The Resolvent Algebra of Non-relativistic Bose Fields: Sectors, Morphisms, Fields and Dynamics

Abstract

It was recently shown [2] that the resolvent algebra of a non-relativistic Bose field determines a gauge invariant (particle number preserving) kinematical algebra of observables which is stable under the automorphic action of a large family of interacting dynamics involving pair potentials. In the present article, this observable algebra is extended to a field algebra by adding to it isometries, which transform as tensors under gauge transformations and induce particle number changing morphisms of the observables. Different morphisms are linked by intertwiners in the observable algebra. It is shown that such intertwiners also induce time translations of the morphisms. As a consequence, the field algebra is stable under the automorphic action of the interacting dynamics as well. These results establish a concrete C*-algebraic framework for interacting non-relativistic Bose systems in infinite space. It provides an adequate basis for studies of long range phenomena, such as phase transitions, stability properties of equilibrium states, condensates, and the breakdown of symmetries.The resolvent algebra: sectors, morphisms, fields and dynamics 5 Gauge transformations, tensors and observablesOn the resolvent algebra R acts the global gauge group U (1) ≃ T by maps γ, which are defined on the basic resolvents according toThese maps are unitarily implemented on Fock space by exponentials of the particle number operator N ,Thus these maps define a group of automorphisms γ T of the resolvent algebra. This can also be seen in the abstract setting since the defining relations of the resolvent algebra remain unchanged under their action [5, Def. 3.1].The action of the gauge group on the resolvent algebra R is not pointwise norm continuous, cf. [5, Thm. 5.3(ii)]. Nevertheless, one can perform a harmonic analysis of the elements of R with regard to this group by exploiting the fact that it acts pointwise continuously on R in the strong operator topology of the Fock representation. Wheras the definition of the harmonics by Fourier integrals relies on this weaker topology, the harmonics themselves are elements of the C*algebra R, as is shown in the subsequent lemma. Let us recall in this context that the resolvent algebra is faithfully represented on Fock space. Lemma 3.1. Let R ∈ R. The integrals R m . = (2π) −1 2π 0 du e −ium e iuN Re −iuN , m ∈ Z , being defined in the strong operator topology on F , are elements of the resolvent algebra, i.e. R m ∈ R, m ∈ Z. For fixed m, the operators R m transform as tensors (harmonics) under the gauge transformations, γ u (R m ) = e ium R m , u ∈ [0, 2π].Remark: Note that there exist elements R ∈ R, such as the basic resolvents, which can not be approximated by the coresponding sums m R m in the norm topology. So harmonic synthesis fails in the C*-algebra R. We will return to this point further below.Proof. Since the polynomials of the basic resolvents are norm dense in R and the map R → R m is norm continuous, it suffices to establish the statement for monomials. So, for j = 1, . . . , k,...

The Unitary Master Ward Identity: Time Slice Axiom, Noether’s Theorem and Anomalies

“…We then prove that the new axiom implies the time slice axiom [40] (primitive causality in [39]), which states that each observable can be expressed in terms of observables in any neighborhood of a Cauchy surface. The remarkable stability of the algebra under a large class of time evolutions compares well to a similar property in non-relativistic quantum field theory [18] in the framework of Resolvent Algebras by Buchholz and Grundling [23].…”

The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies

“…We then prove that the new axiom implies the time slice axiom [39] (primitive causality in [38]), which states that each observable can be expressed in terms of observables in any neighbourhood of a Cauchy surface. The remarkable stability of the algebra under a large class of time evolutions compares well to a similar property in non-relativistic quantum field theory [18] in the framework of Resolvent Algebras by Buchholz and Grundling [23].…”

A note on commutation relations and finite dimensional approximations

“…This algebra was introduced in order to remedy well-known limitations of the Weyl algebra in the description of quantum systems and quantum dynamics. We refer to [16,12,17] for proofs and further results and motivation and to [13,14,15] for additional applications of this algebra to quantum physics.…”