COHERENT STATES OF THE Q-CANONICAL COMMUTATION RELATIONS

Abstract

For the q-deformed canonical commutation relations a(f)a(dagger)(g)= (1 - q)[f, g] 1 qa(dagger)(g)a(f) for f, g in some Hilbert space H we consider representations generated from a vector OMEGA satisfying a(f)OMEGA = [f, phi]OMEGA, where phi is-an-element-of H. We show that such a representation exists if and only if phi less-than-or-equal-to 1. Moreover, for phi < 1 these representations are unitarily equivalent to the Fock representation (obtained for phi = 0). On the other hand representations obtained for different unit vectors phi are disjoint. We show that the universal C*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a natural q-analogue of the Cuntz algebra (obtained for q = 0). We discuss the conjecture that, for d < infinity, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases q = /- 1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.