Approach to statement of problem

updated December 21,  2001

It is doubtless that the true space-time model is necessary for a correct description of physical phenomena. One needs to choose a true geometry from the list of possible space-time geometries. If the list of possible geometries is incomplete, one is doomed to use of  a false space-time geometry. Fortunately, in this case one can introduce additional suppositions, compensating negative features of  the false space-time geometry. Using these additional hypotheses, one can describe correctly several  experimental date and observations. But another experimental data appear not to be described in proper way, and one needs to introduce another additional suppositions to explain new experimental data and observations. In other words, it is impossible to introduce such additional suppositions which would explain all experimental data.

 Such a situation took place many years ago, when  the Ptolemaic conception of the celestial mechanics used a false statement about nonrotating Earth. It was compensated by the means of additional statements about epicycles and differentials and provided quite impressive accuracy in predicting astronomical events. Such conceptions, using incorrect suppositions and compensating them by additional hypotheses, will be referred to as compensating, or Ptolemaic conceptions.

The author of this work strongly believes that the contemporary theoretical physics has the same problem arising from the usage of Minkowski space-time model which is incorrect at small distances. The quantum mechanics with its strange axioms and properties appears to be a compensating construction, which is  necessary to achieve agreement of theory with experimental data (and it does an excellent job on that). But is valid only for nonrelativistic quantum mechanics. Construction of relativistic quantum theory (quantum field theory) needs additional suppositions which sometimes are incompatible with principles of nonrelativistic quantum mechanics. In other words, the quantum theory has typical features of a Ptolemaic construction.

One could eliminate additional suppositions and solve  problems of  the quantum phenomena description, using a true space-time geometry.  But to choose the true space-time geometry, one needs to posses the list of all possible geometries. Then one could choose from this list the most appropriate geometry.  The list of possible geometries is determined by  the method of the geometry construction. There are several methods (geometric conceptions) of constructing the proper Euclidean geometry. They are presented in the table.

Conceptions of geometry (CG)

title of conception of geometry
non-numerical (verbal) information
numerical information
Euclidean CG
Euclidean axioms
empty set
Riemannian CG
continuity, manifold, coordinate system
dimension n, metric tensor g_ik
topological-metric CG
topology, curve 
nonnegative metric S(x,x') and constraints on metric
pure metric CG
empty set
real world function G(x,x')=S(x,x')S(x,x')/2

Any geometric conception contains non-numerical information and numerical one. At some values of numerical information all geometric conceptions generate (CG) the proper Euclidean geometry. Varying the numerical information, one obtains another geometries, associated with the given CG. The more numerical information is contained in the geometric conception, the more powerful class of geometries, generated by this CG. The most powerful class of geometries is generated by the pure metric CG, because it contains only numerical  information. The Euclidean CG does not generate other geometries, because it does not contain numerical information. The Riemannian CG, used usually for construction of the space-time geometry, hold intermediate place between the Euclidean CG and pure metric CG.

The Riemannian CG generates only one uniform isotropic flat geometry. This is the Mincowski geometry, which can be described completely by the world function G_M. The pure metric CG generates a class L of uniform isotropic flat geometries, labelled by a function  D(G_M). The Mincowski geometry belongs to this class (if  D(G_M)=0). The list of geometries, generated by the pure metric CG is the most complete one. It is this list L that is to be used for choosing an appropriate space-time geometry.

Motion of a free particle in any space-time geometry except for Minkowski geometry is stochastic. Real microparticles are stochastic, and it is not reasonable to choose the Minkowski geometry as a space-time geometry, where free motion of particles is deterministic. But what what geometry of the class L should be chosen. There is a lot of them. The answer is as follows. One should choose such a space-time geometry, so that  the statistical description of the stochastic particle motion coincides with experimental data. As far as all experimental data on nonrelativistic motion of microparticles are described by quantum mechanics very well, it is sufficient to choose the space-time geometry in such a way that the statistical description of the stochastic particle motion would coincide with the quantum mechanical description. This constraint determines the distortion function D(G_M) in the form D(G_M)=d=h/(2bc) for G_M>G_0, where h is the quantum constant, c is the speed of the light, and b is a new universal constant. Then the world function


depends on the quantum constant, and quantum constant becomes to be an attribute of the space-time geometry. Using this space-time geometry, one does not need additional (Ptolemaic) suppositions, know as principles of quantum mechanics.

Quantum principles and quantum axiomatics are considered in this work as secondary physical laws generated by a statistical description of the stochastic behavior of particles. In this description the stochastic world lines of particles are conditioned by the space-time properties. This approach reminds the interplay between the thermodynamics and the statistical physics, where thermodynamical principles appear to be generated by the statistical description of the random molecular motion. Such an approach to quantum mechanics is substantiated in the presented papers.

For more formal statement of the problem click here