nLab C* algebraic deformation quantization

Idea

What is called the strict or $C^\ast$-algebraic form of deformation quantization (sometimes just: strict quantization) is an attempt to formalize quantization of phase spaces or more generally of Poisson manifolds by continuously deforming, in a precise sense, their commutative algebras of functions (algebras of observables) to non-commutative C*-algebras whose commutators are, to “first order” in a suitable sense, determined by the given Poisson bracket.

This is in contrast to formal deformation quantization, where one asks not for C*-algebras but just for formal power series algebras. Where formal deformation quantization is perturbative quantization (perturbation in Planck's constant, see Collini (2016)), strict deformation quantization is meant to reflect non-perturbative quantization.

In general, by deformation quantization one means notions of quantization of Poisson manifolds $\big(X, \{-,-\}\big)$ in terms of sequences of non-commutative algebras $A_{\hbar}$ parameterized by specific admissible (formal) values of Planck's constant $\hbar$ including $\hbar = 0$, where $A_0 = C^\infty(X)$ is the ordinary commutative algebra of functions on the underlying smooth manifold. The idea is that $A_{\hbar}$ is the algebra of observables of the corresponding quantum system at that value of $\hbar$, arising from “deforming” the commutative product of $A_0$ in a way that increases with $\hbar$ and is infinitesimally controlled by the given Poisson bracket $\{-,-\}$.

Beware that the term “deformation quantization” is often taken by default to refer to the historically first notion of formal deformation quantization, where $\hbar$ is just a formal variable (i.e. an infinitesimal, but not an actual number) and the underlying vector space of all algebras in question is that of formal power series in $\hbar$.

One might naively imagine that the formal power series appearing in formal deformation quantization have a finite radius of convergence $\epsilon \in \mathbb{R}_+$ thus yielding actual (non-formal) deformation quantizations for $\hbar \lt \epsilon$, but in practice this happens rarely (see the first references below). Indeed, geometric quantization makes manifest that prequantization conditions typically force admissible values of $\hbar$ to form a discrete subspace of $\mathbb{R}_+$ with only an accumulation point at $\hbar = 0$.

Therefore, in strict or $C^\ast$-algebraic deformation quantization the parameter $\hbar$ is typically allowed to take discrete positive real values with an accumulation point at $\hbar = 0$, and where to each such value is associated an actual C*-algebra-of observables. There are a variety of similar but different proposals for what exactly this should mean in detail, see Hawkins (2008a), Section 2 for overview and references.

In its focus on algebras of observables the notion of deformation quantization is roughly dual to geometric quantization, which primarily constructs the spaces of quantum states. In special sitations both notions are compatible, but in general there is a large amount of ambiguity in quantization, between but also within the different approaches.

Typically the $C^\ast$-algebraic deformation takes the quantum algebra to be a suitable convolution algebra of suitably polarized sections over a Lie groupoid that Lie integrates a Poisson Lie algebroid which encodes the original Poisson bracket to be quantized [Hawkins (2008b)], see at geometric quantization of symplectic groupoids.

While there are good examples of strict $C^\ast$-algebraic deformation quantization for toy examples such as low spacetime dimension (notably quantum mechanics) to date no examples of interacting field theories in spacetime dimension $\geq 4$ have a known non-perturbative quantization. (For the case of Yang-Mills theory/QCD the construction of its non-perturbative quantization is one of the open “Millennium Problems” listed by the Clay Mathematics Institute, see at quantization of Yang-Mills theory.)

Properties

Relation to formal deformation quantization

Under favorable circumstances, one can form from a strict $C^\ast$-algebraic deformation quantization given by a continuous field of C*-algebras over a subset of the interval the “differentiation” as $\hbar \in [0,1]$ tends to 0, such that this reproduces a formal deformation quantization.

Conversely, a natural intuition might be that given a formal deformation quantization then the subalgebra of converging power series inside all formal power series has a completion to a C*-algebra which constitutes a strict deformation quantization.

While this seems natural, the only actual example where this is understood to date seems to be the simple case of the standard Poisson structure on $\mathbb{R}^{2n}$ with its Weyl algebra star product. (See this MO discussion).

Examples

• deformation quantization of the 2-sphere

Related concepts

• geometric quantization of symplectic groupoids

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

Converging formal deformation quantization

On convergence of formal power series in formal deformation quantization:

• M. Bordemann, M. Brischle, C. Emmrich, Stefan Waldmann, Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for $\mathbb{C}P^n$ (arXiv:q-alg/9512019)

• H. Omori, Y. Maeda, N. Miyazaki, A. Yoshida, Deformation quantization of Fréchet-Poisson algebras of Heisenberg type 2001 (pdf)

Textbook account:

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

Strict deformation quantization proper

The notion of strict $C^\ast$-algebraic deformation quantization was introduced in

• Marc Rieffel, Quantization and $C^\ast$-algebras, Contemporary mathematics vol. 167 (1994) (pdf)

A brief review with a list of open questions is in

• Marc Rieffel, Questions on quantization, Contemp.Math. 228 (1998) 315-326 (arXiv:quant-ph/9712009)

More details are in

• Marc Rieffel, Deformation quantization and operator algebras, in: Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), 411–423, Proc. Sympos. Pure Math. 51, Part 1, Amer. Math. Soc. 1990, MR91h:46120; (pdf)

• Marc Rieffel, Deformation quantization for actions of $\mathbb{R}^d$, Mem. Amer. Math. Soc. 106 (1993), no. 506, x+93 pp. MR94d:46072

Comparative review of notions of strict deformation quantization:

• Eli Hawkins, Section 2 of: An Obstruction to Quantization of the Sphere, Communications in Mathematical Physics 283 (2008) 675–699 [arXiv:0706.2946, doi:10.1007/s00220-008-0517-2]

Discussion of strict deformation quantization in terms of geometric quantization of symplectic groupoids via polarized twisted groupoid convolution algebras is in

• Eli Hawkins, A Groupoid Approach to Quantization, J. Symplectic Geom. 6 1 (2008) 61-125 [arXiv:math/0612363, euclid]

For the special case of Moyal deformation quantization [Hawkins (2008b), section 6.2] this construction had been suggested without proof in

• Alan Weinstein, in P. Donato et al. (eds.) Symplectic Geometry and Mathematical Physics, (Birkhäuser, Basel, 1991); p. 446.

and a detailed proof was given in

• José Gracia-Bondia, Joseph Varilly, From geometric quantization to Moyal quantization, J.Math.Phys. 36 (1995) 2691-2701 (arXiv:hep-th/9406170)

see also

• Martin Bordemann, Eckhard Meinrenken, Martin Schlichenmaier, Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$, Commun.Math.Phys. 165 (1994) 281-296 (arXiv:hep-th/9309134)

For the special case of Poisson manifolds that are total spaces of Lie algebroids, discussion is in

• Klaas Landsman, B. Ramazan, Quantization of Poisson algebras associated to Lie algebroids (arXiv:math-ph/0001005)

• Klaas Landsman, Strict deformation quantization of a particle in external gravitational and Yang-Mills fields, Journal of Geometry and Physics 12:2, p. 93-132 (web)