Remarks on the naturality of quantization

TL;DR

This paper investigates how the choice of almost complex structure affects the geometric quantization of symplectic manifolds, analyzing the curvature of the associated bundle and its semi-classical limit, linking symplectomorphisms to quantum evolution.

Contribution

It provides a detailed analysis of the dependence of quantization on complex structures and explores the semi-classical behavior of the curvature in this context.

Findings

Curvature of the quantum bundle has a well-defined semi-classical limit.

Parallel transport relates symplectomorphisms to Schrödinger evolution.

Dependence of quantum spaces on complex structures is explicitly characterized.

Abstract

Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's constant, the Hilbert spaces of the quantization form a bundle over U with a natural connection. In this paper we examine the dependence of the Hilbert spaces on the choice of J, by computing the semi-classical limit of the curvature of this connection. We also show that parallel transport provides a link between the action of the group Symp(M) of symplectomorphisms of M and the Schrodinger equation.