].
Among these spatially local observables are the _Hamiltonian local observables_ (def. \ref{HamiltonianLocalObservables} below)
which are [[transgression of variational differential forms|transgressions]] specifically of the
[[Hamiltonian differential forms|Hamiltonian forms]] (def. \ref{HamiltonianForms}).
These inherit a transgression of the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (prop. \ref{LocalPoissonBracket})
to a [[Poisson bracket]] on Hamiltonian local observables (def. \ref{PoissonBracketOnHamiltonianLocalObservables} below).
This is known as the _[[Peierls bracket]]_ (example \ref{PoissonBracketForRealScalarField} below).
+-- {: .num_defn #HamiltonianLocalObservables}
###### Definition
**(Hamiltonian local observables)**
Let $(E, \mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Consider a [[local observable]] (def. \ref{LocalObservables})
$$
\tau_\Sigma(A)
\;\colon\;
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
\longrightarrow
\mathbb{C}
\,,
$$
hence the [[transgression of variational differential forms|transgression]] of a variational horizontal
$p+1$-form $A \in \Omega^{p+1,0}_{\Sigma,cp}(E)$ of compact spacetime support.
Given a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface})
we say that $\tau_\Sigma (A)$ is _[[Hamiltonian]]_ if it is also the transgression of a
[[Hamiltonian differential form]] (def. \ref{HamiltonianForms}), hence if there exists
$$
(H,v) \in \Omega^{p,0}_{\Sigma, Ham}(E)
$$
whose transgression over the Cauchy surface $\Sigma_p$ equals the transgression of $A$ over all of
spacetime $\Sigma$, under the isomorphism (eq:CauchySurfaceIsomorphismOnHistorySpace)
$$
\array{
\Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0 }
&& \underoverset{\simeq}{(-)\vert_{N_\Sigma \Sigma_p}}{\longrightarrow} &&
\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}
\\
& {}_{\mathllap{\tau_\Sigma}(A)}\searrow && \swarrow_{\mathrlap{ \tau_{\Sigma_p}(H) }}
\\
&& \mathbf{\Omega}^2
}
$$
=--
Beware that the [[local observable]] $\tau_{\Sigma_p}(H)$ defined by a [[Hamiltonian differential form]]
$H \in \Omega^{p,0}_{\Sigma,Ham}(E)$ as in def. \ref{HamiltonianLocalObservables} does in general depend
not just on the choice of $H$, but also on the choice $\Sigma_p$ of the Cauchy surface.
The exception are those Hamiltonian forms which are _[[conserved currents]]_:
+-- {: .num_prop #ConservedCharge}
###### Proposition
**([[conserved charges]] -- [[transgression of variational differential forms|transgression]] of [[conserved currents]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
If a [[Hamiltonian differential form]] $J \in \Omega^{p,0}_{\Sigma,Ham}(E)$ (def. \ref{HamiltonianForms}) happens to be a
[[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}) in that its [[total derivative|total spacetime derivative]]
vanishes [[on-shell]]
$$
d J \vert_{\mathcal{E}} \;= \; 0
$$
then the induced Hamiltonian [[local observable]] $\tau_{\Sigma_p}(J)$ (def. \ref{HamiltonianLocalObservables})
is independent of the choice of [[Cauchy surface]] $\Sigma_p$ (def \ref{CauchySurface})
in that for $\Sigma_p, \Sigma'_p \hookrightarrow \Sigma$ any two Cauchy surfaces which are [[cobordism|cobordant]],
then
$$
\tau_{\Sigma_p}(J) = \tau_{\Sigma'_p}(J)
\,.
$$
The resulting [[constant function|constant]] is called the
_[[conserved charge]]_ of the conserved current, traditionally denoted
$$
Q \;\coloneqq\; \tau_{\Sigma_p}(J)
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition the [[transgression of variational differential forms|transgression]] of $d J$
vanishes on the [[on-shell]] [[space of field histories]]. Therefore the result
is given by [[Stokes' theorem]] (prop. \ref{StokesTheorem}).
=--
+-- {: .num_defn #PoissonBracketOnHamiltonianLocalObservables}
###### Definition
**([[Poisson bracket]] of [[Hamiltonian differential form|Hamiltonian]] [[local observables]] on [[covariant phase space]])**
Let $(E, \mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) where the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}).
We say that the _[[Poisson bracket]]_ on Hamiltonian local observables (def. \ref{HamiltonianLocalObservables})
is the [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
of the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (def. \ref{LocalPoissonBracket})
of the corresponding [[Hamiltonian differential forms]] (def. \ref{LocalPoissonBracket})
to the [[covariant phase space]] (def. \ref{CovariantPhaseSpace}).
Explicitly: for $\Sigma_p \hookrightarrow \Sigma$ a choice of [[Cauchy surface]] (def. \ref{CauchySurface})
then the Poisson bracket between two local Hamiltonian observables
$\tau_{\Sigma_p}((H_i, v_i))$ is
$$
\label{PoissonBracketTransgressedToCauchySurface}
\left\{
\tau_{\Sigma_p}((H_1, v_1))
\,,\,
\tau_{\Sigma_p}( (H_2, v_2) )
\right\}
\;\coloneqq\;
\tau_{\Sigma_p}( \, \{ (H_1, v_1), (H_2, v_2) \} \, )
\,,
$$
where on the right we have the transgression of the [[Poisson bracket Lie n-algebra|local Poisson bracket]]
$\{(H_1, v_1), (H_2, v_2)\}$ of [[Hamiltonian differential forms]] on the [[jet bundle]] from prop. \ref{LocalPoissonBracket}.
=--
+-- {: .proof}
###### Proof
We need to see that equation (eq:PoissonBracketTransgressedToCauchySurface) is well defined,
in that it does not depend on the choice of Hamiltonian form $(H_i, v_i)$
representing the local Hamiltonian observable $\tau_{\Sigma_p}(H_i)$.
It is clear that all the transgressions involved depend only on the restriction
of the Hamiltonian forms to the pullback of the jet bundle to
the [[infinitesimal neighbourhood]] $N_\Sigma \Sigma_p$.
Moreover, the Poisson bracket on the jet bundle (eq:LocalPoissonLieBracket)
clearly respects this restriction.
If a Hamiltonian differential form $H$ is in the [[kernel]] of the transgression map
relative to $\Sigma_p$, in that for every smooth collection $\Phi_{(-)} \colon U \to \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ of field histories (according to def. \ref{DifferentialFormsOnDiffeologicalSpaces}) we have (by def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
$$
\int_{\Sigma_p} j^\infty_\Sigma(\Phi_{(-)})^\ast H \;= \;0
\;\;\;
\in \Omega^p(U)
$$
then the fact that the _[[kernel of integration is the exact differential forms]]_ says that
$j^\infty_\Sigma(\Phi_{(-)})^\ast H \in \Omega^p(U \times \Sigma)$ is $d_\Sigma$-[[exact differential form|exact]] and hence in particular
$d_\Sigma$-[[closed differential form|closed]] for all $\Phi_{(-)}$:
$$
d_\Sigma j^\infty(\Phi_{(-)})^\ast H \;=\; 0
\,.
$$
By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} this means that
$$
j^\infty(\Phi_{(-)})^\ast ( d H ) \;= \; 0
$$
for all $\Phi_{(-)}$. Since $H \in \Omega^{p,0}_\Sigma(E)$ is horizontal, the same
proposition (see also example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}) implies that
in fact $H$ is horizontally closed:
$$
d H \;=\; 0
\,.
$$
Now since the field bundle $E \overset{fb}{\to} \Sigma$ is [[trivial bundle|trivial]] by assumption, prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact}
applies and says that this horizontally closed form on the jet bundle is in fact horizontally exact.
In conclusion this shows that the [[kernel]] of the [[transgression of variational differential forms|transgression]]
map $\tau_{\Sigma_p} \;\colon\; \Omega^{p,0}_\Sigma(E) \to C^\infty\left( \Gamma_{\Sigma_p}(E)\right)$
is precisely the space of horizontally exact horizontal $p$-forms.
Therefore the claim
now follows with the statement that horizontally exact [[Hamiltonian differential forms]] constitute a
[[Lie ideal]] for the local Poisson bracket on the jet bundle; this is lemma \ref{HorizontallyExactFormsDropOutOfLocalLieBracket}.
=--
+-- {: .num_example #PoissonBracketForRealScalarField}
###### Example
**([[Poisson bracket]] of the [[real scalar field]])**
Consider the [[Lagrangian field theory]] of the [[free field|free]] [[scalar field]]
(example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}),
and consider the [[Cauchy surface]] defined by $x^0 = 0$.
By example \ref{LocalPoissonBracketForRealScalarField} the [[Poisson bracket Lie n-algebra|local Poisson bracket]]
of the [[Hamiltonian forms]]
$$
Q \coloneqq \phi \iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E)
$$
and
$$
P \coloneqq \eta^{\mu \nu} \phi_{,\mu} \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E)
\,.
$$
is
$$
\{Q,P\} = \iota_{v_Q} \iota_{v_P} \omega = \iota_{\partial_0} dvol_\Sigma
\,.
$$
Upon [[transgression of variational differential forms|transgression]] according to def. \ref{PoissonBracketOnHamiltonianLocalObservables}
this yields the following [[Poisson bracket]]
$$
\left\{
\int_{\Sigma_p} b_1(\vec x) \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(x)
d^p \vec x
\;,\;
\int_{\Sigma_p} b_2(\vec x) \partial_0 \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x)
\right\}
\;=\;
\int_{\Sigma_p} b_1(\vec x) b_2(\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x)
d^p \vec x
\,,
$$
where
$$
\mathbf{\Phi}(x), \partial_0 \mathbf{\Phi}(x)
\;:\;
PhaseSpace(\Sigma_p^t) \to \mathbb{R}
$$
denote the point-evaluation observables (example \ref{PointEvaluationObservables}), which act on a field history $\Phi \in \Gamma_\Sigma(E) = C^\infty(\Sigma)$ as
$$
\mathbf{\Phi}(x) \;\colon\; \Phi \mapsto \Phi(x)
\phantom{AAAAAAAA}
\partial_0 \mathbf{\Phi}(x) \;\colon\; \Phi \mapsto \partial_0 \Phi(x)
\,.
$$
Notice that these point-evaluation functions themselves do not arise
as the transgression of elements in $\Omega^{p,0}(E)$; only their smearings such as $\int_{\Sigma_p} b_1 \phi dvol_{\Sigma_p}$ do.
Nevertheless we may express the above Poisson bracket conveniently via the [[integral kernel]]
$$
\label{PoissonBracketOfScalarFieldPointEvaluationOnMinkowskiSpacetime}
\left\{
\mathbf{\Phi}(t,\vec x), \partial_0\mathbf{\Phi}(t,\vec y)
\right\}
\;=\;
\delta(\vec x - \vec y)
\,.
$$
=--
+-- {: .num_prop #PoissonBracketForDiracField}
###### Proposition
**([[super Lie algebra|super]]-[[Poisson bracket]] of the [[Dirac field]])**
Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{LagrangianDensityForDiracField}) with [[field bundle]] the odd-shifted [[spinor bundle]] $E = \Sigma \times S_{odd}$ (example \ref{DiracFieldBundle}) and with
$$
\theta \Psi_\alpha(x)
\;\colon\;
\mathbb{R}^{0\vert 1}
\longrightarrow
\left[
\Gamma_\Sigma(\Sigma \times S_{odd})_{\delta_{EL}\mathbf{L} = 0},
\mathbb{C}
\right]
$$
the corresponding odd-graded point-evaluation observable (example \ref{PointEvaluationObservables}).
Then consider the [[Cauchy surfaces]] in [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) given by $x^0 = t$ for $t \in \mathbb{R}$. Under [[transgression of variational differential forms|transgression]] to this Cauchy surface via def. \ref{PoissonBracketOnHamiltonianLocalObservables},
the [[Poisson bracket Lie n-algebra|local Poisson bracket]], which by example \ref{LocalPoissonBracketForDiracField} is given by the [[super Lie algebra|super Lie bracket]]
$$
\left\{
\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
\,,\,
\left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
\right\}
\;=\;
\left(\gamma^\mu\right)_\alpha{}^{\beta} \, \iota_{\partial_\mu} dvol_\Sigma
\,,
$$
has [[integral kernel]]
$$
\left\{
\psi_\alpha(t,\vec x)
,
\overline{\psi}^\beta(t,\vec y)
\right\}
\;=\;
(\gamma^0)_{\alpha}{}^\beta \delta(\vec y - \vec x)
\,.
$$
=--
$\,$
This concludes our discussion of the [[phase space]] and the [[Poisson-Peierls bracket]] for well behaved [[Lagrangian field theories]].
In the [next chapter](#Propagators) we discuss in detail the [[integral kernels]] corresponding to the [[Poisson-Peierls bracket]]
for key classes of examples. These are the _[[propagators]]_ of the theory.
|
A first idea of quantum field theory -- Propagators | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Propagators | ## Propagators
{#Propagators}
In this chapter we discuss the following topics:
* _Background_
* _[Fourier analysis and Plane wave modes](#FourierAnalysis)_
* _[Microlocal analysis and UV-Divergences](#MicrolocalAnalysisAndUltravioletDivergence)_
* _[Cauchy principal values](#CauchyPrincipalValues)_
* _Propagators for the free scalar field on Minkowski spacetime_
* _[advanced and regarded propagators](#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime)_
* _[causal propagator](#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)_
* _[Wightman propagator](#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime)_
* _[Feynman propagator](#FeynmanPropagator)_
* _[singular support and wave front sets](#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime)_
* _[Propagators for the Dirac field on Minkowski spacetime](#DiracEquationOnMinkowskiSpacetimePropagators)_
$\,$
In the [previous chapter](#PhaseSpace) we have seen the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) of sufficiently nice [[Lagrangian field theories]], which is the [[on-shell]] [[space of field histories]] equipped with the [[presymplectic form]] [[transgression of variational differential forms|transgressed]] from the [[presymplectic current]]
of the theory; and we have seen that in good cases this induces a bilinear pairing on sufficiently well-behaved [[observables]],
called the _[[Poisson bracket]]_ (def. \ref{PoissonBracketOnHamiltonianLocalObservables}), which reflects the [[infinitesimal symmetries]] of the [[presymplectic current]]. This [[Poisson bracket]] is of central importance for passing to actual [[quantum field theory]],
since, as we will discuss in _[Quantization](#Quantization)_ below, it is the [[infinitesimal]] approximation to the [[quantization]] of a [[Lagrangian field theory]].
We have moreover seen that the [[Poisson bracket]] on the [[covariant phase space]] of a
[[free field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]]
-- the [[Peierls-Poisson bracket]] --
is determined by the [[integral kernel]] of the _[[causal Green function]]_ (prop. \ref{PPeierlsBracket}).
Under the identification of linear on-shell observables with off-shell observables
that are [[generalized solution of a PDE|generalized solutions]] to the [[equations of motion]] (theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}) the convolution with this [[integral kernel]]
may be understood as _propagating_ the values of an off-shell observable through [[spacetime]],
such as to then compare it with any other observable at any spacetime point (prop. \ref{PPeierlsBracket}).
Therefore the [[integral kernel]] of the [[causal Green function]] is also called the _[[causal propagator]]_ (prop. \ref{GreenFunctionsAreContinuous}).
This means that for [[Green hyperbolic differential equation|Green hyperbolic]] [[free field theory|free]] [[Lagrangian field theory]]
the [[Poisson bracket]], and hence the infinitesimal [[quantization]] of the theory, is all encoded in the
[[causal propagator]]. Therefore here we analyze the [[causal propagator]], as well as its variant [[propagators]], in detail.
The main tool for these computations is _[[Fourier transform|Fourier analysis]]_ (reviewed [below](#FourierAnalysis)) by which [[field histories]], [[observables]] and [[propagators]] on [[Minkowski spacetime]] are decomposed as [[superpositions]] of [[plane waves]] of various [[frequencies]], [[wave lengths]] and [[wave vector]]-[[direction of a vector|direction]]. Using this, all [[propagators]] are exhibited as those [[superpositions]]
of [[plane waves]] which satisfy the [[dispersion relation]] of the given [[equation of motion]], relating [[plane wave]] [[frequency]]
to [[wave length]].
This way the [[causal propagator]] is naturally decomposed into its contribution from [[positive real number|positive]]
and from [[negative real number|negative]] [[frequencies]]. The positive frequency part of the [[causal propagator]] is called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below). It turns out (prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below) that this is equivalently the [[sum]] of the [[causal propagator]], which itself is skew-symmetric (cor. \ref{CausalPropagatorIsSkewSymmetric} below), with a symmetric component, or equivalently that the [[causal propagator]] is the skew-symmetrization of the [[Wightman propagator]].
After [[quantization]] of [[free field theory]] discussed [further below](#FreeQuantumFields), we will see that the
Wightman propagator is equivalently the [[2-point function|correlation function]] between two point-evaluation field observables (example \ref{PointEvaluationObservables}) in a _[[vacuum state]]_ of the field theory (a [[state]] in the sense of def. \ref{States}).
Moreover, by def. \ref{AdvancedAndRetardedGreenFunctions} the [[causal propagator]] also decomposes into its
contributions with [[future]] and [[past]] [[support of a distribution|support]], given by the _difference_ between
the [[advanced and retarded propagators]]. These we analyze first, starting with prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} below.
Combining these two decompositions of the [[causal propagator]] (positive/negative frequency as well as positive/negative time) yields one more
propagator, the _[[Feynman propagator]]_ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below).
We will see [below](#Quantization) that the [[quantization]] of a [[free field theory]] is given by a
"[[star product]]" (on [[observables]]) which is given by "[[exponential|exponentiating]]" these [[propagators]].
For that to make sense, certain pointwise products of these [[propagators]], regarded as [[generalized functions]] (prop. \ref{DistributionsAreGeneralizedFunctions}) need to exist. But since the [[propagators]] are [[distributions]] with
[[singularities]], the existence of these products requires that certain potential "[[UV divergences]]" in their [[Fourier transform of distributions|Fourier transforms]] (remark \ref{UltravioletDivergencesFromPaleyWiener} below) are absent ("[[Hörmander's criterion]]", prop. \ref{HoermanderCriterionForProductOfDistributions} below). These [[UV divergences]] are captured by what what is called the _[[wave front set]]_ (def. \ref{WaveFrontSet} below).
The study of [[UV divergences]] of [[distributions]] via their [[wave front sets]] is called _[[microlocal analysis]]_
and provides powerful tools for the understanding of [[quantum field theory]]. In particular the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem}) shows that for [[distributional solutions to a PDE|distributional solutions]] (def. \ref{DistributionalDerivatives}) of [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]], such as the [[propagators]],
their [[singular support]] propagates itself through [[spacetime]] along the [[wave front set]].
Using this theorem we work out the [[wave front sets]] of the [[propagators]] (prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} below).
Via [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) this computation will serve to show
why upon [[quantization]] the [[Wightman propagator]] replaces the [[causal propagator]] in the construction of the [[Wick algebra]]
of [[quantum observables]] of the [[free field theory]] (discussed below in _[Free quantum fields](#FreeQuantumFields)_) and the [[Feynman propagator]] similarly controls the [[quantum observables]] of the [[interaction|interacting]] [[quantum field theory]]
(below in _[Feynman diagrams](#FeynmanDiagrams)_).
$\,$
The following table summarizes the structure of the system of propagators.
(The column "as vacuum expectation value of field operators" will be discussed further below in _[Free quantum fields](#FreeQuantumFields)_).
$\,$
[[!include propagators - table]]
$\,$
$\,$
**[[Fourier transform|Fourier analysis]] and [[plane wave]] modes**
{#FourierAnalysis}
By definition, the [[equations of motion]] of [[free field theories]] (def. \ref{FreeFieldTheory}) are [[linear partial differential equations]]
and hence lend themselves to _[[harmonic analysis]]_, where all [[field histories]] are decomposed into
[[superpositions]] of [[plane waves]] via _[[Fourier transform]]_. Here we briefly survey the relevant definitions
and facts of [[Fourier transform|Fourier analysis]].
In [[formal duality]] to the [[harmonic analysis]] of the [[field histories]] themselves,
also the linear [[observables]] (def. \ref{LinearObservables}) on the [[space of field histories]], hence
the _[[distribution|distributional]] [[generalized functions]]_ (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions})
are subject to _[[Fourier transform of distributions]]_ (def. \ref{FourierTransformOnTemperedDistributions} below).
Throughout, let $n \in \mathbb{N}$ and consider the [[Cartesian space]] $\mathbb{R}^n$ of [[dimension]] $n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}). In the application to [[field theory]], $n = p + 1$ is the
[[dimension]] of [[spacetime]] and $\mathbb{R}^n$ is either [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (def. \ref{MinkowskiSpacetime})
or its [[dual vector space]], thought of as the space of _[[wave vectors]]_ (def. \ref{PlaneWaves} below). For
$x = (x^\mu) \in \mathbb{R}^{p,1}$ and $k = (k_\mu) \in (\mathbb{R}^(p,1))^\ast$ we write
$$
x \cdot k \;=\; x^\mu k_\mu
$$
for the canonical pairing.
+-- {: .num_defn #PlaneWaves}
###### Definition
**([[plane wave]])**
A _[[plane wave]]_ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (def. \ref{MinkowskiSpacetime}) is a
[[smooth function]] with values in the [[complex numbers]] given by
$$
\array{
\mathbb{R}^{p,1} &\longrightarrow& \mathbb{C}
\\
(x^\mu) &\mapsto& e^{i k_\mu x^\mu}
}
$$
for $k = (k_\mu) \in (\mathbb{R}^{p,1})^\ast$ a [[covector]], called the _[[wave vector]]_ of the [[plane wave]].
We use the following terminology:
[[!include plane waves -- table]]
=--
+-- {: .num_defn #SchwartzSpace}
###### Definition
**([[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]])**
A [[complex number|complex]]-valued [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ is said to have _[[rapidly decreasing function|rapidly decreasing]] [[partial derivatives]]_ if for all $\alpha,\beta \in \mathbb{N}^{n}$ we have
$$
\underset{x \in \mathbb{R}^n}{sup} {\vert x^\beta \partial^\alpha f(x) \vert}
\;\lt\; \infty
\,.
$$
Write
$$
\mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)
$$
for the sub-[[vector space]] on the functions with rapidly decreasing partial derivatives regarded as a [[topological vector space]] for the [[Fréchet space]] [[structure]] induced by the [[seminorms]]
$$
p_{\alpha, \beta}(f) \coloneqq \underset{x \in \mathbb{R}^n}{sup} {\vert x^\beta \partial^\alpha f(x) \vert}
\,.
$$
This is also called the _[[Schwartz space]]_.
=--
(e.g. [Hörmander 90, def. 7.1.2](Fourier+transform#Hoermander90))
+-- {: .num_example #CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}
###### Example
**([[compactly supported function|compactly supported]] [[smooth function]] are [[functions with rapidly decreasing partial derivatives]])**
Every [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R}^n)$ has rapidly decreasing partial derivatives (def. \ref{SchwartzSpace}):
$$
C^\infty(\mathbb{R}^n)
\hookrightarrow
\mathcal{S}(\mathbb{R}^n)
\,.
$$
=--
+-- {: .num_prop #ConvolutionProductOnSchwartzSpace}
###### Proposition
**(pointwise product and [[convolution product]] on [[Schwartz space]])**
The [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is closed under the following operatios on smooth functions $f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)$
1. pointwise product:
$$
(f \cdot g)(x) \coloneqq f(x) \cdot g(x)
$$
1. [[convolution product]]:
$$
(f \star g)(x) \coloneqq \underset{y \in \mathbb{R}^n}{\int} f(y)\cdot g(x-y) \, dvol(y)
\,.
$$
=--
+-- {: .proof}
###### Proof
By the [[product law]] of [[differentiation]].
=--
+-- {: .num_prop #RapidlyDecreasingFunctionsAreIntegrable}
###### Proposition
**([[rapidly decreasing functions]] are [[integrable functions|integrable]])**
Every [[rapidly decreasing function]] $f \colon \mathbb{R}^n \to \mathbb{R}$ (def. \ref{SchwartzSpace}) is an [[integrable function]] in that its [[integral]] exists:
$$
\underset{x \in \mathbb{R}^n}{\int} f(x) \, d^n x
\;\lt\;
\infty
$$
In fact for each $\alpha \in \mathbb{N}^n$ the product of $f$ with the $\alpha$-power of the [[coordinate functions]] exists:
$$
\underset{x \in \mathbb{R}^n}{\int}
x^\alpha f(x)\, d^n x
\;\lt\;
\infty
\,.
$$
=--
+-- {: .num_defn #FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}
###### Definition
**([[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**
The _[[Fourier transform]]_ is the [[continuous linear functional]]
$$
\widehat{(-)}
\;\colon\;
\mathcal{S}(\mathbb{R}^n)
\longrightarrow
\mathcal{S}(\mathbb{R}^n)
$$
on the [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}), which is given by [[integration]] against [[plane wave]] functions (def. \ref{PlaneWaves})
$$
x \mapsto e^{- i k \cdot x}
$$
times the standard [[volume form]] $d^n x$:
$$
\label{IntegralExpressionForFourierTransform}
\hat f(k)
\;\colon\;
\int_{x \in \mathbb{R}^n}
e^{- i \, k \cdot x} f(x) \, d^n x
\,.
$$
Here the argument $k \in \mathbb{R}^n$ of the Fourier transform is also called the _[[wave vector]]_.
=--
(e.g. [Hörmander, lemma 7.1.3](Fourier+transform#Hoermander90))
+-- {: .num_prop #FourierInversion}
###### Proposition
**([[Fourier inversion theorem]])**
The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is an [[isomorphism]], with [[inverse function]] the _[[inverse Fourier transform]]_
$$
\widecheck {(-)}
\;\colon\;
\mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n)
$$
given by
$$
\widecheck g (x)
\;\coloneqq\;
\underset{k \in \mathbb{R}^n}{\int}
g(k) e^{i k \cdot x}
\, \frac{d^n k}{(2\pi)^n}
\,.
$$
Hence in the language of [[harmonic analysis]] the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the [[superposition]] of [[plane waves]] (def. \ref{PlaneWaves}) in which the plane wave with [[wave vector]] $k\in \mathbb{R}^n$ appears with [[amplitude]] $g(k)$.
=--
(e.g. [Hörmander, theorem 7.1.5](Fourier+transform#Hoermander90))
+-- {: .num_prop #BasicPropertiesOfFourierTransformOverCartesianSpaces}
###### Proposition
**(basic properties of the [[Fourier transform]])**
The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) satisfies the following properties, for all $f,g \in \mathcal{S}(\mathbb{R}^n)$:
1. (interchanging [[coordinate]] [[multiplication]] with [[partial derivatives]])
$$
\label{FourierTransformInterchangesCoordinateProductWithDerivative}
\widehat{ x^a f } = + i \partial_a \widehat f
\phantom{AAAAA}
\widehat{ - i\partial_a f} = k_a \widehat f
$$
1. (interchanging pointwise multiplication with [[convolution product]], remark \ref{ConvolutionProductOnSchwartzSpace}):
$$
\label{FourierTransformInterchangesPointwiseProductWithConvolution}
\widehat {(f \star g)} = \widehat{f} \cdot \widehat{g}
\phantom{AAAA}
\widehat{ f \cdot g } = (2\pi)^{-n} \widehat{f} \star \widehat{g}
$$
1. ([[unitary operator|unitarity]], [[Parseval's theorem]])
$$
\underset{x \in \mathbb{R}^n}{\int} f(x) g^\ast(x)\, d^n x
\;=\;
\underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \widehat{g}^\ast(k) \, d^n k
$$
1. $$
\label{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor}
\underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \cdot g(k) \, d^n k
\;=\;
\underset{x \in \mathbb{R}^n}{\int} f(x) \cdot \widehat{g}(x) \, d^n x
$$
=--
(e.g [Hörmander 90, lemma 7.1.3, theorem 7.1.6](Fourier+transform#Hoermander90))
The [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace})
serves the purpose to support the [[Fourier transform]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace})
together with its inverse (prop. \ref{FourierInversion}), but for many applications
one needs to apply the Fourier transform to more general functions, and in fact to _[[generalized functions]]_
in the sense of [[distributions]] (via [this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).
But with the [[Schwartz space]] in hand, this generalization is readily obtained by [[formal duality]]:
+-- {: .num_defn #TemperedDistribution}
###### Definition
**([[tempered distribution]])**
A _[[tempered distribution]]_ is a [[continuous linear functional]]
$$
u \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathbb{C}
$$
on the [[Schwartz space]] (def. \ref{SchwartzSpace}) of [[functions with rapidly decaying partial derivatives]]. The [[vector space]] of all tempered distributions is canonically a [[topological vector space]] as the [[dual space]] to the [[Schwartz space]], denoted
$$
\mathcal{S}'(\mathbb{R}^n)
\;\coloneqq\;
\left(
\mathcal{S}(\mathbb{R}^n)
\right)^\ast
\,.
$$
=--
e.g. ([Hörmander 90, def. 7.1.7](Fourier+transform#Hoermander90))
+-- {: .num_example #SomeNonSingularTemperedDistributions}
###### Example
**(some [[non-singular distribution|non-singular]] [[tempered distributions]])**
Every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) induces a [[tempered distribution]] $u_f \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) by [[integration|integrating]] against it:
$$
u_f \;\colon\; g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x)\, d^n x
\,.
$$
This construction is a linear inclusion
$$
\mathcal{S}(\mathbb{R}^n) \overset{\text{dense}}{\hookrightarrow} \mathcal{S}'(\mathbb{R}^n)
$$
of the [[Schwartz space]] into its [[dual space]] of [[tempered distributions]]. This is a [[dense subspace]] inclusion.
In fact already the restriction of this inclusion to the [[compactly supported function|compactly supported]] [[smooth functions]] (example \ref{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}) is a [[dense subspace]] inclusion:
$$
C^\infty_{cp}(\mathbb{R}^n)
\overset{dense}{\hookrightarrow}
\mathcal{S}'(\mathbb{R}^n)
\,.
$$
This means that every [[tempered distribution]] is a [[limit of a sequence|limit]] of a [[sequence]]
of ordinary [[functions with rapidly decreasing partial derivatives]], and in fact even the [[limit of a sequence|limit]]
of a [[sequence]] of [[compactly supported function|compactly supported]] [[smooth functions]] ([[bump functions]]).
It is in this sense that [[tempered distributions]] are "generalized functions".
=--
(e.g. [Hörmander 90, lemma 7.1.8](Fourier+transform#Hoermander90))
+-- {: .num_example #CompactlySupportedDistibutionsAreTemperedDistributions}
###### Example
**([[compactly supported distributions]] are [[tempered distributions]])**
Every [[compactly supported distribution]] is a [[tempered distribution]] (def. \ref{TemperedDistribution}),
hence there is a [[linear map|linear]] [[injection|inclusion]]
$$
\mathcal{E}'(\mathbb{R}^n)
\hookrightarrow
\mathcal{S}'(\mathbb{R}^n)
\,.
$$
=--
+-- {: .num_example #DiracDeltaDistribution}
###### Example
**([[delta distribution]])**
Write
$$
\delta_0(-) \;\in\; \mathcal{E}'(\mathbb{R}^n)
$$
for the [[distribution]] given by point evaluation of functions at the origin of $\mathbb{R}^n$:
$$
\delta_0(-) \;\colon\; f \mapsto f(0)
\,.
$$
This is clearly a [[compactly supported distribution]]; hence a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.
We write just "$\delta(-)$" (without the subscript) for the corresponding [[generalized function]] (example \ref{SomeNonSingularTemperedDistributions}), so that
$$
\underset{x \in \mathbb{R}^n}{\int} \delta(x) f(x) \, d^n x
\;\coloneqq\;
f(0)
\,.
$$
=--
+-- {: .num_example #SquareIntegrableFunctionsInduceTemperedDistributions}
###### Example
**([[square integrable functions]] induce [[tempered distributions]])**
Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th [[Lebesgue space]], e.g. for $p = 2$ this means that $f$ is a [[square integrable function]]. Then the operation of [[integration]] against the [[measure]] $f dvol$
$$
g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x) \, d^n x
$$
is a [[tempered distribution]] (def. \ref{TemperedDistribution}).
=--
(e.g. [Hörmander 90, below lemma 7.1.8](Fourier+transform#Hoermander90))
Property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) of the ordinary [[Fourier transform]] on [[functions with rapidly decreasing partial derivatives]] motivates and justifies the fullowing generalization:
+-- {: .num_defn #FourierTransformOnTemperedDistributions}
###### Definition
**([[Fourier transform of distributions]] on [[tempered distributions]])**
The _[[Fourier transform of distributions]]_ of a [[tempered distribution]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is the [[tempered distribution]] $\widehat u$ defined on a smooth function $f \in \mathcal{S}(\mathbb{R}^n)$ in the [[Schwartz space]] (def. \ref{SchwartzSpace}) by
$$
\widehat{u}(f) \;\coloneqq\; u\left( \widehat f\right)
\,,
$$
where on the right $\widehat f \in \mathcal{S}(\mathbb{R}^n)$ is the [[Fourier transform]] of functions from def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}.
=--
(e.g. [Hörmander 90, def. 1.7.9](Fourier+transform#Hoermander90))
+-- {: .num_example #FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}
###### Example
**([[Fourier transform of distributions]] indeed generalizes [[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**
Let $u_f \in \mathcal{S}'(\mathbb{R}^n)$ be a [[non-singular distribution|non-singular]] [[tempered distribution]] induced, via example \ref{SomeNonSingularTemperedDistributions}, from a [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$.
Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is the [[non-singular distribution]] induced from the [[Fourier transform]] of $f$:
$$
\widehat{u_f} \;=\; u_{\hat f}
\,.
$$
=--
+-- {: .proof}
###### Proof
Let $g \in \mathcal{S}(\mathbb{R}^n)$. Then
$$
\begin{aligned}
\widehat{u_f}(g)
& \coloneqq
u_f\left( \widehat{g}\right)
\\
& =
\underset{x \in \mathbb{R}^n}{\int} f(x) \hat g(x)\, d^n x
\\
& =
\underset{x \in \mathbb{R}^n}{\int} \hat f(x) g(x) \, d^n x
\\
& =
u_{\hat f}(g)
\end{aligned}
$$
Here all equalities hold by definition, except for the third: this is property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.
=--
+-- {: .num_example #FourierTransformOfKleinGordonEquation}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of [[Klein-Gordon equation]] [[derivative of distributions|of]] [[distributions]])**
Let $\Delta \in \mathcal{S}'(\mathbb{R}^{p,1})$ be any [[tempered distribution]] (def. \ref{TemperedDistribution}) on [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and let
$P \coloneqq \eta^{\mu \nu} \frac{\partial}{\partial x^\mu}\frac{\partial}{\partial x^\nu} - \left( \tfrac{m c}{\hbar} \right)^2$
be the [[Klein-Gordon operator]] (eq:KleinGordonOperator). Then the [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of $P \Delta$ is, in [[generalized function]]-notation (remark \ref{LinearObservablesAsGeneralizedFunctions})given by
$$
\widehat {P \Delta}(k)
\;=\;
\left(
- \eta^{\mu \nu}k_\mu k_\nu
- \left( \tfrac{m c}{\hbar}\right)^2
\right)
\widehat(k)
\,.
$$
=--
+-- {: .proof}
###### Proof
Let $r \in \mathcal{S}(\mathbb{R}^n)$ be any [[function with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}).
Then
$$
\begin{aligned}
\widehat {P \Delta}(r)
& =
P \Delta(\widehat r)
\\
& =
\Delta(P^\ast \widehat r)
\\
& =
\Delta(P \widehat r)
\\
& =
\Delta\left( \left(-\eta^{\mu \nu}k_\mu k_\nu - \left( \tfrac{m c}{\hbar}\right)^2\right) \widehat{r} \right)
\end{aligned}
$$
Here the first step is def. \ref{FourierTransformOnTemperedDistributions}, the second is def. \ref{DistributionalDerivatives},
the third is example \ref{FormallySelfAdjointKleinGordonOperator}, while the last step is prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.
=--
+-- {: .num_example #CompactlySupportedDistributionFourierTransform}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of [[compactly supported distributions]])**
Under the identification of [[smooth functions]] of bounded growth with [[non-singular distributions|non-singular]] [[tempered distributions]] (example \ref{SomeNonSingularTemperedDistributions}), the [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of a [[tempered distribution]] that happens to be [[compactly supported distribution|compactly supported]] (example \ref{CompactlySupportedDistibutionsAreTemperedDistributions})
$$
u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$
is simply
$$
\widehat{u}(k) = u\left( e^{- i k \cdot (-)}\right)
\,.
$$
=--
([Hörmander 90, theorem 7.1.14](Fourier+transform#Hoermander90))
+-- {: .num_defn #FourierTransformOfDeltaDistribution}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of the [[delta-distribution]])**
The [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[delta distribution]] (def. \ref{DiracDeltaDistribution}), via example \ref{CompactlySupportedDistributionFourierTransform}, is the [[constant function]] on 1:
$$
\begin{aligned}
\widehat {\delta}(k)
& =
\underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- i k x} \, d x
\\
& =
1
\end{aligned}
$$
This implies by the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) that the [[delta distribution]] itself has equivalently the following expression as a [[generalized function]]
$$
\begin{aligned}
\delta(x)
& =
\widecheck{\widehat {\delta_0}}(x)
\\
& =
\underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \, \frac{d^n k}{ (2\pi)^n }
\end{aligned}
$$
in the sense that for every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) we have
$$
\begin{aligned}
f(x)
& =
\underset{y \in \mathbb{R}^n}{\int}
f(y) \delta(y-x) \, d^n y
\\
& =
\underset{y \in \mathbb{R}^n}{\int}
\underset{k \in \mathbb{R}^n}{\int}
f(y) e^{i k \cdot (y-x)}
\, \frac{d^n k}{(2\pi)^n} \, d^n y
\\
& =
\underset{k \in \mathbb{R}^n}{\int}
e^{- i k \cdot x}
\underset{= \widehat{f}(-k) }{
\underbrace{
\underset{y \in \mathbb{R}^n}{\int}
f(y) e^{i k \cdot y}
\, d^n y
}
}
\,\, \frac{d^n k}{(2\pi)^n}
\\
& =
+
\underset{k \in \mathbb{R}^n}{\int}
e^{i k \cdot x}
\underset{= \widehat{f}(k) }{
\underbrace{
\underset{y \in \mathbb{R}^n}{\int}
f(y) e^{- i k \cdot y}
\, d^n y
}
}
\,\, \frac{d^n k}{(2\pi)^n}
\\
& =
\widecheck{\widehat{f}}(x)
\end{aligned}
$$
which is the statement of the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).
(Here in the last step we used [[change of integration variables]] $k \mapsto -k$ which introduces one sign $(-1)^{n}$ for the new volume form,
but another sign $(-1)^n$ from the re-[[orientation]] of the integration domain. )
Equivalently, the above computation shows that the [[delta distribution]] is the [[neutral element]] for the
[[convolution product of distributions]].
=--
+-- {: .num_prop #PaleyWienerSchwartzTheorem}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]] I)**
Let $u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ be a [[compactly supported distribution]] regarded as a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is a [[non-singular distribution]] induced from a [[smooth function]] that grows at most exponentially.
=--
(e.g. [Hoermander 90, theorem 7.3.1](Paley-Wiener-Schwartz+theorem#Hoermander90))
+-- {: .num_prop #FourierInversionTheoremForDistributions}
###### Proposition
**([[Fourier inversion theorem]] for [[Fourier transform of distributions]])**
The operation of forming the [[Fourier transform of distributions]] $\widehat{u}$ (def. \ref{FourierTransformOnTemperedDistributions}) [[tempered distributions]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is an [[isomorphism]],
with [[inverse]] given by
$$
\widecheck{ u } \;\colon\; g \mapsto u\left( \widecheck{g}\right)
\,,
$$
where on the right $\widecheck{g}$ is the ordinary [[inverse Fourier transform]] of $g$ according to prop. \ref{FourierInversion}.
=--
+-- {: .proof}
###### Proof
By def. \ref{FourierTransformOnTemperedDistributions}
this follows immediately from the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).
=--
We have the following distributional generalization of the basic property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}:
+-- {: .num_prop #FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct}
###### Proposition
**([[Fourier transform of distributions]] interchanges [[convolution of distributions]] with pointwise product)**
Let
$$
u_1 \in \mathcal{S}'(\mathbb{R}^n)
$$
be a [[tempered distribution]] (def. \ref{TemperedDistribution}) and
$$
u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$
be a [[compactly supported distribution]], regarded as a [[tempered distribution]] via example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.
Observe here that the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) implies that the [[Fourier transform of distributions]] of $u_1$ is a [[non-singular distribution]]
$\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ is always defined.
Then the [[Fourier transform of distributions]] of the [[convolution product of distributions]] is the product of the [[Fourier transform of distributions]]:
$$
\widehat{u_1 \star u_2}
\;=\;
\widehat{u_1} \cdot \widehat{u_2}
\,.
$$
=--
(e.g. [Hörmander 90, theorem 7.1.15](Fourier+transform#Hoermander90))
+-- {: .num_remark #ProductOfDistributionsViaFourierTransformOfConvolution}
###### Remark
**([[product of distributions]] via [[Fourier transform of distributions]])**
Prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct} together with the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) suggests to _define_ the [[product of distributions]] $u_1 \cdot u_2$ for [[compactly supported distributions]] $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula
$$
\widehat{ u_1 \cdot u_2 }
\;\coloneqq\;
(2\pi)^n \widehat{u_1} \star \widehat{u_2}
$$
which would complete the generalization of of property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.
For this to make sense, the [[convolution product]] of the [[smooth functions]] on the right needs to exist, which is not guaranteed (prop. \ref{ConvolutionProductOnSchwartzSpace} does not apply here!). The condition that this exists is the [[Hörmander criterion]] on the _[[wave front set]]_ (def. \ref{WaveFrontSet}) of $u_1$ and $u_2$ (prop. \ref{HoermanderCriterionForProductOfDistributions} belwo). This we further discuss in _[Microlocal analysis and UV-Divergences](#MicrolocalAnalysisAndUltravioletDivergence)_ below.
=--
$\,$
**[[microlocal analysis]] and [[ultraviolet divergences]]**
{#MicrolocalAnalysisAndUltravioletDivergence}
A _[[distribution]]_ (def. \ref{LinearObservablesAreTheCompactlySupportedDistributions}) or _[[generalized function]]_ (prop. \ref{DistributionsAreGeneralizedFunctions}) is like a [[smooth function]] which may have "[[singularities]]", namely points at which it values or that of its [[derivative of a distribution|derivatives]] "become infinite". Conversely, [[smooth functions]] are the [[non-singular distributions]] (prop. \ref{DistributionsAreGeneralizedFunctions}). The collection of points around which a distribution is singular (i.e. not [[non-singular distribution|non-singular]]) is called its _[[singular support]]_ (def. \ref{SingularSupportOfADistribution} below).
The [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) decomposes a [[generalized function]] into the [[plane wave]] modes that it is made of (def. \ref{PlaneWaves}). The [[Paley-Wiener-Schwartz theorem]] (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} below) says that the singular nature of a [[compactly supported distribution]] may be read off from this [[Fourier transform|Fourier mode]] decomposition: Singularities correspond to large contributions by Fourier modes of high [[frequency]] and small [[wavelength]], hence to large "[[ultraviolet divergence|ultraviolet]]" (UV) contributions (remark \ref{UltravioletDivergencesFromPaleyWiener} below). Therefore the [[singular support]] of a distribution is the set of points around which the Fourier transform does not sufficiently decay "in the UV".
But since the [[Fourier transform]] is a function of the full [[wave vector]] of the [[plane wave]] modes (def. \ref{PlaneWaves}), not just of the [[frequency]]/[[wavelength]], but also of the [[direction of a vector|direction]] of the wave vector, this means that it contains _[[direction of a vector|directional]] information_ about the singularities: A distribution may have UV-singularities at some point and in some [[wave vector]] [[direction of a vector|direction]], but maybe not in other [[direction of a vector|directions]].
In particular, if the distribution in question is a [[distributional solution to a partial differential equation]] (def. \ref{DistributionalDerivatives}) on [[spacetime]] then the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem} below) says that the [[singular support]] of the solution evolves in spacetime along the direction of those [[wave vectors]] in which the Fourier transform exhibits high UV constributions. This means that these directions are the "wave front" of the distributional solution. Accordingly, the [[singular support]] of a distribution together with, over each of its points, the [[direction of a vector|directions]] of [[wave vectors]] in which the Fourier transform around that point has large UV constributions is called the _[[wave front set]]_ of the distribution (def. \ref{WaveFrontSet} below).
What is called _[[microlocal analysis]]_ is essentially the analysis of [[distributions]] with attention to their [[wave front set]], hence to the [[wave vector]]-[[direction of a vector|directions]] of [[UV divergences]].
In particular the [[product of distributions]] is well defined (only) if the [[wave front sets]] of the distributions to not "collide". And this in fact motivates the definition of the wave front set:
To see this, let $u,v \in \mathcal{D}'(\mathbb{R}^1)$ be two [[distributions]], for simplicity of exposition taken on the [[real line]].
Since the product $u \cdot v$, is, if it exists, supposed to generalize the _pointwise_ product of smooth functions, it must be fixed locally: for every point $x \in \mathbb{R}$ there ought to be a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R})$ with $f(x) = 1$ such that
$$
b^2 u \cdot v = (b u) \cdot (b v)
\,.
$$
But now $b v$ and $b u$ are both [[compactly supported distributions]] (def. \ref{PropductOfDistributionWithASmoothFunction} below), and these have the special property that their [[Fourier transform of distributions|Fourier transforms]] $\widehat{b v}$ and $\widehat{b u}$ are, in particular, [[smooth functions]] (by the [[Paley-Wiener-Schwartz theorem]], prop \ref{PaleyWienerSchwartzTheorem}).
Moreover, the operation of [[Fourier transform]] interchanges pointwise products with [[convolution products]] (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}). This means that if the [[product of distributions]] $u \cdot v$ exists, it must locally be given by the [[inverse Fourier transform]] of the [[convolution product]] of the Fourier transforms $\widehat {b u}$ and $\widehat b v$:
$$
\widehat{ b^2 u \cdot v }(x)
\;=\;
\underset{\underset{k_{max} \to \infty}{\longrightarrow}}{\lim}
\,
\int_{- k_{max}}^{k_{max}} \widehat{(b u)}(k) \widehat{(b v)}(x - k) d k
\,.
$$
(Notice that the converse of this formula holds as a fact by prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct})
This shows that the [[product of distributions]] exists once there is a [[bump function]] $b$ such that the [[integral]] on the right converges as $k_{max} \to \infty$.
Now the [[Paley-Wiener-Schwartz theorem]] says more, it says that the Fourier transforms $\widehat {b u}$ and $\widehat {b u}$ are polynomially bounded. On the other hand, the [[integral]] above is well defined if the [[integrand]] decreases at least quadratically with $k \to \infty$.
This means that for the convolution product to be well defined, either $\widehat {b u}$ has to polynomially decrease faster with $k \to \pm \infty$ than $\widehat {b v}$ grows in the _other_ direction, $k \to \mp \infty$ (due to the minus sign in the argument of the second factor in the [[convolution product]]), or the other way around.
Moreover, the degree of polynomial growth of the [[Fourier transform of distributions|Fourier transform]] increases by one with each [[derivative of distributions|derivative]] (def. \ref{DistributionalDerivatives}). Therefore if the [[product law]] for [[derivatives of distributions]] is to hold generally, we need that either $\widehat{b u}$ or $\widehat{b v}$ decays faster than _any_ polynomial in the opposite of the directions in which the respective other factor does not decay.
Here the set of directions of wave vectors in which the Fourier transform of a distribution localized around any point does not decay exponentially is the _[[wave front set]]_ of a distribution (def. \ref{WaveFrontSet} below). Hence the condition that the product of two distributions is well defined is that for each wave vector direction in the wave front set of one of the two distributions, the opposite direction must not be an element of the wave front set of the other distribution. This is called _[[Hörmander's criterion]]_ (prop. \ref{HoermanderCriterionForProductOfDistributions} below).
We now say this in detail:
+-- {: .num_defn #RestrictionOfDistributions}
###### Definition
**([[restriction of distributions]])**
For $U \subset \mathbb{R}^n$ a [[subset]], and $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]], then the _[[restriction of distributions|restriction]]_ of $u$ to $U$ is the distribution
$$
u\vert_U \in \mathcal{D}'(U)
$$
give by restricting $u$ to test functions whose [[support]] is in $U$.
=--
+-- {: .num_defn #SingularSupportOfADistribution}
###### Definition
**([[singular support of a distribution]])**
Given a [[distribution]] $u \in \mathcal{D}'(\mathbb{R}^n)$, a point $x \in \mathbb{R}^n$ is a _singular point_ if there is no [[neighbourhood]] $U \subset \mathbb{R}^n$ of $x$ such that the [[restriction of distributions|restriction]] $u\vert_U$ (def. \ref{RestrictionOfDistributions}) is a [[non-singular distribution]] (given by a [[smooth function]]).
The set of all singular points is the _[[singular support]]_ $supp_{sing}(u) \subset \mathbb{R}^n$ of $u$.
=--
+-- {: .num_defn #PropductOfDistributionWithASmoothFunction}
###### Definition
**([[product of distributions|product of a distribution]] with a [[smooth function]])**
Let $u \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]], and $f \in C^\infty(\mathbb{R}^n)$ a smooth function. Then the _product_ $f u \in \mathcal{D}'(\mathbb{R}^n)$ is the evident [[distribution]] given on a test function $b \in C^\infty_{cp}(\mathbb{R}^n)$ by
$$
f u
\;\colon\;
u
\mapsto
u(f \cdot b)
\,
$$
=--
+-- {: .num_prop #DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]] II -- decay of [[Fourier transform of distributions|Fourier transform]] of [[compactly supported functions]])**
A [[compactly supported distribution]] $u \in \mathcal{E}'(\mathbb{R}^n)$ is non-[[singular support of a distribution|singular]], hence given by a [[compactly supported function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its [[Fourier transform of a distribution|Fourier transform]] $\hat u$ ([this def.](compactly+supported+distribution#FourierTransformOfCompactlySupportedDistribution)) satisfies the following decay property:
For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $k \in \mathbb{R}^n$ we have that the [[absolute value]] ${\vert \hat v(k)\vert}$ of the Fourier transform at that point is bounded by
$$
\label{DecayEstimateForFourierTransformOfNonSingularDistribution}
{\vert \hat v(k)\vert}
\;\leq\;
C_N \left( 1 + {\vert k\vert} \right)^{-N}
\,.
$$
=--
([Hörmander 90, around (8.1.1)](Paley-Wiener-Schwartz+theorem#Hoermander90))
+-- {: .num_remark #UltravioletDivergencesFromPaleyWiener}
###### Remark
**([[ultraviolet divergences]])**
In words, the [[Paley-Wiener-Schwartz theorem]] II (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) says that the [[singular support|singularities]] of a [[distribution]] "in position space" are reflected in non-decaying contributions of high [[frequencies]] (small [[wavelength]]) in its [[Fourier transform of distributions|Fourier mode]]-decomposition (def. \ref{FourierTransformOnTemperedDistributions}). Since for ordinary [[light waves]] one associates high [[frequency]] with the "ultraviolet", we may think of these as "ultaviolet contributions".
But apart from the [[wavelength]], the [[wave vector]] that the [[Fourier transform of distributions]] depends on also encodes the _[[direction of a vector|direction]]_ of the corresponding [[plane wave]]. Therefore the [[Paley-Wiener-Schwartz theorem]] says in more detail that a [[distribution]] is singular at some point already if along any _one_ [[direction of a vector|direction]] of the [[wave vector]] its local [[Fourier transform of distributions|Fourier transform]] picks up ultraviolet contributions _in that direction_.
It therefore makes sense to record this extra directional information in the singularity structure of a distribution. This is called the [[wave front set]] (def. \ref{WaveFrontSet}) below. The refined study of singularities taking this directional information into account is what is called _[[microlocal analysis]]_.
Moreover, the [[Paley-Wiener-Schwartz theorem]] I (prop. \ref{PaleyWienerSchwartzTheorem}) says that if the ultraviolet contributions diverge more than polynomially with high [[frequency]], then the corresponding would-be [[compactly supported distribution]] is not only singular, but is actually ill defined.
