Representations of the $SU(N)$ $T$-algebra and the loop representation in $1+1$-dimensions

Abstract

We consider the phase-space of Yang-Mills on a cylindrical space-time ($S^1\times \mathbf{R}$) and the associated algebra of gauge-invariant functions, the $T$-variables. We solve the Mandelstam identities both classically and quantum-mechanically by considering the $T$-variables as functions of the eigenvalues of the holonomy and their associated momenta. It is shown that there are two inequivalent representations of the quantum $T$-algebra. Then we compare this reduced phase space approach to Dirac quantization and find it to give essentially equivalent results. We proceed to define a loop representation in each of these two cases. One of these loop representations (for $N=2$) is more or less equivalent to the usual loop representation.

15 pages, LaTeX, 1 postscript figure included, uses epsf.sty, Göteborg ITP 93-39

References in corpus (1)

• Loop Variable Inequalities in Gravity and Gauge Theory

Cited by in corpus (2)

• Towards a Many-Body Treatment of Hamiltonian Lattice SU(N) Gauge Theory
• Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables