Pet. Jorgensen et Rf. Werner, COHERENT STATES OF THE Q-CANONICAL COMMUTATION RELATIONS, Communications in Mathematical Physics, 164(3), 1994, pp. 455-471
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 suc
h a representation exists if and only if \\phi\\ less-than-or-equal-to
1. Moreover, for \\phi\\ < 1 these representations are unitarily equi
valent to the Fock representation (obtained for phi = 0). On the other
hand representations obtained for different unit vectors phi are disj
oint. We show that the universal C-algebra for the relations has a la
rgest proper, closed, two-sided ideal. The quotient by this ideal is a
natural q-analogue of the Cuntz algebra (obtained for q = 0). We disc
uss the conjecture that, for d < infinity, this analogue should, in fa
ct, 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.