## Abstract

We show that the action of the boosts on an infinite system can be described continuously by bundle maps of Hilbert bundles based on the manifolds G/G_{0}, where G is the full relativity group and G_{0} its closed subgroup which can be unitarily implemented on the fibre, which is a Hilbert space. We then develop a general theory of group representations on product bundles M × ℋ, where M is a manifold and ℋ a Hilbert space. We construct certain bundle representations of the Galilei and the Poincaré group and find that they correspond to various classes of elementary excitations. In particular, we define nonrelativistic zero-mass systems and obtain an analogue of the Faraday effect for the passage of hot electrons through matter. Our construction remains valid for the case when G_{0} is the product of a lattice translation group and the time translations. We conclude that many qualitative features of the physics of condensed matter can be directly understood in terms of relativity group action on a bundle space as state space, which also suggests some avenues for further work.

Original language | English |
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Pages (from-to) | 101-126 |

Number of pages | 26 |

Journal | Communications in Mathematical Physics |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 1975 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics