stuff, structure, property

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### Context
#### Category theory
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[[!include category theory - contents]]
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# Stuff, structure, and properties
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## Idea
 {#Idea}

It is common in [[mathematics]] to speak of

1. objects enjoying "extra properties" (for instance a [[ring]] being [[commutative ring]]);

1. objects carrying "extra [[structure]]" (for instance a [[set]] carrying a [[topological space|topology]]);

1. objects being equipped with "extra stuff" (for instance a [[ring]] being equipped with a [[module]]).

In [[model theory]] there is a formalization of what it means to equip a [[set]] with [[mathematical structure]], and one may say what it means for a function between these sets to have the property of preserving this extra structure.
But this formalization captures only parts of the situations in which it is useful to speak of "extra property", "extra structure", and "extra stuff".

Now [[category theory]] is precisely the kind of mathematical meta-theory that allows a nice and general formalization of such matters. Here we discuss such a formalization, due to ([Baez-Bartels-Dolan 98](#BaezBartelsDolan98), see also [Baez-Shulman 06, section 2.4](#BaezShulman06)), phrased in terms of properties of [[functors]] that compare a [[category]] of [[objects]] with extra structure/property/stuff to the underlying category of objects without.

In fact this formalization involves the generalization of what in [[homotopy theory]] is called the [[Postnikov tower]] theory (or [[(n-connected, n-truncated) factorization system]]), generalizing this from [[groupoids]] to [[categories]] ([[directed homotopy type theory|directed homotopy types]]).

For related discussion see also at _[[structure type]]_ and _[[stuff type]]_.

Note that this account operates under the [[principle of equivalence]], where constructions are taken to be invariant under morphisms which are invertible in a maximally weak sense. When working with various amounts of [[strict n-category|strictness]], further requirements are appropriate, some of which are noted below.

## Definitions

To begin with, let $C$ and $D$ be [[groupoids]], and let $F: C \to D$ be a [[functor]].  By fiat, declare $F$ to be a [[forgetful functor]].  Then

* $F$ **forgets nothing** if it is an [[equivalence of categories]];
* $F$ **forgets only properties** if $F$ is [[full and faithful functor|fully faithful]];
* $F$ **forgets at most structure** if $F$ is [[faithful functor|faithful]];
* $F$ **forgets at most stuff** regardless.

Another way to break down the possibilities (used in a $3$-way [[factorization system|factorisation system]]) is as follows:

* $F$ **forgets only properties** if $F$ is [[full functor|full]] and faithful;
* $F$ **forgets purely structure** if $F$ is [[essentially surjective functor|essentially surjective]] (on objects) and faithful;
* $F$ **forgets purely stuff** if $F$ is essentially surjective and full.

{#Strengthening} Depending on author and the strictness of the situation, more properties are required of a functor to forget structure ([Ehresmann 57](#Ehresmann57), [Ehresmann 65](#Ehresmann65), [Adamek-Rosicky-Vitale 09, remark 13.18](#AdamekRosickyVitale)):

1. $F$ is an [[amnestic functor]] ($F$-vertical [[isomorphisms]] are [[identities]]),

1. $F$ is an [[isofibration]].

However, notice that these two conditions violate the [[principle of equivalence]] for [[categories]]. In the terminology of _[[strict categories]]_ one might hence refer to these conditions as expressing "strict extra structure".

### Interpretation in terms of k-surjectivity

The pattern here is best understood in terms of the notion of [[k-surjective functor|essentially k-surjective functor]].

Recall that for a functor between ordinary categories (1-categories)

* essentially surjective $\simeq$ essentially 0-surjective
* full $\simeq$ essentially 1-surjective
* faithful $\simeq$ essentially 2-surjective

and that every 1-functor is essentially $k$-surjective for all $k \geq 3$.

So the above says for a functor $F : C \to D$:

| If it is ...                        | then it ...               | but it ...                             |
| ----------------------------------- | ------------------------- | -------------------------------------- |
| essentially $(k \geq 0)$-surjective | forgets nothing           | remembers everything                   |
| essentially $(k \geq 1)$-surjective | forgets only properties   | remembers at least stuff and structure |
| essentially $(k \geq 2)$-surjective | forgets at most structure | remembers at least stuff               |
| essentially $(k \geq 3)$-surjective | may forget everything     | may remember nothing                   |

It is worth noting that this formalism captures the intuition of how "stuff", "structure", and "properties" are expected to be related:

* stuff may be equipped with structure;
* structure may have (be equipped with) properties.

The $3$-way breakdown looks like this:

| If it is ...                       | then it ...              | but it ...                                  |
| ---------------------------------- | ------------------------ | ------------------------------------------- |
| essentially $(k \ne 0)$-surjective | forgets only properties  | remembers at least stuff and structure      |
| essentially $(k \ne 1)$-surjective | forgets purely structure | remembers at least stuff and properties     |
| essentially $(k \ne 2)$-surjective | forgets purely stuff     | remembers at least structure and properties |

This formalism does *not* capture the intuition so well, and in fact the 'properties' (and 'structure') remembered by a functor that forgets purely structure (or purely stuff) may not match what one expects.

See also the examples below.

### Generalization to higher groupoids

The formulation in terms of $k$-surjectivity induces an immediate generalization of the notions of stuff, structure and properties to the context of [[infinity-category|infinity-groupoids]].  Baez's students speak of "2-stuff," "3-stuff," and so on.   Of course, structure and properties can then be called 0-stuff and $(-1)$-stuff, respectively.

### Generalization to categories and higher categories

The theory is easiest when restricted to groupoids as above; for [[category|categories]], there are several ways to go.  One is to keep the definition as phrased above (a functor between categories forgets only properties if it is fully faithful, forgets at most structure if it is faithful, etc.).  Another is to apply the above definition instead to the functor between the [[core|underlying groupoids]] of the categories in question.

To tell the difference, ask yourself whether the difference between a [[monoid]] and a [[semigroup]] is the *structure* of being equipped with an [[identity element]] or only the *property* that an identity element exists.  Note that an identity element, if it exists, must be unique and must be preserved by semigroup isomorphisms and by monoid homomorphisms but not by semigroup homomorphisms.

A third option is to define a new notion: a functor **forgets at most property-like structure** if it is [[pseudomonic functor|pseudomonic]].  This means that (1) the functor is faithful and (2) its induced functor between underlying groupoids is fully faithful.  Intuitively, property-like structure can be described as consisting of "properties which need not be automatically preserved by morphisms" or "structure which, if it exists, is uniquely determined." Examples include that a category has all limits of a specified sort.

Property-like structure becomes much more prevalent for higher categories.  For example, the forgetful functor from the 2-category of [[cartesian monoidal categories]] (and product-preserving functors) to [[Cat]] is essentially $(k\ge 2)$-surjective, and its induced functor between 2-groupoids is essentially $(k\ge 1)$-surjective; thus it forgets property-like structure.  See also [[lax-idempotent 2-monad]].

Note that property-like structure is known in traditional [[logic]] as __categorical__ structure.  Obviously, this term can be confusing in [[categorial logic]]!

## A factorisation system
 {#AFactorizationSystem}

Just as the category [[Set]] has the best known $2$-way [[factorization system|factorisation system]], in which every [[function]] is factored into a [[surjection]] followed by an [[injection]], so the [[2-category]] [[Cat]] has a [[ternary factorisation system]], in which every functor is factored into parts which forget 'purely' stuff, structure, and properties.

Specifically, given a functor $F: C \to D$, let the __[[1-image]]__ $1 im F$ of $F$ be the category whose objects are objects of $C$ and whose morphisms $x \to y$ are morphisms $F(x) \to F(y)$ in $D$; let the __[[2-image]]__ $2 im F$ of $F$ be the category whose objects are objects of $C$ and whose morphisms $x \to y$ are morphisms $b: F(x) \to F(y)$ in $D$ such that $b = F(a)$ for some $a: x \to y$ in $C$.  (So the only difference bewteen $2 im F$ and $C$ itself is equality of morphisms.)  If you want to be complete, call $C$ itself the __[[3-image]]__ of $F$ and $D$ the __[[0-image]]__.

The situation looks like this:

| This category ... | gets objects from ... | and morphisms from ... | and equality of morphisms from ... |
| ----------------- | --------------------- | ---------------------- | ---------------------------------- |
| $C$               | $C$                   | $C$                    | $C$                                |
| $2 im F$          | $C$                   | $C$                    | $D$                                |
| $1 im F$          | $C$                   | $D$                    | $D$                                |
| $D$               | $D$                   | $D$                    | $D$                                |

Then $F$ can be factored into functors
$$ C \stackrel{F_2}\to 2 im F \stackrel{F_1}\to 1 im F \stackrel{F_0}\to D ,$$
where $F_2$ forgets purely stuff, $F_1$ forgets purely structure, and $F_0$ forgets only properties.  Conversely, this ternary factorization suffices to determine the notions of faithful, full, and essentially surjective functors: see [[ternary factorization system]].

## Examples

### The classical categories of sets with extra structure

The classical examples are the forgetful functors to [[Set]]  that define the classical categories such as [[Top]], [[Grp]], [[Vect]], etc. All these categories are categories of sets equipped with extra structure (e.g. with a topology, with a group structure, etc). Accordingly the obvious functors to [[Set]] are

* faithful
* not full.

Hence indeed, by the above yoga, they forget this extra structure but remember the stuff in question (the underlying set).

### More examples

The embedding of abelian groups into all groups, $F :$ [[Ab]] $\to$ [[Grp]] is faithful and full, but not essentially surjective. Hence it should remember stuff and structure but forget properties. Indeed, the property it forgets is the property "is abelian" which is a property of the group _structure_ sitting on the underlying set of a group. Hence the sequence of functors
$$
  Ab \to Grp \to Set \to pt
$$
(with [[point|pt]] the terminal category) successively forgets first a property (being abelian) then structure (the group structure on a set) then stuff (the underlying set).

Notice that the order here is backwards from the automatic factorisation given by the $3$-way factorisation system described above.  (And in fact, the structure forgotten here is not 'pure'; $Grp \to Set$ is not essentially surjective.)  Indeed, the above factorisation is arbitrary; it comes from seeing an abelian group as a group with an extra property and a group as a set with extra structure, but one may view things differently (for example, that an abelian group is a [[monoid]] with extra property, or a set with two group structures that are related as in the [[Eckmann-Hilton argument]]).

The automatic factorisation is in fact like this (up to [[equivalence of categories]]):
$$
  Ab \to \pt \to \pt \to \pt
$$
because the original functor $Ab \to \pt$ is already essentially surjective and full.  In other words, from the perspective of $\pt$, an abelian group is simply extra stuff.

