Hilbert Space Representations of Generalized Canonical Commutation Relations

We consider Hilbert space representations of a generalization of canonical commutation relations (CCRs):[X-i,X-k] := XjXk - XkXj = i Theta(jk) I (j,k =1,2,....2n) where 's are the elements of an algebra with identity , is the imaginary unit, and Theta(jk) is a real number with antisymmetry Theta(jk) = -Theta(kj) (k,j =1,2,....2n)Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrodinger-type representation of the GCCR by an analogy with the usual Schrodinger 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