Remarks on $q$-CCR relations for $|q|>1$
Volume 78 / 2007
Banach Center Publications 78 (2007), 59-67
MSC: Primary 46L53; Secondary 81S05.
DOI: 10.4064/bc78-0-4
Abstract
In this paper we give a construction of operators satisfying $q$-CCR relations for $q>1$: $$ A(f) A^{*}(g) - A^{*}(g) A(f) = q^{N} \langle f, g \rangle I $$ and also $q$-CAR relations for $q<-1$: $$ B(f) B^{*}(g) + B^{*}(g) B(f) = |q|^{N} \langle f, g \rangle I, $$ where $N$ is the number operator on a suitable Fock space $\mathcal{F}_{q}({\mathcal{H}})$ acting as $$ N x_1\otimes\cdots\otimes x_n = n\, x_1\otimes\cdots\otimes x_n.$$ Some applications to combinatorial problems are also given.
