Abstract Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents $(aQ + bP - i r)^{-1}$ ( a Q + b P - i r ) - 1 for real constants $a,b,r \ne 0$ a , b , r ≠ 0 . This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra $\mathfrak {R}$ R generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results are obtained for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.Abstract Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents $$(aQ + bP - i r)^{-1}$$ ( a Q + b P - i r ) - 1 for real constants $$a,b,r \ne 0$$ a , b , r ≠ 0 . This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra $$\mathfrak {R}$$ R generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results are obtained for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.

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Quantum structures and operator algebras

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Trapped bosons, thermodynamic limit, and condensation: A study in the framework of resolvent algebras

Dorothea Bahns, Detlev Buchholz A

The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose–Einstein condensates. The method is illustrated in the case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet, the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose–Einstein condensates. The method is illustrated in the case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet, the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.

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The Resolvent Algebra of Non-relativistic Bose Fields: Observables, Dynamics and States

Detlev Buchholz A

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Quantum Affine Algebras and Integrable Quantum Systems

Vyjayanthi Chari, Andrew Pressley

NATO ASI series. Series B : PhysicsJan 1, 19974 citations

(aQ + bP - i r)^{-1}

( a Q + b P - i r ) - 1 for real constants

a,b,r \ne 0

a , b , r ≠ 0 . This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra

\mathfrak {R}

R generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results are obtained for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.