ar X iv : f un ct - a n / 93 03 00 2 v 1 1 2 M ar 1 99 3 Coherent States of the q – Canonical Commutation Relations

For the q-deformed canonical commutation relations a(f)a † (g) = (1 − q) f, g1I + q a † (g)a(f) for f, g in some Hilbert space H we consider representations generated from a vector Ω satisfying a(f)Ω = f, ϕΩ, where ϕ ∈ H. We show that such a representation exists if and only if ϕ ≤ 1. Moreover, for ϕ < 1 these representations are unitarily equivalent to the Fock representation (obtained for ϕ = 0). On the other hand representations obtained for different unit vectors ϕ 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 < ∞, 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.