Particle statistics from induced representations of a local current group

G. A. Goldin, R. Menikoff, D. H. Sharp

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124 Scopus citations

Abstract

Representations of the nonrelativistic current group ℒ ∧ script K are studied in the Gel'fand-Vilenkin formalism, where ℒ is Schwartz' space of rapidly decreasing functions, and script 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 script KF (the stability group of a point Fεℒ') which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a script K orbit Δ in ℒ', together with a suitable representation of script KF for FεΔ, sufficient conditions are developed for inducing a representation of ℒ ∧ script 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 script 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 ℒ ∧ script K determines a well-defined representation of script KF; (2) equivalent representations of ℒ ∧ script K determine equivalent representations of script KF; (3) a representation of script KF induces a representation ℒ ∧ script K; and (4) equivalent representations of script KF determine equivalent induced representations.

Original languageEnglish (US)
Pages (from-to)650-664
Number of pages15
JournalJournal of Mathematical Physics
Volume21
Issue number4
DOIs
StatePublished - 1979
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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