Positive representations of general commutation relations allowing Wick ordering.

*(English)*Zbl 0864.46047The 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 \(S_vU(2)\).

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 \(S_vU(2)\).

Reviewer: W.Slowikowski (Aarhus)

##### 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 |