A note on product measures and representations of the canonical commutation relations
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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.
KeywordsNeural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing
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