Such _[[ultraviolet divergences]]_ appear notably when forming a would-be [[product of distributions]] whose two factors have [[wave front sets]] whose UV-contributions "add up". This condition for the appearance/avoidance of [[UV-divergences]] is called _[[Hörmander's criterion]]_ (prop. \ref{HoermanderCriterionForProductOfDistributions} below).
=--
+-- {: .num_defn #WaveFrontSet}
###### Definition
**([[wavefront set]])**
Let $u \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]]. For $b \in C^\infty_{cp}(\mathbb{R}^n)$ a [[compactly supported function|compactly supported]] [[smooth function]], write $b u \in \mathcal{E}'(\mathbb{R}^n)$ for the corresponding product (def. \ref{PropductOfDistributionWithASmoothFunction}), which is now a [[compactly supported distribution]].
For $x\in supp(b) \subset \mathbb{R}^n$, we say that a unit [[covector]] $k \in S((\mathbb{R}^n)^\ast)$ is _regular_ if there exists a [[neighbourhood]] $U \subset S((\mathbb{R}^n)^\ast)$ of $k$ in the [[unit sphere]] such that for all $c k' \in (\mathbb{R}^n)^\ast$ with $c \in \mathbb{R}_+$ and $k' \in U \subset S((\mathbb{R}^n)^\ast)$ the decay estimate (eq:DecayEstimateForFourierTransformOfNonSingularDistribution) is valid for the [[Fourier transform of distributions|Fourier transform]] $\widehat{b u}$ of $b u$; at $c k'$. Otherwise $k$ is _non-regular_. Write
$$
\Sigma(b u)
\;\coloneqq\;
\left\{
k \in S((\mathbb{R}^n)^\ast)
\;\vert\;
k \, \text{non-regular}
\right\}
$$
for the set of non-regular covectors of $b u$.
The _wave front set at $x$_ is the [[intersection]] of these sets as $b$ ranges over [[bump functions]] whose [[support]] includes $x$:
$$
\Sigma_x(u)
\;\coloneqq\;
\underset{
{ b \in C^\infty_{cp}(\mathbb{R}^n) }
\atop
{ x \in supp(b) }
}{\cap}
\Sigma(b u)
\,.
$$
Finally the _[[wave front set]]_ of $u$ is the subset of the [[sphere bundle]] $S(T^\ast \mathbb{R}^n)$ which over $x \in \mathbb{R}^n$ consists of $\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n$:
$$
WF(u)
\;\coloneqq\;
\underset{x \in \mathbb{R}^n}{\cup}
\Sigma_x(u)
\;\subset\;
S(T^\ast \mathbb{R}^n)
$$
Often this is equivalently considered as the full [[conical set]] inside the [[cotangent bundle]] generated by the unit covectors under multiplication with [[positive number|positive]] [[real numbers]].
=--
([Hörmander 90, def. 8.1.2](wavefron+set#Hoermander90))
+-- {: .num_remark #WaveFrontSetIsBundleOverSingularSupport}
###### Remark
**([[wave front set]] is the [[UV divergence]]-[[direction of a vector|direction]]-[[bundle]] over the [[singular support]])**
For $u \in \mathcal{D}'(\mathbb{R}^n)$
The [[Paley-Wiener-Schwartz theorem]] (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) implies that
1. Forgetting the [[direction of a vector|direction]] [[covectors]] in the [[wave front set]] $WF(u)$ (def. \ref{WaveFrontSet}) and remembering only the points where they are based yields the set of singlar points of $u$, hence the [[singular support]] (def. \ref{SingularSupportOfADistribution})
$$
\array{
WF(u)
\\
\downarrow
\\
supp_{sing}(u) &\hookrightarrow& \mathbb{R}^n
}
$$
1. the [[wave front set]] is [[empty set|empty]], precisely if the [[singular support]] is [[empty set|empty]], which is the case precisely if $u$ is a [[non-singular distribution]].
=--
+-- {: .num_example #NonSingularDistributionTrivialWaveFrontSet}
###### Example
**([[wave front set]] of [[non-singular distribution]] is [[empty set|empty]])**
By prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}, the [[wave front set]] (def. \ref{WaveFrontSetIsBundleOverSingularSupport}) of a [[non-singular distribution]] (prop. \ref{DistributionsAreGeneralizedFunctions}) is [[empty set|empty]]. Conversely, a [[distribution]] is [[non-singular distribution|non-singular]] if its wave front set is empty:
$$
u \in \mathcal{D}'\;\text{non-singular}
\phantom{AA}
\Leftrightarrow
\phantom{AA}
WF(u) = \emptyset
$$
=--
+-- {: .num_example #WaveFrontOfDeltaDistribution}
###### Example
**([[wave front set]] of [[delta distribution]])**
Consider the [[delta distribution]]
$$
\delta_0 \in \mathcal{D}'(\mathbb{R}^n)
$$
given by [[evaluation]] at the origin. Its [[wave front set]] (def. \ref{WaveFrontSet}) consists of all the [[direction of a vector|directions]] at the origin:
$$
WF(\delta_0)
\;=\;
\left\{
(0,k)
\;\vert\;
k \in \mathbb{R}^n \setminus \{0\}
\right\}
\subset
\mathbb{R}^n \times \mathbb{R}^n
\simeq
T^\ast \mathbb{R}^n
\,.
$$
=--
+-- {: .proof}
###### Proof
First of all the [[singular support of a distribution|singular support]] (def. \ref{SingularSupportOfADistribution}) of $\delta_0$ is clearly $supp_{sing}(\delta(0)) = \{0\}$, hence by remark \ref{WaveFrontSetIsBundleOverSingularSupport} the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.
At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the [[Fourier transform of distributions|Fourier transform]] $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a [[constant function]] (example \ref{FourierTransformOfDeltaDistribution}) and hence does not decay in any direction.
=--
+-- {: .num_example #WaveFrontSetOfHeavisideDistribution}
###### Example
**([[wave front set]] of [[step function]])**
Let $\Theta \in \mathcal{D}'(\mathbb{R}^1)$ be the [[Heaviside step function]] given by
$$
\Theta(b) \coloneqq \int_0^\infty b(x)\, d x
\,.
$$
Its [[wave front set]] (def. \ref{WaveFrontSet}) is
$$
WF(H) = \{(0,k) \vert k \neq 0\}
\,.
$$
=--
+-- {: .num_prop #WaveFrontSetOfCompactlySupportedDistributions}
###### Proposition
**([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])**
Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two [[compactly supported distributions]]. Then the [[wave front set]] (def. \ref{WaveFrontSet}) of their [[convolution of distributions]] (def. \ref{ConvolutionOfADistributionWithACompactlySupportedDistribution}) is
$$
WF(u \star v)
\;=\;
\left\{
(x + y, k)
\;\vert\;
(x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
\right\}
\,.
$$
=--
([Bengel 77, prop. 3.1](convolution+product+of+distributions#Bengel77))
+-- {: .num_prop #HoermanderCriterionForProductOfDistributions}
###### Proposition
**([[Hörmander's criterion]] for [[product of distributions]])**
Let $u, v \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]]. If their
[[wave front sets]] (def \ref{WaveFrontSet}) do not collide, in that
for $v \in T^\ast_x X$ a [[covector]] contained in one of the two wave front sets then the covector $-v \in T^\ast_x X$ with the opposite [[direction of a vector|direction]] in _not_ contained in the other wave front set, i.e. the [[intersection]] [[fiber product]]
inside the [[cotangent bundle]] $T^\ast X$ of the pointwise [[sum]] of wave fronts with the [[zero section]] is [[empty set|empty]]:
$$
\left( WF(u_1) + WF(u_2) \right) \underset{T^\ast X}{\times} X
\;=\;
\emptyset
$$
i.e.
$$
\array{
&& \emptyset
\\
& \swarrow && \searrow
\\
WF(u_1) + WF(u_2) && (pb) && X
\\
& \searrow && \swarrow_{\mathrlap{0}}
\\
&& T^\ast X
}
$$
then the [[product of distributions]] $u \cdot v$ exists, given, locally, by the [[Fourier inversion theorem|Fourier inversion]] of the [[convolution product]] of their [[Fourier transform of distributions]] (remark \ref{ProductOfDistributionsViaFourierTransformOfConvolution}).
=--
For making use of [[wave front sets]], we need a collection of results about how wave front sets change as we apply certain operations to distributions:
+-- {: .num_prop #RetainsOrShrinksWaveFrontSetDifferentialOperator}
###### Proposition
**([[differential operator]] preserves or shrinks [[wave front set]])**
Let $P$ be a [[differential operator]] (def. \ref{DifferentialOperator}). Then for $u \in \mathcal{D}'$ a [[distribution]], the [[wave front set]] (def. \ref{WaveFrontSet}) of the [[derivative of distributions]] $P u$ (def. \ref{DistributionalDerivatives}) is contained in the original wave front set of $u$:
$$
WF(P u) \subset WF(u)
$$
=--
([Hörmander 90, (8.1.11)](differential+operator#Hoermander90))
+-- {: .num_prop #WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}
###### Proposition
**([[wave front set]] of [[product of distributions]] is inside [[fiber]]-wise [[sum]] of [[wave front sets]])**
Let $u,v \in \mathcal{D}'(X)$ be a [[pair]] of [[distributions]] satisfying [[Hörmander's criterion]], so that their [[product of distributions]] $u \cdot v$ exists by prop. \ref{HoermanderCriterionForProductOfDistributions}. Then the [[wave front set]] (def. \ref{WaveFrontSet}) of the product distribution is contained inside the [[fiber]]-wise [[sum]] of the wave front set elements of the two factors:
$$
WF(u \cdot v)
\;\subset\;
(WF(u) \cup (X \times \{0\}))
+
(WF(v) \cup (X \times \{0\}))
\,.
$$
=--
([Hörmander 90, theorem 8.2.10](product+of+distributions#Hoermander90))
More generally:
+-- {: .num_prop #PartialProductOfDistributionsOfSeveralVariables}
###### Proposition
**(partial [[product of distributions|product of]] [[distributions of several variables]])**
Let
$$
K_1 \in \mathcal{D}'(X \times Y)
\phantom{AAA}
K_2 \in \mathcal{D}'(Y \times Z)
$$
be two [[distributions of two variables]]. For their [[product of distributions]] to be defined over $Y$, [[Hörmander's criterion]] on the [[pair]] of [[wave front sets]] $WF(K_1), WF(K_2)$ needs to hold for the [[wave front set|wave front]] [[wave vectors]] along $X$ and $Y$ taken to be zero.
If this is satisfied, then composition of [[integral kernels]] (if it exists)
$$
(K_1 \circ K_2)(-,-)
\;\coloneqq\;
\underset{Y}{\int}
K_1(-,y) K_2(y,-)
dvol_Y(y)
\;\in\;
\mathcal{D}'(X \times Z)
$$
has [[wave front set]] constrained by
$$
\label{CompositionOfIntegralKernelsWaveFronConstraint}
WF(K_1 \circ K_2)
\;\subset\;
\left\{
(x,z, k_x, k_z)
\;\vert\;
\array{
\left(
(x,y,k_x,-k_y) \in WF(K_1)
\,\,
\text{and}
\,\,
(y,z,k_y, k_z) \in WF(K_2)
\right)
\\
\text{or}
\\
\left(
k_x = 0
\,\text{and}\,
(y,z,0,-k_z) \in WF(K_2)
\right)
\\
\text{or}
\\
\left(
k_z = 0
\,\text{and}\,
(x,y,k_x,0) \in WF(K_1)
\right)
}
\right\}
$$
=--
([Hörmander 90, theorem 8.2.14](product+of+distributions#Hoermander90))
A key fact for identifying [[wave front sets]] is the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem} below). In order to state this we need the following concepts regarding symbols of differential operators:
+-- {: .num_defn #SymbolOfADifferentialOperator}
###### Definition
**([[symbol of a differential operator]])**
Let
$$
D
\;=\;
\underset{n \leq N}{\sum}
D^{\mu_1 \cdots \mu_n}
\frac{\partial}{\partial x^{\mu^1}}
\cdots
\frac{\partial}{\partial x^{\mu^n}}
+
D^0
$$
be a [[differential operator]] on $\mathbb{R}^n$ (def. \ref{DifferentialOperator}). Then its _[[symbol of a differential operator]]_ is the [[smooth function]] on the [[cotangent bundle]] $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ (def. \ref{TangentVectorFields}) given by
$$
\array{
T^\ast \mathbb{R}^n
&\overset{q}{\longrightarrow}&
\mathbb{C}
\\
k &\mapsto&
\underset{n \leq N}{\sum}
D^{\mu_1 \cdots \mu_k} k_{\mu_1} \cdots k_{\mu_n}
}
\,.
$$
The _[[principal symbol]]_ is the top degree [[homogeneous function|homogeneous]] part
$D^{\mu_1 \cdots \mu_k} k_{\mu_1} \cdots k_{\mu_N}$.
=--
+-- {: .num_defn #SymbolOrder}
###### Definition
**([[symbol order]])**
A [[smooth function]] $q$ on the [[cotangent bundle]] $T^\ast \mathbb{R}^n$ (e.g. the [[symbol of a differential operator]], def. \ref{SymbolOfADifferentialOperator} ) is of _order $m$_ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each [[coordinate chart]] $((x^i), (k_i))$ we have that for every [[compact subset]] $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a [[real number]] $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the [[absolute value]] of the [[partial derivatives]] of $q$ is bounded by
$$
\left\vert
\frac{\partial^\alpha}{\partial k_\alpha}
\frac{\partial^\beta}{\partial x^\beta}
q(x,k)
\right\vert
\;\leq\;
C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}
$$
for all $x \in K$ and all cotangent vectors $k$ to $x$.
A [[Fourier transform|Fourier integral]] operator $Q$ is of _[[symbol class]]_ $L^m = L^m_{1,0}$ if it is of the form
$$
Q f (x)
\;=\;
\int \int e^{i k \cdot (x - y)} q(x,y,k) f(y) \, d y \, d k
$$
with symbol $q$ of order $m$, in the above sense.
=--
([Hörmander 71, def. 1.1.1 and first sentence of section 2.1 with (1.4.1)](symbol+order#Hoermander71))
+-- {: .num_prop #PropagationOfSingularitiesTheorem}
###### Proposition
**([[propagation of singularities theorem]])**
Let $Q$ be a [[differential operator]] (def. \ref{DifferentialOperator}) of [[symbol class]] $L^m$ (def. \ref{SymbolOrder}) with real [[principal symbol]] $q$ that is [[homogeneous function|homogeneous]] of degree $m$.
For $u \in \mathcal{D}'(X)$ a [[distribution]] with $Q u = f$, then the [[complement]] of the [[wave front set]] of $u$ by that of $f$ is contained in the set of covectors on which the [[principal symbol]] $q$ vanishes:
$$
WF(u) \setminus WF(f) \;\subset\; q^{-1}(0)
\,.
$$
Moreover, $WF(u)$ is invariant under the [[bicharacteristic flow]] induced by the [[Hamiltonian vector field]] of $q$ with respect to the canonical [[symplectic manifold]] structure on the [[cotangent bundle]] ([here](cotangent+bundle#SymplecticStructure)).
=--
([Duistermaat-Hörmander 72, theorem 6.1.1](propagation+of+singularities+theorem#DuistermaatHoermander72), recalled for instance as [Radzikowski 96, theorem 4.6](propagation+of+singularities+theorem#Radzikowski96))
$\,$
**[[Cauchy principal value]]**
{#CauchyPrincipalValues}
An important application of the [[Fourier transform|Fourier analysis]] of [[distributions]] is the class of distributions known
broadly as _[[Cauchy principal values]]_. Below we will find that these control the detailed nature of the
various [[propagators]] of [[free field theories]], notably the [[Feynman propagator]] is manifestly a [[Cauchy principal value]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} and def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime} below),
but also the [[singular support]] properties of the [[causal propagator]] and the [[Wightman propagator]] are governed by
Cauchy principal values (prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} and prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} below). This way the understanding of
Cauchy principal values eventually allows us to determine the [[wave front set]] of all the propagators (prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski}) below.
Therefore we now collect some basic definitions and facts on [[Cauchy principal values]].
The _Cauchy principal value_ of a [[function]] which is [[integrable function|integrable]] on the [[complement]] of one point is, if it exists, the [[limit of a sequence|limit]] of the [[integrals]] of the function over subsets in the [[complement]] of this point as these integration [[domains]] tend to that point _symmetrically_ from all sides.
One also subsumes the case that the "point" is "at infinity", hence that the function is [[integrable function|integrable]] over every [[bounded subsets|bounded]] [[domain]]. In this case the Cauchy principal value is the [[limit of a sequence|limit]], if it exists, of the [[integrals]] of the function over bounded domains, as their bounds tend _symmetrically_ to infinity.
The operation of sending a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) to Cauchy principal value of its pointwise product with a function $f$ that may be singular at the origin defines a [[distribution]], usually denoted $PV(f)$.
+-- {: .num_defn #CauchyIntegralValueOfIntegralOverRealline}
###### Definition
**([[Cauchy principal value]] of an [[integral]] over the [[real line]])**
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for every [[positive real number]] $\epsilon$ its [[restriction]] to $\mathbb{R}\setminus (-\epsilon, \epsilon)$ is [[integrable function|integrable]]. Then the _[[Cauchy principal value]]_ of $f$ is, if it exists, the [[limit of a sequence|limit]]
$$
PV(f) \coloneqq \underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R} \setminus (-\epsilon, \epsilon)}{\int} f(x) \, d x
\,.
$$
=--
+-- {: .num_defn #CauchyPrincipalValueAsDistributionOnRealLine}
###### Definition
**([[Cauchy principal value]] as [[distribution]] on the [[real line]])**
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for all [[bump functions]] $b \in C^\infty_{cp}(\mathbb{R})$ the Cauchy principal value of the pointwise product function $f b$ exists, in the sense of def. \ref{CauchyIntegralValueOfIntegralOverRealline}. Then this assignment
$$
PV(f)
\;\colon\;
b \mapsto PV(f b)
$$
defines a [[distribution]] $PV(f) \in \mathcal{D}'(\mathbb{R})$.
=--
+-- {: .num_example }
###### Example
Let $f \colon \mathbb{R} \to \mathbb{R}$ be an [[integrable function]] which is symmetric, in that $f(-x) = f(x)$ for all $x \in \mathbb{R}$. Then the principal value integral (def. \ref{CauchyIntegralValueOfIntegralOverRealline}) of $x \mapsto \frac{f(x)}{x}$ exists and is zero:
$$
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}\setminus (-\epsilon, \epsilon)}{\int}
\frac{f(x)}{x} d x
\; = \; 0
$$
This is because, by the symmetry of $f$ and the skew-symmetry of $x \mapsto 1/x$, the the two contributions to the integral are equal up to a sign:
$$
\int_{-\infty}^{-\epsilon} \frac{f(x)}{x} d x
\;=\;
-
\int_{\epsilon}^\infty \frac{f(x)}{x} d x
\,.
$$
=--
+-- {: .num_example #PrincipalValueOfInverseFunctionCharacteristicEquation}
###### Example
The [[Cauchy principal value]] [[distribution]] $PV\left( \frac{1}{x}\right)$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) solves the distributional [[equation]]
$$
\label{DistributionalEquationxfOfxEqualsOne}
x PV\left(\frac{1}{x}\right) = 1
\phantom{AAA}
\in \mathcal{D}'(\mathbb{R}^1)
\,.
$$
Since the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R}^1)$ solves the equation
$$
x \delta(x) = 0
\phantom{AAA}
\in \mathcal{D}'(\mathbb{r}^1)
$$
we have that more generally every [[linear combination]] of the form
$$
\label{GeneralDistributionalSolutionToxfEqualsOne}
F(x)
\coloneqq
PV(1/x) + c \delta(x)
\phantom{AAA}
\in \mathcal{D}'(\mathbb{R}^1)
$$
for $c \in \mathbb{C}$, is a distributional solution to $x F(x) = 1$.
The [[wave front set]] of all these solutions is
$$
WF\left(
PV(1/x) + c \delta(x)
\right)
\;=\;
\left\{
(0,k) \;\vert\; k \in \mathbb{R}^\ast \setminus \{0\}
\right\}
\,.
$$
=--
+-- {: .proof}
###### Proof
The first statement is immediate from the definition: For $b \in C^\infty_c(\mathbb{R}^1)$ any [[bump function]] we have that
$$
\begin{aligned}
\left\langle x PV\left(\frac{1}{x}\right), b \right\rangle
& \coloneqq
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1 \setminus (-\epsilon, \epsilon)}{\int}
\frac{x}{x}b(x) \, d x
\\
& =
\int b(x) d x
\\
& =
\langle 1,b\rangle
\end{aligned}
$$
Regarding the second statement: It is clear that the wave front set is concentrated at the origin. By symmetry of the distribution around the origin, it must contain both [[direction of a vector|directions]].
=--
+-- {: .num_prop}
###### Proposition
In fact (eq:GeneralDistributionalSolutionToxfEqualsOne) is the most general distributional solution to (eq:DistributionalEquationxfOfxEqualsOne).
=--
This follows by the characterization of [[extension of distributions]] to a point, see there at [this prop.](extension+of+distributions#SpaceOfPointExtensions) ([Hörmander 90, thm. 3.2.4](Cauchy+principa+value#HoermanderLPDO1))
+-- {: .num_defn #IntegrationAgainstInverseOfxWithImaginaryOffset}
###### Definition
**(integration against inverse variable with imaginary offset)**
Write
$$
\tfrac{1}{x + i0^\pm}
\;\in\;
\mathcal{D}'(\mathbb{R})
$$
for the [[distribution]] which is the [[limit]] in $\mathcal{D}'(\mathbb{R})$ of the [[non-singular distributions]] which are given by the [[smooth functions]] $x \mapsto \tfrac{1}{x \pm i \epsilon}$ as the [[positive real number]] $\epsilon$ tends to zero:
$$
\frac{1}{
x + i 0^\pm
}
\;\coloneqq\;
\underset{ { \epsilon \in (0,\infty) } \atop { \epsilon \to 0 } }{\lim}
\tfrac{1}{x \pm i \epsilon}
$$
hence the distribution which sends $b \in C^\infty(\mathbb{R}^1)$ to
$$
b \mapsto
\underset{\mathbb{R}}{\int}
\frac{b(x)}{x \pm i \epsilon} \, d x
\,.
$$
=--
+-- {: .num_prop #CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}
###### Proposition
**([[Cauchy principal value]] equals integration with [[imaginary number|imaginary]] offset plus [[delta distribution]])**
The [[Cauchy principal value]] [[distribution]] $PV\left( \tfrac{1}{x}\right) \in \mathcal{D}'(\mathbb{R})$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) is equal to the sum of the integration over $1/x$ with imaginary offset (def. \ref{IntegrationAgainstInverseOfxWithImaginaryOffset}) and a [[delta distribution]].
$$
PV\left(\frac{1}{x}\right)
\;=\;
\frac{1}{x + i 0^\pm}
\pm i \pi \delta
\,.
$$
In particular, by prop. \ref{PrincipalValueOfInverseFunctionCharacteristicEquation} this means that $\tfrac{1}{x + i 0^\pm}$ solves the distributional equation
$$
x \frac{1}{x + i 0^\pm}
\;=\;
1
\phantom{AA}
\in \mathcal{D}'(\mathbb{R}^1)
\,.
$$
=--
+-- {: .proof}
###### Proof
Using that
$$
\begin{aligned}
\frac{1}{x \pm i \epsilon}
& =
\frac{
x \mp i \epsilon
}{
(x + i \epsilon)(x - i \epsilon)
}
\\
& =
\frac{ x \mp i \epsilon }{(x^2 + \epsilon^2)}
\end{aligned}
$$
we have for every [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^1)$
$$
\begin{aligned}
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{b(x)}{x \pm i \epsilon}
d x
& \;=\;
\underset{
(A)
}{
\underbrace{
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{x^2}{x^2 + \epsilon^2} \frac{b(x)}{x}
d x
}
}
\mp i \pi
\underset{(B)}{
\underbrace{
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{1}{\pi}
\frac{\epsilon}{x^2 + \epsilon^2} b(x)
\,
d x
}}
\end{aligned}
$$
Since
$$
\array{
&& \frac{x^2}{x^2 + \epsilon^2}
\\
& {}^{\mathllap{ { {\vert x \vert} \lt \epsilon } \atop { \epsilon \to 0 } }}\swarrow
&&
\searrow^{\mathrlap{ {{\vert x\vert} \gt \epsilon} \atop { \epsilon \to 0 } }}
\\
0 && && 1
}
$$
it is plausible that $(A) = PV\left( \frac{b(x)}{x} \right)$, and similarly that $(B) = b(0)$. In detail:
$$
\begin{aligned}
(A)
& =
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{x}{x^2 + \epsilon^2} b(x)
d x
\\
& =
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{d}{d x}
\left(
\tfrac{1}{2}
\ln(x^2 + \epsilon^2)
\right)
b(x)
d x
\\
& =
-\tfrac{1}{2}
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\ln(x^2 + \epsilon^2)
\frac{d b}{d x}(x)
d x
\\
& =
-\tfrac{1}{2}
\underset{\mathbb{R}^1}{\int}
\ln(x^2)
\frac{d b}{d x}(x)
d x
\\
& =
-
\underset{\mathbb{R}^1}{\int}
\ln({\vert x \vert})
\frac{d b}{d x}(x)
d x
\\
& =
-
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
\ln( {\vert x \vert} )
\frac{d b}{d x}(x)
d x
\\
& =
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
\frac{1}{x}
b(x)
d x
\\
& =
PV\left( \frac{b(x)}{x} \right)
\end{aligned}
$$
and
$$
\begin{aligned}
(B)
& =
\tfrac{1}{\pi}
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\frac{\epsilon}{x^2 + \epsilon^2}
b(x)
\,
d x
\\
& =
\tfrac{1}{\pi}
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\left(
\frac{d}{d x}
\arctan\left( \frac{x}{\epsilon} \right)
\right)
b(x)
\,
d x
\\
& =
-
\tfrac{1}{\pi}
\underset{\epsilon \to 0}{\lim}
\underset{\mathbb{R}^1}{\int}
\arctan\left( \frac{x}{\epsilon} \right)
\frac{d b}{d x}(x)
\,
d x
\\
& =
-
\frac{1}{2}
\underset{\mathbb{R}^1}{\int}
sgn(x)
\frac{d b}{d x}(x)
\,
d x
\\
& =
b(0)
\end{aligned}
$$
where we used that the [[derivative]] of the [[arctan]] function is $\frac{d}{ d x} \arctan(x) = 1/(1 + x^2)$ and that $\underset{\epsilon \to + \infty}{\lim} \arctan(x/\epsilon) = \tfrac{\pi}{2}sgn(x)$ is proportional to the [[sign function]].
=--
+-- {: .num_example #FourierIntegralFormulaForStepFunction}
###### Example
**([[Fourier transform|Fourier integral]] formula for [[step function]])**
The [[Heaviside distribution]] $\Theta \in \mathcal{D}'(\mathbb{R})$ is equivalently the following [[Cauchy principal value]] (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}):
$$
\begin{aligned}
\Theta(x)
& =
\frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+}
\\
& \coloneqq
\underset{ \epsilon \to 0^+}{\lim}
\frac{1}{2 \pi i}
\int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
\,,
\end{aligned}
$$
where the limit is taken over [[sequences]] of [[positive numbers|positive]] [[real numbers]] $\epsilon \in (-\infty,0)$ tending to zero.
=--
+-- {: .proof}
###### Proof
We may think of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ uniquely extended to a [[holomorphic function]] on the [[complex plane]] and consider computing the given real [[line integral]] for fixed $\epsilon$ as a [[contour integral]] in the [[complex plane]].
If $x \in (0,\infty)$ is [[positive number|positive]], then the exponent
$$
i \omega x = - Im(\omega) x + i Re(\omega) x
$$
<div style="float:right;margin:0 10px 10px 0;"><img src="https://ncatlab.org/nlab/files/ContoursForHeavisideFourierTransform.png" width="300">
</div>
has negative [[real part]] for _positive_ [[imaginary part]] of $\omega$. This means that the [[line integral]] equals the complex [[contour integral]] over a contour $C_+ \subset \mathbb{C}$ closing in the [[upper half plane]]. Since $i \epsilon$ has positive [[imaginary part]] by construction, this contour does encircle the [[pole]] of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ at $\omega = i \epsilon$. Hence by the [[Cauchy integral formula]] in the case $x \gt 0$ one gets
$$
\begin{aligned}
\underset{\epsilon \to 0^+}{\lim}
\frac{1}{2 \pi i}
\int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
& =
\underset{\epsilon \to 0^+}{\lim}
\frac{1}{2 \pi i}
\oint_{C_+} \frac{e^{i \omega x}}{\omega - i \epsilon} d \omega
\\
&
=
\underset{\epsilon \to 0^+}{\lim}
\left(e^{i \omega x}\vert_{\omega = i \epsilon}\right)
\\
& =
\underset{\epsilon \to 0^+}{\lim}
e^{- \epsilon x}
\\
& =
e^0 = 1
\end{aligned}
\,.
$$
Conversely, for $x \lt 0$ the real part of the integrand decays as the _[[negative number|negative]]_ imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour $C_- \subset \mathbb{C}$ closing in the lower half plane. Since the integrand has no pole in the lower half plane, in this case the [[Cauchy integral formula]] says that this integral is zero.
=--
Conversely, by the [[Fourier inversion theorem]], the [[Fourier transform]] of the [[Heaviside distribution]] is the Cauchy principal value as in prop. \ref{CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}:
+-- {: .num_example #RelationToFourierTransformOfHeavisideDistribution}
###### Example
**(relation to [[Fourier transform]] of [[Heaviside distribution]] / [[Schwinger parameter|Schwinger parameterization]])**
The [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[Heaviside distribution]] is
the following [[Cauchy principal value]]:
$$
\begin{aligned}
\widehat \Theta(x)
& =
\int_0^\infty e^{i k x} \, dk
\\
& =
i \frac{1}{x + i 0^+}
\end{aligned}
$$
Here the second equality is also known as _complex [[Schwinger parameter|Schwinger parameterization]]_.
=--
+-- {: .proof}
###### Proof
As [[generalized functions]] consider the [[limit of a sequence|limit]] with a decaying component:
$$
\begin{aligned}
\int_0^\infty e^{i k x} \, dk
& =
\underset{\epsilon \to 0^+}{\lim}
\int_0^\infty e^{i k x - \epsilon k} \, dk
\\
& =
-
\underset{\epsilon \to 0^+}{\lim}
\frac{1}{ i x - \epsilon}
\\
& =
i \frac{1}{x + i 0^+}
\end{aligned}
$$
=--
Let now $q \colon \mathbb{R}^{n} \to \mathbb{R}$ be a non-degenerate real [[quadratic form]]
[[analytic continuation|analytically continued]] to a real quadratic form
$$
q \;\colon\; \mathbb{C}^n \longrightarrow \mathbb{C}
\,.
$$
Write $\Delta$ for the [[determinant]] of $q$
Write $q^\ast$ for the induced quadratic form on [[dual vector space]]. Notice that $q$ (and hence $a^\ast$) are assumed non-degenerate but need not necessarily be positive or negative definite.
+-- {: .num_prop #FourierTransformOfPrincipalValueOfPowerOfQuadraticForm}
###### Proposition
**([[Fourier transform of distributions|Fourier transform]] of principal value of power of [[quadratic form]])**
Let $m \in \mathbb{R}$ be any [[real number]], and $\kappa \in \mathbb{C}$ any [[complex]] number. Then the [[Fourier transform of distributions]] of $1/(q + m^2 + i 0^+)^\kappa$ is
$$
\widehat
{
\left(
\frac{1}{(q + m^2 + i0^+)^\kappa}
\right)
}
\;=\;
\frac{
2^{1- \kappa} (\sqrt{2\pi})^{n} m^{n/2-\kappa}
}
{
\Gamma(\kappa) \sqrt{\Delta}
}
\frac{
K_{n/2 - \kappa}\left( m \sqrt{q^\ast - i 0^+} \right)
}
{
\left(\sqrt{q^\ast - i0^+ }\right)^{n/2 - \kappa}
}
\,,
$$
where
1. $\Gamma$ deotes the [[Gamma function]]
1. $K_{\nu}$ denotes the [[modified Bessel function]] of order $\nu$.
Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).
=--
([Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289](Cauchy+principal+value#GelfandShilov66))
+-- {: .num_prop #FourierTransformOfDeltaDistributionappliedToMassShell}
###### Proposition
**([[Fourier transform]] of [[delta distribution]] applied to [[mass shell]])**
Let $m \in \mathbb{R}$, then the [[Fourier transform of distributions]] of the [[delta distribution]] $\delta$ applied to the "mass shell" $q + m^2$ is
$$
\widehat{
\delta(q + m^2)
}
\;=\;
- \frac{i}{\sqrt{{\vert\Delta\vert}}}
\left(
e^{i \pi t /2 }
\frac{
K_{n/2-1}
\left(
m \sqrt{ q^\ast + i0^+ }
\right)
}{
\left(\sqrt{q^\ast + i0^+}\right)^{n/2 - 1}
}
\;-\;
e^{-i \pi t /2 }
\frac{
K_{n/2-1}
\left(
m \sqrt{ q^\ast - i0^+ }
\right)
}{
\left(\sqrt{q^\ast - i0^+}\right)^{n/2 - 1}
}
\right)
\,,
$$
where $K_\nu$ denotes the [[modified Bessel function]] of order $\nu$.
Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).
=--
([Gel'fand-Shilov 66, III 2.11 (7), p 294](Cauchy+principal+value#GelfandShilov66))
$\,$
**[[propagators]] for the [[free field theory|free]] [[scalar field]] on [[Minkowski spacetime]]**
1. _[Advanced and regarded propagators](#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime)_
1. _[Causal propagator](#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)_
1. _[Wightman propagator](#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime)_
1. _[Feynman propagator](#FeynmanPropagator)_
1. _[Singular support and Wave front sets](#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime)_
$\,$
On [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ consider the [[Klein-Gordon operator]] (example \ref{EquationOfMotionOfFreeRealScalarField})
$$
\eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} \Phi - \left( \tfrac{m c}{\hbar} \right)^2 \Phi \;=\; 0 \,.
$$
By example \ref{FourierTransformOfKleinGordonEquation} its [[Fourier transform]] is
$$
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2
\;=\;
(k_0)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
\,.
$$
The [[dispersion relation]] of this equation we write (see def. \ref{PlaneWaves})
$$
\label{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime}
\omega(\vec k)
\;\coloneqq\;
+ c \sqrt{ {\vert \vec k \vert}^2 + \left( \tfrac{m c}{\hbar}\right)^2 }
\,,
$$
where on the right we choose the [[non-negative real number|non-negative]] [[square root]].
$\,$
**[[advanced and retarded propagators]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]]**
{#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime}
+-- {: .num_prop #AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}
###### Proposition
**(mode expansion of [[advanced and retarded propagators]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**
The [[advanced and retarded Green functions]] $G_\pm$ (def. \ref{AdvancedAndRetardedGreenFunctions}) of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) are induced from [[integral kernels]] ("[[propagators]]"), hence [[distributions in two variables]]
$$
\Delta_\pm \in \mathcal{D}'(\mathbb{R}^{p,1}\times \mathbb{R}^{p,1})
$$
by (in [[generalized function]]-notation, prop. \ref{DistributionsAreGeneralizedFunctions})
$$
G_\pm(\Phi)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int}
\Delta_{\pm}(x,y) \Phi(y) \, dvol(y)
$$
where the [[advanced and retarded propagators]] $\Delta_{\pm}(x,y)$ have the following equivalent expressions:
$$
\label{ModeExpansionForMinkowskiAdvancedRetardedPropagator}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$
Here $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]].
=--
+-- {: .proof}
###### Proof
The [[Klein-Gordon operator]] is a [[Green hyperbolic differential operator]] (example \ref{GreenHyperbolicKleinGordonEquation}) therefore its advanced and retarded Green functions exist uniquely (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}).
Moreover, prop. \ref{GreenFunctionsAreContinuous} says that they are [[continuous linear functionals]] with respect to the [[topological vector space]] [[structures]] on [[spaces of smooth sections]] (def. \ref{TVSStructureOnSpacesOfSmoothSections}). In the case of the [[Klein-Gordon operator]] this just means that
$$
G_{\pm}
\;\colon\;
C^\infty_{cp}(\mathbb{R}^{p,1})
\longrightarrow
C^\infty_{\pm cp}(\mathbb{R}^{p,1})
$$
are [[continuous linear functionals]] in the standard sense of [[distributions]]. Therefore the
[[Schwartz kernel theorem]] implies the existence of [[integral kernels]] being [[distributions in two variables]]
$$
\Delta_{\pm} \in \mathcal{D}(\mathbb{R}^{p,1} \times \mathbb{R}^{p,1})
$$
such that, in the notation of [[generalized functions]],
$$
(G_\pm \alpha)(x)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \alpha(y) \, dvol(y)
\,.
$$
These integral kernels are the advanced/retarded "[[propagators]]". We now compute
these [[integral kernels]] by making an Ansatz and showing that it has the defining properties,
which identifies them by the uniqueness statement of prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.
We make use of the fact that the [[Klein-Gordon equation]] is [[invariant]] under the defnining [[action]]
of the [[Poincaré group]] on [[Minkowski spacetime]], which is a [[semidirect product group]] of the [[translation group]]
and the [[Lorentz group]].
Since the [[Klein-Gordon operator]] is invariant, in particular, under [[translations]] in $\mathbb{R}^{p,1}$ it is clear that the propagators, as a [[distribution in two variables]], depend only on the difference of its two arguments
$$
\label{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime}
\Delta_{\pm}(x,y) = \Delta_{\pm}(x-y)
\,.
$$
Since moreover the [[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] ([this prop.](Klein-Gordon+equation#FormallySelfAdjointKleinGordonOperator)) this implies that for $P$ the Klein the equation (eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator)
$$
P \circ G_\pm = id
$$
is equivalent to the equation (eq:AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator)
$$
G_\pm \circ P = id
\,.
$$
Therefore it is sufficient to solve for the first of these two equation, subject to
the defining support conditions. In terms of the [[propagator]] [[integral kernels]] this means that we have to solve the [[distribution|distributional]] equation
$$
\label{KleinGordonEquationOnAdvacedRetardedPropagator}
\left(
\eta^{\mu \nu}
\frac{\partial}{\partial x^\mu}
\frac{\partial}{\partial x^\nu}
-
\left( \tfrac{m c}{\hbar} \right)^2
\right)
\Delta_\pm(x-y)
\;=\;
\delta(x-y)
$$
subject to the condition that the [[support of a distribution|distributional support]] (def. \ref{DistributionalSections}) is
$$
supp\left( \Delta_{\pm}(x-y) \right)
\subset
\left\{
{\vert x-y\vert^2_\eta}\lt 0
\;\,,\;
\pm(x^0 - y^ 0) \gt 0
\right\}
\,.
$$
We make the _Ansatz_ that we assume that $\Delta_{\pm}$, as a distribution in a single variable $x-y$, is a [[tempered distribution]]
$$
\Delta_\pm \in \mathcal{S}'(\mathbb{R}^{p,1})
\,,
$$
hence amenable to [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}). If we do find a [[solution]] this way, it is guaranteed to be the unique solution by prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.
By example \ref{FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}
the [[Fourier transform of distributions|distributional Fourier transform]] of equation (eq:KleinGordonEquationOnAdvacedRetardedPropagator) is
$$
\begin{aligned}
\label{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_{\pm}}(k)
& =
\widehat{\delta}(k)
\\
& =
1
\end{aligned}
\,,
$$
where in the second line we used the [[Fourier transform of distributions|Fourier transform]] of the [[delta distribution]]
from example \ref{FourierTransformOfDeltaDistribution}.
Notice that this implies that the [[Fourier transform]] of the [[causal propagator]] (eq:CausalPropagator)
$$
\Delta_S \coloneqq \Delta_+ - \Delta_-
$$
satisfies the homogeneous equation:
$$
\label{FourierVersionOfPDEForKleinGordonCausalPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_S}(k)
\;=\;
0
\,,
$$
Hence we are now reduced to finding solutions $\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})$ to (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator) such that their [[Fourier inversion theorem|Fourier inverse]] $\Delta_\pm$ has the required [[support of a distribution|support]] properties.
We discuss this by a variant of the [[Cauchy principal value]]:
Suppose the following [[limit of a sequence|limit]] of [[non-singular distributions]] in the [[variable]] $k \in \mathbb{R}^{p,1}$
exists in the space of [[distributions]]
$$
\label{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\frac{1}{ (k_0 \mp i \epsilon)^2 - {\vert \vec k\vert^2} - \left( \tfrac{m c}{\hbar} \right)^2 }
\;\in\;
\mathcal{D}'(\mathbb{R}^{p,1})
$$
meaning that for each [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ the [[limit of a sequence|limit]] in $\mathbb{C}$
$$
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\underset{\mathbb{R}^{p,1}}{\int} \frac{b(k)}{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 }
d^{p+1}k
\;\in\;
\mathbb{C}
$$
exists. Then this limit is clearly a solution to the distributional equation (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator)
because on those bump functions $b(k)$ which happen to be products with $\left(-\eta^{\mu \nu}k_\mu k^\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)$
we clearly have
$$
\label{LimitOfDistributionsForFourierTransformedPropagators}
\begin{aligned}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) b(k)
}{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
}
d^{p+1}k
& =
\underset{\mathbb{R}^{p,1}}{\int}
\underset{= 1}{
\underbrace{
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) }{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 }
}
}
b(k)\, d^{p+1}k
\\
& =
\langle 1, b\rangle
\,.
\end{aligned}
$$
Moreover, if the limiting distribution (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator)
exists, then it is clearly a [[tempered distribution]], hence we may apply [[Fourier inversion theorem|Fourier inversion]]
to obtain [[Green functions]]
$$
\label{AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets}
\Delta_{\pm}(x,y)
\;\coloneqq\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{1}{(2\pi)^{p+1}}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{e^{i k_\mu (x-y)^\mu}}{
(k_0 \mp i \epsilon )^2 - {\vert \vec k\vert}^2 - \left(\tfrac{m c}{\hbar}\right)^2
}
d k_0 d^p \vec k
\,.
$$
To see that this is the correct answer, we need to check the defining support property.
Finally, by the [[Fourier inversion theorem]], to show that the [[limit of a sequence|limit]] (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator) indeed exists it is sufficient to show that the
limit in (eq:AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets) exists.
We compute as follows
$$
\label{TheSupportOfTheCandidateAdvancedRetardedPropagatorIsinTheFutureOrPastRespectively}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i \epsilon)^2 - \left(\omega(\vec k)/c\right)^2
}
\, d k_0 \, d^p \vec k
\\
&=
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
\left(
(k_0 \mp i\epsilon) - \omega(\vec k)/c
\right)
\left(
(k_0 \mp i \epsilon) + \omega(\vec k)/c
\right)
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
d^p \vec k
& \vert & \text{if} \, \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$
where $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]]. The last step is simply the application of [[Euler's formula]] $\sin(\alpha) = \tfrac{1}{2 i }\left( e^{i \alpha} - e^{- i \alpha}\right)$.
Here the key step is the application of [[Cauchy's integral formula]] in the fourth step. We spell this out now for $\Delta_+$, the discussion for $\Delta_-$ is the same, just with the appropriate signs reversed.
1. If $(x^0 - y^0) \gt 0$ thn the expression $e^{ik_0 (x^0 - y^0)}$ decays with _[[positive number|positive]] [[imaginary part]]_ of $k_0$, so that we may expand the [[integration]] [[domain]] into the [[upper half plane]] as
$$
\begin{aligned}
\int_{-\infty}^\infty d k_0
& = \phantom{+}
\int_{-\infty}^0 d k_0 + \int_{0}^{+ i \infty} d k_0
\\
& = + \int_{+i \infty}^0 d k_0 + \int_0^\infty d k_0
\,;
\end{aligned}
$$
Conversely, if $(x^0 - y^0) \lt 0$ then we may analogously expand into the [[lower half plane]].
1. This integration domain may then further be completed to two [[contour integrations]]. For the expansion into the [[upper half plane]] these encircle counter-clockwise the [[poles]] at $\pm \omega(\vec k)+ i\epsilon \in \mathbb{C}$, while for expansion into the [[lower half plane]] no poles are being encircled.
<img src="https://ncatlab.org/nlab/files/ContourForAdvancedPropagator.png" height="280">
1. Apply [[Cauchy's integral formula]] to find in the case $(x^0 - y^0)\gt 0$ the sum of the [[residues]] at these two [[poles]] times $2\pi i$, zero in the other case. (For the retarded propagator we get $- 2 \pi i$ times the residues, because now the contours encircling non-trivial poles go clockwise).
1. The result is now non-singular at $\epsilon = 0$ and therefore the [[limit of a sequence|limit]] $\epsilon \to 0$ is now computed by evaluating at $\epsilon = 0$.
This computation shows a) that the limiting distribution indeed exists, and b) that the [[support of a distribution|support]]
of $\Delta_+$ is in the future, and that of $\Delta_-$ is in the past.
Hence it only remains to see now that the support of $\Delta_\pm$ is inside the [[causal cone]].
But this follows from the previous argument, by using that the [[Klein-Gordon equation]] is invariant under
[[Lorentz transformations]]: This implies that the support is in fact in the [[future]] of _every_ spacelike
slice through the origin in $\mathbb{R}^{p,1}$, hence in the [[closed future cone]] of the origin.
=--
\begin{proposition}
\label{CausalPropagatorIsSkewSymmetric}
**([[causal propagator]] is skew-symmetric)**
\linebreak
Under reversal of arguments the [[advanced and retarded causal propagators]] from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} are related by
$$
\label{AdvancedAndRetardedPropagatorTurnIntoEachOtherUnderSwitchingArguments}
\Delta_{\pm}(y-x) = \Delta_\mp(x-y)
\,.
$$
It follows that the [[causal propagator]] (eq:CausalPropagator) $\Delta \coloneqq \Delta_+ - \Delta_-$ is skew-symmetric in its arguments:
$$
\Delta_S(x-y) = - \Delta_S(y-x)
\,.
$$
\end{proposition}
+-- {: .proof}
###### Proof
By prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} we have with (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator)
$$
\begin{aligned}
\Delta_\pm(y-x)
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) }
\right)
d^p \vec k
& \vert & \text{if} \, \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\Delta_\mp(x-y)
\end{aligned}
$$
Here in the second step we applied [[change of integration variables]] $\vec k \mapsto - \vec k$ (which introduces _no_ sign because in addition to $d \vec k \mapsto - d \vec k$ the integration domain reverses [[orientation]]).
=--
$\,$
**[[causal propagator]]**
{#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
+-- {: .num_prop #ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}
###### Proposition
**(mode expansion of [[causal propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[causal propagator]] (eq:CausalPropagator) for the [[Klein-Gordon equation]] for [[mass]] $m$ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (example \ref{EquationOfMotionOfFreeRealScalarField}) is given, in [[generalized function]] notation, by
$$
\label{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\,,
\end{aligned}
$$
where in the second line we used [[Euler's formula]] $sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)$.
In particular this shows that the [[causal propagator]] is [[real part|real]], in that it is equal to its [[complex conjugation|complex conjugate]]
$$
\label{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal}
\left(\Delta_S(x,y)\right)^\ast = \Delta_S(x,y)
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition and using the expression from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} for the [[advanced and retarded causal propagators]] we have
$$
\begin{aligned}
\Delta_S(x,y)
& \coloneqq
\Delta_+(x,y) - \Delta_-(x,y)
\\
& =
\left\{
\array{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, + (x^0 - y^0) \gt 0
\\
\frac{(-1) (-1) i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
& \vert & \text{if} \, - (x^0 - y^0) \gt 0
}
\right.
\\
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\end{aligned}
$$
For the reality, notice from the last line that
$$
\begin{aligned}
\left(\Delta_S(x,y)\right)^\ast
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{-i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{+i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\\
& =
\Delta_S(x,y)
\,,
\end{aligned}
$$
where in the last step we used the [[change of integration variables]] $\vec k \mapsto - \vec k$
(whih introduces no sign, since on top of $d \vec k \mapsto - d \vec k$ the [[orientation]] of the integration [[domain]] changes).
=--
We consider a couple of equivalent expressions for the causal propagator which are useful for computations:
+-- {: .num_prop #CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral}
###### Proposition
**([[causal propagator]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]] as a [[contour integral]])**
The [[causal propagator]] (prop. \ref{GreenFunctionsAreContinuous}) for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) has the following equivalent expression, as a [[generalized function]], given as a [[contour integral]] along a [[Jordan curve]] $C(\vec k)$ going counter-clockwise around the two [[poles]] at $k_0 = \pm \omega(\vec k)/c$:
$$
\Delta_S(x,y)
\;=\;
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2g }
\,d k_0
\,d^{p} k
\,.
$$
=--
<img src="https://ncatlab.org/nlab/files/ContourForCausalPropagator.png" height="160">
> graphics grabbed from [Kocic 16](advanced+and+retarded+causal+propagators#Kocic16#Kocic16)
+-- {: .proof}
###### Proof
By [[Cauchy's integral formula]] we compute as follows:
$$
\begin{aligned}
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x^\mu - y^\mu)}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
\,d k_0
\,d^{p} k
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 x^0} e^{ i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 + \omega(\vec k)/c )
( k_0 - \omega(\vec k)/c )
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
2\pi i
\int
\left(
\frac{
e^{i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}
{
2 \omega(\vec k)/c
}
-
\frac{
e^{ - i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}{
2 \omega(\vec k)/c
}
\right)
\,d^p \vec k
\\
& =
i
(2\pi)^{-p}
\int
\frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\,d^p \vec k
\,.
\end{aligned}
$$
The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}
=--
+-- {: .num_prop #CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator}
###### Proposition
**([[causal propagator]] as [[Fourier transform]] of [[delta distribution]] on the [[Fourier transform|Fourier transformed]] [[Klein-Gordon operator]])**
The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]]:
$$
\Delta_S(x,y)
\;=\;
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
\,,
$$
where the [[integrand]] is the product of the [[sign function]] of $k_0$ with the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] and a [[plane wave]] factor.
=--
+-- {: .proof}
###### Proof
By decomposing the integral over $k_0$ into its negative and its positive half, and applying the [[change of integration variables]] $k_0 = \pm\sqrt{h}$ we get
$$
\begin{aligned}
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
& =
+ i (2\pi)^{-p} \int \int_0^\infty \delta\left( -k_0^2 + \vec k^2 + \left( \tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0) + i \vec k \cdot (\vec x - \vec y)} d k_0 \, d^p \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_{-\infty}^0 \delta\left( -k_0^2 + \vec k^2 + \left(\tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0)+ i \vec k \cdot (\vec x - \vec y) } d k_0 \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( -h + \omega(\vec k)^2/c^2 \right) e^{ + i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( - h + \omega(\vec k)^2/c^2 \right) e^{ - i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x} d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ - i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x } d^{p} \vec k
\\
& = -(2 \pi)^{-p} \int \frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x-y)^0/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\end{aligned}
$$
The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.
=--
$\,$
**[[Wightman propagator]]**
{#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime}
Prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} exhibits the [[causal propagator]] of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] as the difference of a contribution for [[positive real number|positive]] temporal [[angular frequency]] $k_0 \propto \omega(\vec k)$ (hence positive [[energy]] $\hbar \omega(\vec k)$ and a contribution of negative temporal [[angular frequency]].