More interestingly, we can factor the forgetful functor $Ab \to Set$:
$$
  Ab \to Ab \to Set \setminus \{\empty\} \to Set
$$
Here, the first part is trivial because $Ab \to Set$ is faithful.  The category $Set \setminus \{\empty\}$ is the category of [[inhabited set]]s, that is the category of sets that are capable of being equipped with the structure of an abelian group.  So from the point of view of its underlying set, an abelian group is a set with the *property* that it is inhabited and the *structure* of an abelian group but no additional *stuff*.

For something interesting at every level, take the functor $Ab \times Ab \to Set$ that takes the underlying set of the first abelian group.  This factors as follows:
$$
  Ab \times Ab \to Ab \to Set \setminus \{\empty\} \to Set
$$
So a pair of abelian groups (from the perspective of the underlying set of the first one) consists of the *property* that the set is inhabited, then the *structure* of an abelian group on that set, and finally extra *stuff* consisting of the entire second group.

## Logical interpretation

In [[logic]] a property $P$ is given by a [[predicate]], which we may think of as an operation that takes a thing $x$ to the [[truth value]] of the statement "$x$ has  property $P$".  Note that a truth value is a $(-1)$-[[(-1)-groupoid|groupoid]]; we get structure and stuff by replacing this with a [[set]] (a $0$-[[0-groupoid|groupoid]]) or a [[groupoid]] (a $1$-[[1-groupoid|groupoid]]) and we get $n$-stuff by replacing this with an $n$-[[n-groupoid|groupoid]].

Now, if a property of objects of a category $C$ that respects the [[principle of equivalence]] holds for some object $x$, then it must hold for any object [[isomorphism|isomorphic]] to $x$.  That is, the predicate defining that property is actually a [[functor]] from the [[core]] $C_iso$ of $C$ to the groupoid $TV_iso$ of truth values.  Given such a predicate functor $P$, it\'s immediate how to define a [[full subcategory]] $C_P$ of $C$ consisting of those objects with the property; the [[inclusion functor]] $C_P \hookrightarrow C$ is fully faithful, as it should be for extra property.  Conversely, given a fully faithful functor $F: D \to C$, define a property of objects of $C$ that respects the [[principle of equivalence]] as follows: an object $x$ of $C$ has the property if there is some object $a$ of $D$ such that $x \cong F(a)$.  If you apply this to $C_P \hookrightarrow C$, then you get the predicate $P$ back; if you start with an arbitrary fully faithful $F: D \to C$, define a predicate $P$, and then form $C_F$, you\'ll find that $C_F$ and $D$ are [[equivalence of categories|equivalent]], even as [[bundle]]s over $C$.

Diagrammatically this may be phrased as saying that every fully faithful functor $D \to C$ arises as a [[2-limit|weak pullback]] of the $1$-[[subobject classifier]]

$$
  \array{
    D_{iso} &\to& *
    \\
    \downarrow &\Downarrow& \downarrow^{true}
    \\
    C_{iso} &\stackrel{P}{\to}& \{\true, false \}
  }
$$

Similarly, any structure on objects of $C$ that respects the [[principle of equivalence]] is given by a functor from $C_iso$ to the groupoid $Set_iso$ of sets.  Given such a functor $P$, let $C_P$ be the [[category of elements]] of $P$, which comes with a faithful functor from $C_P$ to $C$.  Conversely, given any faithful functor $F: D \to C$ and an object $x$ of $C$, let $P(x)$ be the [[essential fiber]] of $F$ over $x$, which (because $F$ is faithful) is a [[discrete category]] and hence (equivalent to) a set.  These operations are also invertible, up to equivalence.

Diagrammatically this may be phrased as saying that every faithful functor $D \to C$ arises as a [[2-limit|weak pullback]] of the $2$-subobject classifier (as described at [[generalized universal bundle]] and at [[category of elements]])

$$
  \array{
    D_{iso} &\to& (Set_*)_{iso}
    \\
    \downarrow &\Downarrow& \downarrow
    \\
    C_{iso} &\stackrel{}{\to}& Set_{iso}
  }
  \,.
$$

Next, any stuff on objects of $C$ that respects the [[principle of equivalence]] is given by a functor from $C_iso$ to $Grpd_iso$.  Here $Grpd_iso$ should be taken to be the $2$-[[2-groupoid|groupoid]] whose objects are groupoids, whose morphisms are equivalences, and whose $2$-morphisms are [[natural isomorphism]]s; similarly, the functor $P: C_iso \to Grpd_iso$ should be taken in the weakest sense (often called a [[pseudofunctor]]).  Then the [[Grothendieck construction]] turns $P$ into a category $C_P$ equipped with a functor to $C$; again, the essential fiber converts any functor $F: D \to C$ into such a $P$ (although really you must take the core of the essential fiber to get a groupoid).

Diagrammatically this may be phrased as saying that every functor $D \to C$ arises as a [[2-limit|weak pullback]] of the $3$-subobject classifier (as described at [[generalized universal bundle]] )

$$
  \array{
    D_{iso} &\to& (Grpd_*)_{iso}
    \\
    \downarrow &\Downarrow& \downarrow
    \\
    C_{iso} &\stackrel{}{\to}& Grpd_{iso}
  }
  \,.
$$

If $C$ is a mere $1$-[[1-category|category]], then any $P: C_iso \to 2 Grpd_iso$ is equivalent to some $P: C_iso \to Grpd_iso$, but in general we need to consider $P: C_iso \to n Grpd_iso$ or $P: C_iso \to \infty Grpd_iso$ to study higher forms of $n$-stuff.

Lest we forget, to be even more simple than an extra property, the groupoid of $(-2)$-[[(-2)-groupoid|groupoid]]s is the [[point]] $pt$, and there is exactly one functor $P$ from any $C_iso$ to $pt$, corresponding to the unique (up to equivalence) category equivalent to $C$.

For the [[(∞,1)-topos]] [[∞-Grpd]] of [[∞-groupoid]]s the analog of the subobject classifier is the  [[universal fibration of (∞,1)-categories]] $Z|_{\infty Grpd} \to \infty Grpd$. See also section 6.1.6 _$\infty$-Topoi and Classifying objects_ of [[Higher Topos Theory|HTT]].

## Formulation in homotopy type theory

A map between [[homotopy type theory|homotopy types]] is equivalent to a [[dependent sum]] projection $f: \sum_{x: A} B(x) \to A$. If the types $B(x)$ are [[homotopy n-type|n-types]], $f$ forgets $n$-stuff.

## Related entries

*  [[predicate]]

*  [[structure]], [[structure type]]

*  [[stuff type]]

* [[topological property]], [[uniform property]]

## References

The original proposal for formalizing notions of [[mathematical structures]] via [[category theory]], specifically via the [[forgetful functors]] between the [[groupoids]] which they form:

* {#Ehresmann57} [[Charles Ehresmann]], *Gattungen von lokalen Strukturen*, Jahresbericht der Deutschen Mathematiker-Vereinigung **60** (1958) 49-77 [[dml:146434](https://eudml.org/doc/146434)]

* {#Ehresmann65} [[Charles Ehresmann]], _Catégories et Structures_, Séminaire Ehresmann. Topologie et géométrie différentielle **6** (1964) 1-31 [[numdam:SE_1964__6__A8_0](http://www.numdam.org/item?id=SE_1964__6__A8_0), [dml:112200](https://eudml.org/doc/112200)]

Exposition and re-amplification:

* {#BaezBartelsDolan98} [original UseNet discussion](http://math.ucr.edu/home/baez/qg-spring2004/discussion.html) on `sci.physics.research` in 1998;
* a pedagogical comparison to quadratic, linear, and constant polynomials ([PDF](http://math.ucr.edu/home/baez/qg-spring2004/polynomials.pdf)) by Toby Bartels in 2004;

* {#BaezShulman06} [section 2.4, p. 15](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15) and [section 3.1, p. 17](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=17) of J. Baez and M. Shulman, _[[Lectures on n-Categories and Cohomology]]_ ([arXiv](http://arxiv.org/abs/math/0608420)).

For modern accounts on mathematical structures via [[categorical algebra]] see also at *[[algebraic theory]]*, *[[monad]]*, *[[sketch]]*.

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Sturm-Liouville theory

# Sturm--Liouville theory
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{: toc}

## Idea

Sturm--Liouville theory consists of methods concerning a certain class of [[differential equations]].

## References

* Wikipedia, _[Sturm--Liouville theory](https://en.wikipedia.org/wiki/Sturm–Liouville_theory)_
* W.N. Everitt, _[A Catalogue of Sturm-Liouville differential equations](http://www.math.niu.edu/SL2/papers/birk0.pdf)_

[[!redirects Sturm-Liouville theory]]
[[!redirects Sturm–Liouville theory]]
[[!redirects Sturm--Liouville theory]]

[[!redirects Sturm-Liouville equation]]
[[!redirects Sturm-Liouville equations]]
[[!redirects Sturm–Liouville equation]]
[[!redirects Sturm–Liouville equations]]
[[!redirects Sturm--Liouville equation]]
[[!redirects Sturm--Liouville equations]]

[[!redirects Sturm-Liouville differential equation]]
[[!redirects Sturm-Liouville differential equations]]
[[!redirects Sturm–Liouville differential equation]]
[[!redirects Sturm–Liouville differential equations]]
[[!redirects Sturm--Liouville differential equation]]
[[!redirects Sturm--Liouville differential equations]]


Stéphane Attal

* [personal page](http://math.univ-lyon1.fr/~attal/)

## Selected writings

On [[open quantum systems|open]] [[quantum systems]] ([[quantum probability]], [[quantum noise]], [[quantum channels]], ...):

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems I -- The Hamiltonian Approach*, Lecture Notes in Mathematics **1880**, Springer (2006) [[doi:10.1007/b128449](https://doi.org/10.1007/b128449)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*,  Lecture Notes in Mathematics **1881**, Springer (2006) [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems III -- Recent Developments*, Lecture Notes in Mathematics **1882**, Springer (2006) [[doi:10.1007/b128453](https://doi.org/10.1007/b128453)]

On [[quantum noise]]:

* [[Stéphane Attal]], *Quantum Noises*, in: *Open Quantum Systems II*, Lecture Notes in Mathematics **1881** (2006)  [[doi:10.1007/3-540-33966-3_3](https://doi.org/10.1007/3-540-33966-3_3), [pdf](http://math.univ-lyon1.fr/~attal/Mesarticles/Attal.pdf)]

On [[quantum physics]], [[quantum channels]], [[quantum probability theory]],  and (eventually) [[quantum noise]]:

* [[Stéphane Attal]], *Lectures on Quantum Noises* [[webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

category: people

[[!redirects Stephane Attal]]


Stéphane Gaubert

Stéphane Gaubert is Directeur de Recherche at INRIA, at their Research unit in Saclay, Île-de-France. He is Head of Tropical research team, joint to INRIA and the _Centre de Mathématiques Appliquéees (CMAP)_ of the _École Polytechnique_, UMR 7641 CNRS

* [homepage](http://www.cmap.polytechnique.fr/~gaubert/)

##Selected publications / preprints etc.

