Topologically inequivalent quantizations

We discuss the representations of the algebra of quantization, the canonical commutation relations, in a scalar quantum field theory with spontaneously broken U(1) internal symmetry, when a topological defect of the vortex type is formed via the condensation of Nambu-Goldstone particles. We find that the usual thermodynamic limit is not necessary in order to have the inequivalent representations needed for the existence of physically disjoint, stable phases of the system. This points to a novel notion of spontaneous symmetry breaking, one where the volume can stay finite, an instance that makes our treatment substantially different from the usual semiclassical (NOLGA) approach to vortices. This new type of inequivalence is different from the well-known inequivalence occurring for the quantum particle on the circle. We finally comment on possible applications to quantum gravity.