The [[positive real number|positive]] [[frequency]] contribution to the [[causal propagator]] is called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below), also known as the the _[[vacuum state]] [[2-point function]] of the [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]]_. Notice that the temporal component
of the [[wave vector]] is proportional to the _negative_ [[angular frequency]]
$$
k_0 = -\omega/c
$$
(see at _[[plane wave]]_), therefore the appearance of the [[step function]] $\Theta(-k_0)$ in (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) below:
+-- {: .num_defn #StandardHadamardDistributionOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] or [[vacuum state]] [[2-point function]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**
The _[[Wightman propagator]]_ for the [[Klein-Gordon operator]] at [[mass]] $m$ on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) is the [[tempered distribution|tempered]] [[distribution in two variables]] $\Delta_H \in \mathcal{S}'(\mathbb{R}^{p,1})$ which as a [[generalized function]] is given by the expression
$$
\label{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime}
\begin{aligned}
\Delta_H(x,y)
& \coloneqq
\frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
\\
& =
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\,,
\end{aligned}
$$
Here in the first line we have in the [[integrand]] the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] times a [[plane wave]] and times the [[step function]] $\Theta$ of the temporal component of the [[wave vector]]. In the second line we used the [[change of integration variables]] $k_0 = \sqrt{h}$, then the definition of the [[delta distribution]] and the fact that $\omega(\vec k)$ is by definition the [[non-negative real number|non-negative]] solution to the Klein-Gordon [[dispersion relation]].
=--
(e.g. [Khavkine-Moretti 14, equation (38) and section 3.4](Hadamard+distribution#KhavineMoretti14))
+-- {: .num_prop #OnMinkowskiWightmanIsDistributionalSolutionToKleinGordon}
###### Proposition
**([[Wightman propagator]] on [[Minkowski spacetime]] is [[distributional solution to a PDE|distributional solution]] to [[Klein-Gordon equation]])**
The [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is a [[distributional solution to a PDE|distributional solution]] (def. \ref{DistributionalDerivatives}) to the [[Klein-Gordon equation]]
$$
(\Box_x - m^2)\Delta_H(x,y) = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime} the Wightman propagator is the [[Fourier transform of distributions]] of the [[product of distributions]]
$$
\delta(k_\mu k^\mu + m^2) \Theta(-k_0)
\,,
$$
where in turn the argument of the [[delta distribution]] is just $-1$ times the Fourier transform of the [Klein-Gordon operator]] itself (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}). This is clearly a solution to the equation
$$
(-k_\mu k^\mu - m^2) \, \delta(k_\mu k^\mu + m^2) \Theta(-k_0)
\;=\;
0
\,.
$$
Under [[Fourier inversion theorem|Fourier inversion]] (prop. \ref{FourierInversion}), this is the equation $(\Box_x - m^2)\Delta_H(x,y) = 0$, as in the proof of prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}.
=--
+-- {: .num_prop #ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime}
###### Proposition
**([[contour integral]] representation of the [[Wightman propagator]] for the [[Klein-Gordon operator]] on [[Minkowski spacetime]])
The [[Wightman propagator]] from def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} is
equivalently given by the [[contour integral]]
$$
\label{StandardHadamardPropagatorOnMinkowskiSpacetimeInTermsOfContourIntegral}
\Delta_H(x,y)
\;=\;
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{-i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
\,,
$$
where the [[Jordan curve]] $C_+(\vec k) \subset \mathbb{C}$ runs counter-clockwise, enclosing the point $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$, but not enclosing the point $- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$.
<img src="https://ncatlab.org/nlab/files/ContourForHadamardPropagator.png" height="200">
> graphics grabbed from [Kocic 16](advanced+and+retarded+causal+propagators#Kocic16#Kocic16)
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{ - i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
& =
-i(2\pi)^{-(p+1)}
\int
\oint_{C_+(\vec k)}
\frac{
e^{ -i k_0 x^0} e^{i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
d k_0
d^p \vec k
\\
& =
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{
e^{ - i k_0 (x^0-y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 - \omega_\epsilon(\vec k) )
( k_0 + \omega_\epsilon(\vec k) )
}
d k_0
d^p \vec k
\\
& =
(2\pi)^{-p}
\int
\frac{1}{2 \omega(\vec k)}
e^{-i \omega(\vec k) (x^0-y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
d^p \vec k
\,.
\end{aligned}
$$
The last step is application of [[Cauchy's integral formula]], which says that the [[contour integral]] picks up the [[residue]] of the [[pole]] of the [[integrand]] at $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$. The last line is $\Delta_H(x,y)$, by definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime}.
=--
+-- {: .num_prop #SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(skew-symmetric part of [[Wightman propagator]] is the [[causal propagator]])**
The [[Wightman propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is of the form
$$
\label{DeompositionOfHadamardPropagatorOnMinkowkski}
\begin{aligned}
\Delta_H
& =
\tfrac{i}{2} \Delta_S
+
H
\\
& =
\tfrac{i}{2}
\left(
\Delta_+ - \Delta_-
\right)
+
H
\end{aligned}
\,,
$$
where
1. $\Delta_S$ is the [[causal propagator]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}),
which is real (eq:CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal) and skew-symmetric (prop. \ref{CausalPropagatorIsSkewSymmetric})
$$
(\Delta_S(x,y))^\ast = \Delta_S(x,y)
\phantom{AA}
\,,
\phantom{AA}
\Delta_S(y,x) = - \Delta_S(x,y)
$$
1. $H$ is real and symmetric
$$
\label{RealAndSymmetricH}
(H(x,y))^\ast = H(x,y)
\phantom{AA}
\,,
\phantom{AA}
H(y,x) = H(x,y)
$$
=--
+-- {: .proof}
###### Proof
By applying [[Euler's formula]] to (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) we obtain
$$
\label{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime}
\begin{aligned}
\Delta_H(x,y)
& =
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\\
& =
\tfrac{i}{2}
\underset{= \Delta_S(x,y)}{
\underbrace{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\;+\;
\underset{
\coloneqq H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\end{aligned}
$$
On the left this identifies the [[causal propagator]] by (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime), prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.
The second summand changes, both under complex conjugation as well as under $(x-y) \mapsto (y-x)$,
via [[change of integration variables]] $\vec k \mapsto - \vec k$ (because the [[cosine]] is an even function).
This does not change the integral, and hence $H$ is symmetric.
=--
$\,$
**[[Feynman propagator]]**
{#FeynmanPropagator}
We have seen that the [[positive real number|positive]] [[frequency]] component of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) is the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) given, according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}, by (eq:DeompositionOfHadamardPropagatorOnMinkowkski)
$$
\begin{aligned}
\Delta_H
& =
\tfrac{i}{2} \Delta_S
+
H
\\
& =
\tfrac{i}{2}
\left(
\Delta_+ - \Delta_-
\right)
+
H
\end{aligned}
\,,
$$
There is an evident variant of this combination, which will be of interest:
+-- {: .num_defn #FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The _[[Feynman propagator]]_ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) is the [[linear combination]]
$$
\Delta_F
\coloneqq
\tfrac{i}{2}
\left(
\Delta_+ + \Delta_-
\right)
+
H
$$
where the first term is proportional to the sum of the [[advanced and retarded propagators]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) and the second is the symmetric part of the [[Wightman propagator]] according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}.
Similarly the _[[anti-Feynman propagator]]_ is
$$
\Delta_{\overline{F}}
\coloneqq
\tfrac{i}{2}
\left(
\Delta_+ + \Delta_-
\right)
-
H
\,.
$$
=--
It follows immediately that:
+-- {: .num_prop #SymmetricFeynmanPropagator}
###### Proposition
**([[Feynman propagator]] is symmetric)
The [[Feynman propagator]] $\Delta_F$ and anti-Feynman propagator $\Delta_{\overline{F}}$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) are symmetric:
$$
\Delta_F(x,y) = \Delta_F(y,x)
\,.
$$
=--
+-- {: .proof}
###### Proof
By equation (eq:AdvancedAndRetardedPropagatorTurnIntoEachOtherUnderSwitchingArguments) in cor. \ref{CausalPropagatorIsSkewSymmetric} we have that $\Delta_+ + \Delta_-$
is symmetric, and equation (eq:RealAndSymmetricH) in prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} says that $H$ is symmetric.
=--
+-- {: .num_prop #ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(mode expansion for [[Feynman propagator]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is given by the following equivalent expressions
$$
\begin{aligned}
\Delta_F(x,y)
& =
\left\{
\array{
\frac{1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
\\
\Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
Similarly the [[anti-Feynman propagator]] is equivalently given by
$$
\begin{aligned}
\Delta_{\overline{F}}(x,y)
& =
\left\{
\array{
\frac{-}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{-}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
-\Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
\\
-\Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
=--
+-- {: .proof}
###### Proof
By the mode expansion of $\Delta_{\pm}$ from (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator) and the mode expansion of $H$ from (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have
$$
\begin{aligned}
\Delta_F(x,y)
& =
\left\{
\array{
\underset{
= \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
}{
\underbrace{
\frac{- i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
+
\underset{
= H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \gt 0
\\
\underset{
= 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
}{
\underbrace{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
+
\underset{ = H(x,y) }{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
\\
\Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
where in the second line we used [[Euler's formula]]. The last line follows by comparison with (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) and using that the integral over $\vec k$
is invariant under $\vec k \mapsto - \vec k$.
The computation for $\Delta_{\overline{F}}$ is the same, only now with a minus sign in front of the [[cosine]]:
$$
\begin{aligned}
\Delta_{\overline{F}}(x,y)
& =
\left\{
\array{
\underset{
= \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
}{
\underbrace{
\frac{- i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
-
\underset{
= H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \gt 0
\\
\underset{
= 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
}{
\underbrace{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
-
\underset{ = H(x,y) }{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}
}
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{-1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{-1i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
- \Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
\\
- \Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
}
\right.
\end{aligned}
$$
=--
As before for the [[causal propagator]], there are equivalent reformulations of the [[Feynman propagator]] which are useful for computations:
+-- {: .num_prop #FeynmanPropagatorAsACauchyPrincipalvalue}
###### Proposition
**([[Feynman propagator]] as a [[Cauchy principal value]])**
The [[Feynman propagator]] and [[anti-Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is equivalently given by the following expressions, respectively:
$$
\begin{aligned}
\left.
\array{
\Delta_F(x,y)
\\
\Delta_{\overline{F}}(x,y)
}
\right\}
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\end{aligned}
$$
where we have a [[limit of a sequence|limit]] of [[distributions]] as for the [[Cauchy principal value]] ([this prop](Cauchy+principal+vlue#CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta)).
=--
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
(k_0)^2
-
\underset{
\coloneqq \omega_{\pm\epsilon}(\vec k)^2/c^2
}{\underbrace{ \left( \omega(\vec k)^2/c^2 \pm i \epsilon \right) }}
}
\, d k_0 \, d^p \vec k
\\
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu (x^\mu - y^\mu)}
}{
\left(
k_0 - \omega_{\pm \epsilon}(\vec k)/c
\right)
\left(
k_0 + \omega_{\pm \epsilon}(\vec k)/c
\right)
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{\pm i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \gt 0
\\
\frac{\mp 1}{(2\pi)^p}
\int
\frac{1}{2\omega(\vec k)c}
e^{\mp i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
&\vert&
(x^0 - y^0) \lt 0
}
\right.
\\
& =
\left\{
\array{
\Delta_F(x,y)
\\
\Delta_{\overline{F}}(x,y)
}
\right.
\end{aligned}
$$
Here
1. In the first step we introduced the [[complex number|complex]] [[square root]] $\omega_{\pm \epsilon}(\vec k)$. For this to be compatible with the choice of _non-negative_ square root for $\epsilon = 0$ in (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) we need to choose that complex square root whose [[complex phase]] is one half that of $\omega(\vec k)^2 - i \epsilon$ (instead of that plus [[π]]). This means that $\omega_{+ \epsilon}(\vec k)$ is in the _[[upper half plane]]_ and $\omega_-(\vec k)$ is in the [[lower half plane]].
1. In the third step we observe that
1. for $(x^0 - y^0) \gt 0$ the [[integrand]] decays for [[positive real number|positive]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[upper half plane]];
1. for $(x^0 - y^0) \lt 0$ the integrand decays for [[negative real number|negative]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[lower half plane]]
and then apply [[Cauchy's integral formula]] which picks out $2\pi i$ times the [[residue]] a these poles.
<img src="https://ncatlab.org/nlab/files/ContourForFeynmanPropagator.png" height="300">
Notice that when completing to a contour in the [[lower half plane]] we pick up a minus signs from the fact that now the contour runs clockwise.
1. In the fourth step we used prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}.
=--
It follows that:
\begin{proposition}
\label{GreenFunctionFeynmanPropagator}
**([[Feynman propagator]] is [[Green function]])**
\linebreak
The [[Feynman propagator]] $\Delta_F$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) is proportional to a [[Green function]] for the [[Klein-Gordon equation]] in that
$$
\left( \Box_x - \left( \tfrac{m c}{\hbar}\right)^2 \right) \Delta_{F}(x,y)
=
(+i) \delta(x-y)
\,.
$$
\end{proposition}
+-- {: .proof}
###### Proof
Equation (eq:FeynmanPropagatorInCauchyPrincipalValueForm) in prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} says that the Feynman propagator is the [[Fourier inversion theorem|inverse]] [[Fourier transform of distributions]] of
$$
\widehat{\Delta_F}(k)
\;=\;
(+i)
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{
1
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
$$
This implies the statement as in the proof of prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}, via the analogue of equation (eq:LimitOfDistributionsForFourierTransformedPropagators).
=--
$\,$
**[[singular support]] and [[wave front sets]]**
{#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime}
We now discuss the [[singular support]] (def. \ref{SingularSupportOfADistribution}) and the [[wave front sets]] (def. \ref{WaveFrontSet}) of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]].
+-- {: .num_prop #SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[causal propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**
The [[singular support]] of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]],
regarded via [[translation]] [[invariant|invariance]] as a [[generalized function]] in a single variable (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
is the [[light cone]] of the origin:
$$
supp_{sing}(\Delta_S)
\;=\;
\left\{
x \in \mathbb{R}^{p,1}
\,\vert\,
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by the basic properties of the Fourier transform (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}) this is the [[convolution of distributions]] of the separate
Fourier transforms:
$$
\begin{aligned}
\Delta_S(x)
& \propto
\widehat{
\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
}
\\
&\propto
\widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
\star
\widehat{sgn( k_0 )}
\end{aligned}
$$
By prop. \ref{FourierTransformOfDeltaDistributionappliedToMassShell}, the [[singular support]]
of the first convolution factor is the [[light cone]].
The second factor is
$$
\begin{aligned}
\widehat{sgn(k_0)}
& \propto
\left(2\widehat{\Theta(k_0)} - \widehat{1}\right) \delta(\vec k)
\\
& \propto
\left(2\tfrac{1}{i x^0 + 0^+} - \delta(x^0)\right) \delta(\vec k)
\end{aligned}
$$
(by example \ref{FourierTransformOfDeltaDistribution} and example \ref{RelationToFourierTransformOfHeavisideDistribution}) and hence the [[wave front set]] (def. \ref{WaveFrontSet}) of the second factor is
$$
WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$
(by example \ref{WaveFrontOfDeltaDistribution} and example \ref{PrincipalValueOfInverseFunctionCharacteristicEquation}).
With this the statement follows, via a [[partition of unity]], from [this prop.](convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions).
For illustration we now make this general argument more explicit in the special case of [[spacetime]] [[dimension]]
$$
p + 1 = 3 + 1
$$
by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].
We follow ([Scharf 95 (2.3.18)](causal+perturbation+theory#Scharf95)).
Consider the formula for the [[causal propagator]] in terms of the mode expansion (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime). Since the [[integrand]] here depends on the [[wave vector]] $\vec k$ only via its [[norm]] ${\vert \vec k\vert}$ and the [[angle]] $\theta$ it makes with the given [[spacetime]] [[vector]] via
$$
\vec k \cdot (\vec x - \vec y)
\;=\;
{\vert \vec k\vert} \, {\vert \vec x\vert} \, \cos(\theta)
$$
we may express the [[integration]] in terms of [[polar coordinates]] as follws:
$$
\begin{aligned}
\Delta_S(x - y)
& =
\frac{-1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
\\
& =
\frac{- vol_{S^{p-2}}}{(2\pi)^p}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\underset{ \theta \in [0,\pi] }{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
{\vert \vec k\vert} ({\vert \vec k\vert} \sin(\theta))^{p-2}
\,
d \theta
\wedge
d {\vert \vec k\vert}
\end{aligned}
$$
In the special case of [[spacetime]] [[dimension]] $p + 1 = 3 + 1$ this becomes
$$
\label{StepsInComputingCausalPropagatorIn3plus1Dimension}
\begin{aligned}
\Delta_S(x - y)
& =
\frac{- 2\pi}{(2\pi)^{3}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert}^2 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\underset{
=
\tfrac{1}{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} }
\left(
e^{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
-
e^{-i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
\right)
}{
\underbrace{
\underset{ \cos(\theta) \in [-1,1] }{\int}
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
d \cos(\theta)
}
}
\wedge
d {\vert \vec k \vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert} }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\sin\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\cos\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 1}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left( \omega(\kappa) (x^0 - y^0) /c \right)
\cos\left( \kappa\, {\vert \vec x - \vec y\vert} \right)
\, d \kappa
\\
& =
\frac{- 1}{2(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y} \vert }
\left(
\underset{\coloneqq I_+}{
\underbrace{
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
+
\kappa\, {\vert \vec x - \vec y\vert}
\right)
d\kappa
}
}
+
\underset{ \coloneqq I_- }{
\underbrace{
\underset{ \kappa \in \mathbb{R} }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
-
\kappa\, {\vert \vec x - \vec y\vert}
\right)
\,
d \kappa
}
}
\right)
\,.
\end{aligned}
$$
Here in the second but last step we renamed $\kappa \coloneqq {\vert \vec k\vert}$ and
doubled the integration domain for convenience,
and in the last step we used the [[trigonometric identity]] $\sin(\alpha) \cos(\beta)\;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$.
In order to further evaluate this, we parameterize the remaining components $(\omega/c, \kappa)$ of the [[wave vector]]
by the dual [[rapidity]] $z$, via
$$
\left(\cosh(z)\right)^2 - \left( \sinh(z)\right)^2 = 1
$$
as
$$
\omega(\kappa)/c
\;=\;
\left(
\tfrac{m c}{\hbar}
\right)
\cosh(z)
\phantom{AA}
\,,
\phantom{AA}
\kappa
\;=\;
\left(
\tfrac{m c}{\hbar}
\right)
\sinh(z)
\,,
$$
which makes use of the fact that $\omega(\kappa)$ is non-negative, by construction.
This [[change of integration variables]] makes the integrals under the braces above become
$$
\label{TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski}
I_\pm
\;=\;
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
\,.
$$
Next we similarly parameterize the vector $x-y$ by its [[rapidity]] $\tau$. That parameterization depends on whether
$x-y$ is spacelike or not, and if not, whether it is future or past directed.
First, if $x-y$ is [[spacelike]] in that ${\vert x-y\vert}^2_\eta \gt 0$
then we may parameterize as
$$
(x^0 - y^0)
=
\sqrt{{\vert x-y\vert}^2_\eta} \sinh(\tau)
\phantom{AA}
\,,
\phantom{AA}
{\vert \vec x - \vec y\vert}
=
\sqrt{ {\vert x-y\vert}^2_\eta} \cosh(\tau)
$$
which yields
$$
\begin{aligned}
I_{\pm}
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh(\tau) \cosh(z)
\pm
\cosh(\tau)
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta}
\left(
\sinh\left( \tau \pm z\right)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh\left( z \right)
\right)
\right)
\, d z
\\
& = 0
\,,
\end{aligned}
$$
where in the last line we observe that the integrand is a skew-symmetric function of $z$.
Second, if $x-y$ is [[timelike]] with $(x^0 - y^0) \gt 0$ then we may parameterize as
$$
(x^0 - y^0)
=
\sqrt{ -{\vert x-y\vert}^2_\eta} \cosh(\tau)
\phantom{AA}
\,,
\phantom{AA}
{\vert \vec x - \vec y\vert}
=
\sqrt{ -{\vert x - y\vert}^2_\eta }
\sinh(\tau)
$$
which yields
$$
\label{IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski}
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\sin\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\sqrt{ - {\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\left(
\cosh(\tau)\cosh(z)
\pm
\cosh(\tau)
\sinh(z)
\right)
\right)
\, d z
\\
& =
\int_{-\infty}^\infty
\sin\left(
\sqrt{ - {\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\left(
\cosh(z \pm \tau)
\right)
\right)
\, d z
\\
& =
\pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
\end{aligned}
\,.
$$
Here in the last line we identified the integral representation of the [[Bessel function]] $J_0$ of order 0 (see [here](Bessel+function#eq:J0AsIntSinOfxCoshtdt)). The important point here is that this is a smooth function.
Similarly, if $x-y$ is [[timelike]] with $(x^0 - y^0) \lt 0$ then the same argument yields
$$
I_\pm = - \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
$$
In conclusion, the general form of $I_\pm$ is
$$
I_\pm
=
\pi
sgn(x^0 - y^0)
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right)
\,.
$$
Therefore we end up with
$$
\label{FinalResultOfComputationOf3Plus1dCausalPropagator}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{1}{4 \pi {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y\vert}}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
\sqrt{ -{\vert x-y\vert}^2_\eta }
\tfrac{m c}{\hbar}
\right)
\\
& =
\frac{-1}{2 \pi }
\frac{d}{d (-{\vert x-y\vert}^2_\eta)}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
\sqrt{-{\vert x-y \vert}^2_\eta} \tfrac{m c}{\hbar}
\right)
\\
& =
-\frac{1}{2 \pi }
\frac{d}{d (- \vert x-y\vert^2_{\eta})}
sgn(x^0) \Theta\left( - {\vert x - y\vert}^2_\eta \right)
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\\
& =
\frac{-1}{2\pi}
sgn(x^0)
\left(
\delta\left(
-{\vert x-y\vert}^2_\eta
\right)
\;-\;
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
\frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\right)
\end{aligned}
$$
=--
+-- {: .num_prop #SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[Wightman propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**
The [[singular support]] of the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:
$$
supp_{sing}(\Delta_H)
=
\left\{
x \in \mathbb{R}^{p,1}
\;\vert\;
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by basic properties of the Fourier transform (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}) this is the [[convolution of distributions]] of the separate
Fourier transforms:
$$
\begin{aligned}
\Delta_S(x)
& \propto
\widehat{
\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
}
\\
&\propto
\widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
\star
\widehat{sgn( k_0 )}
\end{aligned}
$$
By prop. \ref{FourierTransformOfDeltaDistributionappliedToMassShell}, the [[singular support]]
of the first convolution factor is the [[light cone]].
The second factor is
$$
\widehat{\Theta(k_0)}
\propto
\tfrac{1}{i x^0 + 0^+} \delta(\vec k)
$$
(by example \ref{FourierTransformOfDeltaDistribution} and example \ref{RelationToFourierTransformOfHeavisideDistribution} and hence the [[wave front set]] (def. \ref{WaveFrontSet}) of the second
factor is
$$
WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$
(by example \ref{WaveFrontOfDeltaDistribution} and example \ref{PrincipalValueOfInverseFunctionCharacteristicEquation}).
With this the statement follows, via a [[partition of unity]], from prop. \ref{WaveFrontSetOfCompactlySupportedDistributions}.
For illustration, we now make this general statement fully explicit in the special case of [[spacetime]] [[dimension]]
$$
p + 1 = 3 + 1
$$
by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].
We follow ([Scharf 95 (2.3.36)](causal+perturbation+theory#Scharf95)).
By (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have
$$
\begin{aligned}
\Delta_H(x,y)
& =
\tfrac{i}{2}
\underset{= \Delta_S(x,y)}{
\underbrace{
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\;+\;
\underset{
\coloneqq H(x,y)
}{
\underbrace{
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\cos\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
}}
\end{aligned}
$$
The first summand, proportional to the [[causal propagator]], which we computed as (eq:FinalResultOfComputationOf3Plus1dCausalPropagator)
in prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} to be
$$
\tfrac{i}{2}\Delta_S(x,y)
\;=\;
\frac{-i}{4\pi}
sgn(x^0)
\left(
\delta\left(
-{\vert x-y\vert}^2_\eta
\right)
\;-\;
\Theta\left( -{\vert x-y\vert}^2_\eta \right)
\frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
J_0\left(
\tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\right)
\,.
$$
The second term is computed in a directly analogous fashion: The integrals $I_\pm$ from
(eq:TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski) are now
$$
I_\pm
\coloneqq
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\left(
(x^0 - y^0) \cosh(z)
\pm
{\vert \vec x - \vec y\vert}
\sinh(z)
\right)
\right)
\, d z
$$
Parameterizing by [[rapidity]], as in the proof of prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}, one finds that for [[timelike]] $x-y$ this is
$$
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\cosh\left( z \right)
\right)
\right)
\, d z
\\
& =
- \pi N_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\end{aligned}
$$
while for [[spacelike]] $x-y$ it is
$$
\begin{aligned}
I_\pm
& =
\int_{-\infty}^\infty
\cos\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\left(
\sinh\left( z \right)
\right)
\right)
\, d z
\\
& =
2 K_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\,,
\end{aligned}
$$
where we identified the integral representations of the [[Neumann function]] $N_0$ (see [here](Bessel+function#N0AsIntSinOfxCoshtdt))
and of the [[modified Bessel function]] $K_0$ (see [here](Bessel+function#eq:K0AsIntSinOfxCoshtdt)).
As for the [[Bessel function]] $J_0$ in (eq:IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski) the key point is that these are [[smooth functions]]. Hence we conclude that
$$
H(x,y)
\;\propto\;
\frac{d}{d \left( {\vert x-y\vert}^2_\eta \right)}
\left(
-\Theta\left( -{\vert x-y\vert}^2_\eta \right)
N_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
+
\Theta\left( {\vert x-y\vert}^2_\eta \right)
\tfrac{2}{\pi}
K_0
\left(
\tfrac{m c}{\hbar}
\sqrt{ {\vert x-y\vert}^2_\eta }
\right)
\right)
\,.
$$
This expression has singularities on the [[light cone]] due to the [[step functions]].
In fact the expression being differentiated is continuous at the light cone
([Scharf 95 (2.3.34)](causal+perturbation+theory#Scharf95)), so that the singularity on the light cone is not a [[delta distribution]]
singularity from the derivative of the step functions. Accordingly it does not cancel the singularity
of $\tfrac{i}{2}\Delta_S(x,y)$ as above, and hence the singular support of $\Delta_H$ is still the whole
light cone.
=--
+-- {: .num_prop #SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**([[singular support]] of [[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[singular support]] of the [[Feynman propagator]] $\Delta_H$ and of the [[anti-Feynman propagator]] $\Delta_{\overline{F}}$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:
$$
\left.
\array{
supp_{sing}(\Delta_F)
\\
supp_{sing}(\Delta_{\overline{F}})
}
\right\}
=
\left\{
x \in \mathbb{R}^{p,1}
\;\vert\;
{\vert x\vert}^2_\eta = 0
\right\}
\,.
$$
=--
(e.g [DeWitt 03 (27.85)](Feynman+propagator#DeWitt03))
+-- {: .proof}
###### Proof
By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the Feynman propagator is equivalently
the [[Cauchy principal value]] of the inverse of the Fourier transformed Klein-Gordon operator:
$$
\Delta_F
\;\propto\;
\widehat{
\frac{1}{-k_\mu k^\mu - \left(\tfrac{m c}{\hbar}\right)^2 + i 0^+}
}
\,.
$$
With this, the statement follows immediately from prop. \ref{FourierTransformOfPrincipalValueOfPowerOfQuadraticForm}.
=--
+-- {: .num_prop #WaveFronSetsForKGPropagatorsOnMinkowski}
###### Proposition
**([[wave front sets]] of [[propagators]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**
The [[wave front set]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded, via [[translation]] [[invariant|invariance]], as [[distributions]] in a single variable, are as follows:
* the [[causal propagator]] $\Delta_S$ (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the lightcone:
$$
WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0 \right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60"> <br/> - <br/> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60">
</center>
* the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $k^0 \gt 0$:
$$
WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0 \right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60">
</center>
* the [[Feynman propagator]] $\Delta_S$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0$
$$
WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \;
\left(
\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0
\right)
\right\}
$$
<center>
<img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60">
</center>
=--
([Radzikowski 96, (16)](Hadamard+distribution#Radzikowski96))
+-- {: .proof}
###### Proof
First regarding the causal propagator:
By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the [[singular support]] of $\Delta_S$ is the [[light cone]].
Since the causal propagator is a solution to the homogeneous Klein-Gordon equation, the
[[propagation of singularities theorem]] (prop. \ref{PropagationOfSingularitiesTheorem}) says that also all [[wave vectors]] in the wave front set are [[lightlike]].
Hence it just remains to show that all non-vanishing lightlike wave vectors based on the lightcone in spacetime
indeed do appear in the wave front set.
To that end, let $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ be a [[bump function]] whose [[compact support]] includes the origin.
For $a \in \mathbb{R}^{p,1}$ a point on the light cone, we need to determine the decay property of the Fourier transform of $x \mapsto b(x-a)\Delta_S(x)$. This is the [[convolution of distributions]] of $\hat b(k)e^{i k_\mu a^\mu}$ with $\widehat \Delta_S(k)$. By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have
$$
\widehat \Delta_{S}(k)
\;\propto\;
\delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right)
sgn(k_0)
\,.
$$
This means that the convolution product is the smearing of the mass shell by $\widehat b(k)e^{i k_\mu a^\mu}$.
Since the mass shell asymptotes to the light cone, and since $e^{i k_\mu a^\mu} = 1$ for $k$ on the light cone (given that $a$ is on the light cone), this implies the claim.
Now for the [[Wightman propagator]]:
By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form
$$
\widehat \Delta_H(k)
\;\propto\;
\delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 )
$$
Moreover, its [[singular support]] is also the light cone (prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}).
Therefore now same argument as before says that the wave front set consists of wave vectors $k$ on the light cone, but now due to the [[step function]] factor $\Theta(-k_0)$ it must satisfy $0 \leq - k_0 = k^0$.
Finally regarding the [[Feynman propagator]]:
by prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime} the Feynman propagator
coincides with the positive frequency Wightman propagator for $x^0 \gt 0$ and with the "negative frequency Hadamard operator"
for $x^0 \lt 0$. Therefore the form of $WF(\Delta_F)$ now follows directly with that of $WF(\Delta_H)$ above.
=--
$\,$
**[[propagators]] for the [[Dirac equation]] on [[Minkowski spacetime]]**
{#DiracEquationOnMinkowskiSpacetimePropagators}
We now discuss how the [[propagators]] for the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{GreenHyperbolicDiracOperator}) follow directly from those for the [[scalar field]] discussed above.
+-- {: .num_prop #DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}
###### Proposition
**([[advanced and retarded propagator]] for [[Dirac equation]] on [[Minkowski spacetime]])**
Consider the [[Dirac operator]] on [[Minkowski spacetime]], which in [[Feynman slash notation]] reads
$$
\begin{aligned}
D
& \coloneqq
-i {\partial\!\!\!/\,}
+ \tfrac{m c}{\hbar}
\\
& =
-i \gamma^\mu \frac{\partial}{\partial x^\mu} + \tfrac{m c}{\hbar}
\end{aligned}
\,.
$$
Its [[advanced and retarded propagators]] (def. \ref{AdvancedAndRetardedGreenFunctions})
are the [[derivatives of distributions]] of the advanced and retarded propagators $\Delta_\pm$ for the [[Klein-Gordon equation]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) by ${\partial\!\!\!/\,} + m$:
$$
\Delta_{D, \pm}
\;=\;
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
\Delta_{\pm}
\,.
$$
Hence the same is true for the [[causal propagator]]:
$$
\Delta_{D, S}
\;=\;
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
\Delta_{S}
\,.
$$
=--
+-- {: .proof}
###### Proof
Applying a [[differential operator]] does not change the [[support]] of a [[smooth function]], hence also not the [[support of a distribution]]. Therefore the uniqueness of the advanced and retarded propagators (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique})
together with the translation-invariance and the anti-[[formally self-adjoint differential operator|formally self-adjointness]] of the [[Dirac operator]] (as for the [[Klein-Gordon operator]] (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
implies that it is sufficent to check that applying the [[Dirac operator]] to the $\Delta_{D, \pm}$
yields the [[delta distribution]]. This follows since the Dirac operator squares to the Klein-Gordon operator:
$$
\begin{aligned}
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)
\Delta_{D, \pm}
& =
\underset{ = \Box - \left(\tfrac{m c}{\hbar}\right)^2}{
\underbrace{
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)
\left(
-i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
\right)
}
}
\Delta_{\pm}
\\
& =
\delta
\end{aligned}
\,.
$$
=--
Similarly we obtain the other [[propagators]] for the [[Dirac field]] from those of the [[real scalar field]]:
+-- {: .num_defn #HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**
The _[[Wightman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the [[positive real number|positive]] [[frequency]] part of the [[causal propagator]] (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}),
hence the [[derivative of distributions]] (def. \ref{DistributionalDerivatives}) of the Wightman propagator for the Klein-Gordon field (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) by the [[Dirac operator]]:
$$
\begin{aligned}
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)\Delta_{H}(x,y)
& =
\frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) ( {k\!\!\!/\,} + \tfrac{m c}{\hbar}) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
\\
& =
\frac{1}{(2\pi)^p}
\int \frac{ \gamma^0 \omega(\vec k)/c + \vec \gamma \cdot \vec k + \tfrac{m c}{\hbar} }{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\,.
\end{aligned}
$$
Here we used the expression (eq:StandardHadamardDistributionOnMinkowskiSpacetime) for the Wightman propagator of the Klein-Gordon equation.
=--
+-- {: .num_defn #FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**
The _[[Feynman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the linear combination
$$
\Delta_{D, F}
\;\coloneqq\;
\Delta_{D,H}
+
i \Delta_{D, -}
$$
of the [[Wightman propagator]] (def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}) and the retarded propagator (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}). By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}
this means that it is the
[[derivative of distributions]] (def. \ref{DistributionalDerivatives}) of the [[Feynman propagator]] of the [[Klein-Gordon equation]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) by the [[Dirac operator]]
$$
\begin{aligned}
\Delta_{D, F}
& =
\left(
-i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
\right)\Delta_{F}(x,y)
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{-i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
\left( {k\!\!\!/\,} + \tfrac{m c}{\hbar} \right) e^{i k_\mu (x^\mu - y^\mu)}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
}
\, d k_0 \, d^p \vec k
\,.
\end{aligned}
$$
=--
$\,$
$\,$
This concludes our discussion of [[propagators]] induced from the [[covariant phase space]] of [[Green hyperbolic differential equation|Green hyperbolic]] [[free field theory|free]] [[Lagrangian field theory]]. These propagators will be the key in for [[quantization]] via [[causal perturbation theory]]. But not all [[free field theories]] have a [[covariant phase space]] of [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]], for instance the [[electromagnetic field]], a priori, does not. Therefore before turning to [[quantization]] in the [next chapter](#GaugeSymmetries) we first discuss how _[[gauge symmetries]]_ [[obstruction|obstruct]] the existence of [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]].
|
A first idea of quantum field theory -- Quantization | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Quantization | ## Quantization
{#Quantization}
In this chapter we discuss the following topics:
* _[Motivation from Lie theory](#MotivationFromLieTheory)_
* _[Geometric quantization](#GeometricQuantization)_
* _[Moyal star products](#StarProducts)_
* _[Moyal star product as deformation quantization](#MoyalStarProductAsDeformationQuantization)_
* _[Moyal star product via geometric quantization](#MoyalStarProductViaGeometricQuantization)_
* _[Example: Wick algebra of normal ordered product on Kähler vector space](#WickAlgebraOfNormalOrderedProductsOnKählerVectorspace)_
* _[Star-product on regular polynomial observables in field theory](#RegularPolynomialFieldTheoryStarProduct)_
$\,$
In the previous chapters we had found the [[Peierls-Poisson bracket]] (theorem \ref{PPeierlsBracket}) on the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) of a [[gauge fixing|gauge fixed]] (def. \ref{GaugeFixingLagrangianDensity}) [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}).
This [[Poisson bracket]] (def. \ref{SuperPoissonAlgebra} below) is a [[Lie bracket]] and hence reflects [[infinitesimal symmetries]] acting on the [[covariant phase space]]. Just as with the [[infinitesimal symmetries of the Lagrangian]] and the [[BRST-complex|BRST]]-[[reduced phase space|reduced]] [[field bundle]] (example \ref{LocalOffShellBRSTComplex}), we may hard-wire these [[Hamiltonian]] symmetries into the very geometry of the phase space by forming their [[homotopy quotient]] given by the corresponding [[Lie algebroid]] (def. \ref{LInfinityAlgebroid}): here this is called the _[[Poisson Lie algebroid]]_. Its [[Lie integration]] to a finite (instead of infinitesimal) structure is called the _[[symplectic groupoid]]_. This is the original [[covariant phase space]], but with its [[Hamiltonian flows]] hard-wired into its [[higher differential geometry]] ([[schreiber:master thesis Bongers|Bongers 14, section 4]]).
Where smooth functions on the plain covariant phase space form the [[commutative algebra|commutative]] [[algebra of observables]] under their pointwise product (def. \ref{Observable}), the smooth functions on this [[symplectic groupoid]]-refinement of the phase space are multiplied by the _[[groupoid convolution product]]_ and as such become a [[noncommutative algebra|non-commutative]] [[algebra of quantum observables]]. This passage from the commutative to the non-commutative algebra of observables is called _[[quantization]]_, here specifically _[[geometric quantization of symplectic groupoids]]_ ([Hawkins 04](geometric+quantization+of+symplectic+groupoids#EH), [[schreiber:master thesis Nuiten|Nuiten 13]]).
Instead of discussing this in generality, we here focus right away on the simple special case relevant for the [[quantization]] of [[gauge fixing|gauge fixed]] [[free field theories|free]] [[Lagrangian field theories]] in the [next chapter](#FreeQuantumFields).
After an informal motivation of [[geometric quantization]] from [[Lie theory]] [below](#MotivationFromLieTheory) (for a self-contained introduction see [[schreiber:master thesis Bongers|Bongers 14]]), we first showcase [[geometric quantization]] by discussing how the archetypical example of [[quantum mechanics]] in the [[Schrödinger representation]] arises from the _[[polarization|polarized]]_ action of the [[Poisson bracket]] [[Lie algebra]] (example \ref{GeometricRepresentationSchroedingerRepresentation} below). With the concept of [[polarization]] thus motivated, we use this to find the polarized [[groupoid convolution algebra]] of the [[symplectic groupoid]] of a free theory (prop. \ref{PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct} below).
The result is the "[[Moyal star product|Moyal]]-[[star product]]" (def. \ref{StarPoduct} below). This is the [[exponentiation]] of the [[integral kernel]] of the [[Poisson bracket]] plus possibly a symmetric shift (prop. \ref{SymmetricContribution} below); it turns out to be (example \ref{MoyalStarProductIsFormalDeformationQuantization} below) a _[[formal deformation quantization]]_ of the original commutative pointwise product (def. \ref{FormalDeformationQuantization} below).
Below we spell out the (elementary) proofs of these statements for the case of [[phase spaces]] which are [[finite dimensional vector spaces]]. But these proofs manifestly depend only on elementary algebraic properties of [[polynomials]] and hence go through in more general contexts as long as these basic algebraic properties are retained.
In the context of [[free field theory|free]] [[Lagrangian field theory]] the analogue of the [[formal power series algebras]] on a linear [[phase space]] is, a priori, the algebra of [[polynomial observables]] (def. \ref{PolynomialObservables}). These are effectively [[polynomials]] in the [[field observables]] $\mathbf{\Phi}^a(x)$ (def. \ref{PointEvaluationObservables}) whose [[coefficients]], however, are [[distributions]] [[distribution of several variables|of several variables]]. By [[microlocal analysis]], such polynomial distributions do satisfy the usual algebraic properties of ordinary polynomials _if_ potential [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}) encoded in their [[wave front set]] (def. \ref{WaveFrontSet}) vanish, according to _[[Hörmander's criterion]]_ (prop. \ref{HoermanderCriterionForProductOfDistributions}).
This criterion is always met on the subspace of _[[regular polynomial observables]]_ and hence every [[propagator]] induces a [[star product]] on these (prop. \ref{PropagatorStarProduct} below). In particular thus the [[star product]] of the [[causal propagator]] of a [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]] is a [[formal deformation quantization]] of its algebra of [[regular polynomial observables]] (cor. \ref{FreeGaugeFixedLagrangianFieldTheoryQuantizationOfRegularObservables} below). In order to extend this to [[local observables]] one may appeal to a certain quantization freedom (prop. \ref{SymmetricContribution} below) and shift the [[causal propagator]] by a symmetric contribution, such that it becomes the [[Wightman propagator]]; this is the topic of the following chapters (remark \ref{TowardsQuantizationExtendingBeyondRegularPolynomialObservables} at the end below).
In conclusion, for [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] the product in the [[algebra of quantum observables]] is given by [[exponential|exponentiating]] [[propagators]]. It is the [[combinatorics]] of these exponentiated propagator expressions that yields the hallmark structures of [[perturbative quantum field theory]], namely the combinatorics of [[Wick's lemma]] for the [[Wick algebra]] of free fields, and the combinatorics of [[Feynman diagrams]] for the [[time-ordered products]]. This is the topic of the following chapters _[Free quantum fields](#FreeQuantumFields)_ and _[Scattering](#Scattering)_. Here we conclude just with discussing the finite-dimensional toy version of the [[normal-ordered product]] in the [[Wick algebra]] (example \ref{WickAlgebraOfASingleMode} below).
$\,$
**motivation from Lie theory**
{#MotivationFromLieTheory}
Quantization of course was and is motivated by experiment, hence by observation of the [[observable universe]]: it just so happens that [[quantum mechanics]] and [[quantum field theory]] correctly account for experimental observations where [[classical mechanics]] and [[classical field theory]] gives no answer or incorrect answers. A historically important example is the phenomenon called the "[[ultraviolet catastrophe]]", a [[paradox]] predicted by classical [[statistical mechanics]] which is _not_ observed in nature, and which is corrected by [[quantum mechanics]].
But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from [[classical mechanics]] to [[quantum mechanics]]. Could one have been led to [[quantum mechanics]] by just pondering the mathematical formalism of [[classical mechanics]]?
The following spells out an argument to this effect. It will work for readers with a background in modern [[mathematics]], notably in [[Lie theory]], and with an understanding of the formalization of classical/prequantum mechanics in terms of [[symplectic geometry]].
So to briefly recall, a system of [[classical mechanics]]/[[prequantum field theory|prequantum mechanics]] is a [[phase space]], formalized as a [[symplectic manifold]] $(X, \omega)$. A symplectic manifold is in particular a [[Poisson manifold]], which means that the [[algebra of functions]] on [[phase space]] $X$, hence the algebra of _classical [[observables]]_, is canonically equipped with a compatible [[Lie bracket]]: the _[[Poisson bracket]]_. This Lie bracket is what controls [[dynamics]] in [[classical mechanics]]. For instance if $H \in C^\infty(X)$ is the function on [[phase space]] which is interpreted as assigning to each configuration of the system its [[energy]] -- the [[Hamiltonian]] function -- then the [[Poisson bracket]] with $H$ yields the [[infinitesimal object|infinitesimal]] time evolution of the system: the [[differential equation]] famous as [[Hamilton's equations]].
Something to take notice of here is the _[[infinitesimal space|infinitesimal]]_ nature of the [[Poisson bracket]]. Generally, whenever one has a [[Lie algebra]] $\mathfrak{g}$, then it is to be regarded as the [[infinitesimal object|infinitesimal]] approximation to a globally defined object, the corresponding [[Lie group]] (or generally [[smooth group]]) $G$. One also says that $G$ is a _[[Lie integration]]_ of $\mathfrak{g}$ and that $\mathfrak{g}$ is the [[Lie differentiation]] of $G$.
Therefore a natural question to ask is: _Since the observables in [[classical mechanics]] form a [[Lie algebra]] under [[Poisson bracket]], what then is the corresponding [[Lie group]]?_
The answer to this is of course "well known" in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this [[Lie group]] which integrates the [[Poisson bracket]] is the "[[quantomorphism group]]", an object that seamlessly leads to the [[quantum mechanics]] of the system.
Before we spell this out in more detail, we need a brief technical aside: of course [[Lie integration]] is not quite unique. There may be different global [[Lie group]] objects with the same [[Lie algebra]].
The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian [[line Lie algebra]] $\mathbb{R}$. This has essentially two different [[Lie groups]] associated with it: the [[simply connected topological space|simply connected]] [[translation group]], which is just $\mathbb{R}$ itself again, equipped with its canonical additive [[abelian group]] structure, and the [[discrete space|discrete]] [[quotient]] of this by the group of [[integers]], which is the [[circle group]]
$$
U(1) = \mathbb{R}/\mathbb{Z}
\,.
$$
Notice that it is the discrete and hence "quantized" nature of the [[integers]] that makes the [[real line]] become a [[circle]] here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is "quantized" about [[quantum mechanics]].
Namely, one finds that the [[Poisson bracket]] [[Lie algebra]] $\mathfrak{poiss}(X,\omega)$ of the classical [[observables]] on [[phase space]] is (for $X$ a [[connected topological space|connected]] [[manifold]]) a [[Lie algebra extension]] of the Lie algebra $\mathfrak{ham}(X)$ of [[Hamiltonian vector fields]] on $X$ by the [[line Lie algebra]]:
$$
\mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X)
\,.
$$
This means that under [[Lie integration]] the [[Poisson bracket]] turns into an [[central extension]] of the group of [[Hamiltonian symplectomorphisms]] of $(X,\omega)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting [[central extension]] by the [[circle group]] $U(1)$. For this non-trivial [[Lie integration]] to exist, $(X,\omega)$ needs to satisfy a quantization condition which says that it admits a [[prequantum line bundle]]. If so, then this $U(1)$-[[central extension]] of the group $Ham(X,\omega)$ of [[Hamiltonian symplectomorphisms]] exists and is called... the _[[quantomorphism group]]_ $QuantMorph(X,\omega)$:
$$
U(1) \longrightarrow QuantMorph(X,\omega) \longrightarrow Ham(X,\omega)
\,.
$$
While important, for some reason this group is not very well known, which is striking because it contains a small [[subgroup]] which is famous in [[quantum mechanics]]: the _[[Heisenberg group]]_.
More precisely, whenever $(X,\omega)$ itself has a [[Hamiltonian action|compatible]] [[group]] structure, notably if $(X,\omega)$ is just a [[symplectic vector space]] (regarded as a group under addition of vectors), then we may ask for the [[subgroup]] of the [[quantomorphism group]] which covers the (left) [[action]] of [[phase space]] $(X,\omega)$ on itself. This is the corresponding [[Heisenberg group]] $Heis(X,\omega)$, which in turn is a $U(1)$-[[central extension]] of the group $X$ itself:
$$
U(1) \longrightarrow Heis(X,\omega) \longrightarrow X
\,.
$$
At this point it is worth pausing for a second to note how the hallmark of [[quantum mechanics]] has appeared as if out of nowhere simply by applying [[Lie integration]] to the [[Lie algebra|Lie algebraic]] structures in [[classical mechanics]]:
if we think of [[Lie integration|Lie integrating]] $\mathbb{R}$ to the interesting [[circle group]] $U(1)$ instead of to the uninteresting [[translation group]] $\mathbb{R}$, then the name of its canonical [[basis]] element $1 \in \mathbb{R}$ is canonically "$i$", the imaginary unit. Therefore one often writes the above [[central extension]] instead as follows:
$$
i \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X,\omega)
$$
in order to amplify this. But now consider the simple special case where $(X,\omega) = (\mathbb{R}^2, d p \wedge d q)$ is the 2-dimensional [[symplectic vector space]] which is for instance the [[phase space]] of the [[particle]] propagating on the line. Then a canonical set of generators for the corresponding [[Poisson bracket]] [[Lie algebra]] consists of the linear functions $p$ and $q$ of classical mechanics textbook fame, together with the _constant_ function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i \mathbb{R}$, hence purely Lie theoretically it is to be called "$i$".
With this notation then the [[Poisson bracket]], written in the form that makes its [[Lie integration]] manifest, indeed reads
$$
[q,p] = i
\,.
$$
Since the choice of [[basis]] element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing [[real number]] without changing this statement. If we write "$\hbar$" for this element, then the [[Poisson bracket]] instead reads
$$
[q,p] = i \hbar
\,.
$$
This is of course the hallmark equation for [[quantum physics]], if we interpret $\hbar$ here indeed as [[Planck's constant]]. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) [[Lie integration]] of the [[Poisson bracket]].
This is only the beginning of the story of quantization, naturally understood and indeed "derived" from applying [[Lie theory]] to [[classical mechanics]]. From here the story continues. It is called the story of _[[geometric quantization]]_. We close this motivation section here by some brief outlook.
The [[quantomorphism group]] which is the non-trivial [[Lie integration]] of the [[Poisson bracket]] is naturally constructed as follows: given the [[symplectic form]] $\omega$, it is natural to ask if it is the [[curvature]] 2-form of a $U(1)$-[[principal connection]] $\nabla$ on [[complex line bundle]] $L$ over $X$ (this is directly analogous to [[Dirac charge quantization]] when instead of a [[symplectic form]] on [[phase space]] we consider the the [[field strength]] 2-form of [[electromagnetism]] on [[spacetime]]). If so, such a connection $(L, \nabla)$ is called a _[[prequantum line bundle]]_ of the [[phase space]] $(X,\omega)$. The [[quantomorphism group]] is simply the [[automorphism group]] of the [[prequantum line bundle]], covering [[diffeomorphisms]] of the phase space (the [[Hamiltonian symplectomorphisms]] mentioned above).
As such, the [[quantomorphism group]] naturally [[action|acts]] on the [[space of sections]] of $L$. Such a [[section]] is like a [[wavefunction]], except that it depends on all of [[phase space]], instead of just on the "[[canonical coordinates]]". For purely abstract mathematical reasons (which we won't discuss here, but see at _[[motivic quantization]]_ for more) it is indeed natural to choose a "[[polarization]]" of [[phase space]] into [[canonical coordinates]] and [[canonical momenta]] and consider only those [[sections]] of the [[prequantum line bundle]] which depend only on the former. These are the actual _[[wavefunctions]]_ of [[quantum mechanics]], hence the _[[quantum states]]_. And the [[subgroup]] of the [[quantomorphism group]] which preserves these polarized sections is the group of exponentiated [[quantum observables]]. For instance in the simple case mentioned before where $(X,\omega)$ is the 2-dimensional [[symplectic vector space]], this is the [[Heisenberg group]] with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.
$\,$
**[[geometric quantization]]**
{#GeometricQuantization}
We had seen that every [[Lagrangian field theory]] induces a [[presymplectic current]] $\Omega_{BFV}$ (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) on the [[jet bundle]] of its [[field bundle]] in terms of which there is a concept of [[Hamiltonian differential forms]] and [[Hamiltonian vector fields]] on the jet bundle (def. \ref{HamiltonianForms}). The concept of [[quantization]] is induced by this _local [[phase space]]_-structure.
In order to disentangle the core concept of [[quantization]] from the technicalities involved in fully fledged [[field theory]], we now first discuss the [[finite number|finite]] [[dimension|dimensional]] situation.
+-- {: .num_example #GeometricRepresentationSchroedingerRepresentation}
###### Example
**([[Schrödinger representation]] via [[geometric quantization]])**
Consider the [[Cartesian space]] $\mathbb{R}^2$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) with canonical [[coordinate functions]] denoted $\{q,p\}$ and to be called the _[[canonical coordinate]]_ $q$ and its [[canonical momentum]] $p$ (as in example \ref{CanonicalMomentum})
and equipped with the [[left invariant differential form|constant]] [[differential 2-form]] given in in (eq:CanonicalMomentumPresymplecticCurrent) by
$$
\label{R2SymplecticForm}
\omega = d p \wedge d q
\,.
$$
This is [[closed differential form|closed]] in that $d \omega = 0$, and invertible in that the contraction of [[tangent vector fields]] into it (def. \ref{ContractionOfFormsWithVectorFields}) is an [[isomorphism]] to [[differential 1-forms]], and as such it is a _[[symplectic form]]_.
A choice of [[presymplectic potential]] for this [[symplectic form]] is
$$
\label{CanonicalSymplecticPotentialOnR2}
\theta \;\coloneqq\; - q \, d p
$$
in that $d \theta = \omega$. (Other choices are possible, notably $\theta = p \, d q$).
