To problem of quantum mechanics interpretation:

invariance groups of Hamilton systems

in reconstruction of non-classical enlargements

A.M.Mukhamedov

KSTU named after A.N.Tupolev

9-15, Tatarstan, Kazan, 420021, RUSSIA

This paper is divided on few section:

Introduction. 1. The equation of the motion of Hamilton systems. 2. Covariance principle for Hamilton systems. 3. Canonical transformations. 4. An illustrative example of canonical variables. 5. Classical realizations of Hamilton systems and their wave functions. 6. Quantum realizations of Hamilton systems and their wave functions. 7. Operational reconstruction of identifiers of realizations. 8. Quantum covariance principle for Hamilton systems. 9. Concluding remarks.

It is known that the problem of interpretation of quantum mechanics was mentioned as the one among the other modern physical problems in the list made up by Nobel Prize winner in physics V.L.Ginzburg.

Though the problem is appeared at the earliest stage of creating of quantum mechanics, its actuality is grown up especially nowadays. During last decade the abovementioned topic was discussed on the pages of scientific journal "Uspekhy Fizicheskikh Nauk". The discussion was stimulated mainly by novel ideas which were proposed to explain cognitive phenomena on the basis of reduction of the quantum wave function. The present paper is aimed to the same mainstream.

The starting idea of the paper can be easily understood from the following explanation. As the quantum formalism, pragmatically developed in order to systematize observable data, contains hardly understandable statements, it seems necessary to find out direct indications obtained on the basis of classical paradigm that are able to supply these statements by particular mechanisms. This means that the analogies, which were recruited in the course of historical creation of quantum mechanics, are useless for this purpose. Interpretation must not be of analogous type. It must be a direct consequence of those classical concepts that are employed for interpretation.

In this paper in order to bring about the idea, there was employed the notion that the quantum mechanical correlations appear as results of enlarged identity relations which can be accepted for classical prototypes of the quantum systems under considerations. Indeed, we do not tell the difference between realizations of the motion obtained with the help of different frames. In Newton mechanics this identification appears as invariance property of master equations under Galilean transformations. But, there are some other cases of the motion, for example the motion of Hamiltonian systems, which are invariant under a larger set of transformations. The canonical transformations give the required case in which the configuration coordinates and momentums are allowed to be confused with each other. If one takes the internal point of view to regard the motion, which is invariant under abovementioned group of transformations, then, being forced to change this viewpoint into external one, which is based on the non-invariant configuration subspace of the phase space, the multivalued cases of representation must happen. It is the enlarged identity relations that give rise to multivalued interpretations of the motion of such systems, and nothing more.

In the paper there put forward a hypothesis according to which the problem of interpretation may be caused by the fact that the structures, used to interpret the motion, are less symmetrical than the motion itself. In this case the quantum correlations must be regarded as conclusive manifestations which prove the possibility for the nature to take an internal position and to make different external outcomes.

The matter of the paper can be traced by the list of content presented at the beginning of the paper.

In the introduction the full representation of the main concepts are articulated. The enlarged invariance principle, which is employed to identify different realizations of the motion, is explained. There are settled down the identifiers as natural prerequisites for operator reconstruction of observables adopted in the quantum theory.

In the two starting paragraphs the particular systems, which will be employed to deal with, are specified. The covariance principle adopted for Hamilton systems are employed as primordial statement in order to generate the enlarged identity relation for different realizations of the motion of Hamilton systems.

The following two paragraphs are intended to give particular examples in order to exclude variant readings.

In the fifth paragraph the classical Hamilton dynamics is reformulated in terms which are usually used in quantum mechanics. Here, the idea of identifiers of the families of classical trajectories defined by action function and its derived object presented by classical wave function are introduced. This part of the subject does not exceed the limits of well-known optic-mechanical analogy.

In the sixth paragraph the transition from the classical theory to the quantum one is produced. Here, the novel idea of a bundle space of identifiers is introduced to deal with the interference phenomena. It is in this connection that the different realizations, which in case of classical mechanics must not interfere of each other, turn to quantum realizations, presented by interfering alternatives.

Though the appearance of interfering alternatives is hardly predictable effect from the classical viewpoint, it, being discovered, forces us to find out the explanations. Copenhagen treatment refuses to do it. More intriguing many-worlds treatment of Everett deduces the explanations out of physical context. In the present paper the explanations keep their physical character. The interpretation of interference is to be a simple consequence of enlarged identity relations imposed to the classically determined systems.

The final two paragraphs are written in order to obtain a complete version of quantum formalism. The principal result of them consists in the fact that the basic quantum equation of Schrödinger is not postulated but does deduced on the base of the bundle space of identifiers introduced in the sixth paragraph. The significance of this derivation consists in the fact that it enables us to trace theoretically some new level of reality that nominally contained in equation of Schrödinger and its solutions.

In this paper the quantum reality is only indicated in terms of quantum covariance principle. The specifications of this principle that uncover such level of reality may be regarded as a new research program which is only proposed in the present paper, but, as is expected, will be developed in the following papers.