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Model conception of quantum phenomena: logical structure and investigation methods

One can construct the model conception of quantum phenomena (MCQP) which relates to the axiomatic conception of quantum phenomena (ACQP), (i.e. to the conventional quantum mechanics) in the same way, as the statistical physics relates to thermodynamics. Such a possibility is based on a new conception of geometry, which admits one to construct such a deterministic space-time geometry, where motion of free particles is primordially stochastic. The space-time geometry can be chosen in such a way that statistical description of random particle motion coincides with the quantum description. Transition from ACQP to MCQP is a result of correction of some mistakes in the foundation of ACQP. Methods of MCQP in investigation of quantum phenomena appear to be more subtle and effective than, that of ACQP. For instance, investigation of the free Dirac equation in framework of MCQP shows that the Dirac particle is in reality a rotator, i.e. two particles rotating around their common center of inertia. In the framework of MCQP one can discover the force field, responsible for pair production, that is impossible in the framework of ACQP.


Abstract
One can construct the model conception of quantum phenomena (MCQP) which relates to the axiomatic conception of quantum phenomena (ACQP), (i.e. to the conventional quantum mechanics) in the same way, as the statistical physics relates to thermodynamics. Such a possibility is based on a new conception of geometry, which admits one to construct such a deterministic space-time geometry, where motion of free particles is primordially stochastic. The space-time geometry can be chosen in such a way that statistical description of random particle motion coincides with the quantum description. Transition from ACQP to MCQP is a result of correction of some mistakes in the foundation of ACQP. Methods of MCQP in investigation of quantum phenomena appear to be more subtle and effective, than that of ACQP. For instance, investigation of the free Dirac equation in framework of MCQP shows that the Dirac particle is in reality a rotator, i.e. two particles rotating around their common center of inertia. In the framework of MCQP one can discover the force field, responsible for pair production, that is impossible in the framework of ACQP.

Introduction
Sometimes investigation of a new class of physical phenomena is carried out by two stages. At first, the simpler axiomatic conception based on simple empiric considerations arises. Next, the axiomatic conception is replaced by the more developed model conception, where axioms of the first stage are obtained as properties of the model. Theory of thermal phenomena was developed according to this scheme. At first, the thermodynamics (axiomatic conception of thermal phenomena, or ACTP) appeared. Next, the statistical physics (model conception of thermal phenomena, or MCTP) appeared. Axioms of thermodynamics were obtained as properties of the chaotic molecule motion.
The contemporary quantum theory is the first (axiomatic) stage in the development of the microcosm physics. Formal evidences of this is an existence of quantum principles. Appearance of the next (model) stage, where the quantum principles are consequences of the model, seems to be unavoidable. The model conception is attractive, because it uses more subtle and effective mathematical methods of investigation. Besides it gives boundaries of the axiomatic conception application. We can see this in example of statistical physics and thermodynamics.
From practical viewpoint, the main difference between the axiomatic and model stages of a theory lies in mathematical methods of description and investigation. Methods of axiomatic conception are more rough are rigid. They can be changed only by a change of axiomatics. This is produced mainly by introduction of additional suppositions. Mathematical methods of model conception are more flexible and adequate, because usually the model parameters are numbers and functions, which can be changed fluently.
Let us imagine, that we do not know methods of statistical physics and try to investigate nature of crystal anisotropy, using only thermodynamical methods. Using experimental data we can calculate thermodynamical potentials and describe macroscopic properties of crystal, but we hardly can calculate anything theoretically. For description of the next crystal we are forced to use experimental data again. By means of methods of statistical physics we can explain and calculate parameters of crystal anisotropy theoretically. At any rate, methods of model conception appears to be more effective in the given case.
Something like that we observe in the theory of elementary particles, when theorists use rough and rigid methods of axiomatic conception (quantum theory), which do not allow to construct a perfect theory. There is a hope that mathematical methods of the model conception of quantum phenomena (MCQP) appear to be more effective, because MCQP does not use the rigid principles of quantum mechanics. Instead quantum principles MCQP uses parameters of space-time geometry which can be changed fluently.
