Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bulletin de la Classe des sciences

Subdynamics in the Lee model, I

Michel De Haan

Published Jan 1, 1977• 1 citation
Publisher

Abstract

In the framework of the theory of subdynamics introduced by the Brussels group, we construct explicitly and in closed form the whole set of subdynamics for a particular model : the Lee model in the one particle sector. The method presented here can be used for applications to other systems in field theory.
DOI
10.3406/barb.1977.58185

Related papers

Dynamics as a subdynamics. The Friedrich Model

Michel De Haan, Claude D. George

De Haan Michel, George Claude D. Dynamics as a subdynamics. The Friedrich Model. In: Bulletin de la Classe des sciences, tome 15, n°1-6, 2004. pp. 41-76.

Bulletin de la Classe des sciences•Jan 1, 2004•0 citations

On the existence of the subdynamics concept

Michel De Haan

The concept of subdynamics projectors Π introduced by the Brussels group in the treatment of systems in statistical mechanics remains central in the construction of the non-unitary transformation theory or in the theory of complex spectral representations. Originally only one of such subdynamics, II, was supposed to persist asymptotically. When the evolution of the systems presents temporal behaviours with long tails, the usefulness of this subdynamics concept, and even its existence, is to be questioned. To catch the nature of the problem, we analyze simple systems with a non exponential decay. We indicate how the corresponding operators may effectively be constructed and we examine the validity of the asymptotic projection.

Bulletin de la Classe des sciences•Jan 1, 1998•0 citations

Field theory reformulated without self-energy parts. Kinetic extension for the Lee model

Michel De Haan

Annals of Physics•Feb 28, 2004•6 citations

On time evolution for large quantum systems in superspace

Nobuhiko Mishima, Tomio Yamakoshi Petrosky

A reformulation in a superspace is given for Van Hove and Janner’s perturbation theory based on the two-resolvent method for nonequilibrium quantum statistical mechanics. This is attained by introducing an ordered superoperator in the superspace. As an application of our formalism, the derivation of an asymptotic generalized master equation in a Markovian form is given. Further, it is investigated how a quantum subdynamics theory similar to the Brussels school’s one can be formally constructed. Our subdynamics theory is formulated on every eigenspace of a superoperator (the ss-Hamiltonian) of which the eigenvalue is related to the intrinsic parameter E of Van Hove and Janner’s generalized master equation.

Journal of Mathematical Physics•May 1, 1978•6 citations

Mathematical problems of irreversible statistical mechanics for quantum systems. II: On the singularities of (Ψ(z)−z)−1 and the pseudo-Markovian equation. Application to Lee model

M. Courbage

We study in the frame of the superspace of the Hilbert–Schmidt operators the contributions of some complex singularities of the analytic continuation of (Ψ(z)−z)−1 to the diagonal part of the solution of the Liouville–von Neumann equation. Under some conditions, the θ̄ operator of the pseudo-Markovian master equation can be explicitly constructed. It is necessary to specify the diagonal representation and the class of initial conditions having regularity properties. This Hilbertian structure does not allow the construction of a closed subspace which reduces the Liouville–von Neumann operator L, giving an exact irreversible subdynamics; more elaborated mathematical structures are therefore necessary. The above methods are illustrated in the case of the Lee model.

Journal of Mathematical Physics•Apr 1, 1982•8 citations

Lee-Friedrichs Model

Lawrence Paul Horwitz

A brief summary of the mathematical structure of the Lee-Friedrichs model is given, including the natural extension of its Hamiltonian to a Gel'fand triple, containing a generalized eigenstate for the complex pole. Contains references to some physical applications and relativistic generalizations.

arXiv:hep-ph/9811383•Nov 18, 1998•1 citation
The concept of subdynamics projectors Π introduced by the Brussels group in the treatment of systems in statistical mechanics remains central in the construction of the non-unitary transformation theory or in the theory of complex spectral representations. Originally only one of such subdynamics, II, was supposed to persist asymptotically. When the evolution of the systems presents temporal behaviours with long tails, the usefulness of this subdynamics concept, and even its existence, is to be questioned. To catch the nature of the problem, we analyze simple systems with a non exponential decay. We indicate how the corresponding operators may effectively be constructed and we examine the validity of the asymptotic projection.
A reformulation in a superspace is given for Van Hove and Janner’s perturbation theory based on the two-resolvent method for nonequilibrium quantum statistical mechanics. This is attained by introducing an ordered superoperator in the superspace. As an application of our formalism, the derivation of an asymptotic generalized master equation in a Markovian form is given. Further, it is investigated how a quantum subdynamics theory similar to the Brussels school’s one can be formally constructed. Our subdynamics theory is formulated on every eigenspace of a superoperator (the ss-Hamiltonian) of which the eigenvalue is related to the intrinsic parameter E of Van Hove and Janner’s generalized master equation.
We study in the frame of the superspace of the Hilbert–Schmidt operators the contributions of some complex singularities of the analytic continuation of (Ψ(z)−z)−1 to the diagonal part of the solution of the Liouville–von Neumann equation. Under some conditions, the θ̄ operator of the pseudo-Markovian master equation can be explicitly constructed. It is necessary to specify the diagonal representation and the class of initial conditions having regularity properties. This Hilbertian structure does not allow the construction of a closed subspace which reduces the Liouville–von Neumann operator L, giving an exact irreversible subdynamics; more elaborated mathematical structures are therefore necessary. The above methods are illustrated in the case of the Lee model.