* [[Stéphane Gaubert]], _Methods and Applications of $(max,+)$ Linear Algebra,_ [Report 3088](https://hal.inria.fr/inria-00073603/document), January 1997, INRIA.

category:people


Stéphane Peigné

* [InSpire pgae](https://inspirehep.net/authors/993937)

## Selected writings

On [[string diagram]]-calculus ([Cvitanović 2008](string+diagram#Cvitanovic08): "bird tracks") for the [[representation theory]] involved in [[quantum chromodynamics]]

* [[Stéphane Peigné]], *Introduction to color in QCD: Initiation to the birdtrack pictorial technique* [[arXiv:2302.07574](https://arxiv.org/abs/2302.07574)]

[[!redirects Stephane Peigne]]


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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the _Stückelberg-Petermann renormalization group_ ([Stückelberg-Petermann 53](#StiecelbergPetermann53)) is the (original) incarnation of the [[renormalization group]] in the perspective of [[causal perturbation theory]]: It is the group (def. \ref{StueckelbergPetermannRenormalizationGroup} below) of perturbative [[interaction vertex redefinitions]] (def. \ref{InteractionVertexRedefinition}) which is such that any two choices of [[S-matrix]] [[renormalization schemes]] $\mathcal{S}, \mathcal{S}'$ are related by a unique [[vertex redefinition]] $\mathcal{Z}$ via [[precomposition]]

$$
  \mathcal{S}'
  \;=\;
  \mathcal{S} \circ \mathcal{Z}
  \,.
$$

This statement (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} below) is also called the _[[main theorem of perturbative renormalization]]_.

If [[scaling transformations]] on [[spacetime]] happen to transform [[renormalization schemes]] into each other, then this [[main theorem of perturbative renormalization]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) directly implies that every [[scaling transformation]] uniquely corresponds to a [[interaction vertex redefinition]], or conversely that the [[vertex redefinitions]] are given as funcitons of scale. As such they are also referred to as _[[running coupling constants]]_ (def. \ref{GellMannLowTransformations} below). Beware that these do not in general form a [[group]] but a [[group cocycle]], the _[[Gell-Mann-Low renormalization cocycle]]_ (prop. \ref{CocyclePropertyOfRunningCouplingConstants} below.

## Definition

The elements of the Stückelberg-Petermann renormalization group are _perturbative [[interaction vertex redefinitions]]_ (def. \ref{InteractionVertexRedefinition} below). These [[action|act]] on [[S-matrix]] [[renormalization schemes|("re"-)normalization schemes]] by [[precomposition]] (def. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}) and this action is [[free action|free]] and [[transitive action|transitive]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} below). In this way these [[vertex redefinitions]] translate between different choices of [[renormalization|("re"-)normalization]], and as such they form the _Stückelerg-Petermann renormalization group_ (def. \ref{StueckelbergPetermannRenormalizationGroup}) 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]] ([this def.](S-matrix#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 ([this def.](S-matrix#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 ([this equation](S-matrix#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]] ([this def.](S-matrix#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 $supp(O_i) \subset \Sigma$ ([this def.](A+first+idea+of+quantum+field+theory#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](#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 [this remark](S-matrix#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]] ([this def.](S-matrix#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 ([this def.](S-matrix#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" ([this prop.](S-matrix#BasicConditionsRenormalization)), then so does $\mathcal{S} \circ \mathcal{Z}$.

=--

(e.g [Dütsch 18, theorem 3.99 (b)](#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]] ([this remark](S-matrix#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 [this prop.](S-matrix#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 a unique [[vertex redefinition]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] ([this def.](S-matrix#VacuumFree)).

Then for $\mathcal{S}, \mathcal{S}'$ any two [[S-matrix]] schemes ([this def.](S-matrix#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

$$
  \mathcal{S}'
  \;=\;
  \mathcal{S} \circ \mathcal{Z}
  \,.
$$

=--

(See any of the references at _[[main theorem of perturbative renormalization]]_.)

+-- {: .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 [this example](S-matrix#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 [this example](S-matrix#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

$$
  \begin{aligned}
    & Z_k( O_1,\cdots, O_k )
    \\
    & =
    T'_k( O_1, \cdots, O_k )
    \;-\;
    \underset{
      (T \circ \mathcal{Z}_{\lt k})_k
    }{
    \underbrace{
    \underset{
      2 \leq n \leq k
    }{\sum}
      \frac{1}{n!}
      (i \hbar)^{k-n}
      \underset{
        { { I_1 \sqcup \cdots \sqcup I_n } \atop { =  \{1, \cdots, k\}, } }
        \atop
        { I_1, \cdots, I_n \neq \emptyset }
      }{\sum}
      T_n
      \left(
        Z_{{\vert I_1\vert}}( (O_{i_1})_{i_1 \in I_1} ),
        \cdots,
        Z_{{\vert I_n\vert}}( (O_{i_n})_{i_n \in I_n} ),
      \right)
    }
    }
  \end{aligned}
$$

(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 [this example](S-matrix#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 [this prop.](S-matrix#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.

=--

+-- {: .num_defn #StueckelbergPetermannRenormalizationGroup}
###### Definition
**([[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]] ([this def.](S-matrix#VacuumFree)).

Then prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition} and prop \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} together says that ("[[main theorem of perturbative renormalization]]"):

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]] ([this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering), [this remark](S-matrix#calSFunctionIsRenormalizationScheme)) satisfying the [[renormalization condition]] "field independence" ([this prop.](S-matrix#BasicConditionsRenormalization)) is a [[torsor]] over this group, meaning that the action is [[regular action|regular]] on this set, in that any two are S-matrix ("re"-)normalization schemes are related by a unique vertex redefinition.

This group is called the (large) _[[Stückelberg-Petermann renormalization group]]_.

Typically one imposes a set of [[renormalization conditions]] ([this def.](S-matrix#RenormalizationConditions))
and then the corresponding [[subgroup]] of [[vertex redefinitions]] preserving these.

=--

## Properties

### Scaling transformations and "running coupling constants"

A priori the Stückelberg-Petermann renormalization group is not about [[scaling transformations]]. But if [[scaling transformations]] happen to produce new S-matrices/renormalization schemes from given ones, then the [[main theorem of perturbative renormalization]] induces for each such scaling transformation a re-definition of interaction Lagrangian densities, this is the [[Gell-Mann-Low renormalization cocycle]] ([Gell-Mann & Low 54](#GellMannLow54), [Brunetti-Dütsch-Fredenhagen 09](#BrunettiDuetschFredenhagen09)) for review see ([Dütsch 18, section 3.5.3](#Duetsch18)).

+-- {: .num_defn #GellMannLowTransformations}
###### Definition
**([[running coupling constants]] under [[scale transformations]])**

Let

$$
  vac
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}_{kin}, \Delta_H )
$$

be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].

Assume that in fact

1. the [[free field]] [[vacuum]] $vac = vac(m)$ depends on a [[mass]] parameter, and with it the choice $\mathcal{S}_{vac(m)}$ of [[renormalization scheme|normalization scheme]],

1. under [[scaling transformations]] on [[local observables]] $\sigma_\rho$ ([Dütsch 18, def. 3.19](#Duetsch18))  we have that with $\mathcal{S}_{vac(m)}$ a [[perturbative S-matrix]] scheme perturbing around $vac(m)$ also

   $$
     \sigma_\rho \circ \left(\mathcal{S}_{vac(m/\rho)}\right) \circ \sigma_\rho^{-1}
   $$

   is a perturbative S-matrix around $L_{kin}(m)$.

In this case the [[main theorem of perturbative renormalization]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) says that there exists for each scale $\rho$ a unique
[[interaction vertex redefinition]] $\mathcal{Z}^m_\rho$ (def. \ref{InteractionVertexRedefinition}) such that

$$
  \begin{aligned}
    &
    \sigma_\rho
      \circ
    \mathcal{S}_{vac(m/\rho)}
      \circ
    \sigma_\rho^{-1}( g S_{int} + j A )
    \\
    & =
    \mathcal{S}_{vac(m)}(\mathcal{Z}^m_\rho(g S_{int} + j A))
  \end{aligned}
$$

for all [[interaction]] [[action functionals]] $g S_{int} + j A$.

These $\mathcal{Z}^m_\rho$ are the _[[Gell-Mann-Low renormalization cocycle]]_ elements.

The [[interaction vertex redefinitions]] $\mathcal{Z}^m_\rho$ as a function of the [[rescaling]]
is known as the _[[running coupling constants]]_.

=--

+-- {: .num_prop #CocyclePropertyOfRunningCouplingConstants}
###### Proposition
**(cocycle property of [[running coupling constants]])**

In the situation of def. \ref{GellMannLowTransformations}, the [[Gell-Mann-Low renormalization cocycles]] ([[running coupling constants]])
$\mathcal{Z}^m_\rho$ satisfy the relation

$$
  \mathcal{Z}^m_{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}^m_{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_2}
  \right)
$$

Hence only for vanishing [[mass]] do these "renormalization cocycles" themselves form an actual [[renormalization group]].

=--

([Brunetti-Dütsch-Fredenhagen 09 (69)](#BrunettiDuetschFredenhagen09), [Dütsch 18 (3.325)](#Duetsch18))

+-- {: .proof}
###### Proof

From the definition we have

$$
  \begin{aligned}
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1 \rho_2}
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \mathcal{Z}^{m/\rho_1}_{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{vac(m/\rho_1\rho_2)}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{vac(m)}
        \circ
      \mathcal{Z}^m_{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1}
      \circ
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
  \end{aligned}
$$

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]] 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.

=--

## References

(See also the references at _[[main theorem of perturbative renormalization]]_.)

