quantaloid

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## Idea

The notion of _quantaloid_ is the [[horizontal categorification]] of that of [[quantale]]: a quantaloid is a quantale with many objects.

## Definition

A **quantaloid** is a category [[enriched category|enriched in]] the [[closed monoidal category|closed]] [[symmetric monoidal category]] of [[suplattices]]. Equivalently, it may be defined as a [[locally posetal 2-category]] in which each [[hom object|hom-]]poset is cocomplete, and which admits right adjoints to composing with an arrow on either side (making it a (bi)closed 2-category).

Taking the view that a quantaloid $Q$ is a closed 2-category, one can study categories enriched in $Q$. This can become particularly interesting if $Q$ is a **$*$-quantaloid**, i.e., comes equipped with an involution $*: Q \to Q$ which is the identity on 0-cells, reverses the direction of 1-cells, and preserves the direction of 2-cells. In that case one can study $*$-enriched categories over $Q$, i.e., $Q$-categories

$$(X, p: X \to Q_0, d: X \times X \to Q_1)$$

for which $d(y, x) = d(x, y)^*$. A famous example, due to R.F.C. Walters, is given below.

## Examples of quantaloids ##

Despite the exotic-sounding name (a portmanteau of [[quantale]] and [[groupoid]], i.e., a many-object quantale), quantaloids are quite commonplace.

* Let $E$ be a Grothendieck topos. Then the bicategory of relations in $E$, $Rel(E)$, is a quantaloid.

A toy form of this example: take any [[frame]] $H$, and consider the bicategory of spans in $H$: this is evidently a quantaloid, $Span(H)$.

Here is a particularly rich source of examples. Let $Q$ be a quantale. Then there is a quantaloid $Q$-$Mat$ of $Q$-valued matrices:

* Objects are sets $X$;

* Morphisms $X \to Y$ are functions $M: X \times Y \to Q$.

Matrix composition is performed according to the usual rule

$$(M N)(x, z) = \sum_{y: Y} M(x, y) \cdot N(y, z)$$

where $\cdot$ is the multiplication of $Q$ and the sum is the sup in $Q$. This class of examples easily internalizes in a topos, and in that sense it subsumes the first example, $Rel(E)$, by taking $Q = \Omega$ (as an internal frame, hence a quantale).

Slightly more generally, if $Q$ is a quantaloid, there is a quantaloid $Q$-$Mat$ of $Q$-matrices, as follows:

* Objects are sets $X$ together with functions $p: X \to Q_0$;

* Morphisms $(X, p) \to (Y, q)$ are matrices $M: X \times Y \to Q_1$ that satisfy the typing constraint $M(x, y): p(x) \to q(y)$.

Composition works exactly as before.

## Examples of $*$-quantaloids ##

The examples in the preceding section carry over straightforwardly:

* For $E$ a Grothendieck topos, $Rel(E)$ is a $*$-quantaloid where the $*$-operator takes a relation from $X$ to $Y$, i.e., a subobject $i: R \hookrightarrow X \times Y$, to its opposite obtained by composing $i$ with the symmetry isomorphism $X \times Y \cong Y \times X$.

Next, let $Q$ be a $*$-[[star-quantale|quantale]]. The quantaloid $Q$-$Mat$ is as before; this becomes a $*$-quantaloid by defining the transpose of a matrix as

* $M^*(x, y) = (M(y, x))^*$

This easily carries over to the case where we start with a $*$-quantaloid $Q$: we similarly obtain a $*$-quantaloid $Q$-$Mat$, by defining transpose as above.

## Connection with Q-valued sets ##

Let $Q$ again be a $*$-quantale; for example, $Q$ could be a frame or it could be a commutative quantale, taking the involution to be the identity.

A **$Q$-valued set** consists of a set $X$ and a morphism $M: X \to X$ in $Q$-$Mat$ which is

* Symmetric: $M^* = M$,

* Idempotent: $M \circ M = M$

In the case where $Q = \Omega$, the subobject classifier in a topos with its locale structure, symmetry and idempotence of a relation is equivalent to the _a priori_ weaker condition of symmetry and transitivity. The reason is that $Q$-$Mat = Rel(E)$ is an [[allegory]], where the modular law holds, one of whose consequences is the inequality $M \leq M M^* M$. As a result, we have

$$M \leq M M^* M \stackrel{sym}{=} M M M \stackrel{trans}{\leq} M M$$

which together with transitivity guarantees idempotence.

Thus, symmetric idempotents in $Rel(E)$ are what are known as [[partial equivalence relation]]s (which differ from equivalence relations by dropping reflexivity), or PERs for short.

To be continued...


quantic nucleus

## Idea

__Quantic nucleus__ is the concept of [[nucleus]]. adapted to the theory of [[quantale]]s.

## Literature

It seems that the notion is first studied in

* [[Susan B. Niefield]], Kimmo I. Rosenthal, _Constructing locales from quantales_, Mathematical Proceedings of the Cambridge Philosophical Society __104__:2 (1988) 215--234 [doi](https://doi.org/10.1017/S0305004100065403)

Textbook treatment

* [[Kimmo Rosenthal|K. I. Rosenthal]], _Quantales and their applications_, Longman 1990

category: algebra

[[!redirects quantic nuclei]]


quantification

# Quantifiers
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## Idea

A quantifier is an operation in [[logic]] that moves a statement from one [[context]] to a related context.

## Definitions

### Quantification in first-order logic

In [[first-order logic]], we have two basic quantifiers, each of which moves a [[predicate]] on one type to a [[proposition]] (a predicate on no types), or more generally moves a predicate on $n + 1$ types to a predicate on $n$ types.

Given a predicate $P$ on a type $T$, the __[[universal quantification]]__ of $P$, denoted $\forall\, x\colon T, P(x)$ (and with many variations in punctuation), is intended to be [[true]] if and only if $P(a)$ is true for every possible element $a$ of $T$.  Similarly, the __[[existential quantification]]__ of $P$ (also called its __particular quantification__), denoted $\exists\, x\colon T, P(x)$, is intended to be true if and only if $P(a)$ is true for at least one element $a$ of $T$.  However, it is quite possible that $P(a)$ may be provable (in a given context $\Gamma$) for every *term* $a$ of type $T$ that *can actually be constructed* (in $\Gamma$), yet $\forall\, x\colon T, P(x)$ cannot be proved; conversely, it is quite possible that $\exists\, x\colon T, P(x)$ can be proved (in a given context) but $P(a)$ cannot be proved for any term $a$ of type $T$ that can actually be constructed.

Therefore, we must define the quantifiers more carefully; one way to do this is as follows:

*  $Q \vdash_{\Gamma}\forall\, x\colon T, P(x)$ if and only if $Q \vdash_{\Gamma, x\colon T} P(x)$;
*  $\exists\, x\colon T, P(x) \vdash_{\Gamma} Q$ if and only if $P(x) \vdash_{\Gamma, x\colon T} Q$.

Here, $\Gamma$ is an arbitrary context, and $\Gamma, x\colon T$ is that context supplemented by a free variable $x$ of type $T$.  Note that $Q$ is (a priori) a statement in the context $\Gamma$ but also needs to make sense in the context $\Gamma, x\colon T$; therefore, our logic needs an appropriate [[weakening rule]] for this to make sense.  This definition also presumes that a statement (in any given context) can be identified precisely by placing it within the [[poset]] of statements (in that context) under entailment ($\vdash$), which is true for ordinary first-order logic.

From this definition, one can derive rules for the quantifiers in [[sequent calculus]] and [[natural deduction]].

### Quantification in first-order logic with equality

If all of our types have an [[equality predicate]], then we can construct some fancier quantifiers.

Given a predicate $P$ on a type $T$ with equality, the __[[uniqueness quantifier]]__ of $P$, denoted $\exists!\, x\colon T, P(x)$, may defined in terms of the universal and existential quantifiers as

$$ \exists!\, x\colon T, P(x) \;\equiv\; \exists\, x\colon T, P(x) \wedge \forall\, y\colon T, P(y) \Rightarrow (x = y) .$$

The intended interpretation is that $\exists!\, x\colon T, P(x)$ is true iff $P(a)$ is true for *exactly* one element $a$ of $T$.

Given a natural number $n$, we can similarly write down quantifiers to say that $P$ holds for exactly $n$ elements, at most $n$ elements, at least $n$ elements, etc. (These all require [[negation]] and therefore there are some slight variations possible when using [[intuitionistic logic]].)

With these examples, one can see quite readily that quantification is about saying how often a statement is true, that is *quantifying* it.

### Lawvere quantifiers: Quantification as adjunction
 {#LawvereQuantifier}

Given two contexts $\Gamma$ and $\Delta$ and an interpretation $f\colon \Gamma \to \Delta$, we obtain an operation $f^*\colon Prop_\Delta \to Prop_\Gamma$ from the propositions in $\Delta$ to the propositions in $\Gamma$.  If this operation has a [[right adjoint]], then this right adjoint is the __[[universal quantifier]]__ along $f$; if it has a [[left adjoint]], then this is the __[[existential quantifier]]__ along $f$.  The ordinary quantifiers in first-order logic are simply special cases of this where $f$ is given by weakening.

To bring this down to earth, let $S$ and $T$ be [[sets]] and let $f\colon S \to T$ be a [[function]].  We will think of each set as defining a context with one free variable for an element of that set; then the propositions in one of those contexts correspond to the [[subsets]] of the corresponding set.  In this way, we are looking at $f^*\colon \mathcal{P}T \to \mathcal{P}S$, the [[preimage]] map between [[power sets]] (often denoted $f^{-1}$).  Then the adjoints $\forall_f$ and $\exists_f$ are maps from $\mathcal{P}S \to \mathcal{P}T$ as follows:

*  $\forall_f\, A \coloneqq \{ y\colon T \;|\; \forall\, x\colon S,\; f(x) = y \;\Rightarrow\; x \in A \}$;

*  $\exists_f\, A \coloneqq \{ y\colon T \;|\; \exists\, x\colon S,\; f(x) = y \;\wedge\; x \in A \}$.

Note that $\exists_f A$ is simply the [[image]] of $f$ restricted to $A$.  Accordingly, one often denotes $\exists_f$ as $f_!$ (if not simply $f$).  When using this notation, one can also denote $\forall_f$ as $f_*$.

When $f$ is the unique function from a set $X$ to the terminal set, $\mathcal{P}1$ is the two-element set and an object in $\mathcal{P}X$ is a predicate on $X$.  The adjoints then map a predicate $Q$ to a truth value:

* $\exists x \in X$ such that $Q(x)$;

* $\forall x \in X, Q(x).$

### Guarded quantification: quantifying over subtypes {#Guarded}

Recall that, given a [[type]] $T$ and a [[predicate]] $Q$ on $T$, the __[[subtype]]__ of $T$ defined by $Q$ is, or would be, a type $S$ whose elements ([[terms]]) $x$ are thought of as elements ([[terms]]) of $T$ such that $Q(x)$ holds. But many [[type theories]] do not include sub-[[type formation]] among their rules.  However, we can mimic quantification over subtypes by using so-called 'guarded' quantifiers.