We shall refer to the contemporary quantum theory as the axiomatic conception of quantum phenomena (ACQP). Logical structure of ACQP distinguishes from the logical structure of MCQP and from that of other well defined physical conceptions. For instance, we compare the logical structure ACQP with the logical structure of such a physical conception as the special relativity theory. The relativity theory appeared in the beginning of the XXth century as a solution of the problem of large velocities, when the particle motion with the speed close to the speed of the light cannot be described correctly in the framework of the classical nonrelativistic physics. Creation of the special relativity theory was a result of a change of the fundamental principle of physics. The space-time model has been changed. The Newtonian space-time model with two independent invariant distances between two events (the temporal interval and spatial interval) was replaced by the space-time geometry with one space-time interval between the two events. Such a solution of the problem of large velocities was sufficiently fundamental, and the scenario of the special relativity development was classical in the sense that it reminds the development scenario of the classical (Newtonian) physics. The last developed in accordance with the Newton's slogan "Hypotheses non fingo" (Not to invent hypotheses). Logical structure of the special relativity theory and that of ACQP are shown in figure 1, where ovals show experimental data and bars show the statement of the conception.
ACQP arises as a solution of the problem of the atomic spectra discreteness. This problem appeared in the beginning of the XXth century. As it became clear later, this problem is a special case of a more general problem of the small mass particle motion. Experiments show that the particles of small mass move stochastically, and this stochasticity depends on the mass of the particle. In the classical (relativistic and nonrelativistic) physics the motion of a free particle is determined only by the space-time geometry. This property should be conserved in the modified theory. One should invent such a space-time geometry, where the motion of a free particle would be stochastic, and this stochasticity would depend on the particle mass. (By definition, the geometry is a set of points, or events with a given distance between any two points). One should note that the geometrical character of the free motion and a dependence on the particle mass are compatible only in that case, when the particle mass is a geometrical characteristic of the particle (the mass geometrization). In the classical (relativistic and nonrelativistic) physics the particle mass is not geometrized, and the mass geometrization is a quite new property of a space-time geometry. Space-time geometries with such unusual properties (stochasticity of free motion and the mass geometrization) were not known in the beginning of the XXth century (they were not known until the end of the XXth century). Besides, at that time the problem of the small mass particles was perceived rather as a problem of the atomic spectra discreteness.
In the beginning of the XXth century the problem of the small mass particles cannot be solved on the fundamental level (by a change of the space-time geometry). The problem was solved on the second (after fundamental) level by introduction of the quantum principles, which replace some of corollaries of the true space-time geometry (see figure 1b). The space-time model remains to be Newtonian (nonrelativistic), but the logical and mathematical corollaries of this model which concern the quantum phenomena (motion of the small mass particles), are blocked. Corollaries of quantum principles are used instead of them. Such a logical construction, when corollaries of a false fundamental statement are replaced by the true corollaries of the true fundamental statement, will be referred to as a compensating conception. The compensating conception contains simultaneously both statements: the false fundamental statement and the true corollaries of the true fundamental statement. The reason, why some true fundamental statement is absent in the compensating conception may be different. This reason is of no importance, but it is important that because of the absence of the true fundamental statement one cannot obtain all its corollaries. One obtains only some of them, whereas other corollaries of the true fundamental statement remained to be unknown.
