On the geometry of quantum indistinguishability

TL;DR

This paper introduces an algebraic framework for understanding quantum indistinguishability on spaces with finite fundamental groups, linking geometric properties to algebraic structures and revisiting foundational quantum principles.

Contribution

It develops a novel algebraic approach using Gelfand-Naimark and Serre-Swan equivalences to analyze quantum systems with finite fundamental groups, providing new insights into indistinguishability and spin-statistics.

Findings

Reformulation of quantum indistinguishability in algebraic terms

Critical analysis of wave function single-valuedness

Potential applications to quantization and phase transitions

Abstract

An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wave functions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.