The original article on the Stückelberg-Petermann renormalization group is

* {#StueckelbergPetermann53} [[Ernst Stückelberg]], [[André Petermann]], _La normalisation des constantes dans la theorie des quanta_, Helv. Phys. Acta 26 (1953), 499–520

The relation of the Stückelberg-Petermann renormalization group to [[scale transformations]] and the [[Gell-Mann-Low renormalization cocycle]] is due to

* {#GellMannLow54} [[Murray Gell-Mann]] and F. E. Low, _Quantum Electrodynamics at Small Distances_, Phys. Rev. 95 (5) (1954), 1300–1312 ([pdf](http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf))

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

Review includes

* {#Duetsch18} [[Michael Dütsch]], section 3.5.1 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* {#Duetschfredenhagen07} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Action Ward Identity and the Stückelberg-Petermann renormalization group_, Prog. Math. 251:113-124,2007

[[!redirects Stückelberg-Petermann renormalization groups]]

[[!redirects Stueckelberg-Petermann renormalization group]]
[[!redirects Stueckelberg-Petermann renormalization groups]]


SU(2)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The [[special unitary group]] $SU(n)$ for $n = 2$.

+-- {: .num_prop}
###### Proposition

As a [[matrix group]] $SU(2)$ is equivalent to the subgroup of the [[general linear group]] $GL(2, \mathbb{C})$ on those of the form

$$
  \left(
    \array{
      u & v
      \\
      - \overline{v} & \overline{u}
    }
  \right)
  \;\;\;
   with
  \;\;
   {\vert u\vert}^2 + {\vert v\vert}^2 = 1
  \,,
$$

where $u,v \in \mathbb{C}$ are [[complex numbers]] and $\overline{(-)}$ denotes [[complex conjugation]].

=--

## Properties

### General

+-- {: .num_prop}
###### Proposition

The underlying [[manifold]] of $SU(2)$ is [[diffeomorphism|diffeomorphic]] to the 3-[[sphere]] $S^3$.

=--

+-- {: .num_prop #ExceptionalIsomorphismOfSU2ToSpin3AndSp1}
###### Proposition

There are [[isomorphisms]] of [[Lie groups]]

1. of $SU(2)$ with the [[spin group]] in [[dimension]] 3 and with the [[quaternionic unitary group]] in one dimension

   $$
     SU(2)
     \;\simeq\;
     Spin(3)
     \;\simeq\;
     Sp(1)
   $$

1. of the [[direct product group]] of $SU(2)$ with itself, to [[Spin(4)]]

   $$
     SU(2) \times SU(2)
     \;\simeq\;
     Spin(3) \times Spin(3)
     \;\simeq\;
     Spin(4)
   $$

   with respect to which the canonical inclusion $Spin(3) \hookrightarrow Spin(4)$ is given by the [[diagonal]] map.

=--

See at _[spin group -- Exceptional isomorphisms](spin%20group#ExceptionalIsomorphisms)_.

+-- {: .num_prop}
###### Proposition

The [[maximal torus]] of $SU(2)$ is the [[circle group]] $U(1)$. In the above [[matrix group]] presentation this is naturally identified with the subgroup of matrices of the form

$$
  \left(
    \array{
      t & 0
      \\
      0 & t^{-1}
    }
  \right)
  \;\;
  with
  t \in U(1) \hookrightarrow \mathbb{C}
  \,.
$$

=--

### Lie algebra

+-- {: .num_prop}
###### Proposition

The [[Lie algebra]] $\mathfrak{su}(2)$ (see also at [[su(2)]]) as a [[complex numbers|complex]] [[matrix Lie algebra]] is the sub Lie algebra on those matrices of the form

$$
  \left(
    \array{
      i z & x + i y
      \\
      - x + i y & - i z
    }
  \right)
  \;\;\;
  with
  \;\;
  x,y,z \in \mathbb{R}
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

The standard [[basis]] elements of $\mathfrak{su}(2)$ given by the above presentation are

$$
  \sigma_1
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     0 & 1
     \\
     -1 & 0
   }
  \right)
$$

$$
  \sigma_2
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     0 & i
     \\
     i & 0
   }
  \right)
$$

$$
  \sigma_3
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     i & 0
     \\
     0 & -i
   }
  \right)
  \,.
$$

These are called the _[[Pauli matrices]]_.

=--

+-- {: .num_prop}
###### Proposition

The [[Pauli matrices]] satisfy the [[commutator]] [[relations]]

$$
  [\sigma_1, \sigma_2] = \sigma_3
$$

$$
  [\sigma_2, \sigma_3] = \sigma_1
$$

$$
  [\sigma_3, \sigma_1] = \sigma_2
  \,.
$$

=--

### Coadjoint orbits

+-- {: .num_prop}
###### Proposition

The [[coadjoint orbits]] of the [[coadjoint action]] of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = r^2$ for some $r \geq 0$.

These are _regular_ coadjoint orbits for $r \gt 0$.

=--

### Finite subgroups

The [[finite subgroup of SU(2)]] have an [[ADE classification]]. See [this theorem](classification+of+finite+rotation+groups#ClassificationOfFiniteSubgroupsOfSO3).

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

### Representation theory
 {#RepresentationTheory}

To record some aspects of the [[linear representation|linear]] [[representation theory]] of [[SU(2)]].

(...)

+-- {: .num_lemma #4Wedge4SpSp1Is3TimesTrivialPlusThree}
###### Lemma

  We have
  $$
    \wedge^2 \mathbf{4}
    \;\simeq\;
    3 \cdot \mathbf{1}
    \;\oplus\;
    1 \cdot \mathbf{3}
    \;\;\;\in\;
    \mathrm{RO}(\mathrm{Sp}(1))
  $$

=--

+-- {: .proof}
###### Proof

  Consider the canonical [[linear basis]]
  \[
    \label{BasisFor4}
    \mathbf{4}
    \;\simeq\;
    \big\langle \{q_0, q_1, q_2, q_3\}\big\rangle_{\mathbb{R}}
  \]
  by [[orthonormal basis|orthonormal]] [[quaternions]]
  $$
    q_0, q_1, q_2, q_3
    \;\in\;
    \mathbb{H}
  $$
  $$
    \begin{aligned}
    &
    q_0 = 1\,,
    \\
    &
    q_i q_i = -1 = - q_0  \;\; \text{for} i \in \{1,2,3\} \,,
    \\
    &
    q_{\sigma(1)} q_{\sigma(3)} =  \mathrm{sgn}(\sigma)  q_{\sigma(3)}
    \,,
    \text{for}  \sigma \in \mathrm{Sym}(3)
    \end{aligned}
  $$
  On this, the [[action]] by [[Sp(1)]]
  $$
    \mathrm{Sp}(1)
    \;\simeq\;
    \{
    q \in \mathbb{H}
    \;\vert\;
    q \bar q = 1
    \}
  $$
  is by left [[quaternion]] [[multiplication]].

  Consider then the following
  [[linear basis]] of $\mathbf{4}\wedge \mathbf{4}$:
  $$
    \mathbf{4}\wedge \mathbf{4}
    \;\simeq\;
    \left\langle
    \left\{
      \array{
        a_1^+
        :=
        q_0 \wedge q_1 + q_2 \wedge q_3,
        \\
        a_2^+
        :=
        q_0 \wedge  q_2 + q_3 \wedge  q_1,
        \\
        a_3^+
        :=
        q_0 \wedge q_3 + q_1 \wedge  q_2,
      }
      \array{
        a_1^-
        :=
        q_0 \wedge q_1 - q_2 \wedge q_3,
        \\
        a_2^-
        :=
        q_0 \wedge q_2 - q_3 \wedge q_1,
        \\
        a_3^-
        :=
        q_0 \wedge q_3 - q_1 \wedge q_2
      }
    \right\}
    \right\rangle
  $$
  with induced $\mathfrak{sp}(1)$ [[Lie algebra action]] given by
  $$
    q_i \cdot ( q_j \wedge q_k )
    \;=\;
    (q_i q_j) \wedge q_k
    +
    q_j \wedge (q_i q_k)
    \,.
  $$
  From this we find
  $$
    \begin{aligned}
    q_1 \cdot a_1^{\pm}
    & = \;
    q_1
      \cdot
    \big(
      q_0 \wedge q_1
      \pm
      q_2 \wedge q_3
    \big)
    \\
    & =\;
    \big(
    \underset{
      = 0
    }{
      \underbrace{
        q_1 \wedge q_1
      }
    }
    +
    \underset{
      = 0
    }{
      \underbrace{
        q_0 \wedge (-q_0)
      }
    }
    \big)
    \pm
    \big(
    \underset{
      = 0
    }{
      \underbrace{
        q_3 \wedge q_3
      }
    }
    +
    \underset{
      = 0
    }{
      \underbrace{
        q_2 \wedge (- q_2)
      }
    }
    \big)
    \\
    & =\;
    0
    \end{aligned}
  $$
  and
  $$
    \begin{aligned}
    q_1 \cdot a_2^{\pm}
    & =\;
    q_1
      \cdot
    \big(
      q_0 \wedge q_2
      \pm
      q_3 \wedge q_1
    \big)
    \\
    & = \;
    \big(
    \underset{
      = q_1 \wedge q_2
    }{
      \underbrace{
        q_1 \wedge q_2
      }
    }
    +
    \underset{
      = q_0 \wedge q_3
    }{
      \underbrace{
        q_0 \wedge q_3
      }
    }
    \big)
    \pm
    \big(
    \underset{
      = q_1 \wedge q_2
    }{
      \underbrace{
        (- q_2) \wedge q_1
      }
    }
    +
    \underset{
      = q_0 \wedge q_3
    }{
      \underbrace{
        q_3 \wedge (- q_0)
      }
    }
    \big)
    \\
    & =\;
    \left\{
      \begin{array}{l}
        2 a_3^+
        \\
        0
      \end{array}
    \right.
    \end{aligned}
  $$
  and
  $$
    \begin{aligned}
    q_1 \cdot a_3^{\pm}
    & =\;
    q_1
      \cdot
    \big(
      q_0 \wedge q_3
      \pm
      q_1 \wedge q_2
    \big)
    \\
    & = \;
    \big(
    \underset{
      = - q_3 \wedge q_1
    }{
      \underbrace{
        q_1 \wedge q_3
      }
    }
    +
    \underset{
      = - q_0 \wedge q_2
    }{
      \underbrace{
        q_0 \wedge (-q_2)
      }
    }
    \big)
    \pm
    \big(
    \underset{
      = - q_0 \wedge q_2
    }{
      \underbrace{
        (- q_0) \wedge q_2
      }
    }
    +
    \underset{
      = - q_3 \wedge q_1
    }{
      \underbrace{
        q_1 \wedge q_3
      }
    }
    \big)
    \\
    & =\;
    \left\{
      \begin{array}{l}
        -2 a_2^+
        \\
        0
      \end{array}
    \right.
    \end{aligned}
    \,.
  $$
  Since everything here is [[invariant]] under [[cyclic permutation]] of
  the three non-zero indices it follows generally that
  $$
    (\tfrac{1}{2} q_i) \cdot a_j^+ \;=\; \underset{k}{\sum} \epsilon_{i j k} a_k^+
    \,,
    \;\;
    (\tfrac{1}{2} q_i) \cdot a_j^- \;=\; 0
    \;\;
    \text{for all} i,j \in \{1,2,3\}
  $$
  But this means that
  $$
    \big\langle \{a^+_1, a^+_2, a^+_3\} \big\rangle
    \;\simeq\;
    \mathbf{3}
    \,,
    \phantom{aa}
    \big\langle \{a^-_i\} \big\rangle
    \;\simeq\;
    \mathbf{1}
    \;\;\;\in
    \mathrm{RO}(\mathrm{Sp}(1))
    \,.
  $$