For
$$
A
\;\colon\;
\mathbb{R}^2
\longrightarrow
\mathbb{C}
$$
a [[smooth function]] (an [[observable]]), we say that a _[[Hamiltonian vector field]]_ for it (as in def. \ref{HamiltonianForms}) is a [[tangent vector field]] $v_A$ (example \ref{TangentVectorFields}) whose contraction (def. \ref{ContractionOfFormsWithVectorFields}) into the [[symplectic form]] (eq:R2SymplecticForm) is the [[de Rham differential]] of $A$:
$$
\label{HamiltonianVectorFieldOnR2}
\iota_{v_A} \omega = d A
\,.
$$
Consider the [[foliation]] of this phase space by constant-$q$-slices
$$
\label{ConstantqSlicesOnR2}
\Lambda_q \subset \mathbb{R}^2
\,.
$$
These are also called the _[[leaves]]_ of a _[[real polarization]]_ of the [[phase space]].
(Other choices of [[polarization]] are possible, notably the constant $p$-slices.)
We says that a smooth function
$$
\psi \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{C}
$$
is _[[polarization|polarized]]_ if its [[covariant derivative]] with [[connection on a bundle]] $i \theta$ along the [[leaves]] vanishes; which for the choice of polarization in (eq:ConstantqSlicesOnR2) means that
$$
\nabla_{\partial_p} \psi = 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\iota_{\partial_p} \left( d \psi + i \theta \psi \right)
=
0
\,,
$$
which in turn, for the choice of [[presymplectic potential]] in (eq:CanonicalSymplecticPotentialOnR2), means that
$$
\frac{\partial}{\partial p} \psi - i q \psi = 0
\,.
$$
The solutions to this [[differential equation]] are of the form
$$
\label{PolarizedFunctionsForBasicExampleOnR2}
\Psi(q,p) = \psi(q) \, \exp(+ i p q)
$$
for $\psi \colon \mathbb{R} \to \mathbb{C}$ any [[smooth function]], now called a _[[wave function]]_.
This establishes a [[linear isomorphism]] between polarized smooth functions and [[wave functions]].
By (eq:HamiltonianVectorFieldOnR2) we have the [[Hamiltonian vector fields]]
$$
v_q = \partial_p
\phantom{AAAA}
v_p = -\partial_q
\,.
$$
The corresponding _[[Poisson bracket]]_ is
$$
\label{R2PoissonBracket}
\begin{aligned}
\{q,p\}
& \coloneqq
\iota_{v_p} \iota_{v_q} \omega
\\
& =
-\iota_{\partial_q} \iota_{\partial_p} d p \wedge d q
=
\\
& =
- 1
\end{aligned}
$$
The action of the corresponding [[quantum operator (in geometric quantization)|quantum operators]] $\hat q$ and $\hat p$ on the polarized functions (eq:PolarizedFunctionsForBasicExampleOnR2) is as follows
$$
\begin{aligned}
\hat q \Psi(q,p)
& =
- i \nabla_{\partial_p}\Psi(q,p) + q \Psi(q,p)
\\
& = -i
\underset{ = 0 }{
\underbrace{
\left(
\underset{
= i q \Psi(q,p)
}{
\underbrace{
\frac{\partial}{\partial p}
\left(
\psi(q) e^{i q p}
\right)
}
}
- i q \Psi(q,p)
\right)
}
}
+
q \Psi(q,p)
\\
& = \left( q \psi(q) \right) e^{i q p}
\end{aligned}
$$
and
$$
\begin{aligned}
\hat p \Psi(q,p)
& =
i \nabla_{\partial_q} \Psi(q,p) + p \Psi(q,p)
\\
& = i \frac{\partial}{\partial q} (\psi(q)e^{i q p}) + p \Psi(q,p)
\\
& =
\left(
i \frac{\partial}{\partial q}\psi(q)
\right) e^{i q p}
+
\underset{ = 0}{
\underbrace{
\underset{
= - p \Psi(q,p)
}{
\underbrace{
\psi(q)
\left(
i \frac{\partial}{\partial q} e^{i q p}
\right)
}
}
+
p \Psi(q,p)
}
}
\\
& =
\left(
i \frac{\partial}{\partial q}\psi(q)
\right) e^{i p q}
\end{aligned}
\,.
$$
Hence under the identification (eq:PolarizedFunctionsForBasicExampleOnR2) we have
$$
\hat q \psi = q \psi
\phantom{AAAA}
\hat p \psi = i \frac{\partial}{\partial q} \psi
\,.
$$
This is called the [[Schrödinger representation]] of the [[canonical commutation relation]] (eq:R2PoissonBracket).
=--
$\,$
**[[Moyal star products]]**
{#StarProducts}
Let $V$ be a [[finite dimensional vector space]] and let $\pi \in V \otimes V$ be an element of the [[tensor product]] (not necessarily skew symmetric at the moment).
We may canonically regard $V$ as a [[smooth manifold]], in which case $\pi$ is canonically regarded as a constant rank-2 [[tensor]]. As such it has a canonical [[action]] by forming [[derivatives]] on the tensor product of the space of [[smooth functions]]:
$$
\pi
\;\colon\;
C^\infty(V) \otimes C^\infty(V)
\longrightarrow
C^\infty(V) \otimes C^\infty(V)
\,.
$$
If $\{\partial_i\}$ is a [[linear basis]] for $V$, identified, as before, with a basis for $\Gamma(T V)$, then in this basis this operation reads
$$
\pi(f \otimes g)
\;=\;
\pi^{i j}
(\partial_i f) \otimes (\partial_j g)
\,,
$$
where $\partial_i f \coloneqq \frac{\partial f}{\partial x^i}$ denotes the [[partial derivative]] of the [[smooth function]] $f$ along the $i$th [[coordinate]], and where we use the [[Einstein summation convention]].
For emphasis we write
$$
\array{
C^\infty(V) \otimes C^\infty(V)
&\overset{prod}{\longrightarrow}&
C^\infty(V)
\\
f \otimes g &\mapsto& f \cdot g
}
$$
for the pointwise product of smooth functions.
+-- {: .num_defn #StarPoduct}
###### Definition
**([[star product]] induced by constant rank-2 [[tensor]])**
Given $(V,\pi)$ as above, then the _[[star product]]_ induced by $\pi$ on the [[formal power series algebra]] $C^\infty(V) [ [\hbar] ]$ in a formal variable $\hbar$ ("[[Planck's constant]]") with [[coefficients]] in the [[smooth functions]] on $V$ is the linear map
$$
(-) \star_\pi (-)
\;\colon\;
C^\infty(V)[ [ \hbar ] ] \otimes C^\infty(V)[ [ \hbar ] ]
\longrightarrow
C^\infty(V)[ [\hbar] ]
$$
given by
$$
(-) \star_\pi (-)
\;\coloneqq\;
prod
\circ \exp\left(
\hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
\right)
$$
Hence
$$
f \star_\pi g
\;\coloneqq\;
1
+
\hbar \pi^{i j}
\frac{\partial f}{\partial x^i} \cdot \frac{\partial g}{\partial x^j}
+
\hbar^2
\tfrac{ 1 }{2}
\pi^{i j}
\pi^{k l}
\frac{\partial^2 f}{\partial x^{i} \partial x^{k}}
\cdot
\frac{\partial^2 g}{\partial x^{j} \partial x^{l}}
+
\cdots
\,.
$$
=--
+-- {: .num_example }
###### Example
**([[star product]] degenerating to pointwise product)**
If $\pi = 0$ in def. \ref{StarPoduct}, then the star product $\star_0 = \cdot$ is the plain pointwise product of functions.
=--
+-- {: .num_exaple #MoyalStarProduct}
###### Example
**([[Moyal star product]])**
If the tensor $\pi$ in def. \ref{StarPoduct} is skew-symmetric, it may be regarded as a constant [[Poisson tensor]] on the smooth manifold $V$. In this case $\star_{\tfrac{1}{2}\pi}$ is called a _[[Moyal star product]]_ and the star-product algebra $C^\infty(V)[ [\hbar] ], \star_\pi)$ is called the _[[Moyal deformation quantization]]_ of the [[Poisson manifold]] $(V,\pi)$.
=--
+-- {: .num_prop #AssociativeAndUnitalStarProduct}
###### Proposition
**([[star product]] is [[associativity|associative]] and [[unitality|unital]])**
Given $(V,\pi)$ as above, then the star product
$(-) \star_\pi (-)$ from def. \ref{StarPoduct}
is [[associativity|associative]] and [[unitality|unital]]
with unit the [[constant function]] $1 \in C^\infty(V) \hookrightarrow C^\infty(V)[ [ \hbar ] ]$.
Hence the [[vector space]] $C^\infty(V)$ equipped with the star product $\pi$ is a [[unital algebra|unital]] [[associative algebra]].
=--
+-- {: .proof}
###### Proof
Observe that the [[product rule]] of [[differentiation]] says that
$$
\partial_i \circ prod
=
prod \circ ( \partial_i \otimes id \;+\; id \otimes \partial_i )
\,.
$$
Using this we compute as follows:
$$
\begin{aligned}
& (f \star_\pi g) \star_\pi h
\\
& =
prod \circ
\exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
\left(
\left( prod \circ \exp( \pi^{k l} \partial_k \otimes \partial_l ) \right) \otimes id
\right)
(f \otimes g \otimes g)
\\
& =
prod \circ
\exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
(prod \otimes id)
\circ
\left(
\exp( \pi^{k l} \partial_k \otimes \partial_l ) \otimes id
\right)
(f \otimes g \otimes g)
\\
& =
prod \circ
(prod \otimes id) \circ
\exp( \pi^{i j} ( \partial_i \otimes id \otimes \partial_j +id \otimes \partial_i \otimes \partial_j )
\circ
\exp( \pi^{k l} \partial_k \otimes \partial_l ) \otimes id
(f \otimes g \otimes g)
\\
& =
prod \circ
(prod \otimes id) \circ
\exp( \pi^{i j} \partial_i \otimes id \otimes \partial_j )
\circ
\exp( \pi^{i j} id \otimes \partial_i \otimes \partial_j )
\circ
\exp( \pi^{k l} \partial_k \otimes \partial_l \otimes id )
(f \otimes g \otimes g)
\\
& =
prod_3
\circ
\exp( \pi^{i j} ( \partial_i \otimes \partial_j \otimes id + \partial_i \otimes id \otimes \partial_j + id \otimes \partial_i \otimes \partial_j) )
\end{aligned}
$$
In the last line we used that the ordinary pointwise product of functions is associative, and wrote $prod_3 \colon C^\infty(V) \otimes C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ for the unique pointwise product of three functions.
The last expression above is manifestly independent of the choice of order of the arguments in the triple star product, and hence it is clear that an analogous computation yields
$$
\cdots = f \star_\pi (g \star_\pi h)
\,.
$$
=--
+-- {: .num_prop #SymmetricContribution}
###### Proposition
**(shift by symmetric contribution is [[isomorphism]] of [[star products]])**
Let $V$ be a vector space, $\pi \in V \otimes V$ a rank-2 [[tensor]] and $\alpha \in Sym(V \otimes V)$ a _symmetric_ rank-2 tensor.
Then the [[linear map]]
$$
\array{
C^\infty(V)
&\overset{\exp\left(\tfrac{1}{2}\alpha \right)}{\longrightarrow}&
C^\infty(V)
\\
f
&\mapsto&
\exp\left( \tfrac{1}{2}\hbar \alpha^{i j} \partial_i \partial_j \right) f
}
$$
constitutes an [[isomorphism]] of star product algebras (prop. \ref{AssociativeAndUnitalStarProduct}) of the form
$$
\exp\left(\hbar\tfrac{1}{2}\hbar\alpha \right)
\;\colon\;
(C^\infty(V)[ [\hbar] ], \star_{\pi})
\overset{\simeq}{\longrightarrow}
(C^\infty(V))[ [\hbar] ], \star_{\pi + \alpha})
\,,
$$
hence identifying the star product induced from $\pi$ with that induced from $\pi + \alpha$.
In particular every star product algebra $(C^\infty(V)[ [\hbar] ],\star_\pi)$ is isomorphic to a Moyal star product algebra $\star_{\tfrac{1}{2}\pi}$ (example \ref{MoyalStarProduct}) with $\tfrac{1}{2}\pi_{skew}^{i j} = \tfrac{1}{2}(\pi^{i j} - \pi^{j i})$ the skew-symmetric part of $\pi$, this isomorphism being exhibited by the symmetric part $2\alpha^{i j} = \tfrac{1}{2}(\pi^{i j} + \pi^{j i})$.
=--
+-- {: .proof}
###### Proof
We need to show that
$$
\array{
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
&
\overset{
\exp\left( \tfrac{1}{2}\hbar \alpha \right)
\otimes
\exp\left( \tfrac{1}{2}\hbar \alpha \right)
}{\longrightarrow}&
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
\\
{}^{\mathllap{\star_{\pi}}}\downarrow
&&
\downarrow^{\mathrlap{\star_{\pi + \alpha}}}
\\
C^\infty(V)[ [\hbar] ]
&\underset{\exp\left( \tfrac{1}{2} \alpha \right) }{\longrightarrow}&
C^\infty(V)[ [\hbar] ]
}
$$
hence that
$$
prod \circ
\exp( \hbar(\pi + \alpha) )
\circ
\left(
\exp\left( \tfrac{1}{2}\alpha\right)
\otimes
\exp\left( \tfrac{1}{2}\alpha \right)
\right)
\;=\;
\exp\left( \tfrac{1}{2}\alpha \right)
\circ
prod \circ \exp( \pi )
\,.
$$
To this end, observe that the [[product rule]] of [[differentiation]] applied twice in a row implies that
$$
\partial_i \partial_j \circ prod
\;=\;
prod \circ
\left(
(\partial_i \partial_j) \otimes id
+
id \otimes (\partial_i \partial_j)
+
\partial_i \otimes \partial_j
+
\partial_j \otimes \partial_i
\right)
\,.
$$
Using this we compute
$$
\begin{aligned}
&
\exp\left( \hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right)
\circ
prod
\circ
\exp( \hbar \pi^{i j} \partial_{i} \otimes \partial_j )
\\
& =
prod
\circ
\exp\left(
\hbar \tfrac{1}{2}\alpha^{i j}
\left(
(\partial_i \partial_j) \otimes id
+
id \otimes (\partial_i \partial_j)
+
\partial_i \otimes \partial_j
+
\partial_j \otimes \partial_i
\right)
\right)
\circ
\exp( \hbar \pi^{i j} \partial_{k} \otimes \partial_l )
\\
&
=
prod
\circ
\exp\left(
\hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
\right)
\circ
\exp\left(
\hbar \tfrac{1}{2} \alpha^{i j} (\partial_i \partial_j) \otimes id
\hbar \tfrac{1}{2} \alpha^{i j} id \otimes (\partial_i \partial_j)
\right)
\\
& =
prod
\circ
\exp\left(
\hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
\right)
\circ
\left(
\exp\left( \tfrac{1}{2} \hbar \alpha \right)
\otimes
\exp\left( \tfrac{1}{2} \hbar \alpha \right)
\right)
\end{aligned}
$$
=--
$\,$
**[[Moyal star product]] as [[deformation quantization]]**
{#MoyalStarProductAsDeformationQuantization}
+-- {: .num_defn #SuperPoissonAlgebra}
###### Definition
**(super-[[Poisson algebra]])**
A super-[[Poisson algebra]] is
1. a [[supercommutative algebra]] $\mathcal{A}$ (here: over the [[real numbers]])
1. a [[bilinear function]]
$$
\{-,-\}
\;\colon\;
\mathcal{A} \otimes \mathcal{A}
\longrightarrow
A
$$
to be called the _[[Poisson bracket]]_
such that
1. $\{-,-\}$ is a [[super Lie algebra|super Lie bracket]] on $\mathcal{A}$, hence it
1. is graded skew-symmetric;
1. satisfies the super-[[Jacobi identity]];
1. for each $A \in \mathcal{A}$ of homogeneous degree, the operation
$$
\left\{ A, -\right\}
\;\colon\;
\mathcal{A}
\longrightarrow
\mathcal{A}
$$
is a graded [[derivation]] on $\mathcal{A}$ of the same degree as $A$.
=--
+-- {: .num_defn #FormalDeformationQuantization}
###### Definition
**([[formal deformation quantization]])**
Let $(\mathcal{A},\{-,-\})$ be a super-[[Poisson algebra]] (def. \ref{SuperPoissonAlgebra}). Then a _[[formal deformation quantization]]_ of $(A,\{-,-\})$ is
* the [[structure]] of an [[associative algebra]] on the [[formal power series algebra]] over $\mathcal{A}$ in a [[variable]] to be called $\hbar$, hence an [[associativity|associative]] and [[unitality|unital]] product
$$
\mathcal{A}[ [\hbar] ]
\otimes
\mathcal{A}[ [\hbar] ]
\longrightarrow
\mathcal{A}[ [\hbar] ]
$$
such that for all $f,g \in \mathcal{A}$ of homogeneous degree we have
1. $f \star g \, mod \hbar = f g$
1. $f \star g - (-1)^{deg(f) deg(g)} g \star f \, \mod \hbar^2 = \hbar \{f,g\}$
meaning that
1. to zeroth order in $\hbar$ the [[star product]] coincides with the given commutative product on $\mathcal{A}$,
1. to first order in $\hbar$ the [[graded commutator]] of the [[star product]] coincides with the given [[Poisson bracket]] on $\mathcal{A}$.
=--
+-- {: .num_example #MoyalStarProductIsFormalDeformationQuantization}
###### Example
**([[Moyal star product]] is [[formal deformation quantization]])**
Let $(V,\pi)$ be a _[[Poisson manifold|Poisson vector space]]_, hence a [[vector space]] $V$, equipped with a skew-symmetric [[tensor]] $\pi \in V \wedge V$.
Then with $V$ regarded as a [[smooth manifold]], the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(X)$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) becomes a [[Poisson algebra]] (def. \ref{SuperPoissonAlgebra}) with [[Poisson bracket]] given by
$$
\{f,g\}
\;\coloneqq\;
\pi^{i j} \frac{\partial f}{\partial x^i} \frac{\partial g}{\partial x^j}
\,.
$$
Moreover, for every symmetric tensor $\alpha \in V \otimes V$, the [[Moyal star product]] associated with $\tfrac{1}{2}\pi + \alpha$
$$
\array{
C^\infty(V)[ [\hbar] ] \otimes C^\infty(V)[ [\hbar] ]
&\overset{\star_{\tfrac{1}{2}\pi + \alpha}}{\longrightarrow}&
C^\infty(V)[ [\hbar] ]
\\
(f,g)
&\mapsto&
((-)\cdot (-)) \circ \exp\left( (\tfrac{1}{2}\pi^{i j} + \alpha^{i j}) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j} \right\}
}
(f,g)
$$
is a [[formal deformation quantization]] (def. \ref{FormalDeformationQuantization}) of this [[Poisson algebra]]-structure.
=--
$\,$
**[[Moyal star product]] via [[geometric quantization]] of [[symplectic groupoid]]**
{#MoyalStarProductViaGeometricQuantization}
+-- {: .num_prop #IntegralRepresentationOfStarProduct}
###### Proposition
**([[integral]] representation of [[star product]])**
If $\pi$ skew-symmetric and invertible, in that there exists $\omega \in V^\ast \otimes V^\ast$ with $\pi^{i j}\omega_{j k} = \delta^i_k$, and if the functions $f,g$ admit [[Fourier transform|Fourier analysis]] (are [[functions with rapidly decreasing partial derivatives]]), then their [[star product]] (def. \ref{StarPoduct}) is equivalently given by the following [[integral]] expression:
$$
\begin{aligned}
\left(f \star_\pi g\right)(x)
&=
\frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} }
\int
e^{ \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\end{aligned}
$$
=--
([Baker 58](Moyal+deformation+quantization#Baker58))
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
\left(f \star_\pi g\right)(x)
& \coloneqq
prod
\circ \exp\left(
\hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
\right)(f, g)
\\
& =
\frac{1}{(2 \pi)^{2n}}
\frac{1}{(2 \pi)^{2n}}
\int
\int
\underbrace{
e^{ i \hbar \pi(k,q) }
}
\underbrace{
e^{i k \cdot (x-y)}
f(y)
}
\underbrace{
e^{i q \cdot (x- \tilde y)}
g(\tilde y)
}
\,
d^{2 n} k
\,
d^{2 n} q
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{1}{(2 \pi)^{2n}}
\int
\delta\left( x - \tilde y + \hbar \pi \cdot k \right)
e^{i k \cdot (x-y)}
f(y)
g(\tilde y)
\,
d^{2 n} k
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{1}{(2 \pi)^{2n}}
\int
\delta\left( x - \tilde y + z \right)
e^{ \tfrac{i}{\hbar} \omega(z, (x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} z
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\\
& =
\frac{(det(\pi)^{2n})}{(2 \pi \hbar)^{2n} }
\int
e^{\tfrac{1}{i \hbar}\omega((x - \tilde y),(x-y))}
f(y)
g(\tilde y)
\,
d^{2 n} y
\,
d^{2 n} \tilde y
\end{aligned}
$$
Here in the first step we expressed $f$ and $g$ both by their [[Fourier transform]] (inserting the Fourier expression of the [[delta distribution]] from [this example](Dirac+distribution#FourierTransformOfDeltaDistribution)) and used that under this transformation the [[partial derivative]] $\pi^{a b} \frac{\partial}{\partial\phi^a}{\frac{\partial}{\phi^b}}$ turns into the product with $i \pi^{i j} k_i k_j$ ([this prop.](Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces)).
Then we identified again the Fourier-expansion of a [[delta distribution]]
and finally we applied the [[change of integration variables]] $k = \tfrac{1}{\hbar}\omega \cdot z$ and then evaluated the [[delta distribution]].
=--
Next we express this as the [[groupoid convolution product]] of polarized sections of the [[symplectic groupoid]]. To this end, we first need the following definnition:
+-- {: .num_defn #SymplecticGroupoidOfSymplecticVectorSpace}
###### Definition
**([[symplectic groupoid]] of [[symplectic vector space]])**
Assume that $\pi$ is the inverse of a [[symplectic form]] $\omega$ on $\mathbb{R}^{2n}$. Then the [[Cartesian product]]
$$
\array{
&& \mathbb{R}^{2n} \times \mathbb{R}^{2n}
\\
& {}^{\mathllap{pr_1}}\swarrow && \searrow^{\mathrlap{pr_2}}
\\
\mathbb{R}^{2n} && && \mathbb{R}^{2n}
}
$$
inherits the symplectic structure
$$
\Omega
\;\coloneqq\;
\left(
pr_1^\ast \omega - pr_2^\ast \omega
\right)
$$
given by
$$
\begin{aligned}
\Omega
& =
\omega_{i j} d x^i \wedge d x^j - \omega_{i j} d y^i \wedge d y^j
\\
& =
\omega_{i j} ( d x^i - d y^i ) \wedge ( d x^j + d y^j )
\end{aligned}
\,.
$$
The [[pair groupoid]] on $\mathbb{R}^{2n}$ equipped with this [[symplectic form]] on its space of [[morphisms]] is a [[symplectic groupoid]].
A choice of potential form $\Theta$ for $\Omega$, hence with $\Omega = d \Theta$, is given by
$$
\Theta
\coloneqq
-\omega_{i j} ( x^i + y^i ) d (x^j - y^j) )
$$
Choosing the [[real polarization]] spanned by $\partial_{x^i} - \partial_{y^i}$ a polarized section is function $F = F(x,y)$ such that
$$
\iota_{\partial_{x^j} - \partial_{y^j}}(d F - \tfrac{1}{i \hbar} \tfrac{1}{4} \Theta F) = 0
$$
hence
$$
\label{PolarizedSectionOnMorphismsOfSymplecticGroupoid}
F(x,y)
=
f\left( \tfrac{x + y}{2} \right) e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{x - y}{2} , \tfrac{x + y}{2} \right)}
\,.
$$
=--
+-- {: .num_prop #PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct}
###### Proposition
**(polarized [[symplectic groupoid|symplectic]] [[groupoid convolution product]] of [[symplectic vector space]] is given by [[Moyal star product]])**
Given a [[symplectic vector space]] $(\mathbb{R}^{2n}, \omega)$, then
the [[groupoid convolution product]] on polarized sections (eq:PolarizedSectionOnMorphismsOfSymplecticGroupoid) on its [[symplectic groupoid]] (def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}), given by [[convolution product]] followed by averaging ([[integration]]) over the polarization [[fiber]], is given by the [[star product]] (def. \ref{StarPoduct}) for the corresponding [[Poisson tensor]] $\pi \coloneqq \omega^{-1}$, in that
$$
\begin{aligned}
\int \int
F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
& =
(f \star_\pi g)((x+y)/2)
\end{aligned}
\,.
$$
=--
([Weinstein 91, p. 446](Moyal+deformation+quantization#Weinstein91), [Garcia-Bondia & Varilly 94, section V](Moyal+deformation+quantization#GBV), [Hawkins 06, example6.2](Moyal+deformation+quantization#EH))
+-- {: .proof}
###### Proof
We compute as follows:
$$
\begin{aligned}
&
\int \int
F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
\\
& \coloneqq
\int \int f((x + t)/2) g( (t + y)/2 )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( x-t, x+t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-y, t + y) }
\, d^{2n} t
\, d^{2n} (x-y)
\\
& =
\int \int
f(t/2) g( (t - (x - y))/2 )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) + (x - y) - t, t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-(x+y), t - (x-y)) }
\, d^{2n} t
\, d^{2n} (x-y)
\\
& =
\int \int
f(t/2) g( \tilde t / 2)
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - \tilde t, t ) - \tfrac{1}{i \hbar} \tfrac{1}{4} \omega((x+y)-t, \tilde t) }
\, d^{2n} t
\, d^{2n} \tilde t
\\
& =
\int \int
f(t) g( \tilde t )
e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - 2 \tilde t, 2 t ) - \tfrac{1}{ii \hbar} \tfrac{1}{4} \omega((x+y)- 2 t, 2 \tilde t) }
\, d^{2n} t
\, d^{2n} \tilde t
\\
\\
& =
\int \int
f(t) g(\tilde t )
e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{1}{2}(x+y) - \tilde t, \tfrac{1}{2}(x + y) - t \right)}
\, d^{2n} t
\, d^{2n} \tilde t
\\
& =
(f \star_\omega g)((x+y)/2)
\end{aligned}
$$
The first line just unwinds the definition of polarized sections from def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}, the following lines each implement a [[change of integration variables]] and
finally in the last line we used prop. \ref{IntegralRepresentationOfStarProduct}.
=--
$\,$
**Example: [[Wick algebra]] of [[normal ordered products]] on [[Kähler vector space]]**
{#WickAlgebraOfNormalOrderedProductsOnKählerVectorspace}
+-- {: .num_defn #AlmostKaehlerVectorSpace}
###### Definition
**([[Kähler vector space]])**
An _[[Kähler vector space]]_ is a [[real vector space]] $V$ equipped with a [[linear complex structure]] $J$ as well as two [[bilinear forms]] $\omega, g \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R}$ such that the following equivalent conditions hold:
1. $\omega(J v, J w) = \omega(v,w)$ and $g(v,w) = \omega(v,J w)$;
1. with $V$ regarded as a [[smooth manifold]] and with $\omega, g$ regarded as constant [[tensors]], then $(V, \omega, g)$ is an [[almost Kähler manifold]].
=--
+-- {: .num_example #StandardAlmostKaehlerVectorSpaces}
###### Example
**(standard [[Kähler vector spaces]])**
Let $V \coloneqq \mathbb{R}^2$ equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is
$$
J = (J^i{}_j)
\coloneqq
\left(
\array{
0 & -1
\\
1 & 0
}
\right)
\,.
$$
Moreover let
$$
\omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right)
$$
and
$$
g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1} \right)
\,.
$$
Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}).
The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.
If we write
$$
x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$
for the standard [[coordinate functions]] on $\mathbb{R}^2$ with
$$
z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$
and
$$
\overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$
for the corresponding complex coordinates, then this translates to
$$
\omega
\in
\Omega^2(\mathbb{R}^2)
$$
being the [[differential 2-form]] given by
$$
\begin{aligned}
\omega
& =
d x \wedge d y
\\
& =
\tfrac{1}{2i} d z \wedge d \overline{z}
\end{aligned}
$$
and with [[Riemannian metric]] [[tensor]] given by
$$
g = d x \otimes d x + d y \otimes d y
\,.
$$
The [[Hermitian form]] is given by
$$
\begin{aligned}
h
& =
g - i \omega
\\
& =
d z \otimes d \overline{z}
\end{aligned}
$$
=--
(for more see at _[[Kähler vector space]]_ [this example](Kähler+vector+space#StandardAlmostKaehlerVectorSpaces)).
+-- {: .num_defn #WickAlgebraOfAlmostKaehlerVectorSpace}
###### Definition
**([[Wick algebra]] of a [[Kähler vector space]])**
Let $(\mathbb{R}^{2n},\sigma, g)$ be a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}). Then its _Wick algebra_ is the [[formal power series]] vector space $\mathbb{C}[ [ \mathbb{R}^{2n} ] ] [ [ \hbar ] ]$ equipped with the [[star product]] (def. \ref{StarPoduct})
which is given by the [[bilinear form]]
$$
\label{InStarProductTensorInvertingHermitianForm}
\pi \coloneqq \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}
\,,
$$
hence:
$$
\begin{aligned}
A_1 \star_\pi A_2
& \coloneqq
((-)\cdot (-)) \circ \exp \left( \hbar\underoverset{k_1, k_2 = 1}{2 n}{\sum}\pi^{a b} \partial_a \otimes \partial_b \right) (A_1 \otimes A_2)
\\
& =
A_1 \cdot A_2 + \hbar \underoverset{k_1, k_2 = 1}{2n}{\sum}\pi^{k_1 k_2}(\partial_{k_1} A_1) \cdot (\partial_{k_2} A_2)
+ \cdots
\end{aligned}
$$
=--
(e.g. [Collini 16, def. 1](star+product#Collini16))
+-- {: .num_prop #StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}
###### Proposition
**([[star product]] [[associative algebra|algebra]] of [[Kähler vector space]] is [[star-algebra]])**
Under [[complex conjugation]] the [[star product]] $\star_\pi$ of a [[Kähler vector space]] structure (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace}) is a [[star algebra]] in that for all $A_1, A_2 \in \mathbb{C}[ [\mathbb{R}^{2n}] ][ [\hbar] ]$ we have
$$
\left( A_1 \star_\pi A_2 \right)^\ast
\;=\;
A_2^\ast \star_\pi A_1^\ast
$$
=--
+-- {: .proof}
###### Proof
This follows directly from that fact that in $\pi = \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}$ the [[imaginary part]] coincides with the skew-symmetric part, so that
$$
\begin{aligned}
(\pi^\ast)^{a b}
& =
-\tfrac{i}{2} (\omega^{-1})^{a b}
+ \tfrac{1}{2} (g^{-1})^{a b}
\\
& =
\tfrac{i}{2} (\omega^{-1})^{b a}
+
\tfrac{1}{2} (g^{-1})^{b a}
\\
& = \pi^{b a}
\,.
\end{aligned}
$$
=--
+-- {: .num_example #WickAlgebraOfASingleMode}
###### Example
**([[Wick algebra]] of a single mode)**
Let $V \coloneqq \mathbb{R}^2 \simeq Span(\{x,y\})$ be the standard [[Kähler vector space]] according to example \ref{StandardAlmostKaehlerVectorSpaces}, with canonical coordinates denoted $x$ and $y$. We discuss its Wick algebra according to def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace} and show that this reproduces the traditional definition of products of "normal ordered" operators as [above](#Idea).
To that end, consider the complex linear combination of the coordinates to the canonical complex coordinates
$$
z \;\coloneqq\; x + i y
\phantom{AAA}
\text{and}
\phantom{AAA}
\overline{z} \coloneqq x - i y
$$
which we use in the form
$$
a^\ast \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x + i y)
\phantom{AAA}
\text{and}
\phantom{AAA}
a \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x - i y)
$$
(with "$a$" the traditional symbol for the _[[amplitude]]_ of a field mode).
Now
$$
\omega^{-1}
=
\frac{\partial}{\partial y} \otimes \frac{\partial}{\partial x}
-
\frac{\partial}{\partial x} \otimes \frac{\partial}{\partial y}
$$
$$
g^{-1}
=
\frac{\partial}{\partial x}
\otimes
\frac{\partial}{\partial x}
+
\frac{\partial}{\partial y}
\otimes
\frac{\partial}{\partial y}
$$
so that with
$$
\frac{\partial}{\partial z}
=
\tfrac{1}{2}
\left(
\frac{\partial}{\partial x}
-i
\frac{\partial}{\partial y}
\right)
\phantom{AAAA}
\frac{\partial}{\partial \overline{z}}
=
\frac{1}{2}
\left(
\frac{\partial}{\partial x}
+
i
\frac{\partial}{\partial y}
\right)
$$
we get
$$
\begin{aligned}
\tfrac{i \hbar}{2}\omega^{-1} + \tfrac{\hbar}{2} g^{-1}
& =
2 \hbar
\frac{\partial}{\partial \overline{z}}
\otimes
\frac{\partial}{\partial z}
\\
& =
\hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}
\end{aligned}
$$
Using this, we find the [[star product]]
$$
A \star_\pi B
\;=\;
prod \circ \exp\left( \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast} \right)
$$
to be as follows (where we write $(-)\cdot (-)$ for the plain commutative product in the [[formal power series algebra]]):
$$
\begin{aligned}
a \star_\pi a & = a \cdot a
\\
a^\ast \star_\pi a^\ast & = a^\ast \cdot a^\ast
\\
a^\ast \star_\pi a & = a^\ast \cdot a
\\
a \star_\pi a^\ast
& =
a \cdot a^\ast + \hbar
\end{aligned}
$$
and so forth, for instance
$$
\array{
(a \cdot a ) \star_\pi (a^\ast \cdot a^\ast)
& =
a^\ast \cdot a^\ast \cdot a \cdot a
+
4 \hbar a^\ast \cdot a
+
\hbar^2
}
$$
If we instead indicate the commutative pointwise product by colons and the star product by plain juxtaposition
$$
:f g: \;\coloneqq\; f \cdot g
\phantom{AAAA}
f g \;\coloneqq\; f \star_\pi
$$
then this reads
$$
\array{
:a a: \, :a^\ast a^\ast:
& =
: a^\ast a^\ast a a :
+
4 \hbar \, : a^\ast a :
+
\hbar^2
}
$$
This is the way the _[[Wick algebra]]_ with its _[[operator product]]_ $\star_\pi$ and _[[normal-ordered product]]_ $:-:$ is traditionally presented.
=--
$\,$
**[[star products]] on [[regular polynomial observables]] in [[field theory]]**
{#RegularPolynomialFieldTheoryStarProduct}
+-- {: .num_prop #PropagatorStarProduct}
###### Proposition
**([[star products]] on [[regular polynomial observables]] induced from [[propagators]])**
Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[field bundle]] $E \overset{fb}{\to} \Sigma$, and let $\Delta \in \Gamma'_\Sigma((E \boxtimes E)^\ast)$ be a [[distribution of two variables]] on [[field histories]].
On the [[off-shell]] [[regular polynomial observables]] with a [[formal power series|formal paramater]] $\hbar$ adjoined consider the bilinear map
$$
PolyObs(E)_{reg}[ [\hbar] ]
\otimes
PolyObs(E)_{reg} [ [ \hbar ] ]
\overset{\star_{\Delta}}{\longrightarrow}
PolyObs(E)_{reg}[ [\hbar] ]
$$
given as in def. \ref{StarPoduct}, but with [[partial derivatives]] replaced by [[functional derivatives]]
$$
A_1 \star_{\Delta} A_2
\;\coloneqq\;
((-)\cdot(-))
\circ
\exp\left(
\int_\Sigma \Delta^{a b}(x,y)
\frac{\delta}{\delta \Phi^a(x)}
\otimes
\frac{\delta}{\delta \Phi^b(y)}
\right)
(A_1 \otimes A_2)
$$
As in prop. \ref{AssociativeAndUnitalStarProduct} this defines a [[unital algebra|unital]] and [[associative algebra]] [[structure]].
If the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] $P\Phi ) = 0$ induced by the [[Lagrangian density]] $\mathbf{L}$ are [[Green hyperbolic differential equations]] and if $\Delta$ is a _homogeneous_ [[propagator]] for these [[differential equations]] in that $P \Delta = 0$, then this [[star product]] algebra descends to the [[on-shell]] [[regular polynomial observables]]
$$
PolyObs(E,\mathbf{L})_{reg}[ [\hbar] ]
\otimes
PolyObs(E, \mathbf{L})_{reg} [ [ \hbar ] ]
\overset{\star_{\Delta}}{\longrightarrow}
PolyObs(E, \mathbf{L})_{reg}[ [\hbar] ]
\,.
$$
=--
+-- {: .proof}
###### Proof
The proof of prop. \ref{AssociativeAndUnitalStarProduct} goes through verbatim in the present case, as long as all [[products of distributions]] that appear when the [[propagator]] is multiplied with the [[coefficients]] of the [[polynomial observables]] are well-defined, in that [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) on the [[wave front sets]] (def. \ref{WaveFrontSet}) of the [[propagator]] and of these [[coefficients]] is met. But the definition the [[coefficients]] of [[regular polynomial observables]] are [[non-singular distributions]], whose wave front set is necessarily empty (example \ref{NonSingularDistributionTrivialWaveFrontSet}), so that their [[product of distributions]] is always well-defined.
=--
+-- {: .num_cor #FreeGaugeFixedLagrangianFieldTheoryQuantizationOfRegularObservables}
###### Corollary
**([[quantization]] of [[regular polynomial observables]] of [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]])**
Consider a [[gauge fixing|gauge fixed]] (def. \ref{GaugeFixingLagrangianDensity}) [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with BV-BRST-extended [[field bundle]] (remark \ref{FieldBundleBVBRST})
$$
E_{\text{BV-BRST}}
\;\coloneqq\;
T^\ast_{\Sigma,inf}[-1]
\left(
E
\times_\Sigma
\mathcal{G}[1]
\times_\Sigma
A
\times_\Sigma
A[-1]
\right)
$$
and with [[causal propagator]] (eq:CausalPropagator)
$$
\Delta
\;\in\;
\Gamma'_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
\,.
$$
Then the [[star product]] $\star_\Delta$ (def. \ref{StarPoduct}) is well-defined on [[off-shell]] (as well as [[on-shell]]) [[regular polynomial observables]] (def. \ref{PolynomialObservables})
$$
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\otimes
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\overset{\star_{\tfrac{i}{2}\Delta}}{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
and the resulting [[non-commutative algebra]] [[structure]]
$$
\left(
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\,,\,
\star_\Delta
\right)
$$
is a [[formal deformation quantization]] (def. \ref{FormalDeformationQuantization}) of the [[Peierls-Poisson bracket]] on the [[covariant phase space]] (theorem \ref{PPeierlsBracket}), restricted to [[regular polynomial observables]].
=--
([Dito 90](deformation+quantization#Dito90), [Dütsch-Fredenhagen 00](deformation+quantization#DuetschFredenhagen00) [Dütsch-Fredenhagen 01](deformation+quantization#DuetschFredenhagen01), [Hirshfeld-Henselder 02](deformation+quantization#HirschfeldHenselder02))
+-- {: .proof}
###### Proof
As in prop. \ref{PropagatorStarProduct}, the vanishing of the [[wave front set]] of the [[coefficients]] of the [[regular polynomial observables]] implies that all arguments go through as for [[star products]] on [[polynomial algebras]] on [[finite dimensional vector spaces]]. By theorem \ref{PPeierlsBracket} the [[causal propagator]] is the [[integral kernel]] of the [[Peierls-Poisson bracket]], so that the tensor $\pi$ from the definition of the [[Moyal star product]] (example \ref{MoyalStarProduct}) now is
$$
\pi = \Delta
\,.
$$
With this the statement follows by example \ref{MoyalStarProductIsFormalDeformationQuantization}.
=--
+-- {: .num_remark #TowardsQuantizationExtendingBeyondRegularPolynomialObservables}
###### Remark
**(extending [[quantization]] beyond [[regular polynomial observables]])**
While cor. \ref{FreeGaugeFixedLagrangianFieldTheoryQuantizationOfRegularObservables} provides a [[quantization]] of the [[regular polynomial observables]] of any [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]], the [[regular polynomial observables]] are too small a subspace of that of all [[polynomial observables]]:
By example \ref{RegularPolynomialLocalObservablesAreNecessarilyLinear} the only [[local observables]] (def. \ref{LocalObservables}) contained among the [[regular polynomial observables]] are the [[linear observables]] (def. \ref{LinearObservables}). But in general it is necessary to consider also non-linear polynomial [[local observables]]. Notably the [[interaction]] [[action functionals]] $S_{int}$ induced from interaction [[Lagrangian densities]] $\mathbf{L}_{int}$ (example \ref{ActionFunctional}) are non-linear polynomial observables.
For example:
* For [[quantum electrodynamics]] on [[Minkowski spacetime]] (example \ref{LagrangianQED}) the [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[electron-photon interaction]] is a cubic [[local observable]]
$$
S_{int}
\;=\;
i
\underset{\Sigma}{\int}
g_{sw}(x)
\,
(\gamma^\mu)^{\alpha}{}_\beta
\,
\overline{\mathbf{\Psi}}_\alpha(x)
\cdot \mathbf{\Psi}^\beta(x)
\cdot \mathbf{A}^a(x)
\, dvol_\Sigma(x)
$$
* For [[scalar field]] [[phi^n theory]] (example \ref{phintheoryLagrangian}) the [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[phi^n interaction]]
$$
S_{int}
\;=\;
\underset{\Sigma}{\int}
g_{sw}
\,
\underset{
n \, \text{factors}
}{
\underbrace{
\mathbf{\Phi}(x)
\cdot
\mathbf{\Phi}(x)
\cdots
\mathbf{\Phi}(x)
\cdot
\mathbf{\Phi}(x)
}
}
\,
dvol_\Sigma(x)
$$
is a [[local observable]] of order $n$.
Therefore one needs to extend the [[formal deformation quantization]] provided by corollary \ref{FreeGaugeFixedLagrangianFieldTheoryQuantizationOfRegularObservables} to a larger subspace of [[polynomial observables]] that includes at least the [[local observables]].
But prop. \ref{SymmetricContribution} characterizes the freedom in choosing a [[formal deformation quantization]]: We may shift the [[causal propagator]] by a symmetric contribution. In view of prop. \ref{PropagatorStarProduct} and in view of of [[Hörmander's criterion]] for the [[product of distributions]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) to be well defined, we are looking for symmetric [[integral kernels]] $H$ such that the sum
$$
\label{ShiftingCausalPropagatorBySymmetricContribution}
\Delta_H = \tfrac{i}{2}\Delta + H
$$
has a _smaller_ [[wave front set]] (def. \ref{WaveFrontSet}) than $\tfrac{i}{2}\Delta$ itself has. The smaller $WF(\tfrac{i}{2}\Delta + H)$, the larger the subspace of [[polynomial observables]] on which the corresponding [[formal deformation quantization]] exists.
Now by prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} the [[Wightman propagator]] $\Delta_H$ is of the form (eq:ShiftingCausalPropagatorBySymmetricContribution) and by prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} its [[wave front set]] is only "half" that of the [[causal propagator]]. It turns out that $\Delta_H$ does yield a [[formal deformation quantization]] of a subspace of [[polynomial observables]] that includes all [[local observables]]: this is the _[[Wick algebra]]_ on [[microcausal polynomial observables]]. We discuss this in detail in the chapter _[Free quantum fields](#FreeQuantumFields)_.
With such a [[formal deformation quantization]] of the [[local observables]] [[free field theory]] in hand, we may then finally obtain also a formal deformation quantization of [[interaction|interacting]] [[Lagrangian field theories]] by [[perturbative quantum field theory|perturbation theory]]. This we discuss in the chapters _[Scattering](#Scattering)_ and _[Quantum observables](#QuantumObservables)_.
=--
$\,$
This concludes our discussion of some basic concepts of [[quantization]]. In the [next chapter](#FreeQuantumFields) we apply this to discuss the [[algebra of quantum observables]] of [[free field theories|free]] [[Lagrangian field theories]]. Further below in the chapter _[Quantum observables](#QuantumObservables)_ we then discuss also the quantization of the [[interaction|interacting]] [[Lagrangian field theories]], [[perturbation theory|perturbatively]].
|
A first idea of quantum field theory -- Quantum electrodynamics | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Quantum+electrodynamics |
## Quantum Electrodynamics
{#QED}
* [[quantum electrodynamics]]
|
A first idea of quantum field theory -- Reduced phase space | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Reduced+phase+space | ## Reduced phase space
{#ReducedPhaseSpace}
In this chapter we discuss these topics:
* Global gauge reduction for strictly [[invariant]] functions ([[action functionals]]):
* _[Derived critical loci inside Lie algebroids](#DerivedCriticalLocusInsideLieAlgebroids)_
* _[Schouten bracket on Lie algebroids](#SchoutenBracketAntibracket)_
* Local gauge reduction for weakly invariant local functions ([[Lagrangian densities]]):
* _[Local antibracket](#LocalJetBundleAntibracket)_
* _[Local BV-BRST complex](#DerivedCriticalLocusOnJetBundle)_
* _[Global BV-BRST complex](#BVBRSTComplexGlobal)_
For a [[Lagrangian field theory]] with [[infinitesimal gauge symmetries]], the _[[reduced phase space]]_ is the [[quotient]] of the [[shell]] (the [[solution]]-locus of the [[equations of motion]]) by the [[action]] of the [[gauge symmetries]]; or rather it is the combined _[[homotopy quotient]]_ by the [[gauge symmetries]] and its _[[homotopy intersection]]_ with the [[shell]]. Passing to the [[reduced phase space]] may lift the [[obstruction]] for a [[gauge theory]] to have a [[covariant phase space]] and hence a [[quantization]].
The [[higher differential geometry]] of [[homotopy quotients]] and [[homotopy intersections]] is usefully modeled by tools from [[homological algebra]], here known as the _[[BV-BRST complex]]_.
In order to exhibit the key structure without getting distracted by the local [[jet bundle]] geometry, we first discuss the simple form in which the reduced phase space would appear after [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
if [[spacetime]] were [[compact space|compact]], so that, by the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction}),
it would be the [[derived critical locus]] ($d S \simeq 0$) of a globally defined [[action functional]] $S$. This "global" version of the [[BV-BRST complex]] is example \ref{ArchetypeOfBVBRSTComplex} below.
The genuine _[[local field theory|local]]_ construction of the derived [[shell]] is in the [[jet bundle]] of the [[field bundle]], where the [[action functional]] appears "de-transgressed" in the form of the [[Lagrangian density]], which however is invariant under gauge transformations generally
only up to horizontally exact terms. This _local_ incarnation of the redcuced phase space is modeled by the genuine _[[local BV-BRST complex]]_, example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below.
Finally, under [[transgression of variational differential forms]] this yields a [[differential]] on the graded [[local observables]] of the field theory. This is the _global [[BV-BRST complex]]_ of the [[Lagrangian field theory]] (def. \ref{ComplexBVBRSTGlobal} below).
$\,$
**[[derived critical loci]] inside [[Lie algebroids]]**
{#DerivedCriticalLocusInsideLieAlgebroids}
By analogy with the algebraic formulation of [[smooth functions]] between [[Cartesian spaces]] (the [[embedding of smooth manifolds into formal duals of R-algebras|embedding of Cartesian spaces into formal duals of R-algebras]], prop. \ref{AlgebraicFactsOfDifferentialGeometry}) it is clear how to define a map ([[homomorphism]]) between [[Lie algebroids]]:
+-- {: .num_defn #HomomorphismBetweenLieAlgebroids}
###### Definition
**([[homomorphism]] between [[Lie algebroids]])**
Given two [[derived Lie algebroids]] $\mathfrak{a}$, $\mathfrak{a}'$ (def. \ref{LInfinityAlgebroid}), then a [[homomorphism]]
between them
$$
f \;\colon\; \mathfrak{a} \longrightarrow \mathfrak{a}'
$$
is a [[dg-algebra]]-[[homomorphism]] between their [[Chevalley-Eilenberg algebras]] going the other way around
$$
CE(\mathfrak{a}) \longleftarrow CE(\mathfrak{a}') \;\colon\; f^\ast
$$
such that this covers an algebra homomorphism on the function algebras:
$$
\array{
CE(\mathfrak{a}) &\overset{f^\ast}{\longleftarrow}& CE(\mathfrak{a}')
\\
\downarrow && \downarrow
\\
C^\infty(X) &\underset{(f\vert_X)^\ast}{\longleftarrow}& C^\infty(Y)
}
\,.
$$
(This is also called a "[[curved sh-map|non-curved sh-map]]".)
=--
+-- {: .num_example #GaugeInvariantFunctionsIntermsOfLieAlgebroids}
###### Example
**([[invariant]] [[functions]] in terms of [[Lie algebroids]])**
Let $\mathfrak{g}$ be a [[super Lie algebra]] equipped with a [[Lie algebra action]] (def. \ref{InfinitesimalActionByLieAlgebra})
$$
\array{
\mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X
\\
& {}_{\mathllap{pr_2}}\searrow && \swarrow_{\mathrlap{rb}}
\\
&& X
}
$$
on a [[supermanifold]] $X$. Then there is a canonical homomorphism of [[Lie algebroids]] (def. \ref{HomomorphismBetweenLieAlgebroids})
$$
\label{ProjectionMapForActionLieAlgebroid}
\array{
X &&& CE(X) &=& C^\infty(X) &\oplus& 0
\\
\downarrow^{\mathrlap{p}}
&\phantom{AAA}&&
\uparrow^{\mathrlap{p^\ast}}
&&
\uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}}
\\
X/\mathfrak{g}
&&&
CE(X/\mathfrak{g})
&=&
C^\infty(X)
&\oplus&
C^\infty(X) \otimes \wedge^\bullet \mathfrak{g}^\ast
}
$$
from the manifold $X$ regarded as a Lie algebroid by example \ref{BasicExamplesOfLieAlgebroids} to the [[action Lie algebroid]] $X/\mathfrak{g}$ (example \ref{ActionLieAlgebroid}), which may be called the _[[homotopy quotient]] [[coprojection]] map_. The dual homomorphism of [[differential graded-commutative superalgebras]] is given simply by the identity on $C^\infty(X)$ and the [[zero map]] on $\mathfrak{g}^\ast$.
Next regard the [[real line]] [[manifold]] $\mathbb{R}^1$ as a Lie algebroid by example \ref{BasicExamplesOfLieAlgebroids}.
Then homomorphisms of Lie algebroids (def. \ref{HomomorphismBetweenLieAlgebroids}) of the form
$$
S \;\colon\; X/\mathfrak{g} \longrightarrow \mathbb{R}^1
\,,
$$
hence _smooth functions on the Lie algebroid_, are equivalently
* ordinary [[smooth functions]] $S \;\colon\; X \longrightarrow \mathbb{R}^1$ on the underlying [[smooth manifold]],
* which are [[invariant]] under the Lie algebra action in that $R(-)(S) = 0$.
In terms of the canonical [[homotopy quotient]] [[coprojection]] map $p$ (eq:ProjectionMapForActionLieAlgebroid) this says that a smooth function on $X$ [[extension]] extends to the [[action Lie algebroid]] precisely if it is [[invariant]]:
$$
\array{
X &\overset{S}{\longrightarrow}& \mathbb{R}^1
\\
{}^{\mathllap{p}}\downarrow
&
\nearrow_{
\mathrlap{
\text{exists precisely if} \; S \; \text{is invariant}
}
}
\\
X/\mathfrak{g}
}
$$
=--
+-- {: .proof}
###### Proof
An $\mathbb{R}$-algebra homomorphism
$$
CE( X/\mathfrak{g} )
\overset{S^\ast}{\longleftarrow}
C^\infty(\mathbb{R}^1)
$$
is fixed by what it does to the canonical [[coordinate function]] $x$ on $\mathbb{R}^1$, which is taken by
$S^\ast$ to $S \in C^\infty(X) \hookrightarrow CE(X/\mathfrak{g})$. For this to be a dg-algebra homomorphism it needs to respect the differentials on both sides. Since the differential on the right is trivial, the condition is that $0 = d_{CE} S = R(-)(f)$:
$$
\array{
\left\{ S \right\}
&\overset{S^\ast}{\longleftarrow}&
\left\{ x \right\}
\\
{}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
&&
\downarrow^{\mathrlap{d_{CE(\mathbb{R}^1)} = 0 } }
\\
\left\{
R(-)(S)
=
0
\right\}
&\underset{S^\ast}{\longleftarrow}&
\left\{ 0 \right\}
}
$$
=--
Given a gauge invariant function, hence a function $S \colon X/\mathfrak{g} \to \mathbb{R}$ on a Lie algebroid (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}), its [[exterior derivative]] $d S$ should be a
[[section]] of the [[cotangent bundle]] of the Lie algebroid. Moreover, if all field variations are
infinitesimal (as in def. \ref{LocalObservablesOnInfinitesimalNeighbourhood}) then it should in fact be
a section of the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of the [[zero section]] inside the [[cotangent bundle]], the _infinitesimal cotangent bundle_ $T^\ast_{inf}(X/\mathfrak{g})$ of the Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle} ebelow).
