nLab
deformation quantization

Idea

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.

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 $\hslash$ (physically: Planck's constant) of the product in the deformed algebra of observables.

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

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.

As any other quantization, the deformation quantization has as an 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 $\hslash$, 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 $ℝ[[\hslash ]]$ of a formal parameter $\hslash$ (thought of as Planck's constant) such that the limit $\hslash \to 0$ 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 $\hslash =1$ such that all expressions in $\hslash$ 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 then discussed in

• Deformation quantization of field theory to factorization algebras of observables

Explicit definition of deformation of Poisson manifolds/Poisson algebras

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

Definition

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

(1)$$a*b=\sum _{n=0}^{\mathrm{\infty }}{B}_n(a,b)t^n$$a \ast b = \sum_{n=0}^\infty B_n(a,b) t^n

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

Definition

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

Formulation as lifts from $P_n$-algebras to ${\mathrm{BD}}_n$-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 of field theory to factorization algebras of observables

(…)

(Costello-Gwilliam, chapter 5)

(…)

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 03).

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). 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.

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

(2)$$(f\circ g)(v_1,\dots v_{p+q-1})=f(v_1,\dots ,v_{i-1},g(v_i,\dots ,v_{i+q-1}),v_{i+q},\dots ,v_{p+q-1}).$$(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 $f\in C^{*}(V,V)$, 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

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

where $\mid f\mid =p$ 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 $(C^{*}(V,V),d_{\mu })$ 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 $\theta (a,b)=\mathrm{ab}+c(a,b)$. What we want is that

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

write this out and we get the equation

(5)$$d_{\mu }c+c\circ c=0,$$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 $C^{*}(A,A)[[t]]$. One should note that $d_{\mu }$ is actually a derivation of the Lie bracket, hence we have a dg-Lie algebra.

Theorem

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

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

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

(7)$$[\cdot ,\cdot {]}_G\mapsto [\cdot ,\cdot {]}_S.$$[ \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 ${\mathrm{HH}}^{*}(A,A)$ of $A$, $d_{\mu }P\mapsto 0$ and $[P,P{]}_S=0$, so we have a solution to M-C in $H^{*}(A,A)[[t]]$.

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 $L_1\leftarrow M\to L_2$. 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). $C^{*}(A,A)[[t]]$ is quasi-isomorphic to $H^{*}(A,A)[[t]]$. ($A=C^{\mathrm{\infty }}(M)$)

Hence there is a solution to M-C in $C^{*}(A,A)[[t]]$, 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

(8)$$(f\cup g)(a_1,\dots ,a_{p+q})=\mu (f(a_1,\dots ,a_p),g(a_{p+1},\dots ,a_{p+q}))$$(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 ${\mathrm{HH}}^{*}(A,A)$, 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 $G_{\mathrm{\infty }}$-algebra, which is a Gerstenhaber algebra up to coherent homotopy.

Examples

• Moyal quantization

Related concepts

• reductions deformations resolutions in physics

• quantization

  • deformation quantization, geometric quantization

  • path integral

  • semiclassical approximation

• Jordan algebra

duality between algebra and geometry in physics:

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

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ		smooth function
tangent vector		jet		germ of curve		curve
Lie algebra		formal group		local Lie group		Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

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).

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

Deformation quantization in quantum field theory in the context of AQFT is discussed in

• 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)

The relation 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)

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, $*$-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