## 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 ℝ^{s}. For the case of N identical particles, information about particle statistics is contained in a representation of script K_{F} (the stability group of a point Fεℒ') which factors through the permutation group S_{N}. Starting from a quasi-invariant measure μ concentrated on a script K orbit Δ in ℒ', together with a suitable representation of script K_{F} 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 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 ℒ ∧ script K determines a well-defined representation of script K_{F}; (2) equivalent representations of ℒ ∧ script K determine equivalent representations of script K_{F}; (3) a representation of script K_{F} induces a representation ℒ ∧ script K; and (4) equivalent representations of script K_{F} determine equivalent induced representations.

Original language | English (US) |
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Pages (from-to) | 650-664 |

Number of pages | 15 |

Journal | Journal of Mathematical Physics |

Volume | 21 |

Issue number | 4 |

DOIs | |

State | Published - 1979 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics