Quantization map - Knowlpedia

Definition

Let $(M,\omega)$ be a symplectic manifold, let $\mathcal A\subseteq C^\infty(M,\mathbb R)$ be a Poisson subalgebra containing the constants, and let $\mathscr D$ be a dense subspace of a complex Hilbert space $\mathscr H$ . A quantization map on $\mathcal A$ is a real-linear map

Q_\hbar:\mathcal A\longrightarrow \operatorname{End}(\mathscr D)

whose values are symmetric operators and which satisfies specified classical-to-quantum compatibility conditions. In the exact Dirac form these include

Q_\hbar(1)=I|_{\mathscr D}, \qquad [Q_\hbar(f),Q_\hbar(g)]=i\hbar Q_\hbar(\{f,g\}).

The domain $\mathcal A$ , the common operator domain $\mathscr D$ , and any extra irreducibility or self-adjointness requirements are part of the quantization data.

Covariance under symmetries

Suppose a symplectic Lie group action $\alpha:G\to\operatorname{Symp}(M,\omega)$ is represented by a strongly continuous unitary representation $U_\hbar$ preserving $\mathscr D$ . The quantization is covariant when

U_\hbar(g)Q_\hbar(f)U_\hbar(g)^{-1} =Q_\hbar\!\left(f\circ\alpha(g^{-1})\right).

For a Hamiltonian action with comoment map $J:\mathfrak g\to C^\infty(M)$ , infinitesimal compatibility is commonly expressed by

Q_\hbar(J_X)=i\hbar\,dU_\hbar(X),

with signs adjusted consistently if different Hamiltonian-vector-field or derived-representation conventions are used.

Scope and obstructions

The displayed axioms are a template, not a claim that such a map exists on all smooth observables. Even on linear symplectic space, the exact bracket rule cannot be extended from the basic observables to the full polynomial Poisson algebra while retaining the usual additional requirements; this is the Groenewold–Van Hove obstruction. Geometric, deformation, and semiclassical quantization respond by changing the domain, codomain, or exactness demanded of the correspondence.

References

M. J. Gotay, H. B. Grundling, and G. M. Tuynman, “Obstruction Results in Quantization Theory,” Journal of Nonlinear Science 6 (1996), 469–498. arXiv record. Relevant: §§2–3, definitions of prequantization and quantization of a Poisson algebra.
N. M. J. Woodhouse, Geometric Quantization, 2nd ed., Oxford University Press, 1992. Publisher record. Relevant: Chapters 1–5, classical observables, prequantization, and polarizations.