nLab formal deformation quantization

Idea

The notion of formal deformation quantization (often taken to be the default notion of deformation quantization) is one formalization of the general idea of quantization of a classical mechanical system/classical field theory to a quantum mechanical system/quantum field theory, at least perturbatively in small (really: infinitesimal) values of Planck's constant $\hbar$.

Formal deformation quantization focuses on the algebras of observables of a physical system (hence on the Heisenberg picture): it provides rules for how to deform the commutative algebra of classical observables to a non-commutative algebra of quantum observables. (This is in contrast to geometric quantization, which focuses on the spaces of states and hence on the Schrödinger picture.)

Usually and traditionally, deformation quantization refers to (just) formal deformations, in the sense that it produces formal power series expansions in a formal parameter $\hbar$ (physically: Planck's constant) of the product in the deformed algebra of observables.

classical mechanics	semiclassical approximation	…	formal deformation quantization	quantum mechanics
$\mathcal{O}(\hbar^0)$	$\mathcal{O}(\hbar^1)$	$\mathcal{O}(\hbar^n)$	$\mathcal{O}(\hbar^\infty)$

(This is related to perturbation theory, which is about formal power series in coupling constants, instead of in Planck's constant.)

But there are refinements of this to C-star algebraic deformation quantization which studies the proper deformation to a genuine C-star algebra of observables. (This in turn is related to genuine geometric quantization via the notion of geometric quantization of symplectic groupoids.)

One can, therefore, argue that only strict deformation quantization is genuine quantization. For instance in (Gukov-Witten 09, section 1.4) it says

Generally speaking, physics is based on $[$ strict $]$ quantization, rather than $[$ formal $]$ deformation quantization, although conventional quantization sometimes leads to problems that can be treated by deformation quantization.

As other methods of quantization, deformation quantization has as input a description of a classical mechanical system, which is in this case most often a smooth Poisson manifold. The deformation quantization replaces the algebra of smooth functions on the Poisson manifold with the same vector space, but equipped with new noncommutative associative unital product whose commutator agrees, up to order $\hbar$, with the underlying Poisson bracket. Of course the proper study of quantization of Poisson manifolds studied the appropriate notion at the level of sheaves of algebras. Gluing local solutions to the quantization problem furthermore involves stacks and specifically gerbes.

Definition

If the result of deformation quantization is an algebra over the power series ring $\mathbb{R}[ [ \hbar ] ]$ of a formal parameter $\hbar$ (thought of as Planck's constant) such that the limit $\hbar \to 0$ reproduces the starting point of the deformation, then one speaks of

• Formal deformation quantization

In much of the literature this is regarded as the default meaning of “deformation quantization”. But this is really the case corresponding to perturbation theory in quantum field theory. A “genuine” or “strict” deformation quantization

• Strict deformation quantization

is supposed to result in a non-formal deformation, which in terms of the above formal power series at least means that one can set $\hbar = 1$ such that all expressions in $\hbar$ converge, but which in general is taken to mean something stronger, such as that there is a continuous field of C-star algebraic deformation quantization.

Formal deformation quantization

We first give the traditional

• Explicit definition of traditional deformation quantization

of a Poisson manifold/Poisson algebra. Thought of in terms of physics this describes a quantization of a system of quantum mechanics, as opposed to full quantum field theory.

More abstractly, this may be formulated and generalized in terms of lifts of algebras over an operad over a P-n operad? to a BD-n operad? and hence an E-n operad, for $n = 1$. This we discuss in

• Formulation as lifts from P-n algebras to BD-n algebras

In this formulation one sees that for genral $n$ the construction applies to $n$-dimensional quantum field theory (with quantum mechanics for $n = 1$ be 1-dimensional quantum field theory, for instance the sigma-model “on the worldline” of a particle). A formulation of deformation quantization to local quantum field theory formulated in terms of factorization algebras of observables over spacetime/worldvolume is indicated in (Costello-Gwilliam, chapter 5).

Explicit definition of deformation of Poisson manifolds/Poisson algebras

Let $M$ be a Poisson manifold and let $A = C^\infty(M)$ be the Poisson algebra of smooth functions.

