Physics geometrization in microcosm: discrete spacetime and relativity theory
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Abstract
The presented paper is a review of papers on the microcosm physics geometrization in the last twenty years. These papers develop a new direction of the microcosm physics. It is socalled geometric paradigm, which is alternative to the quantum paradigm, which is conventionally used now. The hypothesis on discreteness of the spacetime geometry appears to be more fundamental, than the hypothesis on quantum nature of microcosm. Discrete spacetime geometry admits one to describe quantum effects as pure geometric effects. Mathematical technique of the microcosm physics geometrization (geometric paradigm) is based on the physical geometry, which is described completely by the world function. Equations, describing motion of particles in the microcosm, are algebraic (not differential) equations. They are written in a coordinateless form in terms of world function. The geometric paradigm appeared as a result of overcoming of inconsistency of the conventional elementary particle theory. In the suggested skeleton conception the state of an elementary particle is described by its skeleton (several spacetime points). The skeleton contains all information on the particle properties (mass, charge, spin, etc.). The skeleton conception is a monistic construction, where elementary particle motion is described in terms of skeleton and world function and only in these terms. The skeleton conception can be constructed only on the basis of the physical geometry. Unfortunately, most mathematicians do not accept the physical geometries, because these geometries are nonaxiomatizable. It is a repetition of the case, when mathematicians did not accept the nonEuclidean geometries of LobachevskyBolyai. As a result this review is a review of papers of one author. This situation has some positive sides, because it appears to be possible a consideration not only of papers, but also of motive for writing some papers.
1 Introduction
The conventional paradigm of the microcosm physics development may be classified as quantum paradigm. The quantum paradigm is based on hypothesis of continuous spacetime geometry equipped by quantum principles of particle motion. There is an alternative geometric paradigm based on hypothesis on a discrete spacetime. There is no necessity to use quantum principles in the geometric paradigm, because all quantum effects can be explained by existence of elementary length of the discrete geometry. The elementary length appears to be proportional to the quantum constant .
The hypothesis on discreteness of the spacetime geometry looks more reasonable and natural, than the hypothesis on mysterious quantum nature of microcosm. One of reasons, why the geometric paradigm is not used in the contemporary physics is the circumstance, that the discrete geometry has not been developed properly. One believed that the discrete geometry is a geometry on a lattice. Any lattice point set cannot be uniform and isotropic, and such a point set is not adequate for the spacetime.
In reality, the discrete spacetime geometry can be defined on the same point set, where the spacetime geometry of Minkowski is given. In other words, a discrete geometry may be uniform and isotropic. This unexpected circumstance admits one to use a discrete geometry as a spacetime geometry. The discrete geometry is such a geometry, where there are no close points. Mathematically it means
(1.1) 
Here is the point set, where the geometry is given, and is a distance between the points , . The quantity is the elementary length of the discrete geometry . The geometry on a lattice can satisfy the property (1.1), but such a geometry cannot be uniform and isotropic. The discrete spacetime geometry has a set of new unexpected properties, which were unknown in the twentieth century. This fact was one of reasons, why the physics geometrization in microcosm has not been developed in the twentieth century.
This paper is a short review of the physics geometrization development in the last two decades. The physics geometrization began in the end of the nineteenth century. Different stages of the physics geometrization are: (1) connection of the conservation laws with the properties of the spacetime geometry (uniformity and isotropy), (2) the special relativity theory,(3) the general relativity theory, (4) the KaluzaKlein spacetime geometry. Most physicists do not believe in the physics geometrization in microcosm. They believe in the quantum nature of physical phenomena in microcosm, and they do not know properties of a discrete geometry, which admits one to explain quantum phenomena as geometrical effects. It is a reason, why practically nobody deal with the physics geometrization now. By necessity this review of papers on the physics geometrization in microcosm is a review of papers of one author.
It should note that we distinguish between a conception and a theory. A conception does not coincide with a theory. For instance, the skeleton conception of elementary particles distinguishes from a theory of elementary particles. A conception investigates connections between concepts of a theory. For instance, the skeleton conception of elementary particles investigates the structure of a possible theory of elementary particles. It investigates, why an elementary particle is described by its skeleton (several spacetime points), which contains all information on the elementary particle. The skeleton conceptions explains, why dynamic equations are coordinateless algebraic equations and why the dynamic equations a written in terms of the world functions. However, the skeleton conception does not answer the question, which skeleton corresponds to a concrete elementary particle and what is the world function of the real spacetime. In other words, the skeleton conception deals with physical principles, but not with concrete elementary particles. The conception cannot be experimentally tested. However, if the world function of the real spacetime geometry has been determined and correspondence between a concrete elementary particle and its skeleton has been established, the skeleton conception turns to the elementary particle theory. The theory of elementary particle (but not a conception) can be tested experimentally.
In other words, it is useless to speak on experimental test of the skeleton conception, because it deals only with physical principles. Discussing properties of a conception, one should discuss only properties of the concept and logical connection between them, but not to what extent they agree with experimental data.
We consider in the review the following problems

Conceptual defects of the quantum paradigm, which manifest themselves, in particular, in incorrect use of the relativity principles at a description of indeterministic particles.

Explanation of quantum effects as a statistical description of the indeterministic particle motion.

