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Physical Review A

Structure of Berry’s phase: Some group-theoretical examples

Stefan Giller, Cezary Gonera, Piotr Kosiński, Paweł Maślanka

Published Aug 1, 1993• 6 citations
Publisher

Abstract

The group-theoretical treatment of Berry's aholonomy developed in a paper of Giller, Kosi\ifmmode \acute{n}\else \'{n}\fi{}ski, and Szijmanowski [Int. J. Mod. Phys. A 4, 1453 (1989)] is applied to some specific examples. The general structure of the corresponding Berry matrices is determined. All irreducible representations of SO(n) are found for which the Berry matrices parametrized by the points of ${\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$\ensuremath{\simeq}SO(n)/SO(n-1) are determined by the Riemannian connection on ${\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$.

DOI
10.1103/physreva.48.907

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The group-theoretical treatment of Berry's aholonomy developed in a paper of Giller, Kosi\ifmmode \acute{n}\else '{n}\fi{}ski, and Szijmanowski [Int. J. Mod. Phys. A 4, 1453 (1989)] is applied to some specific examples. The general structure of the corresponding Berry matrices is determined. All irreducible representations of SO(n) are found for which the Berry matrices parametrized by the points of Sn\ensuremath−1{\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}Sn\ensuremath−1\ensuremath{\simeq}SO(n)/SO(n-1) are determined by the Riemannian connection on Sn\ensuremath−1{\mathit{S}}^{\mathit{n}\mathrm{\ensuremath{-}}1}Sn\ensuremath−1.
A Short Introduction to: Connections on U \mathbb{U} U (1) Bundles and Berry’s Phase.
A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)] is generalized within a group-theoretical framework. The construction of Berry and Robbins is reformulated 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 (generalized) 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).