Coherent States of the q--Canonical Commutation Relations

Abstract

For the $q$-deformed canonical commutation relations $a(f)a^{†}(g)=(1-q)⟨f,g⟩1+q{a}^{†}(g)a(f)$ for $f,g$ in some Hilbert space $\mathcal{H}$ we consider representations generated from a vector $\Omega$ satisfying $a(f)\Omega =⟨f,\varphi ⟩\Omega$, where $\varphi \in \mathcal{H}$. We show that such a representation exists if and only if $\Vert \varphi \Vert \le 1$. Moreover, for $\Vert \varphi \Vert <1$ these representations are unitarily equivalent to the Fock representation (obtained for $\varphi =0$). On the other hand representations obtained for different unit vectors $\varphi$ 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 $d<\infty$, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting cases $q=\pm 1$ we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.

19 pages, Plain TeX

References in corpus (1)

• The Free Quon Gas Suffers Gibbs' Paradox

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