Obstruction Results in Quantization Theory

TL;DR

This paper investigates the conditions under which obstructions to quantization arise in Poisson algebras, extending known results for Euclidean space, the sphere, and the torus, and clarifying when consistent quantizations are possible.

Contribution

It provides a comprehensive analysis of obstructions in quantization for various Poisson algebras, highlighting new insights into when such obstructions occur or are absent.

Findings

Obstructions exist for Euclidean space and the sphere.

No obstruction is found for the torus.

The paper clarifies conditions influencing quantization obstructions.

Abstract

We define the quantization structures for Poisson algebras necessary to generalise Groenewold and Van Hove's result that there is no consistent quantization for the Poisson algebra of Euclidean phase space. Recently a similar obstruction was obtained for the sphere, though surprising enough there is no obstruction to the quantization of the torus. In this paper we want to analyze the circumstances under which such obstructions appear. In this context we review the known results for the Poisson algebras of Euclidean space, the sphere and the torus.