These unknown corollaries appear to be needed, when we consider a new set of physical phenomena. One can guess right the unknown corollary of the absent true fundamental statement, provided this corollary is located on the upper level of the conception statements, i.e. in the case, when this corollary is close to the experimental data. But in this case the corollary is far from the true fundamental statement (on the bottom level) and from the quantum (compensating) principles (on the second level) (see figure 1b). If we succeeded to guess right the unknown corollary and to explain some experimental data, the obtained statement looks as something unrelated with the quantum principles. This statement is perceived as a new additional supposition (hypothesis) which is needed for explanation of experimental data. The more the new experimental data, the more new hypotheses we need. Thus, we can develop the compensating conception only inventing new hypotheses, located on the upper level near the experimental data. Discovery of corollaries, located near the fundamental level, is difficult, because the true fundamental statement is unknown and one cannot deduce its corollaries. At the same time it is difficult to guess right these corollaries, using experimental information, because they connected with the experimental data by a long logical chains. The farther from the fundamental level are the new hypotheses located, the less is the predictability of the physical conception in the relation of the new experiments. The closer is the new hypotheses to the experimental data, the easier they can be guessed right. Thus, if the true fundamental statement is unknown, the development of the physical conception is carried out in such a way, that the number of additional hypotheses increases, whereas the predictability of the compensating conception in the relation of new experiments decreases. To improve the predictability of the compensating conception, it is necessary to obtain the true fundamental statement. New experimental data cannot help in this search, because as a rule the logical chains connecting the true fundamental statement with the experimental data are too long. One can obtain this fundamental statement only by the logical way, using the whole set of experimental data.
Studying the history of the quantum theory development in application to relativistic phenomena and to the theory of elementary particles, one can discover characteristic evidence of the compensating conception. The number of additional hypotheses increases, whereas the predictability of the conception in the relation to new experiments decreases. Besides, there are additional circumstances conditioned by the long work with the compensating conception (ACQP). These circumstances impede transition from ACQP to MCQP. As a rule the contemporary researchers dealing with the relativistic quantum theory evaluate a new hypothesis as follows. If the new hypothesis explains some new experimental data, this hypothesis deserves to be accepted and investigated. If the author of a new hypothesis cannot manifest any experiment which can be explained by his new hypothesis, such a hypothesis does not deserve even to be investigated. Such an approach to the test of new hypotheses is justified only in framework of the compensating conception, where all new hypotheses are located on the upper level, and logical chain between the hypothesis and the experimental data is short. However, this approach to the test of the hypothesis is inadequate, if one suggests the fundamental statement, located on the bottom level. In this case the logical chain between the fundamental statement and experimental data is very long. To explain experimental data and to test this chain, one needs to develop the investigation methods and mathematical technique, corresponding to the new fundamental statement. All this is a very difficult problem, and to test the new hypothesis, which pretends to be a fundamental statement of the conception, one should use mainly such an argument, as absence of other additional hypotheses. Another criterion of the fundamental statement is the fact that it is introduced instead of a false fundamental statement, but not in addition to it. For instance, if the space-time geometry is considered to be a fundamental statement of the physical conception, the transition from ACQP to MCQP is to be accompanied mainly by a change of the space-time geometry. In this case one should speak rather on a correction of a mistake in the foundation of the physical conception. Of course, ultimately an explanation of all experimental data and prediction of experimental results is the final criterion of the fundamental statement validity, but this explanation may appear only after a long development of the mathematical technique, corresponding to the new fundamental statement.
According to MCQP quantum phenomena are a result of stochastic behavior of microparticles. Statistical description of this stochastic motion leads to a description of the quantum phenomena. This idea is very old. It was reasonable after the successful explanation of thermal phenomena as a chaotic motion of molecules. There were many attempts of this idea realization, but all these attempts had failed. As a result a sceptic relation to this idea appeared. Now most of physicists believe that quantum principles describe correctly the origin of physical phenomena in microcosm, and the microcosm physics has a quantum nature.
Logical connection between the ACQP and MCQP is shown in figure 2. Problems of the logical links of MCQP in transition from ACQP to MCQP are marked by the symbol "?". As one can see from this scheme that there are three obstacles on the way of creation MCQP: (1) inadequate space-time geometry, which describes the particle motion in microcosm as a deterministic, although experiments show that this motion is random in reality, (2) inadequate statistical description, when particles and antiparticles are considered to be objects of statistical description, whereas objects of statistical description are to be world lines, which are primary physical objects in relativistic theory, (3) integration of dynamic equations for ideal fluid, which is necessary for transformation of wave function and spin (which are fundamental objects of ACQP) into a method of the ideal fluid description, i.e. in attributes of a description of physical objects.
