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 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.
It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less
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.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
The microscopic theory of irreversible processes that we developed is summarized and illustrated, using as a simple example the Friedrichs model. Our approach combines the Poincaré's point of view (dynamical interpretation of irreversibility) with the Gibbs-Einstein ensemble point of view. It essentially consists in a nonunitary transformation theory based on the symmetry properties of the Liouville equation and dealing with continuous spectrum. The second law acquires a microscopic content in terms of a Liapounov function which is a quadratic functional of the density operator. In our new representation of dynamics, which is defined for a restricted set of observables and states, this functional takes a universal form. We obtain, in this way, a semi-group description, the generator of which contains a part directly related to the microscopic entropy production. The Friedrichs model gives us a simple field theoretical example for which the entropy production can be evaluated. The thermodynamical meaning of life-times is explicitly displayed. The transition from pure states to mixtures, as well as the occurrence of long tails in thermodynamic systems, are also briefly discussed.