=--

\linebreak

+-- {: .num_lemma #4Wedge4Wedge4ofSp1IsFour}
###### Lemma

  We have
  $$
    \wedge^3 \mathbf{4}
    \;\simeq\;
    \mathbf{4}
    \;\;
    \in RO(Sp(1))
    \,.
  $$

=--

+-- {: .proof}
###### Proof

  Consider the following [[linear basis]]

  $$
    \wedge^3 \mathbf{4}
    \;\simeq_{\mathbb{R}}\;
    \left\langle
    \left\{
    \begin{array}{l}
      b_0 := + q_1 \wedge q_2 \wedge q_3
      \\
      b_1 := - q_0 \wedge q_2 \wedge q_3
      \\
      b_2 := + q_0 \wedge q_1 \wedge q_3
      \\
      b_3 := - q_0 \wedge q_1 \wedge q_2
    \end{array}
    \right\}
    \right\rangle
    \,.
  $$

  We claim that in terms of these basis elements
  and those of (eq:BasisFor4) the [[isomorphism]] is given by
  $b_0 \mapsto q_0$, $b_i \mapsto q_i$.
  This follows by direct inspection. For instance for the
  [[Lie algebra action]] of $q_1$ we find:

  $$
    \begin{aligned}
      q_1 \cdot b_0
      & \; = \;
      q_1 \cdot ( q_1 \wedge q_2 \wedge q_3 )
      \\
      & \; = \;
      \underset{
        = b_1
      }{
        \underbrace{
          (- q_0) \wedge q_2 \wedge q_3
        }
      }
      \;+\;
      \underset{
        = 0
      }{
        \underbrace{
          q_1 \wedge q_3 \wedge q_3
        }
      }
      \;+\;
      \underset{
        = 0
      }{
        \underbrace{
          q_1 \wedge q_2 \wedge (- q_2)
        }
      }
      \\
      &
      \;=\;
      b_1
    \end{aligned}
  $$

  $$
    \begin{aligned}
      q_1 \cdot b_1
      & \;=\;
      q_1 \cdot ( - q_0 \wedge q_2 \wedge q_3 )
      \\
      & \;=\;
      -
      \underset{
        b_0
      }{
        \underbrace{
          q_1 \wedge q_2 \wedge q_3
        }
      }
      -
      \underset{
        = 0
      }{
      \underbrace{
        q_0 \wedge q_3 \wedge q_3
      }
      }
      -
      \underset{
        = 0
      }{
        \underbrace{
          q_0 \wedge q_2 \wedge (-q_2)
        }
      }
      \\
      & \;=\;
      - b_0
    \end{aligned}
  $$

  $$
    \begin{aligned}
      q_1 \cdot b_2
      & \;=\;
      q_1 \cdot (q_0 \wedge q_1 \wedge q_3)
      \\
      &
      \;=\;
      \underset{
        = 0
      }{
        \underbrace{
          q_1 \wedge q_1 \wedge (- q_2)
        }
      }
      +
      \underset{
        = 0
      }{
        \underbrace{
          q_0 \wedge (- q_0) \wedge q_3
        }
      }
      +
      \underset{
        = b_3
      }{
        \underbrace{
          q_0 \wedge q_1 \wedge (- q_2)
        }
      }
      \\
      & \;=\;
      b_3
    \end{aligned}
  $$

  $$
    \begin{aligned}
    q_1 \cdot b_3
    & \;=\;
    q_1 \cdot (- q_0 \wedge q_1 \wedge q_2)
    \\
    & \;=\;
    -
    \underset{
      = 0
    }{
      \underbrace{
        q_1 \wedge q_1 \wedge q_2
      }
    }
    -
    \underset{
      = 0
    }{
      \underbrace{
        q_0 \wedge (- q_0) \wedge q_2
      }
    }
    -
    \underset{
      = b_2
    }{
      \underbrace{
        q_0 \wedge q_1 \wedge q_3
      }
    }
    \\
    & \;=\;
    - b_2
    \end{aligned}
  $$

=--

## Related concepts

* [[su(2)]]

* [[complex Hopf fibration]]

* [[Euler angles]]

* [[SU(3)]], [[SU(4)]], [[SU(5)]]

* [[Pauli quantum gate]]

* [[Solovay-Kitaev theorem]]

[[!include low dimensional rotation groups -- table]]

## References

* [[Howard Georgi]], §3  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]

On the [[coadjoint orbits]] of $SU(2)$:

* [[Alexandre Kirillov]], p. 183 in: _[[Lectures on the Orbit Method]]_, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004)

See also

* [[Qiaochu Yuan]], _[The representation theory of SU(2)](https://qchu.wordpress.com/2011/06/26/the-representation-theory-of-su2/)_

* Wikipedia, _Representation theory of SU(2)_

[[!redirects Spin(3)]]

[[!redirects Spin3]]

[[!redirects Sp(1)]]
[[!redirects Sp1]]


su(2)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie algebra]] $\mathfrak{su}(2)$ is the special case of [[special unitary Lie algebras]] $\mathfrak{su}(n)$ for $n = 2$, underlying the [[Lie group]] [[SU(2)]] (the [[special unitary group]] $SU(n)$ for $n =2$).

## Idea

The Lie algebra $\mathfrak{su}(2)$ is equivalently given as follows:

1. the Lie algebra on 3 [[generators]] $\{\sigma_1, \sigma_2,\sigma_3\}$ subject to the following [[relations]] on their [[Lie bracket]]:

   $$
     [e_i, e_j] = \underset{k}{\sum} \epsilon_{i j k} e_k
   $$

1. the Lie algebra spanned by ($i$ times) the three [[Pauli matrices]] with [[Lie bracket]] their [[commutator]] in their [[matrix algebra]].

## Properties

### Pauli matrix presentation

+-- {: .num_prop}
###### Proposition

The [[Lie algebra]] $\mathfrak{su}(2)$ as a [[complex numbers|complex]] [[matrix Lie algebra]] is the sub Lie algebra on those matrices of the form

$$
  \left(
    \array{
      i z & x + i y
      \\
      - x + i y & - i z
    }
  \right)
  \;\;\;
  with
  \;\;
  x,y,z \in \mathbb{R}
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

The standard [[basis]] elements of $\mathfrak{su}(2)$ given by the above presentation are

$$
  \sigma_1
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     0 & 1
     \\
     -1 & 0
   }
  \right)
$$

$$
  \sigma_2
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     0 & i
     \\
     i & 0
   }
  \right)
$$

$$
  \sigma_3
   \coloneqq
  \frac{1}{2}
  \left(
   \array{
     i & 0
     \\
     0 & -i
   }
  \right)
  \,.
$$

These are called the _[[Pauli matrices]]_.

=--

+-- {: .num_prop}
###### Proposition

The [[Pauli matrices]] satisfy the [[commutator]] [[relations]]

$$
  [\sigma_1, \sigma_2] = \sigma_3
$$

$$
  [\sigma_2, \sigma_3] = \sigma_1
$$

$$
  [\sigma_3, \sigma_1] = \sigma_2
  \,.
$$

=--

Another common basis in use is the _[[Cartan-Weyl basis]]_

$$
  \begin{aligned}
    \sigma^\pm & \coloneqq \sigma_2 \pm  i \sigma_3
    \\
    \sigma^0 & \coloneqq \sigma_1
  \end{aligned}
$$

### Complexification
 {#Complexification}

The [[complexification]] of $\mathfrak{su}(2)$ is the [[special linear Lie algebra]] $\mathfrak{sl}(2, \mathbb{C})$ (see at [[sl(2)]]) (...)

## Related concepts

* [[fuzzy 2-sphere]]

  [[fuzzy funnel]], [[Dp-D(p+2)-brane bound state]]

* [[SU(2)]]

* [[Pauli quantum gate]]

## References

Textbook accounts:

* {#Pfeifer} Walter Pfeifer, _The Lie algebra $\mathfrak{su}(2)$_, In: _The Lie Algebras $\mathfrak{su}(N)$, Birkhäuser, Basel (2003) ([doi:10.1007/978-3-0348-8097-8_3](https://doi.org/10.1007/978-3-0348-8097-8_3), [pdf](https://link.springer.com/content/pdf/10.1007/978-3-0348-8097-8_3.pdf))

* [[Howard Georgi]], §3 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]

See also

* [[Qiaochu Yuan]], _[The representation theory of SU(2)](https://qchu.wordpress.com/2011/06/26/the-representation-theory-of-su2/)_

* Wikipedia, _Representation theory of SU(2)_

[[!redirects su2]]


su(2)-anyon

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], many or all [[anyon]]-species of (potential) practical interest (such as for [[topological quantum computation]]) are thought to be characterized by [[affine Lie algebras]] $\widehat{\mathfrak{g}}^k$ (at some [[level (Chern-Simons theory)|level]] $k$), in that their [[wavefunctions]] are, essentially, $\widehat{g}$-[[conformal blocks]] and their [[braid group representation|braiding]] is described by $G$-[[Chern-Simons theory]] at [[level (Chern-Simons theory)|level]] $k$ (possibly fractional, see at *[[logarithmic CFT]]* [here](logarithmic+CFT#FractionalLevelWZWModelReferences)).

If here $\mathfrak{g} = $ [[su(2)|$\mathfrak{su}(2)$]], then one also speaks of "SU(2)-anyons" (with varying conventions on capitalization, etc.).
With "Majorana anyons" ($k = 2$) and "Fibonacci anyons" ($k = 3$) this class subsumes most or all anyon species which seem to have a realistic chance of existing in nature.