To motivate the definition \ref{LieAlgebroidInfinitesimalCotangentBundle} below of _infinitesimal cotangent bundle of a Lie algebroid_
recall from example \ref{InfinitesimalNeighbourhood} that the [[algebra of functions]]
on the infinitesimal cotangent bundle should be fiberwise the [[formal power series algebra]] in the [[linear functions]].
But a fiberwise linear function on a [[cotangent bundle]] is by definition a [[vector field]].
Finally observe that [[derivations of smooth functions are vector fields|vector fields are equivalently derivations of smooth functions]] (prop. \ref{AlgebraicFactsOfDifferentialGeometry}). This leads to the following definition:
+-- {: .num_defn #LieAlgebroidInfinitesimalCotangentBundle}
###### Definition
**([[automorphism ∞-Lie algebra|infinitesimal cotangent Lie algebroid]])**
Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some manifold $X$.
Then its _infinitesimal cotangent bundle_ $T^\ast_{inf} \mathfrak{a}$ is the [[Lie ∞-algebroid]] over $X$
whose underlying [[graded module]] over $C^\infty(X)$ is the [[direct sum]] of the original module
with the [[derivations]] of the graded algebra underlying $CE(\mathfrak{a})$:
$$
(T^\ast_{inf} \mathfrak{a})^\ast_\bullet
\;\coloneqq\;
\mathfrak{a}^\ast_\bullet \oplus Der(CE(\mathfrak{a}))_\bullet
$$
with [[differential]] on the summand $\mathfrak{a}$ being the original differential and
on $Der(CE(\mathfrak{a}))$ being the graded [[commutator]] with the differential $d_{CE(\mathfrak{a})}$ on $CE(\mathfrak{a})$
(which is itself a graded derivation of degree +1):
$$
\array{
\mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} }
&\mathrlap{ \vert_{\mathfrak{a}^\ast} }&
&
\coloneqq & d_{CE(\mathfrak{a})}
\\
\mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} }
& \mathrlap{ \vert_{Der(\mathfrak{a})} } &
\phantom{ \vert_{Der(\mathfrak{a})} }
&
\coloneqq & [d_{CE(\mathfrak{a})},-]
}
$$
Just as for ordinary [[cotangent bundles]] (def. \ref{Differential1FormsOnCartesianSpaces}) there is a canonical homomorphism of Lie algebroids (def. \ref{HomomorphismBetweenLieAlgebroids}) from the infinitesimal cotangent Lie algebroid down to the base Lie algebroid:
$$
\label{CotangentLieAlgebrpoidProjection}
\array{
T^\ast_{inf} \mathfrak{a}
&\phantom{AAA}&&
CE(T^\ast_{inf} \mathfrak{g})
&=&
CE(\mathfrak{a})
&\oplus&
\wedge^{\bullet \geq 1}_{CE(\mathfrak{a})} Der(\mathfrak{a})
\\
\downarrow^{\mathrlap{cb}} &&& \uparrow^{\mathrlap{cb^\ast}}
&& \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}}
\\
\mathfrak{a}
&&&
CE(\mathfrak{a}) &=& CE(\mathfrak{a}) &\oplus& 0
}
$$
given dually by the identity on the original generators.
=--
+-- {: .num_example #CotangentBundleOfActionLieAlgebroid}
###### Example
**([[automorphism ∞-Lie algebra|infinitesimal cotangent bundle]] of [[action Lie algebroid]])**
Let $X/\mathfrak{g}$ be an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) whose [[Chevalley-Eilenberg differential]] is given in local coordinates by (eq:DifferentialOnActionLieAlgebroid)
$$
d_{CE(X/\mathfrak{g})}
\;=\;
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}
+
c^\alpha R_a^\alpha \frac{\partial}{\partial \phi^a}
\,.
$$
Then its infinitesimal cotangent Lie algebroid $T^\ast_{inf} (X/\mathfrak{g})$ (def. \ref{LieAlgebroidInfinitesimalCotangentBundle})
has the generators
$$
\array{
&
\left(
\frac{\partial}{\partial c^\alpha}
\right)
&
\left(
\phi^a
\right)
,
\left(
\frac{\partial}{\partial \phi^a}
\right)
&
\left(
c^\alpha
\right)
\\
deg =
&
-1
&
0
&
+1
}
$$
and we find that CE-differential on the new derivation generators is given by
$$ \label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates}
\begin{aligned}
d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
\left(
\frac{\partial}{\partial c^\alpha}
\right)
& \coloneqq
\left[d_{CE(X/\mathfrak{g})}, \frac{\partial}{\partial c^\alpha} \right]
\\
& =
R_\alpha^a \frac{\partial}{\partial \phi^a}
+
\gamma^\beta{}_{\alpha \gamma} c^\gamma \frac{\partial}{\partial c^\beta}
\end{aligned}
$$
and
$$
\label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates}
\begin{aligned}
d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
\left(
\frac{\partial}{\partial \phi^a}
\right)
& \coloneqq
\left[
d_{CE(X/\mathfrak{g})},
\frac{\partial}{\partial \phi^a}
\right]
\\
& =
-
c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial}{\partial \phi^b}
\end{aligned}
\,.
$$
To amplify that the [[derivations]] _on_ $CE(X/\mathfrak{g})$, such as $\frac{\partial}{\partial \phi^a}$ and $\frac{\partial}{\partial c^\alpha}$, are now [[coordinate functions]] _in_ $CE(T^\ast_{inf}(X/\mathfrak{g}))$ one writes them as
$$
\label{AntiNotationForDerivations}
\phi^\ddagger_a \;\coloneqq\; \frac{\partial}{\partial \phi^a}
\phantom{AAAAA}
c\ddagger_\alpha \;\coloneqq\; \frac{\partial}{\partial c^\alpha}
\,.
$$
so that the generator content then reads as follows:
$$
\label{GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid}
\array{
&
\left(
c^\ddagger_\alpha
\right)
&
\left(
\phi^a
\right)
,
\left(
\phi^\ddagger_a
\right)
&
\left(
c^\alpha
\right)
\\
deg =
&
-1
&
0
&
+1
}
\,.
$$
In this notation the full action of the CE-differential for $T^\ast_{inf}(X/\mathfrak{g})$ is therefore the following:
$$
\label{CEDifferentialOnGeneratorsForInfinitesimalCotangentBundleOfActionLieAlgebroid}
\array{
& d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
\\
\phi^a
&\mapsto&
c^\alpha R^a_\alpha
\\
c^\alpha
& \mapsto&
\tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
\\
\phi^\ddagger_a
&\mapsto&
- c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
\\
c^\ddagger_\alpha
&\mapsto&
R_\alpha^a \phi^\ddagger_a
+
\gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_\beta
}
$$
=--
With a concept of [[cotangent bundles]] for [[Lie algebroids]] in hand, we want to see next that their [[sections]] are [[differential 1-forms]] on a [[Lie algebroid]] in an appropriate sense:
+-- {: .num_prop #ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}
###### Proposition
**([[exterior differential]] of [[invariant]] function is [[section]] of [[automorphism ∞-Lie algebra|infinitesimal cotangent bundle]])**
For $\mathfrak{a}$ a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some $X$;
and $S \;\colon\;\mathfrak{a} \longrightarrow \mathbb{R}$ a [[invariant]] smooth function on it (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids})
there is an induced [[section]] $d S$ of the infinitesimal cotangent Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle})
bundle projection (eq:CotangentLieAlgebrpoidProjection):
$$
\array{
&& T^\ast_{inf} \mathfrak{a}
\\
& {}^{\mathllap{d S}}\nearrow & \downarrow^{\mathrlap{cb}}
\\
\mathfrak{a}
&=&
\mathfrak{a}
}
\,,
$$
given dually by the [[homomorphism]] of [[differential graded-commutative superalgebras]]
$$
(d S)^\ast \;\colon\; CE(T^\ast_{inf} \mathfrak{a}) \longrightarrow CE(\mathfrak{a})
$$
which sends
1. the generators in $\mathfrak{a}^\ast$ to themselves;
1. a [[vector field]] $v$ on $X$, regarded as a degree-0 [[derivation]] to
$d S(v) = v(S) \in C^\infty(X)$;
1. all other derivations to zero.
=--
+-- {: .proof}
###### Proof
We discuss the proof in the special case that $\mathfrak{a} = X/\mathfrak{g}$ is an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) hence where $T^\ast_{inf}(\mathfrak{a}) = T^\ast_{inf}(X/\mathfrak{g})$ is as in example \ref{CotangentBundleOfActionLieAlgebroid}. The general case is directly analogous.
Since $(d S)^\ast$ has been defined on generators, it is uniquely a homomorphism of graded algebras.
It is clear that if $(d S)^\ast$ is indeed a [[homomorphism]] of [[differential graded-commutative superalgebras]] in that it also respects the CE-differentials, then it yields a section as claimed, because by definition it is the identity on $\mathfrak{a}^\ast$.
Hence all we need to check is that $(d S)^\ast$ indeed respects the CE-differentials.
On the original generators in $\mathfrak{a}^\ast$ this is immediate, since on these the CE-differential on both sides are by definition the same.
On the derivation $\phi^\ddagger_a \coloneqq \frac{\partial}{ \partial \phi^a}$ we find from (eq:CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates)
$$
\array{
\left\{
\frac{\partial S}{\partial \phi^a}
\right\}
&\overset{(d S)^\ast}{\longleftarrow}&
\left\{ \phi^\ddagger_a \right\}
\\
{}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
&&
\downarrow^{\mathrlap{d_{CE(T^\ast_{inf} (X/\mathfrak{g}))}}}
\\
\left\{
-c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial S}{\partial \phi^b}
\right\}
&\underset{(d S)^\ast}{\longleftarrow}&
\left\{
-c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
\right\}
}
$$
{#NoticeThatTheLeftVerticalMap} Notice that the left vertical map is indeed as shown, due to the invariance of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}), which allows an "[[integration by parts]]":
$$
\begin{aligned}
d_{CE(X/\mathfrak{g})}\left( \frac{\partial S}{\partial \phi_a} \right)
& =
c^\alpha R_\alpha^{b} \frac{\partial}{\partial \phi^b} \frac{\partial}{\partial \phi^a} S
\\
& =
\frac{\partial}{\partial \phi^a}
\left(
c^\alpha
\underset{
= 0
}{
\underbrace{
R_\alpha^b \frac{\partial S}{\partial \phi^b}
}
}
\right)
\;-\;
c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a}
\frac{\partial S}{\partial \phi^b}
\end{aligned}
$$
Similarly, on the derivation $c^\ddagger_\alpha \coloneqq \frac{\partial}{\partial c^\alpha}$ we find from (eq:CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates) and using the invariance of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids})
$$
\array{
\left\{
0
\right\}
&\overset{(d S)^\ast}{\longleftarrow}&
\left\{
c^\ddagger_\alpha
\right\}
\\
{}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
&&
\downarrow^{\mathrlap{d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}}}
\\
\left\{
0 = R_\alpha^a \frac{\partial S}{\partial \phi^a}
\right\}
&\underset{(d S)^\ast}{\longleftarrow}&
\left\{
R_\alpha^a \phi^\ddagger_a
+
\gamma^\beta{}_{\alpha \gamma} c^\gamma
c^\ddagger_\alpha
\right\}
}
\,.
$$
This shows that the differentials are being respected.
=--
Next we describe the [[vanishing locus]] of $d S$, hence the [[critical locus]] of $S$. Notice that if $d S$ is regarded as an ordinary [[differential 1-form]] on an ordinary [[smooth manifold]] $X$, then its ordinary [[vanishing locus]]
$$
X_{d S = 0}
\;=\;
\left\{
x \in X \;\vert\;
d S(x) = 0
\right\}
$$
is simply the [[fiber product]] of $d S$ with the [[zero section]] of the [[cotangent bundle]], hence the [[universal property|universal]] space that makes the following [[commuting diagram|diagram commute]]:
$$
\array{
X_{d S = 0}
&\overset{\phantom{AAA}}{\hookrightarrow}&
X
\\
\downarrow && \downarrow^{\mathrlap{0}}
\\
X &\underset{d S}{\longrightarrow}& T^\ast_{inf} X
}
\,.
$$
This is just the [[category theory|general abstract]] way to express the [[equation]] $d S = 0$.
In this [[category theory|general abstract]] form the concept of [[critical locus]] generalizes to [[invariant]] functions on [[super L-infinity algebra|super]] [[Lie algebroids]], where the vanishing of $d S$ is regarded only _up to [[homotopy]]_, namely up to [[infinitesimal symmetry]] transformations by the [[Lie algebra]] $\mathfrak{g}$. In this [[homotopy theory|homotopy-theoretic]] refinement we speak of the _[[derived critical locus]]_. The following definition simply states what this comes down to in components. For a detailed derivation see at _[[derived critical locus]]_ and for general introduction to [[higher differential geometry]] and [[higher Lie theory]] see at _[[schreiber:Higher Structures|Higher structures in Physics]]_.
+-- {: .num_defn #DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}
###### Definition
**([[derived critical locus]] of [[invariant]] function on [[Lie ∞-algebroid]])**
Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some $X$, let
$$
S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R}
$$
be an [[invariant]] function (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}) and consider
the [[section]] of its infinitesimal [[cotangent bundle]] $T^\ast_{inf} \mathfrak{a}$ (def. \ref{CotangentBundleOfActionLieAlgebroid}) corresponding to its exterior derivative
via prop. \ref{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}:
$$
\array{
\mathfrak{a} && \overset{d S}{\longrightarrow} && T^\ast_{inf} \mathfrak{a}
\\
& {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{cb}}
\\
&& \mathfrak{a}
}
$$
Then the _[[derived critical locus]]_ of $S$ is the [[derived Lie algebroid]] (def. \ref{LInfinityAlgebroid})
to be denoted $\mathfrak{a}_{d S \simeq 0}$ which is the [[homotopy pullback]] of the section $d S$ along the [[zero section]]:
$$
\array{
\mathfrak{a}_{d S \simeq 0}
&\longrightarrow&
\mathfrak{a}
\\
\downarrow &(pb)& \downarrow^{\mathrlap{0}}
\\
\mathfrak{a}
&\underset{d S}{\longrightarrow}&
T^\ast_{inf} \mathfrak{a}
}
\,.
$$
This means equivalently (details are at _[[derived critical locus]]_) that the Chevalley-Eilenberg algebra of $\mathfrak{a}_{d S \simeq 0}$ is like that of the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) except for two changes:
1. all [[derivations]] are shifted down in degree by one;
rephrased in terms of [[graded manifold]] (remark \ref{dgManifolds}) this means that the [[graded manifold]] underlying $\mathfrak{a}_{d S \simeq 0}$ is $T^\ast_{inf}[-1]\mathfrak{a}$;
1. the [[Chevalley-Eilenberg differential]] on the derivations coming from [[tangent vector fields]] $v$ on $X$ is that of the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ plus $d S(v) = v(S)$.
=--
We now make the general concept of [[derived critical locus]] inside an [[L-∞ algebroid]] (def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}) explicit in our running example of an [[action Lie algebroid]]; the reader not concerned with the general idea of [[homotopy pullbacks]] may consider the following example as the definition of derived critical locus for the purposes of our running examples:
+-- {: .num_example #ArchetypeOfBVBRSTComplex}
###### Example
**([[derived critical locus]] inside [[action Lie algebroid]])**
Consider an [[invariant]] function (def. \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids})
on an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid})
$$
S \;\colon\; X/\mathfrak{g} \overset{\phantom{AAA}}{\longrightarrow} \mathbb{R}
$$
for the case that the underlying [[supermanifold]] $X$ is a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) with global [[coordinates]] $(\phi^a)$ as in example \ref{CotangentBundleOfActionLieAlgebroid}. Then the [[derived critical locus]] (def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid})
$$
(X/\mathfrak{g})_{d S \simeq 0}
$$
is, in terms of its [[Chevalley-Eilenberg algebra]] $CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)$ (def. \ref{LInfinityAlgebroid}) given as follows:
Its generators are those of $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ as in (eq:GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid), except for a shift of degree of the [[derivation]]-generators down by one:
$$
\array{
&
\left(
c^\ddagger_{\alpha}
\right)
&
\left(
\phi^\ddagger_a
\right)
&
\left(
\phi^a
\right)
&
\left(
c^\alpha
\right)
\\
deg = &
-2
&
-1
&
0
&
+1
}
$$
Rephrased in terms of [[graded manifold]] (remark \ref{dgManifolds}) this means that the [[graded manifold]] underlying the derived critical locus is the _shifted infinitesimal cotangent bundle_ of the graded manifold $\mathfrak{g}[1] \times X$ (eq:ActionLieAlgebroidGradedManifold) which underlies the [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}):
$$
\label{ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid}
(X/\mathfrak{g})_{d S \simeq 0}
\;=_{grmfd}\;
T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times X \right)
$$
and if $X = \mathbb{R}^{b\vert s}$ is a [[super Cartesian space]] this becomes more specifically
$$
\begin{aligned}
(\mathbb{R}^{p \vert q}/\mathfrak{g})_{d S \simeq 0}
& =_{grmfd}
T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times \mathbb{R}^{p \vert q} \right)
\\
& =_{\phantom{grmfd}}
\underset{
(c^\alpha)
}{
\underbrace{
\mathfrak{g}[1]
}}
\times
\underset{ (\phi^a) }{
\underbrace{
\mathbb{R}^{p\vert q}
}}
\times
\underset{ (\phi^\ddagger_a) }{
\underbrace{
(\mathbb{R}^{p \vert q})^\ast_{inf}[-1]
}}
\times
\underset{ (c^\ddagger_\alpha) }{
\underbrace{
\mathfrak{g}^\ast[-2]
}}
\end{aligned}
$$
Moreover, on these generators the CE-differential is given by
$$
\label{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid}
\array{
& d_{CE\left((X/\mathfrak{g})_{d S \simeq 0}\right)}
\\
\phi^a
&\mapsto&
c^\alpha R^a_\alpha
\\
c^\alpha
& \mapsto&
\tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
\\
\phi^\ddagger_a
&\mapsto&
\underset{
new
}{
\underbrace{
\frac{\partial S}{\partial \phi^a}
}}
-
c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
\\
c^\ddagger_\alpha
&\mapsto&
R_\alpha^a \phi^\ddagger_a
+
\gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b
}
$$
which is just the expression for the differential (eq:CEDifferentialOnGeneratorsForInfinitesimalCotangentBundleOfActionLieAlgebroid) in $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ from example \ref{CotangentBundleOfActionLieAlgebroid}, except for the fact that (the derivations are shifted down in degree and) the new term $\frac{\partial S}{\partial \phi^a}$ over the brace.
=--
The following example illustrates how the concept of [[derived critical locus]] $X_{d S \simeq 0}$ of $S$ is a [[homotopy theory|homotopy theoretic]] version of the ordinary concept of [[critical locus]] $X_{d S = 0}$:
+-- {: .num_example #OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}
###### Example
**(ordinary [[critical locus]] is [[cochain cohomology]] of [[derived critical locus]] in degree 0)**
Let $X$ be an [[superpoint]] (def. \ref{SuperCartesianSpace}) or more generally the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of a point in a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) with [[coordinate functions]] $(\phi^a)$, so that its [[algebra of functions]] $C^\infty(X)$ is a truncated [[polynomial algebra]] or [[formal power series algebra]] in the [[variables]] $\phi^a$.
Consider for simplicity the special case that $\mathfrak{g} = 0$ so that there is no [[Lie algebra action]] on $X$.
Then the [[Chevalley-Eilenberg algebra]] of the [[derived critical locus]] $X_{d S \simeq 0}$ of $S$ (example \ref{ArchetypeOfBVBRSTComplex}) has generators
$$
\begin{aligned}
&
&
\left(
\phi^\ddagger_a
\right)
&
\left(
\phi^a
\right)
&
\\
deg = &
&
-1
&
0
&
\end{aligned}
$$
and [[differential]] given by
$$
\array{
& d_{CE\left( X_{d S \simeq 0} \right)}
\\
\phi^a &\mapsto& 0
\\
\phi^\ddagger_a &\mapsto& \frac{\partial S}{\partial \phi^a}
}
\,.
$$
Hence the [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]] of the derived critical locus indegree 0 is the [[quotient]] of $C^\infty(X)$ by the ideal which is generated by $\left( \frac{\partial S}{\partial \phi^a} \right)$
$$
H^0\left(
CE\left( X_{d S \simeq 0} \right)
\right)
\;=\;
C^\infty(X)/\left( \frac{\partial S}{\partial \phi^a} \right)
\,.
$$
But under the assumption that $X$ is a [[superpoint]] or [[infinitesimal neighbourhood]] of a point, this quotient algebra is just the [[algebra of functions]] on the ordinary [[critical locus]] $X_{d S = 0}$.
(The quotient says that every function on $X$ which vanishes where $\frac{\partial S}{\partial \phi^a}$ vanishes is [[zero]] in the quotient. This means that the quotient algebra consists of the functions on $X$ modulo the [[equivalence relation]] that identifies two if they agree on the critical locus $X_{d S = 0}$, which is the functions on $X_{d S = 0}$.)
Hence the [[derived critical locus]] yields the ordinary [[critical locus]] in [[cochain cohomology]]:
$$
H^0\left(
CE\left( X_{d S \simeq 0} \right)
\right)
\;\simeq\;
C^\infty\left( X_{d S = 0} \right)
\,.
$$
However, it is not in general the case that the [[derived critical locus]] is a [[resolution]] of the ordinary [[critical locus]], in that all its cohomology in [[negative number|negative]] degree vanishes. Instead, the cohomology of the [[Chevalley-Eilenberg algebra]] of a [[derived critical locus]] in [[negative number|negative]] degree detects [[Lie algebra action]] and more generally [[L-∞ algebra action]] on $X$ under which $S$ is invariant. If this action is incorporated into $X$ by passing to the [[action Lie algebroid]] $X/\mathfrak{g}$ and then forming the [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ in there, as in example \ref{ArchetypeOfBVBRSTComplex}.
This issue we discuss in detail in the chapter _[Gauge fixing](#GaugeFixing)_, see prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres} below.
=--
In order to generalize the statement of example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} to the case that a [[Lie algebra action]] is taken into account, we need to realize the [[Chevalley-Eilenberg algebra]] of a [[derived critical locus]] in a [[Lie algebroid]] is the [[total complex]] of a [[double complex]]:
+-- {: .num_prop #DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}
###### Proposition
**([[Chevalley-Eilenberg algebra]] of [[derived critical locus]] is [[total complex]] of [[BV-BRST formalism|BV-BRST]] [[bicomplex]])**
Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}. Then its [[Chevalley-Eilenberg differential]] (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid) may be decomposed as the sum of two anti-commuting differential
$$
d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)}
\;=\;
s_{BRST} + s_{BS}
$$
which are defined on the generators of the [[Chevalley-Eilenberg algebra]] as follows:
$$
\label{ExplicitBVandBRSTDifferentialInCotangentBundleOfActionLieAlgebroid}
\array{
& s_{BV}
\\
\phi^a
&\mapsto&
0
\\
c^\alpha
& \mapsto&
0
\\
\phi^\ddagger_a
&\mapsto&
\frac{\partial S}{\partial \phi^a}
\\
c^\ddagger_\alpha
&\mapsto&
R_\alpha^a \phi^\ddagger_a
\\
\phantom{A}
\\
& s_{BRST}
\\
\phi^a
&\mapsto&
c^\alpha R^a_\alpha
\\
c^\alpha
& \mapsto&
\tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
\\
\phi^\ddagger_a
&\mapsto&
-
c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
\\
c^\ddagger_\alpha
&\mapsto&
\gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b
}
$$
If we moreover decompose the degree of the generators into two degrees
$$
\array{
&
\left(
c^\ddagger_{\alpha}
\right)
&
\left(
\phi^\ddagger_a
\right)
&
\left(
\phi^a
\right)
&
\left(
c^\alpha
\right)
\\
deg_{gh} =
&
0
&
0
&
0
&
+1
\\
deg_{af} =
&
-2
&
-1
&
0
&
0
}
$$
then these two differentials constitute a [[bicomplex]]
$$
\array{
CE^{0,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{1,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{2,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
\cdots
\\
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\\
CE^{0,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{1,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{2,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
\cdots
\\
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\\
CE^{0,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{1,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
CE^{2,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
&\overset{s_{BRST}}{\longrightarrow}&
\cdots
\\
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\big\uparrow^{\mathrlap{s_{BV}}}
&&
\\
\vdots && \vdots && \vdots
}
$$
whose [[total complex]] is the [[Chevalley-Eilenberg dg-algebra]] of the derived critical locus
$$
\begin{aligned}
CE\left(
(X/\mathfrak{g})_{d S \simeq 0}
\right)
& =
\underset{ gh, af }{\bigoplus}
CE^{gh,af}\left(
(X/\mathfrak{g})_{d S \simeq 0}
\right)
\\
d_CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right){}
& =
s_{BV} + s_{BRST}
\end{aligned}
\,.
$$
=--
+-- {: .proof}
###### Proof
It is clear from the definition that the graded [[derivations]] $s_{BV}$ and $s_{BRST}$ have (i.e. increase) bidegree as follows:
$$
\array{
& s_{BRST} & s_{BV}
\\
deg_{gh} = & +1 & 0
\\
deg_{af} = & 0 & +1
}
\,.
$$
This implies that in
$$
\begin{aligned}
0 & =
\left( d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)} \right)^2
\\
& =
\left( s_{BV} + s_{BRST}\right)^2
\\
& =
\underset{ = 0 }{
\underbrace{
\left( s_{BV}\right)^2
}}
+
\underset{ = 0 }{
\underbrace{
\left( s_{BRST} \right)^2
}}
+
\underset{ = 0 }{
\underbrace{
\left[ s_{BV}, s_{BRST} \right]
}
}
\end{aligned}
$$
all three terms have to vanish separately, as shown, since they each have different bidegree (the last term denotes the graded commutator, hence the [[anticommutator]]). This is the statement to be proven.
Notice that the nilpotency of $s_{BV}$ is also immediately checked explicitly, due to the [[invariant|invariance]] of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}):
$$
\begin{aligned}
s_{BV} \left( s_{BV} \left( c^\ddagger_\alpha \right) \right)
& =
s_BV\left( R_\alpha^a \phi^\ddagger_a \right)
\\
& =
R_\alpha^a \frac{\partial S}{\partial \phi^a}
\\
& = 0
\end{aligned}
$$
=--
As a corollary of prop. \refDerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure{} we obtain the generalization of example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} to non-trivial $\mathfrak{g}$-actions:
+-- {: .num_prop #CochainCohomologyOfBVBRSTComplexInDegreeZero}
###### Proposition
**([[cochain cohomology]] of [[BV-BRST complex]] in degree 0 is the [[invariant]] function on the [[critical locus]])**
Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}.
Then if the vertical [[differential]] (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure})
$$
\array{
CE^{\bullet, \bullet+1}\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
\\
\uparrow^{\mathrlap{s_{BV}}}
\\
CE^{\bullet, \bullet}\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
}
$$
has vanishing [[cochain cohomology]] in [[negative number|negative]] $af$-degree
$$
\label{VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree}
H^{\bullet \leq 1}(s_{BV}) = 0
$$
then the [[cochain cohomology]] of the full [[Chevalley-Eilenberg dg-algebra]] is given by the cochain cohomology of $s_{BRST}$ on $H^0(s_{BV})$:
$$
H^k\left(
CE\left(
(X/\mathfrak{g})_{d S \simeq 0}
\right)
\right)
\;\simeq\;
H^k\left( H^0(s_{BV}), s_{BRST} \right)
\,.
$$
Moreover if $X$ is inside the [[infinitesimal neighbourhood]] of a point as in example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} then the full cochain cohomology in degree 0 is the space of those functions on the ordinary [[critical locus]] $X_{d S = 0}$ which are $\mathfrak{g}$-[[invariant]]:
$$
H^0
\left(
CE\left(
(X/\mathfrak{g})_{d S \simeq 0}
\right)
\right)
\;=\;
\left\{
X_{d S = 0}
\overset{f}{\to}
\mathbb{R}
\;\vert\;
\left(R_\alpha^a \frac{\partial f}{\partial \phi^a} = 0\right)
\right\}
$$
=--
+-- {: .proof}
###### Proof
The first statement follows from the [[spectral sequence]] [[spectral sequence of a double complex|of the double complex]]
$$
H^{gh}
\left(
H^{af}
\left(
CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
\right)
\right)
\;\Rightarrow\;
H^{gh + af}\left(
CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
\right)
\,.
$$
Under the given assumption the second page of this [[spectral sequence]] is concentrated on the row $af = 0$. This implies that all differentials on this page vanish, so that the sequence collapses on this page. Moreover, since the spectral sequence consists of [[vector spaces]] ([[modules]] over the [[real numbers]]) the [extension problem](spectral+sequence#ExtensionProblem) is trivial, and hence the claim follows.
Now if $X$ is inside the [[infinitesimal neighbourhood]] of a point, then example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} says that $H^0(s_{BV})$ in $deg_{gh} = 0$ consists of the functions on the ordinary critical locus and hence the abvove result implies that
$$
\begin{aligned}
H^0\left( CE\left( (X/\mathfrak{g})_{d S \simeq 0}\right) \right)
& =
ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) }
\,/\,
\underset{= 0}{
\underbrace{
im(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right)}
}
}
\\
& = ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) }
\\
& =
\left\{
X_{d S = 0} \overset{f}{\longrightarrow} \mathbb{R}
\,\vert\,
\left(
R_\alpha^a \frac{\partial S}{\partial \phi^a}
=
0
\right)
\right\}
\end{aligned}
$$
=--
This means that under condition (eq:VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree) the construction of a [[derived critical locus]] inside an [[action Lie algebroid]] provides a [[resolution]] of the space of those functions which are
1. _[[restriction|restricted]]_ to the [[critical locus]] (a [[homotopy intersection]]);
1. _[[invariant]]_ under the [[Lie algebra action]] (a [[homotopy quotient]]).
We apply this general mechanism [below](#DerivedCriticalLocusOnJetBundle) to [[Lagrangian field theory]], where it serves to provide a [[resolution]] by the _[[BV-BRST complex]]_ of the space of [[observables]] which are
1. [[on-shell]],
1. _[[gauge invariance|gauge invariant]]_.
But in order to control this application, we first establish the tool of the _[[Schouten bracket]]/[[antibracket]]_.
$\,$
**[[Schouten bracket]]/[[antibracket]]**
{#SchoutenBracketAntibracket}
Since the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ has function algebra given by tensor products of [[tangent vector fields]]/[[derivations]], we expect that a graded analogue of the [[Lie bracket]] of ordinary [[tangent vector fields]] exists on the [[Chevalley-Eilenberg algebra]] $CE\left( T^\ast_{inf} \mathfrak{a}\right)$. This is indeed the case, and crucial for the theory:
+-- {: .num_defn #SchoutenBracketAndAntibracket}
###### Definition
**([[Schouten bracket]] and [[antibracket]] for [[action Lie algebroid]])**
Consider a [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ inside an [[action Lie algebroid]] $X/\mathfrak{g}$ as in example \ref{ArchetypeOfBVBRSTComplex}.
Then the graded [[commutator]] of graded [[derivations]] of the [[Chevalley-Eilenberg algebra]] of $X/\mathfrak{g}$
$$
[-,-]
\;\colon\;
Der(CE(X/\mathfrak{g}))
\otimes
Der(CE(X/\mathfrak{g}))
\longrightarrow
Der(CE(X/\mathfrak{g}))
$$
uniquely [[extension|extends]], by the graded [[Leibniz rule]], to a graded bracket of degree $(1,even)$ on the CE-algebra of the [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$
$$
\left\{ -,-\right\}
\;\colon\;
CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
\otimes
C\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
\longrightarrow
CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
$$
such that this is a graded [[derivation]] in both arguments.
This is called the _[[Schouten bracket]]_.
There is an elegant way to rewrite this in terms of components: With the notation (eq:AntiNotationForDerivations) for the coordinate-derivations the [[Schouten bracket]] is equivalently given by
$$
\label{Antibracket}
\begin{aligned}
\left\{ f,g \right\}
& =
\phantom{+}
\frac{\overset{\leftarrow}{\partial} f}{\partial \phi^\ddagger_a}
\frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^a}
-
\frac{\overset{\leftarrow}{\partial} f}{\partial \phi^a}
\frac{\overset{\rightarrow}{\partial} g}{\partial \phi^\ddagger_a}
\\
&
\phantom{=}
+
\frac{\overset{\leftarrow}{\partial} f}{\partial c^\ddagger_\alpha}
\frac{\overset{\rightarrow}{\partial} g}{\partial {c}^{\alpha}}
-
\frac{\overset{\leftarrow}{\partial} f}{\partial c^{\alpha}}
\frac{\overset{\rightarrow}{\partial} g}{\partial c^\ddagger_\alpha}
\end{aligned}
\,,
$$
where the arrow over the [[partial derivative]] indicates that we we pick up signs via the [[Leibniz rule]] either as usual, going through products from left to right (for $\overset{\rightarrow}{\partial}$) or by going through the products from right to left (for $\overset{\leftarrow}{\partial}$).
In this form the [[Schouten bracket]] is called the _[[antibracket]]_.
=--
(e. g. [Henneaux 90, (53d)](antibracket#Henneaux90), [Henneaux-Teitelboim 92, section 15.5.2](antibracket#HenneauxTeitelboim92))
The power of the [[Schouten bracket]]/[[antibracket]] rests in the fact that it makes the [[Chevalley-Eilenberg differential]] on a [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ become a [[Hamiltonian vector field]], for "[[Hamiltonian]]" the sum of $S$ with the [[Chevalley-Eilenberg differential]] of $X/\mathfrak{g}$:
+-- {: .num_example #ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}
###### Example
**([[Chevalley-Eilenberg differential]] of [[derived critical locus]] is [[Hamiltonian vector field]] for the [[Schouten bracket]]/[[antibracket]])**
Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}.
Then the CE-differential (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid) of the [[derived critical locus]] $X/\mathfrak{g}\vert_{S \simeq 0}$ is simply the [[Schouten bracket]]/[[antibracket]] (def. \ref{SchoutenBracketAndAntibracket}) with the [[sum]]
$$
\label{BVBRSTFunctionForActionLieAlgebroid}
S_{\text{BV-BRST}}
\;\coloneqq\;
S - d_{CE(X/\mathfrak{g})}
$$
of the [[Chevalley-Eilenberg differential]] of $X/\mathfrak{g}$ and the function $-S$:
$$
d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) }(-)
\;=\;
\left\{
- S + d_{CE(X/\mathfrak{g})}
\,,\,
(-)
\right\}
\,.
$$
In coordinates, using the expression for $d_{CE(X/\mathfrak{g})}$ from (eq:DifferentialOnActionLieAlgebroid) and using the notation for derivations from (eq:AntiNotationForDerivations) this means that
$$
d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)}(-)
\;=\;
\left\{
- S
+
c^\alpha R_\alpha^a \phi^\ddagger_a
-
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_\alpha
\,,\,
(-)
\right\}
\,.
$$
=--
+-- {: .proof}
###### Proof
This is a simple straightforward computation, but we spell it out for illustration of the general principle. The result is to be compared with (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid):
for $\phi^a$:
$$
\begin{aligned}
\left\{
- S
+
c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
-
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_\alpha
\,,\,
\phi^a
\right\}
&
=
\left\{
c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} \,,\, \phi^a
\right\}
\\
& =
c^\alpha R_\alpha^{a'}
\underset{
\delta_{a'}^a
}{
\underbrace{
\left\{
\phi^\ddagger_{a'} \,,\, \phi^a
\right\}
}
}
\\
& =
c^\alpha R_\alpha^{a}
\end{aligned}
$$
for $c^\alpha$:
$$
\begin{aligned}
\left\{
- S
+
c^\alpha R_\alpha^{a} \phi^\ddagger_{a}
-
\tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_{\alpha'}
\,,\,
c^\alpha
\right\}
& =
\left\{
\tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_{\alpha'}
\,,\,
c^\alpha
\right\}
\\
& =
\tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
\underset{
\delta_{\alpha'}^\alpha
}{
\underbrace{
\left\{
c^\ddagger_{\alpha'}
\,,\,
c^\alpha
\right\}
}
}
\\
& =
\tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma
\end{aligned}
$$
for $\phi^\ddagger_a$:
$$
\begin{aligned}
\left\{
- S
+
c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
-
\tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_{\alpha}
\,,\,
\phi^\ddagger_a
\right\}
& =
-
\underset{
= -\frac{\partial S}{\partial \phi^a}
}{
\underbrace{
\left\{
S \,,\, \phi^{\ddagger}_a
\right\}
}
}
+
\left\{
c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
\,,\,
\phi^\ddagger_a
\right\}
\\
& =
\frac{\partial S}{\partial \phi^a}
+ c^\alpha
\underset{
= -\frac{\partial R_\alpha^{a'}}{\partial \phi^a}
}{
\underbrace{
\left\{ R_\alpha^{a'} \,,\, \phi^\ddagger_a \right\}
}
}
\phi^\ddagger_{a'}
\\
& =
\frac{\partial S}{\partial \phi^a}
-
c^\alpha \frac{\partial R_\alpha^{a'}}{\partial \phi^a}
\phi^\ddagger_{a'}
\end{aligned}
$$
for $c^\ddagger_\alpha$:
$$
\begin{aligned}
\left\{
- S
+
c^{\alpha'} R_{\alpha'}^{a} \phi^\ddagger_{a}
-
\tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_{\alpha'}
\,,\,
c^\ddagger_\alpha
\right\}
& =
\left\{
c^{\alpha'} R_{\alpha'}^a \phi^{\ddagger}_a
\,,\,
c^\ddagger_{\alpha}
\right\}
\;+\;
\left\{
\tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_{\alpha'}
\,,\,
c^\ddagger_\alpha
\right\}
\\
& =
\left\{
c^{\alpha'}
\,,\,
c^\ddagger_{\alpha}
\right\}
R_{\alpha'}^a \phi^{\ddagger}_a
\;+\;
\tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma}
\underset{ = - c^\beta \delta_{\alpha}^\gamma + \delta_{\alpha}^\beta c^\gamma}{
\underbrace{
\left\{
c^\beta c^\gamma
\,,\,
c^\ddagger_\alpha
\right\}
}}
c^\ddagger_{\alpha'}
\\
& =
R_\alpha^a \phi^\ddagger_{a}
+
\gamma^{\alpha'}{}_{\alpha \gamma} c^\gamma c^\ddagger_{\alpha'}
\end{aligned}
$$
Hence these values of the [[Schouten bracket]]/[[antibracket]] indeed all agree with the values of the CE-differential from (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid).
=--
As a corollary we obtain:
+-- {: .num_prop #ClassicalMasterEquation}
###### Proposition
**([[classical master equation]])**
Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}.
Then the [[Schouten bracket]]/[[antibracket]] (def. \ref{SchoutenBracketAndAntibracket}) of the function $S_{\text{BV-BRST}}$ S_{\text{BV-BRST}}
$$
S_{\text{BV-BRST}}
\;\coloneqq\;
S - d_{CE\left( X/\mathfrak{g}\right)}
$$
with itself vanishes:
$$
\left\{
S_{\text{BV-BRST}}
\,,\,
S_{\text{BV-BRST}}
\right\}
\;=\;
0
\,.
$$
Conversely, given a shifted [[cotangent bundle]] of the form $T^\ast[-1](X \times \mathfrak{g}[1])$ (eq:ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid), then the [[mathematical structure|struture]] of a
[[differential]] of degree +1 on its [[algebra of functions]] is equivalent to a degree-0 element $S \in C^\infty(T^\ast[-1](X \times \mathfrak{g}[1]))$ such that
$$
\left\{
S, S
\right\}
\;=\;
0
\,.
$$
Since therefore this equation controls the structure of [[derived critical loci]] once the underlying manifold $X$ and
[[Lie algebra]] $\mathfrak{g}$ is specified, it is also called the _[[master equation]]_ and here specifically the
_[[classical master equation]]_.
=--
$\,$
This concludes our discussion of plain [[derived critical loci]] inside [[Lie algebroids]]. Now we turn to applying these considerations about to [[Lagrangian densities]] on a [[jet bundle]], which are [[invariant]] under [[infinitesimal gauge symmetries]] generally
only up to a [[total spacetime derivative]]. By example \ref{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket} it is clear that this is best understood by first considering the refinement of the [[Schouten bracket]]/[[antibracket]] to this situation.
$\,$
**[[local BV-BRST complex|local]] [[antibracket]]**
{#LocalJetBundleAntibracket}
If we think of the invariant function $S$ in def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} as being the [[action functional]] (example \ref{ActionFunctional})
of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over a [[compact space|compact]] [[spacetime]] $\Sigma$,
with $X$ the [[space of field histories]] (or rather an [[infinitesimal neighbourhood]] therein),
hence with $\mathfrak{g}$ a Lie algebra of [[gauge symmetries]] acting on the field histories, then
the [[Chevalley-Eilenberg algebra]] $CE\left((X/\mathfrak{g})_{d S \simeq 0}\right)$ of the [[derived critical locus]] of $S$ is called the
_[[BV-BRST complex]]_ of the theory.
In applications of interest, the spacetime $\Sigma$ is _not_ [[compact space|compact]]. In that case one may still appeal to a construction on the [[space of field histories]]
as in example \ref{ArchetypeOfBVBRSTComplex} by considering the action functional for all [[adiabatic switching|adiabatically switched]] $b \mathbf{L}$ Lagrangians, with $b \in C_{cp}^\infty(\Sigma)$.
This approach is taken in ([Fredenhagen-Rejzner 11a](BV-BRST+formalism#FredenhagenRejzner11a)).
Here we instead consider now the "local lift" or "de-transgression" of the above construction from the
[[space of field histories]] to the [[jet bundle]] of the field bundle of the theory, refining the
[[BV-BRST complex]] (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) to the _[[local BV-BRST complex]]_ (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below), corresponding to the [[local BRST complex]] from example \ref{LocalOffShellBRSTComplex} ([Barnich-Brandt-Henneaux 00](local+BRST+cohomology#BarnichBrandtHenneaux00)).
This requires a slight refinement of the construction that leads to example \ref{ArchetypeOfBVBRSTComplex}:
In contrast to the [[action functional]] $S = \tau_\Sigma(g\mathbf{L})$ (example \ref{ActionFunctional}),
the [[Lagrangian density]] $\mathbf{L}$ is not strictly _invariant_ under [[infinitesimal gauge transformations]], in general,
rather it may change up to a horizontally exact term (by the very definition \ref{GaugeParameters}). The same is then true, in general, for its
[[Euler-Lagrange variational derivative]] $\delta_{EL} \mathbf{L}$ (unless we have already restricted to the [[shell]], by prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}, which however here we do not explicitly, but only via passing to [[cochain cohomology]] as in example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}).
This means that the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$
is, [[off-shell]], not a section of the infinitesimal cotangent bundle (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) of the
gauge action Lie algebroid on the jet bundle.
But it turns out that it still is a section of local refinement of the cotangent bundle, which is twisted by horizontally exact terms (prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid} below). To see the required twist, it is most convenient to make use of a local version of the [[antibracket]] (def. \ref{LocalAntibracket} below), via local refinement of example \ref{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}. As a result we may form the _local_ [[derived critical locus]] as in def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} but now with the invariance of the [[Lagrangian density]] only up to [[total spacetime derivatives]] taken into account. Its [[Chevalley-Eilenberg algebra]] is called the _[[local BV-BRST complex]]_ (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below).
The following is the direct refinement of the concept of the underlying [[graded manifold]] of the infinitesimal [[cotangent bundle]] of an [[action Lie algebroid]] in example \ref{CotangentBundleOfActionLieAlgebroid} to the case where the base manifold is generalized to a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and the [[Lie algebra]] to a [[gauge parameter bundle]] (def. \ref{GaugeParameters}):
+-- {: .num_defn #InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}
###### Definition
**([[infinitesimal neighbourhood]] of [[zero section]] in [[cotangent bundle]] of [[fiber product]] of [[field bundle]] with shifted [[gauge parameter bundle]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be
a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}), inducing the [[Lie algebroid]]
$$
E / ( \mathcal{G} \times_\Sigma T \Sigma )
\;=\;
\left(
J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
,
s_{BRST} )
\right)
$$
whose [[Chevalley-Eilenberg algebra]] is the _[[local BRST complex]]_ of the field theory (example \ref{LocalOffShellBRSTComplex}).
Then we write
$$
T^\ast_{\Sigma,inf}\left(
E \times_\Sigma (\mathcal{G}[1])
\right)
\,,
\phantom{AAA}
T^\ast_{\Sigma,inf}[-1]\left(
E \times_\Sigma (\mathcal{G}[1])
\right)
$$
for, on the left, the [[infinitesimal neighbourhood]] of the [[zero section]] of the [[vertical cotangent bundle]] of the [[graded manifold|graded]] [[fiber product]] of the [[field bundle]] with the fiber-wise shifted [[gauge parameter bundle]], as well as its shifted version on the right, as in (eq:ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid).
In [[local coordinates]] this means the following: Assuming that the [[field bundle]] $E$ and the [[gauge parameter bundle]] $\mathcal{G}$ are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with fiber coordinates $(\phi^a)$ and $(c^\alpha)$, respectively, then $T^\ast_{\Sigma,inf}\left(E \times_\Sigma (\mathcal{G}[1])\right)$ is the trivial graded vector bundle with fiber coordinates
$$
\label{coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle}
\array{
T^\ast_{\Sigma,inf}\left(
E \times_\Sigma (\mathcal{G}[1])
\right)
& \phantom{AAAAA}&
T^\ast_{\Sigma,inf}[-1]\left(
E \times_\Sigma (\mathcal{G}[1])
\right)
\\
& \phantom{A}
\\
\array{
& (c^\ddagger_\alpha), & (\phi^\ddagger_a),(\phi^a), & (c^\alpha)
\\
deg = & -1 & 0 & 1
}
& \phantom{AA}&
\array{
& (c^\ddagger_\alpha), & (\phi^\ddagger_a)\, & (\phi^a), & (c^\alpha)
\\
deg = & -2 & -1 & 0 & 1
}
}
$$
and such that smooth functions on $T^\ast_{\Sigma,inf}\left(E \times_\Sigma (\mathcal{G}[1])\right)$ are [[formal power series]] in $c^\ddagger_\alpha$ (necessarily due to degree reasons) and in $\phi^\ddagger_a$ (reflecting the [[infinitesimal neighbourhood]] of the [[zero section]]).
Here the shifted cotangents to the fields are called the _[[antifields]]_:
* $\phi^\ddagger_a$ is _[[antifield]]_ to the [[field (physics)|field]] $\phi^a$
* $c^\ddagger_\alpha$ is _[[antifield]]_ to the [[ghost field]] $c^\alpha$.
=--
The following is the direct refinement of the concept of the [[Schouten bracket]] on an [[action Lie algebroid]] from def. \ref{SchoutenBracketAndAntibracket} to the case where the base manifold is generalized to the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and the [[Lie algebra]] to the [[jet bundle]] of a [[gauge parameter bundle]] (def. \ref{GaugeParameters}):
+-- {: .num_defn #LocalAntibracket}
###### Definition
**([[local antibracket]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}),
inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]]
$$
E / ( \mathcal{G} \times_\Sigma T \Sigma )
\;=\;
\left(
J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
,
s_{BRST} )
\right)
$$
whose [[Chevalley-Eilenberg algebra]] is the _[[local BRST complex]]_ of the field theory with shifted infinitesimal [[vertical cotangent bundle]]
$$
\label{BVBRSTGradedFieldBundle}
E_{\text{BV-BRST}}
\;\coloneqq\;
T^\ast_{\Sigma,inf}[-1]\left(
E \times_\Sigma (\mathcal{G}[1])
\right)
$$
of its underlying graded bundle
from def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}.
Then on the horizontal $p+1$-forms on this bundle (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) which in terms of the [[volume form]] may all be decomposed as (eq:LagrangianFunctionViaVolumeForm)
$$
H
\;=\;
h \, dvol_\Sigma
\;\in\;
\Omega^{p+1}_\Sigma\left( \,T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \, \right)
$$
the _[[local antibrackets]]_
$$
\{-,-\}'
,
\{-,-\}
\;\colon\;
\Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
\,\otimes\,
\Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
\longrightarrow
\Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
$$
are the functions which are given in the [[local coordinates]] (eq:coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle) as follows:
The first version is
$$
\begin{aligned}
\left\{ f\, dvol_\Sigma \,,\,g \, dvol_\Sigma \right\}'
& \coloneqq
\phantom{+}
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f }{\delta \phi^\ddagger_a}
\frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {\phi^a}^{\phantom{\ddagger}}}
-
\frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {\phi^a}^{\phantom{\ddagger}}}
\frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta \phi^\ddagger_a}
\right)
dvol_\Sigma
\\
&
\phantom{=} +
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha}
\frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {c^\alpha}^{\phantom{\ddagger}}}
-
\frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {c^\alpha}^{\phantom{\ddagger}}}
\frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta c^\ddagger_\alpha}
\right)
dvol_\Sigma
\,.
\end{aligned}
$$
This is of the form of the [[Schouten bracket]] (eq:Antibracket) but with [[Euler-Lagrange derivatives]] (eq:EulerLagrangeEquationGeneral) instead of [[partial derivatives]],
The second version is this:
$$
\label{LocalCommutatorOfDerivationsOnJetBundle}
\begin{aligned}
\left\{
f \, dvol_\Sigma,
g \, dvol_\Sigma
\right\}
& \coloneqq
\phantom{+}
\left(
\left(
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^a}
\right)
\right)
\left(
\frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^\ddagger_{a,\mu_1 \cdots \mu_k}}
\right)
-
\left(
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^\ddagger_a}
\right)
\right)
\left(
\frac{\overset{\rightarrow}{\partial} g}{\partial \phi^a_{,\mu_1 \cdots \mu_k}}
\right)
\right) \, dvol_\Sigma
\\
&
\phantom{\coloneqq}
+
\left(
\left(
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\alpha}
\right)
\right)
\left(
\frac{\overset{\rightarrow}{\partial} g}{\partial {c}^\ddagger_{\alpha,\mu_1 \cdots \mu_k}}
\right)
-
\left(
\frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha}
\right)
\right)
\left(
\frac{\overset{\rightarrow}{\partial} g}{\partial c^\alpha_{,\mu_1 \cdots \mu_k}}
\right)
\right) \, dvol_\Sigma
\end{aligned}
$$
where again $\frac{\delta_{EL}}{\delta \phi^a}$ denotes the [[Euler-Lagrange variational derivative]] (eq:EulerLagrangeEquationGeneral)
=--
([Barnich-Henneaux 96 (2.9) and (2.12)](local+BRST+cohomology#BarnichHenneaux96), reviewed in [Barnich 10 (4.9)](BRST+complex#Barnich10))
+-- {: .num_prop #BasicPropertiesOfTheLocalAntibracket}
###### Proposition
**(basic properties of the [[local antibracket]])**
The [[local antibracket]] from def. \ref{LocalAntibracket} satisfies the following properties:
1. The two versions differ by a [[total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):
$$
\{f,g\}
=
\{f,g\}' + d(...)
