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 con-densation of Nambu-Goldstone particles. We find that the usual thermodynamic limit is not necessary in order to have the in-equivalent representations needed for the existence of physically disjoint, stable phases of the system. This points to a novel no-tion of spontaneous symmetry breaking, one where the volume can stay finite, an instance that makes our treatment substan-tially 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. (c) 2021 Elsevier Inc. All rights reserved.