Definition

A $\ast$-product (star product) on $A$ is a product on the power series $A [ [ t ] ]$ that is (1) bilinear over $\mathbb{R}[ [ t ] ]$, (2) associative, and (3) for $a,b \in A$ it can be written out as a formal power series

$$a \ast b = \sum_{n=0}^\infty B_n(a,b) t^n$$

where $B_n$ are bilinear maps on $A$ such that $B_0(a,b) = ab$.

Definition

A (formal) deformation quantization of $M$ is a star product on $A = C^\infty(M)$ such that the Poisson bracket $\{a,b\} = B_1(a,b) - B_1(b,a)$ for $a,b \in A$; by bilinearity over $\mathbb{R}[ [ t ] ]$, this characterizes it.

Formulation as lifts from $P_n$-algebras to $BD_n$-algebras and $E_n$-algebras

(…)

(Costello-Gwilliam, section 2.3, 2.4)

(…)

algebraic deformation quantization

dimension	classical field theory	Lagrangian BV quantum field theory	factorization algebra of observables
general $n$	P-n algebra	BD-n algebra?	E-n algebra
$n = 0$	Poisson 0-algebra	BD-0 algebra? = BD algebra	E-0 algebra? = pointed space
$n = 1$	P-1 algebra = Poisson algebra	BD-1 algebra?	E-1 algebra? = A-∞ algebra

Deformation quantization in perturbative quantum field theory

Deformation quantization in perturbative quantum field theory is discussed for free field theory via Moyal deformation quantization yielding Wick algebras in (Dütsch-Fredenhagen 00, Hirschfeld-Henselder 02), and for interacting perturbative quantum field theory (perturbative AQFT) via Fedosov deformation quantization in (Collini 16), see also (Hawkins-Rejzner 16).

Strict deformation quantization

(…) C-star algebraic deformation quantization (…)

Properties

Existence results

Vladimir Drinfel'd has sketched a proof (and gave main ingredients) to show that every Poisson Lie group can be deformation quantized to a Hopf algebra; this proof has been completed by Etingof and Kazhdan. Maxim Kontsevich proved a certain formality theorem (formality is here in the sense of formal dg-algebra in rational homotopy theory) whose main corollary (and motivation) was the statement that every Poisson manifold has a deformation quantization (Kontsevich 97).

For symplectic manifolds and those Poisson manifolds that have a regular foliation by symplectic leafs, the theory of deformation quantization is much simpler; Boris Fedosov gave a construction of star products on symplectic manifolds using symplectic connections on smooth manifolds (Fedosov 94), see at Fedosov's deformation quantization. An analogous argument was given by Roman Bezrukavnikov and Dmitry Kaledin in the context of an algebraic symplectic form (BK 04).

Caution: the following are rough notes from a talk by J.D.S. Jones (Cambridge, 8.1.2013); there are probably many typos and sign errors.

Deformation of Poisson manifolds

Theorem

(Kontsevich). Every Poisson manifold has a (formal) deformation quantization.

This was shown in (Kontsevich 97). There the deformed product is constructed by a kind of Feynman diagram perturbation series. Later this was identified as the perturbation series of the Poisson sigma-model for the given Poisson manifold. See there for more details.

Deformation of Algebraic Varieties

(Not from the notes of J.D.S. Jones)

Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this:

Theorem

(Swan, Yekutieli). There is a canonical isomorphism

$$\operatorname{Ext}^i_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \operatorname{H}^{i - q}(X, \bigwedge^q (\mathcal{T}_X)) .$$

Here $\mathcal{T}_X$ is the tangent sheaf of $X$, and $X$ is embedded diagonally in $X^2$.

This is a consequence of the following result. Let $\mathcal{C}_{cd, X}$ be the sheaf of continuous Hochschild cochains of $X$. It is a bounded below complex of quasi-coherent $\mathcal{O}_{X^2}$-modules.

Theorem

(Yekutieli).

1. There is a canonical isomorphism

   $$\operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \mathcal{C}_{cd, X}$$

   in the derived category of $\mathcal{O}_{X^2}$-modules.

