The Hilbert space of the Chern–Simons theory on a cylinder: a loop quantum gravity approach

Abstract

As a laboratory for loop quantum gravity, we consider the canonical quantization of the three-dimensional Chern–Simons theory on a noncompact space with the topology of a cylinder. Working within the loop quantization formalism, we define at the quantum level the constraints appearing in the canonical approach and completely solve them, thus constructing a gauge and diffeomorphism invariant physical Hilbert space for the theory. This space turns out to be infinite dimensional, but separable.