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International Journal of Bifurcation and Chaos

WHY IRREVERSIBILITY? THE FORMULATION OF CLASSICAL AND QUANTUM MECHANICS FOR NONINTEGRABLE SYSTEMS

Ilya Prigogine

Published Feb 1, 1995• 20 citations
Publisher

Abstract

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.

DOI
10.1142/s0218127495000028

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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.

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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.
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.
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.
This paper develops a constraint-based account of dynamical models in physics, showing that reversible dynamics arise as highly restricted boundary cases of more general irreversible frameworks. Using open quantum systems (Lindblad master equation) and classical statistical mechanics (Boltzmann equation), we prove that reversible models form a proper subclass of irreversible models (R⊂I\mathcal{R} \subset \mathcal{I}R⊂I), obtained by imposing bijectivity, time-reversal invariance, and strict isolation. This structural inclusion clarifies how unitary Schrödinger evolution and Liouvillian dynamics emerge from dissipative, many-to-one processes, and reframes debates on the arrow of time, coarse-graining, and open-system foundations in terms of constraint hierarchies rather than fundamental reversibility.
Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time‐asymmetric boundary conditions of time‐symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann‐Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time‐asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.
Arguments are presented for the viewpoint that irreversibility is inherent in quanturn mechanics, irrespective of the number of particles participating. This fact is related to the notion of transition–ramifying evolution–which is characteristic of quantum mechanics. It is in contrast to classical mechanics based on the notion of orbit, where the irreversibility arises only as the result of a large number of particles or a certain complexity of the system. The statistical assumption needed for the derivation of irreversibility in classical mechanics is absorbed into the uncertainty principle of quantum mechanics, and is revived as inherent irreversibility. The statements conform with the parallelism of quantum mechanics and classical statistical mechanics emphasized in a previous report.
Abstract We discuss the spectral property of unstable dynamical systems in both classical and quantum mechanics. An important class of unstable dynamical systems corresponds to the Large Poincare Systems (LPS). Conventional perturbation technique leads then to divergences. We introduce methods for the elimination of Poincare divergences to obtain a solution of the spectral problem analytic in the coupling constant. To do so, we have to enlarge the class of permissible transformations, to include non-unitary transformations as well as to extend the Hilbert space. A simple example refers to the Friedrichs model, which was studied independently by George Sudarshan and his co-workers. However, our main interest is the irreducible representations in the Liouville space. In these representations the central quantity is the density matrix, and the eigenfunctions of the Liouville operator cannot be expressed in terms of the wave functions. We suggest that this situation corresponds to quantum chaos. Indeed, classical chaos does not mean that Newton's equation becomes "wrong" but that trajectories loose their operational meaning. Similarly, whenever we have an irreducible representation in the Liouville space this means that the wave function description looses its operational meaning. Additional statistical features appear. A simple example corresponds to persistent interactions in the scattering problem which cannot be treated in the frame of usual S-matrix theory.