Canonical Quantization of Two Dimensional Gauge Fields

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 Aμ (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, χ0(W ), is the first excited state Ψ1(Aμ) on h. 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.