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Positive representations of general commutation relations allowing Wick ordering. (English) Zbl 0864.46047
The paper analyses representations in Hilbert space of a wide class of involutive algebras which are obtained from generators \(a_j\) with the relations \(a_ia^*_j=\delta_{ij}+\Sigma_{kl}T^{kl}_{ij}a^*_la_k\), where the \(T^{kl}_{ij}\) are complex coefficients constrained only by a hermiticity condition so that the relations respect the involution. The ‘structure constants’ \(T^{kl}_{ij}\) determine an operator \(\widetilde T:{\mathcal H}^\dagger\otimes{\mathcal H}\to{\mathcal H}\otimes{\mathcal H}^\dagger\), where \(\mathcal H\) is a Hilbert space with basis \(\{e_i\}\) labelled as generators, and \({\mathcal H}^\dagger\) is the conjugate of \(\mathcal H\). The relations \(f^\dagger\otimes g=\langle f,g\rangle 1+\widetilde T(f^\dagger\otimes g)\), \(f^\dagger\in{\mathcal H}^\dagger\), \(g\in{\mathcal H}\), then define an ideal in the algebra of all tensors over \(\mathcal H\) and \({\mathcal H}^\dagger\) with tensor multiplication as the product. The quotient of the tensor algebra by this ideal is an abstract involutive algebra denoted by \({\mathcal W}(T)\) and called ‘Wick algebra’. The introduced relations make it possible to write any polynomial in generators \(a_j\) and their adjoints in ‘Wick ordered form’ in which all starred generators are to the left of all unstarred ones.
The paper treats mostly the ‘positive representations’ of \({\mathcal W}(T)\), i.e. representations of \(a_i\) as operators on a Hilbert space such that \(a^*_i\) is a restriction of the operator adjoint to \(a_i\). The authors are mainly interested in representations by bounded operators. For any choice of \(T^{kl}_{ij}\), \({\mathcal W}(T)\) has a unique the so-called Fock representation constructed from a cyclic vector \(\Omega\) with the property that \(a_j\Omega=0\) for all generators \(a_i\). The Fock representation carries a natural Hermitian scalar product, not necessarily positive semidefinite; the paper presents a number of criteria for positive semidefiniteness of this scalar product.
The reader will find fascinating mathematics following these introductory concepts. We mention a few subjects: coherent representations, a characterization of the Fock representation, boundedness and positivity, the coefficients \(T^{kl}_{ij}\) as an operator, bounds for small \(T\), the universal bounded representation, Wick ideals and quadratic Wick ideals, Wick algebra relations as differential calculus. The examples given in the paper comprise the \(q\)-canonical commutation relations introduced by Greenberg, Bozejko and Speicher, Pusz and Woronowicz, as well as the quantum group \(S_vU(2)\).

MSC:
46N50 Applications of functional analysis in quantum physics
81S05 Commutation relations and statistics as related to quantum mechanics (general)
46K05 General theory of topological algebras with involution
46L60 Applications of selfadjoint operator algebras to physics
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