Model conception of quantum phenomena (MCQP) could be constructed only after overcoming of these three obstacles. But each of the said obstacles was a very difficult problem. Besides, it was necessary to realize that on the way to creation of MCQP there are these obstacles. To overcome the first obstacle it was necessary to construct a new conception of geometry (T-geometry), which contains such a space-time geometry, where the particle motion be random and the particle mass be geometrized.
The true space-time geometry is such a geometry, where motion of microparticles. is primordially stochastic, although the geometry in itself is not random (intervals between the events in such a geometry are deterministic, but not random). To construct such a geometry (T-geometry), one needs to go outside the framework of Riemannian geometry, which is the most general contemporary geometry fitting for the space-time description. Being flat, uniform and isotropic, the true geometry of the absolute space-time distinguishes from the Minkowski geometry only in some correction containing the quantum constant . This correction is essential only for short space-time intervals, i.e. only in microcosm. Formally, introduction of this correction is equivalent to introduction of some fundamental length, but it is a transverse length (thickness of the world line).
Statistical description of a random motion of particles generated by the spacetime geometry leads to the quantum mechanical description (the quantum constant appears in the theory via geometry), in the same way as the statistical description of chaotic molecule motion leads to thermodynamics. Essential difference between the two statistical descriptions lies in the difference between the statistical description for relativistic and nonrelativistic cases. In the case of the statistical physics both regular and random components of the particle velocity are nonrelativistic, whereas in the case of the geometric stochasticity the random velocity component is always relativistic, although the regular component may be nonrelativistic. As a result even the nonrelativistic quantum mechanics appears to be a hidden relativistic theory. There is an essential difference between the relativistic and nonrelativistic statistical descriptions. The fact is that the nonrelativistic statistical description may be probabilistic, i.e. it can be carried out in terms of the probability density, whereas the relativistic statistical description cannot be produced in terms of the probability theory. The origin lies in the fact that physical objects to be statistically described are different in the relativistic and nonrelativistic theories.
In the nonrelativistic theory the physical object is a particle, i.e. a pointlike (zerodimensional) object in three-dimensional space, and the particle world line describes a history of the pointlike object. In other words, the particle is primary and its world line is secondary. In the consequent relativistic theory the situation is inverse. The physical object is the world line, i.e. the one-dimensional line in the space-time, whereas the particle and the antiparticle are derivative objects (intersections of the world line with the surface t =const). In other words, in the consequent relativistic theory the world line is primary, whereas the particle and antiparticle are secondary. The term 'WL' will be used for the world line considered to be the primary physical object. The difference in the choice of the primary physical object is conditioned by different relation to the existence of absolute simultaneity in relativistic and nonrelativistic physics. Statistical description must be a description of primary physical objects. Foundation for such a description is the density of physical objects in the three-dimensional space (for particles) or in the space-time (for WLs). Density ρ (x) of particles at the point x is defined by the relation where dN is the number of particles in the 3-volume dV . The particle density is defined as a proportionality coefficient between dN and dV . The density j k (x) of WLs is defined by the relation where dN is the flux of WLs through three-dimensional area dS k in the space-time in vicinity of the point x. The quantity j k (x) is the proportionality coefficient between dN and dS k in vicinity of the point x. The quantity ρ (x) is a nonnegative 3-scalar. It can serve as a basis for introduction of the probability density, whereas the 4-vector j k (x) is not a nonnegative quantity, and it cannot serve as a basis for introduction of the probability density, which must be nonnegative quantity.
It is a common practice to think that terms "probabilistic description" and "statistical description" are synonyms. It is a delusion, because the probabilistic description is a description, founded on a use of the probability theory, whereas the statistical description is a description, dealing with many similar or almost similar objects. Such a set of similar objects is called statistical ensemble. Statistical description is an investigation and description of the statistical ensemble properties. Statistical description without a use of the probability density is possible. It is necessary only to investigate the statistical ensemble without a use of the probability theory. We shall consider statistical ensembles of dynamic or stochastic systems and use essentially the circumstance, that the statistical ensemble is a dynamic system, even if its elements are stochastic systems. Use of the statistical ensemble as some means for calculation of statistical averages is not necessary. Such an approach may be qualified as the dynamic conception of statistical description (DCSD). It is appropriate in any case (relativistic and nonrelativistic). Interpretation of DCSD is carried out in terms of the ideal fluid and world lines of its particles. Construction of DCSD is a result of overcoming of the second obstacle.
