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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.)
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 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 |
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:
The notion of strict $C^\ast$-algebraic deformation quantization was introduced in:
Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Communications in Mathematical Physics 122 (1989) 531–562 [doi:10.1007/BF01256492]
(motivated by noncommutative tori)
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) [pdf, pdf MR91h:46120]
Marc Rieffel, Deformation quantization for actions of $\mathbb{R}^d$, Mem. Amer. Math. Soc. 106 506 (1993) [ams:memo-106-506, MR94d:46072]
Marc Rieffel, Quantization and $C^\ast$-algebras, Contemporary Mathematics 167 (1994) [pdf, pdf]
See also:
Specifically on Lie-group algebras as strict deformation quantization of Lie-Poisson structures:
and generalization to groupoid algebras of Lie groupoids integrating given Lie algebroids:
Nicolaas P. Landsman, Ex. 2 in: Lie Groupoid $C^\ast$-Algebras and Weyl Quantization, Communications in Mathematical Physics 206 (1999) 367–381 [doi:10.1007/s002200050709]
Nicolaas P. Landsman, B. Ramazan, Ex. 11.1 in: Quantization of Poisson algebras associated to Lie algebroids, in: Groupoids in Analysis, Geometry, and Physics, Contemporary Mathematics 282 (2001) 159-192 [arXiv:math-ph/0001005, ams:conm/282]
see also:
Review:
Marc Rieffel, Questions on quantization, Contemp. Math. 228 (1998) 315-326 [arXiv:quant-ph/9712009]
Nicolaas P. Landsman, Classical and quantum representation theory [arXiv:hep-th/9411172]
Comparative review of notions of strict deformation quantization:
Discussion of strict deformation quantization in terms of geometric quantization of symplectic groupoids via polarized twisted groupoid convolution algebras is in
For the special case of Moyal deformation quantization [Hawkins (2008b), section 6.2] this construction had been suggested without proof in
and a detailed proof was given in
see also
On continuous fields of Weyl algebras as continuous deformation quantizations of symplectic vector spaces:
On group algebras of (underlying discrete) Heisenberg groups as strict deformation quantizations of pre-symplectic topological vector spaces by continuous fields of Weyl algebras:
Ernst Binz, Reinhard Honegger, Alfred Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic strict deformation quantization, International Journal of Pure and Applied Mathematics 38 1 (2007) [ijpam:2007-38-1/6, pdf]
Reinhard Honegger, Alfred Rieckers, Heisenberg Group Algebra and Strict Weyl Quantization, Chapter 23 in: Photons in Fock Space and Beyond, Volume I: From Classical to Quantized Radiation Systems, World Scientific (2015) [chapter:doi;10.1142/9789814696586_0023, book:doi:10.1142/9251-vol1]
Last revised on January 13, 2024 at 13:32:42. See the history of this page for a list of all contributions to it.