\,.
$$
1. The primed version is strictly graded skew-symmetric:
$$
\left\{f \, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\}'
\;=\;
- (-1)^{deg(f) deg(g)} \, \left\{g \, dvol_\Sigma \,,\, f\, dvol_\Sigma \right\}
$$
1. The unprimed version $\{-,-\}$ strictly satisfies the graded [[Jacobi identity]]; in that it is a graded [[derivation]] in the second argument, of degree one more than the degree of the first argument:
$$
\label{LocalAntibracketGradedDerivationInSecondArgument}
\left\{ f\, dvol_\Sigma, \left\{ g\, dvol_\Sigma \,,\, h\, dvol_\Sigma \right\}\right\}
\;=\;
\underset{
=
\left\{
\left\{ f\, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\}'
\,,
h\, dvol_\Sigma
\right\}
}{
\underbrace{
\left\{
\left\{
f\, dvol_\Sigma
\,,\,
g\, dvol_\Sigma
\right\}
\,,\,
h\, dvol_\Sigma
\right\}
}
}
\;+\;
(-1)^{(deg(f)+1) deg(g)}
\left\{
g\, dvol_\Sigma
\,,\,
\left\{
f\, dvol_\Sigma
\,,\,
h\, dvol_\Sigma
\right\}
\right\}
$$
and the first term on the right is equivalently given by the primed bracket, as shown under the brace;
1. the [[horizontal derivative|horizontally]] [[exact differential form|exact]] [[horizontal differential forms]] are an [[ideal]] for either bracket, in that for $f dvol_\Sigma = d(\cdots)$ or $g dvol_\Sigma = d(\cdots)$ we have
$$
\{ f dvol_\Sigma, g \, dvol_\Sigma \}' = 0
\phantom{AAA}
\{ f dvol_\Sigma, g \, dvol_\Sigma \} = d(\cdots)
$$
for all $f$, $g$ of homogeneous degree $deg(f)$ and $deg(g)$, respectively.
=--
([Barnich-Henneaux 96 (B.6) and footnote 9](local+BRST+cohomology#BarnichHenneaux96)).
+-- {: .proof}
###### Proof
That the two expressions differ by a horizontally exact terms follows by the very definition of the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral). Also the graded skew symmetry of the primed bracket is manifest.
The third point requires some computation ([Barnich-Henneaux 96 (B.9)](local+BRST+cohomology#BarnichHenneaux96)).
Finally that $\{-,-\}'$ vanishes when at least one of its arguments is horizontally exact follows from the fact that already the [[Euler-Lagrange derivative]] vanishes on this argument (example \ref{TrivialLagrangianDensities}). This implies that $\{-,-\}$ is horizontally exact when at least one of its arguments is so, by the first item.
=--
The following is the local refinement of prop. \ref{ClassicalMasterEquation}:
+-- {: .num_defn #ClassicalMasterEquationLocal}
###### Remark
**(local [[classical master equation]])**
The third item in prop. \ref{BasicPropertiesOfTheLocalAntibracket} implies that the following
conditions on a [[Lagrangian density]] $\mathbf{K} \in \Omega^{p+1}_\Sigma( T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] ) )$
whose degree is even
$$
\mathbf{K} = K\, dvol_\Sigma
\,,
\phantom{AAA}
deg(L) \in 2 \mathbb{Z}
$$
are equivalent:
1. forming the [[local antibracket]] (def. \ref{LocalAntibracket}) with $\mathbf{K}$ is a [[differential]]
$$
\left(\left\{ \mathbf{K},-\right\}\right)^2 = 0
\,,
$$
1. the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]:
$$
\left\{ \mathbf{K}, \mathbf{K}\right\} = d(...)
$$
1. the other variant of the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]:
$$
\left\{ \mathbf{K}, \mathbf{K}\right\}' = d(...)
$$
This condition is also called the _local [[classical master equation]]_.
=--
$\,$
**[[derived critical locus]] on [[jet bundle]] -- the [[local BV-BRST complex]]**
{#DerivedCriticalLocusOnJetBundle}
With the local version of the [[antibracket]] in hand (def. \ref{LocalAntibracket}) it is now straightforward to refine the construction of a [[derived critical locus]] inside an [[action Lie algebroid]] (example \ref{ArchetypeOfBVBRSTComplex}) to the "derived" [[shell]] (eq:ShellInJetBundle) inside the formal dual of the [[local BRST complex]] (example \ref{LocalOffShellBRSTComplex}).
The result is a [[derived Lie algebroid]] whose [[Chevalley-Eilenberg algebra]] is called the _[[local BV-BRST complex]]_.
This is example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below.
The following definition \ref{LocalInfinitesimalCotangentLieAlgebroid} is the local refinement of
def. \ref{LieAlgebroidInfinitesimalCotangentBundle}:
+-- {: .num_defn #LocalInfinitesimalCotangentLieAlgebroid}
###### Definition
**(local infinitesimal cotangent Lie algebroid)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be
a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}),
inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]]
$$
E / ( \mathcal{G} \times_\Sigma T \Sigma )
\;=\;
\left(
J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
,
s_{BRST} )
\right)
$$
whose [[Chevalley-Eilenberg algebra]] is the _[[local BRST complex]]_ of the field theory.
Consider the case that both the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (def. \ref{FieldsAndFieldBundles})
as well as the [[gauge parameter]] bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ are [[trivial vector bundles]]
(example \ref{TrivialVectorBundleAsAFieldBundle}) over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime})
with [[field (physics)|field]] coordinates $(\phi^a)$ and [[gauge parameter]] coordinates $(c^\alpha)$.
Then the vertical infinitesimal cotangent Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) has
coordinates as in (eq:GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid) as well as all the corresponding jets
and including also the
horizontal differentials:
$$
\array{
&
\left(
c^\ddagger_{\alpha,\mu_1 \cdots \mu_k}
\right)
&
\left(
\phi^a_{,\mu_1 \cdots \mu_k}
\right)
,
\left(
\phi^\ddagger_{a,\mu_1 \cdots \mu_k}
\right)
&
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
\right),
\left(
d x^\mu
\right)
\\
deg =
&
-1
&
0
&
+1
}
\,.
$$
In terms of these coordinates [[BRST differential]] $s_{BRST}$, thought of as a prolonged [[evolutionary vector field]]
on $E \times_\Sigma \mathcal{G}$,
corresponds to the smooth function on the shifted cotangent bundle given by
$$
\label{BRSTFunctionForClosed}
L_{BRST}
\;=\;
\left(
\underset{k \in \mathbb{N}}{\sum}
c^\alpha_{,\mu_1 \cdots \mu_k}
R_\alpha^{a \mu_1 \cdots \mu_k}
\right)
\phi^\ddagger_a
\;+\;
\tfrac{1}{2}
\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
c^\ddagger_\alpha
\;\in\;
C^\infty\left(
T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] )
\right)
\,,
$$
to be called the _[[BRST complex|BRST]] [[Lagrangian function]]_ and the product with the [[spacetime]] [[volume form]]
$$
L_{BRST} \, dvol_\Sigma
\;\in\;
\Omega^{p+1,0}_\Sigma(E \times_\Sigma \mathcal{G}[1])
$$
as the _[[BRST complex|BRST]] [[Lagrangian density]]_.
We now define the [[Chevalley-Eilenberg differential]] on smooth functions on $T^\ast_{inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) )$ to be
given by the [[local BV-BRST complex|local]] [[antibracket]] $\{-,-\}$ (eq:LocalCommutatorOfDerivationsOnJetBundle)
with the BRST Lagrangian density (eq:BRSTFunctionForClosed)
$$
d_{CE(T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) ))}
\;\coloneqq\;
\left\{
L_{BRST} dvol_\Sigma, -
\right\}
$$
This defines an $L_\infty$-algebroid to be denoted
$$
T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) )
\,.
$$
=--
The local refinement of prop. \ref{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid} is now this:
+-- {: .num_prop #EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}
###### Proposition
**([[Euler-Lagrange form]] is [[section]] of local cotangent bundle of [[jet bundle]] [[gauge symmetry|gauge]]-[[action Lie algebroid]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParameters})
which are closed (def. \ref{GaugeParametersClosed}),
inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]] $E / ( \mathcal{G} \times_\Sigma T \Sigma )$
and via def. \ref{LocalInfinitesimalCotangentLieAlgebroid} its local cotangent [[Lie ∞-algebroid]]
$T^\ast_{inf}_\Sigma(E / ( \mathcal{G} \times_\Sigma T \Sigma ))$.
Then the [[Euler-Lagrange variational derivative]]
(prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) constitutes
a [[section]] of the local cotangent Lie ∞-algebroid (def. \ref{LocalInfinitesimalCotangentLieAlgebroid})
$$
\array{
&& T^\ast_{\Sigma,inf}\left( E/(\mathcal{G} \times_\Sigma T \Sigma) \right)
\\
& {}^{\mathllap{ \delta_{EL} \mathbf{L} }}\nearrow & \downarrow^{\mathrlap{cb}}
\\
E/(\mathcal{G} \times_\Sigma T \Sigma)
&=&
E/(\mathcal{G} \times_\Sigma T \Sigma)
}
$$
given dually
$$
CE(E/(\mathcal{G} \times_\Sigma T\Sigma))
\overset{(\delta_{EL}\mathbf{L})^\ast}{\longleftarrow}
CE(T^\ast_{inf}(E/(\mathcal{G}\times_\Sigma T \Sigma)))
$$
by
$$
\array{
\left\{
\phi^a_{,\mu_1 \cdots \mu_k}
\right\}
&\longleftarrow&
\left\{
\phi^a_{,\mu_1 \cdots \mu_k}
\right\}
\\
\left\{
c^\alpha_{,\mu_1 \cdots \mu_k}
\right\}
&\longleftarrow&
\left\{
c^\alpha_{,\mu_1 \cdots \mu_k}
\right\}
\\
\left\{
\frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
\frac{\delta_{EL} L}{\delta \phi^a}
\right)
\right\}
&\longleftarrow&
\left\{
\phi^\ddagger_{a,\mu_1 \cdots \mu_k}
\right\}
\\
\left\{
0
\right\}
&\longleftarrow&
\left\{
c^\ddagger_{\alpha,\mu_1 \cdots \mu_k}
\right\}
}
$$
=--
+-- {: .proof}
###### Proof
The proof of this proposition is a special case of the
observation that the differentials involved are part of the local BV-BRST differential;
this will be a direct consequence of the proof of prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below.
=--
The local analog of def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} is now the following
definition \ref{DerivedProlongedShell} of the "derived prolonged shell" of the theory (recall the ordinary [[prolonged shell]]
$\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ from (eq:ProlongedShellInJetBundle)):
+-- {: .num_defn #DerivedProlongedShell}
###### Definition
**(derived reduced [[prolonged shell]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be
a bundle of closed irreducible [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}),
inducing via prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}
a section $\delta_{EL} L$ of the local cotangent Lie algebroid of the jet bundle gauge-action Lie algebroid.
Then the _derived prolonged shell_ $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ is the [[derived critical locus]]
of $\delta_{EL} L$, hence the [[homotopy pullback]] of $\delta_{EL} L$
along the zero section of the local cotangent Lie $\infty$-algebroid:
$$
\array{
(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}
&\longrightarrow&
E/( \mathcal{G} \times_\Sigma T \Sigma )
\\
\downarrow
&(pb)&
\downarrow^{\mathrlap{0}}
\\
E/(\mathcal{G} \times_\Sigma T \Sigma)
&\underset{\delta_{EL} L}{\longrightarrow}&
T^\ast_{\Sigma,inf}
\left(
E/( \mathcal{G} \times_\Sigma T \Sigma )
\right)
}
$$
=--
As before, for the purpose of our running examples the reader may take the following example as the
definition of the derived reduced prolonged shell (def. \ref{DerivedProlongedShell}).
This is local refinement of example \ref{ArchetypeOfBVBRSTComplex}:
+-- {: .num_example #LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}
###### Example
**([[local BV-BRST complex]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Assume that both are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with field coordinates as in prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}.
Then the [[Chevalley-Eilenberg algebra]] of the derived prolonged shell
$(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ (def. \ref{DerivedProlongedShell}) is
$$
CE\left(
(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}
\right)
\;=\;
\left(
C^\infty\left(
T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] \times_\Sigma T^\ast \Sigma[1] )
\right)
\,,\,
\underset{
= s
}{
\underbrace{
\left\{
\left(- L + L_{BRST}\right) dvol_\Sigma
\,,
(-)
\right\}
}
}
\;+\;
d
\right)
$$
where the underlying graded algebra is the [[algebra of functions]] on the (-1)-shifted [[vertical cotangent bundle]] of the [[fiber product]] of the [[field bundle]] with the (+1)-shifted [[gauge parameter bundle]] (as in example \ref{ArchetypeOfBVBRSTComplex}) and the shifted cotangent bundle of $\Sigma$, and where the [[Chevalley-Eilenberg differential]] is the sum of the [[horizontal derivative]] $d$ with the _[[BV-BRST differential]]_
$$
\label{LocalAntibracketVersionOfBVBRSTDifferential}
s
\;\coloneqq\;
\left\{
\left(- L + L_{BRST}\right) dvol_\Sigma
\,,
(-)
\right\}
$$
which is the [[local antibracket]] (def. \ref{LocalAntibracket}) with the _[[BV-BRST Lagrangian density]]_
$$
\left( -L + L_{BRST}\right)
\;\in\;
\Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1] \right)\right)
$$
which itself is the sum of (minus) the given [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with the BRST Lagrangian (eq:BRSTFunctionForClosed).
The action of the [[BV-BRST differential]] on the generators is as follows:
$$
\array{
& & \array{ \text{BV-BRST differential} \\ s } &
\\
\text{field}
&
\phi^a
&\mapsto&
\underset{ = s_{BRST}(\phi^a) }{
\underbrace{
\left(
\underset{k \in \mathbb{N}}{\sum} c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{a \mu_1 \cdots \mu_k}
\right)
}
}
&
\text{gauge symmetry}
\\
\text{
ghost field
}
&
c^\alpha
&\mapsto&
\underset{ = s_{BRST}(c^\alpha) }{
\underbrace{
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
}
}
&
\text{Lie bracket}
\\
\text{antifield}
&
\phi^\ddagger_\alpha
&\mapsto&
\phantom{-}
\underset{ = s_{BV}(\phi^\ddagger_a) }{
\underbrace{
\frac{\delta_{EL} L}{\delta \phi^a}
}}
&
\text{equations of motion}
\\
&&&
\underset{ = s_{BRST}(\phi^\ddagger_a) }{
\underbrace{
-
\left(
\underset{k \in \mathbb{N}}{\sum}
\frac{\delta_{EL}}{\delta \phi^a}
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
R_\alpha^{b \mu_1 \cdots \mu_k}
\phi^\ddagger_b
\right)
\right)
}
}
&
\\
\array{
\text{antifield of}
\\
\text{ghost field}
}
&
c^\ddagger_\alpha
&\mapsto&
\underset{ = s_{BV}(c^\ddagger_\alpha) }{
\underbrace{
-
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
}
}
&
\text{Noether identities}
\\
&&& +
\underset{ = s_{BRST}(c^\ddagger_\alpha) }{
\underbrace{
\gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'}
}
}
}
$$
and this extends to jets of generator by $s \circ d + d \circ s = 0$.
This is called the _[[local BV-BRST complex]]_.
By introducing a bigrading as in prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}
$$
\array{
&
\left(
c^\ddagger_{\alpha, \mu_1 \cdots \mu_k}
\right)
&
\left(
\phi^\ddagger_{a, \mu_1 \cdots \mu_k}
\right)
&
\left(
\phi^a_{,\mu_1 \cdots \mu_k}
\right)
&
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
\right)
\\
deg_{gh} =
&
0
&
0
&
0
&
+1
\\
deg_{af} =
&
-2
&
-1
&
0
&
0
}
$$
this splits into the [[total complex]] of a [[bicomplex]] with
$$
s \;=\; s_{BV} + s_{BRST}
$$
with
$$
\array{
& s_{BRST} & s_{BV}
\\
deg_{gh} = & +1 & 0
\\
deg_{af} = & 0 & +1
}
$$
as shown in the above table. Under this decomposition, the _[[classical master equation]]_
$$
s^2 = 0
\phantom{AAAA}
\Leftrightarrow
\phantom{AAAA}
\left\{ \left( -L + L_{BRST}\right) dvol_\Sigma \,,\, \left( -L + L_{BRST}\right) dvol_\Sigma \right\} = 0
$$
is equivalent to three conditions:
$$
\array{
\left( s_{BV} \right)^2 = 0 && \text{Noether's second theorem}
\\
\left( s_{BRST} \right)^2 = 0 && \text{closure of gauge symmetry}
\\
\left[ s_{BV}, s_{BRST} \right] = 0 &&
\left\{
\array{
\text{ gauge symmetry preserves the shell },
\\
\text{ gauge symmetry acts on Noether identities }
}
\right.
}
$$
=--
(e.q. [Barnich 10 (4.10)](BRST+complex#Barnich10))
+-- {: .proof}
###### Proof
Due to the construction in def. \ref{DerivedProlongedShell} the [[BRST differential]] by itself is already assumed to square to the
$$
\left(s_{BRST}\right)^2 = 0
$$
The remaining conditions we may check on 0-jet generators.
The condition
$$
\left( s_{BV} \right)^2 = 0
$$
is non-trivial only on the [[antifields]] of the [[ghost fields]]. Here we obtain
$$
\begin{aligned}
s_{BV} s_{BV} c^\ddagger_\alpha
& =
-\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
\\
& =
-\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \frac{\delta_{EL} L}{\delta \phi^a}
\right)
\end{aligned}
$$
That this vanishes is the statement of [[Noether's theorem|Noether's second theorem]] (prop. \ref{NoetherIdentities}).
Next we check
$$
s_{BV} \circ s_{BRST} + s_{BRST} \circ s_{BV}
=
0
$$
on generators. On the [[field (physics)|fields]] $\phi^a$ and the [[ghost fields]] $c^\alpha$ this is trivial (both summands vanish separately).
On the [[antifields]] we get on the one hand
$$
\begin{aligned}
s_{BRST} s_{BV} \phi^{\ddagger}_a
& =
s_{BRST} \frac{\delta_{EL} L}{\delta \phi^a}
\\
& =
\underset{k}{\sum}
\underset{q}{\sum}
\frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}}
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
R_\alpha^{b \mu_1 \cdots \mu_k}
\right)
\frac{\partial}{\partial \phi^b_{,\nu_1 \cdots \nu_q}}
\frac{\delta_{EL} L}{\delta \phi^a}
\end{aligned}
$$
and on the other hand
$$
\begin{aligned}
s_{BV} s_{BRST} \phi^\ddagger_a
& =
-
s_{BV}
\underset{k}{\sum}
\frac{\delta_{EL}}{\delta \phi^a}
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
R_\alpha^{b \mu_1 \cdots \mu_k}
\phi^\ddagger_b
\right)
\\
& =
+
\underset{k}{\sum}
\underset{q}{\sum}
(-1)^q
\frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}}
\left(
\frac{\partial}{\partial \phi^a_{,\mu_1 \cdots \mu_q}}
\left(
c^\alpha_{,\mu_1 \cdots \mu_k}
R_\alpha^{b \mu_1 \cdots \mu_k}
\right)
\frac{\delta_{EL} L}{\delta \phi^b}
\right)
\end{aligned}
$$
That the sum of these two terms indeed vanishes is equation (eq:TowardsProofThatSymmetriesPreserveTheShell) in the proof of the
on-shell invariance of the [[equations of motion]] under [[infinitesimal symmetries of the Lagrangian]] (prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion})
Finally, on antifields of ghostfields we get
$$
\begin{aligned}
s_{BV} s_{BRST} c^\ddagger_\alpha
& =
s_{BV} \gamma^{\alpha'}{}_{\alpha \beta} c^\beta c^\ddagger_{\alpha'}
\\
& =
-
\gamma^{\alpha'}{}_{\alpha \beta} c^\beta
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_{\alpha'}^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
\end{aligned}
$$
as well as
$$
\begin{aligned}
s_{BRST} s_{BV} c^\ddagger_\alpha
& =
s_{BRST}
\left(
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
\right)
\\
& =
R
\left(
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
\right)
\;-\;
\left(
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k}
\left(
\underset{q \in \mathbb{N}}{\sum}
\frac{\delta_{EL}}{\delta \phi^a}
\left(
c^{\alpha'}_{,\nu_1 \cdots \nu_q}
R_{\alpha'}^{b \nu_1 \cdots \nu_q}
\phi^\ddagger_b
\right)
\right)
\right)
\right)
\\
& +
R
\left(
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
\right)
\right)
\;-\;
\left(
\underset{k \in \mathbb{N}}{\sum}
(-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
\left(
R_\alpha^{a \mu_1 \cdots \mu_k}
\left(
\underset{q,r \in \mathbb{N}}{\sum}
(-1)^{r}
\frac{d^r}{d x^{\rho_1} \cdots d x^{\rho_r}}
\left(
c^{\alpha'}_{,\nu_1 \cdots \nu_q}
\frac{\partial R_{\alpha'}^{b \nu_1 \cdots \nu_q}}{\partial \phi^a_{,\rho_1 \cdots \rho_r}}
\phi^\ddagger_b
\right)
\right)
\right)
\right)
\\
& = (R \cdot N_R)_a^b (\phi^\ddagger_b)
\end{aligned}
$$
where in the last line we identified the [[Lie algebra action]] of [[infinitesimal symmetries of the Lagrangian]] on [[Noether operators]] from def. \ref{NoetherOperator}. Under this identification, the fact that
$$
\left(
s_{BRST}s_{BV}
+
s_{BV} s_{BRST}
\right)
c^\ddagger_\alpha
=
0
$$
is relation (eq:LieActionOnNoetherOperatorGivesLieBracketUnderNoetherTheorem) in prop. \ref{LieAlgebraActionOfInfinitesimalSymmetriesOfTheLagrangianOnNoetherOperators}.
=--
+-- {: .num_example #DerivedProlongedShellInAbsenceOfExplicitGaugeSymmetries}
###### Example
**(derived prolonged shell in the absence of explicit gauge symmetry -- the [[local BV-complex]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrde
rJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
with vanishing [[gauge parameter bundle]] (def. \ref{GaugeParameters}) (possibly because there are no non-trivial [[infinitesimal gauge symmetries]], such as for the [[scalar field]], or because none were chose), hence with no [[ghost fields]] introduced. Then the
local [[derived critical locus]] of its [[Lagrangian density]] (def. \ref{DerivedProlongedShell})
is the plain [[local BV-complex]] of def. \ref{BVComplexOfOrdinaryLagrangianDensity}.
$$
s = s_{BV}
\,.
$$
=--
+-- {: .num_example #LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}
###### Example
**([[local BV-BRST complex]] of [[vacuum]] [[electromagnetism]] on [[Minkowski spacetime]])**
Consider the [[Lagrangian field theory]] of [[free field theory|free]] [[electromagnetism]]
on [[Minkowski spacetime]] (example \ref{ElectromagnetismLagrangianDensity})
with [[gauge parameter]] as in example \ref{InfinitesimalGaugeSymmetryElectromagnetism}.
With the [[field (physics)|field]] and [[gauge parameter]] coordinates as chosen in these examples
$$
\left(
(a_\mu), c
\right)
$$
then the [[local BV-BRST complex]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm})
has generators
$$
\array{
& c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
\\
deg =
&
-2 & -1 & 0 & 1
}
$$
together with their [[total spacetime derivatives]],
and the local BV-BRST differential $s$ acts on these generators as follows:
$$
s
\;\colon\;
\left\{
\array{
(a^\dagger)^\mu &\mapsto& f^{\nu \mu}_{,\nu} & \text{(equations of Motion -- vacuum Maxwell equations)}
\\
c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{(Noether identity)}
\\
a_\mu &\mapsto& c_{,\mu} & \text{(infinitesimal gauge transformation)}
}
\right.
$$
=--
More generally:
+-- {: .num_example #LocalBVBRSTComplexOfYangMillsTheory}
###### Example
**([[local BV-BRST complex]] of [[Yang-Mills theory]])**
For $\mathfrak{g}$ a [[semisimple Lie algebra]], consider $\mathfrak{g}$-[[Yang-Mills theory]] on [[Minkowski spacetime]] from example \ref{YangMillsLagrangian}, with [[local BRST complex]] as in example \ref{YangMillsLocalBRSTComplex}, hence with
[[BRST Lagrangian]] (eq:BRSTFunctionForClosed) given by
$$
L_{BRST}
=
\left(
c^\alpha_{,\mu}
-
\gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu
\right)
(a^\ddagger)_\alpha^\mu
\;+\;
\tfrac{1}{2}
\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha
\,.
$$
Then its [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) has
[[BV-BRST differential]] $s = \left\{ -L + L_{BRST} \,,\, - \right\}$ given on 0-jets as follows:
$$
\array{
& & s &
\\
\text{field}
&
a_\mu^\alpha
&\mapsto&
c^\alpha_{,\mu}
-
\gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu
&
\text{gauge symmetry}
\\
\text{
ghost field
}
&
c^\alpha
&\mapsto&
\tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
&
\text{Lie bracket}
\\
\text{antifield}
&
(a^\ddagger)^\mu_\alpha
&\mapsto&
\phantom{-}
\left(
\frac{d}{d x^\mu} f^{\mu \nu \alpha'}
+
\gamma^{\alpha'}{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma}
\right)
k_{\alpha' \alpha}
&
\text{equations of motion}
\\
&&&
-
\gamma^{\alpha'}{}_{\beta \alpha}c^\beta
(a^\ddagger)_{\alpha'}^\mu
&
\\
\text{anti ghostfield}
&
c^\ddagger_\alpha
&\mapsto&
\gamma^{\alpha'}{}_{\alpha \gamma} a^\gamma_\mu (a^\ddagger)^\mu_{\alpha'}
+ \frac{d}{d x^\mu} (a^\ddagger)^\mu_\alpha
&
\text{Noether identities}
\\
&&& + \gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'}
}
$$
=--
(e.g. [Barnich-Brandt-Henneaux 00 (2.8)](local+BRST+cohomology#BarnichBrandtHenneaux00))
$\,$
So far the discussion yields just the [[algebra of functions]] on the derived reduced prolonged shell.
We now discuss the derived analog of the full [[variational bicomplex]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
to the derived reduced shell.
$\,$
**(derived variational bicomplex)**
The analog of the [[de Rham complex]] of a [[derived Lie algebroid]] is called the _[[Weil algebra]]_:
+-- {: .num_defn #WeilAlgebra}
###### Definition
**([[Weil algebra]] of a [[Lie algebroid]])**
Given a [[derived Lie algebroid]] $\mathfrak{a}$ over some $X$ (def. \ref{LInfinityAlgebroid}), its [[Weil algebra]] is
$$
W(\mathfrak{a})
\;\coloneqq\;
\left(
Sym_{C^\infty(X)}( \Gamma(T^\ast_{inf} X) \oplus \mathfrak{a}_\bullet \oplus \mathfrak{a}[1]_\bullet )
\;,\;
\mathbf{d}_W \coloneqq \mathbf{d} + d_{CE}
\right)
\,,
$$
where $\mathbf{d}$ acts as the de Rham differential $\mathbf{d} \colon C^\infty(X) \to \Gamma(T^\ast_{inf} X)$
on functions, and as the degree shift operator $\mathbf{d} \colon \mathfrak{a}_\bullet \to \mathfrak{a}[1]_\bullet$
on the graded elements.
=--
| [[smooth manifolds]] | [[derived Lie algebroids]] |
|----------------------|----------------------------|
| [[algebra of functions]] | [[Chevalley-Eilenberg algebra]] |
| algebra of [[differential forms]] | [[Weil algebra]] |
+-- {: .num_example #ClassicalWeilAlgebra}
###### Example
**(classical [[Weil algebra]])**
Let $\mathfrak{g}$ be a [[Lie algebra]] with corresponding
[[Lie algebroid]] $B \mathfrak{g}$ (example \ref{BasicExamplesOfLieAlgebroids}).
Then the Weil algebra (def. \ref{WeilAlgebra}) of $B \mathfrak{g}$ is
the traditional Weil algebra of $\mathfrak{g}$ from classical [[Lie theory]].
=--
+-- {: .num_defn #BVVariationalBicomplex}
###### Definition
**([[variational BV-bicomplex]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
equipped with a [[gauge parameter bundle]] $\mathcal{G}$ (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}).
Consider the [[Lie algebroid]] $E/(\mathcal{G} \times_\Sigma T \Sigma)$ from example \ref{LocalOffShellBRSTComplex},
whose [[Chevalley-Eilenberg algebra]] is the [[local BRST complex]] of the theory.
Then its [[Weil algebra]] $W(E/(\mathcal{G} \times_\Sigma T \Sigma))$ (def. \ref{WeilAlgebra})
has as differential the [[variational derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
plus the [[BRST differential]]
$$
\begin{aligned}
d_{W}
& = \mathbf{d} - (d - s_{BRST})
\\
& =
\delta + s_{BRST}
\end{aligned}
\,.
$$
Therefore we speak of the _[[variational BRST-bicomplex]]_ and write
$$
\Omega^\bullet_\Sigma( E/(\mathcal{G} \times_\Sigma T \Sigma) )
\,.
$$
Similarly, the Weil algebra of the derived prolonged shell $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ (def. \ref{DerivedProlongedShell})
has differential
$$
\begin{aligned}
d_W
& = \mathbf{d} - (d - s)
\\
& = \delta + s
\end{aligned}
\,.
$$
Since $s$ is the [[BV-BRST differential]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm})
this defines the "BV-BRST [[variational bicomplex]]".
=--
$\,$
**global [[BV-BRST complex]]**
{#BVBRSTComplexGlobal}
Finally we may apply [[transgression of variational differential forms]] to turn the [[local BV-BRST complex]] on smooth functions on the [[jet bundle]] into a global [[BV-BRST complex]] on graded [[local observables]] on the graded [[space of field histories]].
+-- {: .num_defn #ComplexBVBRSTGlobal}
###### Definition
**(global [[BV-BRST complex]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
equipped with a [[gauge parameter bundle]] $\mathcal{G}$ (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Then on the [[local observables]] (def. \ref{LocalObservables}) on the [[space of field histories]] (def. \ref{FieldsAndFieldBundles}) of the [[graded manifold|graded]] [[field bundle]]
$$
E_{\text{BV-BRST}}
=
T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1])
$$
underlying the [[local BV-BRST complex]] (eq:BVBRSTGradedFieldBundle), consider the [[linear map]]
$$
\label{LocalAntibracketTransgressed}
\array{
LocObs(E_{\text{BV-BRST}})
\otimes
LocObs(E_{\text{BV-BRST}})
&\overset{\{-,-\}}{\longrightarrow}&
LocObs(E_{\text{BV-BRST}})
\\
\tau_\Sigma(\alpha), \tau_\Sigma(\beta)
&\mapsto&
\tau_\Sigma( \{\alpha, \beta\} )
}
$$
where $\alpha, \beta \in \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})$ (def. \ref{SpacetimeSupport}), where $\tau_\Sigma$ denotes
[[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}), and where on the right $\{-,-\}$ is the [[local antibracket]] (def. \ref{LocalAntibracket}).
This is well-defined, in that this formula indeed depends on the [[horizontal differential forms]] $\alpha$ and $\beta$ only through the [[local observables]] $\tau_\Sigma(\alpha), \tau_\Sigma(\beta)$ which they induce. The resulting bracket is called the (global) _[[antibracket]]_.
Indeed the formula makes sense already if at least one of $\alpha, \beta$ have compact spacetime support (def. \ref{SpacetimeSupport}), and hence
the [[transgression]] of the [[BV-BRST differential]] (eq:LocalAntibracketVersionOfBVBRSTDifferential) is a well-defined
[[differential]] on the graded [[local observables]]
$$
\left\{
-\tau_\Sigma \mathbf{L}
+ \tau_\Sigma \mathbf{L}_{BRST}
\;,\, -
\right\}
\;\colon\;
LocObs(E_{\text{BV-BRST}})
\longrightarrow
LocObs(E_{\text{BV-BRST}})
\,,
$$
where by example \ref{ActionFunctional} we may think of the first argument on the left as the BV-BRST [[action functional]]
without [[adiabatic switching]], which makes sense inside the [[antibracket]] when acting on functionals with compact spacetime support.
Hence we may suggestively write
$$
\label{GlobalBVBRSTDifferential}
\left\{
-S + S_{BRST}
\;,\;-
\right\}
\;\coloneqq\;
\left\{
-\tau_\Sigma \mathbf{L}
+ \tau_\Sigma \mathbf{L}_{BRST}
\;,\, -
\right\}
$$
for this (global) _[[BV-BRST differential]]_.
This uniquely extends as a graded [[derivation]] to [[multilocal observables]] (def. \ref{LocalObservables}) and from there along the [[dense subspace]] inclusion (eq:InclusionOfPolynomialLocalObservablesIntoPolynomialObservables)
$$
PolyMultiLocObs(E_{\text{BV-BRST}})
\overset{\text{dense}}{\hookrightarrow}
PolyObs(E_{\text{BV-BRST}})
$$
to a differential on [[off-shell]] [[polynomial observables]] (def. \ref{PolynomialObservables}):
$$
\{-S' + S'_{BRST}\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})
\longrightarrow
PolyObs(E_{\text{BV-BRST}})
$$
This [[differential graded-commutative superalgebra]]
$$
\label{GlobalBVComplexdgAlgebra}
\left(
\left(
\underset{
\text{vector space}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}})
}}
,
\underset{
\text{product}
}{
\underbrace{
(-)\cdot(-)
}}
\right)
,
\underset{
\text{differential}
}{
\underbrace{
\{-S' + S'_{BRST}, -\}
}}
\right)
$$
is the _global [[BV-BRST complex]]_ of the given [[Lagrangian field theory]] with the chosen [[gauge parameters]].
=--
+-- {: .proof}
###### Proof
We need to check that the global [[antibracket]] (eq:LocalAntibracketTransgressed) is well defined:
By the last item of prop. \ref{BasicPropertiesOfTheLocalAntibracket} the horizontally exact horizontal differential forms
form a "[[Lie ideal]]" for the [[local antibracket]]. With this the proof that the transgressed bracket is well defined
is the same as the proof that the global [[Poisson bracket]] on the [[Hamiltonian differential form|Hamiltonian]]
[[local observables]] is well defined, def. \ref{PoissonBracketOnHamiltonianLocalObservables}.
=--
+-- {: .num_example #BVDifferentialGlobal}
###### Example
**(global BV-differential in components)**
In the situation of def. \ref{ComplexBVBRSTGlobal},
assume that the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates on their [[fiber product]] bundle collectively denoted $(\phi^A)$.
Then the first summand of the global BV-BRST differential (def. \ref{ComplexBVBRSTGlobal}) is given by
$$
\label{ComponentsOfGlobalBVDifferential}
\begin{aligned}
\left\{ -S', -\right\}
& =
\int_\Sigma
j^{\infty}\left(\mathbf{\Phi}\right)^\ast
\left(
\frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A}
\right)(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\\
& =
\underset{A}{\sum}
(-1)^{deg(\phi^A)}
\int_\Sigma
(P_{A B}\mathbf{\Phi}^A)(x)
\frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
\,
dvol_\Sigma(x)
\end{aligned}
$$
where
1. $P \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(E^\ast)$
is the [[differential operator]] (eq:DifferentialOperatorEulerLagrangeDerivative) from def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}, corresponding to the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]].
1. $deg(\phi^A) \coloneqq n_{(\Phi^A)} + \sigma_{\Phi^A} \;\in\; \mathbb{Z}/2$
is the sum of the cohomological degree and of the super-degree of $\Phi^A$ (as in def. \ref{differentialgradedcommutativeSuperalgebra}, def. \ref{A+first+idea+of+quantum+field+theoryDifferentialFormOnSuperCartesianSpaces}).
It follows that the [[cochain cohomology]] of the global [[BV-differential]] $\{-S',-\}$ (eq:GlobalBVComplexdgAlgebra) in $deg_{af} = 0$ is the space of [[on-shell]] [[polynomial observables]]:
$$
\label{OnShellPolynomialObservablesAsBVCohomology}
\underset{
\text{off-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
}}/im(\{-S',-\})
\;\simeq\;
\underset{
\text{on-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
}}
\,.
$$
=--
+-- {: .proof}
###### Proof
By definition, the part $\mathbf{L}'$ of the gauge fields Lagrangian density is independent of [[antifields]], so that the [[local antibracket]] with $\mathbf{L}'$ reduces to
$$
\left\{ -\mathbf{L}',-\right\}
\;=\;
\frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}'}{\delta \phi^A}
\frac{\delta}{\delta \phi^{\ddagger}_A}
$$
With this the expression for $\{-S',-\}$ follows directly from the definition of the global antibracket (def. \ref{ComplexBVBRSTGlobal}) and the [[Euler-Lagrange equations]] (eq:DifferentialOperatorEulerLagrangeDerivative)
$$
(P \Phi)_A = j^\infty_\Sigma(\Phi)\left( \frac{\delta_{EL} L}{\delta \phi^A} \right)
\,.
$$
where the sign $(-1)^{deg(\phi^A)}$ is the relative sign between $\frac{\delta_{EL} L}{\delta \phi^A} = \frac{\overset{\rightarrow}{\delta}_{EL} L'}{\delta \phi^A}$ and $\frac{\overset{\leftarrow}{\delta}_{EL} L'}{\delta \phi^A}$ (def. \ref{SchoutenBracketAndAntibracket}):
By the assumption that $L'$ defines a [[free field theory]], $\mathbf{L}'$ is quadratic in the fields, so that from $deg(\mathbf{L}) = 0$ it follows that the derivations from the left and from the right differ by the relative sign
$$
\begin{aligned}
(-1)^{
\left(
n_{(\phi^A)} n_{(\phi^A)}
+
\sigma_{(\phi^A)} \sigma_{(\phi^A)}
\right)
}
& =
(-1)^{
\left(
n_{(\phi^A)} + \sigma_{(\phi^A)}
\right)
}
\\
& =
(-1)^{deg(\phi^A)}
\end{aligned}
\,.
$$
From this the identification (eq:OnShellPolynomialObservablesAsBVCohomology) follows by (eq:PolynomialOnShellObservablesArePolynomialOffShellobservableModuloTheEquationsOfMotion) in theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}.
=--
$\,$
This concludes our discussion of the [[reduced phase space]] of a [[Lagrangian field theory]] exhibited, [[formal dual|dually]] by its
[[local BV-BRST complex]]. In the [next chapter](#GaugeFixing) we finally turn to the key implication of this construction:
the [[gauge fixing]] of a [[Lagrangian field theory|Lagrangian]] [[gauge theory]] which makes the collection of [[field (physics)|fields]]
and [[auxiliary fields]] ([[ghost fields]] and [[antifields]]) jointly have a (differential-graded) [[covariant phase space]].
|
A first idea of quantum field theory -- Renormalization | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Renormalization |
## Renormalization
{#Renormalization}
In this chapter we discuss the following topics:
* _[Epstein-Glaser normalization](#EpsteinGlaserRenormalization)_
* _[Stückelberg-Petermann re-normalization](#SPRenormalizationGroup)_
* _[UV-Regularization via Counterterms](#UVRegularizationViaZ)_
* _[Wilson-Polchinski effective QFT flow](#EffectiveQFTFlowWislonian)_
* _[Renormalization group flow](#RGFlowGeneral)_
* _[Gell-Mann & Low RG Flow](#ScalingTransformatinRGFlow)_
$\,$
In the [previous chapter](#InteractingQuantumFields) we have seen that the construction of [[interacting quantum field theory|interacting]] [[perturbative quantum field theories]] is given by perturbative [[S-matrix schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), equivalently by [[time-ordered products]] (def. \ref{TimeOrderedProduct}) or equivalently by [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}). These are uniquely fixed away from coinciding interaction points (prop. \ref{TimeOrderedProductAwayFromDiagonal}) by the given [[local observable|local]] [[interaction]] (prop. \ref{TimeOrderedProductAwayFromDiagonal}), but involve further choices of interactions whenever interaction vertices coincide (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}). This choice is called the choice of [[renormalization|("re"-)normalization]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) in [[perturbative QFT]].
In this rigorous discussion no "infinite divergent quantities" (as in the original informal discussion due to [[Schwinger-Tomonaga-Feynman-Dyson]]) that need to be "re-normalized" to finite well-defined quantities are ever considered,
instead finite well-defined quantities are considered right away, and the available space of choices is determined.
Therefore making such choices is rather a _normalization_ of the [[time-ordered products]]/[[Feynman amplitudes]]
(as prominently highlighted in [Scharf 95, see title, introduction, and section 4.3](causal+perturbation+theoryscatt#Scharf95)).
Actual re-normalization is the the change of such normalizations.
The construction of [[perturbative QFTs]] may be explicitly described by an [[induction|inductive]] [[extension of distributions]]
of [[time-ordered products]]/[[Feynman amplitudes]] to coinciding interaction points. This is called
* _[Epstein-Glaser renormalization](#EpsteinGlaserRenormalization)_.
This inductive construction has the advantage that it gives accurate control over the space
of available choices of ("re"-)normalizations (theorem \ref{ExistenceRenormalization} below) but it
leaves the nature of the "new interactions" that are to be chosen at coinciding interaction points somwewhat implicit.
Alternatively, one may [[vertex redefinition|re-define the interactions]] explicitly (by adding "[[counterterms]]", remark \ref{TermCounter} below),
depending on a chosen [[UV cutoff]]-scale (def. \ref{CutoffsUVForPerturbativeQFT} below), and construct the [[limit of a sequence|limit]] as the "cutoff is removed" (prop. \ref{UVRegularization} below). This is called ("re"-)normalization by
* _[UV-Regularization via Counterterms](#UVRegularizationViaZ)_.
This still leaves open the question how to choose the [[counterterms]]. For that it serves to understand
the _[[relative effective action]]_ induced by the choice of [[UV cutoff]] at any given cutoff scale (def. \ref{EffectiveActionRelative} below). This is the perspective of _[[effective quantum field theory]]_ (remark \ref{pQFTEffective} below).
The [[infinitesimal]] change of these [[relative effective actions]] follows a universal [[differential equation]], known as
_[[Polchinski's flow equation]]_ (prop. \ref{FlowEquationPolchinski} below). This makes the problem of ("re"-)normalization
be that of solving this [[differential equation]] subject to chosen initial data. This is the perspective on
("re"-)normalization called
* _[Wilson-Polchinski effective QFT flow](#EffectiveQFTFlowWislonian)_.
The [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem} below) states that different [[S-matrix schemes]] are precisely related by [[vertex redefinitions]]. This yields the
* _[Stückelberg-Petermann renormalization group](#SPRenormalizationGroup)_.
If a sub-collection of [[renormalization schemes]] is parameterized by some [[group]] $RG$,
then the [[main theorem of perturbative renormalization|main theorem]] implies [[vertex redefinitions]]
depending on pairs of elements of $RG$ (prop. \ref{FlowRenormalizationGroup} below). This is known as
* _[Renormalization group flow](#RGFlowGeneral)_
Specifically [[scaling transformations]] on [[Minkowski spacetime]] yield such a collection of [[renormalization schemes]] (prop. \ref{RGFlowScalingTransformations} below);
the corresponding [[renormalization group flow]] is known as
* _[Gell-Mann & Low RG flow](#ScalingTransformatinRGFlow)_.
The [[infinitesimal]] behaviour of this flow is known as the _[[beta function]]_, describing the _[[running of the coupling constants]]_ with scale (def. \ref{CouplingRunning} below).
$\,$
**[[Epstein-Glaser renormalization|Epstein-Glaser normalization]]**
{#EpsteinGlaserRenormalization}
The construction of [[perturbative quantum field theories]] around a given [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free field]] [[vacuum]]
is equivalently, by prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}, the construction of [[S-matrices]] $\mathcal{S}(g S_{int} + j A)$ in the sense of [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) for the given [[local observable|local]] [[interaction]]
$g S_{int} + j A$. By prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} the construction of these
[[S-matrices]] is [[induction|inductively]] in $k \in \mathbb{N}$ a choice of [[extension of distributions]] (remark \ref{TimeOrderedProductOfFixedInteraction} and def. \ref{ExtensionOfDistributions} below) of the corresponding $k$-ary [[time-ordered products]] of the [[interaction]] to the locus of coinciding interaction points. An inductive construction of the
[[S-matrix]] this way is called _[[Epstein-Glaser renormalization|Epstein-Glaser-("re"-)normalization]]_ (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).
By paying attention to the [[scaling degree of distributions|scaling degree]] (def. \ref{ScalingDegree} below)
one may precisely characterize the space of choices in the [[extension of distributions]] (prop. \ref{SpaceOfPointExtensions} below):
For a given [[local observable|local]] [[interaction]] $g S_{int} + j A$ it is inductively in $k \in \mathbb{N}$ a
[[finite dimensional vector space|finite-dimensional]] [[affine space]]. This conclusion is theorem \ref{ExistenceRenormalization} below.
$\,$
+-- {: .num_prop #RenormalizationIsInductivelyExtensionToDiagonal}
###### Proposition
**([[renormalization|("re"-)normalization]] is [[induction|inductive]] [[extension of distributions|extension]] of [[time-ordered products]] to [[diagonal]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}).
Assume that for $n \in \mathbb{N}$, [[time-ordered products]]
$\{T_{k}\}_{k \leq n}$ of arity $k \leq n$ have been constructed
in the sense of def. \ref{TimeOrderedProduct}.
Then the time-ordered product $T_{n+1}$ of arity $n+1$ is uniquely fixed on the [[complement]]
$$
\Sigma^{n+1} \setminus diag(n)
\;=\;
\left\{
(x_i \in \Sigma)_{i = 1}^n
\;\vert\;
\underset{i,j}{\exists} (x_i \neq x_j)
\right\}
$$
of the [[image]] of the [[diagonal]] inclusion $\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}$
(where we regarded $T_{n+1}$ as a [[generalized function]] on $\Sigma^{n+1}$ according to remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}).
=--
This statement appears in ([Popineau-Stora 82](renormalization#PopineauStora82)), with (unpublished) details in ([Stora 93](renormalization#Stora93)), following personal communication by [[Henri Epstein]] (according to [Dütsch 18, footnote 57](renormalization#Duetsch18)). Following this, statement and detailed proof appeared in ([Brunetti-Fredenhagen 99](renormalization#BrunettiFredenhagen99)).
+-- {: .proof}
###### Proof
We will construct an [[open cover]] of $\Sigma^{n+1} \setminus \Sigma$ by subsets $\mathcal{C}_I \subset \Sigma^{n+1}$ which are [[disjoint unions]]
of [[inhabited set|non-empty]] sets that are in [[causal order]], so that by [[causal factorization]] the
time-ordered products $T_{n+1}$ on these subsets are uniquely given by $T_{k}(-) \star_H T_{n-k}(-)$.
Then we show that these unique products on these special subsets do coincide on [[intersections]].
This yields the claim by a [[partition of unity]].
We now say this in detail:
For $I \subset \{1, \cdots, n+1\}$ write $\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I$.
For $I, \overline{I} \neq \emptyset$, define the subset
$$
\mathcal{C}_I
\;\coloneqq\;
\left\{
(x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1}
\;\vert\;
\{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I}
\right\}
\;\subset\;
\Sigma^{n+1}
\,.
$$
Since the [[causal order]]-relation involves the [[closed future cones]]/[[closed past cones]], respectively, it is clear that these are [[open subsets]]. Moreover it is immediate that they form an [[open cover]] of the [[complement]] of the [[diagonal]]:
$$
\underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } } }{\cup}
\mathcal{C}_I
\;=\;
\Sigma^{n+1} \setminus diag(\Sigma)
\,.
$$
(Because any two distinct points in the [[globally hyperbolic spacetime]] $\Sigma$ may be causally separated by a [[Cauchy surface]],
and any such may be deformed a little such as not to intersect any of a given finite set of points. )
Hence the condition of [[causal factorization]] on $T_{n+1}$ implies that
[[restriction of distributions|restricted]] to any $\mathcal{C}_{I}$ these have to be given (in the condensed [[generalized function]]-notation from remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}) on
any unordered tuple $\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I$ with corresponding induced tuples
$\mathbf{I} \coloneqq \{x_i\}_{i \in I}$ and $\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}$ by
$$
\label{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}
T_{n+1}( \mathbf{X} )
\;=\;
T(\mathbf{I}) T(\overline{\mathbf{I}})
\phantom{AA}
\text{for}
\phantom{A}
\mathcal{X} \in \mathcal{C}_I
\,.
$$
This shows that $T_{n+1}$ is unique on $\Sigma^{n+1} \setminus diag(\Sigma)$ if it exists at all,
hence if these local identifications glue to a global definition of $T_{n+1}$. To see that this is the case,
we have to consider any two such subsets
$$
I_1, I_2
\subset
\{1, \cdots, n+1\}
\,,
\phantom{AA}
I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset
\,.
$$
By definition this implies that for
$$
\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}
$$
a tuple of spacetime points which decomposes into causal order with respect to both these subsets,
the corresponding mixed intersections of tuples are spacelike separated:
$$
\mathbf{I}_1 \cap \overline{\mathbf{I}_2}
\;
{\gt\!\!\!\!\lt}
\;
\overline{\mathbf{I}_1} \cap \mathbf{I}_2
\,.
$$
By the assumption that the $\{T_k\}_{k \neq n}$ satisfy causal factorization, this implies that the corresponding
time-ordered products commute:
$$
\label{TimeOrderedProductsOfMixedIntersectionsCommute}
T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
\,
T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
\;=\;
T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
\,
T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
\,.
$$
Using this we find that the identifications of $T_{n+1}$
on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal),
agree on the intersection: in that for $ \mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have
$$
\begin{aligned}
T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} )
& =
T( \mathbf{I}_1 \cap \mathbf{I}_2 )
T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
\,
T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
\\
& =
T( \mathbf{I}_1 \cap \mathbf{I}_2 )
\underbrace{
T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
}
T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
\\
& =
T( \mathbf{I}_2 )
T( \overline{\mathbf{I}_2} )
\end{aligned}
$$
Here in the first step we expanded out the two factors using (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal) for $I_2$,
then under the brace we used (eq:TimeOrderedProductsOfMixedIntersectionsCommute) and in the last step we used again
(eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal), but now for $I_1$.
To conclude, let
$$
\label{PartitionCausalOfUnityForComplementOfDiagonal}
\left(
\chi_I
\in
C^\infty_{cp}(\Sigma^{n+1}),
\,
supp(\chi_I) \subset \mathcal{C}_i
\right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }
$$
be a [[partition of unity]] subordinate to the [[open cover]] formed by the $\mathcal{C}_I$:
Then the above implies that
setting for any $\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)$
$$
\label{TimeOrderedProductsAwayFromDiagonalByInduction}
T_{n+1}(\mathbf{X})
\;\coloneqq\;
\underset{
{ I \in \{1, \cdots, n+1\} }
\atop
{ I, \overline{I} \neq \emptyset }
}{\sum}
\chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} )
$$
is well defined and satisfies causal factorization.
=--
+-- {: .num_remark #TimeOrderedProductOfFixedInteraction}
###### Remark
**([[time-ordered products]] of fixed [[interaction]] as [[distributions]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and assume that the [[field bundle]] is a
[[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle})
and let
$$
g S_{int} + j A
\;\in\;
LoObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
$$
be a polynomial [[local observable]] as in def. \ref{FormalParameters}, to be regarded as a [[adiabatic switching|adiabatically switched]]
[[interaction]] [[action functional]]. This means that there is a [[finite set]]
$$
\left\{
\mathbf{L}_{int,i}, \mathbf{\alpha}_{i'} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
\right\}_{i,i'}
$$
of [[Lagrangian densities]] which are monomials in the field and jet coordinates,
and a corresponding finite set
$$
\left\{
g_{sw,i} \in C^\infty_{cp}(\Sigma)\langle g \rangle
\,,\,
j_{sw,i'} \in C^\infty_{cp}(\Sigma)\langle j \rangle
\right\}
$$
of [[adiabatic switchings]], such that
$$
g S_{int} + j A
\;=\;
\tau_{\Sigma}
\left(
\underset{i}{\sum}
g_{sw,i} \mathbf{L}_{int,i}
\;+\;
\underset{i'}{\sum}
j_{sw,i'} \mathbf{\alpha}_{i'}
\right)
$$
is the [[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
of the sum of the products of these [[adiabatic switching]] with these [[Lagrangian densities]].
