Topologically inequivalent quantizations

TL;DR

This paper explores how topological defects in quantum field theory lead to inequivalent representations of the quantization algebra, revealing a new finite-volume topological phase structure with potential implications for quantum gravity.

Contribution

It introduces a novel type of inequivalence in quantum field representations caused by topological structures at finite volume, without requiring the thermodynamic limit.

Findings

Topological defects induce inequivalent quantization representations.

Finite-volume topological phases can exist without thermodynamic limit.

Potential links between topological phases and quantum gravity are discussed.

Abstract

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.