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Introduction to Representations of the Canonical Commutation and Anticommutation Relations

Jan Dereziński

Since the early days of quantum mechanics it has been noted that the position operator x and the momentum operator D := −i∇ satisfy the following commutation relation:

Jan 1, 200539 citations

Quantum Indistinguishability: Spin-statistics without Relativity or Field Theory?

Michael Berry, Jonathan M. Robbins

WORLD SCIENTIFIC eBooksJul 23, 20170 citations

Quantum indistinguishability from general representations of SU(2n)

J. M. Harrison, J. M. Robbins

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).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).

arXiv:math-ph/0302037Journal of Mathematical PhysicsMar 17, 20044 citations

Quantum Indistiguishability: Spin Statistics without Relativity or Field Theory?

Michael Berry, Robbins, JM

Jan 1, 20000 citations

Quantum indistinguishability: Spin-statistics without relativity or field theory?

Michael Berry

We review the formulation of quantum mechanics for identical spinning particles with wavefunctions that are singlevalued when permuted configurations are identified. The identification requires the spins to be smoothly permuted along with position variables, so spin is represented in a position-dependent ‘transported basis’, rather than the usual fixed basis. The simplest transported basis, constructed in terms of spins represented as pairs of commuting harmonic oscillators, gives the correct connection between spin and statistics. More complicated constructions can give the wrong exchange sign. The theory is generalized to incorporate additional properties such as isospin, color and strangeness. Some remarks about the relation between this approach and those based on relatitivity and/or field theory are given.We review the formulation of quantum mechanics for identical spinning particles with wavefunctions that are singlevalued when permuted configurations are identified. The identification requires the spins to be smoothly permuted along with position variables, so spin is represented in a position-dependent ‘transported basis’, rather than the usual fixed basis. The simplest transported basis, constructed in terms of spins represented as pairs of commuting harmonic oscillators, gives the correct connection between spin and statistics. More complicated constructions can give the wrong exchange sign. The theory is generalized to incorporate additional properties such as isospin, color and strangeness. Some remarks about the relation between this approach and those based on relatitivity and/or field theory are given.

AIP conference proceedingsJan 1, 20009 citations

On Description of the Exchange and Spin States within the Feynman Representation of Quantum Statistics

С. В. Шевкунов

Doklady PhysicsNov 1, 19996 citations

Quantum indistinguishability: alternative constructions of the transported basis

Abstract

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Quantum indistinguishability: alternative constructions of the transported basis

Abstract

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