Discrete geometry as a special case of a physical geometry and properties of physical geometries.

Elementary particle dynamics in physical spacetime geometry and skeleton conception of particle dynamics.
Idea of the physics geometrization is based on the following circumstance. Description of the particle motion contains two essential elements: the spacetime geometry and the dynamic laws. The two categories are connected. One can investigate the two categories only together, and the boundary between the laws of geometry and the laws of dynamics is not fixed rigidly. One can shift this boundary. For instance, one can choose a very simple spacetime geometry, then the laws of dynamics appear to be rather complicated. One may try to use a complicated spacetime geometry, which is chosen in such a way, that the dynamic laws be very simple. For instance, maybe, there exists such a spacetime geometry, where the elementary particles move freely. Interaction between particles is realized via the spacetime geometry. The KaluzaKlein geometry is an example of such a spacetime geometry, where the electromagnetic field is a property of the spacetime geometry. If one uses the spacetime geometry of Minkowski (instead of the KaluzaKlein geometry), the electromagnetic interaction of particles is explained as a result of interaction with the electromagnetic field,
The spacetime of Minkowski is uniform and isotropic, and one can easily write the conservation laws of energymomentum and of angular momentum in the Minkowski spacetime. One cannot write the conservation laws in the KaluzaKlein spacetime with electromagnetic field, because this spacetime is not uniform and isotropic, in general. Such a difference is conditioned by the circumstance, that in the spacetime of Minkowski the electromagnetic field is a substantive essence, whereas in the KaluzaKlein spacetime the electromagnetic field is only a property of the spacetime geometry.
What point of view is true? We believe, that one should use both approaches. In the geometrical approach the number of essences is less (in the limit of a complete geometrization there is only one essence), and it is easier to establish physical (and geometrical) principles responsible for description of different sides of a physical phenomenon. On the other hand, when the physical principles and connection between different sides of a physical phenomenon have been established, one may consider the different sides of a physical phenomenon as different essences. Such an approach admits one to describe concrete physical phenomena easier and more convenient, considering them as a result of interaction of different essences.
Developing the physics geometrization, one tries to work with physical principles, assuming that the good old classical principles are true. We stand aback from introducing new physical principles on the basis of consideration of single physical phenomena. We believe that classical physical principles are valid, although they are applied sometimes incorrectly. We have succeeded to discover several mistakes in application of classical principles of physics. Some of mistakes were connected with our imperfect knowledge of geometry and, in particular, with imperfect knowledge of a discrete geometry.
At the complete geometrization of physics the spacetime geometry is chosen in such a way, that all particles move freely. The force fields and their interaction with particles appear only in the case, when the spacetime geometry is chosen incorrectly. In this case, when the chosen spacetime geometry differs from the true geometry, the deviation of geometries generates appearance of force fields. The complete geometrization of physics is known for classical (gravitational an electromagnetic) interactions . However, it is not yet known in microcosm. The reason of this circumstance lies mainly in the fact, that our knowledge of geometry is imperfect. The complete geometrization of physics is possible only at a perfect description of the spacetime geometry.
A geometry as a science on disposition of geometrical objects in space or in the event space (spacetime) is described completely by the distance between any two points and , or by the world function . The geometry, which is described completely by the world function will be referred to as a physical geometry. After complete physics geometrization the particle dynamics turns to a monistic conception, which is described completely in terms of one quantity (world function). Any conception, which contains several basic concepts (quantities), needs an agreement between all concepts, used in the conception. Achievement of such an agreement is a very difficult problem. One can see this in the example of a geometry. The physical geometry is a monistic conception, because it is described by means of only world function. One uses a few concepts (manifold, coordinate system, metric tensor) in the conventional description of Riemannian geometries, and a Riemannian geometry appears to be a less general conception, than a physical geometry.
Albert Einstein dreamed on creation of a united field theory. Such a theory was to be a monistic conception, and this circumstance was the most attractive feature of such a theory. However, a monistic theory on the basis of a geometry seems to be more attractive, than a monistic theory based on a united field, because the main object of a physical geometry (world function) is a simpler object, than a force field of the united field theory.
Problems of the physical geometrization appeared, when physicists began to investigate physical phenomena in microcosm. We cannot know exactly the microcosm spacetime geometry. It is rather natural, that the spacetime geometry in microcosm may appear to be discrete. Contemporary researchers consider a discrete geometry as a geometry on a lattice point set. In particular, there is a special section in the ArXiv publications, entitled High Energy Physics  Lattice. A lattice point set cannot be uniform and isotropic. In accordance with this circumstance the discrete spacetime geometry (geometry on a lattice) is considered to be not uniform and isotropic.
In reality a discrete spacetime geometry is not a geometry on a lattice. The discrete spacetime geometry may be given on a continual point set. In particular, it can be given on the same manifold, where the geometry of Minkowski is given. It is connected with the fact, that the geometry discreteness is a property of the geometry, but not a property of the point set, where the geometry is given. A discrete geometry satisfies the condition (1.1).
Geometry on a lattice satisfies the condition (1.1), but such a geometry is a special kind of a discrete geometry, which cannot be uniform and isotropic.
