Canonical Quantization of Constants of Motion

TL;DR

This paper introduces a new quantization method called decomposable Weyl quantization that preserves classical constants of motion in quantum systems, linking classical reduction and quantum diagonalization.

Contribution

It develops a novel quantization approach ensuring preservation of constants of motion, with a spectral decomposition method and applications to spectral theory and geometric quantization.

Findings

Spectral decomposition of quantum constants of motion obtained directly from quantization.

Characterization of when quantization preserves constants of motion via Wigner transforms.

Application of the method to spectral theory and insights into deformation and geometric quantization.

Abstract

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.