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Connecting Spin and Statistics in Quantum Mechanics

Arthur Jabs, Paul Ehrenfest

Published Nov 20, 2012• 37 citations
OpenAlex

Abstract

Abstract. The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spinquantization axis, is exchanged along with the other parameters. The spin factor (−1) 2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required.

Related papers

Spin and statistics

Arthur A. Broyles

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.

American Journal of Physics•Apr 1, 1976•41 citations

Non-relativistic proofs of the spin-statistics connection

Anil Shaji, E. C. G. Sudarshan

A recent paper by Peshkin [1] has drawn attention again to the problem of understanding the spin statistics connection in non-relativistic quantum mechanics. Allen and Mondragon [2] has pointed out correctly some of the flaws in Peshkin's arguments which are based on the single valuedness under rotation of the wave functions of systems of identical particles. We examine carefully the claim that is made in the title of [2] that there can be ``no spin-statistics connection in non-relativistic quantum mechanics''. We show that we can derive the spin statistics connection for non-relativistic quantum field theories which have equations of motion of the Hamiltonian type based on SU(2) invariance of the Lagrangian. The formalism and machinary of non-relativistic quantum field theory as opposed to usual quantum mechanics is necessary for constructing our proof.

arXiv:quant-ph/0306033•Jun 4, 2003•7 citations

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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Robert T. Deck, Jearl D. Walker

We demonstrate that the exchange of two identical particles can be effected by a rotation of a coordinate system defined by the two particles through an angle π. It is then a consequence of the intimate connection between the generator of a space rotation and the total angular momentum operator, that the effect of the particle exchange on the ket corresponding to the physical system is dependent upon the integer or half-integer character of the particle spins. From this it can be concluded that the connection between spin and statistics that derives from the symmetry of the identical particle system under particle exchange is explained (in the most fundamental way) by the equivalence between the exchange of two identical particles and a rotation in configuration space.

Physica Scripta•Jan 1, 2001•1 citation

From Primitives to the Spin-Statistics Theorem

Allen Proxmire

The spin‑statistics theorem is one of the deepest individual results in physics. Particles with integer spin obey Bose–Einstein statistics: their wavefunctions are symmetric under exchange, and any number can occupy the same state. Particles with half‑integer spin obey Fermi–Dirac statistics: their wavefunctions are antisymmetric, and no two can occupy the same state. Compactly: η = (−1)^{2s}, where η is the exchange phase (+1 bosons, −1 fermions) and s is the spin. This is empirically the most thoroughly tested relationship in particle physics. Every confirmation of Pauli exclusion in atomic structure, every superconductor, every superfluid, every neutron star, every electron‑degenerate white dwarf, every blackbody distribution, every Bose–Einstein condensate confirms it. It determines why atoms have shell structure and why chemistry exists. It determines why matter is stable against gravitational collapse below the Chandrasekhar limit. The structural integrity of essentially everything in the material world traces back to this one Z₂ × Z₂ relationship. Despite its empirical centrality, the theorem is notoriously hard to derive cleanly. Pauli’s 1940 argument used QFT, specifically requiring positivity of energy and microcausality. Streater–Wightman formalizes this with Lorentz invariance, locality, positivity of energy, and uniqueness of the vacuum. Burgoyne, Lüders–Zumino, and Weinberg all require similar axiomatic foundations with microcausality as a postulate or close relative. The dependence on QFT is not incidental — the cleanest microcausality‑based arguments need the field‑operator algebra to make their case. It leaves a question hanging: must the theorem really depend on QFT? This walkthrough shows how the Event Density framework derives spin‑statistics from substrate primitives plus 3+1D spatial topology plus the algebraic uniqueness of the Cl(3,1) frame, *without QFT*. The argument runs through: the configuration‑space topology π₁(Q₂) = Z₂ in 3+1D; the geometric theorem identifying the exchange‑class generator with the 2π‑rotation generator; the algebraic uniqueness of Cl(3,1) realizing the rotational double cover; and a closure step using the framework’s individuation primitive to tie the topological exchange phase to spinor‑module representation theory. The result is η = (−1)^{2s} forced unconditionally at primitive level. The structural beauty: two Z₂ dichotomies — the exchange‑phase (η = ±1, forced by topology) and the integer/half‑integer spin parity (forced by the Lorentz representation ladder) — collapse into a single forced equality, with the identification between them supplied by Cl(3,1) plus the individuation pairing. The framework reproduces spin‑statistics without QFT, at the level of substrate primitives plus 3+1D topology.