Specifically, given a type $T$, a predicate $Q$ on $T$, and a predicate $P$ on $T$, the __guarded quantifications__ of $P$ relative to $Q$ are these [[propositions]]:

*  $\underset{x\colon T}{\forall}
   \big( Q(x) \Rightarrow P(x) \big)$,

*  $\underset{x\colon T}{\exists}
   \big( Q(x) \wedge P(x) \big)$.

We have already had an example of these on this page, in the discussion of the Lawvere quantifiers along a function ([above](#LawvereQuantifier)).

When the notation for $Q$ makes the type $T$ obvious, and especially when the variable appears only at beginning in this notation, it is quite common to suppress mention of $T$ and write "$\forall Q(x), P(x)$" and "$\exists Q(x), P(x)$".  This is particularly common in untyped theories where the domain type "$T$" would be notationally suppressed anyway; for example, this is almost always how one writes guarded quantifiers in [[material set theory]].

### Bounded quantification: Size issues and predicativity

When [[type theory]] is used as a [[foundation of mathematics]], we have a freely generated [[universe]] of types (see also at _[[type of types]]_) and can also consider quantification over these. Similarly, when [[set theory]] is used as a foundation, we have a type of all sets and can consider quantification over this type. Such quantification is called __unbounded__ and often forbidden in the [[axiom of separation]] (if it can be stated in the language at all), especially in [[predicative mathematics]] but also in some impredicative foundations, such as [[ETCS]]. Even if we allow unbounded quantification, if we wish to refer to large objects using the language of [[proper classes]], this introduces the possibility of quantification over all classes, and this is forbidden even in [[ZFC]] and [[SEAR]] (although it is allowed in Morse--Kelley class theory).

In contrast, quantification over variables from *small* types (for a suitable notion of smallness) is called __bounded quantification__.

In the usual formulations of [[material set theory]], there is only one type (the type of sets), which is not small.  However, we can consider quantification over small subtypes using the trick of guarded quantification (above).  So the axiom of separation in so-called 'bounded' variations of ZFC allows only statements in which all quantifiers are guarded by a set.

In classical mathematics, at least, formulas with unbounded quantifiers can be classified into the [[Lévy hierarchy]], whose bottom level $\Delta_0$ consists of the formulas with only bounded quantifiers.

### Internalised quantifiers

... [[Heyting category]] ...

### Categorified quantifiers

... [[dependent product]] ... [[dependent sum]] ...

## Examples

* [[existential quantifier]], [[dependent sum]]

* [[universal quantifier]], [[dependent product]]

* [[uniqueness quantifier]]

## Related concepts
 {#RelatedConcepts}

* [[de dicto and de re]]

* [[Lévy hierarchy]]

* [[elimination of quantifiers]]

* [[generalized quantifier]]

## References

The identification of universal/existential quantification with [[adjoints]] of [[substitution]]/[[context extension]] ([[dependent product]]/[[dependent sum]]) originates around

* [[William Lawvere]], *Adjointness in Foundations*, Dialectica **23** (1969) 281-296,

  reprinted in: Reprints in Theory and Applications of Categories **16** (2006) 1-16 [[tac:tr16](https://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html), [pdf](https://www.emis.de/journals/TAC/reprints/articles/16/tr16.pdf)]

* [[William Lawvere]],  *Quantifiers and sheaves*, Actes Congrès intern. Math. **1** (1970) 329-334 [[pdf](https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1970-quantifiers-and-sheaves.pdf), [[Lawvere-QuantifiersAndSheaves.pdf:file]]]

A definition of guarded and bounded quantification for the membership relation could be found in

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

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quantity > history

< [[quantity]]


quantization

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## Idea

By _quantization_ is meant some process that

1. reads in a system of [[classical mechanics]], or rather of _[[prequantum field theory|prequantum]]_ data in the form of an [[Lagrangian]]/[[action functional]] and or a [[phase space]] equipped with [[symplectic manifold|symplectic structure]]

1. and returns a corresponding system of [[quantum mechanics]].

### Motivation from classical mechanics and Lie theory
 {#MotivationFromClassicalMechanicsAndLieTheory}

We indicate here a systematic motivation of quantization by looking at [[classical mechanics]] formalized as [[symplectic geometry]] from the point of view of [[Lie theory]] ([Fiorenza-Rogers-Schreiber 13](#FiorenzaRogersSchreiber13), [Nuiten 13](#Nuiten13)).

$\,$

Quantization of course was and is motivated by experiment, hence by observation of the [[observable universe]]: it just so happens that [[quantum mechanics]] and [[quantum field theory]] correctly account for experimental observations where [[classical mechanics]] and [[classical field theory]] gives no answer or incorrect answers. A historically important example is the phenomenon called the "[[ultraviolet catastrophe]]", a [[paradox]] predicted by classical [[statistical mechanics]] which is _not_ observed in nature, and which is corrected by [[quantum mechanics]].

But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from [[classical mechanics]] to [[quantum mechanics]]. Could one have been led to [[quantum mechanics]] by just pondering the mathematical formalism of [[classical mechanics]]? (Hence more precisely: is there a natural _[[schreiber:Synthetic Quantum Field Theory]]_?)

The following spells out an argument to this effect. It will work for readers with a background in modern [[mathematics]], notably in [[Lie theory]], and with an understanding of the formalization of classical/prequantum mechanics in terms of [[symplectic geometry]].

So to briefly recall, a system of [[classical mechanics]]/[[prequantum field theory|prequantum mechanics]] is a [[phase space]], formalized as a [[symplectic manifold]] $(X, \omega)$. A symplectic manifold is in particular a [[Poisson manifold]], which means that the [[algebra of functions]] on [[phase space]] $X$, hence the algebra of _classical [[observables]]_, is canonically equipped with a compatible [[Lie bracket]]: the _[[Poisson bracket]]_. This Lie bracket is what controls [[dynamics]] in [[classical mechanics]]. For instance if $H \in C^\infty(X)$ is the function on [[phase space]] which is interpreted as assigning to each configuration of the system its [[energy]] -- the [[Hamiltonian]] function -- then the [[Poisson bracket]] with $H$ yields the [[infinitesimal object|infinitesimal]] time evolution of the system: the [[differential equation]] famous as [[Hamilton's equations]].

Something to take notice of here is the _[[infinitesimal space|infinitesimal]]_ nature of the [[Poisson bracket]]. Generally, whenever one has a [[Lie algebra]] $\mathfrak{g}$, then it is to be regarded as the [[infinitesimal object|infinitesimal]] approximation to a globally defined object, the corresponding [[Lie group]] (or generally [[smooth group]]) $G$. One also says that $G$ is a _[[Lie integration]]_ of $\mathfrak{g}$ and that $\mathfrak{g}$ is the [[Lie differentiation]] of $G$.

Therefore a natural question to ask is: _Since the observables in [[classical mechanics]] form a [[Lie algebra]] under [[Poisson bracket]], what then is the corresponding [[Lie group]]?_

The answer to this is of course "well known" in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this [[Lie group]] which integrates the [[Poisson bracket]] is the "[[quantomorphism group]]", an object that seamlessly leads to the [[quantum mechanics]] of the system.

Before we spell this out in more detail, we need a brief technical aside: of course [[Lie integration]] is not quite unique. There may be different global [[Lie group]] objects with the same [[Lie algebra]].

The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian [[line Lie algebra]] $\mathbb{R}$. This has essentially two different [[Lie groups]] associated with it: the [[simply connected topological space|simply connected]] [[translation group]], which is just $\mathbb{R}$ itself again, equipped with its canonical additive [[abelian group]] structure, and the [[discrete space|discrete]] [[quotient]] of this by the group of [[integers]], which is the [[circle group]]

$$
  U(1) = \mathbb{R}/\mathbb{Z}
  \,.
$$

Notice that it is the discrete and hence "quantized" nature of the [[integers]] that makes the [[real line]] become a [[circle]] here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is "quantized" about [[quantum mechanics]].

Namely, one finds that the [[Poisson bracket]] [[Lie algebra]] $\mathfrak{poiss}(X,\omega)$ of the classical [[observables]] on [[phase space]] is (for $X$ a [[connected topological space|connected]] [[manifold]]) a [[Lie algebra extension]] of the Lie algebra $\mathfrak{ham}(X)$ of [[Hamiltonian vector fields]] on $X$ by the [[line Lie algebra]]:

$$
  \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X)
  \,.
$$

This means that under [[Lie integration]] the [[Poisson bracket]] turns into an [[central extension]] of the group of [[Hamiltonian symplectomorphisms]] of $(X,\omega)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting [[central extension]] by the [[circle group]] $U(1)$. For this non-trivial [[Lie integration]] to exist, $(X,\omega)$ needs to satisfy a quantization condition which says that it admits a [[prequantum line bundle]]. If so, then this $U(1)$-[[central extension]] of the group $Ham(X,\omega)$ of [[Hamiltonian symplectomorphisms]] exists and is called... the _[[quantomorphism group]]_ $QuantMorph(X,\omega)$:

$$
  U(1) \longrightarrow QuantMorph(X,\omega) \longrightarrow Ham(X,\omega)
  \,.
$$

While important, for some reason this group is not very well known, which is striking because it contains a small [[subgroup]] which is famous in [[quantum mechanics]]: the _[[Heisenberg group]]_.

More precisely, whenever $(X,\omega)$ itself has a [[Hamiltonian action|compatible]] [[group]] structure, notably if $(X,\omega)$ is just a [[symplectic vector space]] (regarded as a group under addition of vectors), then we may ask for the [[subgroup]] of the [[quantomorphism group]] which covers the (left) [[action]] of [[phase space]] $(X,\omega)$ on itself. This is the corresponding [[Heisenberg group]] $Heis(X,\omega)$, which in turn is a $U(1)$-[[central extension]] of the group $X$ itself:

$$
  U(1) \longrightarrow Heis(X,\omega) \longrightarrow X
  \,.
$$

At this point it is worth pausing for a second to note how the hallmark of [[quantum mechanics]] has appeared as if out of nowhere simply by applying [[Lie integration]] to the [[Lie algebra|Lie algebraic]] structures in [[classical mechanics]]:

if we think of [[Lie integration|Lie integrating]] $\mathbb{R}$ to the interesting [[circle group]] $U(1)$ instead of to the uninteresting [[translation group]] $\mathbb{R}$, then the name of its canonical [[basis]] element $1 \in \mathbb{R}$ is canonically "$i$", the imaginary unit. Therefore one often writes the above [[central extension]] instead as follows:

$$
  i \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X,\omega)
$$

in order to amplify this. But now consider the simple special case where $(X,\omega) = (\mathbb{R}^2, d p \wedge d q)$ is the 2-dimensional [[symplectic vector space]] which is for instance the [[phase space]] of the [[particle]] propagating on the line. Then a canonical set of generators for the corresponding [[Poisson bracket]] [[Lie algebra]] consists of the linear functions $p$ and $q$ of classical mechanics textbook fame, together with the _constant_ function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i \mathbb{R}$, hence purely Lie theoretically it is to be called "$i$".

