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AIP conference proceedings

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

Michael Berry

Published Jan 1, 2000• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.1337708

Related papers

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: alternative constructions of the transported basis

Michael Berry, Jonathan M. Robbins

Our earlier arguments (Berry M V and Robbins J M 1997 Proc. R. Soc. Lond. A 453 1771-90) leading to the spin-statistics relation are summarized and then revisited. Constructions are described that satisfy all our previous requirements but lead to the wrong exchange sign (one such alternative is the replacement of commutators by anticommutators in the Schwinger representation of the spins of the particles); we suggest why these might be unacceptable.

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Identical Particles in Quantum Mechanics Revisited

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The treatment of identical particles in quantum mechanics rests on two (related) principles: the spin‐statistics connection and the Symmetrization Postulate. In light of recent theories (such as q‐deformed commutators) that allow for ‘small’ violations of the spin‐statistics connection and the Symmetrization Postulate, we revisit the issue of how quantum mechanics deals with identical particles and how it supports or fails to support various philosophical stances concerning individuality. As a consequence of the expanded possibilities for quantum statistics, we argue that permutation symmetry is best formulated as a formal property of the state function describing the system of particles rather than as a property of the individual particles.1 Introduction2 Philosophical background2.1 Important terminology2.1.1 Identity2.1.2 Indistinguishability2.1.3 Indiscernibility2.2 When are particles indistinguishable?2.3 The Principle of the Identity of Indiscernibles and quantum mechanics2.4 The Principle of the Identity of Indiscernibles and logic2.5 Particle history2.6 Transcendental individuality3 Some quantum formalism3.1 The Principle of Permutation Invariance and the Symmetrization Postulate3.2 The configuration‐space approach3.3 Commutators and anticommutators, and identical particle statistics3.4 Q‐mutators4 Identical particle statistics: a holistic point of view5 Conclusions

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Identity, Geometry, Permutation And The Spin-Statistics Theorem

Michael J. York

We examine historic formulations of the spin-statistics theorem from a point of view that distinguishes between the observable consequences and the ``symmetrization postulate''. In particular, we make a critical analysis of concepts of particle identity, state distinguishability and permutation, and particle ``labels''. We discuss how to construct unique state vectors and the nature of the full state descriptions required for this -- in particular the elimination of arbitrary $2π$ rotations on fermion spin quantization frames and argue that the failure to do this renders the conventional symmetrization postulate (and previous ``proofs'' of it) at best {\em incomplete}. We discuss particle permutation in a general way for any pairs of particles, whether identical or not, and make an essential distinction between exchange and pure permutation. We prove a revised symmetrization postulate that allows us to construct state vectors that are naturally symmetric under pure permutation, {\em for any spin}. The significance of particle labels (which, in the exchange operation, are not permuted along with other variables) is then that they stand in for any asymmetry (order dependence) that is present in the full state descriptions necessary for unique state vectors but not explicit in the regular state variables. {\em The exchange operation is then the physical transformation that reverses any asymmetry implicit in the labels}. We point out a previously unremarked geometrical asymmetry between all pairs of particles that is present whenever we choose a common frame of reference. We compute the exchange phase for various state vectors using different spin quantization frames, and prove the Pauli Exclusion Principle and its generalization to arbitrary spin.

arXiv:quant-ph/9908078•Aug 25, 1999•6 citations
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.
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.
The treatment of identical particles in quantum mechanics rests on two (related) principles: the spin‐statistics connection and the Symmetrization Postulate. In light of recent theories (such as q‐deformed commutators) that allow for ‘small’ violations of the spin‐statistics connection and the Symmetrization Postulate, we revisit the issue of how quantum mechanics deals with identical particles and how it supports or fails to support various philosophical stances concerning individuality. As a consequence of the expanded possibilities for quantum statistics, we argue that permutation symmetry is best formulated as a formal property of the state function describing the system of particles rather than as a property of the individual particles.1 Introduction2 Philosophical background2.1 Important terminology2.1.1 Identity2.1.2 Indistinguishability2.1.3 Indiscernibility2.2 When are particles indistinguishable?2.3 The Principle of the Identity of Indiscernibles and quantum mechanics2.4 The Principle of the Identity of Indiscernibles and logic2.5 Particle history2.6 Transcendental individuality3 Some quantum formalism3.1 The Principle of Permutation Invariance and the Symmetrization Postulate3.2 The configuration‐space approach3.3 Commutators and anticommutators, and identical particle statistics3.4 Q‐mutators4 Identical particle statistics: a holistic point of view5 Conclusions
A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a πππ-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.
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.
We examine historic formulations of the spin-statistics theorem from a point of view that distinguishes between the observable consequences and the ``symmetrization postulate''. In particular, we make a critical analysis of concepts of particle identity, state distinguishability and permutation, and particle ``labels''. We discuss how to construct unique state vectors and the nature of the full state descriptions required for this -- in particular the elimination of arbitrary 2π2π2π rotations on fermion spin quantization frames and argue that the failure to do this renders the conventional symmetrization postulate (and previous ``proofs'' of it) at best {\em incomplete}. We discuss particle permutation in a general way for any pairs of particles, whether identical or not, and make an essential distinction between exchange and pure permutation. We prove a revised symmetrization postulate that allows us to construct state vectors that are naturally symmetric under pure permutation, {\em for any spin}. The significance of particle labels (which, in the exchange operation, are not permuted along with other variables) is then that they stand in for any asymmetry (order dependence) that is present in the full state descriptions necessary for unique state vectors but not explicit in the regular state variables. {\em The exchange operation is then the physical transformation that reverses any asymmetry implicit in the labels}. We point out a previously unremarked geometrical asymmetry between all pairs of particles that is present whenever we choose a common frame of reference. We compute the exchange phase for various state vectors using different spin quantization frames, and prove the Pauli Exclusion Principle and its generalization to arbitrary spin.