Positive Representations of General Commutation Relations Allowing Wick Ordering

Abstract

We consider the problem of representing in Hilbert space commutation relations of the formwhere the T kℓ ij 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 kℓ ij , 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 → exp(it)a j .

A Family of *-Algebras Allowing Wick Ordering: Fock Representations and Universal Enveloping C*-Algebras

“…The universal bounded representation for q ij -CCR was studied in [9,10]. The following proposition follows from the main result of paper [10].…”

“…It was shown in [10] that for sufficiently small coefficients we have strict positivity of the Fock inner product. This result is a corollary of the stability of the universal enveloping C * -algebra for the Wick algebra around the zero base point (see Sec.…”

“…They give a canonical realisation of a given deformed relation like the Fock representation, or a realisation by differential operators. When the relations can be realised by bounded operators, it is useful to study the universal enveloping C * -algebras for them and the stability of isomorphism classes of these C * -algebras on parameters (see for example [9,10]). The stability question [10] refers to how the C * -isomorphism classes depend on variations in the deformation variables; in some cases there are open regions in parameter space where the C * -isomorphism class is constant.…”

“…The complete proofs of all results presented here can be found in [4,10,11,18,19]. For detailed information about * -representations of finitely generated * -algebras see [20].…”

On commutation relations for quons

“…The commutation relations for Hecke braiding has been studied by Kempf [23]. It is interesting that from algebraic point of view that all deformations of commutation relations for CAO can be described in terms of the so-called Wick algebras [25]. Note that Wick algebras are some special examples of quantum Weyl algebras [26], see also Refers [27,28].…”