An Obstruction to Quantization of the Sphere

TL;DR

This paper demonstrates that under a natural axiom, the set of quantization parameters for the symplectic sphere cannot be connected, highlighting an obstruction to continuous quantization.

Contribution

It introduces a new axiom for strict deformation quantization that proves the parameter set for quantizing the sphere is inherently disconnected.

Findings

The set of allowed quantization parameters for the sphere is never connected.

Previous examples showed disconnected parameter sets; this work explains why.

A natural axiom enforces the disconnection of the parameter set.

Abstract

In the standard example of strict deformation quantization of the symplectic sphere $S^2$, the set of allowed values of the quantization parameter $\hbar$ is not connected; indeed, it is almost discrete. Li recently constructed a class of examples (including $S^2$) in which $\hbar$ can take any value in an interval, but these examples are badly behaved. Here, I identify a natural additional axiom for strict deformation quantization and prove that it implies that the parameter set for quantizing $S^2$ is never connected.