With this notation then the [[Poisson bracket]], written in the form that makes its [[Lie integration]] manifest, indeed reads

$$
  [q,p] = i
  \,.
$$

Since the choice of [[basis]] element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing [[real number]] without changing this statement. If we write "$\hbar$" for this element, then the [[Poisson bracket]] instead reads

$$
  [q,p] = i \hbar
  \,.
$$

This is of course the hallmark equation for [[quantum physics]], if we interpret $\hbar$ here indeed as [[Planck's constant]]. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) [[Lie integration]] of the [[Poisson bracket]].

This is only the beginning of the story of quantization, naturally understood and indeed "derived" from applying [[Lie theory]] to [[classical mechanics]]. From here the story continues. It is called the story of _[[geometric quantization]]_. We close this motivation section here by some brief outlook.

The [[quantomorphism group]] which is the non-trivial [[Lie integration]] of the [[Poisson bracket]] is naturally constructed as follows: given the [[symplectic form]] $\omega$, it is natural to ask if it is the [[curvature]] 2-form of a $U(1)$-[[principal connection]] $\nabla$ on [[complex line bundle]] $L$ over $X$ (this is directly analogous to [[Dirac charge quantization]] when instead of a [[symplectic form]] on [[phase space]] we consider the  the [[field strength]] 2-form of [[electromagnetism]] on [[spacetime]]). If so, such a connection $(L, \nabla)$ is called a _[[prequantum line bundle]]_ of the [[phase space]] $(X,\omega)$. The [[quantomorphism group]] is simply the [[automorphism group]] of the [[prequantum line bundle]], covering [[diffeomorphisms]] of the phase space (the [[Hamiltonian symplectomorphisms]] mentioned above).

As such, the [[quantomorphism group]] naturally [[action|acts]] on the [[space of sections]] of $L$. Such a [[section]] is like a [[wavefunction]], except that it depends on all of [[phase space]], instead of just on the "[[canonical coordinates]]". For purely abstract mathematical reasons (which we won't discuss here, but see at _[[motivic quantization]]_ for more) it is indeed natural to choose a "[[polarization]]" of [[phase space]] into [[canonical coordinates]] and [[canonical momenta]] and consider only those [[sections]] of the [[prequantum line bundle]] which depend only on the former. These are the actual _[[wavefunctions]]_ of [[quantum mechanics]], hence the _[[quantum states]]_. And the [[subgroup]] of the [[quantomorphism group]] which preserves these polarized sections is the group of exponentiated [[quantum observables]]. For instance in the simple case mentioned before where $(X,\omega)$ is the 2-dimensional [[symplectic vector space]], this is the [[Heisenberg group]] with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.

### Quantization of field theory

An interesting subclass of [[quantum field theory|quantum field theories]], especially of [[perturbative quantum field theories]], is thought to arise from [[prequantum field theory]] via a process of _quantization_.   This process reads in certain -- usually [[differential geometry|differential geometric]] -- data ([[Lagrangian field theory]]), interprets this data as specifying the dynamics of some [[physical system]], and spits out the [[perturbative quantum field theory|perturbative]] [[quantum field theory]] that encodes the time evolution of this system.

For more on this see at _[[Wick algebra]]_ and _[[causal perturbation theory]]_.

Historically, it was an approximation to the true time evolution that was originally found and studied, by Newton, Maxwell, Einstein and others. This is now known as "[[classical physics]]". The true dynamics in turn is "[[quantum physics]]".

In view of this, quantization is often understood as a [[retraction|right inverse]] to the procedure that sends the full quantum dynamics to its [[classical limit]]. As such it is not well defined, i.e. unique, when it exists. Additional structures sometimes make it unique.

[[!include deformation quantization - table]]

There have been found some formalizations of at least some aspects of quantization, notably there is _[[algebraic deformation quantization]]_ and _[[geometric quantization]]_.

[[!include Isbell duality - table]]

These traditional formulations are geared towards [[mechanics]] (as opposed to [[field theory]]). Even in that restricted application, it seems that a more comprehensive understanding is still missing

> The goal is to get closer to a systematic theory of [[quantization]].

> ([Gukov-Witten 08, p.4](#GukovWitten08))

In the context of [[field theory]] the conceptual issues become even more severe. For example the class of [[field theories]] called _[[Yang-Mills theory]]_ is a core ingredient in the [[standard model of particle physics]] but the _[[quantization of Yang-Mills theory]]_ (see there for more) poses famous open problems -- of which of course that there isn't yet a comprehensive theory of what this even means is not the least.

## Quantization as an index map
 {#QuantizationAsAnIndexMap}

Here we survey aspects of quantization which give rise to an _[[index]]_ map or something similar, that is, a [[push-forward in generalized cohomology]]. More on this is in ([Nuiten 13](#Nuiten13)) and at _[[motivic quantization]]_.

The general pattern here is this:

* The [[kinetic action]] part of the [[action functional]] of a [[prequantum field theory]] is really part of the [[measure]]/[[orientation in generalized cohomology]] on the [[moduli space]] of [[field (physics)|fields]]

* The (gauge coupling-)[[interaction]] in the [[action functional]] of a [[prequantum field theory]] induces a [[cocycle]] in some [[cohomology]] which serves as a [[twisted cohomology|twist]] for the [[generalized cohomology theory]] in which the above [[integration]]/push-forward takes place.

* the [[push-forward in generalized cohomology|push-forward]] itself is the [[path integral quantization]], the construction of [[partition functions]].

### Path integral by integration against the Wiener measure

For instance the archetypical example of the quantization of the [[charged particle]] propagating on a [[Riemannian manifold]] $X$ proceeds (in [[Wick rotation|Wick rotated]] form as it appears in the [[worldline formalism]] for corresponding QFT) like so: write $[\exp(-S_{kin}(\gamma)) D\gamma]$ for the [[Wiener measure]] on paths in $X$, then the [[integral kernel]] of the time evolution propagator of the quantum particle is given by the [[path integral]] (see there for more)

$$
  \int_{\gamma \in P_{x_{in}, x_{out}}X} tra_{\nabla}(\gamma) \; [\exp(- S_{kin}(\gamma)) D\gamma]
$$

which is the [[integration]] of the [[parallel transport]]/[[holonomy]] functional (of the given [[connection on a bundle|connection]] that models the [[background gauge field]]) against the [[Wiener measure]].

### Quantization by partition function/index in supersymmetric quantum mechanics

More generally, for given any [[phase space]] equipped with a [[prequantum line bundle]] $(L_\omega, \nabla)$, the corresponding _[[geometric quantization]]_ is, if $X$ admits a compatible [[spin^c structure]], the [[index]] of the [[spin^c Dirac operator]] coupled to this bundle, hence the [[space of quantum states]] is

$$
  \mathcal{H} = index( D_{\nabla} )
  \,.
$$

For more on this see at _[geometric quantization -- As Index of the Spin^c Dirac operator](geometric%20quantization#AsIndexOfSpinCDiracOperator)_.

Here the [[Dirac operator]] represents a class in [[K-homology]]

$$
  [D] \in KK(X, \mathbb{C})
$$

and this is now the "[[measure]]" against which the [[K-theory]] class

$$
  [L_\omega] \in KK(\mathbb{C}, X)
$$

of the [[prequantum line bundle]] is being integrated, by the [[index]] map hence the [[composition]] in [[KK-theory]]:

$$
  [\mathcal{H}] \simeq
  (\mathbb{C} \stackrel{L_\omega}{\to} C(X) \stackrel{D}{\to} \mathbb{C} ) \in KK(\mathbb{C}, \mathbb{C})
  \,.
$$

By the discussion at _[[index]]_ this is equivalently the [[partition function]] of an auxiliary [[supersymmetric quantum mechanics]] system with [[background gauge field]] $\nabla$:

$$
  \mathcal{H} \simeq sTr( \exp(-t D^2) ) \;\;\;\; \forall t \gt 0
  \,.
$$

The appearance of an _auxiliary_ and already quantized field theory here may seem circular, but is in fact part of a deeper pattern of quantization by the [[holographic principle]], where rich quantum theories arise as [[boundary field theories]] of higher dimensional [[topological field theories]]. In the above case the higher dimensional theory is secretly the non-perturbative version of the [[Poisson sigma-model]] associated with the original [[phase space]] [[Poisson manifold]]. More details on this are indicated at _[[extended geometric quantization of 2d Chern-Simons theory]]_. In one dimension higher the [[Witten genus]] arises this way as the quantization of the [[heterotic string]] 2d QFT regarded as the [[boundary field theory]] of the [[M2-brane]] in [[Horava-Witten theory]]. (See ([Nuiten13](#Nuiten13)) for more.)

### Cohomological formulation in BV-formalism

The choice of [[spin^c structure]] in the above is really the choice of a [[Poincaré  duality]] (see at [[Poincaré duality algebra]] for more) which exhibits the [[orientation in generalized cohomology]] by a combined [[Atiyah duality]]/[[Thom isomorphism]].

The analogue of such a choice of [[Poincaré duality]] between [[cohomology]] and [[homology]] in [[BV-formalism]] is the choice of a [[volume form]] that identified the [[de Rham complex]] with the [[BV-complex]] of [[multivector fields]]. See at _[BV-BRST formalism -- Homological integration](BV-BRST+formalism#HomologicalIntegration)_.

Again, here the [[kinetic action functional]] is part of the [[measure]], now under the duality part of the [[BV-operator]]. The [[quantum master equation]] is now the analog of the orientability condition. Homological BV-quantization is then obtain by assing to the [[homology groups]] of this [[BV-operator]], hence again by "pushing it to the point".

This is the setup in which one can derive [[Feynman diagram]] rules form cohomologicalquantization, see at _[Feynman diagram -- Refereces -- In homological BV-quantization](Feynman+diagram#ReferencesInHomologicalBVQuantization)_.