Let be the world function of the geometry Minkowski
(1.2) 
where is the distance (interval) between the points with inertial coordinates and . The world function
(1.3) 
describes a discrete geometry , which satisfies the restriction (1.1), although the geometry is given on the same point set , where the geometry of Minkowski is given. The geometry appears to be uniform and isotropic.
However, one cannot use coordinates for description of the geometry . It does not that one cannot introduce coordinates. Substituting (1.2) in (1.3), one obtains representation of the world function in terms of coordinates. However, the points, which have close coordinates, are not close in the sense that the distance between them is greater, than
(1.4) 
It means that coordinate lines and differentiation along them have no relation to the discrete geometry , given on the manifold of Minkowski. It does not mean, that the discrete geometry does not exist. It means only, that capacities of the coordinate description method are restricted, and one needs to use the coordinateless method of description, which are used at description of physical geometries [1, 2, 3]
Besides, the discrete geometry appears to be multivariant and nonaxiomatizable [4]. Such properties of a geometry can be obtained only at a use of the coordinateless description method. In the discrete spacetime a particle cannot be described by a world line, because any world line is a set of connected infinitesimal segments of a straight line. However, in the discrete geometry there are no segments, whose length is shorter, than the elementary length . It means, that instead of world line one has a world chain
(1.5) 
consisting of geometrical vectors , of finite length . The geometrical vector (vector) is an ordered set of two points and . The first point is the origin of the vector, whereas the second point is the end of the vector. Such a definition of the vector is used in physics. However, mathematicians prefer another definition. They define a vector as an element of a linear vector space.
Remark. We used the special term ”geometrical vector”, because conventionally the term ” vector” means some manycomponent quantity (components of the vector in some coordinate system). In general, a vector is defined in the contemporary geometry as an element of the linear vector space. In this case the vector can be decomposed over basic vectors of a coordinate system and represented as a set of the vector coordinates. Such a definition is convenient, when one speaks about vector field, having several components. In the proper Euclidean geometry the concept of a geometrical vector coincides with the conventional concept of a vector as an element of the linear vector space. In the Euclidean geometry the vector can be decomposed over basic vectors. It can be represented as a set of coordinates. However, in the discrete geometry, described by the world function (1.3), a geometric vector cannot be represented as a sum of its projections onto basic vectors, because in the discrete geometry (1.3) one cannot introduce a linear vector space even locally. However, the definition of a vector as a set of two points does not contain a reference to a coordinate system and to special properties of the Euclidean geometry (such as linear vector space). The definition of the geometrical vector is more general, and according to the logic rules the term ”vector” should be used with respect to the geometric vector. Another term, for instance, ”linear vector” should be used for the vector, defined as an element of the linear vector space.
The discrete geometry is obtained from the geometry of Minkowski by means of a deformation of the geometry of Minkowski, when the world function is replaced by the world function [5]. World chains in the discrete spacetime geometry appear to be stochastic. Let the elementary length have the form
(1.6) 
where is the quantum constant is the speed of the light, and is the universal constant, connecting the geometric mass (length of the chain link) with the particle mass by means of
(1.7) 
Then statistical description of the stochastic world chains leads to the Schrödinger equation [6]. As a result the quantum effects can be described as geometrical effects of the discrete spacetime geometry. Quantum principles cease to be prime physical principles. They become secondary principles, which should not be applied always and everywhere. In particular, there is no necessity of the gravitational field quantization.
In the discrete spacetime geometry the relativity theory appears to be incomplete. The fact is that, the transition from the nonrelativistic physics to the relativistic one is followed only by a modification of dynamic equations, describing the particle motion. Description of the particle state remains the same as in the nonrelativistic physics. The particle state is described as a point in the phase space of coordinates and momenta. The particle momentum is defined as a tangent vector to the particle world line , .
(1.8) 
where is a parameter along the world line. In the discrete spacetime geometry there are no world lines, and the limit (1.8) does not exist. This limit does not exist also in the case, when the particle is indeterministic and its world line (if it exists) is random (stochastic). In the physics of usual scale, when characteristic lengths much more, than the elementary length , restricting the link length of the world chain. In this case it is admissible to use the limit (1.8) as a good approximation. However, in the microcosm physics such an approximation appears to be unsatisfactory, because characteristic lengths of physical phenomena appear of the order of the elementary length . As a result the concepts of the elementary particle theory, based on the particle state as point of the phase space appear to be incomplete.
Consequent relativistic description of particles in microcosm does not use the phase space and its points. Instead, one uses a skeleton conception of the elementary particle description, where a particle is described by its skeleton , which consists of rigidly connected points . In the case of pointlike particle its skeleton consists of two points , which define the particle momentum vector. In the given case all characteristics of the particle (mass, charge, momentum) are defined geometrically by the two points . In the case of a more complicated particle, described by the skeleton , there are invariants , , describing geometrically all characteristics of the particle. The question about nature of connection between the points of the skeleton does not arise, because the discrete spacetime geometry may have a restricted divisibility. Such a question is conditioned by the hypothesis on continuous spacetime geometry.
In the beginning of the twentieth century it was natural to think, that the quantum particles are simply indeterministic (stochastic) particles, something like Brownian particles. There were attempts to obtain quantum mechanics as a statistical description of stochastically moving particles [7, 8]. However, these attempts failed, because a probabilistic conception of the statistical description was used.