In order to discuss the [[S-matrix]] $\mathcal{S}(g S_{int} + j A)$ and hence the
[[time-ordered products]] of the special form $T_k\left( \underset{k \, \text{factors}}{\underbrace{g S_{int} + j A, \cdots, g S_{int} + j A }} \right)$ it is sufficient to restrict attention to the [[restriction]] of each $T_k$ to the subspace of
[[local observables]] induced by the finite set of [[Lagrangian densities]] $\{\mathbf{L}_{int,i}, \mathbf{\alpha}_{i'}\}_{i,i'}$.
This restriction is a [[continuous linear functional]] on the corresponding space of [[bump functions]] $\{g_{sw,i}, j_{sw,i'}\}$,
hence a [[distribution|dstributional]] [[section]] of a corresponding [[trivial vector bundle]].
In terms of this, prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} says that the
choice of [[time-ordered products]] $T_k$ is [[induction|inductively]] in $k$ a choice of [[extension of distributions]] to the [[diagonal]].
If $\Sigma = \mathbb{R}^{p,1}$ is [[Minkowski spacetime]] and we impose the [[renormalization condition]] "translation invariance" (def. \ref{RenormalizationConditions}) then each $T_k$ is a distribution on $\Sigma^{k-1} = \mathbb{R}^{(p+1)(k-1)}$ and the [[extension of distributions]] is from the complement of the origina $0 \in \mathbb{R}^{(p+1)(k-1)}$.
=--
Therefore we now discuss [[extension of distributions]] (def. \ref{ExtensionOfDistributions} below) on [[Cartesian spaces]] from the complement of the origin to the origin. Since the space of choices of such extensions turns out to depend on the _[[scaling degree of distributions]]_,
we first discuss that (def. \ref{ScalingDegree} below).
+-- {: .num_defn #RescaledDistribution}
###### Definition
**([[scaling degree of distributions|rescaled distribution]])**
Let $n \in \mathbb{N}$. For $\lambda \in (0,\infty) \subset \mathbb{R}$ a [[positive number|positive]] [[real number]] write
$$
\array{
\mathbb{R}^n
&\overset{s_\lambda}{\longrightarrow}&
\mathbb{R}^n
\\
x &\mapsto& \lambda x
}
$$
for the [[diffeomorphism]] given by multiplication with $\lambda$, using the canonical [[real vector space]]-structure of $\mathbb{R}^n$.
Then for $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]] on the [[Cartesian space]] $\mathbb{R}^n$ the _rescaled distribution_ is the [[pullback of a distribution|pullback]] of $u$ along $m_\lambda$
$$
u_\lambda
\coloneqq
s_\lambda^\ast u
\;\in\;
\mathcal{D}'(\mathbb{R}^n)
\,.
$$
Explicitly, this is given by
$$
\array{
\mathcal{D}(\mathbb{R}^n)
&\overset{ \langle u_\lambda, - \rangle}{\longrightarrow}&
\mathbb{R}
\\
b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle
}
\,.
$$
Similarly for $X \subset \mathbb{R}^n$ an [[open subset]] which is invariant under $s_\lambda$, the rescaling of a distribution $u \in \mathcal{D}'(X)$ is is $u_\lambda \coloneqq s_\lambda^\ast u$.
=--
+-- {: .num_defn #ScalingDegree}
###### Definition
**([[scaling degree of a distribution]])**
Let $n \in \mathbb{N}$ and let $X \subset \mathbb{R}^n$ be an [[open subset]] of [[Cartesian space]] which is invariant under [[rescaling]] $s_\lambda$ (def. \ref{RescaledDistribution}) for all $\lambda \in (0,\infty)$, and let $u \in \mathcal{D}'(X)$ be a [[distribution]] on this subset. Then
1. The _[[scaling degree of a distribution|scaling degree]]_ of $u$ is the [[infimum]]
$$
sd(u)
\;\coloneqq\;
inf
\left\{
\omega \in \mathbb{R}
\;\vert\;
\underset{\lambda \to 0}{\lim}
\lambda^\omega u_\lambda
= 0
\right\}
$$
of the set of [[real numbers]] $\omega$ such that the [[limit of a sequence|limit]] of the rescaled distribution $\lambda^\omega u_\lambda$ (def. \ref{RescaledDistribution}) vanishes. If there is no such $\omega$ one sets $sd(u) \coloneqq \infty$.
1. The _[[degree of divergence of a distribution|degree of divergence]]_ of $u$ is the difference of the scaling degree by the [[dimension]] of the underlying space:
$$
deg(u) \coloneqq sd(u) - n
\,.
$$
=--
+-- {: .num_example #NonSingularDistributionsScalingDegree}
###### Example
**([[scaling degree of distributions|scaling degree]] of [[non-singular distributions]])**
If $u = u_f$ is a [[non-singular distribution]] given by [[bump function]] $f \in C^\infty(X) \subset \mathcal{D}'(X)$, then its [[scaling degree of a distribution|scaling degree]] (def. \ref{ScalingDegree}) is non-[[positive number|positive]]
$$
sd(u_f) \leq 0
\,.
$$
Specifically if the first non-vanishing [[partial derivative]] $\partial_\alpha f(0)$ of $f$ at 0 occurs at order ${\vert \alpha\vert} \in \mathbb{N}$, then the scaling degree of $u_f$ is $-{\vert \alpha\vert}$.
=--
+-- {: .proof}
###### Proof
By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that
$$
\begin{aligned}
\left\langle
\lambda^{\omega}
(u_f)_\lambda,
n
\right\rangle
& =
\lambda^{\omega-n}
\underset{\mathbb{R}^n}{\int}
f(x) g(\lambda^{-1} x)
d^n x
\\
& =
\lambda^{\omega}
\underset{\mathbb{R}^n}{\int}
f(\lambda x) g(x)
d^n x
\end{aligned}
\,,
$$
where in last line we applied [[change of integration variables]].
The limit of this expression is clearly zero for all $\omega \gt 0$, which shows the first claim.
If moreover the first non-vanishing [[partial derivative]] of $f$ occurs at order ${\vert \alpha \vert} = k$, then [[Hadamard's lemma]] says that $f$ is of the form
$$
f(x)
\;=\;
\left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
(\partial_\alpha f(0))
\underset{i}{\prod} (x^i)^{\alpha_i}
+
\underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 } }{\sum}
\underset{i}{\prod} (x^i)^{\beta_i}
h_{\beta}(x)
$$
where the $h_{\beta}$ are [[smooth functions]]. Hence in this case
$$
\begin{aligned}
\left\langle
\lambda^{\omega}
(u_f)_\lambda,
n
\right\rangle
& =
\lambda^{\omega + {\vert \alpha\vert }}
\underset{\mathbb{R}^n}{\int}
\left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
(\partial_\alpha f(0))
\underset{i}{\prod} (x^i)^{\alpha_i}
b(x)
d^n x
\\
& \phantom{=}
+
\lambda^{\omega + {\vert \alpha\vert} + 1}
\underset{\mathbb{R}^n}{\int}
\underset{i}{\prod} (x^i)^{\beta_i}
h_{\beta}(x)
b(x)
d^n x
\end{aligned}
\,.
$$
This makes manifest that the expression goes to zero with $\lambda \to 0$ precisely for $\omega \gt - {\vert \alpha \vert}$, which means that
$$
sd(u_f) = -{\vert \alpha \vert}
$$
in this case.
=--
+-- {: .num_example #DerivativesOfDeltaDistributionScalingDegree}
###### Example
**([[scaling degree of a distribution|scaling degree]] of [[derivative of a distribution|derivatives]] of [[delta-distributions]])**
Let $\alpha \in \mathbb{N}^n$ be a multi-index and $\partial_\alpha \delta \in \mathcal{D}'(X)$ the corresponding [[partial derivative|partial]] [[derivative of distributions|derivatives]] of the [[delta distribution]] $\delta_0 \in \mathcal{D}'(\mathbb{R}^n)$ [[support of a distribution|supported]] at $0$. Then the [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree}) of $\partial_\alpha \delta_0$ is the total order the derivatives
$$
deg\left(
{\, \atop \,} \partial_\alpha\delta_0{\, \atop \,}
\right)
\;=\;
{\vert \alpha \vert}
$$
where ${\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i$.
=--
+-- {: .proof}
###### Proof
By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that
$$
\begin{aligned}
\left\langle
\lambda^\omega (\partial_\alpha \delta_0)_\lambda,
b
\right\rangle
& =
(-1)^{{\vert \alpha \vert}}
\lambda^{\omega-n}
\left(
\frac{
\partial^{{\vert \alpha \vert}}
}{
\partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n
}
b(\lambda^{-1}x)
\right)_{\vert x = 0}
\\
& =
(-1)^{{\vert \alpha \vert}}
\lambda^{\omega - n - {\vert \alpha\vert}}
\frac{
\partial^{{\vert \alpha \vert}}
}{
\partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n
}
b(0)
\end{aligned}
\,,
$$
where in the last step we used the [[chain rule]] of [[differentiation]]. It is clear that this goes to zero with $\lambda$ as long as $\omega \gt n + {\vert \alpha\vert}$. Hence $sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}$.
=--
+-- {: .num_example #FeynmanPropagatorOnMinkowskiScalingDegree}
###### Example
**([[scaling degree of a distribution|scaling degree]] of [[Feynman propagator]] on [[Minkowski spacetime]])**
Let
$$
\Delta_F(x)
\;=\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu x^\mu}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
}
\, d k_0 \, d^p \vec k
$$
be the [[Feynman propagator]] for the massive [[free field|free]] [[real scalar field]] on $n = p+1$-dimensional [[Minkowski spacetime]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}). Its [[scaling degree of a distribution|scaling degree]] is
$$
\begin{aligned}
sd(\Delta_{F})
& = n - 2
\\
& = p -1
\end{aligned}
\,.
$$
=--
([Brunetti-Fredenhagen 00, example 3 on p. 22](renormalization#BrunettiFredenhagen00))
+-- {: .proof}
###### Proof
Regarding $\Delta_F$ as a [[generalized function]] via the given [[Fourier transform of distributions|Fourier-transform]] expression, we find by [[change of integration variables]] in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing [[mass]], which scales homogeneously:
$$
\begin{aligned}
\underset{\lambda \to 0}{\lim}
\left(
\lambda^\omega
\;
\Delta_F(\lambda x)
\right)
& =
\underset{\lambda \to 0}{\lim}
\left(
\lamba^{\omega}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu \lambda x^\mu}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
}
\, d k_0 \, d^p \vec k
\right)
\\
& =
\underset{\lambda \to 0}{\lim}
\left(
\lambda^{\omega-n}
\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu \lambda x^\mu}
}{
- (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
}
\, d k_0 \, d^p \vec k
\right)
\\
& =
\underset{\lambda \to 0}{\lim}
\left(
\lambda^{\omega-n + 2 }
\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu \lambda x^\mu}
}{
- k_\mu k^\mu + i \epsilon
}
\, d k_0 \, d^p \vec k
\right)
\,.
\end{aligned}
$$
=--
+-- {: .num_prop #ScalingDegreeOfDistributionsBasicProperties}
###### Proposition
**(basic properties of [[scaling degree of distributions]])**
Let $X \subset \mathbb{R}^n$ and $u \in \mathcal{D}'(X)$ be a [[distribution]] as in def. \ref{RescaledDistribution}, such that its [[scaling degree of a distribution|scaling degree]] is finite: $sd(u) \lt \infty$ (def. \ref{ScalingDegree}). Then
1. For $\alpha \in \mathbb{N}^n$, the [[partial derivative|partial]] [[derivative of distributions]] $\partial_\alpha$ increases scaling degree at most by ${\vert \alpha\vert }$:
$$
deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}
$$
1. For $\alpha \in \mathbb{N}^n$, the [[product of distributions]] with the smooth coordinate functions $x^\alpha$ decreases scaling degree at least by ${\vert \alpha\vert }$:
$$
deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}
$$
1. Under [[tensor product of distributions]] their scaling degrees add:
$$
sd(u \otimes v) \leq sd(u) + sd(v)
$$
for $v \in \mathcal{D}'(Y)$ another distribution on $Y \subset \mathbb{R}^{n'}$;
1. $deg(f u) \leq deg(u) - k$ for $f \in C^\infty(X)$ and $f^{(\alpha)}(0) = 0$ for ${\vert \alpha\vert} \leq k-1$;
=--
([Brunetti-Fredenhagen 00, lemma 5.1](renormalization#BrunettiFredenhagen00), [Dütsch 18, exercise 3.34](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
The first three statements follow with manipulations as in example \ref{NonSingularDistributionsScalingDegree} and example \ref{DerivativesOfDeltaDistributionScalingDegree}.
For the fourth...
=--
+-- {: .num_prop #ScalingDegreeOfProductDistribution}
###### Proposition
**([[scaling degree of distributions|scaling degree]] of [[product of distributions|product distribution]])**
Let $u,v \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]] such that
1. both have finite [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree})
$$
deg(u), deg(v) \lt \infty
$$
1. their [[product of distributions]] is well-defined
$$
u v \in \mathcal{D}'(\mathbb{R}^n)
$$
(in that their [[wave front sets]] satisfy [[Hörmander's criterion]])
then the product distribution has [[degree of divergence of a distribution|degree of divergence]] bounded by the sum of the separate degrees:
$$
deg(u v)
\;\leq\;
deg(u) + deg(v)
\,.
$$
=--
With the concept of [[scaling degree of distributions]] in hand, we may now discuss [[extension of distributions]]:
+-- {: .num_defn #ExtensionOfDistributions}
###### Definition
**([[extension of distributions]])**
Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of [[open subsets]] of some [[Cartesian space]]. This induces the operation of [[restriction of distributions]]
$$
\mathcal{D}'(\hat X)
\overset{\iota^\ast}{\longrightarrow}
\mathcal{D}'(X)
\,.
$$
Given a [[distribution]] $u \in \mathcal{D}'(X)$, then an _[[extension]]_ of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that
$$
\iota^\ast \hat u
\;=\;
u
\,.
$$
=--
+-- {: .num_prop #ExtensionUniqueNonPositiveDegreeOfDivergence}
###### Proposition
**(unique [[extension of distributions]] with negative [[degree of divergence of a distribution|degree of divergence]])**
For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] on the [[complement]] of the origin, with [[negative number|negative]] [[degree of divergence of a distribution|degree of divergence]] at the origin
$$
deg(u) \lt 0
\,.
$$
Then $u$ has a _unique_ [[extension of distributions]] $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence
$$
deg(\hat u) = deg(u)
\,.
$$
=--
([Brunetti-Fredenhagen 00, theorem 5.2](renormalization#BrunettiFredenhagen00), [Dütsch 18, theorem 3.35 a)](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
Regarding uniqueness:
Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has [[support of a distribution|support]] at the origin $\{0\} \subset \mathbb{R}^n$. By prop. \ref{PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions} this implies that it is a linear combination of [[derivative of a distribution|derivatives]] of the [[delta distribution]] [[support of a distribution|supported]] at the origin:
$$
{\hat u}^\prime - \hat u
\;=\;
\underset{
{\alpha \in \mathbb{N}^n}
}{\sum}
c^\alpha \partial_\alpha \delta_0
$$
for constants $c^\alpha \in \mathbb{C}$. But by example \ref{DerivativesOfDeltaDistributionScalingDegree} the [[degree of divergence of a distribution|degree of divergence]] of these [[point-supported distributions]] is non-negative
$$
deg( \partial_\alpha \delta_0)
=
{\vert \alpha\vert} \geq 0
\,.
$$
This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.
Regarding existence:
Let
$$
b \in C^\infty_{cp}(\mathbb{R}^n)
$$
be a [[bump function]] which is $\leq 1$ and [[constant function|constant]] on 1 over a [[neighbourhood]] of the origin. Write
$$
\chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)
$$
<center>
<img src="https://ncatlab.org/nlab/files/PointExtensionOfDistributions.png" >
</center>
> graphics grabbed from [Dütsch 18, p. 108](renormalization#Duetsch18)
and for $\lambda \in (0,\infty)$ a [[positive real number]], write
$$
\chi_\lambda(x)
\coloneqq
\chi(\lambda x)
\,.
$$
Since the [[product of distributions|product]] $\chi_\lambda u$ has [[support of a distribution]] on a [[complement]] of a [[neighbourhood]] of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:
$$
\chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n)
\,.
$$
By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases. This means that if the following [[limit of a sequence|limit]] exists
$$
\hat u
\;\coloneqq\;
\underset{\lambda \to \infty}{\lim}
\chi_\lambda u
$$
then it is an extension of $u$.
To see that the limit exists, it is sufficient to observe that we have a [[Cauchy sequence]], hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference
$$
(\chi_{n+1} u - \chi_n u)(b)
\;=\;
u(b)( \chi_{n+1} + \chi_n )
$$
becomes arbitrarily small.
It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in ([Brunetti-Fredenhagen 00, p. 24](extension+of+distributions#BrunettiFredenhagen00)).
=--
+-- {: .num_prop #SpaceOfPointExtensions}
###### Proposition
**(space of [[point-extensions of distributions]])
For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] of [[scaling degree of a distribution|degree of divergence]] $deg(u) \lt \infty$.
Then $u$ does admit at least one [[extension of distributions|extension]] (def. \ref{ExtensionOfDistributions}) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same [[degree of divergence of a distribution|degree of divergence]] as $u$
$$
deg(\hat u) = deg(u)
\,.
$$
Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of [[partial derivatives|partial]] [[derivatives of distributions]] of order $\leq deg(u)$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:
$$
{\hat u}^\prime
-
\hat u
\;=\;
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha\vert} \leq deg(u) }
}{\sum}
q^\alpha
\partial_\alpha \delta_0
\,,
$$
for a finite number of constants $q^\alpha \in \mathbb{C}$.
=--
This is essentially ([Hörmander 90, thm. 3.2.4](renormalization#Hoermander90)). We follow ([Brunetti-Fredenhagen 00, theorem 5.3](renormalization#BrunettiFredenhagen00)), which was inspired by ([Epstein-Glaser 73, section 5](#EpsteinGlaser73)). Review of this approach is in ([Dütsch 18, theorem 3.35 (b)](renormalization#Duetsch18)), see also remark \ref{WExtensions} below.
+-- {: .proof}
###### Proof
For $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], and $\rho \in \mathbb{N}$, we say that _$f$ vanishes to order $\rho$_ at the origin if all [[partial derivatives]] with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:
$$
\partial_\alpha f (0) = 0
\phantom{AAA}
{\vert \alpha\vert} \leq \rho
\,.
$$
By [[Hadamard's lemma]], such a function may be written in the form
$$
\label{ForVanishingOrderRhoHadamardExpansion}
f(x)
\;=\;
\underset{
{\alpha \in \mathbb{N}^n}
\atop
{ {\vert \alpha \vert} = \rho + 1 }
}{\sum}
x^\alpha
r_\alpha(x)
$$
for [[smooth functions]] $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.
Write
$$
\mathcal{D}_\rho(\mathbb{R}^n)
\hookrightarrow
\mathcal{D}(\mathbb{R}^n)
\coloneqq
C^\infty_{cp}(\mathbb{R}^n)
$$
for the subspace of that of all [[bump functions]] on those that vanish to order $\rho$ at the origin.
By definition this is equivalently the joint [[kernel]] of the [[partial derivative|partial]] [[derivatives of distributions]] of order ${\vert \alpha\vert}$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:
$$
b \in \mathcal{D}_\rho(\mathbb{R}^n)
\phantom{AA}
\Leftrightarrow
\phantom{AA}
\underset{
{\alpha \in \mathbb{N}^n}
\atop
{ {\vert \alpha\vert} \leq \rho }
}
{\forall}
\left\langle \partial_\alpha \delta_0, b \right\rangle
= 0
\,.
$$
Therefore every [[continuous linear map|continuous linear]] [[projection]]
$$
\label{ForExtensionOfDistributionsProjectionMaps}
p_\rho
\;\colon\;
\mathcal{D}(\mathbb{R}^n)
\longrightarrow
\mathcal{D}_\rho(\mathbb{R}^n)
$$
may be obtained from a choice of _dual basis_ to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions
$$
\left\{
w^\beta \in C^\infty_{cp}(\mathbb{R}^n)
\right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }
$$
such that
$$
\left\langle
\partial_\alpha \delta_0
\,,\,
w^\beta
\right\rangle
\;=\;
\delta_\alpha^\beta
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\partial_\alpha w^\beta(0)
\;=\;
\delta_\alpha^\beta
\phantom{AAAA}
\text{for}\, {\vert \alpha\vert} \leq \rho
\,,
$$
by setting
$$
\label{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector}
p_\rho
\;\coloneqq\;
id
\;-\;
\left\langle
\underset{
{\alpha \in \mathbb{N}^n}
\atop
{ {\vert \alpha\vert} \leq \rho }
}{\sum}
w^\alpha \partial_\alpha \delta_0
\,,\,
(-)
\right\rangle
\,,
$$
hence
$$
p_\rho
\;\colon\;
b
\mapsto
b
-
\underset{
{\alpha \in \mathbb{N}^n}
\atop
{ {\vert \alpha\vert} \leq \rho }
}{\sum}
(-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0)
\,.
$$
Together with [[Hadamard's lemma]] in the form (eq:ForVanishingOrderRhoHadamardExpansion) this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as
$$
\label{ForExtensionOfDistributionsTestFunctionDecomposition}
\begin{aligned}
b(x)
& =
p_\rho(b)(x)
\;+\;
(id - p_\rho)(b)(x)
\\
& =
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha \vert} = \rho + 1 }
}{\sum}
x^\alpha r_\alpha(x)
\;+\;
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha \vert} \leq \rho }
}{\sum}
(-1)^{{\vert \alpha \vert}}
w^\alpha \partial_\alpha b(0)
\end{aligned}
$$
Now let
$$
\rho
\;\coloneqq\;
deg(u)
\,.
$$
Observe that (by prop. \ref{ScalingDegreeOfDistributionsBasicProperties}) the [[degree of divergence of a distribution|degree of divergence]] of the [[product of distributions]] $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is [[negative number|negative]]
$$
\begin{aligned}
deg\left(
x^\alpha u
\right)
& =
\rho - {\vert \alpha \vert}
\leq
-1
\end{aligned}
$$
Therefore prop. \ref{ExtensionUniqueNonPositiveDegreeOfDivergence} says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by (eq:ForExtensionOfDistributionsTestFunctionDecomposition):
$$
\label{ExtensionOfDitstributionsPointFixedAndChoice}
\begin{aligned}
\left\langle
\hat u
\,,\,
b
\right\rangle
& =
\left\langle
\hat u
,
p_\rho(b)
\right\rangle
+
\left\langle
\hat u
,
(id - p_\rho)(b)
\right\rangle
\\
& =
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha \vert} = \rho + 1 }
}{\sum}
\underset{
\text{unique}
}{
\underbrace{
\left\langle
\widehat{x^\alpha u}
\,,\,
r_\alpha
\right\rangle
}
}
\;+\;
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha\vert} \leq \rho }
}{\sum}
\underset{
{ q^\alpha }
\atop
{ \text{choice} }
}{
\underbrace{
\langle
\hat u
\,,\,
w^\alpha
\rangle
}
}
\left\langle
\partial_\alpha \delta_0
\,,\,
b
\right\rangle
\end{aligned}
$$
That says that $\hat u$ is of the form
$$
\hat u
\;=\;
\underset{
\text{unique}
}{
\underbrace{
\widehat{
u \circ p_\rho
}
}
}
+
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha\vert} \leq \rho }
}{\sum}
c^\alpha
\,
\partial_\alpha \delta_0
$$
for a finite number of constants $c^\alpha \in \mathbb{C}$.
Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form
$$
{\hat u}^\prime - \hat u
\;=\;
\underset{
{ \alpha \in \mathbb{N}^n }
\atop
{ {\vert \alpha\vert} \leq \rho }
}{\sum}
( (c')^\alpha - c^\alpha )
\,
\partial_\alpha \delta_0
\,,
$$
where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.
It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.
By example \ref{DerivativesOfDeltaDistributionScalingDegree} the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.
Therefore to conclude it is now sufficient to show that
$$
deg\left(
\widehat{
u \circ p_\rho
}
\right)
\;=\;
\rho
\,.
$$
This is shown in ([Brunetti-Fredenhagen 00, p. 25](extension+of+distributions#BrunettiFredenhagen00)).
=--
+-- {: .num_remark #WExtensions}
###### Remark
**("W-extensions")**
Since in [Brunetti-Fredenhagen 00, (38)](renormalization#BrunettiFredenhagen00) the projectors (eq:SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector) are denoted "$W$", the construction of [[extensions of distributions]] via the proof of prop. \ref{SpaceOfPointExtensions} has come to be called "W-extensions" (e.g [Dütsch 18](renormalization#Duetsch18)).
=--
In conclusion we obtain the central theorem of [[causal perturbation theory]]:
+-- {: .num_theorem #ExistenceRenormalization}
###### Theorem
**(existence and choices of [[renormalization|("re"-)normalization]] of [[S-matrices]]/[[perturbative QFTs]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]], according to def. \ref{VacuumFree}, such that the underlying [[spacetime]] is
[[Minkowski spacetime]] and the [[Wightman propagator]] $\Delta_H$ is translation-invariant.
Then:
1. an [[S-matrix scheme]] $\mathcal{S}$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) around this vacuum exists;
1. for $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a [[local observable]]
as in def. \ref{FormalParameters}, regarded
as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]], the space of possible choices
of [[S-matrices]]
$$
\mathcal{S}(g S_{int} + j A)
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
$$
hence of the corresponding [[perturbative QFTs]], by prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}, is, [[induction|inductively]] in $k \in \mathbb{N}$, a [[finite dimensional vector space|finite dimensional]] [[affine space]],
parameterizing the [[extension of distributions|extension]] of the [[time-ordered product]] $T_k$ to the locus of
coinciding interaction points.
=--
+-- {: .proof}
###### Proof
By prop. \ref{FeynmanPropagatorOnMinkowskiScalingDegree} the [[Feynman propagator]] is finite [[scaling degree of a distribution]],
so that by prop. \ref{ScalingDegreeOfProductDistribution} the binary [[time-ordered product]] away from the diagonal
$T_2(-,-)\vert_{\Sigma^2 \setminus diag(\Sigma)} = (-) \star_{F} (-)$ has finite scaling degree.
By prop. \ref{ScalingDegreeOfProductDistribution} this implies that in the inductive
description of the time-ordered products by prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal},
each induction step is the [[extension of distributions]] of finite [[scaling degree of a distribution]] to the point.
By prop. \ref{SpaceOfPointExtensions} this always exists.
This proves the first statement.
Now if a polynomial local interaction is fixed, then via remark \ref{TimeOrderedProductOfFixedInteraction}
each induction step involved extending a finite number of distributions, each of finite scaling degree.
By prop. \ref{SpaceOfPointExtensions} the corresponding space of choices is in each step
a finite-dimensional affine space.
=--
$\,$
**[[Stückelberg-Petermann renormalization group]]**
{#SPRenormalizationGroup}
A genuine re-normalization is the passage from one [[S-matrix]] [[renormalization scheme|("re"-)normalization scheme]]
$\mathcal{S}$ to another such scheme $\mathcal{S}'$. The [[induction|inductive]] [[Epstein-Glaser renormalization|Epstein-Glaser ("re"-normalization)]] construction (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) shows that the difference between
any $\mathcal{S}$ and $\mathcal{S}'$ is inductively in $k \in \mathbb{N}$ a choice of extra term in the
[[time-ordered product]] of $k$ factors, equivalently in the [[Feynman amplitudes]] for [[Feynman diagrams]] with
$k$ [[vertices]], that contributes when all $k$ of these vertices coincide in [[spacetime]] (prop. \ref{SpaceOfPointExtensions}).
A natural question is whether these additional interactions that appear when several interaction vertices coincide
may be absorbed into a re-definition of the original interaction $g S_{int} + j A$. Such an
_[[interaction vertex redefinition]]_ (def. \ref{InteractionVertexRedefinition} below)
$$
\mathcal{Z}
\;\colon\;
g S_{int} + j A
\;\mapsto\;
g S_{int} + j A
\;+\;
\text{higher order corrections}
$$
should perturbatively send [[local observables|local]] interactions to local interactions with higher order corrections.
The _[[main theorem of perturbative renormalization]]_ (theorem \ref{PerturbativeRenormalizationMainTheorem} below)
says that indeed under mild conditions every re-normalization $\mathcal{S} \mapsto \mathcal{S}'$ is induced by such an [[interaction vertex redefinition]] in that
there exists a _unique_ such redefinition $\mathcal{Z}$ so that for every local interaction $g S_{int} + j A$ we have
that [[scattering amplitudes]] for the interaction $g S_{int} + j A$ computed with the [[renormalization scheme|("re"-)normalization scheme]]
$\mathcal{S}'$ equal those computed with $\mathcal{S}$ but applied
to the [[interaction vertex redefinition|re-defined interaction]] $\mathcal{Z}(g S_{int} + j A)$:
$$
\mathcal{S}'
\left(
{\, \atop \,}
g S_{int} + j A
{\, \atop \,}
\right)
\;=\;
\mathcal{S}\left( {\, \atop \,} \mathcal{Z}(g S_{int} + j A) {\, \atop \,} \right)
\,.
$$
This means that the [[interaction vertex redefinitions]] $\mathcal{Z}$ form a [[group]] under [[composition]]
which [[action|acts]] [[transitive action|transitively]] and [[free action|freely]], hence [[regular action|regularly]], on the
set of [[S-matrix]] [[renormalization schemes|("re"-)normalization schemes]];
this is called the _[[Stückelberg-Petermann renormalization group]]_ (theorem \ref{PerturbativeRenormalizationMainTheorem} below).
$\,$
+-- {: .num_defn #InteractionVertexRedefinition}
###### Definition
**([[perturbative interaction vertex redefinition]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).
A _[[perturbative interaction vertex redefinition]]_ (or just _[[vertex redefinition]]_, for short) is an [[endofunction]]
$$
\mathcal{Z}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\longrightarrow
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$
on [[local observables]] with formal parameters adjoined (def. \ref{FormalParameters}) such that there exists a sequence $\{Z_k\}_{k \in \mathbb{N}}$ of [[continuous linear functionals]], symmetric in their arguments, of the form
$$
\left(
{\, \atop \,}
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
{\, \atop \,}
\right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}}
\longrightarrow
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$
such that for all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$ the following conditions hold:
1. (perturbation)
1. $Z_0(g S_{int + j A}) = 0$
1. $Z_1(g S_{int} + j A) = g S_{int} + j A$
1. and
$$
\begin{aligned}
\mathcal{Z}(g S_{int} + j A)
& =
Z \exp_\otimes( g S_{int} + j A )
\\
&
\coloneqq
\underset{k \in \mathbb{N}}{\sum}
\frac{1}{k!}
Z_k(
\underset{
k \, \text{args}
}{
\underbrace{
g S_{int} + j A , \cdots, g S_{int} + j A
}
}
)
\end{aligned}
$$
1. (field independence) The [[local observable]] $\mathcal{Z}(g S_{int} + j A)$ depends on the [[field histories]] only through its argument $g S_{int} + j A $, hence by the [[chain rule]]:
$$
\label{FieldIndependenceVertexRedefinition}
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\mathcal{Z}(g S_{int} + j A)
\;=\;
\mathcal{Z}'_{g S_{int} + j A}
\left(
\frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A)
\right)
$$
=--
The following proposition should be compared to the axiom of _[[causal additivity]]_ of the [[S-matrix]] scheme (eq:CausalAdditivity):
+-- {: .num_prop #InteractionVertexRedefinitionAdditivity}
###### Proposition
**(local additivity of [[vertex redefinitions]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}) and let $\mathcal{Z}$ be a [[vertex redefinition]] (def. \ref{InteractionVertexRedefinition}).
Then for all [[local observables]] $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle$ with spacetime support denoted $supp(O_i) \subset \Sigma$ (def. \ref{SpacetimeSupport}) we have
1. (local additivity)
$$
\begin{aligned}
&
\left(
supp(O_1)
\cap
supp(O_2)
=
\emptyset
\right)
\\
&
\Rightarrow
\phantom{AA}
\mathcal{Z}( O_0 + O_1 + O_2)
=
\mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2)
\end{aligned}
\,.
$$
1. (preservation of spacetime support)
$$
supp
\left(
{\, \atop \,}
\mathcal{Z}(O_0 + O_1)
-
\mathcal{Z}(O_0)
{\, \atop \,}
\right)
\;\subset\;
supp(O_1)
$$
hence in particular
$$
supp
\left(
{\, \atop \,}
\mathcal{Z}(O_1)
{\, \atop \,}
\right) = supp(O_1)
$$
=--
([Dütsch 18, exercise 3.98](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
Under the inclusion
$$
LocObs(E_{\text{BV-BRST}})
\hookrightarrow
PolyObs(E_{\text{BV-BRST}})
$$
of [[local observables]] into [[polynomial observables]] we may think of each $Z_k$ as a [[generalized function]], as for [[time-ordered products]] in remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}.
Hence if
$$
O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j )
$$
is the [[transgression of variational differential forms|transgression]] of a [[Lagrangian density]] $\mathbf{L}$ we get
$$
Z_k(
(O_1 + O_2 + O_3)
,
\cdots
,
(O_1 + O_2 + O_3)
)
=
\underset{
j_1, \cdots, j_k \in \{0,1,2\}
}{\sum}
\underset{\Sigma^{k}}{\int}
Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
\,.
$$
Now by definition $Z_k(\cdots)$ is in the subspace of [[local observables]], i.e. those [[polynomial observables]] whose [[coefficient]] [[distributions]] are [[support of a distribution|supported]] on the [[diagonal]], which means that
$$
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
\frac{\delta}{\delta \mathbf{\Phi}^b(y)}
Z_{k}(\cdots)
=
0
\phantom{AA}
\text{for}
\phantom{AA}
x \neq y
$$
Together with the axiom "field independence" (eq:FieldIndependenceVertexRedefinition) this means that the support of these generalized functions in the [[integrand]] here must be on the [[diagonal]], where $x_1 = \cdots = x_k$.
By the assumption that the spacetime supports of $O_1$ and $O_2$ are disjoint, this means that only the summands with $j_1, \cdots, j_k \in \{0,1\}$ and those with $j_1, \cdots, j_k \in \{0,2\}$ contribute to the above sum. Removing the overcounting of those summands where all $j_1, \cdots, j_k \in \{0\}$ we get
$$
\begin{aligned}
& Z_k\left(
{\, \atop \,}
(O_1 + O_2 + O_3)
,
\cdots
,
(O_1 + O_2 + O_3)
{\, \atop \,}
\right)
\\
& =
\underset{
j_1, \cdots, j_k \in \{0,1\}
}{\sum}
\underset{\Sigma^{k}}{\int}
Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
\\
& \phantom{=}
-
\underset{
j_1, \cdots, j_k \in \{0\}
}{\sum}
\underset{\Sigma^{k}}{\int}
Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
\\
& \phantom{=}
-
\underset{
j_1, \cdots, j_k \in \{0,2\}
}{\sum}
\underset{\Sigma^{k}}{\int}
Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
\\
& =
Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right)
-
Z_k\left( {\, \atop \,} O_0, \cdots, O_0 {\, \atop \,} \right)
+
Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right)
\end{aligned}
\,.
$$
This directly implies the claim.
=--
As a corollary we obtain:
+-- {: .num_prop #CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}
###### Proposition
**([[composition]] of [[S-matrix]] scheme with [[vertex redefinition]] is again [[S-matrix]] scheme)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}) and let $\mathcal{Z}$ be a [[vertex redefinition]] (def. \ref{InteractionVertexRedefinition}).
Then for
$$
\mathcal{S}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ]
$$
and [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), the [[composition|composite]]
$$
\mathcal{S} \circ \mathcal{Z}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\overset{\mathcal{Z}}{\longrightarrow}
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\overset{\mathcal{S}}{\longrightarrow}
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ]
$$
is again an [[S-matrix]] scheme.
Moreover, if $\mathcal{S}$ satisfies the [[renormalization condition]] "field independence" (prop. \ref{BasicConditionsRenormalization}), then so does $\mathcal{S} \circ \mathcal{Z}$.
=--
(e.g [Dütsch 18, theorem 3.99 (b)](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
It is clear that [[causal order]] of the spacetime supports implies that they are in particular [[disjoint subset|disjoint]]
$$
\left(
{\, \atop \,}
supp(O_1)
{\vee\!\!\!\wedge}
supp(O_2)
{\, \atop \,}
\right)
\phantom{AA}
\Rightarrow
\phantom{AA}
\left(
{\, \atop \,}
supp(O_1)
\cap
supp(O_)
\;=\;
\emptyset
{\, \atop \,}
\right)
$$
Therefore the local additivity of $\mathcal{Z}$ (prop. \ref{InteractionVertexRedefinitionAdditivity})
and the [[causal factorization]] of the [[S-matrix]] (remark \ref{DysonCausalFactorization})
imply the causal factorization of the composite:
$$
\begin{aligned}
\mathcal{S}
\left(
{\, \atop \,}
\mathcal{Z}(O_1 + O_2)
{\, \atop \,}
\right)
& =
\mathcal{S}
\left(
{\, \atop \,}
\mathcal{Z}(O_1)
+
\mathcal{Z}(O_2)
{\, \atop \,}
\right)
\\
& =
\mathcal{S}
\left(
{\, \atop \,}
\mathcal{Z}(O_1)
{\, \atop \,}
\right)
\,
\mathcal{S}
\left(
{\, \atop \,}
\mathcal{Z}(O_2)
{\, \atop \,}
\right)
\,.
\end{aligned}
$$
But by prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} this implies in turn [[causal additivity]]
and hence that $\mathcal{S} \circ \mathcal{Z}$ is itself an S-matrix scheme.
Finally that $\mathcal{S} \circ \mathcal{Z}$ satisfies "field indepndence" if $\mathcal{S}$ does is immediate by the [[chain rule]], given that $\mathcal{Z}$ satisfies this condition by definition.
=--
+-- {: .num_prop #AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}
###### Proposition
**(any two [[S-matrix]] [[renormalization schemes]] differ by unique [[vertex redefinition]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).
Then for $\mathcal{S}, \mathcal{S}'$ any two [[S-matrix]] schemes (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) which both satisfy the [[renormalization condition]] "field independence", the there exists a unique [[vertex redefinition]] $\mathcal{Z}$ (def. \ref{InteractionVertexRedefinition}) relating them by [[composition]], i. e. such that
$$
\mathcal{S}'
\;=\;
\mathcal{S} \circ \mathcal{Z}
\,.
$$
=--
+-- {: .proof}
###### Proof
By applying both sides of the equation to linear combinations of local observables of the form $\kappa_1 O_1 + \cdots + \kappa_k O_k$ and then taking [[derivatives]] with respect to $\kappa$ at $\kappa_j = 0$ (as in example \ref{TimeOrderedProductsFromSMatrixScheme}) we get that the equation in question implies
$$
(i \hbar)^k
\frac{
\partial^k
}{
\partial \kappa_1 \cdots \partial \kappa_k
}
\mathcal{S}'( \kappa_1 O_1 + \cdots + \kappa_k O_k )
\vert_{\kappa_1, \cdots, \kappa_k = 0}
\;=\;
(i \hbar)^k
\frac{
\partial^k
}{
\partial \kappa_1 \cdots \partial \kappa_k
}
\mathcal{S} \circ \mathcal{Z}( \kappa_1 O_1 + \cdots + \kappa_k O_k )
\vert_{\kappa_1, \cdots, \kappa_k = 0}
$$
which in components means that
$$
\begin{aligned}
T'_k( O_1, \cdots, O_k )
& =
\underset{
2 \leq n \leq k
}{\sum}
\frac{1}{n!}
(i \hbar)^{k-n}
\underset{
{ { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } }
\atop
{ I_1, \cdots, I_n \neq \emptyset }
}{\sum}
T_n
\left(
{\, \atop \,}
Z_{{\vert I_1\vert}}\left( (O_{i_1})_{i_1 \in I_1} \right),
\cdots,
Z_{{\vert I_n\vert}}\left( (O_{i_n})_{i_n \in I_n} \right),
{\, \atop \,}
\right)
\\
& \phantom{=}
+
Z_k( O_1,\cdots, O_k )
\end{aligned}
$$
where $\{T'_k\}_{k \in \mathbb{N}}$ are the [[time-ordered products]] corresponding to $\mathcal{S}'$ (by example \ref{TimeOrderedProductsFromSMatrixScheme}) and $\{T_k\}_{k \in \mathcal{N}}$ those correspondong to $\mathcal{S}$.
Here the sum on the right runs over all ways that in the composite $\mathcal{S} \circ \mathcal{Z}$ a $k$-ary operation arises as the composite of an $n$-ary time-ordered product applied to the ${\vert I_i\vert}$-ary components of $\mathcal{Z}$, for $i$ running from 1 to $n$; except for the case $k = n$, which is displayed separately in the second line
This shows that if $\mathcal{Z}$ exists, then it is unique, because its coefficients $Z_k$ are [[induction|inductively]] in $k$
given by the expressions
$$
\label{MainTheoremPerturbativeRenormalizationInductionStep}
\begin{aligned}
& Z_k( O_1,\cdots, O_k )
\\
& =
T'_k( O_1, \cdots, O_k )
\;-\;
\underset{
(T \circ \mathcal{Z}_{\lt k})_k
}{
\underbrace{
\underset{
2 \leq n \leq k
}{\sum}
\frac{1}{n!}
(i \hbar)^{k-n}
\underset{
{ { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } }
\atop
{ I_1, \cdots, I_n \neq \emptyset }
}{\sum}
T_n
\left(
Z_{{\vert I_1\vert}}( (O_{i_1})_{i_1 \in I_1} ),
\cdots,
Z_{{\vert I_n\vert}}( (O_{i_n})_{i_n \in I_n} ),
\right)
}
}
\end{aligned}
$$
(The symbol under the brace is introduced as a convenient shorthand for the term above the brace.)
Hence it remains to see that the $Z_k$ defined this way satisfy the conditions in def. \ref{InteractionVertexRedefinition}.
The condition "perturbation" is immediate from the corresponding condition on $\mathcal{S}$ and $\mathcal{S}'$.
Similarly the condition "field independence" follows immediately from the assumoption that $\mathcal{S}$ and $\mathcal{S}'$ satisfy this condition.
It only remains to see that $Z_k$ indeed takes values in [[local observables]]. Given that the [[time-ordered products]] a priori take values in the larrger space of [[microcausal polynomial observables]] this means to show that the spacetime support of $Z_k$ is on the [[diagonal]].
But observe that, as indicated in the above formula, the term over the brace may be understood as the coefficient at order $k$ of the [[exponential series]]-expansion of the [[composition|composite]] $\mathcal{S} \circ \mathcal{Z}_{\lt k}$, where
$$
\mathcal{Z}_{\lt k}
\;\coloneqq\;
\underset{
n \in \{1, \cdots, k-1\}
}{\sum}
\frac{1}{n!} Z_n
$$
is the truncation of the [[vertex redefinition]] to degree $\lt k$. This truncation is clearly itself still
a vertex redefinition (according to def. \ref{InteractionVertexRedefinition}) so that the composite $\mathcal{S} \circ \mathcal{Z}_{\lt k}$
is still an [[S-matrix]] scheme (by prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition})
so that the $(T \circ \mathcal{Z}_{\lt k})_k$ are [[time-ordered products]] (by example \ref{TimeOrderedProductsFromSMatrixScheme}).
So as we solve $\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}$ inductively in degree $k$, then for the induction step in degree $k$ the expressions $T'_{\lt k}$ and $(T \circ \mathcal{Z})_{\lt k}$ agree and are both time-ordered products.
By prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} this implies that $T'_{k}$ and $(T \circ \mathcal{Z}_{\lt k})_{k}$ agree away from the diagonal. This means that their difference $Z_k$ is supported on the diagonal, and hence is indeed local.
=--
In conclusion this establishes the following pivotal statement of [[perturbative quantum field theory]]:
+-- {: .num_theorem #PerturbativeRenormalizationMainTheorem}
###### Theorem
**([[main theorem of perturbative renormalization]] -- [[Stückelberg-Petermann renormalization group]] of [[vertex redefinitions]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).
1. the [[vertex redefinitions]] $\mathcal{Z}$ (def. \ref{InteractionVertexRedefinition}) form a [[group]] under [[composition]];
1. the set of [[S-matrix]] [[renormalization schemes|("re"-)normalization schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), remark \ref{calSFunctionIsRenormalizationScheme}) satisfying the [[renormalization condition]] "field independence" (prop. \ref{BasicConditionsRenormalization}) is a [[torsor]] over this group,
hence equipped with a [[regular action]] in that
1. the set of [[S-matrix schemes]] is [[inhabited set|non-empty]];
1. any two [[S-matrix]] [[renormalization scheme|("re"-)normalization schemes]] $\mathcal{S}$, $\mathcal{S}'$ are related by a _unique_
[[vertex redefinition]] $\mathcal{Z}$ via [[composition]]:
$$
\mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z}
\,.
$$
This group is called the _[[Stückelberg-Petermann renormalization group]]_.
Typically one imposes a set of [[renormalization conditions]] (def. \ref{RenormalizationConditions})
and considers the corresponding [[subgroup]] of [[vertex redefinitions]] preserving these conditions.
=--
+-- {: .proof}
###### Proof
The [[group]]-[[structure]] and [[regular action]] is given by prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition} and prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}.
The existence of S-matrices follows is the statement of [[Epstein-Glaser renormalization|Epstein-Glaser ("re"-)normalization]]
in theorem \ref{ExistenceRenormalization}.
=--
$\,$
**[[UV-regularization|UV-Regularization]] via [[counterterms]]**
{#UVRegularizationViaZ}
While [[Epstein-Glaser renormalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal})
gives a transparent picture on the space of choices in [[renormalization|("re"-)normalization]]
(theorem \ref{ExistenceRenormalization}) the physical nature of the higher interactions that it introduces at
coincident interaction points (via the [[extensions of distributions]] in prop. \ref{SpaceOfPointExtensions}) remains more implicit. But the
[[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}),
which re-expresses the _difference_ between any two such choices as an [[interaction vertex redefinition]],
suggests that already the choice of [[renormalization|("re"-)normalization]] itself should have an incarnation in terms
of [[interaction vertex redefinitions]].
This may be realized via a construction of [[renormalization|("re"-)normalization]] in terms
of _[[UV-regularization]]_ (prop. \ref{UVRegularization} below): For any choice of "[[UV-cutoff]]", given by an
approximation of the [[Feynman propagator]] $\Delta_F$ by [[non-singular distributions]] $\Delta_{F,\Lambda}$
(def. \ref{CutoffsUVForPerturbativeQFT} below) there is a unique "[[effective S-matrix]]" $\mathcal{S}_\Lambda$
induced at each cutoff scale (def. \ref{SMatrixEffective} below). While
the "UV-limit" $\underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda$ does not in general exist,
it may be "regularized" by applying suitable [[interaction vertex redefinitions]] $\mathcal{Z}_\Lambda$;
if the higher-order corrections that these introduce serve to "[[counterterms|counter]]"
(remark \ref{TermCounter} below) the coresponding UV-divergences.
This perspective of [[renormalization|("re"-)normalization via]] via _[[counterterms]]_ is often regarded as
the primary one.
Its elegant proof in prop. \ref{UVRegularization} below, however relies on the [[Epstein-Glaser renormalization]]
via inductive [[extensions of distributions]] and uses the same kind of argument as in the
proof of the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem} via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) that establishes the [[Stückelberg-Petermann renormalization group]].
$\,$
+-- {: .num_defn #CutoffsUVForPerturbativeQFT}
###### Definition
**([[UV cutoffs]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] over [[Minkowski spacetime]] $\Sigma$ (according to def. \ref{VacuumFree}), where $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ is the corresponding [[Wightman propagator]] inducing the [[Feynman propagator]]
$$
\Delta_F \in \Gamma'_{\Sigma \times \Sigma}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})
$$
by $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$.
Then a choice of _[[UV cutoffs]] for [[perturbative QFT]]_ around this vacuum is a
collection of [[non-singular distributions]] $\Delta_{F,\Lambda}$ parameterized by [[positive real numbers]]
$$
\array{
(0, \infty)
&\overset{}{\longrightarrow}&
\Gamma_{\Sigma \times \Sigma,cp}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})
\\
\Lambda &\mapsto& \Delta_{F,\Lambda}
}
$$
such that:
1. each $\Delta_{F,\Lambda}$ satisfies the following basic properties
1. (translation invariance)
$$
\Delta_{F,\Lambda}(x,y) = \Delta_{F,\Lambda}(x-y)
$$
1. (symmetry)
$$
\Delta^{b a}_{F,\Lambda}(y, x)
\;=\;
\Delta^{a b}_{F,\Lambda}(x, y)
$$
i.e.
$$
\Delta_{F,\Lambda}^{b a}(-x)
\;=\;
\Delta_{F,\Lambda}^{a b}(x)
$$
1. the $\Delta_{F,\Lambda}$ interpolate between zero and the Feynman propagator, in that, in the [[Hörmander topology]]:
1. the [[limit of a sequence|limit]] as $\Lambda \to 0$ exists and is zero
$$
\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}
\;=\;
0
\,.
$$
1. the [[limit of a sequence|limit]] as $\Lambda \to \infty$ exists and is the [[Feynman propagator]]:
$$
\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}
\;=\;
\Delta_F
\,.
$$
=--
([Dütsch 10, section 4](renormalization#Duetsch10))
+-- {: .num_example}
###### Example
**(relativistic momentum cutoff)**
Recall from [this prop.](Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue)
that the [[Fourier transform of distributions]] of the [[Feynman propagator]] for the [[real scalar field]] on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ is,
$$
\begin{aligned}
\widehat{\Delta}_F(k)
& =
\frac{+i}{(2\pi)^{p+1}}
\frac{
1
}{
- \eta(k,k) - \left( \tfrac{m c}{\hbar} \right)^2 + i 0
}
\end{aligned}
$$
To produce a [[UV cutoff]] in the sense of def. \ref{CutoffsUVForPerturbativeQFT} we would like to set this function to zero for [[wave numbers]] $\vert \vec k\vert$ (hence [[momenta]] $\hbar\vert \vec k\vert$) larger than a given $\Lambda$.
This needs to be done with due care: First, the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) says that $\Delta_{F,\Lambda}$ to be a test function and hence compactly supported, its [[Fourier transform of distributions|Fourier transform]] $\widehat{\Delta}_{F,\Lambda}$ needs to be smooth and of bounded growth. So instead of multiplying $\widehat{\Delta}_F$ by a [[step function]] in $k$, we may multiply it with an exponential damping.
=--
([Keller-Kopper-Schophaus 97, section 6.1](UV+regularization#KellerKopperSchophaus97), [Dütsch 18, example 3.126](renormalization#Duetsch18))
+-- {: .num_defn #SMatrixEffective}
###### Definition
**([[effective S-matrix scheme]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
We say that the _[[effective S-matrix scheme]]_ $\mathcal{S}_\Lambda$ at cutoff scale $\Lambda \in [0,\infty)$
$$
\array{
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
&\overset{\mathcal{S}_{\Lambda}}{\longrightarrow}&
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
\\
O &\mapsto& \mathcal{S}_\Lambda(O)
}
$$
is the [[exponential series]]
$$
\label{EffectiveSMatrixScheme}
\begin{aligned}
\mathcal{S}_\Lambda(O)
& \coloneqq
\exp_{F,\Lambda}\left( \frac{1}{i \hbar} O \right)
\\
& =
1
+
\frac{1}{i \hbar} O
+
\frac{1}{2} \frac{1}{(i \hbar)^2} O \star_{F,\Lambda} O
+
\frac{1}{3!} \frac{1}{(i \hbar)^3} O \star_{F,\Lambda} O \star_{F,\Lambda} 0
+
\cdots
\end{aligned}
\,.
$$
with respect to the [[star product]] $\star_{F,\Lambda}$ induced by the $\Delta_{F,\Lambda}$ (def. \ref{PropagatorStarProduct}).
This is evidently defined on all [[polynomial observables]] as shown, and restricts to an endomorphism on
[[microcausal polynomial observables]] as shown, since the contraction coefficients $\Delta_{F,\Lambda}$ are [[non-singular distributions]],
by definition of [[UV cutoff]].
