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AIP conference proceedings

A group-theoretic approach to constructions of non-relativistic spin-statistics

J M Harrison

Published Jan 1, 2000• 1 citation
Publisher

Abstract

We give a group-theoretical generalization of Berry and Robbins’ treatment of identical particles with spin. The original construction, which leads to the correct spin-statistics relation, is seen to arise from particular irreducible representations—the totally symmetric representations—of the group SU(4). Here we calculate the exchange signs and corresponding statistics for all irreducible representations of SU(4).

DOI
10.1063/1.1337714

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We give a group-theoretical generalization of Berry and Robbins’ treatment of identical particles with spin. The original construction, which leads to the correct spin-statistics relation, is seen to arise from particular irreducible representations—the totally symmetric representations—of the group SU(4). Here we calculate the exchange signs and corresponding statistics for all irreducible representations of SU(4).
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).
An exchange operator is defined in terms of rotation and displacement operators. This operator contains a rotation around the z axis that produces a factor (−1)2s in front of the wave function. By postulating that such a rotation must produce the same value for the wave function if the particles are identical, we obtain the usual connection between spin and statistics. Particle states lying in the irreducible representations of the three-dimensional rotation group are seen to allow only Fermi and Bose statistics.
It is shown that the symmetry under parity of the wavefunctions of two identical particles with an arbitrary spin sss in three spatial dimensions accounts for the appropriate wavefunction exchange statistics under the permutations of particles. The standard properties of the angular momentum in non-relativistic quantum mechanics account for the sign factor (−1)2s(-1)^{2s}(−1)2s that the wavefunctions acquire under the permutation of coordinates of the two particles, without any additional requirements, directly relating spin and the particle exchange statistics in the non-relativistic context.