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Communications in Mathematical Physics

, Volume 22, Issue 1, pp 23–26 | Cite as

A note on product measures and representations of the canonical commutation relations

  • Ole A. Nielsen
Article

Abstract

There is a well-known theorem which states that a non-zero σ-finite left quasi-invariant measure on a σ-compact locally compact groupG must be equivalent to left Haar measure. It is shown in this paper that there is a natural generalization of this fact to the case in which the groupG is replaced by a product space, one factor of which is a group. With the aid of this generalization, an easy proof of the following fact, due to H. Araki, is given: the representations of the canonical commutation relations constructed in the usual measure-theoretic manner are ray continuous.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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References

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    Araki, H.: On representations of the canonical commutation relations. Commun. math. Phys.,20, 9–25 (1971).Google Scholar
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    Loomis, L. H.: An introduction to abstract harmonic analysis. New York: Van Nostrand 1953.Google Scholar
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    —— Induced representations of locally compact groups. I, Ann. Math.55, 101–139 (1952).Google Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • Ole A. Nielsen
    • 1
  1. 1.Queen's UniversityKingstonCanada

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