2. There is a canonical quasi-isomorphism of complexes of $\mathcal{O}_{X^2}$-modules

   $$\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \to \mathcal{C}_{cd, X} .$$

3. Therefore there is a canonical isomorphism

   $$\operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$$

   in the derived category of $\mathcal{O}_{X^2}$-modules.

The relation to deformation quantization is this: $\mathcal{C}_{cd, X}$ is a shift by $1$ of the sheaf of $\mathcal{D}_{poly, X}$ of polydifferential operators (viewed only as a complex of quasi-coherent $\mathcal{O}_{X^2}$-modules). Similarly, $\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$ is the shift by $1$ of the sheaf $\mathcal{T}_{poly, X}$ of polyvector fields. Thus item 2 in the theorem above says that there is a canonical $\mathcal{O}_{X^2}$-linear quasi-isomorphism

$$\mathcal{T}_{poly, X} \to \mathcal{D}_{poly, X} .$$

Trying to replicate the global formality theorem of Kontsevich, one would like to upgrade this to an $\mathrm{L}_{\infty}$ quasi-isomorphism. However, it seems that in general this cannot be done directly, but only after a suitable resolution.

Here is the result. (See also Van den Bergh.) Any quasi-coherent sheaf $\mathcal{M}$ on $X$ admits a canonical flasque resolution called the mixed resolution:

$$\mathcal{M} \to \operatorname{Mix}(\mathcal{M}) .$$

This “mixes” the jet resolution with the Cech resolution (corresponding to an affine open covering of $X$ that we suppress). In particular there are quasi-isomorphisms of sheaves of DG Lie algebras

$$\mathcal{T}_{poly, X} \to \operatorname{Mix}(\mathcal{T}_{poly, X})$$

and

$$\mathcal{D}_{poly, X} \to \operatorname{Mix}(\mathcal{D}_{poly, X}) .$$

Theorem

(Yekutieli). There is an $\mathrm{L}_{\infty}$ quasi-isomorphism

$$\Psi : \operatorname{Mix}(\mathcal{T}_{poly, X}) \to \operatorname{Mix}(\mathcal{D}_{poly, X})$$

whose $1$-st order term commutes with the HKR quasi-isomorphism above. It is independent of choices up to quasi-isomorphism.

A Poisson deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of Poisson $\mathbb{k}[[\hbar]]$-algebras on $X$, with an isomorphism $\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X$ called an augmentation. Likewise an associative deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of associative unital (but noncommutative) $\mathbb{k}[[\hbar]]$-algebras on $X$, with an augmentation to $\mathcal{O}_X$.

Theorem 4 implies:

Theorem

(Yekutieli). Assume that the cohomology groups $\operatorname{H}^{1}(X, \mathcal{O}_X)$ and $\operatorname{H}^{2}(X, \mathcal{O}_X)$ vanish. Then there is a canonical bijection

$$\mathrm{quant} : \quad \frac{ \{ \text{ Poisson deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} \quad \xrightarrow{\, \simeq \,} \quad \frac{ \{ \text{ associative deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}}$$

called quantization. It preserves first order brackets.

When $X$ is affine this is Theorem 0.1 of this paper. For the full statement see Corollary 11.2 of this paper.

For twisted (or stacky) deformations there is a corresponding (but much more difficult to state and prove). See the paper and the survey.

Gerstenhaber’s deformation theory by Hochschild (co)homology

Let $V$ be a $k$-vector space and consider $C^p(V,V) = \Hom(V^{\otimes p}, V)$. We define a “circle operator” $\circ$ as follows: for $f \in C^p(V,V)$ and $g \in C^q(V,V)$, we define $f \circ g \in C^{p+q-1}(V,V)$ as the map

$$(f \circ g)(v_1, \ldots v_{p+q-1}) = f(v_1, \ldots, v_{i-1}, g(v_i, \ldots, v_{i+q-1}), v_{i+q}, \ldots, v_{p+q-1}).$$

For $f \in C^\ast(V,V)$, let $A_f(g,h) = (f \circ g) \circ h - f \circ (g \circ h)$. (This is graded symmetric.) It follows that the commutator of $\circ$ is given by

$$[f,g] = f \circ g - (-1)^{(|f| - 1)(|g|-1)} g \circ f$$

where $|f| = p$ when $f \in C^p(V,V)$. This defines a graded Lie bracket of degree -1.

