Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

Skeide, Michael

doi:10.1007/s002200050310

Cited by 32 publications

(35 citation statements)

References 10 publications

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“…What we mean here is different, namely that, even if before the limit atom and field observables commute, after the limit they develop nontrivial commutation relations (in fact an operator generalization of the free commutation relations). In the case of QED Skeide [Ske96] proved that the interacting Fock space structure can be deduced from the Hilbert module structure alone. If this result is a general phenomenon, or if it is specific to the QED Hilbert module, is an open problem.…”

Section: The 90'smentioning

confidence: 99%

Quantum probability: an historical survey

Accardi¹

2000

Probability on Algebraic Structures

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1What is quantum probability Quantum Probability (QP) is a new branch of mathematics interconnecting classical probability, functional analysis, pure algebra, quantum physics and information and communication engineering. The mid seventies is the period that marks the beginning of QP as an autonomous discipline. Since then this cross disciplinary nature has accompanied the development of QP and still now it is one of its points of strength, making it an original new trend in contemporary mathematics as well as one of the earliest pioneers of non-commutative mathematics: a field now flourishing with the more recent development of quantum groups, non-commutative geometry, quantum computer, . . .The developments in this area are taking place at such an high pace and such a broad horizon to make impossible any attempt of a detailed synthesis in a single paper. In what follows I shall give a quick historical survey of QP pointing out some of the main problems which have driven these developments and of the main applications to quantum physics. The early periodSeveral notions, tools and single results, now stably entered in the domain of QP, were developed simultaneously with the birth of quantum mechanics. However these developments occurred independently one another, in several different fields such as functional analysis, the foundations of quantum theory, 1 Abstract of an invited talk to the Annual meeting of the Japan Mathematical Society ,

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Section: The 90'smentioning

confidence: 99%

Quantum probability: an historical survey

Accardi¹

2000

Probability on Algebraic Structures

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“…It allows us to find dilations for (bounded) generators of CP-semigroups on arbitrary C*-algebras B whose form was found by Christenson and Evans [CE79]. Recently, Goswami and Sinha [GS99] introduced a calculus on a symmetric Fock module [Ske98a] and used it to solve the dilation problem for Christensen Evans generators.…”

Section: Introductionmentioning

confidence: 97%

Quantum Stochastic Calculus on Full Fock Modules

Skeide

2000

Journal of Functional Analysis

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We develop a quantum stochastic calculus on full Fock modules over arbitrary Hilbert B B-modules. We find a calculus of bounded operators where all quantum stochastic integrals are limits of Riemann Stieltjes sums. After having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations, we find necessary and sufficient conditions for unitarity of a subclass of solutions. As an application we find dilations of a conservative CP-semigroup (quantum dynamical semigroup) on B with arbitrary bounded (Christensen Evans) generator. We point out that in the case B=B(G) the calculus may be interpreted as a calculus on the full Fock space tensor initial space G with arbitrary degree of freedom dilating CP-semigroups with arbitrary Lindblad generator. Finally, we show how a calculus on the boolean Fock module reduces to our calculus. As a special case this includes a calculus on the boolean Fock space.

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Section: Introductionmentioning

confidence: 99%

“…Further investigation of this Boltzmannian algebra, which governs the limiting dynamics, is interesting in itself. The stochastic limit of this model at zero temperature was considered in [7][8][9].The simplest Boltzmannian algebra is generated by the relationsSuch relations have been investigated in mathematics [10][11][12][13][14][15]; they were obtained in the stochastic limit of the model of a particle interacting with a quantum field [7] and in the large-N limit of quantum chromodynamics with the gauge group SU(N) [16]. The free temperature algebra below can be schematically described as the Boltzmannian algebra with the generators b(k), bt(k), and p, which satisfy the relations …”

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confidence: 99%

A new algebra in the stochastic approximation for the model of a particle interacting with a quantum field

1998

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L. Accardi, l I. V. Volovich, 2 and S. V. Kozyrev 3When the stochastic approximation is used to calculate correlation functions in the model of a partic•e interacting with a quantum field, a new algebra with temperature-dependent commutation relations appears. This algebra generalizes the free ( Boltzmann) algebra. IntroductionWe consider the model of a particle interacting with a quantum field. Such models have been extensively studied in elementary particle physics, in solid-state physics, in quantum optics, etc. [1][2][3][4]. We investigate correlation functions using the stochastic approximation method of Van Hove and Friedrichs. Accardi, Frigerio, and Lu applied this method to quantum optic models [5]. One of these models was analyzed [6] in the dipole approximation. The method consists in using a scaling limit, where the asymptotic behavior of the correlation function is considered at large times and small coupling constants. Then, the limiting dynamics are integrable, in a sense, for a series of problems, and explicit expressions can be obtained for the correlation functions [6]. The limit is called "stochastic" because free correlators become "&correlated" in time in this limit. (That is, we have the white-noise random process.)Our main result is that in the temperature stochastic limit, a new mathematic structure arises in the model of particle interacting with a quantum field. We call this structure the free temperature algebra. It is a Boltzmannian algebra.We consider correlation functions for special operators (the collective variables). In the stochastic limit, the theory is simplified and is described by the free temperature algebra; the correlators correspond to some states of the free temperature algebra. Further investigation of this Boltzmannian algebra, which governs the limiting dynamics, is interesting in itself. The stochastic limit of this model at zero temperature was considered in [7][8][9].The simplest Boltzmannian algebra is generated by the relationsSuch relations have been investigated in mathematics [10][11][12][13][14][15]; they were obtained in the stochastic limit of the model of a particle interacting with a quantum field [7] and in the large-N limit of quantum chromodynamics with the gauge group SU(N) [16]. The free temperature algebra below can be schematically described as the Boltzmannian algebra with the generators b(k), bt(k), and p, which satisfy the relations

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Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

Cited by 32 publications

References 10 publications

Quantum probability: an historical survey

Quantum probability: an historical survey

Quantum Stochastic Calculus on Full Fock Modules

A new algebra in the stochastic approximation for the model of a particle interacting with a quantum field

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