{#NonRealizationOfMZZInNanowires} Notably [[Majorana anyons]] (in the guise of "[[Majorana zero modes]]" in super/semi-conducting nanowires) are (or were until recently, see [arXiv:2106.11840v4](https://arxiv.org/abs/2106.11840v4), [p. 3](https://arxiv.org/pdf/2106.11840v4.pdf#page=3)) at the focus of attention of an intense effort to finally provide a practical proof of principle for the old idea of [[topological quantum computation]] (following the plan laid out in [Das Sarma, Freedman & Nayak 15](#DasSarmaFreedmanNayak15)). After initial claims had to be retracted [in 2021](https://www.nature.com/articles/d41586-021-00612-z) [[doi:10.1038/s41586-021-03373-x](https://doi.org/10.1038/s41586-021-03373-x), [doi:10.5281/zenodo.4587841](https://doi.org/10.5281/zenodo.4587841), [doi:10.5281/zenodo.4545812](https://doi.org/10.5281/zenodo.4545812), [TU Delft press release](https://www.delta.tudelft.nl/article/majorana-not-fraud-confirmation-bias#)] and again [in 2022](http://retractionwatch.com/2022/04/24/authors-retract-second-majorana-paper-from-nature/) [[doi:10.1038/s41586-022-04704-2](https://doi.org/10.1038/s41586-022-04704-2)] (further claims are under criticism, see e.g. [doi:10.5281/zenodo.6344447](https://doi.org/10.5281/zenodo.6344447), [doi:10.5281/zenodo.6325378](https://doi.org/10.5281/zenodo.6325378) and the list [here](https://twitter.com/spinespresso/status/1503352928656138241)) there is a new claim of detection by [Nayak 22](#Nayak22) & [MicrosoftQuantum 22](#MicrosoftQuantum22), but see [Frolov & Mourik 22a](#FrolovMourik22a), [22b](#FrolovMourik22a) and [Frolov 22](#Frolov22).

In any case, Majorana anyons are known not to be universal (not all [[quantum gates]] may be approximated with Majorana braiding). The simplest universal $\mathfrak{su}(2)$-anyon species are the Fibonacci anyons at [[level (Chern-Simons theory)|level]] $k = 3$ (e.g. [Simeon 2021](#Simeon21), also [Kolganov, Mironov & Morozon (2023)](#KolganovMironovMorozon23)).

## Related concepts

* [[parafermion]]

* [[Majorana dimer]]

## References

### General

Early consideration of $\mathfrak{su}(2)$-anyons is *implicit*  in the context of [[Laughlin wavefunctions]] due to

* {#ReadRezayi99} [[Nicholas Read]], [[Edward Rezayi]], *Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level*, Phys. Rev. B **59** (1999) 8084 $[$[doi:10.1103/PhysRevB.59.8084](https://doi.org/10.1103/PhysRevB.59.8084)$]$

Early discussion of [[topological quantum computation]] in $SU(2)$-[[Chern-Simons theory]]:

* {#FreedmanKitaevLarsenWang03} [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], *Topological quantum computation*,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

* [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics. 2002, Vol 227, Num 3, pp 605-622  ([arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108))

More concrete discussion of these phenomena in terms of [[anyons]]:

* S. Trebst, M. Troyer, Z. Wang and A. W. W. Ludwig, *A short introduction to Fibonacci anyon models*, Prog. Theor. Phys. Supp. **176** 384 (2008)
[[arXiv:0902.3275](https://arxiv.org/abs/0902.3275), [doi:10.1143/PTPS.176.384](https://doi.org/10.1143/PTPS.176.384)]

* C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang,  *Anyonic quantum spin chains: Spin-1 generalizations and topological stability*, Phys. Rev. B **87** (2013), 235120
[[doi:10.1103/PhysRevB.87.235120](https://doi.org/10.1103/PhysRevB.87.235120), [arXiv:1303.4290](https://arxiv.org/abs/1303.4290)]

* {#DasSarmaFreedmanNayak15} [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], *Majorana zero modes and topological quantum computation*, npj Quantum Inf **1** 15001 (2015) [[doi:10.1038/npjqi.2015.1](https://doi.org/10.1038/npjqi.2015.1)]

*  [[Emil Génetay-Johansen]], [[Tapio Simula]], *Fibonacci anyons versus Majorana fermions -- A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models*, PRX Quantum **2** 010334 (2021) [[arXiv:2008.10790](https://arxiv.org/abs/2008.10790), [doi:10.1103/PRXQuantum.2.010334](https://doi.org/10.1103/PRXQuantum.2.010334)]

Discussion of Fibonacci anyons:

* {#Simeon21} Ryan Simeon, *Universality of Fibonacci anyons in topological quantum computing* (2021) [[pdf](https://homes.psd.uchicago.edu/~sethi/Teaching/P243-W2021/Final%20Papers/Simeon-PHYS243_final.pdf), [[Simeon-UniversalityOfFibonacciAnyons.pdf:file]]]

Discussion of universality at higher [[Chern-Simons level|level]] $k$ (and also for $SU(N)$-anyons with $N \gt 2$):

* {#KolganovMironovMorozon23} [[Nikita Kolganov]], [[Sergey Mironov]], [[Andrey Morozov]], *Large $k$ topological quantum computer*, Nuclear Physics B
**987** (2023) 116072 [[arXiv:2105.03980](https://arxiv.org/abs/2105.03980), [doi:10.1016/j.nuclphysb.2023.116072](https://doi.org/10.1016/j.nuclphysb.2023.116072)]

### Experimental realization

#### (Non-)Observation of Majorana zero modes

The general strategy of realizing [[Majorana zero modes]] in supercondocuting/semiconducting nanowires is due to

* [[Alexei Kitaev]], *Unpaired Majorana fermions in quantum wires*,  Physics-Uspekhi, **44** 10S (2001) 131-136 [[doi:10.1070/1063-7869/44/10S/S29](https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29)]

* Roman M. Lutchyn, Jay D. Sau, [[Sankar Das Sarma]], *Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures*, Phys. Rev. Lett. **105** (2010) 077001 [[doi:10.1103/PhysRevLett.105.077001](https://doi.org/10.1103/PhysRevLett.105.077001)]

* Yuval Oreg, Gil Refael, and Felix von Oppen, *Helical Liquids and Majorana Bound States in Quantum Wires*, Phys. Rev. Lett. **105** (2010) 177002 [[doi:10.1103/PhysRevLett.105.177002](https://doi.org/10.1103/PhysRevLett.105.177002)]

reviewed in:

* Pasquale Marra: *Majorana nanowires for topological quantum computation: A tutorial* [[arXiv:2206.14828](https://arxiv.org/abs/2206.14828)]

On the general problem of distinguishing the expected effect from noise:

* [[Sankar Das Sarma]], Haining Pan, *Disorder-induced zero-bias peaks in Majorana nanowires* Phys. Rev. B **103** 195158 (2021) [[arXiv:2103.05628](https://arxiv.org/abs/2103.05628), [doi:10.1103/PhysRevB.103.195158](https://doi.org/10.1103/PhysRevB.103.195158)]

> we believe that similar confirmation bias applies to many other topological discovery claims in the literature during 2000–2020 where a precise knowledge of what one is looking for has been the key factor in the discovery claim, with the experimental quantization results themselves not being sufficiently compelling. [...] Our results certainly apply to most of the Majorana experiments during 2012–2021 in the literature.

Non-retracted claims of experimental realization of something in the direction of Majorana zero modes:

* {#MMBDRSC19} Gerbold C. Ménard, Andrej Mesaros, Christophe Brun, François Debontridder, Dimitri Roditchev, Pascal Simon, Tristan Cren, *Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer*,  Nat Commun **10** 2587 (2019) $[$[doi:10.1038/s41467-019-10397-5](https://doi.org/10.1038/s41467-019-10397-5)$]$

* {#Nayak22} [[Chetan Nayak]], *Microsoft has demonstrated the underlying physics required to create a new kind of qubit*, Microsoft Research Blog (March 2022)

* {#MicrosoftQuantum22} Microsoft Quantum, *InAs-Al Hybrid Devices Passing the Topological Gap Protocol* [[arXiv:2207.02472](https://arxiv.org/abs/2207.02472)]

but see commentary in:

* {#FrolovMourik22a} [[Sergey M. Frolov]], [[Vincent Mourik]], *We cannot believe we overlooked these Majorana discoveries* [[arXiv:2203.17060](https://arxiv.org/abs/2203.17060), [doi:10.5281/zenodo.6364928](https://doi.org/10.5281/zenodo.6364928), conclusion on: [p. 7](https://arxiv.org/ftp/arxiv/papers/2203/2203.17060.pdf#page=7)]

* {#FrolovMourik22b} [[Sergey M. Frolov]], [[Vincent Mourik]]: *Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review* [[video](https://www.youtube.com/watch?v=RnYghkDaHH0), conclusion at: [1:01:31](https://www.youtube.com/watch?v=RnYghkDaHH0&t=3691s)]

  > [1:01:52](https://youtu.be/RnYghkDaHH0?t=3712) The signal is fully consistent, from what we see, with *not* having discovered any Majorana or topological superconductivity here. At the same time, the amount of data is extremely narrow.

* {#Frolov22} [[Sergey M. Frolov]], *Superconductors and semiconductors, nanowires and majorana, research and integrity* [[video](https://www.youtube.com/watch?v=FR7J4yH67GA), general caution: [55:34](https://youtu.be/FR7J4yH67GA?t=3334), concrete criticism: [1:01:41](https://youtu.be/FR7J4yH67GA?t=3701)]

  > [1:01:50](https://youtu.be/FR7J4yH67GA?t=3710): The claims of the discovery of Majorana have been overblown and are false. Majorana has not been discovered in nanowires. I don't believe in any other system it has been discovered either.

On how this could happen:

* Elizabeth Gibney, *Inside Microsoft’s quest for a topological quantum computer* (Interview with Alex Bocharov), Nature (2016) [[doi:10.1038/nature.2016.20774](https://doi.org/10.1038/nature.2016.20774)]

  > [Bocharov:] We’re people-centric, rather than problem-centric.

