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

Abstract

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

p-Adic wavelets and their applications

“…For the case when the environment field is in the Fock state (i.e. the state is given by the vacuum average) the corresponding model was considered in [3], see also [2], [4]. Other models with non-linear interaction and free statistics in the stochastic limit were studied in [5], [6].…”