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 Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F 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 𝒦F. 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 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.