nLab deformation quantization



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Broadly, by deformation quantization one means notions of quantization of Poisson manifolds (X,{,})\big(X, \{-,-\}\big) in terms of families of non-commutative algebras A A_{\hbar} parameterized by specific admissible (formal) values of Planck's constant \hbar including =0\hbar = 0, where A 0=C (X)A_0 = C^\infty(X) is the ordinary commutative algebra of functions on the underlying smooth manifold. The idea is that A 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 0A_0 in a way that increases with \hbar and is infinitesimally controlled by the given Poisson bracket {,}\{-,-\}.

Deformation quantization is often taken by default to refer to the historically first notion of

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 ϵ +\epsilon \in \mathbb{R}_+ thus yielding actual (non-formal) deformation quantizations for <ϵ\hbar \lt \epsilon, but in practice this happens rarely (see reference here). 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 =0\hbar = 0 (namely, in geometric quantization the inverse 1/1/\hbar is typically constrained to come in integral multiples).

Therefore, beyond formal deformation quantization there are notions of

where \hbar is allowed to take discrete positive real values with an accumulation point at =0\hbar = 0, and where to each such value is associated an actual C*-algebra-of observables.

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.

Last revised on March 20, 2023 at 12:07:44. See the history of this page for a list of all contributions to it.