## Examples

* [[quantum harmonic oscillator]]

* [[quantization of the 2-sphere]]

* [[quantization of Chern-Simons theory]]

* [[quantization of loop groups]]

## Related entries

* **quantization**

  * [[deformation quantization]]

    * [[Moyal quantization]]

  * [[geometric quantization]], [[higher geometric quantization]]

    * [[geometric quantization of symplectic groupoids]]

    * [[geometric quantization by push-forward]]

  * [[Weyl quantization]], [[Born-Jordan quantization]]

  * [[path integral quantization]]

  * [[stochastic quantization]]

  * [[semiclassical approximation]]

  * [[light-cone gauge quantization]]

  * [[quantization via the A-model]]

* [[second quantization]]

* [[fiber bundles in physics]]

* [[motivic quantization]]

* [[perturbation theory]]

  * [[formal deformation quantization]]

  * [[BV quantization]]

* overview: [[reductions deformations resolutions in physics]]

## References

* [[Nik Weaver]], *Mathematical Quantization*, Routledge (2011) [[ISBN 9781584880011](https://www.routledge.com/Mathematical-Quantization/Weaver/p/book/9781584880011)]

* S. Twareque Ali, Miroslav Engliš, _Quantization Methods: A Guide for Physicists and Analysts_, Rev. Math. Phys. **17** (2005) 391-490 [[arXiv:math-ph/0405065](http://arxiv.org/abs/math-ph/0405065)]

Specifically on [[geometric quantization]]:

* {#BatesWeinstein} Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, AMS (1997) [[pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)]

See also references at *[[quantum mechanics]]*.

A proposal for a full formalization of the notion of quantization for "finite" theories such as [[Dijkgraaf-Witten theory]] is in

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ ([arXiv:0905.0731](http://arxiv.org/abs/0905.0731))

A historical discussion by [[Zoran Škoda|one]] of the $n$labizants is here: [mathlight:quantization](http://mathlight.wordpress.com/2010/03/19/quantization). See also Urs's manifesto at [[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]].

* {#HenneauxTeitelboim} [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_, Princeton University Press 1992

* [[Nikolai Reshetikhin]], _Lectures on quantization of gauge systems_, [arxiv/1008.1411](http://arxiv.org/abs/1008.1411)

The [[quantization via the A-model]]-method is described in

* {#GukovWitten08} [[Sergey Gukov]], [[Edward Witten]], _Branes and quantization_, ([arxiv/0809.0305](http://arxiv.org/abs/0809.0305))

The perspective of [[geometric quantization|geometric]] pre-quantization as a canonical construction in higher [[Lie theory]] is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))
  {#FiorenzaRogersSchreiber13}

A discussion of quantization as cohomological/[[motivic quantization]] of [[higher prequantum field theory]] is in

* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local boundary prequantum field theory]]_ Master thesis, Utrecht 2013
 {#Nuiten13}

Some discussion of quantization in terms of [[Bohr toposes]] is in

* Kunji Nakayama, _Sheaves in Quantum Topos Induced by Quantization_ ([arXiv:1109.1192](http://arxiv.org/abs/1109.1192))

[[!redirects quantizations]]

[[!redirects quantisation]]
[[!redirects quantisations]]


Quantization as a Kan Extension

> This page used to point to some notes on the idea of formalizing [[quantization]] (the supposed process that takes [[prequantum field theory]]/[[classical mechanics|classical field theory]] to [[quantum field theory]]) by essentially realising the [[path integral]] as a [[Kan extension]].

> The content that used to be here has been superseded. Discussion of quantization in terms of Kan extension and [[pushforward in generalized cohomology]] is now at _[[motivic quantization]]_, with notes being at _[[schreiber:master thesis Nuiten]]_ and _[[schreiber:Quantization via Linear homotopy types]]_ .


quantization of 3d Chern-Simons theory

{:bluebox: .un_remark style="border:solid #000000;background: #E6DF13;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

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#### Physics
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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

This entry discusses the full ([[non-perturbative quantum field theory|non-perturbative]]) [[quantization]] of the [[prequantum field theory|prequantum]] data of standard 3d [[Chern-Simons theory]] (induced from a suitable [[Lie group]] and [[invariant polynomial]]/[[second Chern class]] [[action functional]]) to a [[3d TQFT]].

(For the [[perturbative QFT|perturbative quantization]] of [[Chern-Simons theory]] see [there](Chern-Simons+theory#PerturbativeQuantization)).

Existing literature knows three sectors of this problem, which overlap but do not coincide

1. [[path integral quantization]]. This may be made precise sense of in [[perturbation theory]] where it involves lots of interesting structure such as [[analytic torsion]] ([Witten 89](Chern-Simons#theory#Witten89)). However, being just [[perturbation theory]] it is just an approximation to the full answer.

1. [[geometric quantization]] yields the full (non-perturbative quantization) in [[codimension]] 1, but does not say anything about codimension 0.

1. The [[Reshetikhin-Turaev construction]] produces a [[3d TQFT]] from algebraic data that is naturally associated with the prequantum data defining Chern-Simons theory (such as the category of [[positive energy representations]] of the [[loop group]] of the given [[gauge group]] $G$, or else of a [[quantum group]] [Sawin 06](#Sawin06)), but it is not a priori clear that this 3d quantum field theory is genuinely the result of quantizing the Chern-Simons [[action functional]].

The known relation between the second and the third point here is the following:

That the complex-geometric [[modular functor]] obtained from [[geometric quantization]] of Chern-Simons theory as in ([Axelrod-Pietra-Witten 91](#AxelrodPietraWitten91), [Hitchin 90](#Hitchin90)) coincides with that of [[conformal blocks]] of the [[WZW model]] was shown in ([Laszlo 98](#Laszlo98),see also [Andersen 11](#Andersen11), [Andersen 12](#Andersen12)).
That this in turn indeed satisfies the required [[sewing law]] (and hence really is a modular functor in the strong sense) was shown in ([Tsuchiya-Ueno-Yamada](#TsuchiyaUenoYamada)). By [deprojectivization](modular%20functor#TopologicalLift) these constructions yield a topological modular functor of the form also obtained from the [[Reshetikhin-Turaev construction]].

These (topological) modular functors are fixed by their [[genus]]-0 data ([Andersen-Ueno 06](#AndersenUeno06)) which is equivalently the datum of a (weakly) [[modular tensor category]]. Hence for matching [[geometric quantization]] of 3d Chern-Simons theory to the [[Reshetikhin-Turaev construction]] one has to match the [[modular tensor categories]] obtained from the [[conformal blocks]] of the [[WZW model]] in genus-0 to that associated with the coresponding [[quantum groups]]. This works ([Ostrik 14](#Ostrik14)).

## Something more

> random notes, needs to be brought into shape

### Space of Chern-Simons quantum states

Chern-Simons [[action functional]]

$$
  \exp(\tfrac{i}{\hbar}S_{CS}) \colon \mathbf{Fields}_{CS} \longrightarrow \mathbf{B}^{n+1} U(1)_{conn}
$$

given a [[closed manifold]] $\Sigma_n$ then choice of [[complex structure]] $\mathbf{\Sigma}_n$ on $\Sigma_n$ is supposed to naturally induce a [[complex structure]] on the space of (on-shell) fields over $\Sigma_n$

$$
  \mathbf{Fields}_{CS}(\mathbf{\Sigma}_n)
$$

Moreover, [[transgression]] of $\exp(\tfrac{i}{\hbar}S)$ to $\mathbf{Fields}_{CS}(\Sigma)$ is supposed to yield a [[holomorphic line bundle]] with connection with respect to that complex structure

$$
  \exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n))
  \colon
  \mathbf{Fields}_{CS}(\mathbf{\Sigma}_n)
  \longrightarrow
  \mathbf{B}U(1)_{conn}
  \,.
$$

This is the [[prequantum line bundle]] of the Chern-Simons theory, already equipped with a [[Kähler polarization]].

Accordingly, the [[geometric quantization]] of the CS action functional assigns to $\Sigma_n$ the [[Hilbert space]] $\mathcal{H}_{\mathbf{\Sigma}}$ of holomorphic sections of $\exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n))$.

As the complex structure $\mathbf{\Sigma}_n$ on $\Sigma_n$ varies over the [[moduli stack of complex structures]] $\mathcal{M}_{\Sigma}$, these vector spaces $\mathcal{H}_{\mathbf{\Sigma}_n}$ form a [[vector bundle]] with projective flat connection (the _[[Hitchin connection]]_) on the moduli stack

$$
  \array{
    \mathcal{H}
    \\
    \downarrow
    \\
    \mathcal{M}_{\Sigma}
  }
$$

The assignment

$$
  \Sigma_n \mapsto \mathcal{H}_{\Sigma}
$$

natural in [[diffeomorphisms]] of $\Sigma$ is called the _[[modular functor]]_, this we focus on more [below](#ModularFunctorAndEquivariantEllipticCohomology)

One such section $\Psi$ is to be singled out. For instance if $\exp(\tfrac{i}{\hbar}S(\Sigma_n))$ is a [[theta characteristic]] then there is up to scale a unique holomorphic section. This singling-out is formalized by the [[FRS-formalism]]. See there for more.

### Space of WZW-model pre-correlators

Under the [[AdS3-CFT2 and CS-WZW correspondence]] the states of Chern-Simons theory also correspond to [[partition functions]] of the [[gauged WZW model]] and hence to [[generating functions]] for [[correlation functions]] of the actual [[WZW model]].

The [[field (physics)|fields]] $\mathbf{Fields}(\Sigma_n)$ may also be thought of as the [[sources]] of a (higher) [[gauged WZW model]] on $\Sigma$.

The [[holographic principle]] says that the [[quantum state]]/[[wavefunction]]

$$
  \Psi \in \Gamma(\mathbf{Fields}_{CS}(\mathbf{\Sigma}_n), L)
$$

is also the [[generating function]] for the [[correlators]] of the [[WZW model]] on $\Sigma_n$, meaning that its [[functional derivatives]]

+-- {: bluebox}

$$
  \frac{\delta}{\delta A^{a_1}(z_1)}
  \cdots
  \frac{\delta}{\delta A^{a_n}(z_n)}
  \Psi
  =
  \left\langle J^{a_1}(z_1) \cdots J^{a_n}(z_n) \right\rangle
$$

=--

with respect to the Chern-Simons-[[field (physics)|fields]], hence the WZW [[sources]],  are the [[n-point functions]] of the WZW model for [[current algebra]] insertions, as indicated

(e.g. [Gawędzki 99 (4.23), 5.1](#Gawedzki99))

### Modular functor and equivariant elliptic cohomology
 {#ModularFunctorAndEquivariantEllipticCohomology}

Discussion of [[equivariant elliptic cohomology]] (see there at _[interpretation in QFT](equivariant+elliptic+cohomology#InterpretationInQuantumFieldTheory)_) shows that the construction of the [[modular functor]] refines from equipping $\Sigma$ with [[complex structure]] to equipping it with [[arithmetic geometry|arithmetic structure]]. Hence for $\Sigma$ a [[torus]] it refines from structures of [[elliptic curves]] over the complex numbers to general arithmetic elliptic curves (over the [[integers]]) and in fact to [[derived elliptic curves]] (over the [[sphere spectrum]]). Hence eventually the theory of [[geometric quantization]] needs to be refined to admit [[polarizations]] in [[arithmetic geometry]]. See at _[[differential cohesion and idelic structure]]_ for more on this.

