Particle statistics from induced representations of a local current group

Abstract

Representations of the nonrelativistic current group S‐K are studied in the Gel’fand–Vilenkin formalism, where S is Schwartz’ space of rapidly decreasing functions, and K is a group of diffeomorphisms of R^s. For the case of N identical particles, information about particle statistics is contained in a representation of K_F (the stability group of a point F∈S′) which factors through the permutation group S_N. Starting from a quasi‐invariant measure μ concentrated on a K orbit Δ in S′, together with a suitable representation of K_F for F∈Δ, sufficient conditions are developed for inducing a representation of S‐K. The Hilbert space for the induced representation consists of square‐integrable functions on a covering space of Δ, which transform in accordance with a representation of K_F. The Bose and Fermi N‐particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of S‐K determines a well‐defined representation of K_F; (2) equivalent representations of S‐K determine equivalent representations of K_F; (3) a representation of K_F induces a representation of S‐K; and (4) equivalent representations of K_F determine equivalent induced representations.