Fernand Mayne

Liapounov function for the friedrichs model

The Friedrichs model of an unstable particle provides us with a non-trivial example of a dynamical system with a continuous spectrum. The non-unitary transformation operator which leads in our approach to a semi-group description of the time evolution of the system has been constructed in earlier papers. It is used here to evaluate the Liapounov function which plays the role of a microscopic entropy. The results, obtained without any recourse to perturbation expansions, validate resummation procedures introduced previously in a perturbative treatment. Various types of initial conditions are considered. It is shown explicitly why the distinction between pure states and mixtures is lost in the asymptotic time limit. The result is of relevance for the discussion of the quantum mechanical measurement problem.

Vacuum subdynamics in large quantum systems

We present a constructive procedure to deal with large quantum systems in the thermodynamic limit. Starting with a discrete spectrum, we perform a complete decomposition of the evolution into one-dimensional subdynamics. We then go to the limit of a continuous spectrum after collecting them into global subdynamics of given degrees of correlation. Previously obtained results for the vacuum subdynamics are justified. The procedure is applied to the problem of potential scattering.

Scattering theory and subdynamics

Abstract Recently Prigogine, Henin and George have introduced the concept of subdynamics: the evolution of a dynamical system can be decomposed under very general conditions into independent evolutions taking place in orthogonal subspaces. The aim of this paper is to show that for scattering theory in the usual limits of large smooth wave packets, only the evolution in one of the subspaces (II) is relevant. The cases of potential scattering and interacting fields describing stable particles are treated. The problem of production of unstable states will be considered in a forthcoming paper.

Physical description of decay processes

A dynamical formulation of quasi-particles corresponding to complex poles of Greens functions has been provided in the frame of “Liapounov” transformations in a superoperator formalism. In this formalism, two different superoperators are associated with the hamiltonian, on the one hand an energy superoperator H, and on the other, an evolution superoperator L. In the case of systems with continuous spectrum, it is possible to diagonalize the transform of H, the diagonal elements being the energies found by the Greens functions methods. This can be achieved without diagonalizing at the same time, the transform of L: the inverse lifetimes and the collision processes are obtained in this way.

Theory of transport coefficients. IV. Electrical conductivity in the presence of a strong magnetic field

Synopsis The transverse electrical conductivity of an electron-phonon system under a strong static magnetic field is calculated up to the second order in the electron-phonon interaction starting from the Nalcano-Kubo formula and using the Feynman diagram techniques. The result is in agreement with that obtained by Titeica from intuitive grounds. It is found that the initial electron correlations contained in the canonical distribution exp [− β ( H 0 + H I )] contribute to the conductivity in contradistinction to the case of a magnetic field-free system, where the (static) conductivity in general can be expressed in terms of asymptotic cross sections defined between states free from correlation. This fact necessarily implies the impossibility of properly describing the magnetoresistance on the basis of Boltzmann — like equations.

On the application of non-equilibrium statistical mechanics to quantum field theory

Abstract The class of problems which belong to Heitlers general theory of damping phenomena may be reformulated in a new way using the recent development of non-equilibrium statistical mechanics. As examples cross sections of Moller scattering through a real boson and as well as higher order contribution to the life time of an unstable fermion are indicated.