Hall's coherent states, the Cameron-Martin theorem, and the quantization of Yang-Mills theory on a circle

TL;DR

This paper explores the quantum reduction of Yang-Mills theory on a circle, employing advanced mathematical tools like Wiener measures, Cameron-Martin theorem generalizations, and Hall's coherent states to understand the physical degrees of freedom.

Contribution

It introduces a novel quantum reduction framework for Yang-Mills theory on a circle, integrating Wiener measures, loop group analysis, and coherent states in a unified approach.

Findings

Finite-dimensional classical reduced phase space identified.

Quantum reduction involves Wiener measure and loop group analysis.

Hall's coherent states facilitate the quantization process.

Abstract

We discuss the classical and quantum reduction to the space of physical degrees of freedom of Yang--Mills theory on a circle (so that space-time is a cylinder). Although the classical reduced phase space is finite-dimensional, the quantum reduction procedure is mathematically fascinating, involving firstly the Wiener measure on a loop group, secondly a generalization of the Cameron--Martin theorem to loop groups, and thirdly Hall's coherent states for compact Lie groups. Our approach is based on a quantum analogue of the classical Marsden--Weinstein symplectic reduction process.