Statistical description is used in physics for description of indeterministic particles (or systems), when there are no dynamic equations, or initial conditions are indefinite. One considers statistical ensemble of indeterministic particles, i.e. many independent similar particles. It appears, that there are dynamic equations for the statistical ensemble of indeterministic particles, although there are no dynamic equations for a single indeterministic particle, which is a constituent of this statistical ensemble . Consideration of the statistical ensemble as a dynamic system is the dynamic conception of the statistical description (DCSD). It is a primordial conception of statistical description. A use of DCSD is founded on independence of constituents of the statistical ensemble. Random components of motion are compensated due to their independence, whereas regular components of motion are accumulated. As a result the statistical ensemble, considered as a dynamic system, describes a mean motion of an indeterministic particle.
In the nonrelativistic physics one uses the probabilistic conception of the statistical description (PCSD). PCSD is used successfully, for instance, for description of Brownian motion. In PCSD one traces the motion of a point in the phase space. The point represents the state of indeterministic particle, and motion of the point in the phase space is described by the transition probability. Attempts of obtaining the quantum mechanics as a result of statistical description in the framework PCSD failed [7, 8], whereas the statistical description in the framework of DCSD appeared to be successful [9, 10, 11]. This fact is explained by a use of the dynamic conception of statistical description (DCSD), which does not use a concept of the phase space.
In the relativistic case the ensemble state is described by a 4vecotor , which described the density of world lines in the vicinity of the point . The ensemble state does not contain a reference to the phase space. In the nonrelativistic case the ensemble state is described by a 3scalar , which describes the particle density in vicinity of the point of the phase space. PCSD is based on a use of the nonnegative quantity , which is used as a probability density of the particle position at the point of the phase space.
Nonrelativistic quantum mechanics is a relativistic construction in reality, because the stochastic component of the quantum particle motion may be relativistic. At such a situation and one has to use the dynamic conception of statistical description (DCSD), which does not use the nonrelativistic concept of the phase space. Besides, one may not use the limit (1.8) in the definition for the particle momentum of stochastic world lines, which can have no tangent vectors.
Indeed, in terms of DCSD one succeeded to obtain the quantum mechanics as a statistical description of stochastically moving particles [9, 10, 11]. This use of dynamic conception of statistical description was not a stage of the physics geometrization. DCSD was simply an overcoming of the incompleteness of the relativity theory, when relativistic dynamic equations are combined with nonrelativistic concept of the particle state. However, the explanation of quantum mechanics effects as a result of statistical description of stochastic particle motion arose the question on the nature of stochasticity of such a stochastic particle motion.
Primarily the particle motion stochasticity was interpreted as a result of interaction with an ether. However, further the idea has been appeared, that the spacetime geometry in itself may play the role of the ether. In other words, the spacetime geometry is to determine the free particle motion. If the free particle motion is stochastic, the spacetime geometry cannot be geometry of Minkowski, because in the spacetime geometry of Minkowski a free particle motion is deterministic. The real spacetime geometry is to be uniform and isotropic, but it is to distinguish from the geometry of Minkowski. It is to be multivariant. It means that at the point there are many vectors , , ,…, which are equivalent to the vector at the point . But vectors , ,,…are not equivalent between themselves. It means that the equivalence relation is intransitive. Such a geometry cannot be axiomatizable, because in any axiomatizable geometry the equivalence relation is to be transitive. Nonaxiomatizable geometries were not known in seventieth of the twentieth century. The discrete geometry (1.3) was not known also, because in that time the discrete geometry was perceived as a geometry on a lattice point set.
Idea of the physical geometry as a geometry described completely by the world function appeared only in ninetieth of the twentieth century [12]. The close idea of the distance (metrical) geometry appeared earlier [13, 14]. But such a geometry cannot be used as a spacetime geometry.
One uses the discrete geometry (1.3) to explain the stochasticity of free particle motion [6]. However, this geometry was used as a simplest multivariant generalization of the geometry of Minkowski, but not as a discrete spacetime geometry. The fact, that the spacetime geometry (1.3) is discrete, has been remarked several years later. It is rather natural, that starting from idea of a discrete spacetime geometry, one comes to a geometry on a lattice, because one cannot obtain the geometry (1.3), if concepts of physical geometry are unknown.
Application of a physical geometry for description of the spacetime has serious consequences for microcosm physics. It appears, that quantum principles are not primary principles of nature. The relativity theory appeared to be not completed. One needs to revise concept of the particle state. The mathematical technique of description of the microcosm physical phenomena changed essentially. Dynamic equations become finite difference equations instead of differential equations. Description of particle motion and that of gravitational field becomes coordinateless, and it was a progress in the particle motion description.