Between DCSD and axiomatic conception of quantum phenomena (ACQP), i.e. conventional quantum mechanics, there is a connection. To obtain this connection it was necessary to obtain fundamental objects of ACQP (wave function and spin) as attributes of ideal fluid, which appears in DCSD. Connection between the fluid and the Schrödinger equation is known since the beginning of the quantum mechanics construction [1,2]. In after years many authors developed this interplay known as hydrodynamic interpretation of quantum mechanics [3,4,5,6,7,8,9,10,11]. But this interpretation was founded ultimately on the wave function as a fundamental object of dynamics. It cannot go outside the framework of quantum principles, because the connection between the hydrodynamic interpretation and the quantum mechanics was one-way connection. One could obtain the irrotational fluid flow from the dynamic equation for the wave function (Schrödinger equation), but one did not know how to transform dynamic equations for a fluid to the dynamic equation for a wave function. In other words, we did not know how to describe rotational fluid flow in terms of the wave function. In terms of the wave function we could describe only irrotational fluid flow.
To describe arbitrary fluid flow in terms of a wave function, one needs to integrate conventional dynamic equations for a fluid (Euler equations). Indeed, the Schrödinger equation may be reduced to the hydrodynamic equations for the density ρ and the velocity v of some quantum fluid. Substituting ψ = √ ρ exp (i ϕ) in (1.3) and separating real and imaginary parts of the equation, we obtain expressions for time derivatives ∂ 0 ρ and ∂ 0 ϕ. To obtain expression for the time derivative ∂ 0 v of the velocity v = m ∇ϕ, we need to differentiate dynamic equation for ∂ 0 ϕ, forming combination The reverse transition from hydrodynamic equations to dynamic equations for the wave function needs a general integration of hydrodynamic equations. This integration is simple in the partial case of irrotational flow, but it is a rather complicated mathematical problem in the general case, when a result of integration has to contain three arbitrary functions of three arguments. Without producing this integration, one cannot derive description of a fluid in terms of the wave function, and one cannot manipulate dynamic equations, transforming them from representation in terms of ρ, v to representation in terms of wave function and back. This problem has not been solved for years. It had been solved in the end of eighties, and the first application of this integration can be found in [12] Statistical ensemble of discrete dynamic or stochastic systems is a continuous dynamic system, i.e. some ideal fluid. Integration of hydrodynamic equations allows one to show that the wave function and spin are a way of description of an ideal fluid [13]. This was an overcoming of the third obstacle. In other words, the wave function appears as a property of some model (but not as a fundamental object whose properties are defined by axiomatics). Under some conditions the irrotational flow of the statistical ensemble (fluid) is described by the Schrödinger equation [14,15]. Thus, one can connect MCQP with the conventional quantum description and show that in the nonrelativistic case the description in terms of MCQP agrees with description in terms of the conventional quantum mechanics.
Note that the said obstacles were overcame in other order, than they are listed above. It took about thirty years for overcoming the first obstacle and construction of T-geometry. In the first paper [16] the system of differential equations for the world function of Riemannian geometry [17] was obtained. These equations do not contain metric tensor. They contain only the world function and their derivatives with respect to both arguments. These equations put the following question. Let the world function do not satisfy these equations. What is then? Do we obtain non-Riemannian geometry, or does no geometry exist? Then we could not answer this question, because as well as other scientists, we believed that the straight line must be one-dimensional geometrical object (curve) in any geometry. Only thirty years later we succeeded to answer this question and to construct non-Riemannian geometry (T-geometry) [18]. It appeared to be possible, because we had realized that in some geometries the straight line can be a non-one-dimensional object (surface). Moreover, it is the general case of geometry, whereas the Riemannian geometry with one-dimensional straight lines (geodesic) is a special (degenerate) case. The string theory suggests this idea of non-one-dimensional straight line. The second obstacle was overcame the first [19,20,21,22]. The third obstacle was overcame only in the end of eighties. Finally the first obstacle was overcame last [23].