Example

Let $\mu \in C^2(V,V)$ ($\mu : V \otimes V \to V$). Note that $\mu$ is associative iff $\mu \circ \mu = 0$ iff $[\mu, \mu] = 0$. Let $d_\mu : C^{p}(V,V) \to C^{p+1}(V,V)$ be defined by $d_\mu(x) = \mu \otimes x \pm x \otimes \mu = [\mu, x]$. We have $d_\mu \circ d_\mu = 0$ so $(C^\ast(V,V), d_\mu)$ becomes a differential graded algebra. In fact this is the Hochschild cochain complex of the associative algebra $A = (V, \mu)$.

Apply this example to the construction of deformation quantization. The star product is uniquely determined by $\theta : A \otimes A \to A[ [ t ] ]$ given by $\theta(a,b) = ab + c(a,b)$. What we want is that

$$(\mu + c) \otimes (\mu + c) = 0;$$

write this out and we get the equation

$$d_\mu c + c \circ c = 0,$$

or $d_\mu c + \frac{1}{2}[c,c] = 0$; this is the Maurer-Cartan equation. Hence we are looking for solutions of the M-C equation but in the Hochschild complex $C^\ast(A,A)[ [ t ] ]$. One should note that $d_\mu$ is actually a derivation of the Lie bracket, hence we have a dg-Lie algebra.

Theorem

(HKR theorem). $HH^p(A,A) = \Gamma(M, \Lambda^p TM)$.

(Note that $C^p(C^\infty(M),C^\infty(M))$ should be interpreted as $\Hom_{diff}(C^\infty(M), C^\infty(M))$.) Under this isomorphism the Poisson bracket is mapped to the Poisson tensor:

$$\{ \cdot , \cdot \} \in HH^2(A,A) \quad \mapsto \quad P \in \Gamma(M, \Lambda^2 TM).$$

The bracket in Hochschild cohomology (Gerstenhaber bracket) goes to the Schouten bracket:

$$[ \cdot , \cdot ]_G \quad \mapsto \quad [ \cdot, \cdot ]_S.$$

For vector fields $\xi$ and $\eta$, the Schouten bracket satisfies (1) $[\xi,\eta]_S = [\xi,\eta]$ (the Lie bracket), and (2) $[\alpha, \beta \wedge \gamma] = [\alpha,\beta] \wedge \gamma \pm [\alpha,\gamma] \wedge \beta$; note that this completely determines it (everything is locally given by wedges…).

In the Hochschild cohomology $HH^\ast(A,A)$ of $A$, $d_\mu P \mapsto 0$ and $[P,P]_S = 0$, so we have a solution to M-C in $H^\ast(A,A)[ [ t ] ]$.

In terms of differential graded Lie algebras

Definition

Let $L_1$ and $L_2$ be differential graded Lie algebras (dgL). A quasi-isomorphism $f : L_1 \to L_2$ is a homomorphism of dgLs that induces an isomorphism on homology. $L_1$ and $L_2$ are quasi-isomorphic if there exists $M$ with quasi-isomorphisms $L_1 \leftarrow M \rightarrow L_2$. It can be verified that this is an equivalence relation.

Theorem

(Kontsevich). If $L_1$ is quasi-isomorphic to $L_2$ then there is a solution to the M-C equation in $L_1$ iff there is a solution to the M-C equation in $L_2$.

Theorem

(Kontsevich formality). $C^\ast(A,A)[ [ t ] ]$ is quasi-isomorphic to $H^\ast(A,A)[ [ t ] ]$. ($A = C^\infty(M)$)

Hence there is a solution to M-C in $C^\ast(A,A)[ [ t ] ]$, and hence there is a deformation quantization (!).