## Related concepts

* [[Donaldson-Uhlenbeck-Yau theorem]]

* [[Narasimhan-Seshadri theorem]]

* [[Harder-Narasimhan theorem]]

* [[equivariant elliptic cohomology]]

## References
 {#References}

### Non-perturbative geometric quantization
  {#ReferencesNonPerturbative}

Discussion of [[non-perturbative QFT|non-perturbative]] [[geometric quantization]] of [[Chern-Simons theory]]:

Basics are recalled for instance in

* Fernando Falceto, [[Krzysztof Gawedzki]], _Chern-Simons States at Genus One_, Commun.Math.Phys. 159 (1994) 549-580 ([arXiv:hep-th/9211003](http://arxiv.org/abs/hep-th/9211003))

* {#Gawedzki99} [[Krzysztof Gawedzki]], *Conformal field theory: a case study*, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) *Conformal Field Theory -- New Non-perturbative Methods In String And Field Theory*, CRC Press (2000) [[arXiv:hep-th/9904145](https://arxiv.org/abs/hep-th/9904145), [doi:10.1201/9780429502873](https://doi.org/10.1201/9780429502873)]

* Yasuhiro Abe, _Application of abelian holonomy formalism to the elementary theory of numbers_ ([arXiv:1005.4299](http://arxiv.org/abs/1005.4299))

The [[geometric quantization]] of 3d CS theory in codimension 1 is due to

* {#AxelrodPietraWitten91} [[Scott Axelrod]], S. Della Pietra, [[Edward Witten]], _Geometric quantization of Chern-Simons gauge theory_, Jour. Diff. Geom. 33 (1991), 787-902.  ([EUCLID](http://projecteuclid.org/euclid.jdg/1214446565))

* {#Hitchin90} [[Nigel Hitchin]], _Flat connections and geometric quantization_, Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841))

(see also at _[[Hitchin connection]]_).

The [[3d TQFT]] candidate for quantum CS theory in the form of the [[Reshetikhin-Turaev construction]] and the corresponding [[modular tensor category]] data is discussed in

* {#ReshetikhinTuraev91} [[Nicolai Reshetikhin]], [[Vladimir Turaev]], _Invariants of 3-manifolds via link polynomials and quantum groups_. Invent. Math. 103 (1991), no. 3, 547–597. ([pdf](http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf))

* {#BakalovKirillov} B. Bakalov & [[Alexandre Kirillov]], _Lectures on tensor categories and modular functors_ AMS, University Lecture Series, (2000) ([web](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html)).

* {#Sawin06} [[Stephen Sawin]], _Quantum groups at roots of unity and modularity_ J. Knot Theory Ramifications 15 (2006), no. 10, 1245–1277 ([arXiv:0308281](http://arxiv.org/abs/math/0308281))

* {#Ostrik14} [[Victor Ostrik]], _[MO comment August 2014](http://mathoverflow.net/a/178304/381)_

The relation between the [[modular functor]] obtained from the [[conformal blocks]] of the [[WZW model]] and from geometric quantization of CS theory is discussed in

* {#Laszlo98} [[Yves Laszlo]], _Hitchin's and WZW connection are the same_, J. Differential Geom. 49 (1998), no. 3, 547–576 ([pdf](http://www.emis.de/journals/NYJM/JDG/archive/vol.49/3_5.pdf))

* {#TsuchiyaUenoYamada} Tsuchiya; K Ueno; Yamada, _Conformal field theory on universal family of stable curves with gauge symmetries_, Integrable systems in quantum field theory and statistical mechanics, 459–566.

* {#AndersenUeno06} [[Jørgen Andersen]], K. Ueno, _Modular functors are determined by their genus zero data_, Journal of Quantum Topology ([arXiv:math/0611087](http://arxiv.org/abs/math/0611087))

Discussion specific to [[special unitary group|special unitary]] [[gauge group]] is in

* [[Jørgen Andersen]], K. Ueno, _Abelian Conformal Field theories and Determinant Bundles_, International Journal of Mathematics, 18 919 - 993 (2007).

* [[Jørgen Andersen]], K. Ueno, _Geometric Construction of Modular Functors from Conformal Field Theory_, Journal of Knot theory and its Ramifications, 16 127 -- 202, (2007).

* {#Andersen11} [[Jørgen Andersen]], K. Ueno, _Construction of the Reshetikhin-Turaev TQFT via Conformal Field Theory_ ([arXiv:1110.5027](http://arxiv.org/abs/1110.5027))

* {#Andersen12} [[Jørgen Andersen]], _A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications_ ([arXiv:1206.2785](http://arxiv.org/abs/1206.2785))

For discussion of the state of the proof see also

* {#MODiscussion} MO, _[Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?](http://mathoverflow.net/questions/86792/why-hasnt-anyone-proved-that-the-two-standard-approaches-to-quantizing-chern-si)_

and in particular [this reply](http://mathoverflow.net/a/185401/381) by [[Andre Henriques]].

Detailed review in the case of abelian Chern-Simons theory includes

* Spencer D. Stirling, _Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories_ ([arXiv:http://arxiv.org/abs/0807.2857v1](http://arxiv.org/abs/0807.2857))

Discussion in terms of [[Weyl quantization]] of [[Wilson lines]] and details on the role of [[theta functions]] is in

Discussion of [[quantization of Chern-Simons theory]] in terms of [[Weyl quantization]] and [[skein relations]] is in

* [[Jørgen Andersen]], _Deformation quantization and geometric quantization of abelian moduli spaces_, Commun. Math. Phys., 255 (2005), 727–745

* {#GelcaUribe02} [[Razvan Gelca]], [[Alejandro Uribe]], _The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same_, 	Commun.Math.Phys. 233 (2003) 493-512 ([arXiv:math-ph/0201059](http://arxiv.org/abs/math-ph/0201059))

* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))

Another approach is

* [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], [[Jacob Lurie]], _[TQFT from compact Lie groups -- 3d Chern-Simons as a fully extended TQFT](http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups#3dCSFullyExtended)_.

[[!include perturbative quantization of Chern-Simons theory -- references]]

[[!redirects quantization of Chern-Simons theory]]

[[!redirects perturbative quantization of 3d Chern-Simons theory]]

[[!redirects quantum Chern-Simons theory]]


Quantization of Gauge Systems

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#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

This entry is about the textbook:

* [[Marc Henneaux]], [[Claudio Teitelboim]]:

  \linebreak

  **Quantization of Gauge Systems**

  \linebreak

  Princeton University Press (1992)

  [ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems)

  [doi:10.2307/j.ctv10crg0r](https://doi.org/10.2307/j.ctv10crg0r)

  [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)

on the [[BRST-BV formalism]] for describing [[gauge theories]] in the generality of [[constrained mechanics|constrained]] [[Hamiltonian mechanics]] of [[field theories]] deriving from a [[Lagrangian density]].

#Contents#
* table of contents
{:toc}

## 1) Constrained Hamiltonian systems

* [[constrained mechanics]]

* [[first-class constraint]]

## 2) Geometry of the constraint surface

## 3) Gauge invariance of the action

* [[gauge group]]

* [[gauge invariance]]

## 4) Generally covariant systems

* [[general covariance]]

## 5) First-class constraints: further developments

* [[first-class constraint]]

## 6) Fermi degrees of freedom: classical mechanics over a Grassmann algebra

## 7) Constrained systems with fermi variables

## 8) Graded differential algebra -- algebraic structure of the BRST symmetry

* [[graded differential algebra]]

* [[BRST complex]]

## 9) BRST construction in the irreducible case

## 10) BRST construction in the reducible case

## 11) Dynamics of the ghosts -- gauge-fixed action

* [[ghost field]]

## 12) The BRST transformation in field theory

## 13) Quantum mechanics of constrained systems: standard operator methods

* [[quantum mechanics]]

* [[operator algebra]]

## 14) BRST operator method -- quantum BRST cohomology

## 15) Path integral for unconstrained systems

* [[path integral]]

### 15.1 Path Integral Method of Bose Systems -- Basic Feature

#### 15.1.4 Equations of motion -- Schwinger-Dyson-Equation

* [[equation of motion]]

* [[Euler-Lagrange equation]]

* [[Schwinger-Dyson equation]]

### 15.5 A first bite at the antifield formalism

* [[antifields]]

#### 15.5.2 Antibracket

* [[antibracket]]

#### 15.5.3 Schwinger-Dyson operator

* [[BV-operator]], [[Schwinger-Dyson operator]]

## 16) Path integral for constrained systems

## 17) Antifield formalism: classical theory

### 17.1) Covariant phase space

* [[covariant phase space]]

* [[Peierls bracket]]

### 17.2) Koszul-Tate resolution and longitudinal $d$

* [[Koszul-Tate resolution]]

### 17.3) BRST symmetry -- master equation

* [[master equation]]

### 17.4) Gauge invariance of the solution of the master equation

## 18) Antifield formalism and path integral

## 19) Free Maxwell theory, abelian two-form gauge field

* [[Maxwell theory]]

* [[B-field]]

## 20) Complementary material

category: reference


quantization of loop groups

#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[compact Lie group]] and write $L G$ for its [[loop group]]. See there for details and notation.

We discuss the [[quantization]] of loop groups in the sense of [[geometric quantization]] of their canonical [[prequantum bundle]].

Let $G$ be a [[compact Lie group]].
Let $T \hookrightarrow G$ be the inclusion of a [[maximal torus]]. There is a [[fiber sequence]]

$$
  \array{
    G/T &\to& L G / T
    \\
    && \downarrow
    \\
    && L G / G & \simeq \Omega G
  }
  \,.
$$

+-- {: .num_remark }
###### Remark

By the discussion at _[[orbit method]]_, if $G$ is a [[semisimple Lie group]], then $G/T$ is isomorphic to the [[coadjoint orbit]] of an element $\langle \lambda , -\rangle \in \mathfrak{g}^*$ for which $T \simeq G_\lambda$ is the [[stabilizer subgroup]].

If moreover $G$ is [[simply connected topological space|simply connected]], then the [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]]

$$
  \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1))
  \,.
$$

=--

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

The [[irreducible representation|irreducible]] [[projective representation|projective]] [[positive energy representations]] of $L G$ correspond precisley to the possible [[geometric quantizations]] of $L G / T$ (as in the [[orbit method]]).

More in detail:

The degree-2 [[integral cohomology]] of $L G / T$ is

$$
  H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T
  \,.
$$

Writing $L_{n,\lambda}$ for the corresponding [[complex line bundle]] with level $n \in \mathbb{Z}$ and weight $\lambda \in \hat T$ we have that

1. the space of [[holomorphic sections]] of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;

1. every such arises this way;

1. and is non-zero precisely if $(n,\lambda)$ is positive  in the sense that for each positive [[coroot]] $h_\alpha$ of $G$

   $$
     0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle
    \,.
   $$

=--

This appears for instance as ([Segal, prop. 4.2](#Segal)).