Transition from the conventional description in terms of differential equations to coordinateless description in terms of the world function appears rather unexpected. It is connected with degenerative character of the proper Euclidean geometry with respect to physical geometry. It means that some geometrical concepts and some geometrical objects, which are different in a physical geometry appear coinciding in the Euclidean geometry. For instance, the geometrical vector defined as the ordered set of two points and is a vector in a physical geometry and in the Euclidean geometry. Projections of vector on basic coordinate vectors , are defined by the relation
(1.9) 
Here is the scalar product of two vectors and , which is defined in terms of the world function by the relation
(1.10) 
The expression of the scalar product (1.10) via the world function is the same in a physical geometry and in the proper Euclidean one. In the physical geometry the relation (1.10) is a definition of the scalar product, whereas in the Euclidean geometry the relation (1.10) is obtained as a corollary of the cosine theorem, but in both cases the expression (1.10) is true. The scalar product has conventional linear properties in the Euclidean geometry, but these properties are absent, in general, in the physical geometry. As a result components of the geometrical vector do not determine the vector in the physical geometry, although they determine the vector in the proper Euclidean geometry. It means, that the vector and its components , mean the same quantity in the Euclidean geometry, whereas they are, in general, different quantities in a physical geometry.
In a like way the expression for a circular cylinder , determined by points on the cylinder axis and by the point on cylinder surface, is a set of points , satisfying the relation
(1.11) 
where is the area of the triangle, determined by vertices . The area is calculated by means of the Heron formula via distances between the points . Let the point , where is a segment of a straight line between the points . This segment is defined as a set of points by the relation
(1.12) 
Then in the proper Euclidean geometry . However, in a physical geometry, in general, . In other words, many different cylinders of a physical geometry degenerate in the proper Euclidean geometry into one cylinder, defined by its axis and by the point on the surface of the cylinder. This fact takes place, because the segment of the straight line (1.12) is onedimensional in the case of the proper Euclidean geometry, but it is, in general, a manydimensional surface in the case of a physical geometry.
Onedimensionality of in the Euclidean geometry is formulated in terms of the world function as follows. Any section of the segment at the point consists of one point . Section is defined as a set of points
(1.13) 
In the proper Euclidean geometry , whereas in the case of a physical geometry this equality does not take place, in general.
Thus, the physical geometry degenerates, in general, at a transition from a physical geometry to the proper Euclidean geometry. Different geometrical objects and concepts may coincide. On the contrary at transition from the proper Euclidean geometry to a physical geometry some geometrical objects split into different geometrical objects. Transition from a general case to a special one, followed by a degeneration, is perceived easily, whereas a transition from a special case to a general one, followed by a splitting of geometrical objects and of geometrical concepts, is perceived hard.
2 Relativistic invariance
The relativistic invariance is presented usually as an invariance of dynamic equation with respect to the Poincare group of inertial coordinate transformations. Nonrelativistic dynamic equations are considered to be invariant with respect to Galilean group of inertial coordinates transformation. Is it possible to formulate difference between relativistic physics and nonrelativistic one in invariant terms, i.e. without a reference to coordinate system and the laws of their transformation? Yes, it is possible.
In the relativistic physics the spacetime geometry is described by means of one structure , which is known as the squared spacetime interval, or the world function. In the nonrelativistic physics the event space (spacetime) is described by two invariant geometrical structures. Such a twostructure description is not a spacetime geometry, because the spacetime geometry is described by one structure . If there exist another spacetime structure, such a construction should be referred to as a fortified geometry, i.e. a geometry with additional geometric structure. This additional structure is the time structure which is a difference of absolute times between the points and . One can construct another geometrical structure , which is a difference between of absolute spatial positions of points and . The structure is not an independent structure. The spatial structure can be constructed of two structures and . In any case in the nonrelativistic physics there are two independent geometrical structures. In the relativistic physics there is only one structure .
Usually one uses the time structure and the spatial structure in the nonrelativistic physics. However, one may use geometrical structures and . In this case one can investigate additional restrictions, imposed by time structure on the spacetime geometry of Minkowski. Geometrical structures of the spacetime determine a motion group of the spacetime, and this motion group determines group of invariance of dynamic equations. Thus, the difference between the relativistic physics and nonrelativistic one is determined by the number of geometrical structures. This difference may be formulated in coordinateless form. The transformation laws of dynamic equations are only corollaries of these geometrical structures existence.
3 Statistical description of the stochastic particle motion
Statistical description of stochastic (indeterministic) particles was an origin of the physical geometry, because, it put the question on a nature of this indeterminism, which can be explained only by a more general uniform spacetime geometry, than the geometry of Minkowski.
As we have mentioned in the introduction, a statistical description of indeterministic particles was made at first by means of the dynamical conception of statistical description (DCSD). This approach is founded on a use of relativistic concept of particle state [9, 10, 11]. Another method of the stochastic particles description has been used later, when a statistical ensemble (instead of a single particle) has been considered as a basic element of the particle dynamics [15]. The concept of a single particle and the concept of the phase space are not used in this method. This method goes around the nonrelativistic concept of the particle state. It does not use the concept of the particle state. It uses only concept of the ensemble state, which is insensitive to the problem of the limit (1.8) existence. From formal viewpoint this method uses DCSD, but not PCSD.
The action for the statistical ensemble of free indeterministic particles is written in the form
(3.1) 
Independent variables label constituents of the statistical ensemble. The dependent variable describes the regular component of the particle motion. The variable describes the mean value of the stochastic velocity component, is the quantum constant. The second term in (3.1) describes the kinetic energy of the stochastic velocity component. The third term describes interaction between the stochastic component and the regular component . The operator
(3.2) 
is defined in the space of coordinates . Dynamic equations for the dynamic system are obtained as a result of variation of the action (3.1) with respect to dynamic variables and .