The Deligne conjecture

We have $(C^*(A,A), d_\mu)$, the Gerstenhaber bracket, and we also have a cup product

$$(f \cup g) (a_1, \ldots, a_{p+q}) = \mu(f(a_1,\ldots,a_p), g(a_{p+1},\ldots,a_{p+q}))$$

for $f : A^{\otimes p} \to A$, $g : A^{\otimes q} \to A$; this satisfies also $d_\mu(f \cup g) = (d_\mu f) \cup g \pm f \cup d_\mu g$. The Deligne conjecture gives a relationship between these things.

In $HH^*(A,A)$, we have:

1. $[\cdot, \cdot]$ is a graded Lie bracket of degree -1.
2. The cup product $\cup$ is graded commutative.
3. The Jacobi identity for $[\cdot,\cdot]$.
4. $[a, b \cup c] = [a,b] \cup c \pm [a,c] \cup b$.

Such a thing is called a Gerstenhaber algebra. Note that we do not have these relations in $C^*(A,A)$, they are only true modulo boundaries.

Theorem

(Deligne conjecture). $C^*(A,A)$ is a $G_\infty$-algebra, which is a Gerstenhaber algebra up to coherent homotopy.

Deformation by universal enveloping algebras

It is a classical fact [Gutt 1983, §4 (4.2), cf. Gutt 2011, §2.2] that the universal enveloping algebra of a Lie algebra provides a deformation quantization of the corresponding Lie-Poisson structure (example below). Remarkably, this statement generalizes to more general polynomial Poisson algebras (def. below) for a suitable generalized concept of universal enveloping algebra (def. below): it is always true up to third order in $\hbar$, and sometimes to higher order (Penkava-Vanhaecke 00, theorem 3.2, prop. below). In particular it also holds true for restrictions of Poisson bracket Lie algebras to their Heisenberg Lie algebras (example below).

Definition

(polynomial Poisson algebra)

A Poisson algebra $((A,\cdot), \{-,-\})$ is called a polynomial Poisson algebra if the underlying commutative algebra $(A,\cdot)$ is a polynomial algebra, hence a symmetric algebra

$$Sym(V) \coloneqq T(V)/(x \otimes y - y \otimes x \vert x,y \in V)$$

on some vector space $V$. Here

$$T(V) \coloneqq \underset{n \in \mathbb{N}}{\oplus} V^{\otimes^n}$$

denotes the tensor algebra of $V$. We write

$$\mu \;\colon\; T(V) \longrightarrow Sym(V)$$

for the canonical projection map (which is an algebra homomorphism) and

$$\sigma \;\colon\; Sym(V) \longrightarrow T(V)$$

for its linear inverse (symmetrization, which is not in general an algebra homomorphism).

Notice that by its bi-derivation property the Poisson bracket on a polynomial Poisson algebra is fixed by its restriction to linear elements

$$\{-,-\} \;\colon\; V \otimes V \longrightarrow Sym(V) \,.$$

Example

(polynomial Lie-Poisson structure)

Let $(C^\infty(\mathbb{R}^n), \pi)$ be a Poisson manifold whose underlying manifold is a Cartesian space $\mathbb{R}^n$. Then the restriction of its Poisson algebra $( C^\infty(\mathbb{R}^n, \cdot), \pi^{i j} \partial_i(-) \cdot \partial_j(-) )$ to the polynomial functions $\mathbb{R}[x^1, \cdots, x^n ] \ookrightarrow C^\infty(\mathbb{R}^n)$ is a polynomial Poisson algebra according to def. .

In particular if $(\mathfrak{g}, [-,-])$ is a Lie algebra and $(\mathfrak{g}^\ast, \{-,-\})$ the corresponding Lie-Poisson manifold, then the corresponding polynomial Poisson algebra is $(Sym(\mathfrak{g}), \{-,-\})$ where the restriction of the Poisson bracket to linear polynomial elements coincides with the Lie bracket:

$$\{x,y\} = [x,y] \,.$$

Definition

(universal enveloping algebra of polynomial Poisson algebra)

