Quantum indistinguishability from general representations of SU(2n)

TL;DR

This paper generalizes the spin-statistics relation in nonrelativistic quantum mechanics using group theory, showing how SU(2n) representations influence the connection between spin and quantum statistics.

Contribution

It reformulates the Berry-Robbins construction in terms of vector bundles derived from SU(2n) representations, revealing how different representations affect the spin-statistics connection.

Findings

The Berry-Robbins construction corresponds to symmetric SU(2n) representations.

General SU(2n) representations can break the usual spin-statistics link.

A formula using Littlewood-Richardson theorem determines allowed spin-statistics combinations.

Abstract

A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. Lond. A (1997) 453, 1771-1790] is generalised within a group-theoretical framework. The construction of Berry and Robbins is re-formulated in terms of certain locally flat vector bundles over n-particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group SU(2n). The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spin-statistics connection is typically broken for general SU(2n) representations, which may admit, for a given value of spin, both bose and fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zero-weight representations of a (generalised) Weyl group of SU(2n). A formula for this decomposition is obtained using the Littlewood-Richardson theorem for the decomposition of representations of U(m+n) into representations of U(m)*U(n).