The action for a single indeterministic particle has the form
(3.3) 
This action is not correctly defined, because operator is defined on 3Dspace of coordinates , whereas in the action functional (3.3) the variable is used only on onedimensional set. It means that there are no dynamic equations for the particle , and the particle is a stochastic (indeterministic) system. However, the action functional (3.1) is well defined, and dynamic equations exist for the statistical ensemble , although dynamic equations do not exist for constituents of this statistical ensemble.
After proper change of variables the dynamic equations are reduced to the equation [15]
(3.7) 
where is the twocomponent complex wave function
(3.8) 
are Pauli matrices
(3.9) 
If components and are linear dependent , , then . Two last terms of the equation (3.7) vanish, and the equation turns to the Schrödinger equation
(3.10) 
Thus, the Schrödinger equation and interpretation of the quantum mechanics appear from the dynamical system , described by the action functional (3.1). This fact seems rather unexpected, because in quantum mechanics the wave function is considered as a specific quantum object, which has no analog in classical physics. In reality, the wave function is simply a way of description of ideal continuous medium [16]. One may describe an ideal fluid in terms of hydrodynamic variables: density and velocity . One may describe an ideal fluid in terms of the wave function. The statistical ensemble is a dynamic system of the type of continuous medium. The two representations of dynamic equations for the dynamic system can be transformed one into another.
Generalization of the action (3.3) on the stochastic relativistic charged particle, moving in an electromagnetic field, has the form [17]
(3.11)  
(3.12) 
where are dependent variables. are independent variables, and The quantities are dependent variables, describing stochastic component of the particle motion, is the potential of electromagnetic field. The dynamic system, described by the action (3.11), (3.12) is a statistical ensemble of indeterministic particles, which looks as some continuous medium. The variables are connected with the stochastic component of the particle 4velocity by the relation
(3.13) 
In the nonrelativistic approximation one may neglect the temporal component with respect to the spatial one Setting and in (3.11), (3.12), we obtain the action (3.1) instead of (3.11), (3.12).
After a proper change of variables one obtains dynamic equation for the action (3.11), (3.12). This dynamic equation has the form [17]
(3.14)  
where designations (3.8), (3.9) are used. In the case, when the wave function is onecomponent, vector const, and the dynamic equation (3.14) turns to the KleinGordon equation
(3.15) 
Transformation of hydrodynamic equations (3.4) into dynamic equations in terms of the wave function is based on the fact, that a wave function is a method of description of hydrodynamic equations [16]. Transformation of hydrodynamic equations, described in terms of hydrodynamic variables (density and velocity ), to a description in terms the wave function rather is bulky, because it uses a partial integration of dynamic equations. These integration leads to appearance of arbitrary integration functions . The wave function is constructed of these integration functions [16].
One can explain the situation as follows. It is well known, that the Schrödinger equation can be written in the hydrodynamic form of MadelungBohm [19, 20]. The wave function is presented in the form
(3.16) 
Substituting (3.16) in the Schrödinger equation (3.10), one obtains two real equations for dynamical variables and . Taking gradient from the equation for and introducing designation
(3.17) 
one obtains four equations of the hydrodynamic type
(3.18) 
where is the Bohm potential, defined by the relation
(3.19) 
Hydrodynamic equations (3.18) can be easily obtained from equations (3.4), (3.5). To obtain representation of equations (3.18), (3.19) in terms of wave function, one needs to integrate these equations, because they have been obtained by means of differentiation of the Schrödinger equation. This integration can be easily produced, if the condition (3.17) takes place and the fluid flow is nonrotational.
In the general case of vortical flow the integration is more complicated. Nevertheless this integration has been produced [16], and one obtains a more complicated equation (3.7), where two last terms describe vorticity of the flow. The Schrödinger equation (3.10) is a special case of the more general equation (3.7).
Note that the equation (3.7) is not linear, although it is invariant with respect to transformation
(3.20) 
which admits one to normalize the wave function to any nonnegative quantity. This property describes independence of the statistical ensemble on the number of its constituents.
Representation of quantum mechanics as a statistical description of classical indeterministic particles admits one to interpret all quantum relations in terms of statistical description. This interpretation distinguishes in some clauses from conventional (Copenhagen) interpretation of quantum mechanics.
In any statistical description there are two different kinds of measurement, which have different properties. Massive measurement (Mmeasurement) is produced over all constituents of the statistical ensemble. A result of Mmeasurement of the quantity is a distribution of the quantity , which can be predicted as a result of solution of dynamic equations for the statistical ensemble.
Single measurement (Smeasurement) is produced over one of constituents of the statistical ensemble. A result of Smeasurement of the quantity is some random value of the quantity , which cannot be predicted by the theory. In the Copenhagen interpretation of the quantum mechanics the wave function is supposed to describe a single particle (but not a statistical ensemble of particles). As a result there is only one type of measurement, which is considered sometimes as a Mmeasurement and sometimes as a Smeasurement. As far as Mmeasurement and Smeasurement have different properties, such an identification is a source of numerous contradictions and paradoxes [21].
Representation of quantum mechanics as a statistical description of the indeterministic particles motion has two important consequences: (1) elimination of quantum principles as laws of nature, (2) problem of primordial stochastic motion of free particles.
