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arXiv:0909.5159

Is there a dynamical cause of the spin-statistics connection?

Aditya Gilra

Published Sep 28, 2009• 0 citations
arXivHTMLPDF

Abstract

Extant proofs of the spin-statistics connection (SSC) are kinematical. C S Unnikrishnan has suggested that a dynamical interaction leading to the SSC would involve spin and perforce gravity, the only known universal force. For the scattering of two identical particles, he considers [arXiv: gr-qc/0406043] the interaction of their spins with the gravito-magnetic field generated by their scattering motion through cosmic matter-energy. There the direct and particles-exchanged scattering amplitudes accumulate different quantum phases which provide the relevant bosonic/fermionic sign between them without applying the ad hoc SSC rule. Here it is shown that the scattering probabilities given by the standard implementation of SSC in quantum mechanics are actually not obtained from the above interaction for most initial spin states of the scattering particles. Instead, an unrealized peculiar dynamical interaction is required. Further, a spin-gravito-magnetic interaction as above (with caveats) would result in a large unmeasured spin-orbit coupling type effect on atomic energy levels. A comparison with a typical rotation based proof is also provided.

DOI
10.48550/arxiv.0909.5159

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Extant proofs of the spin-statistics connection (SSC) are kinematical. C S Unnikrishnan has suggested that a dynamical interaction leading to the SSC would involve spin and perforce gravity, the only known universal force. For the scattering of two identical particles, he considers [arXiv: gr-qc/0406043] the interaction of their spins with the gravito-magnetic field generated by their scattering motion through cosmic matter-energy. There the direct and particles-exchanged scattering amplitudes accumulate different quantum phases which provide the relevant bosonic/fermionic sign between them without applying the ad hoc SSC rule. Here it is shown that the scattering probabilities given by the standard implementation of SSC in quantum mechanics are actually not obtained from the above interaction for most initial spin states of the scattering particles. Instead, an unrealized peculiar dynamical interaction is required. Further, a spin-gravito-magnetic interaction as above (with caveats) would result in a large unmeasured spin-orbit coupling type effect on atomic energy levels. A comparison with a typical rotation based proof is also provided.
Recently, it was proposed that the spin-statistics connection arises due to a quantum dynamical phase involving cosmic gravity (Unnikrishnan C S, gr-qc/0406043). There it was assumed that the gravitational quantum phase accumulated on the wavefunctions of each of two identical particles undergoing scattering in the presence of cosmic matter with the direction of their momentum changing by some angle is equivalent to the particles forward scattering with the entire universe rotating oppositely by the same angle. However, if one considers the gravitational phase accumulated by an electron in orbit to be analogous to that acquired by a scattering particle, the accumulated phase inferred from the fine-structure splitting in an atom is too small to explain the spin statistics connection. Though, the desirability of a dynamical proof of the spin-statistics connection remains.
Chapter 5 addresses the question, “What explains CPT invariance and the spin–statistics connection (SSC)?” It first considers why we should care about explaining CPT invariance and the SSC. It then argues that the CPT and spin–statistics theorems cannot be said to provide the sought after explanations. It offers instead an explanation of the SSC in interacting non-relativistic theories. This explanation explains by virtue of a derivation from non-fundamental principles in non-interacting relativistic quantum field theories (RQFTs), and an explication of intertheoretic relations between interacting RQFTs, non-interacting RQFTs, and interacting non-relativistic theories. The chapter ends by arguing that the presence of both the SSC and CPT invariance in realistic interacting RQFTs cannot be explained by even this type of explanation, and that currently, the existence problem precludes a full understanding of these properties in realistic interacting RQFTs.
The traditional standard quantum mechanics (SQM) is unable to solve the spin-statistics problem, i.e. to justify the utterly important "Pauli exclusion principle". The present paper presents a simple and complete solution of the spin-statistics problem on the basis of the "conformal quantum geometrodynamics (CQG)", a theory that was found to reproduce successfully all relevant processes of the SQM based on Dirac's or Schrödinger's equations, including Heisenberg's uncertainty relations and non-local Einstein–Podolsky–Rosen (EPR) correlations. When applied to a system made of many identical particles, an additional property of all elementary particles enters naturally into play: the "intrinsic helicity". This property, not considered in the SQM, determines the correct spin-statistics connection (SSC) observed in nature.
Abstract We demonstrate that the spin-statistics connection -- fermions carry half-integer spin and obey Fermi-Dirac statistics, bosons carry integer spin and obey Bose-Einstein statistics -- emerges as a topological theorem within the Hopf soliton framework, requiring no independent axiom. The configuration space of a single Hopf soliton with charge H≠0H \neq 0H=0 is the mapping space Q=MapsH(S3,S2)\mathcal{Q} = \text{Maps}_H(S^3, S^2)Q=MapsH​(S3,S2), whose fundamental group π1(Q)=Z2\pi_1(\mathcal{Q}) = \mathbb{Z}_2π1​(Q)=Z2​ renders 2π2\pi2π rotation topologically non-trivial. The Finkelstein-Rubinstein construction then selects the double-valued (fermionic) quantization: wavefunctions on Q\mathcal{Q}Q acquire a sign under 2π2\pi2π rotation, yielding spin-1/2. Exchange of two identical H≠0H \neq 0H=0 solitons is topologically equivalent to 2π2\pi2π rotation of one, enforcing antisymmetric wavefunctions -- the Pauli exclusion principle derived from topology. The photon (H=0H=0H=0) has simply connected configuration space (π1=0\pi_1 = 0π1​=0), automatically giving integer spin and Bose-Einstein statistics. In 2+1 dimensions, the fundamental group becomes Z\mathbb{Z}Z (the braid group), producing anyonic statistics. The geometric (Berry) phase under adiabatic rotation reproduces the FR sign rule exactly. We compare this topological proof with the Streater-Wightman axiomatic proof and identify the structural reasons for their agreement. Keywords: spin-statistics theorem, Finkelstein-Rubinstein quantization, Hopf soliton, configuration space topology, fundamental group, Pauli exclusion principle, anyons, Berry phase, braid group, topological quantum field theory Keywords physics spin statistics fermion boson topology soliton hopf finkelstein rubinstein Type Preprint License CC BY 4.0 Date 2026-03-26 Subject Theoretical Physics DOI 10.5281/zenodo.19163319 © 2026 Alexander Novickis. Licensed under Creative Commons Attribution 4.0 International.
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 connection between the intrinsic angular momentum (spin) of particles and quantum statistics is established by considering the response of identical particles to a common background radiation field. For this purpose, the Hamiltonian analysis previously performed in stochastic electrodynamics to derive the quantum description of a one-particle system is extended to a system of two identical bound particles subject to the same field. Depending on the relative phase of the response of the particles to a common field mode, two types of particles are distinguished by their symmetry or antisymmetry with respect to particle exchange. While any number of identical particles responding in phase can occupy the same energy state, there can only be two particles responding in antiphase. The calculation of bipartite correlations between the response functions reveals maximum entanglement as a consequence of the parallel response of the particles to the common field. The introduction of an internal rotation parameter leads to a direct link between spin and statistics and to a physical rationale for the Pauli exclusion principle.