•May 12, 2026•0 citations

Statistics of Indistinguishable Particles

C. Wittig

The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.

The Journal of Physical Chemistry A•Mar 19, 2009•2 citations
Abstract. The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spinquantization axis, is exchanged along with the other parameters. The spin factor (−1) 2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required.
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.
A recent paper by Peshkin [1] has drawn attention again to the problem of understanding the spin statistics connection in non-relativistic quantum mechanics. Allen and Mondragon [2] has pointed out correctly some of the flaws in Peshkin's arguments which are based on the single valuedness under rotation of the wave functions of systems of identical particles. We examine carefully the claim that is made in the title of [2] that there can be ``no spin-statistics connection in non-relativistic quantum mechanics''. We show that we can derive the spin statistics connection for non-relativistic quantum field theories which have equations of motion of the Hamiltonian type based on SU(2) invariance of the Lagrangian. The formalism and machinary of non-relativistic quantum field theory as opposed to usual quantum mechanics is necessary for constructing our proof.
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.
We demonstrate that the exchange of two identical particles can be effected by a rotation of a coordinate system defined by the two particles through an angle π. It is then a consequence of the intimate connection between the generator of a space rotation and the total angular momentum operator, that the effect of the particle exchange on the ket corresponding to the physical system is dependent upon the integer or half-integer character of the particle spins. From this it can be concluded that the connection between spin and statistics that derives from the symmetry of the identical particle system under particle exchange is explained (in the most fundamental way) by the equivalence between the exchange of two identical particles and a rotation in configuration space.
The spin‑statistics theorem is one of the deepest individual results in physics. Particles with integer spin obey Bose–Einstein statistics: their wavefunctions are symmetric under exchange, and any number can occupy the same state. Particles with half‑integer spin obey Fermi–Dirac statistics: their wavefunctions are antisymmetric, and no two can occupy the same state. Compactly: η = (−1)^{2s}, where η is the exchange phase (+1 bosons, −1 fermions) and s is the spin. This is empirically the most thoroughly tested relationship in particle physics. Every confirmation of Pauli exclusion in atomic structure, every superconductor, every superfluid, every neutron star, every electron‑degenerate white dwarf, every blackbody distribution, every Bose–Einstein condensate confirms it. It determines why atoms have shell structure and why chemistry exists. It determines why matter is stable against gravitational collapse below the Chandrasekhar limit. The structural integrity of essentially everything in the material world traces back to this one Z₂ × Z₂ relationship. Despite its empirical centrality, the theorem is notoriously hard to derive cleanly. Pauli’s 1940 argument used QFT, specifically requiring positivity of energy and microcausality. Streater–Wightman formalizes this with Lorentz invariance, locality, positivity of energy, and uniqueness of the vacuum. Burgoyne, Lüders–Zumino, and Weinberg all require similar axiomatic foundations with microcausality as a postulate or close relative. The dependence on QFT is not incidental — the cleanest microcausality‑based arguments need the field‑operator algebra to make their case. It leaves a question hanging: must the theorem really depend on QFT? This walkthrough shows how the Event Density framework derives spin‑statistics from substrate primitives plus 3+1D spatial topology plus the algebraic uniqueness of the Cl(3,1) frame, *without QFT*. The argument runs through: the configuration‑space topology π₁(Q₂) = Z₂ in 3+1D; the geometric theorem identifying the exchange‑class generator with the 2π‑rotation generator; the algebraic uniqueness of Cl(3,1) realizing the rotational double cover; and a closure step using the framework’s individuation primitive to tie the topological exchange phase to spinor‑module representation theory. The result is η = (−1)^{2s} forced unconditionally at primitive level. The structural beauty: two Z₂ dichotomies — the exchange‑phase (η = ±1, forced by topology) and the integer/half‑integer spin parity (forced by the Lorentz representation ladder) — collapse into a single forced equality, with the identification between them supplied by Cl(3,1) plus the individuation pairing. The framework reproduces spin‑statistics without QFT, at the level of substrate primitives plus 3+1D topology.
The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.