Spin and statistics in classical mechanics

Abstract

The spin–statistics connection is obtained for classical particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and that exhibits classical analogs of both spin and statistics. Classical realizations of Lie groups are constructed in a canonical formalism generalized to include Grassmann variables. The theory of irreducible canonical realizations of the Poincaré group is developed in this framework, with particular emphasis on the rotation subgroup. The behavior of irreducible realizations under time inversion and charge conjugation is obtained. The requirement that the Lagrangian retain its form under the combined operation of charge conjugation and time reversal leads directly to the spin–statistics connection by an adaptation of Schwinger’s 1951 proof to irreducible canonical realizations of the Poincaré group of spin j: Generalized spin coordinates and momenta satisfy fundamental Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.