It is worth to note that all obstacles had been overcame on the strictly logical foundation, i.e. only logical constructions based on already known principles of classical physics. New hypotheses and principles were not used, and this is not characteristic for the microcosm physics, developed in XXth century. In other words, the possibility of the MCQP construction was contained in principles of classical physics. It was necessary only to use them consistently. Unfortunately, it was unwarranted additional suppositions and absence of logical consistency, that was the stumbling block for construction of MCQP.
In the next two sections we consider modification of the quantum phenomena theory, generated by the overcoming of the said obstacles.

Geometry
There are two different approaches to geometry: mathematical and physical ones. In the mathematical approach a geometry is a logical construction founded on a system of axioms about points and straights. Practically any system of axioms, containing concepts of a point and of a straight, may be called a geometry. Well known mathematician Felix Klein [24] supposed that only such a construction on a point set is a geometry, where all points of the set have the same properties (homogeneous geometry). For instance, Felix Klein insisted that Euclidean geometry and Lobachevsky geometry are geometries, because they are homogeneous, whereas the Riemannian geometries are not geometries at all. As a rule the Riemannian geometries are not homogeneous, and their points have different properties. According to the Felix Klein viewpoint, they should be called as "Riemannian topographies" or as "Riemannian geographies". It is a matter of habit and taste how to call the geometry. But Felix Klein was quite right in the relation, that he suggested to differ between the Euclidean geometry and the Riemannian one. The fact is that the principle of the Riemannian geometry construction is quite different from that of the Euclidean geometry construction. The Euclidean geometry is constructed on the basis of axioms, whereas the Riemannian geometry is constructed as a deformation of the Euclidean geometry.
At the physical approach the geometry is the science on mutual disposition of points and geometric objects in the space, or events in the space-time. The mutual disposition is described by the metric ρ (distance between two points), or by the world function σ = 1 2 ρ 2 [17]. It is question of the secondary importance, whether all points have the same properties or not, and what axioms are satisfied by the metric.
The Riemannian geometry is obtained as a result of the proper Euclidean geometry deformation, when the infinitesimal Euclidean interval ds 2 E is replaced by the Riemannian interval ds 2 = g ik dx i dx k . Such a change is a deformation of the Euclidean space. Such an approach to geometry, when a geometry is a result of the proper Euclidean geometry deformation will be referred to as a physical approach to geometry. The obtained geometry will be referred to as a physical geometry. The physical geometry has no own axiomatics. It uses "deformed" Euclidean axiomatics. The physical geometry describes mutual disposition of points in the space, or of events in the space-time. It is described by setting the distance between any two points. The metric ρ is the only characteristic of a physical geometry. The world function σ = 1 2 ρ 2 is more convenient for description of the physical geometry, because it is real even for the space-time, where ρ = √ 2σ may be imaginary. Construction of any physical geometry is determined by the deformation principle [25]. It works as follows. The proper Euclidean geometry G E can be described in terms and only in terms of the world function σ E , provided σ E satisfies some constraints formulated in terms of σ E [25]. It means that all geometric objects O E can be described σ-immanently (i.e. in terms of σ E and only of σ Relations between geometric objects are described σ-immanently by some expressions R E = R E (σ E ). Any physical geometry G A can be obtained from the proper Euclidean geometry by means of a deformation, when the Euclidean world function σ E is replaced by some other world function σ A in all definitions of Euclidean geometric objects O E = O E (σ E ) and in all Euclidean relations R E = R E (σ E ) between them. As a result we have the following change The set of all geometric objects O A and all relations R A between them forms a physical geometry, described by the world function σ A . Index 'E' in the relations of physical geometry G A means that axiomatics of the proper Euclidean