nLab deformation quantization

Idea

Broadly, by deformation quantization one means notions of quantization of Poisson manifolds $\big(X, \{-,-\}\big)$ in terms of families 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 $\{-,-\}$.

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 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 $\hbar = 0$ (namely, in geometric quantization the inverse $1/\hbar$ is typically constrained to come in integral multiples).

Therefore, beyond formal deformation quantization there are notions of

• strict deformation quantizations

where $\hbar$ is 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.

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.

Related entries

• geometric quantization

• algebraic quantization

  • formal deformation quantization

  • strict deformation quantization