Given a polynomial Poisson algebra $(Sym(V), \{-,-\})$ (def. ), say that its universal enveloping algebra $\mathcal{U}(V,\{-,-\})$ is the associative algebra which is the quotient of the tensor algebra of $V$ with a formal variable $\hbar$ adjoined by the two-sided ideal which is generated by the the $\hbar$-Poisson bracket relation on linear elements:

$$\mathcal{U}(V,\{-,-\}) \;\coloneqq\; T(V)/( x \otimes y - y \otimes x - \hbar \{x,y\} \vert x,y \in V ) \,.$$

This comes with the quotient projection linear map which we denote by

$$\rho \;\colon\; T(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\hbar\{-,-\}) \,.$$

(Penkava-Vanhaecke 00, def. 3.1)

Example

(universal enveloping algebra of Lie algebra)

In the case of a polynomial Lie-Poisson structure $(Sym(\mathfrak{g}), [-,-])$ (example ) the universal enveloping algebra $\mathcal{U}(\mathfrak{g},[-,-])$ from def. (for $\hbar = 1$) coincides with the standard universal enveloping algebra of the Lie algebra $(\mathfrak{g}, [-,-])$.

The combined linear projection maps from def. and def. we denote by

$$\tau \coloneqq \rho \circ (\sigma/[ [ \hbar ] ]) \;\colon\; Sym(V)[ [ \hbar ] ] \longrightarrow \mathcal{U}(V,\{-,-\}) \,.$$

Proposition

(universal enveloping algebra provides formal deformation quantization at least up to order 3)

Let $( Sym(V), \{-,-\} )$ be a polynomial Poisson algebra (def. ) such that the canonical linear map to its universal enveloping algebra (def. ) is injective up to order $n \in \mathbb{N}\cup \{\infty\}$

$$\tau/(\hbar^{n+1}) \;\colon\; Sym(V)[ [ \hbar ] ]/(\hbar^{n+1}) \hookrightarrow \mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1}) \,.$$

Then the restriction of the product on $\mathcal{U}(V,\hbar\{-,-\})/(\hbar^{n+1})$ to $Sym(V)/(\hbar^{n+1})$ is a formal deformation quantization of $(Sym(V), \{-,-\})$ to order $n$ (hence a genuine deformation quantization in the case that $n = \infty$).

Moreover, this is always the case for $n = 3$, hence for every polynomial Poisson algebra its universal enveloping algebra always provides a deformation quantization of order $3$ in $\hbar$.

(Penkava-Vanhaecke 00, theorem 3.2 with section 2)

Example

(formal deformation quantization of Lie-Poisson structures by universal enveloping algebras)

In the following cases the map $\tau$ in prop. is injective to arbitrary order, hence in these cases the universal enveloping algebra provides a genuine formal deformation quantization:

1. The case that the Poisson bracket is linear in that it restricts as

   $$\{-,-\} \;\colon\; V \otimes V \longrightarrow V \hookrightarrow Sym(V) \,.$$

   This is the case of the Lie-Poisson structure from example and the universal enveloping algebra that provides its deformation quantization is the standard one (example ).

2. More generally, the case that the Poisson bracket restricted to linear elements has linear and constant contribution in that it restricts as

   $$\{-,-\} \;\colon\; V \otimes V \longrightarrow \mathbb{R} \oplus V \hookrightarrow Sym(V) \,.$$

   This includes notably the Poisson structures induced by symplectic vector spaces, in which case the restriction

   $$\{-,-\} \;\colon\; (\mathbb{R} \oplus V) \otimes (\mathbb{R} \oplus V) \longrightarrow (\mathbb{R} \oplus V)$$

   is the Lie bracket of the associated Heisenberg Lie algebra.

This is Penkava & Vanhaecke 2000, p. 26. The first statement in itself is a classical fact (reviewed e.g. in Gutt 2011).

Motivic Galois group action on the space of quantizations

In (Kontsevich 99) it was indicated that a quotient group of the motivic Galois group apparently equivalent to the Grothendieck-Teichmüller group naturally acts on the space of formal deformation quantizations of a finite dimensional manifold. See also at cosmic Galois group.