=--
([Dütsch 10, (4.2)](renormalization#Duetsch10))
+-- {: .num_prop #UVRegularization}
###### Proposition
**([[renormalization|("re"-)normalization]] via [[UV regularization]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a polynomial [[local observable]] as in def. \ref{FormalParameters}, regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Let moreover $\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)}$ be a [[UV cutoff]] (def. \ref{CutoffsUVForPerturbativeQFT}); with $\mathcal{S}_\Lambda$ the induced [[effective S-matrix schemes]] (eq:EffectiveSMatrixScheme).
Then
1. there exists a $[0,\infty)$-parameterized [[interaction vertex redefinition]] $\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}$ (def. \ref{InteractionVertexRedefinition})
such that the [[limit of a sequence|limit]] of [[effective S-matrix schemes]] $\mathcal{S}_{\Lambda}$ (eq:EffectiveSMatrixScheme) applied to the $\mathcal{Z}_\Lambda$-[[vertex redefinition|redefined interactions]]
$$
\mathcal{S}_\infty
\;\coloneqq\;
\underset{\Lambda \to \infty}{\lim}
\left(
\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda
\right)
$$
exists and is a genuine [[S-matrix scheme]] around the given vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering});
1. every [[S-matrix scheme]] around the given vacuum arises this way.
These $\mathcal{Z}_\Lambda$ are called _[[counterterms]]_ (remark \ref{TermCounter} below) and the composite
$\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ is called a _[[UV regularization]]_ of the [[effective S-matrices]] $\mathcal{S}_\Lambda$.
Hence [[UV-regularization]] via [[counterterms]] is a method of [[renormalization|("re"-)normalization]] of [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).
=--
This was claimed in ([Brunetti-Dütsch-Fredenhagen 09, (75)](renormalization#BrunettiDuetschFredenhagen09)), a proof was indicated in ([Dütsch-Fredenhagen-Keller-Rejzner 14, theorem A.1](renormalization#DuetschFredenhagenKellerRejzner14)).
+-- {: .proof}
###### Proof
Let $\{p_{\rho_{k}}\}_{k \in \mathbb{N}}$ be a sequence of projection maps as in (eq:ForExtensionOfDistributionsProjectionMaps) defining an [[Epstein-Glaser renormalization|Epstein-Glaser ("re"-)normalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) of [[time-ordered products]] $\{T_k\}_{k \in \mathbb{N}}$ as [[extensions of distributions]] of the $T_k$, regarded as distributions via remark \ref{TimeOrderedProductOfFixedInteraction}, by the choice $q_k^\alpha = 0$ in (eq:ExtensionOfDitstributionsPointFixedAndChoice).
We will construct that $\mathcal{Z}_\Lambda$ in terms of these projections $p_\rho$.
First consider some convenient shorthand:
For $n \in \mathbb{N}$, write $\mathcal{Z}_{\leq n} \coloneqq \underset{1 \in \{1, \cdots, n\}}{\sum} \frac{1}{n!} Z_n$.
Moreover, for $k \in \mathbb{N}$ write $(T_\Lambda \circ \mathcal{Z}_{\leq n})_k$ for the $k$-ary coefficient in the
expansion of the composite $\mathcal{S}_\Lambda \circ \mathcal{Z}_{\leq n}$,
as in equation (eq:MainTheoremPerturbativeRenormalizationInductionStep) in the proof of the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}, via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}).
In this notation we need to find $\mathcal{Z}_\Lambda$ such that for each $n \in \mathbb{N}$ we have
$$
\label{CountertermsInductionAssumption}
\underset{\Lambda \to \infty}{\lim}
\left(
T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda}
\right)_n
\;=\;
T_n
\,.
$$
We proceed by [[induction]] over $n \in \mathbb{N}$.
Since by definition $T_0 = const_1$, $T_1 = id$ and $Z_0 = const_0$, $Z_1 = id$ the statement is trivially true for
$n = 0$ and $n = 1$.
So assume now $n \in \mathbb{N}$ and $\{Z_{k}\}_{k \leq n}$ has
been found such that (eq:CountertermsInductionAssumption) holds.
Observe that with the chosen renormalizing projection $p_{\rho_{n+1}}$ the time-ordered product $T_{n+1}$ may be expressed as follows:
$$
\label{RenormalizedSMatrixAsLimitOfEffectiveSMatricesEvaluatedOnProjection}
\begin{aligned}
T_{n+1}(O, \cdots, O)
& =
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F}
\left(
{\, \atop \,}
T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_k}(O \otimes \cdots \otimes O)
\right\rangle
\\
& =
\left\langle
\underset{\Lambda \to \infty}{\lim}
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_k}(O \otimes \cdots \otimes O)
\right\rangle
\end{aligned}
\,.
$$
Here in the first step we inserted the causal decomposition (eq:TimeOrderedProductsAwayFromDiagonalByInduction)
of $T_{n+1}$ in terms of the $\{T_k\}_{k \leq n}$
away from the diagonal, as in the proof of prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal},
which is admissible because the image of $p_{\rho_{n+1}}$ vanishes on the diagonal. In the second step
we replaced the star-product of the Feynman propagator $\Delta_F$ with the limit over the star-products
of the regularized propagators $\Delta_{F,\Lambda}$, which converges by the nature of the [[Hörmander topology]]
(which is assumed by def. \ref{CutoffsUVForPerturbativeQFT}).
Hence it is sufficient to find $Z_{n+1,\Lambda}$ and $K_{n+1,\Lambda}$ such that
$$
\label{CountertermsAndCorrectionTerm}
\begin{aligned}
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\Lambda}
\right)_{n+1}
\,,\,
(-, \cdots, -)
\right\rangle
& =
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_{k}}\left( -, \cdots, - \right)
\right\rangle
\\
& \phantom{=}
+
K_{n+1,\Lambda}(-, \cdots, -)
\end{aligned}
$$
subject to these two conditions:
1. $\mathcal{Z}_{n+1,\Lambda}$ is local;
1. $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$.
Now by expanding out the left hand side of (eq:CountertermsAndCorrectionTerm) as
$$
(T_\Lambda \circ \mathcal{Z}_\Lambda)_{n+1}
\;=\;
Z_{n+1,\Lambda}
\;+\;
(T_\Lambda \circ Z_{\leq n, \Lambda})_{n+1}
$$
(which uses the condition $T_1 = id$) we find the unique solution of (eq:CountertermsAndCorrectionTerm) for $Z_{n+1,\Lambda}$, in terms of the $\{Z_{\leq n,\Lambda}\}$
and $K_{n+1,\Lambda}$ (the latter still to be chosen) to be:
$$
\label{CountertermOrderByOrderInTermsOfCorrectionTerm}
\begin{aligned}
\left\langle
Z_{n+1,\Lambda}
,
(-,\cdots, -)
\right\rangle
& =
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_{n+1}}(-, \cdots, -)
\right\rangle
\\
& \phantom{=}
-
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\leq n,\Lambda}
\right)_{n+1}
\,,\,
(-, \cdots, -)
\right\rangle
\\
& \phantom{=}
+
\left\langle
K_{n+1, \Lambda},
(-, \cdots, -)
\right\rangle
\end{aligned}
\,.
$$
We claim that the following choice works:
$$
\label{LocalityCorrection}
\begin{aligned}
K_{n+1, \Lambda}(-, \cdots, -)
& \coloneqq
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda}
\right)_{n+1}
\,,\,
p_{\rho_{n+1}}(-, \cdots, -)
\right\rangle
\\
& \phantom{=}
-
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_{n+1}}(-, \cdots, -)
\right\rangle
\end{aligned}
\,.
$$
To prove this, we need to show that 1) the resulting $Z_{n+1,\Lambda}$ is local and 2) the limit of $K_{n+1,\Lambda}$ vanishes as $\Lambda \to \infty$.
First regarding the locality of $Z_{n+1,\Lambda}$: By inserting (eq:LocalityCorrection) into (eq:CountertermOrderByOrderInTermsOfCorrectionTerm)
we obtain
$$
\begin{aligned}
\left\langle
Z_{n+1,\Lambda}
\,,\,
(-,\cdots,-)
\right\rangle
& =
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\leq n}
\right)_{n+1}
\,,\,
p(-, \cdots, -)
\right\rangle
-
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\leq n}
\right)_{n+1}
\,,\,
(-, \cdots, -)
\right\rangle
\\
& =
\left\langle
\left(
T_{\Lambda} \circ \mathcal{Z}_{\leq n}
\right)_{n+1}
\,,\,
( p_{\rho_{n+1}} - id)(-, \cdots, -)
\right\rangle
\end{aligned}
$$
By definition $p_{\rho_{n+1}} - id$ is the identity on test functions (adiabatic switchings) that vanish at the diagonal. This means that $Z_{n+1,\Lambda}$ is [[support of a distribution|supported]] on the diagonal, and is hence local.
Second we need to show that $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$:
By applying the analogous causal decomposition (eq:TimeOrderedProductsAwayFromDiagonalByInduction)
to the regularized products, we find
$$
\label{InductionStepForCounterterms}
\begin{aligned}
&
\left\langle
(T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1}
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\\
& =
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\,.
\end{aligned}
$$
Using this we compute as follows:
$$
\label{CorrectionTermForCountertermsVanishesAsCutoffIsRemoved}
\begin{aligned}
&
\left\langle
\underset{\Lambda \to \infty}{\lim}
(T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1}
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\\
& =
\left\langle
\underset{\Lambda \to \infty}{\lim}
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\!\!\chi_i(\mathbf{X})\,
\left(
{\, \atop \,}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
{\, \atop \,}
\right)
\star_{F,\Lambda}
\left(
{\, \atop \,}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
{\, \atop \,}
\right)
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\\
& =
\left\langle
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\chi_i(\mathbf{X})\,
\underset{
T_{{\vert \mathbf{I}\vert}}(\mathbf{I})
}{
\underbrace{
\left(
\underset{\Lambda \to \infty}{\lim}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
\right)
}}
\left(
\underset{\Lambda \to \infty}{\lim}
\star_{F,\Lambda}
\right)
\underset{
T_{{\vert \overline{\mathbf{I}}\vert}}(\overline{\mathbf{I}})
}{
\underbrace{
\left(
\underset{\Lambda \to \infty}{\lim}
(T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
\right)
}}
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\\
& =
\left\langle
\underset{\Lambda \to \infty}{\lim}
\underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum}
\chi_i(\mathbf{X})\,
T_{ { \vert \mathbf{I} \vert } }( \mathbf{I} )
\star_{F,\Lambda}
T_{ {\vert \overline{\mathbf{I}} \vert} }( \overline{\mathbf{I}} )
\,,\,
p_{\rho_{n+1}}(-,\cdots,-)
\right\rangle
\end{aligned}
\,.
$$
Here in the first step we inserted (eq:InductionStepForCounterterms); in the second step we used that in the [[Hörmander topology]] the [[product of distributions]] preserves limits in each variable and in the third step we used the induction assumption (eq:CountertermsInductionAssumption)
and the definition of [[UV cutoff]] (def. \ref{CutoffsUVForPerturbativeQFT}).
Inserting this for the first summand in (eq:LocalityCorrection) shows that $\underset{\Lambda \to \infty}{\lim} K_{n+1, \Lambda} = 0$.
In conclusion this shows that a consistent choice of [[counterterms]] $\mathcal{Z}_\Lambda$ exists to produce
_some_ S-matrix $\mathcal{S} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)$.
It just remains to see that for _every_ other S-matrix $\widetilde{\mathcal{S}}$ there exist counterterms
$\widetilde{\mathcal{Z}}_\lambda$ such
that $\widetilde{\mathcal{S}} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \widetilde{\mathcal{Z}}_\Lambda)$.
But by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem})
we know that there exists a [[vertex redefinition]] $\mathcal{Z}$ such that
$$
\begin{aligned}
\widetilde{\mathcal{S}}
& = \mathcal{S} \circ \mathcal{Z}
\\
& =
\underset{\Lambda \to \infty}{\lim}
\left(
\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda
\right)
\circ
\mathcal{Z}
\\
& =
\underset{\Lambda \to \infty}{\lim}
(
\mathcal{S}_\Lambda
\circ
(
\underset{
\widetilde{\mathcal{Z}}_\Lambda
}{
\underbrace{
\mathcal{Z}_\Lambda
\circ
\mathcal{Z}
}
}
)
)
\end{aligned}
$$
and hence with counterterms $\mathcal{Z}_\Lambda$ for $\mathcal{S}$ given, then counterterms for any $\widetilde{\mathcal{S}}$ are given by the composite $\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}$.
=--
+-- {: .num_remark #TermCounter}
###### Remark
**([[counterterms]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
Consider
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
$$
a [[local observable]], regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Then prop. \ref{UVRegularization} says that there exist [[vertex redefinitions]] of this [[interaction]]
$$
\mathcal{Z}_\Lambda(g S_{int} + j A)
\;\in\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
$$
parameterized by $\Lambda \in [0,\infty)$, such that the [[limit of a sequence|limit]]
$$
\mathcal{S}_\infty(g S_{int} + j A)
\;\coloneqq\;
\underset{\Lambda \to \infty}{\lim}
\mathcal{S}_\Lambda\left( \mathcal{Z}_\Lambda( g S_{int} + j A )\right)
$$
exists and is an [[S-matrix]] for [[perturbative QFT]] with the given [[interaction]] $g S_{int} + j A$.
In this case the difference
$$
\begin{aligned}
S_{counter, \Lambda}
& \coloneqq
\left(
g S_{int} + j A
\right)
\;-\;
\mathcal{Z}_{\Lambda}(g S_{int} + j A)
\;\;\;\;\;\in\;
Loc(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g^2, j^2, g j\rangle
\end{aligned}
$$
(which by the axiom "perturbation" in def. \ref{InteractionVertexRedefinition} is at least of second order in the [[coupling constant]]/[[source field]], as shown)
is called a choice of _[[counterterms]]_ at cutoff scale $\Lambda$. These are new interactions which are added to the given interaction at cutoff scale $\Lambda$
$$
\mathcal{Z}_{\Lambda}(g S_{int} + j A)
\;=\;
g S_{int} + j A
\;+\;
S_{counter,\Lambda}
\,.
$$
In this language prop. \ref{UVRegularization} says that for every free field vacuum and every choice of local interaction, there is a choice of counterterms to the interaction that defines a corresponding [[renormalization|("re"-)normalized]] [[perturbative QFT]], and every [[renormalization|(re"-)normalized]] [[perturbative QFT]] arises from some choice of counterterms.
=--
$\,$
**[[effective quantum field theory|Wilson-Polchinski effective QFT flow]]**
{#EffectiveQFTFlowWislonian}
We have seen [above](#UVRegularizationViaZ) that a choice of [[UV cutoff]] induces [[effective S-matrix schemes]] $\mathcal{S}_\Lambda$
at cutoff scale $\Lambda$ (def. \ref{SMatrixEffective}). To these one may associated non-local [[relative effective actions]]
$S_{eff,\Lambda}$ (def. \ref{EffectiveActionRelative} below) which are such that their effective [[scattering amplitudes]]
at scale $\Lambda$ coincide with the true scattering amplitudes of a genuine [[local observable|local]] interaction
as the cutoff is removed. This is the Wilsonian picture of _[[effective quantum field theory]]_ at a given cutoff scale (remark \ref{pQFTEffective} below). Crucially the "flow" of the [[relative effective actions]] with the cutoff scale satisfies
a [[differential equation]] that in itself is independent of the full UV-theory; this is
_[[Polchinski's flow equation]]_ (prop. \ref{FlowEquationPolchinski} below). Solving this equation for given choice
of initial value data is hence another way of choosing [[renormalization|("re"-)normalization]] constants.
$\,$
+-- {: .num_prop #EffectiveSmatrixSchemeInvertible}
###### Proposition
**([[effective S-matrix schemes]] are [[inverse|invertible functions]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
Write
$$
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
\hookrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
$$
for the subspace of the space of [[formal power series]] in $\hbar, g, j$ with [[coefficients]] [[polynomial observables]]
on those which are at least of first order in $g,j$, i.e. those that vanish for $g, j = 0$ (as in def. \ref{FormalParameters}).
Write moreover
$$
1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
\hookrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
$$
for the subspace of polynomial observables which are the sum of 1 (the multiplicative unit) with an
observable at least linear n $g,j$.
Then the [[effective S-matrix schemes]] $\mathcal{S}_\Lambda$ (def. \ref{SMatrixEffective}) [[restriction|restrict]] to
[[linear isomorphisms]] of the form
$$
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
\underoverset{\simeq}{\phantom{AA}\mathcal{S}_\Lambda \phantom{AA} }{\longrightarrow}
1
+
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
\,.
$$
=--
([Dütsch 10, (4.7)](renormalization#Duetsch10))
+-- {: .proof}
###### Proof
Since each $\Delta_{F,\Lambda}$ is symmetric (def. \ref{CutoffsUVForPerturbativeQFT})
if follows by general properties of [[star products]] (prop. \ref{SymmetricContribution})
just as for the genuine [[time-ordered product]] on [[regular polynomial observables]] (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) that
eeach the "effective time-ordered product" $\star_{F,\Lambda}$ is [[isomorphism|isomorphic]]
to the pointwise product $(-)\cdot (-)$ (def. \ref{Observable})
$$
A_1 \star_{F,\Lambda} A_2
\;=\;
\mathcal{T}_\Lambda
\left(
\mathcal{T}_\Lambda^{-1}(A_1)
\cdot
\mathcal{T}_\Lambda^{-1}(A_2)
\right)
$$
for
$$
\mathcal{T}_\Lambda
\;\coloneqq\;
\exp
\left(
\tfrac{1}{2}\hbar
\underset{\Sigma}{\int}
\Delta_{F,\Lambda}^{a b}(x,y)
\frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
\right)
$$
as in (eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism).
In particular this means that the [[effective S-matrix]] $\mathcal{S}_\Lambda$ arises from the [[exponential series]] for the pointwise product by [[conjugation]] with $\mathcal{T}_\Lambda$:
$$
\mathcal{S}_\Lambda
\;=\;
\mathcal{T}_\Lambda \circ \exp_\cdot\left( \frac{1}{i \hbar}(-) \right) \circ \mathcal{T}_\Lambda^{-1}
$$
(just as for the genuine S-matrix on [[regular polynomial observables]] in def. \ref{OnRegularObservablesPerturbativeSMatrix}).
Now the exponential of the pointwise product on
$1 + PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$
has as [[inverse function]] the [[natural logarithm]] [[power series]], and
since $\mathcal{T}$ evidently preserves powers of $g,j$ this [[conjugation|conjugates]]
to an inverse at each UV cutoff scale $\Lambda$:
$$
\label{InverseOfEffectiveSMatrixByLogarithm}
\mathcal{S}_\Lambda^{-1}
\;=\;
\mathcal{T}_\Lambda
\circ
\ln\left( i \hbar (-) \right)
\circ
\mathcal{T}_\Lambda^{-1}
\,.
$$
=--
+-- {: .num_defn #EffectiveActionRelative}
###### Definition
**([[relative effective action]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
Consider
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$
a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Then for
$$
\Lambda,\, \Lambda_{vac} \;\in\; (0, \infty)
$$
two [[UV cutoff]]-scale parameters, we say the _[[relative effective action]]_ $S_{eff, \Lambda, \Lambda_0}$ is the image of this interaction under the [[composition|composite]]
of the [[effective S-matrix scheme]] $\mathcal{S}_{\Lambda_0}$ at scale $\Lambda_0$ (eq:EffectiveSMatrixScheme) and the [[inverse function]] $\mathcal{S}_\Lambda^{-1}$ of the [[effective S-matrix scheme]] at scale $\Lambda$ (via prop. \ref{EffectiveSmatrixSchemeInvertible}):
$$
\label{RelativeEffectiveActionComposite}
S_{eff,\Lambda, \Lambda_0}
\;\coloneqq\;
\mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\Lambda_0}(g S_{int} + j A)
\phantom{AAA}
\Lambda, \Lambda_0 \in [0,\infty)
\,.
$$
For chosen [[counterterms]] (remark \ref{TermCounter}) hence for chosen [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}) this makes sense also for $\Lambda_0 = \infty$ and we write:
$$
\label{RelativeEffectiveActionRelativeToInfinity}
S_{eff,\Lambda}
\;\coloneqq\;
S_{eff,\Lambda, \infty}
\;\coloneqq\;
\mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\infty}(g S_{int} + j A)
\phantom{AAA}
\Lambda \in [0,\infty)
$$
=--
([Dütsch 10, (5.4)](renormalization#Duetsch10))
+-- {: .num_remark #pQFTEffective}
###### Remark
**([[effective quantum field theory]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}), and let $\mathcal{S}_\infty = \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ be a corresponding [[UV regularization]] (prop. \ref{UVRegularization}).
Consider a [[local observable]]
$$
g S_{int} + j A
\;\in\;
LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle
$$
regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].
Then def. \ref{CutoffsUVForPerturbativeQFT} and def. \ref{EffectiveActionRelative} say that
for any $\Lambda \in (0,\infty)$ the [[effective S-matrix]] (eq:EffectiveSMatrixScheme) of the
[[relative effective action]] (eq:RelativeEffectiveActionComposite) equals the genuine [[S-matrix]] $\mathcal{S}_\infty$
of the genuine [[interaction]] $g S_{int} + j A$:
$$
\mathcal{S}_\Lambda( S_{eff,\Lambda} )
\;=\;
\mathcal{S}_\infty\left( g S_{int} + j A \right)
\,.
$$
In other words the [[relative effective action]] $S_{eff,\Lambda}$
encodes what the actual [[perturbative QFT]] defined by $\mathcal{S}_\infty\left( g S_{int} + j A \right)$
_effectively_ looks like at [[UV cutoff]] $\Lambda$.
Therefore one says that $S_{eff,\Lambda}$ defines _[[effective quantum field theory]]_ at [[UV cutoff]] $\Lambda$.
Notice that in general $S_{eff,\Lambda}$ is _not a [[local observable|local]] [[interaction]]_ anymore:
By prop. \ref{EffectiveSmatrixSchemeInvertible} the [[image]] of the [[inverse]] $\mathcal{S}^{-1}_\Lambda$ of the [[effective S-matrix]]
is [[microcausal polynomial observables]] in $1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle$
and there is no guarantee that this lands in the subspace of [[local observables]].
Therefore [[effective quantum field theories]] at finite [[UV cutoff]]-scale $\Lambda \in [0,\infty)$ are in general
_not_ [[local field theories]], even if their [[limit of a sequence|limit]] as $\Lambda \to \infty$ is, via prop. \ref{UVRegularization}.
=--
+-- {: .num_prop #EffectiveActionAsRelativeEffectiveAction}
###### Proposition
**([[effective action]] is [[relative effective action]] at $\Lambda = 0$)**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
Then the [[relative effective action]] (def. \ref{EffectiveActionRelative}) at $\Lambda = 0$ is the actual [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) in the sense of the the [[Feynman perturbation series]] of
[[Feynman amplitudes]] $\Gamma(g S_{int} + j A)$ (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) for [[connected graph|connected]] [[Feynman diagrams]] $\Gamma$:
$$
\begin{aligned}
S_{eff,0}
& \coloneqq\;
S_{eff,0,\infty}
\\
& = S_{eff} \;\coloneqq\;
\underset{\Gamma \in \Gamma_{conn}}{\sum}
\Gamma(g S_{int} + j A)
\,.
\end{aligned}
$$
More generally this holds true for any $\Lambda \in [0, \infty) \sqcup \{\infty\}$
$$
\begin{aligned}
S_{eff,0,\Lambda}
& =
\underset{\Gamma \in \Gamma_{conn}}{\sum}
\Gamma_\Lambda(g S_{int} + j A)
\,,
\end{aligned}
$$
where $\Gamma_\Lambda( g S_{int} + j A)$ denotes the evident version of the [[Feynman amplitude]] (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) with [[time-ordered products]] replaced by
effective time ordered product at scale $\Lambda$ as in (def. \ref{SMatrixEffective}).
=--
([Dütsch 18, (3.473)](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
Observe that the [[effective S-matrix scheme]] at scale $\Lambda = 0$ (eq:EffectiveSMatrixScheme)
is the [[exponential series]] with respect to the pointwise product (def. \ref{Observable})
$$
\mathcal{S}_0(O) = \exp_\cdot( O )
\,.
$$
Therefore the statement to be proven says equivalently that the [[exponential series]]
of the [[effective action]] with respect to the pointwise product is the [[S-matrix]]:
$$
\exp_\cdot\left( \frac{1}{i \hbar} S_{eff} \right)
\;=\;
\mathcal{S}_\infty\left( g S_{int} + j A \right)
\,.
$$
That this is the case is the statement of prop. \ref{LogarithmEffectiveAction}.
=--
The definition of the [[relative effective action]] $\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty}$ in def. \ref{EffectiveActionRelative}
invokes a choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}).
While (by that proposition and the [[main theorem of perturbative renormalization]], theorem \ref{PerturbativeRenormalizationMainTheorem} )this is guaranteed to exist,
in practice one is after methods for constructing this without specifying it a priori.
But the collection [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ for $\Lambda_0 \lt \infty$
"flows" with the cutoff-parameters $\Lambda$ and in particular also with $\Lambda_0$ (remark \ref{GroupoidOfEFTs} below)
which suggests that examination of this flow yields information about full theory at $\mathcal{S}_\infty$.
This is made precise by _[[Polchinski's flow equation]]_ (prop. \ref{FlowEquationPolchinski} below),
which is the [[infinitesimal]] version of the "Wilsonian RG flow" (remark \ref{GroupoidOfEFTs}).
As a [[differential equation]] it is _independent_ of the choice of $\mathcal{S}_{\infty}$ and hence may
be used to solve for the Wilsonian RG flow without knowing $\mathcal{S}_\infty$ in advance.
The freedom in choosing the initial values of this differential equation corresponds to
the [[renormalization|("re"-)normalization freedom]] in choosing the [[UV regularization]] $\mathcal{S}_\infty$.
In this sense "Wilsonian RG flow" is a method of [[renormalization|("re"-)normalization]] of [[perturbative QFT]]
(def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).
+-- {: .num_remark #GroupoidOfEFTs}
###### Remark
**(Wilsonian [[groupoid]] of [[effective quantum field theories]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).
Then the [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ (def. \ref{EffectiveActionRelative}) satisfy
$$
S_{eff, \Lambda', \Lambda_0}
\;=\;
\left(
\mathcal{S}_{\Lambda'}^{-1}
\circ
\mathcal{S}_\Lambda
\right)
\left(
S_{eff, \Lambda, \Lambda_0}
\right)
\phantom{AAA}
\text{for}
\,
\Lambda,\Lambda' \in [0,\infty)
\,,\,
\Lambda_0 \in [0,\infty) \sqcup \{\infty\}
\,.
$$
This is similar to a [[group]] of UV-cutoff scale-transformations. But since the [[composition]] operations are only sensible when the UV-cutoff labels match, as shown, it is really a [[groupoid]] [[groupoid action|action]].
This is often called the _Wilsonian RG_.
=--
We now consider the [[infinitesimal]] version of this "flow":
+-- {: .num_prop #FlowEquationPolchinski}
###### Proposition
**([[Polchinski's flow equation]])**
Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}), such that $\Lambda \mapsto \Delta_{F,\Lambda}$ is [[differentiable function|differentiable]].
Then for _every_ choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}) the
corresponding [[relative effective actions]] $S_{eff,\Lambda}$ (def. \ref{EffectiveActionRelative}) satisfy the following [[differential equation]]:
$$
\frac{d}{d \Lambda}
S_{eff,\Lambda}
\;=\;
-
\frac{1}{2} \frac{1}{i \hbar}
\frac{d}{d \Lambda'}
\left(
S_{eff,\Lambda}
\star_{F,\Lambda'}
S_{eff,\Lambda}
\right)\vert_{\Lambda' = \Lambda}
\,,
$$
where on the right we have the [[star product]] induced by $\Delta_{F,\Lambda'}$ (def. \ref{PropagatorStarProduct}).
=--
This goes back to ([Polchinski 84, (27)](#Polchinski84)). The rigorous formulation and proof is due to ([Brunetti-Dütsch-Fredenhagen 09, prop. 5.2](renormalization#BrunettiDuetschFredenhagen09), [Dütsch 10, theorem 2](renormalization#Duetsch10)).
+-- {: .proof}
###### Proof
First observe that for any [[polynomial observable]] $O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have
$$
\begin{aligned}
&
\frac{1}{(k+2)!}
\frac{d}{d \Lambda}
(
\underset{
k+2 \, \text{factors}
}{
\underbrace{
O
\star_{F,\Lambda}
\cdots
\star_{F,\Lambda}
O
}
}
)
\\
& =
\frac{1}{(k+2)!}
\frac{d}{d \Lambda}
\left(
prod
\circ
\exp\left(
\hbar
\underset{1 \leq i \lt j \leq k}{\sum}
\left\langle
\Delta_{F,\Lambda}
,
\frac{\delta}{\delta \mathbf{\Phi}_i}
\frac{\delta}{\delta \mathbf{\Phi}_j}
\right\rangle
\right)
(
\underset{
k + 2 \, \text{factors}
}{
\underbrace{
O \otimes \cdots \otimes O
}
}
)
\right)
\\
& =
\underset{
= \frac{1}{2} \frac{1}{k!}
}{
\underbrace{
\frac{1}{(k+2)!}
\left(
k + 2
\atop
2
\right)
}}
\left(
\frac{d}{d \Lambda}
O
\star_{F,\Lambda}
O
\right)
\star_{F,\Lambda}
\underset{
k \, \text{factors}
}{
\underbrace{
O
\star_{F,\Lambda}
\cdots
\star_{F,\Lambda}
O
}
}
\end{aligned}
$$
Here $\frac{\delta}{\delta \mathbf{\Phi}_i}$ denotes the functional derivative of the $i$th tensor factor of $O$, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of $k+2$ tensor factors, where we use that the [[star product]] $\star_{F,\Lambda}$ is commutative (by symmetry of $\Delta_{F,\Lambda}$) and associative (by prop. \ref{AssociativeAndUnitalStarProduct}).
With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) = \mathcal{S}(g S_{int} + j A)$ (eq:RelativeEffectiveActionRelativeToInfinity) we compute as follows:
$$
\begin{aligned}
0
& =
\frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A)
\\
& =
\frac{d}{d \Lambda}
\mathcal{S}_\Lambda(S_{eff,\Lambda})
\\
& =
\left(
\frac{1}{i \hbar}
\frac{d}{d \Lambda} S_{eff,\Lambda}
\right)
\star_{F,\Lambda}
\mathcal{S}_\Lambda(S_{eff,\Lambda})
+
\left(
\frac{d}{d \Lambda}
\mathcal{S}_{\Lambda}
\right)
\left( S_{eff, \Lambda} \right)
\\
& =
\left(
\frac{1}{i \hbar}
\frac{d}{d \Lambda} S_{eff,\Lambda}
\right)
\star_{F,\Lambda}
\mathcal{S}_\Lambda(S_{eff,\Lambda})
\;+\;
\frac{1}{2}
\frac{d}{d \Lambda'}
\left(
\frac{1}{i \hbar} S_{eff,\Lambda}
\star_{F,\Lambda'}
\frac{1}{i \hbar} S_{eff, \Lambda}
\right)
\vert_{\Lambda' = \Lambda}
\star_{F,\Lambda}
\mathcal{S}_\Lambda
\left(
S_{eff, \Lambda}
\right)
\end{aligned}
$$
Acting on this equation with the multiplicative inverse $(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )$ (using that $\star_{F,\Lambda}$ is a commutative product, so that exponentials behave as usual) this yields the claimed equation.
=--
$\,$
**[[renormalization group flow]]**
{#RGFlowGeneral}
In [[perturbative quantum field theory]] the construction of the [[scattering matrix]] $\mathcal{S}$, hence of the [[interacting field algebra of observables]] for a given [[interaction]] $g S_{int}$ [[perturbation theory|perturbing]] around a given [[free field theory|free field]] [[vacuum]], involves choices of _normalization_ of [[time-ordered products]]/[[Feynman diagrams]] (traditionally called _[[renormalization|"re"-normalizations]]_) encoding new [[interactions]] that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.
Whenever a [[group]] $RG$ [[action|acts]] on the space of [[observables]] of the theory such that [[conjugation]] by this action takes [[renormalization scheme|("re"-)normalization schemes]] into each other, then these choices of [[renormalization|("re"-)normalization]] are parameterized by -- or "flow with" -- the elements of $RG$. This is called _renormalization group flow_ (prop.
\ref{FlowRenormalizationGroup} below); often called _RG flow_, for short.
The archetypical example here is the [[group]] $RG$ of [[scaling transformations]] on [[Minkowski spacetime]] (def. \ref{ScalingTransformations} below), which induces a [[renormalization group flow]] (prop. \ref{RGFlowScalingTransformations} below) due to the particular nature of the [[Wightman propagator]] resp. [[Feynman propagator]] on [[Minkowski spacetime]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below). In this case the choice of [[renormalization|("re"-)normalization]] hence "flows with scale".
Now the _[[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) states that (if only the basic [[renormalization condition]] called "field independence" is satisfied) any two choices of [[renormalization scheme|("re"-)normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique [[interaction vertex redefinition]] $\mathcal{Z}$, as
$$
\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}
\,.
$$
Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an [[interaction vertex redefinition]] as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] as changing under such an [[interaction vertex redefinition]], and hence then one has [[coupling constants]] $g_j$ that are parameterized by elements $\rho$ of $RG$:
$$
\mathcal{Z}_{\rho_{vac}}^\rho
\;\colon\;
\{g_j\}
\mapsto
\{g_j(\rho)\}
$$
This dependendence is called _running of the coupling constants_ under the renormalization group flow (def. \ref{CouplingRunning} below).
One example of [[renormalization group flow]] is that induced by [[scaling transformations]] (prop. \ref{RGFlowScalingTransformations} below). This is the original and main example of the concept ([Gell-Mann & Low 54](#GellMannLow54))
In this case the [[running of the coupling constants]] may be understood as expressing how "more" [[interactions]] (at higher energy/shorter [[wavelength]]) become visible (say to [[experiment]]) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be [[differentiable function|differentiable]]; its [[logarithm|logarithmic]] [[derivative]] (denoted "$\psi$" in [Gell-Mann & Low 54](#GellMannLow54)) is known as the _[[beta function]]_ ([Callan 70](#Callan70), [Symanzik 70](#Symanzik70)):
$$
\beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho}
\,.
$$
The [[running of the coupling constants]] is not quite a [[representation]] of the [[renormalization group flow]],
but it is a "twisted" representation, namely a [[group cocycle|group 1-cocycle]] (prop. \ref{CocycleRunningCoupling} below). For the case of [[scaling transformations]] this may be called the _[[Gell-Mann-Low renormalization cocycle]]_ ([Brunetti-Dütsch-Fredenhagen 09](renormalization#BrunettiDuetschFredenhagen09)).
$\,$
+-- {: .num_prop #FlowRenormalizationGroup}
###### Proposition
**([[renormalization group flow]])**
Let
$$
vac
\;\coloneqq\;
(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
$$
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].
Consider a [[group]] $RG$ equipped with an [[action]] on the [[Wick algebra]] of [[off-shell]] [[microcausal polynomial observables]] with formal parameters adjoined (as in def. \ref{FormalParameters})
$$
rg_{(-)}
\;\colon\;
RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ]
\,,
$$
hence for each $\rho \in RG$ a [[continuous linear map]] $rg_\rho$ which has an [[inverse]] $rg_\rho^{-1} \in RG$ and is a [[homomorphism]] of the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] of the given vauum $vac$)
$$
rg_\rho( A_1 \star_H A_2 )
\;=\;
rg_\rho(A_1) \star_H rg_\rho(A_2)
$$
such that the following conditions hold:
1. the action preserves the subspace of [[off-shell]] polynomial [[local observables]], hence it [[restriction|restricts]] as
$$
rg_{(-)}
\;\colon\;
RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
\longrightarrow
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
$$
1. the action respects the [[causal order]] of the spacetime support (def. \ref{SpacetimeSupport}) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have
$$
\left(
supp(O_1)
\,{\vee\!\!\!\wedge}\,
supp(O_2)
\right)
\phantom{A}
\Rightarrow
\phantom{A}
\left(
supp(rg_\rho(O_1))
\,{\vee\!\!\!\wedge}\,
supp(rg_\rho(O_2))
\right)
$$
for all $\rho \in RG$.
Then:
The operation of [[conjugation]] by this action on [[observables]] induces an [[action]] on the [[set]] of [[S-matrix]] [[renormalization schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}, remark \ref{calSFunctionIsRenormalizationScheme}),
in that for
$$
\mathcal{S}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
$$
a perturbative [[S-matrix scheme]] around the given [[free field theory|free field]] [[vacuum]] $vac$, also the [[composition|composite]]
$$
\mathcal{S}^\rho
\;\coloneqq\;
rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
$$
is an [[S-matrix]] scheme, for all $\rho \in RG$.
More generally, let
$$
vac_\rho
\;\coloneqq\;
(E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )
$$
be a collection of [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] parameterized by elements $\rho \in RG$,
all with the same underlying [[field bundle]]; and
consider $rg_\rho$ as above, except that it is not an [[automorphism]] of any [[Wick algebra]],
but an [[isomorphism]] between the [[Wick algebra]]-structures on various vacua, in that
$$
\label{IntertwiningWickProductsActionRG}
rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 )
\;=\;
rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)
$$
for all $\rho, \rho_{vac} \in RG$
Then if
$$
\{ \mathcal{S}_{\rho} \}_{\rho \in RG}
$$
is a collection of [[S-matrix schemes]], one around each of the [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]]
$vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the [[composition|composite]]
$$
\label{RGConjugateSmatrix}
\mathcal{S}_{\rho_{vac}}^\rho
\;\coloneqq\;
rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}
$$
is an [[S-matrix scheme]] around the vacuum labeled by $\rho_{vac}$.
Since therefore each element $\rho \in RG$ in the [[group]] $RG$ picks a different choice of [[renormalization|normalization]]
of the [[S-matrix]] scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$
a _[[renormalization group flow|re-normalization group flow]]_.
=--
([Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1](renormalization#BrunettiDuetschFredenhagen09), [Dütsch 18, section 3.5.3](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom "perturbation" (in def. \ref{LagrangianFieldTheoryPerturbativeScattering}).
In order to verify the axiom "[[causal additivity]]",
observe, for convenience, that by prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} it is sufficient to check [[causal factorization]].
So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$
two local observables whose spacetime support is in [[causal order]].
$$
supp(O_1)
\;{\vee\!\!\!\wedge}\;
supp(O_2)
\,.
$$
We need to show that the
$$
\mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2)
=
\mathcal{S}_{\rho_{vac}}^\rho(O_1)
\star_{H,\rho_{vac}}
\mathcal{S}_{vac_e}^\rho(O_2)
$$
for all $\rho, \rho_{vac} \in RG$.
Using the defining properties of $rg_{(-)}$ and the [[causal factorization]] of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:
$$
\begin{aligned}
\mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2)
& =
rg_\rho
\circ
\mathcal{S}_{\rho^{-1} \rho_{vac}}
\circ
rg_\rho^{-1}( O_1 + O_2 )
\\
& =
rg_\rho
\left(
{\, \atop \,}
\mathcal{S}_{\rho^{-1}\rho_{vac}}
\left(
rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2)
\right)
{\, \atop \,}
\right)
\\
& =
rg_\rho
\left(
{\, \atop \,}
\left(
\mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right)
\right)
\star_{H, \rho^{-1} \rho_{vac}}
\left(
\mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right)
\right)
{\, \atop \,}
\right)
\\
& =
rg_\rho
\left(
{\, \atop \,}
\mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right)
{\, \atop \,}
\right)
\star_{H, \rho_{vac}}
rg_\rho
\left(
{\, \atop \,}
\mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right)
{\, \atop \,}
\right)
\\
& =
\mathcal{S}^\rho_{\rho_{vac}}( O_1 )
\,
\star_{H, \rho_{vac}}
\,
\mathcal{S}_{\rho_{vac}}^\rho(O_2)
\,.
\end{aligned}
$$
=--
+-- {: .num_defn #CouplingRunning}
###### Definition
**([[running coupling constants]])**
Let
$$
vac
\coloneqq
vac_e
\;\coloneqq\;
(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
$$
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]], let $\mathcal{S}$ be an [[S-matrix]] scheme
around this vacuum and let $rg_{(-)}$ be a [[renormalization group flow]] according to prop. \ref{FlowRenormalizationGroup}, such that each re-normalized [[S-matrix scheme]] $\mathcal{S}_{vac}^\rho$ satisfies the [[renormalization condition]] "field independence".
Then by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}, via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) there is for every [[pair]] $\rho_1, \rho_2 \in RG$ a unique [[interaction vertex redefinition]]
$$
\mathcal{Z}_{\rho_{vac}}^{\rho}
\;\colon\;
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
\longrightarrow
LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$
which relates the corresponding two [[S-matrix]] schemes via
$$
\label{SMatrixScemesRelatedByRunningFunction}
\mathcal{S}_{\rho_{vac}}^{\rho}
\;=\;
\mathcal{S}_{\rho_{vac}}
\circ
\mathcal{Z}_{\rho_{vac}}^\rho
\,.
$$
If one thinks of an [[interaction]] vertex, hence a [[local observable]] $g S_{int}+ j A$, as specified by the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding [[interaction]] [[Lagrangian densities]] $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as
$$
g S_{int}
\;=\;
\underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)
$$
(where $\tau_\Sigma$ denotes [[transgression of variational differential forms]]) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j$ of the [[renormalization group flow]] parameterized by $\rho$. The corresponding functions
$$
\mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int})
\;\colon\;
(g_j)
\mapsto
(g_j(\rho))
$$
are then called _[[running coupling constants]]_.
=--
([Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1](renormalization#BrunettiDuetschFredenhagen09), [Dütsch 18, section 3.5.3](renormalization#Duetsch18))
+-- {: .num_prop #CocycleRunningCoupling}
###### Proposition
**([[running coupling constants]] are [[group cocycle]] over [[renormalization group flow]])**
Consider [[running coupling constants]]
$$
\mathcal{Z}_{\rho_{vac}}^{\rho}
\;\colon\;
(g_j)
\mapsto
(g_j(\rho))
$$
as in def. \ref{CouplingRunning}. Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied
by the "running functions" (eq:SMatrixScemesRelatedByRunningFunction):
$$
\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
\;=\;
\mathcal{Z}_{\rho_{vac}}^{\rho_1}
\circ
\left(
\sigma_{\rho_1}
\circ
\mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2}
\circ
\sigma_{\rho_1}^{-1}
\right)
\,.
$$
=--
([Brunetti-Dütsch-Fredenhagen 09 (69)](renormalization#BrunettiDuetschFredenhagen09), [Dütsch 18, (3.325)](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
Directly using the definitions, we compute as follows:
$$
\begin{aligned}
\mathcal{S}_{\rho_{vac}}
\circ
\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
& =
\mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 }
\\
& =
\sigma_{\rho_1}
\circ
\underset{
=
\mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
=
\mathcal{S}_{\rho_1^{-1} \rho_{vac}}
\circ
\mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2}
}{
\underbrace{
\sigma_{\rho_2}
\circ
\mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}}
\circ
\sigma_{\rho_2}^{-1}
}}
\circ
\sigma_{\rho_1}^{-1}
\\
& =
\underset{
=
\mathcal{S}_{\rho_{vac}}
\circ
\mathcal{Z}_{\rho_{vac}}^{\rho_1}
\circ
\sigma_{\rho_1}
}{
\underbrace{
\sigma_{\rho_1}
\circ
\mathcal{S}_{\rho_1^{-1} \rho_{vac}}
\circ
\overset{ = id }{
\overbrace{
\sigma_{\rho_1}^{-1}
\circ
\sigma_{\rho_1}
}
}
}}
\circ
\mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
\circ
\sigma_{\rho_1}^{-1}
\\
& =
\mathcal{S}_{\rho_{vac}}
\circ
\mathcal{Z}_{\rho_{vac}}^{\rho_1}
\circ
\underbrace{
\sigma_{\rho_1}
\circ
\mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
\circ
\sigma_{\rho_1}^{-1}
}
\end{aligned}
$$
This demonstrates the equation between vertex redefinitions to be shown after [[composition]] with an S-matrix scheme. But by the uniqueness-clause in the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from [[vertex redefinitions]] to S-matrix schemes is [[injective function|injective]]. This implies the equation itself.
=--
$\,$
**[[Gell-Mann-Low renormalization cocycles|Gell-Mann & Low RG flow]]**
{#ScalingTransformatinRGFlow}
We discuss (prop. \ref{RGFlowScalingTransformations} below) that, if the field species involved have well-defined [[mass dimension]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below) then [[scaling transformations]] on [[Minkowski spacetime]] (example \ref{ScalingTransformations} below) induce a [[renormalization group flow]] (def. \ref{FlowRenormalizationGroup}). This is the original and main example of [[renormalization group flows]] ([Gell-Mann& Low 54](#GellMannLow54)).
+-- {: .num_example #ScalingTransformations}
###### Example
**([[scaling transformations]] and [[mass dimension]])**
Let
$$
E \overset{fb}{\longrightarrow} \Sigma
$$
be a [[field bundle]] which is a [[trivial vector bundle]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.
For $\rho \in (0,\infty) \subset \mathbb{R}$ a [[positive real number]],
write
$$
\array{
\Sigma
&\overset{\rho}{\longrightarrow}&
\Sigma
\\
x &\mapsto& \rho x
}
$$
for the operation of multiplication by $\rho$ using the [[real vector space]]-[[structure]] of the [[Cartesian space]] $\mathbb{R}^{p+1}$ underlying [[Minkowski spacetime]].
By [[pullback of differential forms|pullback]] this acts on [[field histories]] ([[sections]] of the [[field bundle]]) via
$$
\array{
\Gamma_\Sigma(E)
&\overset{\rho^\ast}{\longrightarrow}&
\Gamma_\Sigma(E)
\\
\Phi &\mapsto& \Phi(\rho(-))
}
\,.
$$
Let then
$$
\rho
\mapsto
vac_\rho
\;\coloneqq\;
(E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )
$$
be a 1-parameter collection of [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to def. \ref{VacuumFree}, and consider a decomposition into a set $Spec$ of field species (def. \ref{VerticesAndFieldSpecies}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] $\Delta_{F,\rho,sp}$ for that species _scales homogeneously_ in that there exists
$$
dim(sp) \in \mathbb{R}
$$
such that for all $\rho$ we have (using [[generalized functions]]-notation)
$$
\label{FeynmanPropagatorScalingBehaviour}
\rho^{ 2 dim(sp) } \Delta_{F, 1/\rho, sp}( \rho x )
\;=\;
\Delta_{F,sp, \rho = 1}(x)
\,.
$$
Typically $\rho$ rescales a [[mass]] parameter, in which case $dim(sp)$ is also called the _[[mass dimension]]_ of the field species $sp$.
Let finally
$$
\array{
PolyObs(E)
&
\overset{
\sigma_\rho
}{\longrightarrow}
&
PolyObs(E)
\\
\mathbf{\Phi}_{sp}^a(x)
&\mapsto&
\rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x )
}
$$
be the [[function]] on [[off-shell]] [[polynomial observables]] given on [[field observables]] $\mathbf{Phi}^a(x)$ by [[pullback of differential forms|pullback]] along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the [[mass dimension]], and extended from there to all [[polynomial observables]] as an [[associative algebra|algebra]] [[homomorphism]].
This constitutes an [[action]] of the [[group]]
$$
RG
\coloneqq
\left(
\mathbb{R}_+, \cdot
\right)
$$
of [[positive real numbers]] (under [[multiplication]]) on [[polynomial observables]], called the group of _[[scaling transformations]]_ for the given choice of field species and [[mass]] parameters.
=--
([Dütsch 18, def. 3.19](renormalization#Duetsch18))
+-- {: .num_example #ScalarFieldMassDimensionOnMinkowskiSpacetime}
###### Example
**([[mass dimension]] of [[scalar field]])**
Consider the [[Feynman propagator]] $\Delta_{F,m}$ of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$ for [[mass]] parameter $m \in (0,\infty)$; a [[Green function]] for the [[Klein-Gordon equation]].
Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of [[scaling transformations]] $\rho \in \mathbb{R}_+$ on [[Minkowski spacetime]] (def. \ref{ScalingTransformations}) act on the mass parameter by inverse multiplication
$$
(\rho , \Delta_{F,m})
\mapsto
\Delta_{F,\rho^{-1}m}(\rho (-))
\,.
$$
Then we have
$$
\Delta_{F,\rho^{-1}m}(\rho (-))
\;=\;
\rho^{-(p+1) + 2}
\Delta_{F,1}(x)
$$
and hence the corresponding [[mass dimension]] (def. \ref{ScalingTransformations}) of the [[real scalar field]] on $\mathbb{R}^{p,1}$ is
$$
dim(\text{scalar field})
=
(p+1)/2 - 1
\,.
$$
=--
+-- {: .proof}
###### Proof
By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the [[Feynman propagator]] in question is given by the [[Cauchy principal value]]-formula (in [[generalized function]]-notation)
$$
\begin{aligned}
\Delta_{F,m}(x)
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu x^\mu}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\,.
\end{aligned}
$$
By applying [[change of integration variables]] $k \mapsto \rho^{-1} k$ in the [[Fourier transform of distributions|Fourier transform]] this becomes
$$
\begin{aligned}
\Delta_{F,\rho^{-1}m}(\rho x)
& =
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu \rho x^\mu}
}{
- k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\\
& =
\rho^{-(p+1)}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu x^\mu}
}{
- \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\\
& =
\rho^{-(p+1)+2}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{+i}{(2\pi)^{p+1}}
\int
\int_{-\infty}^\infty
\frac{
e^{i k_\mu x^\mu}
}{
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
}
\, d k_0 \, d^p \vec k
\\
& =
\rho^{-(p+1) + 2}
\Delta_{F,m}(x)
\end{aligned}
$$
=--
+-- {: .num_prop #RGFlowScalingTransformations}
###### Proposition
**([[scaling transformations]] are [[renormalization group flow]])**
Let
$$
vac \coloneqq vac_m
\coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})
$$
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to def. \ref{VacuumFree} equipped with a decomposition into a set $Spec$ of field species (def. \ref{VerticesAndFieldSpecies}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] the corresponding field species has a well-defined [[mass dimension]] $dim(sp)$ (def. \ref{ScalingTransformations})
Then the [[action]] of the [[group]] $RG \coloneqq (\mathbb{R}_+, \cdot)$ of [[scaling transformations]] (def. \ref{ScalingTransformations}) is a [[renormalization group flow]] in the sense of prop. \ref{FlowRenormalizationGroup}.
=--
([Dütsch 18, exercise 3.20](renormalization#Duetsch18))
+-- {: .proof}
###### Proof
It is clear that rescaling preserves [[causal order]] and the [[renormalization condition]] of "field indepencen".
The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]] we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that
$$
\sigma_\rho
\left(
A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2
\right)
\;=\;
\sigma_\rho(A_1)
\star_{H,\rho_{vac}}
\sigma_\rho(A_2)
\,.
$$
By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$.
Moreover, by the nature of the [[star product]] on [[polynomial observables]], which is given by iterated contractions with the [[Wightman propagator]], it is sufficient to check this for one such contraction.
Observe that the scaling behaviour of the [[Wightman propagator]] $\Delta_{H,m}$ is the same as the behaviour (eq:FeynmanPropagatorScalingBehaviour) of the correspponding [[Feynman propagator]]. With this we directly compute as follows:
$$
\begin{aligned}
\sigma_\rho (\mathbf{\Phi}(x))
\star_{F, \rho_{vac} m}
\sigma_\rho (\mathbf{\Phi}(y)
& =
\rho^{-2 dim }
\mathbf{\Phi}(\rho^{-1} x)
\star_{F, \rho_{vac} m}
\mathbf{\Phi}(\rho^{-1} y)
\\
& =
\rho^{-2 dim }
\Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y))
\\
& =
\Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
\\
& =
rg_{\rho}\left(
\Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
\right)
\\
& =
rg_{\rho}
\left(
\mathbf{\Phi}(x)
\star_{F, \rho^{-1} \rho_{vac} m}
\mathbf{\Phi}(y)
\right)
\end{aligned}
\,.
$$
=--
$\,$
This concludes our discussion of [[renormalization]].
|