Towards a deterministic quantum chaos

A.M.Mukhamedov

Kazan State Technical University of A.N.Tupolev's name

Kazan, Russia

Being endowed with a structure of a bundle space built over the space-time a space of quantum states is considered as framework for constructing particular histories of quantum motion. An idea of causality fields presented by non-integrable Pfaff systems is put forward to reproduce ambiguities of chaotic motion. Some analogy between dynamic chaos and modified version of quantum theory designed for description of the results of individual experiments is pointed out.

 

An idea of dynamical chaos as a state that reproduces new dynamical degrees of freedom has been exploring in periodicals for many decades before. In spite of a long term that has been already passed since the first statements of this kind, it must be established the fact that the idea is far beneath from its formalized realization. Moreover, nowadays the idea of dynamical degrees of freedom looks like an expression that takes the place of feeling on account of complicated patterns of chaotic motion. Whether the dynamical degrees of freedom really appear during chaotic motion and whether the idea of such dynamical coordinates can help us in constructing models of chaos? The answers to these questions are not found yet. But it seems clear that without recourse of formalized approaches the required answers cannot be found at all.

In recent papers of the author there was proposed some version of mathematical formalism of chaotic motion that utilizes an idea of additional dynamical degrees of freedom as some variables that are to be treated on an equal footing with spatial coordinates and a time. It turned out that the method allows further generalization onto the area of dynamical descriptions of chaotic phenomena of microcosm. The fact that unpredictable behavior of chaotic motion will obtain entirely deterministic explanation becomes the most attractive feature of the method. So, in the present paper a new attempt was made to develop the study of chaotic motion towards a new area of investigations that uniformly treats both classical chaos and quantum indeterminism.

It is well known that under definite circumstances the motion of wave functions can be reduced to the motion of parameters of order presenting new dynamical variables. But the question is to learn the fact whether these new variables must be regarded as functions of spatial coordinates and a time or they can be treated as independent variables that give the start for new degrees of freedom of the space-time. The author holds an opinion that the four well known spatio-temporal dimensions determine large scaled structure of the space-time only. A small scaled structure of the space-time seems to be more complex. New spatio-temporal degrees of freedom must exist, but their habitat is applied to the small scales of microcosm only.

Nowadays the physics of microcosm is described by quantum theory developed with the help of classical space-time. An alternative idea of description of microphysics by means of (multidimensional) fractal structures imposed on the space-time was recently proposed. Spatio-temporal fractal E^(¥) having four macroscopic dimensions and infinite number of microscopic ones was put forward to explain observable microscopic phenomena. In spite of the fact that the idea has provided new outcomes and that the number of the papers devoted to E-infinity theory has grown up to much, this idea can be treated as a scheme permitting to give up a claim of classical four dimensionality of the space-time only. At the same time the meaning of E^(¥) consists in the fact that this scheme potentially involves some explanations of the very mechanism of appearing of quantum effects.

Indeed, let us suppose that the additional degrees of freedom reproduce quantum random manifestations. We shall examine such possibility in the following. In this case the needs for statistical support for the present quantum theory can be called in question. At the same time this issue enables us to search entirely deterministic schemes of description for seemingly chaotic and randomness dynamics of microcosm. Acting in advance one can suppose that the effect of deterministic randomness will correspond to a grater extent the hopes of those physicists of the last century which were not agree with the probabilistic version of quantum mechanics. In this case such hopes will obtain real shapes.