Identical Particles in Quantum Mechanics: Operational and Topological Considerations

TL;DR

This paper explores the existence of anyons in quantum mechanics, demonstrating they are compatible with the Symmetrization Postulate and arise from topological effects in two-dimensional spaces.

Contribution

It introduces a novel approach for quantizing multiply-connected spaces, explaining how anyonic behavior can emerge without particle identity.

Findings

Anyons are compatible with the Symmetrization Postulate.

Topological properties of configuration space influence particle behavior.

A new quantization method for multiply-connected spaces was developed.

Abstract

This dissertation reports our investigation into the existence of anyons, which interpolate between bosons and fermions, in light of the Symmetrization Postulate, which states that only the two extremes exist. The Symmetrization Postulate can be understood as asserting that there are only two consistent ways of combining the behavior of distinguishable particles to obtain the behavior of identical ones. We showed that anyonic behavior then arises because of the way in which the probability amplitudes of distinguishable particles in two dimensions are affected by the topology of the space. These can then be combined in one of the ways arising from the Symmetrization Postulate, to form identical anyons. Therefore, anyons do not invalidate the Symmetrization Postulate, but are entirely compatible with it. In order to show that anyons can arise without particle identity, we investigated how distinguishable particles can gain particular properties from the topology of their configuration space. We managed to do so, and in the process, discovered a novel approach for quantizing multiply-connected spaces, in a way that is more suited to anyons than the standard approach.