  $SU(N)$ 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.