This has been formalized as follows. The formal deformation quantization of (Kontsevich 97) is all induced by the Kontsevich formality theorem, which states that over suitable manifolds/varieties $X$ there is an equivalence of L-∞ algebras

$$\mathcal{X}(X) \stackrel{\simeq}{\to} C^\bullet(X)$$

identifying the multivector fields on $X$ with the Hochschild cohomology complex (of its function algebra). Every choice of such an equivalence induces one formal deformation quantization of a Poisson manifold $X$, and the two quantizations induced by two equivalent (homotopic) equivalences are in turn equivalent.

Therefore one may regard the ∞-groupoid $Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ as the “space of formal deformation quantizations” of $X$.

In (Dolgushev 1109, theorem 6.2, Dolgushev 1111, theorem 3.1) it is shown that the set $\pi_0 Maps^{L_\infty}_{equiv}(\mathcal{X}(X), C^\bullet(X))$ of connected components of this space is, up to a choice of basepoint, the Grothendieck-Teichmüller group, hence is a torsor over that group.

(This is based on identifications of the GRT Lie algebra with the degree-0 chain cohomology of the graph complex, due to Thomas Willwacher. See at Grothendieck-Teichmüller group – relation to the graph complex.

Aspects of the generalization of this statement to more general spaces then $\mathbb{R}^n$ are discussed in Dolgushev-Rogers-Willwacher 12.

For discussion of motivic structures in geometric quantization see at motivic quantization.

Examples

• Moyal deformation quantization

• Fedosov deformation quantization

Related concepts

• reductions deformations resolutions in physics

• quantization

  • deformation quantization, geometric quantization

  • path integral

  • semiclassical approximation

• Jordan algebra

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

The concept of formal deformation quantization originates with:

• François Bayen, Moshé Flato, Christian Fronsdal, André Lichnerowicz, Daniel Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures., Annals of Physics 111 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

• François Bayen, Moshé Flato, Christian Fronsdal, André Lichnerowicz, Daniel Sternheimer, Deformation theory and quantization. II. Physical applications, Annals of Physics 111 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]

Review:

• Alan Weinstein, Deformation quantization, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [numdam:SB_1993-1994__36__389_0]

• Daniel Sternheimer, Deformation Quantization: Twenty Years After, AIP Conf. Proc. 453 (1998) 107-145 [arXiv:math/9809056]

See also:

• Simone Gutt, An explicit $\ast$-product on the cotangent bundle of a Lie group, Letters in Mathematical Physics 7 (1983) 249–258 [doi:10.1007/BF00400441]

• Simone Gutt, Deformation quantization of Poisson manifolds, Geometry and Topology Monographs 17 (2011) 171-220 [gtm:17, pdf, arXiv:math/0003107]

Monographs:

• Stefan Waldmann, Poisson-Geometrie und Deformationsquantisierung, Springer (2007) [doi:10.1007/978-3-540-72518-3]

The Fedosov deformation quantization prescription for deformation quantization of symplectic manifolds and varieties and also of Poisson manifolds that have a regular foliation by symplectic leaves is discussed in

• Boris Fedosov, Formal quantization, Some Topics of Modern Mathematics and their Applications to Problems of Mathematical Physics (in Russian), Moscow (1985), 129-136.

• Boris Fedosov, Index theorem in the algebra of quantum observables, Sov. Phys. Dokl. 34 (1989), 318-321.

• Boris Fedosov, A simple geometrical construction of deformation quantization J. Differential Geom. Volume 40, Number 2 (1994), 213-238. (EUCLID)

For algebraic forms this is discussed in

• Roman Bezrukavnikov, Dmitry Kaledin, Fedosov quantization in algebraic context Mosc. Math. J., Volume 4, Number 3, (2004) 559–592 (AMS pdf)

More discussion of this approach is in

• Claudio Emmrich, Alan Weinstein, The differential geometry of Fedosov’s quantization (arXiv:hep-th/9311094)

A direct and general formula for the deformation quantization of any Poisson manifold was given in

• Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216, (arXiv:q-alg/9709040).

see also

• Peter Banks, Erik Panzer, Brent Pym, Multiple zeta values in deformation quantization (arXiv:1812.11649)

• Kelvin Ritland: Deformation quantization generates all multiple zeta values [arXiv:2409.18450]

This secretly uses the Poisson sigma-model (see there for more details) induced by the given target Poisson Lie algebroid.