4 Deformation principle
The idea, that a geometry is described completely by means of a distance function (or world function) is very old. At first it was a metric space, described by metric (distance). The metric has been restricted by a set of conditions such as the triangle axiom and nonnegativity of the metric. Condition of nonnegativity of metric does not permit to apply the metric space for description of the spacetime. The main defect of the metric geometry and the distance geometry [13, 14] is impossibility of construction of geometrical objects in terms of the world function or in terms of the metric. Construction of geometrical objects in terms of the world function is to be possible, because it is supposed that the geometry is described completely by the world function and in terms of the world function. Furthermore, a physical geometry is to admit a coordinateless description.
Such a situation is possible, if one defines concepts of a geometry and those of a geometrical objects correctly.
Definition 4.1: The physical geometry is a point set with the singlevalued function on it
(4.1) 
Definition 4.2: Two physical geometries and are equivalent if the point set , or .
Remark: Coincidence of point sets and is not necessary for equivalence of geometries and . If one demands coincidence of and in the case equivalence of and , then an elimination of one point from the point set turns the geometry into geometry , which appears to be not equivalent to the geometry . Such a situation seems to be inadmissible, because a geometry on a part of the point set appears to be not equivalent to the geometry on the whole point set .
According to definition the geometries and on parts of , and are equivalent to the geometry , whereas the geometries and are not equivalent, in general, if and . Thus, the relation of equivalence is intransitive, in general. The spacetime geometry may vary in different regions of the spacetime. It means, that a physical body, described as a geometrical object, may evolve in such a way, that it appears in regions with different spacetime geometry.
Definition 4.3: A geometrical object of the geometry is a subset of the point set . This geometrical object is a set of roots of the function
where
(4.2)  
(4.3)  
(4.4) 
Here are points which are parameters, determining the geometrical object
(4.5) 
is an arbitrary function of arguments and of parameters . The set of the geometric object parameters will be referred to as the skeleton of the geometrical object. The subset will be referred to as the envelope of the skeleton. One skeleton may have many envelopes. When a particle is considered as a geometrical object, its motion in the spacetime is described mainly by the skeleton . The shape of the envelope is of no importance in the first approximation.
Remark: Arbitrary subset of the point set is not a geometrical object, in general. It is supposed, that physical bodies may have a shape of a geometrical object only, because only in this case one can identify identical physical bodies (geometrical objects) in different spacetime geometries.
Existence of the same geometrical objects in different spacetime regions, having different geometries, arises the question on equivalence of geometrical objects in different spacetime geometries. Such a question was not arisen before, because one does not consider such a situation, when the physical body moves from one spacetime region to another spacetime region, having another spacetime geometry. In general, mathematical technique of the conventional spacetime geometry is not applicable for simultaneous consideration of several different geometries of different spacetime regions.
We can perceive the spacetime geometry only via motion of physical bodies in the spacetime, or via construction of geometrical objects corresponding to these physical bodies. As it follows from the definition 4.3 of the geometrical object, the function as a function of its arguments (of world functions of different points) is the same in all physical geometries. It means, that a geometrical object in the geometry is obtained from the same geometrical object in the geometry by means of the replacement in the definition of this geometrical object.
As far as the physical geometry is determined by its geometrical objects construction, a physical geometry can be obtained from some known standard geometry by means a deformation of the standard geometry . Deformation of the standard geometry is realized by the replacement in all definitions of the geometrical objects in the standard geometry. The proper Euclidean geometry is an axiomatizable geometry. It has been constructed by means of the Euclidean method as a logical construction. The proper Euclidean geometry is a physical geometry. It may be used as a standard geometry . Construction of a physical geometry as a deformation of the proper Euclidean geometry will be referred to as the deformation principle. The most physical geometries are nonaxiomatizable geometries. They can be constructed only by means of the deformation principle.
Description of the elementary particle motion in the spacetime contains only the particle skeleton . The form of the function (4.2) is of no importance in the first approximation. In the elementary particle dynamics only equivalence of vectors , is essential. These vectors are defined by the particle skeleton .
The equivalence of two vectors and is defined by the relations
(4.6) 
where
(4.7) 
and the scalar product is defined by the relation (1.10)
(4.8) 
Skeletons and may belong to the same geometrical object, if
(4.9) 
i.e. lengths of all vectors and are equal. However, it is not sufficient for equivalence of skeletons and .
Skeletons and are equivalent
(4.10) 
In other words, the equality of skeletons needs equality of the lengths of vectors and and equality of their mutual orientations.
5 Multivariance
The physical geometry has the property, called multivariance. It means that at the point there are many vectors , , which are equivalent to the vector at the point , but they are not equivalent between themselves. The proper Euclidean geometry has not the property of multivariance. In the proper Euclidean geometry there is only one vector at the point , which is equivalent to the vector at the point .
Multivariance is connected formally with the definition of the vector equivalence via algebraic relations (4.6)  (4.8). If vector is given, and it is necessary to determine the equivalent vector at the point , one needs to solve two equations (4.6) with respect to the point . If the two equations have a unique solution, one has only one equivalent vector (singlevariance). If there are many solutions, one has many vectors , , , which are equivalent to vector (multivariance). It is possible such a case, when there are no solutions. In this case one has zerovariance.
