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International Journal of Modern Physics A

SPIN–STATISTICS THEOREMS WITHOUT RELATIVITY OR FIELD THEORY

A. P. Balachandran, Alan Daughton, Z.-C. GU, Rafael D. Sorkin, Giuseppe Marmo, Ajit Mohan Srivastava

Published Jul 10, 1993• 39 citations
Publisher

Abstract

Conventional proofs of the spin–statistics theorem either use relativistic quantum field theory or concern field theory solitons. We here construct new proofs of this theorem for spinning particles. They are extensions of previous work on spinless particles and require neither relativity nor field theory. They do, however, assume certain continuity conditions involving the existence of antiparticles. Non-Abelian statistics, such as parastatistics and braid statistics of order 2 or more, are excluded in our framework.

DOI
10.1142/s0217751x93001223

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International Journal of Modern Physics A•Nov 30, 1991•1 citation
Conventional proofs of the spin–statistics theorem either use relativistic quantum field theory or concern field theory solitons. We here construct new proofs of this theorem for spinning particles. They are extensions of previous work on spinless particles and require neither relativity nor field theory. They do, however, assume certain continuity conditions involving the existence of antiparticles. Non-Abelian statistics, such as parastatistics and braid statistics of order 2 or more, are excluded in our framework.
Here we explore the possibility to obtain a non-relativistic proof of the spin-statistics theorem. First, we examine the structure of axioms and theorems involved in a relativistic Schwinger-like proof of the spin-statistics relation. Second, starting from this structure we identify the relativistic assumptions. Last, we reformulate these assumptions in order to obtain a Galilean proof. We conclude that the spin-statistics theorem cannot be deduced into the framework of Galilean quantum field theories for three space dimensions because two of the assumptions needed to prove the theorem are incompatible. We analyze, however, the conditions under which a non-relativist proof could still be deduced.
ABS>Use of a quantum field theory obeying parastatistics to treat free particles is further examined. Relativistic quantum field theories independent of locality are noted to hold for paralocal fields, and the TCP theorem is generalized to paralocal fields. The following generalization of the connection of spin and statistics is demonstrated: given the choice between parabose and parafermi statistics, integral spin flelds must obey parabose, and odd halfintegral spin fields must obey parafermi statistics (D.C.W.)
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.