Why irreversibility? The formulation of classical and quantum mechanics for nonintegrable systems
Ilya Prigogine
Abstract Nonintegrable Poincaré systems with a continuous spectrum (so‐called large Poincaré systems, LPS) lead to the appearance of diffusive terms in the frame of classical or quantum dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory theory and of wave‐function formalism (Schrödinger's equation). These diffusive terms correspond to well‐defined classes of dynamical processes (i.e., so‐called vacuum‐vacuum transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include, already, in the fundamental dynamical description the two basic aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulation of classical or quantum mechanics. Instead of being primitive concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville–von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature that we inherited from the nineteenth century, based, on the one hand, on dynamical time‐reversible laws and, on the other, on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory, avoiding the conventional dual structure based on Schrödinger's equation, on the one hand, and on the “collapse” of the wave function, on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as many‐world theory, coarse graining, or averaging over the environment). There is a striking parallelism between classical and quantum theory. For large Poincaré systems (LPS), we have, in general, both a “collapse” of trajectories and of wave functions. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we have known that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of “laws of physics” dealing no more with certitudes but with probabilities. This formulation is appropriate to describe an open, evolving universe. © 1995 John Wiley & Sons, Inc.
International Journal of Quantum Chemistry•Jan 5, 1995•27 citations