Canonical Group Quantization, Rotation Generators and Quantum Indistinguishability

TL;DR

This paper employs canonical group quantization to construct angular momentum operators for systems with non-trivial topology, elucidating their relation to quantum indistinguishability and the spin-statistics connection.

Contribution

It introduces a formalism for deriving angular momentum operators on topologically non-trivial spaces, connecting to existing approaches and advancing understanding of quantum indistinguishability.

Findings

Operators for a sphere correspond to monopole angular momentum.

Operators for the projective plane relate to indistinguishable particles.

Formalism clarifies the spin-statistics connection in non-relativistic quantum mechanics.

Abstract

Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the obtained angular momentum operators are the quantum version of Poincare's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the obtained operators represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. We explicitly show how our formalism relates to the one developed by Berry and Robbins. The relevance of the proposed formalism for an advance in our understanding of the spin-statistics connection in non-relativistic quantum mechanics is discussed.