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Journal of Mathematical Physics

A connection between spin and statistics based on Galilean invariant delta functions

Masanori Kobayashi, M. de Montigny, F. C. Khanna

Published Aug 1, 2010• 1 citation
Publisher

Abstract

We define Galilean invariant delta functions by using the Takahashi formulation, which describes particles with arbitrary spin. The direct calculation of the Galilean invariant delta functions in the (N+1)-dimensional Galilean covariant extended manifolds reveals that, for N odd, there exists a surface on which these delta functions vanish. This feature corresponds to the spacelike separation in the relativistic quantum field theory, and it enables us to discuss a connection between spin and statistics in the Galilean covariant theories. These results show that a spin-statistics connection holds for even-dimensional Galilean covariant extended manifolds; that is, for N-dimensional space-times with odd N≥5. Likewise, we observe that there is no such connection for odd-dimensional Galilean covariant extended manifolds; that is, for N-dimensional space-times with even N≥4. Thus, our results support the claim that there is no connection between spin and statistics in the usual nonrelativistic theory, for which N=4.

DOI
10.1063/1.3456059

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We define Galilean invariant delta functions by using the Takahashi formulation, which describes particles with arbitrary spin. The direct calculation of the Galilean invariant delta functions in the (N+1)-dimensional Galilean covariant extended manifolds reveals that, for N odd, there exists a surface on which these delta functions vanish. This feature corresponds to the spacelike separation in the relativistic quantum field theory, and it enables us to discuss a connection between spin and statistics in the Galilean covariant theories. These results show that a spin-statistics connection holds for even-dimensional Galilean covariant extended manifolds; that is, for N-dimensional space-times with odd N≥5. Likewise, we observe that there is no such connection for odd-dimensional Galilean covariant extended manifolds; that is, for N-dimensional space-times with even N≥4. Thus, our results support the claim that there is no connection between spin and statistics in the usual nonrelativistic theory, for which N=4.
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.
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 traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper [E. Santamato and F. D. De Martini, Found. Phys. 45 (2015) 858] we presented a complete proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics” (CQG). In this paper, by the same theory, the proof of the spin-statistics theorem (SST) is extended to the relativistic domain in the scenario of curved spacetime. No relativistic quantum field operators are used in the present proof and the particle exchange properties are drawn from rotational invariance rather than from Lorentz invariance. Our relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. As in the nonrelativistic case, we find once more that the “intrinsic helicity” of the elementary particles enters naturally into play. It is therefore this property, not considered in the standard quantum mechanics (SQM), which determines the correct spin-statistics connection observed in Nature.
The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. Firstly we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Lévy-Leblond equations, and define the Galilean generating functional and Green’s functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential λ(ΨΨ) 2, and we derive expressions for the 2- and 4-point Green’s functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in (3 + 1) space-time. This is particularly apparent when we represent the results with diagrams in the extended (4 + 1) manifold, since they usually encompass more diagrams in Galilean (3 + 1) space-time. 1
The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important \qo{Pauli Exclusion Principle} but by the adoption of the complex standard relativistic quantum field theory. In a recent paper (Ref. [1]) we presented a proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the \qo{Conformal Quantum Geometrodynamics}. In the present paper, by the same theory the proof of the Spin-Statistics Theorem is extended to the relativistic domain in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the Standard Quantum Mechanics, which determines the correct spin-statistics connection observed in Nature [1]. The present proof of the Spin-Statistics Theorem is simpler than the one presented in Ref. [1], because it is based on symmetry group considerations only, without having recourse to frames attached to the particles.