Positive representations of general commutation relations allowing Wick ordering

Abstract

We consider the problem of representing in Hilbert space commutation relations of the form $$ a_ia_j^*=δ_{ij}{\bold1} + \sum_{k\ell}T_{ij}^{k\ell} a_\ell^*a_k \quad,$$ where the $T_{ij}^{k\ell}$ are essentially arbitrary scalar coefficients. Examples comprise the $q$-canonical commutation relations introduced by Greenberg, Bozejko, and Speicher, and the twisted canonical (anti-)commutation relations studied by Pusz and Woronowicz, as well as the quantum group S$_ν$U$(2)$. Using these relations, any polynomial in the generators $a_i$ and their adjoints can uniquely be written in ``Wick ordered form'' in which all starred generators are to the left of all unstarred ones. In this general framework we define the Fock representation, as well as coherent representations. We develop criteria for the natural scalar product in the associated representation spaces to be positive definite, and for the relations to have representations by bounded operators in a Hilbert space. We characterize the relations between the generators $a_i$ (not involving $a_i^*$) which are compatible with the basic relations. The relations may also be interpreted as defining a non-commutative differential calculus. For generic coefficients $T_{ij}^{k\ell}$, however, all differential forms of degree 2 and higher vanish. We exhibit conditions for this not to be the case, and relate them to the ideal structure of the Wick algebra, and conditions of positivity. We show that the differential calculus is compatible with the involution iff the coefficients $T$ define a representation of the braid group. This condition is also shown to imply improved bounds for the positivity of the Fock representation. Finally, we study the KMS states of the group of gauge transformations defined by $a_j\mapsto\exp(it)a_j$.