Hilbert Space Representations of Generalized Canonical Commutation Relations

Abstract EN

We consider Hilbert space representations of a generalization of canonical commutation relations (CCRs):[Xj,Xk]:=XjXk−XkXj=iΘjkI (j,k=1,2,…,2n), where Xj's are the elements of an algebra with identity I, i is the imaginary unit, and Θjk is a real number with antisymmetry Θjk=−Θkj (k,j=1,2,…,2n).

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 n 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.