See also:

* {#Frolov21} [[Sergey M. Frolov]], *So, You Think You Discovered a New State of Matter?*, Physics **14** 68 (2021) [[physics.aps:v14/68](https://physics.aps.org/articles/v14/68)]

* [[Sergey M. Frolov]], *Quantum computing’s reproducibility crisis: Majorana fermions*, Nature **592** (2021) 350-352 [[doi:10.1038/d41586-021-00954-8](https://doi.org/10.1038/d41586-021-00954-8)]

#### Other

Proposal to realize [[Fibonacci anyons]] on [[quasicrystal]]-states:

* Marcelo Amaral, [[David Chester]], Fang Fang, Klee Irwin, *Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing*, Symmetry **14** 9 (2022) 1780 [[arXiv:2207.08928](https://arxiv.org/abs/2207.08928), [doi:10.3390/sym14091780](https://doi.org/10.3390/sym14091780)]

\linebreak

[[!include Laughlin wavefunctions as conformal blocks -- references]]

[[!redirects su(2)-anyons]]

[[!redirects SU(2)-anyon]]
[[!redirects SU(2)-anyons]]

[[!redirects su2-anyon]]
[[!redirects su2-anyons]]

[[!redirects Majorana anyon]]
[[!redirects Majorana anyons]]

[[!redirects Fibonacci anyon]]
[[!redirects Fibonacci anyons]]

[[!redirects Ising anyon]]
[[!redirects Ising anyons]]


SU(2)-structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[reduction of a structure group]] to [[SU(2)]], of particular interest on [[5-manifolds]]

In [[dimension]] 5 [[Sasaki-Einstein manifolds|Sasaki-Einstein structure]] is a special case of $SU(2)$-structure (e.g. [CMD 14](#CMD14), [Fino 18b](#Fino18b))

## Properties

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## References

* {#ContiSalamon05} Diego Conti, [[Simon Salamon]], _Generalized Killing spinors in dimension 5_, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5319-5343 ([arXiv:math/0508375](https://arxiv.org/abs/math/0508375))

* Luis C. de Andrés, Marisa Fernández, Anna Fino, Luis Ugarte, _Contact 5-manifolds with SU(2)-structure_ ([arXiv:0706.0386](https://arxiv.org/abs/0706.0386))

* Daniël Prins, _On flux vacua, SU(n)-structures and generalised complex geometry_ ([arXiv:1602.05415](https://arxiv.org/abs/1602.05415))

* {#Fino18a} Anna Fino, _Contact manifolds and $SU(2)$-structure_, talk at _Holonomy Groups and Applications in String Theory_, Hamburg July 2008 ([pdf](https://www.math.uni-hamburg.de/home/leistner/workshop08/Slides/Fino-hamburg08.pdf))

* {#Fino18b} Anna Fino, _Hypo contact and Sasakian structures on Lie groups_, talk at _Workshop on CR and Sasakian Geometry_, Luxembourg– 24 - 26 March 2008 ([pdf](http://math.uni.lu/CRSasaki/slides/fino.pdf))

* Lucio Bedullia, Luigi Vezzoni, _Torsion of $SU(2)$-structures and Ricci curvature in dimension 5_, Differential Geometry and its Applications
Volume 27, Issue 1, February 2009, Pages 85-99 ([doi:10.1016/j.difgeo.2008.06.008](https://doi.org/10.1016/j.difgeo.2008.06.008))

* {#CMD14} Beniamino Cappelletti-Montano, Giulia Dileo, _Nearly Sasakian geometry and $SU(2)$-structures_ ([arXiv:1410.0942](https://arxiv.org/abs/1410.0942))


SU(3)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[special unitary group]] in 3 complex [[dimensions]].

## Properties

### Subgroups and Supgroups

+-- {: .num_prop }
###### Proposition

We have the following [[commuting diagram]] of [[subgroup]] inclusions, where each [[square]] exhibits a [[pullback]]/[[fiber product]], hence an [[intersection]] of subgroups:

\begin{center}
\begin{xymatrix}
    \mathrm{SU}(2)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{SU}(3)
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{G}_2
    \ar@{^{(}->}[r]
    \ar@{^{(}->}[d]
    \ar@{}[dr]|-{
      \mbox{
        \tiny
        (pb)
      }
    }
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[d]
    \\
    \mathrm{Spin}(5)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(6)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(7)
    \ar@{^{(}->}[r]
    &
    \mathrm{Spin}(8)
\end{xymatrix}
\end{center}

Here in the bottom row we have the [[Lie groups]]

[[Spin(5)]] $\hookrightarrow$ [[Spin(6)]] $\hookrightarrow$ [[Spin(7)]] $\hookrightarrow$ [[Spin(8)]]

and in the top row we have

[[SU(2)]] $\hookrightarrow$ [[SU(3)]] $\hookrightarrow$ [[G2]] $\hookrightarrow$ [[Spin(7)]]

=--

This is a re-statement of
[Onishchik 93, Table 2, p. 144](#Onishchik93):

\begin{center}
\begin{imagefromfile}
  "file_name": "SO8SubgroupIntersection.jpg",
  "width": 620
\end{imagefromfile}
\end{center}

[[!include coset space structure on n-spheres -- table]]

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

* [[Calabi-Yau manifold]]

* [[U(1)]], [[SU(2)]], [[SU(4)]], [[SU(5)]]

## References

* {#Onishchik93} A. L. Onishchik (ed.) _Lie Groups and Lie Algebras_

  * *I.* A. L. Onishchik, E. B.  Vinberg, _Foundations of Lie Theory_,

  * *II.* V. V. Gorbatsevich, A. L. Onishchik, _Lie Transformation Groups_

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

* [[Howard Georgi]], §7 & §9  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]


SU(5)

#Contents#
* table of contents
{:toc}

## Idea

The [[special unitary group]] in [[complex numbers|complex]] [[dimension]] 5.

## Related concepts

* [[SU(2)]], [[SU(3)]], [[SU(4)]]

* [[GUT]]

## References

* [[Howard Georgi]], §18  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards [[grand unified theory]]


SU2-instantons from the correct maths to the traditional physics story

The traditional story in the [[physics]] textbooks (copied endlessly from one textbook to the next, over generations) of [[SU(2)]]-[[instantons]] ([[BPST instantons]]) tends to fail to highlight some key global points, without which the whole construction really collapses. The following text means to explain the correct description (using the mathematics of [[Cech cohomology]] cocycles via the [[clutching construction]] on the [[one-point compactification]] of [[Minkowski spacetime]]) but presented and phrased such that the [[folklore]] physics story becomes fully visible -- including its crucial fix.

$\,$

**Instantons**

For $G$ any [[gauge group]] (a [[Lie group]]), a $G$-[[Yang-Mills instanton]] on some 4-dimensional [[spacetime]] (a [[pseudo-Riemannian manifold]]) $X$ is a $G$-[[principal bundle]] on $X$ (whose class is going to be the "instanton sector"), equipped with a $G$-[[principal connection]] $\nabla$ (this is the actual [[gauge field]]) such that this is [[self-dual Yang-Mills theory|self-dual]], in that its [[curvature form]] $F_\nabla$ satisfies $F_\nabla = \star F_{\nabla}$, where $\star$ denotes the [[Hodge star]] operator for the given [[Riemannian metric|metric]] (which represents the field of [[gravity]]).

The [[ADHM construction]] gives an essentially unique self-dual [[principal connection]] $\nabla$ on every [[isomorphism class]] of $SU(2)$-[[principal bundles]] (ie.: "in every instanton sector"). Therefore classifying and counting instantons amounts to classifying and counting $G$-[[principal bundles]].

This is what makes instantons a "topological" structure in the parlance of physics, meaning that they do not depend on [[Riemannian metric]] information, after all.

We now take the [[spacetime]] to be [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ and the [[gauge group]] to be the [[special unitary group]] $SU(2)$. This is the case of "[[BPST-instantons]]". Of course other choices are possible and may lead to richer situations, but this simple case is what the physics textbooks tend to focus on (not the least, of course, because these choices are relevant for [[phenomenology]] of the [[weak nuclear force]] as seen in accelerator experiments) and it already serves to highlight key points.
Since, as we just said, we may disregard all metric properties, this means now that we regard spacetime to be just the [[Cartesian space]] $\mathbb{R}^4$.

Or almost. At this point one needs to be careful with the [[boundary conditions]] in order to get the topology right.

$\,$

**Vanishing at infinity**

The physical [[energy]] condition on an instanton is that the [[field strength]] $F_\nabla$ vanishes "at infinity". Mathematically, a [[continuous function]] on a [[locally compact topological space|locally compact]] [[topological space]] $X$ _[[vanishing at infinity|vanishes at infinity]]_ precisely if it extends to the "[[one-point compactification]]" $X^+ \coloneqq X \cup \{\infty\}$ of $X$ -- where we literally adjoin the "point at infinity" "$\infty$" and glue it to $X$ by defining a suitable [[topological space|topology]] on $X \cup \{\infty\}$.

Hence we may formalize the boundary condition by saying that our $SU(2)$-[[principal connection]] actually exists on the [[one-point compactification]] of $\mathbb{R}^4$. This is the [[4-sphere]]

$$
  S^4
  \simeq
  \mathbb{R}^4 \cup \{\infty\}
  \,.
$$

It is this passage from $\mathbb{R}^4$ to $S^4$ which takes care of the subtleties that often tend to be glossed over.

With the boundary conditions "at infinity" taken care of this way, an $SU(2)$-instanton now is the [[isomorphism class]] of an $SU(2)$-[[principal bundle]]  on the [[4-sphere]] $S^4$.

$\,$

**Instanton number**

There is a [[classifying space]] for $SU(2)$-[[principal bundles]], denoted $B SU(2)$. This being a [[classifying space]] means that for $X$ any [[paracompact topological space]], we have that the [[isomorphism classes]] of $SU(2)$-[[principal bundles]] on $X$ are in [[bijection]] with [[homotopy classes]] of [[continuous functions]] from $X$ to $B SU(2)$.

$$
  \{X \to B SU(2)\}_{/homotopy}
  \;\simeq\;
  \{ SU(2)\text{-principal bundles}\}_{/\text{gauge transformations}}
  \,.
$$

Now for $X = S^4$, it follows that

$$
   \{S^4 \to B SU(2)\}_{/homotopy}
   \simeq
   \pi_4(B SU(2))
   \simeq
   \mathbb{Z}
$$

is the 4th [[homotopy group]] of the [[classifying space]] $B SU(2)$.

This is an [[integer]], and so this integer labels [[isomorphism classes]] of instantons on $S^4$ (hence on $\mathbb{R}^4$). We see below that [[Chern-Weil theory]] identifies this number with the [[instanton number]].