A popular exposition of formal deformation quantization with emphasis of the stringy Poisson sigma-model-construction:

• Anton Kapustin, Quantum geometry: The reunion of math and physics [pdf]

The classification of the space of such formal deformation quantization is discussed in

• Vasily Dolgushev, Stable Formality Quasi-isomorphisms for Hochschild Cochains I (arXiv:1109.6031)

• Vasily Dolgushev, Exhausting formal quantization procedures (arXiv:1111.2797)

• Vasily Dolgushev, Christopher Rogers, Thomas Willwacher, Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields (arXiv:1211.4230)

Deformation quantization of polynomial Poisson algebras via universal enveloping algebra (generalizing that of Lie-Poisson structures) is discussed in

• Michael Penkava, Pol Vanhaecke, Deformation Quantization of Polynomial Poisson Algebras, Journal of Algebra 227 1 (2000) 365-393 [arXiv:math/9804022, doi:10.1006/jabr.1999.8239]

Deformation quantization of algebraic varieties is in

• Amnon Yekutieli, Deformation Quantization in Algebraic Geometry, Advances in Mathematics 198 (2005) 383-432.

  Erratum: Advances in Mathematics 217 (2008) 2897-2906.

See also

• MO discussion , Kontsevich’s conjectures on the Grothendieck-Teichmüller group?

Deformation quantization in perturbative quantum field theory is discussed for free field theory is due to

• Joseph Dito, Star-product approach to quantum field theory: The free scalar field. Letters in Mathematical Physics, 20(2):125–134, 1990 (spire)

and further expanded on in

• Michael Dütsch, Klaus Fredenhagen, section 5.1 of Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic quantum field theory and deformation quantization, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (arXiv:hep-th/0101079)

• A. C. Hirshfeld, P. Henselder, Star Products and Perturbative Quantum Field Theory, Annals Phys. 298 (2002) 382-393 (arXiv:hep-th/0208194)

and for interacting perturbative quantum field theory (perturbative AQFT) in

• Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)

The relation to geometric quantization is discussed in

• Eli Hawkins, The Correspondence between Geometric Quantization and Formal Deformation Quantization (arXiv:math/9811049)

• Christoph Nölle, Geometric and deformation quantization (arXiv:0903.5336)

and some remarks on the relation are also in section 1.4 of

• Sergei Gukov, Edward Witten, Branes and Quantization, Adv. Theor. Math. Phys. 13 (2009) 1–73, (arXiv:0809.0305, euclid)

which is about quantization via the A-model.

The formulation of deformation quantization as lifts from P-n operads? over BD-n operads? to E-n operads is discussed in section 2.3 and 2.4 of

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory (wiki, early/partial draft pdf)

See also

• Wikipedia

• F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Quantum mechanics as a deformation of classical mechanics, Lett. Math. Phys. 1 (1975/77), MR674337, doi; Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61–110, MR496157; Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), no. 1, 111–151, MR496158

• M. Flato, A. Lichnerowicz, D. Sternheimer, Deformations of Poisson brackets, Dirac brackets and applications, J. Math. Phys. 17 (1976), no. 9, 1754–1762, MR420723, doi

• D. Arnal, J.-C. Cortet, $\ast$-products in the method of orbits for nilpotent groups, J. Geom. Phys. 2 (1985), no. 2, 83–116, doi

On the stack of deformation quantizations:

• Maxim Kontsevich, Deformation quantization of algebraic varieties (arXiv:math/0106006)

• Pietro Polesello, Pierre Schapira, Stacks of quantization-deformation modules on complex symplectic manifolds (arXiv:math/0305171)

• Amnon Yekutieli, Twisted Deformation Quantization of Algebraic Varieties ,

  arxiv:0905.0488

The action of a motivic Galois group (“cosmic Galois group”) on the space of deformation quantization was observed in

• Maxim Kontsevich, Operads and Motives in Deformation Quantization, Lett. Math. Phys.48:35-72,1999 (arXiv:math/9904055)

See also at motives in physics.