The Wigner semi-circle law in quantum electro dynamics

Accardi, Luigi; Lu, Yun Gang

doi:10.1007/bf02099625

Cited by 58 publications

(55 citation statements)

References 14 publications

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“…The presence of the Mq~ller wave operator represents the main new feature, with respect to the case of a free particle, studied in [3]. For this reason in the following section we describe the main idea of the emergence of this new feature.…”

Section: Hi(t)=i(f~a Eitns(--ipmentioning

confidence: 99%

“…By a standard procedure [3] (introducing a cut-off and changing the sum to integration) we can rewrite the evolved interaction Hamiltonian as…”

Section: Hl(t) := Eit(hs®i+i®hr) Hie -It(hs®i+i®hr)mentioning

confidence: 99%

“…Finally, in Section 5 we deduce the explicit form of the module stochastic equation for the limit operator. The proofs of the last three sections have not been included because, although lengthy, they do not introduce substantial new ideas with respect to [3].…”

Section: Hi(t)=i(f~a Eitns(--ipmentioning

confidence: 99%

“…Moreover, the quantum noise is i) bosonic, if we introduce the (quasi) dipole approximation; ii) an entangled extension of the free white noise, if we drop the dipole approximation. In previous investigations [1][2][3] we have considered a free particle, i.e. a particle driven by the kinetic energy minimally coupled to the EM field.…”

Section: Introductionmentioning

confidence: 99%

“…(cf. [3]). In (1.2) p denotes a fixed component of momentum on the orthogonal space to k. In the interaction picture, the evolution of this system is determined by the operator U.: L) := eit(Hs®l+l®HR) e -it(HS®I+I®HR+xH1), (1.3) where the coefficient 3. is called a coupling constant.…”

Section: Introductionmentioning

confidence: 99%

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Free probability and quantum electrodynamics

Accardi¹,

Lu²

2004

Reports on Mathematical Physics

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The stochastic limit of a flee particle coupled to the quantum electromagnetic field without dipole approximation leads to many new features such as: interacting Fock space, Hilbert module commutation relations, disappearance of the crossing diagrams, etc. In the present paper we begin to study how the situation is modified if a free particle is replaced by a particle in a potential which is the Fourier transform of a bounded measure.We prove that the stochastic limit procedure converges and that the overall picture is similar to the free case with the important difference that the structure of the limit Hilbert module is strongly dependent on the wave operator of the particle.

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Section: Hi(t)=i(f~a Eitns(--ipmentioning

confidence: 99%

“…By a standard procedure [3] (introducing a cut-off and changing the sum to integration) we can rewrite the evolved interaction Hamiltonian as…”

Section: Hl(t) := Eit(hs®i+i®hr) Hie -It(hs®i+i®hr)mentioning

confidence: 99%

Section: Hi(t)=i(f~a Eitns(--ipmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Free probability and quantum electrodynamics

Accardi¹,

Lu²

2004

Reports on Mathematical Physics

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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

Skeide

1998

Communications in Mathematical Physics

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The Wigner semi-circle law in quantum electro dynamics

Cited by 58 publications

References 14 publications

Free probability and quantum electrodynamics

Free probability and quantum electrodynamics

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

Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

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