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Randomness in Quantum Physics Revisited.

Michail A. Zak

Published Jul 1, 2014• 0 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 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 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 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.
In a prior paper, the author described an instability in a nonlinear wavefunction model. Proposed in connection with the Measurement Problem, the model contained an external potential creating a ``classical'' instability. However, it is interesting to ask whether such models possess an intrinsic randomness -- even ``chaos" -- independent of external potentials. In this work, I investigate the criterion analytically and simulate from a small (``3 qubit") model, demonstrating that the Lyapunov exponent -- a standard measure of ``chaos" -- is positive. I also extend the instability criterion to models in the continuum. These results suggest that the boundary between classical and wavefunction physics may also constitute the threshold of chaos, and present an alternative to Max Born's ad hoc probability hypothesis: random outcomes in experiments result not from ``wave-particle duality" or ``the existence of the quantum," but from sensitive dependence on initial conditions, as is common in the other sciences.