Physics – High Energy Physics – High Energy Physics - Lattice
Scientific paper
1993-05-26
Int.J.Mod.Phys. A9 (1994) 3153-3178
Physics
High Energy Physics
High Energy Physics - Lattice
32 pages, 3 figures uuencoded, Plain TeX
Scientific paper
$SU(N)$ 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.
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