## References

The standard textbook on loop groups is

* Andrew Pressley, [[Graeme Segal]], _Loop groups_ Oxford University Press (1988)
  {#PressleySegal}

A review talk is

* [[Graeme Segal]], _Loop groups_ ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter08.pdf))
 {#Segal}


quantization of M2-brane to matrix model -- references

### Quantization of the M2-brane sigma-model to a matrix model
 {#LightConeQuantizationOfM2BranesToBFSSMatrixModel}

The [[Poisson bracket]]-formulation of the [[classical mechanics|classical]] [[light-cone gauge]] [[Hamiltonian]] for the _[[bosonic modality|bosonic]]_ [[relativistic membrane]] and the corresponding matrix [[commutator]] regularization is due to:

* {#Hoppe82} [[Jens Hoppe]], _Quantum theory of a massless relativistic surface and a two-dimensional bound state problem_, MIT 1982 ([dspace:1721.1/15717](http://hdl.handle.net/1721.1/15717), [[HoppeMembrane82.pdf:file]])

Some exact solutions:

* [[Jens Hoppe]], *Exact algebraic M(em)brane solutions* $[$[arXiv:2107.00569](https://arxiv.org/abs/2107.00569)$]$

* [[Jens Hoppe]], *The fast non-commutative sharp drop* $[$[arXiv:2302.13146](https://arxiv.org/abs/2302.13146)$]$

On the regularized [[light-cone gauge quantization]] of the [[Green-Schwarz sigma model]] for the [[M2-brane]] on ([[super Minkowski spacetime|super]]) [[Minkowski spacetime]], yielding the [[BFSS matrix model]]:

Original articles:

* {#deWitHoppeNicolai88} [[Bernard de Wit]], [[Jens Hoppe]], [[Hermann Nicolai]], _On the Quantum Mechanics of Supermembranes_, Nucl. Phys. B305 (1988) 545. ([pdf](http://pubman.mpdl.mpg.de/pubman/item/escidoc:153408:1/component/escidoc:153407/353961.pdf), [[deWitHoppeNicolai88.pdf:file]], [spire:261702](http://inspirehep.net/record/261702))

Observation that the spectrum is continuous:

* [[Bernard de Wit]], W. Lüscher, [[Hermann Nicolai]], _The supermembrane is unstable_, Nucl. Phys. B320 (1989) 135 ([spire:266584](http://inspirehep.net/record/266584/), doi:10.1016/0550-3213(89)90214-9)

Review:

* {#NicolaiHelling98} [[Hermann Nicolai]], [[Robert C. Helling]], _Supermembranes and M(atrix) Theory_, In _Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry_ 29-74 ([arXiv:hep-th/9809103](http://arxiv.org/abs/hep-th/9809103), [spire:476366](http://inspirehep.net/record/476366))

* [[Jens Hoppe]], _Membranes and Matrix Models_ ([arXiv:hep-th/0206192](https://arxiv.org/abs/hep-th/0206192))

* {#DasguptaNicolaiPlefka02} Arundhati Dasgupta, [[Hermann Nicolai]], [[Jan Plefka]], _An Introduction to the Quantum Supermembrane_, Grav. Cosmol. 8:1, 2002; Rev. Mex. Fis. 49S1:1-10, 2003 ([arXiv:hep-th/0201182](http://arxiv.org/abs/hep-th/0201182))

* [[Gijs van den Oord]], _On Matrix Regularisation of Supermembranes_, 2006 ([pdf](http://web.science.uu.nl/itf/Teaching/2006/vandenOord.pdf))

* [[Meer Ashwinkumar]], [[Lennart Schmidt]], [[Meng-Chwan Tan]], Section 2 of: _Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes_ ([arXiv:2103.06666](https://arxiv.org/abs/2103.06666))

The generalization to [[pp-wave spacetimes]] (leading to the [[BMN matrix model]]):

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], Section 2 of: _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], Section 2 of: _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

See also

* {#DuffInamiPopeSezginStelle88} [[Mike Duff]], T. Inami, [[Christopher Pope]], [[Ergin Sezgin]],  [[Kellogg Stelle]], _Semiclassical Quantization of the Supermembrane_, Nucl.Phys. B297 (1988) 515-538 ([spire:247064](http://inspirehep.net/record/247064))

* {#KabatTaylor97} Daniel Kabat, [[Washington Taylor]], section 2 of: _Spherical membranes in Matrix theory_, Adv. Theor. Math. Phys. 2: 181-206, 1998 ([arXiv:hep-th/9711078](https://arxiv.org/abs/hep-th/9711078))

* [[Nathan Berkovits]], _Towards Covariant Quantization of the Supermembrane_ ([arXiv:hep-th/0201151](http://arxiv.org/abs/hep-th/0201151))

* Qiang Jia, _On matrix description of D-branes_ ([arXiv:1907.00142](https://arxiv.org/abs/1907.00142))

A new kind of [[perturbation series]] for the quantized super-membrane:

* [[Olaf Lechtenfeld]], [[Hermann Nicolai]], *A perturbative expansion scheme for supermembrane and matrix theory* ([arXiv:2109.00346](https://arxiv.org/abs/2109.00346))

Relation to the string [[dilaton]] under [[double dimensional reduction]]:

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Fundamental Membranes and the String Dilaton*,  J. High Energ. Phys. **2022** 219 (2022) $[$[arXiv:2208.05822](https://arxiv.org/abs/2208.05822), doi:10.1007/JHEP09(2022)219$]$

[[!redirects quantization of M2-brane on Minkowski spacetime to BFSS matrix model -- references]]


quantization of the 2-sphere

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{:toc}

## Idea

The 2-[[sphere]] with its standard [[volume form]] is canonically a [[Poisson manifold]] and in fact a [[symplectic manifold]]. As such it may be regarded as a simple example of a [[phase space]] of a [[mechanical system]]. The [[quantization]] of this system yields the corresponding [[quantum mechanical system]]. This turns out to be that of the degrees of freedom of a [[rotor]] or [[spinor]] in 3-dimensional [[Euclidean space]] (just the "internal" rotor/spin degrees of freedom, no translational motion).

## Formalizations

* in [[geometric quantization]]: [[geometric quantization of the 2-sphere]]

* in [[deformation quantization]]: [[deformation quantization of the 2-sphere]]


quantization of Yang-Mills theory

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## Idea
 {#Idea}

Despite its fundamental role in the [[standard model of particle physics]], various details of the [[non-perturbative quantum field theory|non-perturbative]] [[quantization]] of the class of [[field theories]] known as [[Yang-Mills theory]] for [[non-abelian group|non-abelian]] [[gauge group]] (such as [[QCD]] but not [[QED]]) are still open, such as derivations of the phenomena

* [[quark]]  [[confinement]]

and notably

* the [[mass gap problem]], the question of why the [[bound states]] of massless [[quarks]] ([[hadrons]]) have [[mass]]

From [Jaffe-Witten](#JaffeWitten):

> By the 1950s, when Yang–Mills theory was discovered, it was already known that the [[quantum field theory|quantum version]] of [[electromagnetism|Maxwell theory]] -- known as [[quantum electrodynamics|Quantum Electrodynamics]] or QED -- gives an extremely accurate account of [[electromagnetic fields]] and [[Lorentz force|forces]]. In fact, QED improved the accuracy for certain earlier quantum theory predictions by several orders of magnitude, as well as predicting new splittings of energy levels.

> So it was natural to inquire whether [[non-abelian group|non-abelian]] [[gauge theory]] described other [[forces]] in [[observable universe|nature]], notably the [[weak nuclear force|weak force]] (responsible among other things for certain forms of radioactivity) and the [[strong nuclear force|strong or nuclear force]] (responsible among other things for the binding of [[protons]] and [[neutrons]] into nuclei). The massless nature of [[classical field theory|classical]] Yang–Mills waves was a serious obstacle to applying Yang–Mills theory to the other forces, for the [[weak nuclear force|weak]] and [[strong nuclear force|nuclear forces]] are short range and many of the particles are massive. Hence these phenomena did not appear to be associated with long-range fields describing massless particles.

> In the 1960s and 1970s, physicists overcame these obstacles to the physical interpretation of non-abelian gauge theory. In the case of the weak force, this was accomplished by the Glashow–Salam–Weinberg [[electroweak theory]]  with gauge group $H = $ [[special unitary group|SU(2)]] $\times$ [[circle group|U(1)]]. By elaborating the theory with an additional "[[Higgs field]]", one avoided the massless nature of [[classical field theory|classical]] Yang–Mills waves. The [[Higgs field]] transforms in a two-dimensional [[representation]] of $H$; its non-zero and approximately constant value in the [[vacuum state]] reduces the [[structure group]] from $H$ to a $U(1)$ [[subgroup]] ([[diagonal|diagonally]] embedded in $SU(2) \times U(1)$. This [[theory (physics)|theory]] describes both the [[electromagnetism|electromagnetic]] and [[weak nuclear force|weak forces]], in a more or less unified way; because of the reduction of the structure group to $U(1)$, the long-range fields are those of [[electromagnetism]] only, in accord with what we see in [[observable universe|nature]].

> The solution to the problem of massless Yang–Mills fields for the [[strong nuclear force|strong interactions]] has a completely different nature.  That solution did not come from adding fields to Yang–Mills theory, but by discovering a remarkable property of the quantum Yang–Mills theory itself, that is, of the quantum theory whose [[classical field theory|classical]] [[Lagrangian density|Lagrangian]] has been given $[$above$]$. This property is called "[[asymptotic freedom]]". Roughly this means that at short distances the field displays [[quantum physics|quantum]] behavior very similar to its classical behavior; yet at long distances the [[classical field theory|classical theory]] is no longer a good guide to the quantum behavior of the [[field (physics)|field]].

> [[asymptotic freedom|Asymptotic freedom]], together with other experimental and theoretical discoveries made in the 1960s and 1970s, made it possible to describe the [[strong nuclear force|nuclear force]] by a [[non-abelian group|non-abelian]] [[gauge theory]] in which the [[gauge group]] is $G = $ [[special unitary group|SU(3)]]. The additional [[field (physics)|fields]] describe, at the [[classical field theory|classical level]], "[[quarks]]," which are [[spinor|spin 1/2 objects]] somewhat analogous to the [[electron]], but transforming in the [[fundamental representation]] of $SU(3)$.  The non-abelian gauge theory of the strong force is called [[quantum chromodynamics|Quantum Chromodynamics]] (QCD).

> The use of [[QCD]] to describe the [[strong nuclear force|strong force]] was motivated by a whole series of experimental and theoretical discoveries made in the 1960s and 1970s, involving the [[symmetries]] and high-energy behavior of the strong interactions. But [[classical field theory|classical]] [[non-abelian group|non-abelian]] [[gauge theory]] is very different from the [[observable universe|observed world]] of [[strong nuclear force|strong interactions]]; for [[QCD]] to describe the strong force successfully, it must have at the quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory:

> (1) It must have a "[[mass gap]];" namely there must be some constant $\Delta \gt 0$ such that every excitation of the [[vacuum]] has [[energy]] at least $\Delta$.