But this counting of instantons works more generally, if we use a suitable counting function. First of all, there is a [[topological space]] whose _only_ non-trivial [[homotopy group]] is $\pi_4$, and such that this is the group of integers. This is the [[Eilenberg-MacLane space]] $K(\mathbb{Z},4)$:

$$
  \{S^4 \to K(\mathbb{Z},4)\}_{/homotopy}
  \simeq
  \mathbb{Z}
  \;\;\;\;\;
  \text{and}
  \;\;\;\;\;
  \{S^{d\neq 4} \to K(\mathbb{Z},4)\}_{/homotopy}
  \simeq
  0
  \,.
$$

This space has the following remarkable property: [[homotopy classes]] of [[continuous functions]] into it compute [[ordinary cohomology]] with [[integer]] [[coefficients]]:

$$
  \left\{
    X \longrightarrow K(\mathbb{Z},4)
  \right\}_{/homotopy}
  \;\simeq\;
  H^4(X,\mathbb{Z})
  \,.
$$

Now there is a [[continuous function]]

$$
  c_2 \;\colon\; B SU(2) \longrightarrow K(\mathbb{Z},4)
$$

called the _universal [[second Chern class]]_. This hence sends $SU(2)$-[[principal bundles]], classified by some map

$$
  \chi  \;\colon\; X \longrightarrow B SU(2)
$$

to classes in degree-4 cohomology

$$
  c_2(\chi)
  \;\colon\;
  X \longrightarrow B SU(2) \overset{c_2}{\longrightarrow} K(\mathbb{Z},4)
  \,.
$$

This [[cohomology class]]

$$
  c_2(\chi) \in H^4(X,\mathbb{Z})
$$

is hence called the [[second Chern class]] of the $SU(2)$-[[principal bundle]].

This is one in a whole sequence of _[[characteristic classes]]_ which are carried by $SU(2)$-[[principal bundles]], the _[[Chern classes]]_.

But in the special case that the base space $X$ is 4-[[dimension|dimensional]], we have that only a single one of these classes may be non-trivial, namely the [[second Chern class]] $c_2$. Therefore this class completely characterizes $SU(2)$-[[principal bundles]] in 4d.

In conclusion: _Where a [[BPST-instanton]] is manifested by an [[special unitary group|SU(2)]]-[[principal bundle]] on a 4-dimensional [[manifold]], the [[instanton number]] is the [[second Chern class]] of this bundle._

$\,$

**Constructing instantons from gauge transformations**

We may construct the bundles, that are classified this way, explicitly by using [[Cech cohomology]]. This says that we get such a bundle by

1. choosing an [[open cover]] $\{U_i \to X\}$ of $X$ (by "[[charts]]")

1. choosing [[transition functions]] $g_{i j} \colon U_i \cap U_j \to SU(2)$ on each double overlap of two charts

1. such that on triple overlaps the cocycle condition $g_{i j} \cdot g_{j k} = g_{i k}$ holds.

Now comes the major fact which makes this general theory look like the structures that appear in the physics books: the _[[clutching construction]]_.

Namely, for general $X$ one needs the charts $\{U_i \to X\}$ to form a [[good open cover]] in order to guarantee that all isomorphism classes of gauge bundles are captured by the construction via transition functions.

But the _[[clutching construction]]_ says that whenever $X$ happens to be a [[sphere]], then it is _sufficient_ to cover it by two [[hemispheres]] that overlap a little:

$$
  U_+ \coloneqq \text{upper hemisphere with a little rim}
$$

$$
  U_- \coloneqq \text{lower hemisphere with a little rim}
  \,.
$$

But since everything is topological now, it doesn't matter that these charts are literally _hemi_-spheres in the metric sense. In order to get the standard picture we instead make $U_+$ maximally large and take it to cover all of $S^4$ except the "north pole" (which is really the "point at infinity", due to the [[one-point compactification]] above), while we take $U_-$ to be a tiny [[open neighbourhood]] of that point, sitting there as a tiny ice cap around the north pole. So

$$
  U_+ \coloneqq S^4 - \{\text{point at infinity}\}
$$

$$
  U_- \coloneqq \text{tiny neighbourhood of the point at infinity}
  \,.
$$

Topologically this is [[homeomorphism|homeomorphic]] to the situation before, and hence just as good.

So now back to the general prescription of building [[principal bundles]] via [[Cech cohomology]], we are to choose transition functions on all overlaps of charts. But thanks to the [[clutching construction]], there is now just a single such overlap, namely

$$
  U_+ \cap U_- = \text{tiny annulus around the point at infinity}
  \,.
$$

Moreover, there is no non-trivial triple overlap, hence no cocycle condition that our transition function is to satisfy.

In conclusion, [[Cech cohomology]] and the [[clutching construction]] jointly now say that $SU(2)$-instantons are classified by maps

$$
  U_+ \cap U_- \longrightarrow SU(2)
$$

But notice here that the intersection of two hemi-4-spheres that overlap slightly is just a 3-sphere times a slight thickening:

$$
  U_+ \cap U_-
  \simeq
  S^3 \times \{-\epsilon, \epsilon\}
  \,.
$$

Moreover, the thickening direction here is trivial, and so one finds that instantons are classified by homotopy classes of maps

$$
  g \;\colon\; S^3 \longrightarrow SU(2)
$$

This is where this "gauge transformation at infinity" in the physics textbooks really comes from. The key point here is that it is indeed gauge transforming, namely the restriction of our bundle to $U_+$ is gauge transformed to its restriction to the "neighbourhood of infinity" $U_-$.

But, as topological spaces, $SU(2) \simeq S^3$ and so

$$
  \{ U_+ \cap U_- \to SU(2) \}_{/homotopy}
  \simeq
  \pi_3(S^3)
  \simeq
  \mathbb{Z}
  \,.
$$

This way we get the same classification from [[Cech cohomology]] that we got from [[classifying space]] theory, as it must be. But now we know how that [[second Chern class]] may be concretely embodied in the "gauge transformation at infinity".

$\,$

**Gauge fields vanishing at infinity**

Now we bring in connections. As discussed before, we may just as well consider any [[principal connection]]. In the [[Cech cohomology]] picture and still using the [[clutching construction]] this now means to choose

1. $\mathfrak{su}(2)$-valued 1-forms $A_{\pm}$ on $U_\pm$

1. such that on the tiny intersection of the two charts at infinity we have

   $$
     A_- = g A_* g^{-1} + g d g^{-1}
   $$

Now observe that in the given situation of using the [[clutching construction]] on a [[sphere]], we may always _choose_

$$
  A_- = 0
  \,.
$$

Because then the above just says that $A_+$ on $U_+$ becomes gauge trivial "at infinity", _by the given gauge transformation_ $g \colon S^3 \to SU(2)$ (the one whose winding number counts our instantons).

(In the present case this is really simple; but also generally there is a [[partition of unity]]-argument to construct connections on any bundle in generalization of this simple situation, see [here](connection+on+a+bundle#ExistenceOfConnections).)

$\,$

**Counting instantons by integrating $tr(F_\nabla \wedge F_\nabla)$**

So far we have derived the physics picture of an instanton: An $SU(2)$-gauge field which becomes gauge trivial "at infinity", witnessed by a gauge transformation on the "annulus at infinity" $S^3 \to SU(2)$, whose winding number is the instanton number. But the key point is that we see that the little neighbourhood of infinity $U_-$ is part of the picture, and that is necessary now to understand the Chern 4-form.

Namely to every $\mathfrak{su}(2)$-valued 1-form $A$ we may assign the ordinary (abelian)[[differential 4-form]]

$$
  \langle F_A \wedge F_A\rangle
  \coloneqq
  tr(F_A \wedge F_A)
  \,,
$$

where $F_A$ is the [[curvature form]] of $A$.

Now a general fact of [[Chern-Weil theory]] is that the 4-form built this way from a single $A$ is always exact, a potential is given by the [[Chern-Simons form]] $CS(A)$:

$$
  d CS(A) = \langle F_A \wedge F_A\rangle
  \,.
$$

But beware that this is only true on _a single chart_. And just because our chart $U_+$ covers "everything except infinity", we must not forget that there is a second chart $U_-$, the "neighbourhood of infinity".

Namely the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ that is defined on the whole of the 4-sphere $S^4$, this 4-form is only _locally exact_ (as every closed 4-form is, by the [[Poincare lemma]]). Generally, we define it chart-wise by

$$
  \langle F_{\nabla} \wedge F_\nabla \rangle |_{U_+}
   \coloneqq
  \langle F_{A_+} \wedge F_{A_+}\rangle
$$

and

$$
  \langle F_{\nabla} \wedge F_\nabla \rangle |_{U_-}
   \coloneqq
  \langle F_{A_-} \wedge F_{A_-}\rangle
  \,.
$$

This does indeed give a globally defined 4-form, no matter what the local connection forms $A_{\pm}$ are, as long as they satisfy the required condition that they are related by a [[gauge transformation]] on the overlap $U_+ \cap U_-$. Because the 4-form is gauge invariant.

So the 4-form thus defined

$$
  \langle F_{\nabla} \wedge F_{\nabla} \rangle
   \in
  \Omega^4_{cl}(S^4)
$$

has

1. a potential 3-form $CS(A_+)$ when restricted to $U_+$;

1. a potential 3-form $CS(A_-)$ when restricted to $U_-$;

but it does not have a potential 3-form on all of $S^4$, unless the instanton number vanishes.

Put this way this should be very obvious now. But it is easy to get confused about this, due to the sheer convenience of the [[clutching construction]] used above: we actually were allowed to choose $A_- = 0$!

This might make it superficially look like there is only a single local gauge potential $A_+$ around, and that the 4-form is locally exact. But this is not the case: there are two local gauge potentials on two different charts, and just because one of them happens to be equal to zero still does not mean that the extension of the 4-form to all of the 4-sphere has a global potential. It has a potential $CS(A_+)$ only after restricting it to $U_+$, even if this means "only removing the point at infinity".

With it thus being understood that $\langle F_\nabla \wedge F_\nabla\rangle$ is not globally exact, it becomes believable that the integral

$$
  \int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle
  \;\in\;
  \mathbb{N}
$$

is generally non-vanishing, and is in fact yet another incarnation of the same integer that we had before, the instanton number. That this is so is given to us by [[Chern-Weil theory]].

$\,$

**Outlook: Chern-Simons 2-Gerbes**

In fact the full story is nicer still. Namely the local [[Chern-Simons 3-forms]] $CS(A_\pm)$ together with the gauge transformation at infinity form a [[Cech cohomology]] cocycle for a [[circle 3-bundle with connection]] (a [[bundle 2-gerbe]]). This is the [[Chern-Simons 2-gerbe]] of the gauge field. And the fourth incarnation of the instanton number is: the [[Dixmier-Douady class]] of this 2-gerbe