Constrained quantization and $\theta$-angles

TL;DR

This paper introduces a rigorous quantization method for constrained gauge systems, demonstrating how $ heta$-angles naturally arise, with a detailed example in two-dimensional Minkowski gauge theory on a cylinder.

Contribution

It applies a new quantization approach based on symplectic reduction and Wiener measure, elucidating the emergence of $ heta$-angles in gauge theories.

Findings

Quantization via symplectic reduction expressed through gauge group integrals.

Construction of the inner product using Wiener measure on loop groups.

Illustration of $ heta$-angle emergence in a specific gauge theory example.

Abstract

We apply a new and mathematically rigorous method for the quantization of constrained systems to two-dimensional gauge theories. In this method, which quantizes Marsden-Weinstein symplectic reduction, the inner product on the physical state space is expressed through a certain integral over the gauge group. The present paper, the first of a series, specializes to the Minkowski theory defined on a cylinder. The integral in question is then constructed in terms of the Wiener measure on a loop group. It is shown how $\th$-angles emerge in the new method, and the abstract theory is illustrated in detail in an example.