> (2) It must have "[[quark]] [[confinement]]," that is, even though the theory is described in terms of elementary [[field (physics)|fields]], such as the quark fields, that transform non-trivially under [[special unitary group|SU(3)]], the physical particle states -- such as the [[proton]], [[neutron]], and [[pion]] --are [[special unitary group|SU(3)]]-[[invariant]].

> (3) It must have "[[chiral symmetry breaking]]," which means that the vacuum is potentially invariant (in the [[limit of a sequence|limit]], that the quark-bare masses vanish) only under a certain [[subgroup]] of the full [[symmetry group]] that [[action|acts]] on the [[quark]] [[field (physics)|fields]].

> The first point is necessary to explain why the [[strong nuclear force|nuclear force]] is strong but short-ranged; the second is needed to explain why we never see individual [[quarks]]; and the third is needed to account for the "[[current algebra]]" theory of soft [[pions]] that was developed in the 1960s.

> Both [[experiment]] -- since [[QCD]] has numerous successes in confrontation with experiment -- and [[lattice gauge theory|computer simulations]], carried out since the late 1970s, have given strong encouragement that QCD does have the properties cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled [[lattice gauge theory]]).  But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.

This is the problem of [[non-perturbative quantum field theory|non-perturbative]] quantization of Yang-Mills theory

## Related concepts

* [[Nakanishi-Lautrup field]], [[antighost field]]

## References

### Non-perturbative theory and mass gap
 {#ReferencesNonperturbativeTheoryAndMassGap}

> See at _[[mass gap]]_ for more.

A survey and problem description of the [[non-perturbative quantum field theory|non-perturbative]] quantization of Yang-Mills theory, in the context of the Clay Mathematics Institute's list of "[Millennium Problems](http://www.claymath.org/millennium-problems)" ("[Yang–Mills and Mass Gap](https://www.claymath.org/millennium/yang-mills-the-maths-gap)") is in

* {#JaffeWitten} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ ([pdf](http://www.claymath.org/sites/default/files/yangmills.pdf))

and a report on the progress (essentially none) is in

* {#Douglas04} [[Michael Douglas]], *Report on the Status of the Yang-Mills Millennium Prize Problem* (2004) [[pdf](http://www.claymath.org/sites/default/files/ym2.pdf)]

Notes reviewing more technical details of the problem are in

* Jay Yablon, _The Origins of QCD Confinement in Yang-Mills Gauge Theory_, January 2008 ([pdf](http://jayryablon.files.wordpress.com/2008/01/qcd-
confinement-handout-10.pdf))

Useful discussion of details of the problem description is in

* [[Arnold Neumaier]], [PO comment](https://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete?show=21846#a21846) 2014

### Perturbative theory

The quantization of Yang-Mills theory in the guise of [[EW]] and [[QCD]] as [[perturbative quantum field theory]] is carried out rigorously via [[causal perturbation theory]] in

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

The [[locally covariant perturbative quantum field theory]] of Yang-Mill [[AQFT on curved spacetime|on curved spacetimes]] is constructed in

* {#Hollands07} [[Stefan Hollands]], _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys. 20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

Discussion of the [[adiabatic limit]] is in

* {#Duch17} [[Pawel Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))


quantization via the A-model

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## Idea

A method of [[quantization]] of [[mechanical systems]] represented by [[symplectic manifolds]] that produces the [[space of states]] of this system as the space of states of [[open strings]] in the [[A-model]].

More in detail, the [[symplectic manifold]] serves as the [[target space]] for the 2-dimensional [[sigma-model]] [[TQFT]] called the [[A-model]]. If the symplectic manifold arises as the [[complexification]] of a [[phase space]] of some [[physical system]], then one finds that the [[boundary]] [[path integral]] of the A-model on [[coisotropic submanifold|coisotropic]] [[branes]] computes the path integral and the [[geometric quantization]] of this physical system.

The setup is reminiscent of how the [[deformation quantization]] of the phase space is computed by the 2-dimensional [[Poisson sigma-model]]. See at _[[holographic principle]]_ for more on the general pattern.

> The goal is to get closer to a systematic theory of [[quantization]].

> ([Gukov-Witten 08, p.4](#GukovWitten08))

## Applications

When the phase space $M$ is the coadjoint orbit for a real form $G_\mathbb{R}$, with the complexification being the complex coadjoint orbit $Y$ for $G_\mathbb{C}$. The particular cases when $G_\mathbb{C} = SL(2,\mathbb{C})$ are covered in section 3 of ([Gukov-Witten 08](#GukovWitten08)). These provide applications to representation theory like in the [[orbit method]].

## Related concepts

* [[deformation quantization]]

* [[geometric quantization]]

## References

The main original article is

* {#GukovWitten08} [[Sergei Gukov]], [[Edward Witten]], _Branes and Quantization_, Adv. Theor. Math. Phys. 13 (2009) 1–73, ([arXiv:0809.0305](http://arxiv.org/abs/0809.0305), [euclid](http://projecteuclid.org/euclid.atmp/1282054099))

Review includes

* [[Constantin Teleman]], _[Branes and Representations](https://math.berkeley.edu/~teleman/classes/brepresentations)_,2016

Earlier, suggestions that the A-model is related to [[deformation quantization]] had been given in

* [[Paul Bressler]], [[Yan Soibelman]], _Mirror symmetry and deformation quantization_ ([arXiv:hep-th/0202128](http://arxiv.org/abs/hep-th/0202128))

* [[Anton Kapustin]], _A-branes and Noncommutative Geometry_ ([arXiv:hep-th/0502212](http://arxiv.org/abs/hep-th/0502212))

which then have been extended and made more precise in

* Vasily Pestun, _Topological strings in generalized complex space_, Adv.Theor.Math.Phys.11:399-450,2007  ([arXiv:hep-th/0603145](http://arxiv.org/abs/hep-th/0603145))

and in

* [[Marco Gualtieri]], _Branes on Poisson varieties_ ([arXiv:0710.2719](http://arxiv.org/abs/0710.2719))

in the context of [[generalized complex geometry]].

Tha the space of states of open strings in the A-model can be interpreted as a quantization of the corresponding symplectic manifold had first been noticed in examples in

* Marco Aldi, [[Eric Zaslow]], _Coisotropic Branes, Noncommutativity, and the Mirror Correspondence_, JHEP 0506 (2005) 019 ([arXiv:hep-th/0501247](http://arxiv.org/abs/hep-th/0501247))

The canonical coisotropic brane was used to elucidate these matters further in

* [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

is the context of the [[Kapustin-Witten TQFT]].

A relation to the [[path integral]] is discussed in

* [[Edward Witten]], _A New Look At The Path Integral Of Quantum Mechanics_ ([arXiv:1009.6032](http://uk.arxiv.org/abs/1009.6032))

Relation to the [[B-model]] via [[mirror symmetry]] is discussed in

* [[Sergei Gukov]], _Quantization via Mirror Symmetry_ ([arXiv:1011.2218](http://arxiv.org/abs/1011.2218))

[[!redirects brane quantization]]


quantized enveloping algebra

__Quantized enveloping algebras__ are certain deformations of the universal enveloping algebras of semisimple Lie algebras considered as Hopf algebras. They were first introduced by Drinfeld and Jimbo.

There are several variants. The deformation can be understood in $h$-adic sense (Drinfeld's approach, $U_h(g)$) where in the classical case $h$ tends to $0$, or one can introduce a parameter $q$ which in the classical case tends to $1$. In the latter case, the parameter can be formal,
say we work over $k[q,q^{-1}]$ where $k$ is the ground ring what is accomplished by defining so called rational form $U_q(g)$ of the quantized enveloping algebra. There is also an integral form over $\mathbb{Z}[q,q^{-1}]$, introduced by Lusztig. The parameter can be also specialized to a value in the field where the simplest case is when $q$ is transcendental.
In the case when
$q = \epsilon$ is a primitive root of unity several variants exist (e.g. restricted and nonrestricted variants). The behaviour usually differs notably between the even and odd primitive roots of unity.

Special care should be made when defining the quantized enveloping algebras for $q = 0$. This crystal limit is very important as it lead to the discovery of the canonical/crystal bases of Lusztig and Kashiwara.

__Quantized coordinate rings__ of quantum groups are essentially dual to the corresponding quantized enveloping algebras.

* [[Vladimir Drinfel'd|V. G. Drinfel'd]], _Quantum groups_, Proceedings of the International Congress of Mathematicians 986, Vol. 1, 798–820, AMS 1987, [djvu:1.3M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu), [pdf:2.5M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf)

* Michio Jimbo, [_A $q$-difference analogue of $U(\mathfrak{g})$ and the Yang-Baxter equation_](https://doi.org/10.1007/BF00704588), Lett. Math. Phys. 10 (1985), no. 1, 63–69.

* J.C. Jantzen, _Lectures on quantum groups_, Grad. Stud. Math. 6, Amermer. Math. Soc. 1996

* Ch. 4 of [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* [[George Lusztig]], _Introduction to quantum groups_

* V. Chari, A. Pressley, _A guide to quantum groups_, Camb. Univ. Press 1994

[[!redirects quantized enveloping algebras]]


quantized function algebra

A noncommutative algebra $A$ is called the quantum/quantized coordinate ring if it is a deformation of a (commutative) coordinate ring of some affine or, in graded case, projective variety. The expression __quantized coordinate ring__ may be considered a synonym, but can also denote more generally a deformation of some other function algebra, say a $C^\ast$-algebra of functions on some compact Hausdorff topological space.

Most often these terms are used in quantum group theory. If $G$ is an algebraic group, then its coordinate ring is usually denoted by $\mathcal{O}(G)$ or $Fun(G)$ and is a Hopf algebra. There is a pairing between $\mathcal{O}(G)$ and $U(g)$ where $g$ is the Lie algebra of $G$ hence the consideration of the function algebra is essentially dual to the consideration of the universal enveloping algebra $U(g)$ of $g$.
If $q$ is a deformation parameter, then there is a 1-parametric deformation $\mathcal{O}(G_q)$ (also denoted $\mathcal{O}_q(G)$) called the quantized function algebra on $G_q$ (or quantized coordinate ring of $G_q$) which is also a Hopf algebra. One considers $G_q$ as a quantum group; the approach via $\mathcal{O}(G_q)$ is dual to the approach via the quantized enveloping algebra $U_q(g)$.

Most well known approach to $\mathcal{O}(G_q)$ is the FRT construction

* N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i analiz 1, 178 (1989) (Russian), English translation in Leningrad Math. J. 1.

and via the matrix elements of representations (using the analogue of the Peter-Weyl theorem)

* A. Joseph, Quantum groups and their primitive ideals, Springer 1995.