Multivariance of the spacetime geometry leads to splitting of one world chain into many stochastic world chains. As a result the multivariance of the spacetime geometry in microcosm leads to appearance of quantum effects.
Zerovariance appears in the case of manypoint skeletons. It is interesting in that relation, that it may forbid existence of elementary particles with manypoint skeletons.
6 Discreteness of the spacetime geometry
The world function (1.3) describes a completely discrete geometry. However, the spacetime geometry may discrete only partly. In the discrete geometry one may introduce the point density with respect to point density in the geometry of Minkowski. The discrete geometry may be described by the relative points density
(6.1) 
For close points the relative point density of the discrete geometry vanishes, and this circumstance is considered as a discreteness of the geometry. However, the discreteness may not be complete.
Let us consider the spacetime geometry with the world function
(6.2) 
The relative point density in the geometry (6.2) has the form
(6.3) 
If the relative point density in the region, where is much less, than . If , the relative point density (6.3) tends to (6.1). The geometry (6.2) should be qualified as a partly discrete spacetime geometry. We shall refer to the geometry (6.2) as a granular geometry. In the granular spacetime geometry the relative density of points, separated by small distance (less, than ), is much less than the relative density of other points. The granular geometry, described by the world function
(6.4)  
is a generalization of the the geometry (6.2).
7 Elementary particle dynamics
Dynamics of elementary particles in the granular spacetime geometry is considered in [22]. The state of an elementary particle is described by its skeleton , consisting of spacetime points. Such description of the particle state is complete in the sense, that it does not need parameters of the particle (mass, charge, spin, etc.). All this information is described by the disposition of points in the skeleton. It means a geometrization of parameters of the elementary particles. Besides, the conventional description of the particle state as a point in the phase space is nonrelativistic. The granular geometry is multivariant, in general. The particle motion is stochastic, and the limit (1.8), which determines the particle momentum, does not exist. Thus, to satisfy the relativity principles, we are forced to describe the particle state by its skeleton.
Evolution of the particle state is described by the world chain , consisting of connected skeletons ,
(7.1) 
Connection between skeletons of the world chain arises, because the second point of the th skeleton coincides with the first point of the th skeleton. The vector will be referred to as the leading vector, determining the shape of the world chain. All skeletons of the chain are similar in the sense, that
(7.2) 
Definition: Two vectors and are equivalent , if
(7.3) 
If the particle is free, then the skeleton motion is progressive (i.e. motion without rotation), and orientation of adjacent skeletons , is the same.
(7.4)  
Equations (7.2), (7.4) means that the adjacent skeletons of the world chain are equivalent eqv, The adjacent skeletons are equivalent, if corresponding vectors of adjacent skeletons are equivalent
(7.5) 
One obtains difference dynamic equations (7.5) (or (7.2), (7.4)), which describe evolution of the particle state. Introducing a coordinate system, one obtains dynamic variables, whose values are to be determined by dynamic equations (7.5). Here is the dimension of the spacetime (the number of coordinates, describing the point position). In particular, in the case of a pointlike particle, whose state is described by two points , , the number of dynamic equations , whereas in the 4Dspacetime the number of variable . In the multivariant spacetime the dynamic equations have many solutions. As a result the world chain appears to be multivariant (stochastic).
In the Riemannian spacetime and in the spacetime of Minkowski the world chain can be approximated by a world line, provided characteristic lengths of the problem are much larger, than the lengths of the world chain links. In this case the dynamic equations (7.5) are reduced to ordinary differential equations. If the world line is timelike [22], the solution of dynamic equations appears to be unique. If the vectors are spacelike, dynamic equations have many solutions even in the Riemannian spacetime. It is connected with the circumstance, that the Riemannian geometry as well as the geometry of Minkowski is multivariant with respect to spacelike vectors. At the conventional approach the spacelike world lines are not considered at all. Such world lines are inadmissible by definition (It is a postulate).
One attempted to obtain differential dynamic equations for a pointlike particle in [23]. At first one obtained equation for free pointlike particle in the spacetime of Minkowski. It is only one equation, whereas in the conventional approach one has three equations for the velocity components This equation has the form
(7.6) 
Let us introduce designation
(7.7) 
where is the angle between vectors and . The equation (7.6) takes the form
(7.8) 
If the world line is timelike , and cos, then the bracket in (7.8) is positive and one concludes from (7.8), that
(7.9) 
One obtains three equations from one equation (7.9)
(7.10) 
If the world line is spacelike, then , and the bracket in (7.9) vanishes at
(7.11) 
The acceleration becomes indefinite at this value of the angle between and . It should be interpreted as impossibility of spacelikeworld lines. At the conventional approach such an impossibility of spacelike world lines is simply postulated.
Such a result is rather evident, because the spacetime of Minkowski is singlevariant with respect to timelike vectors and it is multivariant with respect to spacelike vectors. For timelike vectors one can obtain three dynamic equations (7.10) from one equation (7.8). For spacelike particles it is impossible.
Another example is considered in the paper [23]. Motion of pointlike particle in the gravitational field of a massive sphere of the mass is considered. In the Newtonian approximation the world function between the points with coordinates and has the form
(7.12) 
where is the gravitational constant, and