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Randomness in physics revisited

Michail A. Zak

Published Jan 1, 2016• 5 citations
OpenAlex

Abstract

There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into deterministic (chaos) and true (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.

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There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into deterministic (chaos) and true (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.
There has been proven that mathematical origins of randomness in quantum and Newtonian physics are coming from the same source that is dynamical instability. However in Newtonian physics this instability is measured by positive finite Liapunov exponents averaged over infinite time period, while in quantum physics the instability is accompanied by a loss of the Lipchitz condition and represented by an infinite divergence of trajectories in a singular point. Although from a mathematical viewpoint such a difference is significant, from physical viewpoint it does not justify division of randomness into “deterministic “(chaos) and “true” (quantum physics). The common origin of randomness in Newtonian and quantum physics presents a support of the correspondence principle that is being searched by quantum chaos theory.
The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.
The true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems. In return one should accept the modified form of quantum formalism (presented by the Schroedinger equation) which, however, does not contradict the ordinary form and the main postulates of quantum mechanics. It introduces the principle of the fundamental dynamic multivaluedness (redundance) extending the quantum paradigm to complex dynamical behaviour. Moreover, a causal solution to the well-known problems of the foundation of quantum mechanics, those of quantum indeterminacy and wave reduction, is also found using the same method. The concept of the fundamental dynamic uncertainty thus established is universal in character and provides a unified scheme of the complete description of arbitrary complex system of any origin (physics/9806002). This scheme incorporates, in particular, universal definitions of complexity, (non)integrability, and general solution, as well as the physically complete notion of probability. One obtains thus a self-consistent hierarchic picture of the world characterised by a (high) non-zero complexity and containing the intrinsic causal randomness, where the causally extended quantum mechanics can be consistently interpreted as several lowest levels of complexity (quant-ph/990215,16).
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The chaos exhibited by a quantum mechanical system is revealed in various aspects such as level spacing, avoiding of level crossing on changing a parameter value, irregular nodal lines of wave function, random matrix and so on. On the other hand since the wave function, the propagator or any physical quantity represented by the average of a phase function is usually quasi-periodic in time development, no exponential divergence of its value takes place between the neighboring two initial values. Furthermore the energy of a kicked oscillator ceases to increase after a certain period of time. These observations lead one to suspect the existence of chaos in quantal systems. It is shown that the nature of quasi-periodicity in quantum system is shared with the distribution function of classical hamiltonian system. The action integral which appears in the formulation of the propagator is the key quantity which can connect the classical and quantal systems, and is considered as serving to search for the origin of quantum chaos. The roles of the action integral in the determination of the energy levels and the behavior of the wave function are discussed.