Some special cases can be introduced via construction of [[universal coacting bialgebra]]s, for example the [[quantum linear group]]s in the approach of

* [[Yu. I. Manin]], Quantum groups and non-commutative geometry, CRM, Montreal 1988.

[[!redirects quantum coordinate ring]]
[[!redirects quantized coordinate ring]]


quantomorphism group

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## Idea

Given a ([[presymplectic form|pre]])[[symplectic manifold]] $(X,\omega)$, its _quantomorphism group_ is the [[Lie group]] that [[Lie integration|integrates]] the [[Lie algebra|Lie bracket]] inside the [[Poisson algebra]] of $(X, \omega)$.
This is a [[circle group]]-[[central extension]] of the group of [[Hamiltonian symplectomorphisms]]. It extends and generalizes the [[Heisenberg group]] of a [[symplectic vector space]].

(Warning on terminology: A more evident name for the quantomorphism group might seem to be "Poisson group". But this already means something different, see _[[Poisson Lie group]]_.)

### Traditional construction

Over a [[symplectic manifold]] $(X, \omega)$ an explicit construction of the corresponding quantomorphism group is obtained by choosing $(P \to X, \nabla)$ a [[prequantum circle bundle]], regarded with an [[Ehresmann connection]] 1-form $A$ on $P$, and then defining

$$
  QuantomorphismGroup \hookrightarrow Diff(P)
$$

to be the [[subgroup]] of the [[diffeomorphism group]] $P \stackrel{\simeq}{\to} P$ on those [[diffeomorphisms]] that preserve $A$. In other words, the quantomorphism group is the group of equivalences of [[connection on a bundle|bundles with connection]] that need not cover the identity [[diffeomorphism]] on the base manifold $X$.

Notice that the tuple $(P,A)$ is a [[regular contact manifold]] (see the discussion there), and so the quantomorphism group is equivalently that of [[contactomorphisms]] $(P,A) \to (P,A)$ of weight 0.

This is an infinite-dimensional [[Lie group]]. References discussing its [[infinite-dimensional manifold]]-structure are collected [below](#SmoothManifoldStructure). But the group has immediately the structure of a group in [[diffeological spaces]] (making it a [[smooth group]]) ([Souriau 79](#Souriau79)).

### In higher geometry
 {#InHigherGeometry}

This perspective lends itself to a more abstract description in [[higher differential geometry]]: we may regard the [[prequantum circle bundle]] as being modulated by a morphism

$$
  \nabla : X \to \mathbf{B} U(1)_{conn}
$$

in the [[cohesive (∞,1)-topos]] $\mathbf{H} = $ [[Smooth∞Grpd]], with [[domain]] the given symplectic manifold and [[codomain]] the smooth [[moduli stack]] for [[circle n-bundle with connection|circle bundles with connection]]. This in turn may be regarded as an object $\nabla \in \mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ in the [[slice (∞,1)-topos]]. Then the quantomorphism group is the [[automorphism group]]

$$
  \mathbf{QuantMorph}(X,\nabla) \coloneqq
  \underset{\mathbf{B}U(1)_{conn}}{\prod} \mathbf{Aut}(\nabla)
$$

in $\mathbf{H}$, or rather its [[differential concretification]] ([FRS 13](#FRS13)).

From this it is clear what the quantomorphism [[∞-group]] of an [[n-plectic ∞-groupoid]] should be: for

$$
  \nabla : X \to \mathbf{B}^n U(1)_{conn}
$$

the morphism modulating a [[prequantum circle n-bundle]], the corresponding quantomorphism $n$-group is again $Aut(\nabla)$, now formed in $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$

## Properties

### Smooth structure

The quantomorphism group for a [[symplectic manifold]] may naturally be equipped with the structure of a [[group object]] in [[ILH manifolds]] ([Omori](Omori), [Ratiu-Schmid](RatiuSchmid)), as well as in [[convenient manifolds]] ([Vizman, prop. ](#Vizman)).

### Group extension

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

For $(X,\omega)$ a [[connected topological space|connected]] [[symplectic manifold]] there is a [[central extension of groups]]

$$
  1 \to U(1) \to QuantomorphismGroup(X,\omega) \to HamiltonianSymplectomorphisms(X,\omega) \to 1
  \,.
$$

=--

This is due to ([Kostant](#Kostant)). It appears also a [Brylinski, prop. 2.4.5](#Brylinski93).

[[!include geometric quantization extensions - table]]

## Examples

### Covering an affine symplectic group
 {#CoveringAnAffineSymplecticGroup}

Given a [[symplectic vector space]] $(V,\omega)$ one may consider the restriction of its [[quantomorphism group]] to the [[affine symplectic group]] $ASp(V,\omega)$ ([Robbin-Salamon 93, corollary 9.3](#RobbinSalamon93))

$$
  \array{
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow
     \\
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
$$

Sometimes (e.g. [Robbin-Salamon 93, p. 30](#RobbinSalamon93)) this $ESp(V,\omega)$ is called the _extended symplectic group_, but maybe to be more specific one should at the very least say "[[extended affine symplectic group]]" or "extended inhomogeneous symplectic group" ([ARZ 06, prop. V.1](#ARZ06)).

Notice that the further restriction to $V$ regarded as the [[translation group]] over itself is the [[Heisenberg group]] $Heis(V,\omega)$

$$
  \array{
     Heis(V,\omega)
     &\hookrightarrow&
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     V
     &\hookrightarrow&
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
$$

The group $ESp(V,\omega)$ is that of those [[quantomorphisms]] which come from [[quadratic Hamiltonians]]. Those elements covering elements in the [[symplectic group]] instead of the [[affine symplectic group]] come from [[homogeneously quadratic Hamiltonians]] (e.g. [Robbin-Salamon 93, prop. 10.1](#RobbinSalamon93)). In fact $ESp$ is the [[semidirect product]] of the [[metaplectic group]] $Mp(V,\omega)$ with the [[Heisenberg group]] ([ARZ 06, prop. V.1](#ARZ06), see also [Low 12](#Low12))

$$
  ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega)
  \,.
$$

## Related concepts

* [[conserved current]]

* [[Hamiltonian action]], [[classical anomaly]]

[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!include higher Atiyah groupoid - table]]

## References
 {#References}

### General

Original accounts are

* [[Jean-Marie Souriau]], _Structure des systemes dynamiques_ Dunod, Paris (1970)

  Translated and reprinted as (see section V.18 for the quantomorphism group):

  [[Jean-Marie Souriau]], _Structure of dynamical systems - A symplectic view of physics_, Brikhäuser (1997) doi:[10.1007/978-1-4612-0281-3](https://doi.org/10.1007/978-1-4612-0281-3)

* {#Kostant} [[Bertram Kostant]], _Quantization and unitary representations_, in _Lectures in modern analysis and applications III_. Lecture Notes in Math. 170 (1970), Springer Verlag, 87—208 doi:[10.1007/BFb0079068](https://doi.org/10.1007/BFb0079068)

A textbook account is in

* {#Brylinski93} [[Jean-Luc Brylinski]], section II.4 _Loop spaces, characteristic classes and geometric quantization_, Birkhäuser (1993)

and in

* Rudolf Schmid, _Infinite-dimensional Lie groups with applications to mathematical physics_, J. Geom. Symmetry Phys., Volume 1 (2004), 54-120. [Project Euclid](https://projecteuclid.org/euclid.jgsp/1495505067)

The description in terms of automorphism in the slice $\infty$-topos over the moduli stack of (higher) connections is in

* {#FRS13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

and in section 4.4.17 of

* {#Sch} _[[schreiber:differential cohomology in a cohesive topos]]_

Another expository account, which spells out the details of the isomorphism from smooth functions with the Poisson bracket to infinitesimal quantomorphisms is given in

* [[Jennifer Vaughan]], _[Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization](https://hdl.handle.net/1807/77389)_, PhD dissertation, University of Toronto, 2016.

### Smooth manifold structure
 {#SmoothManifoldStructure}

The [[diffeological space]]-structure ([[diffeological group]], [[smooth group]] structure) on the quantomorphism group is at least implicit in

* {#Souriau79} [[Jean-Marie Souriau]], _Groupes différentiels_, in _Differential Geometrical Methods in Mathematical Physics_ (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=607688))

The [[ILH manifold|ILH group]] structure on the quantomorphism group is discussed in

* {#Omori} H. Omori, _Infinite dimensional Lie transformation groups_, Springer lecture notes in mathematics 427 (1974)

* {#RatiuSchmid} T. Ratiu, R. Schmid, _The differentiable structure of three remarkable diffeomorphism groups_, Math. Z.  177 (1981)

The [[convenient manifold|regular convenient Lie group]] structure is discussed in

* {#Vizman} Cornelia Vizman, _Some remarks on the quantomorphism group_ ([[VizmanQuantomorphism.pdf:file]])

A [[metric]]-structure on quantomorphism groups is discussed in

* Y. Eliashberg,; L. Polterovich, _Partially ordered groups and geometry of contact transformations_. Geom.Funct.Anal.10(2000),no.6, 1448-1476. doi:[10.1007/PL00001656](https://doi.org/10.1007/PL00001656), arXiv:[math/9910065](https://arxiv.org/abs/math/9910065)

### Examples

The quantomorphisms over elements of the [[symplectic group]] of a [[symplectic vector space]] are discussed in

* {#Segal63} [[Irving Segal]], _Transforms for operators and symplectic automorphisms over a locally compact abelian group_, Math. Scand. 13 (1963) 31-43

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], _Feynman path integrals on phase space and the metaplectic representation_ in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993) ([[RobbinSalamonMetaplectic.pdf:file]])

* {#ARZ06} [[Sergio Albeverio]], J. Rezende and J.-C. Zambrini, _Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics_, Journal of Mathematical Physics 47, 062107 (2006) ([pdf](http://gfm.cii.fc.ul.pt/people/jczambrini/JMathPhys-47-062107.pdf))

* {#Low12} Stephen G. Low, _Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group_, J. Math. Phys. 55, 022105 (2014) ([arXiv:1207.6787](http://arxiv.org/abs/1207.6787))

[[!redirects quantomorphism]]
[[!redirects quantomorphisms]]

[[!redirects quantomorphism groups]]

[[!redirects quantomorphism ∞-group]]
[[!redirects quantomorphism ∞-groups]]
[[!redirects quantomorphism infinity-group]]
[[!redirects quantomorphism infinity-groups]]

[[!redirects quantomorphism n-group]]
[[!redirects quantomorphism n-groups]]

[[!redirects higher quantomorphism group]]
[[!redirects higher quantomorphism groups]]


quantum 2-group

#Contents#
* table of contents
{:toc}

## Idea

A pairing of the notions of [[quantum group]] and [[2-group]].

## Related concepts

* [[trialgebra]]

## References

* [[Shahn Majid]], _Strict quantum 2-groups_ ([arXiv:1208.6265](http://arxiv.org/abs/1208.6265))

[[!redirects quantum 2-groups]]