Yang-Mills theory and the Segal-Bargmann transform

TL;DR

This paper applies a specialized Segal-Bargmann transform to rigorously analyze the Hamiltonian in Yang-Mills theory on a space-time cylinder, confirming its reduction to the Laplacian on the gauge group.

Contribution

It introduces a rigorous approach using a variant of the Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a cylinder.

Findings

The transform provides a rigorous framework for the Hamiltonian on the gauge-invariant subspace.

The classical Segal-Bargmann transform reduces to the generalized transform for the structure group.

Confirms the Hamiltonian's reduction to the Laplacian on the gauge group.

Abstract

We use a variant of the classical Segal-Bargmann transform to understand the canonical quantization of Yang-Mills theory on a space-time cylinder. This transform gives a rigorous way to make sense of the Hamiltonian on the gauge-invariant subspace. Our results are a rigorous version of the widely accepted notion that on the gauge-invariant subspace the Hamiltonian should reduce to the Laplacian on the compact structure group. We show that the infinite-dimensional classical Segal-Bargmann transform for the space of connections, when restricted to the gauge-invariant subspace, becomes the generalized Segal-Bargmann transform for the the structure group.