CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GAUGE FIELDS

Abstract

SU (N) gauge fields on a cylindrical space-time are canonically quantized via two routes revealing almost equivalent but different quantizations. After removal of all continuous gauge degrees of freedom, the canonical coordinate Aµ (in the Cartan subalgebra [Formula: see text]) is quantized. The compact route, as in lattice gauge theory, quantizes the Wilson loop W, projecting out gauge-invariant wave functions 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 noncompact spectrum. States of the two quantizations with corresponding energy are shifted relative to each other, such that the ground state on G, χ0(W), is the first excited state Ψ1(Aµ) on [Formula: see text]. The ground state Ψ0(Aµ) 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.