Nonintegrable Poincaré systems with continuous spectrum (so-called Large Poincaré Systems, LPS) lead to the appearance of diffusive terms in the framework of dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory dynamics and of wave functions. 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 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 formulations of classical or quantum mechanics. Instead of being irreducible 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 we inherited from the 19th 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 the many-world theory, coarse graining or averaging over the environment). There is a striking parallelism between classical and quantum theory. For LPS we have, in general, both a “collapse” of trajectories and of wave functions for LPS. 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 know 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. The formulation is appropriate to describe an open, evolving universe.
Schrodinger Wave Equation
