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Coherent states of the \(q\)-canonical commutation relations. (English) Zbl 0838.46056

Summary: For the \(q\)-deformed canonical commutation relations \(a(f) a^†(g)= (1- q)\langle f, g\rangle \text{\textbf{1}}+ qa^†(g) a(f)\) for \(f\), \(g\) in some Hilbert space \(\mathcal H\) we consider representations generated from a vector \(\Omega\) satisfying \(a(f)\Omega= \langle f, \varphi\rangle \Omega\), where \(\varphi\in {\mathcal H}\). We show that such a representation exists if and only if \(|\varphi|\leq 1\). Moreover, for \(|\varphi|< 1\) 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.

MSC:

46L60 Applications of selfadjoint operator algebras to physics
81S05 Commutation relations and statistics as related to quantum mechanics (general)
46L40 Automorphisms of selfadjoint operator algebras
46L30 States of selfadjoint operator algebras
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