arXiv subjects
Publications and source records attributed to J. Hallin.
The kinematics of SL(2,R) Yang-Mills theory on a circle is considered, for reasons that are spelled out. The gauge transformations exhibit hyperbolic fixed points, and this results in a physical configuration space with a non-Hausdorff "network" topology. The ambiguity encountered in canonical quantization is then much more pronounced than in the compact case, and can not be resolved through the kind of appeal made to group theory in that case.
We consider the phase-space of Yang-Mills on a cylindrical space-time ($S^1 \times {\bf R}$) 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.
We consider $SU(N)$ Yang-Mills on a circle (cylindrical space-time) and quantize the eigenvalues of the holonomy. In this way the Mandelstam identities associated with the holonomy are trivially solved. Furthermore we indicate that there are exactly two physically inequivalent representations of the algebra of gauge-invariant operators, resulting in different spectra.