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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 $\hbar$ (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.
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
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
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.
We first give the traditional
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
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
Let $M$ be a Poisson manifold and let $A = C^\infty(M)$ be the Poisson algebra of smooth functions.
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
where $B_n$ are bilinear maps on $A$ such that $B_0(a,b) = ab$.
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.
(…)
(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 |
(…)
(Costello-Gwilliam, chapter 5)
(…)
(…) C-star algebraic deformation quantization (…)
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). 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.
(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.
Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this:
(Swan, Yekutieli). There is a canonical isomorphism
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.
(Yekutieli).
There is a canonical isomorphism
in the derived category of $\mathcal{O}_{X^2}$-modules.
There is a canonical quasi-isomorphism of complexes of $\mathcal{O}_{X^2}$-modules
Therefore there is a canonical isomorphism
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
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:
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 algebras
and
(Yekutieli). There is an $\mathrm{L}_{\infty}$ quasi-isomorphism
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:
(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
called quantization. It preserves first order brackets.
A proof of this theorem when $X$ is affine is here. For the full statement see 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.
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
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
where $|f| = p$ when $f \in C^p(V,V)$. This defines a graded Lie bracket of degree -1.
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
write this out and we get the equation
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.
(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:
The bracket in Hochschild cohomology (Gerstenhaber bracket) goes to the Schouten bracket:
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 $H^\ast(A,A)[ [ t ] ]$.
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.
(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 $L_2$.
(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 (!).
We have $(C^*(A,A), d_\mu)$, the Gerstenhaber bracket, and we also have a cup product
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:
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.
(Deligne conjecture). $C^*(A,A)$ is a $G_\infty$-algebra, which is a Gerstenhaber algebra up to coherent homotopy.
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 ober suitable manifolds/varieties $X$ there is an equivalence of L-∞ algebras
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.
deformation quantization, geometric quantization
duality between algebra and geometry in physics:
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 | 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 |
The concept of algebraic deformation quantization originates around
A 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
More discussion of this approach is in
A direct and general formula for the deformation quantization of any Poisson manifold was given in
This secretly uses the Poisson sigma-model (see there for more details) induced by the given target Poisson Lie algebroid.
The classification of the space of such formal deformation quantization is discussed in
See also
Deformation quantization in quantum field theory in the context of AQFT is discussed in
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)
and some remarks on the relation are also in section 1.4 of
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
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
See also at motives in physics.