The resolvent algebra: A new approach to canonical quantum systems

Buchholz, Detlev; Grundling, Hendrik

doi:10.1016/j.jfa.2008.02.011

Cited by 65 publications

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“…Theorem 1 in [31, p 216]), thus the observable H S does not exist in this representation. Also observe, that this definition of regularity coincides with the one used in [5,8].…”

Section: Kinematics Algebras and Regular Representationssupporting

confidence: 66%

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Dynamics for QCD on an Infinite Lattice

Grundling

Rudolph

2016

Commun. Math. Phys.

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We prove the existence of the dynamics automorphism group for Hamiltonian QCD on an infinite lattice in R 3 , and this is done in a C*-algebraic context. The existence of ground states is also obtained. Starting with the finite lattice model for Hamiltonian QCD developed by Kijowski, Rudolph (cf. [15,16]), we state its field algebra and a natural representation. We then generalize this representation to the infinite lattice, and construct a Hilbert space which has represented on it all the local algebras (i.e. kinematics algebras associated with finite connected sublattices) equipped with the correct graded commutation relations. On a suitably large C*-algebra acting on this Hilbert space, and containing all the local algebras, we prove that there is a one parameter automorphism group, which is the pointwise norm limit of the local time evolutions along a sequence of finite sublattices, increasing to the full lattice. This is our global time evolution. We then take as our field algebra the C*-algebra generated by all the orbits of the local algebras w.r.t. the global time evolution. Thus the time evolution creates the field algebra. The time evolution is strongly continuous on this choice of field algebra, though not on the original larger C*-algebra. We define the gauge transformations, explain how to enforce the Gauss law constraint, show that the dynamics automorphism group descends to the algebra of physical observables and prove that gauge invariant ground states exist.

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“…Theorem 1 in [31, p 216]), thus the observable H S does not exist in this representation. Also observe, that this definition of regularity coincides with the one used in [5,8].…”

Section: Kinematics Algebras and Regular Representationssupporting

confidence: 66%

“…It remains to prove that ω ∞ A Sn = 1 for all n. We will follow the proof of Lemma 7.3 in [5]. Let m > n ∈ N and on H Sm ⊂ H consider the operators H n := H Sn , H m := H Sm and H m\n := H Sm\Sn , and use analogous notation for λ m := λ grnd Sm etc.…”

Section: Using the Identification Above Of Hmentioning

confidence: 99%

Dynamics for QCD on an Infinite Lattice

Grundling

Rudolph

2016

Commun. Math. Phys.

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“…3.1].The action of the gauge group on the resolvent algebra R is not pointwise norm continuous, cf. [5, Thm. 5.3(ii)].…”

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

Buchholz

2020

Commun. Math. Phys.

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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,...

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“…With the basic rules for Fourier transforms, ρ(·) can be shown to be a *-homomorphism on a dense subset, and hence on all of C 0 (R). Furthermore, from [2,Corollary 4.4] it straightforwardly follows that ρ((1/(iλ − ·))ˆ) = (iλ − φ(x)) −1 = R(λ, x). This completes the proof.…”

Section: Weyl Quantizationmentioning

confidence: 99%

“…This simple change turns out to give the resolvent algebra a much richer structure, and makes it better suited for modelling dynamics, compared to the Weyl algebra. The resolvent algebra, introduced and thoroughly investigated by Buchholz and Grundling in [2], appears to be useful for many aspects of quantum mechanics and quantum field theory, but has left us one important question. This question, posed by Buchholz in a personal communication, concerns the classical limit of the resolvent algebra, or, equivalently (at least within a C*-algebraic framework), its emergence from strict deformation quantization theory.…”

Section: Introductionmentioning

confidence: 99%

Quantization and the resolvent algebra

Nuland

2019

Journal of Functional Analysis

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We introduce a novel commutative C*-algebra C R (X) of functions on a symplectic vector space (X, σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of C R (X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1]. We also define a Berezin-type quantization map on all of C R (X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra C R (X), generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum. * This is an author generated post-print of the following article,

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The resolvent algebra: A new approach to canonical quantum systems

Cited by 65 publications

References 18 publications

Dynamics for QCD on an Infinite Lattice

Dynamics for QCD on an Infinite Lattice

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

Quantization and the resolvent algebra

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