Hilbert Space Representations of Generalized Canonical Commutation Relations

We consider Hilbert space representations of a generalization of canonical com-mutation relations (CCR): [Xj, Xk]: = XjXk ¡XkXj = iΘjkI (j, k = 1, 2,..., 2n), where Xj ’s are elements of an algebra with identity I, i is the imaginary unit, and Θjk is a real number with Θjk = ¡Θkj (j, k = 1,..., 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 analogy with the usual Schrödinger representation of the CCR with n degrees of freedom. Also we intro-duce a Weyl type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR