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WORLD SCIENTIFIC eBooks

Indistinguishability for quantum particles: spin, statistics and the geometric phase

Michael Berry, Jonathan M. Robbins

Published Jul 23, 2017• 5 citations
Publisher
DOI
10.1142/9789813221215_0008

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Indistinguishability for quantum particles: spin, statistics and the geometric phase

Michael Berry, Jonathan M. Robbins

The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.

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

Michael Berry, Jonathan M. Robbins

WORLD SCIENTIFIC eBooks•Jul 23, 2017•0 citations
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand–Naimark and Serre–Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wavefunctions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand–Naimark and Serre–Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wavefunctions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.
The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.