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.
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.
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
Abstract In this paper the thermodynamical properties of any system whatsoever are deduced from quantum mechanics. Two fundamental irreversible processes are considered: the conversion of other forms of energy into heat, and the flow of heat from one temperature to another. By the proof of a generalized H -theorem, it is shown that in each case the entropy, correctly defined, must increase, and the system tend towards a state of equilibrium. A simple but rigorous proof of Boltzmann’s law is given from which the thermodynamics of reversible processes may be inferred. An appendix includes the exact solution of the most general time-dependent perturbation problem of quantum mechanics.
The second law of thermodynamics, for quantum systems, is formulated, on the microscopic level. As for classical systems, such a formulation is only possible when specific conditions are satisfied (continuous spectrum, nonvanishing of the collision operator, etc.). The unitary dynamical group can then be mapped into two contractive semigroups, reaching equilibrium either for t --> +infinity or for t --> -infinity. The second law appears as a symmetry-breaking selection principle, limiting the observables and density functions to the class that tends to thermodynamic equilibrium in the future (for t --> +infinity). The physical content of the dynamical structure is now displayed in terms of the appropriate semigroup, which is realized through a nonunitary transformation. The superposition principle of quantum mechanics has to be reconsidered as irreversible processes transform pure states into mixtures and unitary transformations are limited by the requirement that entropy remains invariant. In the semigroup representation, interacting fields lead to units that behave incoherently at equilibrium. Inversely, nonequilibrium constraints introduce correlations between these units.