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

On the existence of the subdynamics concept

Michel De Haan

Published Jan 1, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.3406/barb.1998.27887

Related papers

Single Subdynamics for the Friedrichs Model

Michel De Haan, Claude D. George

The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self-energy processes (already found in quantum electrodynamics), which allows for the widening of the dynamics through the introduction of specific additional variables leading to the recognition of such processes. A projection technique modeled on that introduced long ago by the Brussels group defines a subdynamics that is shown to contain the starting formulation in its entirety. (It corresponds to a restricted class of initial conditions, namely the reversible ones.) A new kinetic description is obtained in which, e.g., collision processes for privileged components arise from the original interaction. This method is applied successively to various models, and we focus in this paper on the possible presence of temporal behaviour excluded in ordinary quantum mechanics. Among the differences between this framework and that of ordinary quantum mechanics, the possibility appears for the purely exponential decay of a particle, while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.

Progress of Theoretical Physics•Jun 1, 2003•7 citations

The Second Law and the Extension of Quantum Mechanics

Michel De Haan

The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics is a challenging problem. Our solution requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self‐energy processes (already found in quantum electrodynamics) which enables the widening of the dynamics by the introduction of specific additionnal variables leading to the recognition of such processes. A projection technique modeled on the one introduced long ago by the Brussels group defines a subdynamics which is shown to contain entirely the starting formulation (it corresponds now to a restricted class of initial conditions, namely the reversible one). A new kinetic description is obtained in which e.g. collision processes for privileged components arise from the original interaction. This method is applied successively to various models and we focus in this paper on the possible presence of temporal behaviours excluded in ordinary quantum mechanics. Among those differences, the possibility appears of a purely exponential decay for a particle while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.

AIP conference proceedings•Jan 1, 2002•0 citations

Commutator of an arbitrary superoperator with the « asymptotic » evolution operator of a many-body system

J. B. Aderca

In the frame of the theory of subdynamics a general picture of the time evolution of a many-body system has been developed by the Brussels school of statistical mechanics. In this paper, we avoid the perturbation scheme so that we get compact notations and demonstrations. We derive here a general expression for the commutator of an arbitrary superoperator W with the " asymptotic " evolution superoperator Σ(t) = ΠU(t) where [formula] is the complete evolution superoperator and where Π is the usual superprojector of the theory of subdynamics. In the frame of relativistic non-equilibrium statistical mechanics, the application of this commutator to all the generators of the Poincaré group is then given and we get back in particular the fundamental property of commutation with Π of the Lorentz-Liouville superoperator.

Bulletin de la Classe des sciences•Jan 1, 1979•0 citations

Single Subdynamics for potential scattering

Michel De Haan, Claude D. George

A subdynamics theory has been presented recently in which quan¬ tities characteristic of an irreversible behaviour, such as that of a collision process, appear in the generator of evolution. The original dynamics, with degrees of freedom in large number, is first widened once self-energy parts have been recognized. The redundancies are then constrained by the requirement that the evolution takes place entirely in a subdynamics associated with a vacuum defined by reference to the observational situation of interest. This scheme, resulting in no loss of physical information, is examined in detail in the simple case of potential scattering, after that incoming, outgoing and intermediate degrees of freedom are distinguished. The relation with previous approaches is analysed.

Bulletin de la Classe des sciences•Jan 1, 2002•1 citation

On some basic theorems in subdynamics of correlations

Claude D. George

The possibility of describing the evolution of large dynamical systems in terms of a superposition of subdynamics, and of formulating a transformation theory of the Liouville-von Neumann equation for such systems rests on a set of relations between dynamical operators. We present a proof of these relations.

Bulletin de la Classe des sciences•Jan 1, 1973•0 citations

An elementary construction of the projection operator for subdynamics

Dragomir M Davidović, Z. Marić, Mirjana Božić

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento•Oct 1, 1981•2 citations
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.
The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self-energy processes (already found in quantum electrodynamics), which allows for the widening of the dynamics through the introduction of specific additional variables leading to the recognition of such processes. A projection technique modeled on that introduced long ago by the Brussels group defines a subdynamics that is shown to contain the starting formulation in its entirety. (It corresponds to a restricted class of initial conditions, namely the reversible ones.) A new kinetic description is obtained in which, e.g., collision processes for privileged components arise from the original interaction. This method is applied successively to various models, and we focus in this paper on the possible presence of temporal behaviour excluded in ordinary quantum mechanics. Among the differences between this framework and that of ordinary quantum mechanics, the possibility appears for the purely exponential decay of a particle, while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.
The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics is a challenging problem. Our solution requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self‐energy processes (already found in quantum electrodynamics) which enables the widening of the dynamics by the introduction of specific additionnal variables leading to the recognition of such processes. A projection technique modeled on the one introduced long ago by the Brussels group defines a subdynamics which is shown to contain entirely the starting formulation (it corresponds now to a restricted class of initial conditions, namely the reversible one). A new kinetic description is obtained in which e.g. collision processes for privileged components arise from the original interaction. This method is applied successively to various models and we focus in this paper on the possible presence of temporal behaviours excluded in ordinary quantum mechanics. Among those differences, the possibility appears of a purely exponential decay for a particle while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.
In the frame of the theory of subdynamics a general picture of the time evolution of a many-body system has been developed by the Brussels school of statistical mechanics. In this paper, we avoid the perturbation scheme so that we get compact notations and demonstrations. We derive here a general expression for the commutator of an arbitrary superoperator W with the " asymptotic " evolution superoperator Σ(t) = ΠU(t) where [formula] is the complete evolution superoperator and where Π is the usual superprojector of the theory of subdynamics. In the frame of relativistic non-equilibrium statistical mechanics, the application of this commutator to all the generators of the Poincaré group is then given and we get back in particular the fundamental property of commutation with Π of the Lorentz-Liouville superoperator.
A subdynamics theory has been presented recently in which quan¬ tities characteristic of an irreversible behaviour, such as that of a collision process, appear in the generator of evolution. The original dynamics, with degrees of freedom in large number, is first widened once self-energy parts have been recognized. The redundancies are then constrained by the requirement that the evolution takes place entirely in a subdynamics associated with a vacuum defined by reference to the observational situation of interest. This scheme, resulting in no loss of physical information, is examined in detail in the simple case of potential scattering, after that incoming, outgoing and intermediate degrees of freedom are distinguished. The relation with previous approaches is analysed.