Inequivalent Quantizations of Yang-Mills Theory on a Cylinder

TL;DR

This paper explores different inequivalent ways to quantize Yang-Mills theories on a 1+1 dimensional cylinder, highlighting how gauge group actions influence the structure of quantum states and the role of theta states.

Contribution

It clarifies the conditions under which theta states relate to the fundamental group of the configuration space in Yang-Mills theory on a cylinder.

Findings

Different quantization routes lead to inequivalent theories.

The relationship between theta states and the fundamental group depends on gauge group action.

Non-free gauge group actions affect the structure of the quantum configuration space.

Abstract

Yang-Mills theories on a 1+1 dimensional cylinder are considered. It is shown that canonical quantization can proceed following different routes, leading to inequivalent quantizations. The problem of the non-free action of the gauge group on the configuration space is also discussed. In particular we re-examine the relationship between ``$\theta$-states" and the fundamental group of the configuration space. It is shown that this relationship does or does not hold depending on whether or not the gauge transformations not connected to the identity act freely on the space of connections modulo connected gauge transformations.