  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 phases of the
system. This is a new type of inequivalence, due to the nontrivial topological
structure of the phase space, that appears at finite volume. We regard this as
a first step towards a unifying view of topological and thermodynamic phases,
and offer here comments on the possible application of this scenario to quantum
gravity.
