Inequivalent Quantizations of Yang-mills Theory on a Cylinder

Yang-Mills theories on a 1+1 dimensional cylinder are considered. It is shown that canonical quantization can proceed following different routes, leading to inequivalent quantizations. The problem of the non-free action of the gauge group on the configuration space is also discussed. In particular we re-examine the relationship between “θ-states” and the fundamental group of the configuration space. It is shown that this relationship does or does not hold depending on whether or not the gauge transformations not connected to the identity act freely on the space of connections modulo connected gauge transformations. To the present day, the understanding of the canonical quantization of non-abelian gauge theories in 3+1 dimensions and the knowledge of their physical degrees of freedom are lacking the necessary rigor. Recent papers [1, 2] have dealt with the somewhat simpler task of quantizing SU(N) or U(N) Yang-Mills theory on a 1+1 dimensional cylinder. These cases are interesting not only because they may cast some light on the features of the 3+1 dimensional case, but also because they deal with the quantization of a gauge theory on a compact spatial manifold. Indeed, it is known [3] that on compact spaces, the action of the gauge group on the space of connections may not be free, causing the phase space to be no longer a manifold but only an orbifold. In relation to both Yang-Mills theories and gravity, some simplified finite-dimensional examples have been worked out in [5] and [6]. In this paper, we describe some novel features that arise in the quantization of pure Yang-Mills theory on a cylinder. It will be shown that this model admits inequivalent quantizations, the ambiguities arising because of two different reasons: 1. Let G denote the gauge group, i.e. the group of all gauge transformations that act on the phase space, and G0 the subgroup of G connected to the identity, which is generated by Gauss law. As in any gauge theory, G0 should leave a physical state invariant. Towards this end, one will have to go to a reduced phase space and/or impose Gauss law on the physical states. In the problem at hand it will be shown that there are inequivalent ways of carrying out this procedure. 2. In our problem, the action of the gauge group is not free and the reduced phase space is an orbifold. After suitably treating the singular points [5], it will be shown that there are different self-adjoint extensions of the relevant operators, and hence different quantum theories. The fact that the action of the gauge group is not free has also another consequence.