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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Geometry of spin and statistics in classical and quantum mechanics

Jouko Mickelsson

Published Oct 1, 1984• 12 citations
Publisher

Abstract

The spin and statistics connection is examined in classical point-particle mechanics in the spirit of the field-theoretic homotopy proof for the spin-statistics theorem by Finkelstein and Rubinstein. The theorem is proven when the geometry of the configuration space is determined by a parallelization describing a soliton field. It is indicated how the characteristic features of the system can be preserved in the corresponding quantized (nonrelativistic) system.

DOI
10.1103/physrevd.30.1843

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The spin and statistics connection is examined in classical point-particle mechanics in the spirit of the field-theoretic homotopy proof for the spin-statistics theorem by Finkelstein and Rubinstein. The theorem is proven when the geometry of the configuration space is determined by a parallelization describing a soliton field. It is indicated how the characteristic features of the system can be preserved in the corresponding quantized (nonrelativistic) system.
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 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.
In this work we consider non-relativistic quantum mechanics, obtained from a classical configuration space 𝒬 of indistinguishable particles. Following an approach proposed in [8], wave functions are regarded as elements of suitable projective modules over C ( 𝒬 ) . We take furthermore into account the G -Theory point of view (cf. [HPRS,S]) where the role of group action is particularly emphasized. As an example illustrating the method, the case of two particles is worked out in detail. Previous works (cf. [BR1,BR2]) aiming at a proof of a spin-statistics theorem for non-relativistic quantum mechanics are re-considered from the point of view of our approach, enabling us to clarify several points.
We study the relation of the spin-statistics theorem to the geometric structures on phase space, which are introduced in quantization procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the nonrelativistic domain (in fact for any symmetry group including internal symmetries) by requiring that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates — the exchange then arises from suitable change of coordinate system. We then examine our construction in the geometric and the coherent-state-path-integral quantization schemes. In the appendix we apply our results to exotic systems exhibiting continuous "spin" and "fractional statistics." This gives novel and unusual forms of the spin-statistics relation.
We review the theory of statistics of identical systems in terms of the properties of the classical configuration space. Special emphasis is placed on the influence of the connectivity of this space on quantum statistics. The theory shows that exotic statistical systems, such as those obeying fractional statistics, can occur in all dimensions. Particle statistics in one dimension involves novel features particular to this dimension and these are briefly described. The proof of a topological spin-statistics theorem which avoids field theory or relativity is also outlined.