Comments on a Full Quantization of the Torus

TL;DR

This paper critically examines a geometric quantization of the torus, highlighting issues with physical acceptability due to unbounded spectra of classically bounded observables, thus questioning the quantization's physical relevance.

Contribution

It analyzes Gotay's full quantization of the torus, revealing fundamental limitations and arguing against its physical viability due to spectral unboundedness.

Findings

Quantization produces unbounded operators for bounded classical observables

The representation effectively lifts the compactness constraints of the torus

Questions the physical acceptability of Gotay's quantization approach

Abstract

Gotay showed that a representation of the whole Poisson algebra of the torus given by geometric quantization is irreducible with respect to the most natural overcomplete set of observables. We study this representation and argue that it cannot be considered as physically acceptable. In particular, classically bounded observables are quantized by operators with unbounded spectrum. Effectively, the latter amounts to lifting the constraints that compactify both directions in the torus.