The spin-statistics connection in classical field theory

Abstract

The spin-statistics connection is obtained for a simple formulation of a classical field theory containing even and odd Grassmann variables. To that end, the construction of irreducible canonical realizations of the rotation group corresponding to general causal fields is reviewed. The connection is obtained by imposing local commutativity on the fields and exploiting the parity operation to exchange spatial coordinates in the scalar product of classical field evaluated at one spatial location with the same field evaluated at a distinct location. The spin-statistics connection for irreducible canonical realizations of the Poincaré group of spin j is obtained in the form: Classical fields and their conjugate momenta satisfy fundamental field-theoretic Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.

Demonstration of the Spin-Statistics Connection in Elementary Quantum Mechanics

“…That question is not addressed here. The demonstration given here renders in (largely) elementary language a proof a proof originally devised for (j, 0) or (0, j) irreducible representations of the Poincaré group, sometimes called Weinberg fields 7,8,9,10 . It exploits the properties of two-particle states constructed from identical noninteracting states of massive particles corresponding to Weinberg fields.…”