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Journal of Mathematics
Volume 2013, Article ID 308392, 7 pages
Research Article

Hilbert Space Representations of Generalized Canonical Commutation Relations

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received 19 November 2012; Accepted 10 December 2012

Academic Editor: Stefan Siegmund

Copyright © 2013 Asao Arai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.