8-manifold

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###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

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###Context###
#### Spheres
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[[!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

#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

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+-- {: .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

+-- {: .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

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+-- {: .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


* [[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

## 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

## 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]].