Parity and the spin-statistics connection

Proofs of the spin-statistics theorem tend, broadly speaking, to fall into two classes. The first class, historically, depends upon analytic properties of field operator commutators [1-4]. The second class invokes topological arguments. Proofs in this latter class variously use homotopies in configuration space for identical particles [5-9] or arguments involving adiabatic exchange of particles carrying topological markers [10,11]. The proof by Schwinger [12] stands apart from both classes in exploiting the action of time-reversal on the Lagrangian density of a field. The use, on the one hand, of identical particle exchange in the topological theorems, and, on the other, of a discrete symmetry applied to a scalar invariant in Schwinger's proof, suggests using another discrete symmetry, parity, to examine the effect of exchanging particle coordinates by passive transformations. This note presents a simple demonstration of the spin-statistics connection based upon that idea. The proof uses elementary parity and angular momentum properties of quantum fields only and is, in essence, algebraic.