Canonical Quantization of Two Dimensional Gauge Fields

TL;DR

This paper compares two methods of canonical quantization for 2D $SU(N)$ gauge fields on a cylinder, revealing differences in spectra and implications for lattice gauge theory and 2D QCD.

Contribution

It introduces and contrasts two quantization approaches, showing how gauge invariance and spectra relate, with implications for understanding 2D gauge theories.

Findings

Compact and non-compact spectra are related but shifted.

Ground state on the group manifold appears as an excited state in the Cartan subalgebra.

Implications for lattice gauge theory and 2D QCD representations.

Abstract

$SU(N)$ gauge fields on a cylindrical spacetime are canonically quantized via two routes revealing almost equivalent but different quantizations. After removing all continuous gauge degrees of freedom, the canonical coordinate $A_\mu$ (in the Cartan subalgebra $\h$) is quantized. The compact route, as in lattice gauge theory, quantizes the Wilson loop $W$, projecting out gauge invariant wavefunctions on the group manifold $G$. After a Casimir energy related to the curvature of $SU(N)$ is added to the compact spectrum, it is seen to be a subset of the non-compact spectrum. States of the two quantizations with corresponding energy are shifted relative each other, such that the ground state on $G$, $\chi_0(W)$, is the first excited state $\Psi_1(A_\mu)$ on $\h$. The ground state $\Psi_0(A_\mu)$ does not appear in the character spectrum as its lift is not globally defined on $G$. Implications for lattice gauge theory and the sum over maps representation of two dimensional QCD are discussed.