geometry was used for construction of geometric objects O E = O E (σ E ) and of relations between them R E = R E (σ E ). The same axiomatics is used for all geometric objects O A = O E (σ A ) and relations between them R A = R E (σ A ) in the geometry G A . But now this axiomatics has another form, because of deformation σ E → σ A . It means that the proper Euclidean geometry G E is the basic geometry for all physical geometries G obtained by means of a deformation of the proper Euclidean geometry. If the basic geometry is fixed (it is this case that will be considered further), the geometry on the arbitrary set Ω of points is called T-geometry (tubular geometry). The T-geometry is determined [18,26] by setting the world function σ: σ : Ω × Ω → R, σ (P, P ) = 0, ∀P ∈ Ω (2.1) In general, no other constraints are imposed, although one can impose any additional constraints to obtain a special class of T-geometries. T-geometry is symmetric, if in addition σ (P, Q) = σ (Q, P ) , ∀P, Q ∈ Ω (2.2) Consequent application of only deformation principle admits one to obtain any physical geometry (T-geometry), which appears to be automatically as consistent as the Euclidean geometry, which lies in its foundation. The Riemannian geometries form a special class of T-geometries, determined by the constraint, imposed on the world function where σ R is the world function of the Riemannian space, and L [xx ′ ] means a geodesic segment between the points x and x ′ . Riemannian geometry is determined by the dimension n and n (n + 1) /2 functions g ik of one point x, whereas the class of all possible T-geometries is essentially more powerful, because it is determined by one function σ of two points x and x ′ . In general, the deformation principle allows one to obtain such geometries, where non-one-dimensional tubes play the role of the straight lines. The real space-time geometry is of such a kind. But creators of the Riemannian geometry supposed that a geometry with tubes instead of straights was impossible. The restriction (2.3) on Riemannian geometries was introduced to forbid deformation transforming one-dimensional Euclidean straights to many-dimensional tubes. But in nonhomogeneous physical geometry one fails to suppress the tubular character of straights. In the Riemannian geometry one succeeds to make this, only refusing from the consequent application of the deformation principle and using additional means of the geometry construction. As a result the Riemannian geometry appears to be not quite consequent construction. This is displayed, in particular, in the lack of absolute parallelism, whereas in any physical geometry constructed in accordance with the deformation principle the absolute parallelism takes place.(see details in [25]).
The tubular character of timelike straights in the real space-time generates the stochastic character of the free particles motion in such a space-time, because the straight (tube) T P 0 P 1 , passing through the points P 0 and P 1 , is determined by the relation In the proper Euclidean space the vectors −−→ P 0 P 1 and − − → P 0 R are linear dependent (collinear), if and only if the second order Gram determinant F 2 (P 0 , P 1 , R) vanishes.
Here −−→ P 0 P 1 . − − → P 0 R denotes the scalar product of two vectors −−→ P 0 P 1 and − − → P 0 R, which is defined by the relation Relations (2.5), (2.6) express the collinearity condition via the world function σ of the proper Euclidean space. In other words, the collinearity of −−→ P 0 P 1 , − − → P 0 R is defined σ-immanently. By definition this relation can be used in arbitrary T-geometry, i.e. for any world function σ. The quantity S P 0 P 1 R is the area of Euclidean triangle with vertices at points P 0 , P 1 , R. It is connected with the Gram determinant by the relation Thus, two relations (2.4) contain two equivalent conditions of collinearity. It is worth to note that conventionally the concept of linear dependence (collinearity) is introduced in the framework of linear space. To introduce the linear space, one needs a set of restrictions (fixed dimension, continuity, coordinate system, etc.). It appears unexpectedly that all these restrictions are not necessary. The concept of linear dependence can be introduced on arbitrary set of points, where the world function (metric) is given. The concept of linear dependence appears to be independent of whether or not the linear space can be introduced on this set. Concept of linear dependence describes some very general geometric property, which concerns the linear space only indirectly.
According to this definition the tube T P 0 P 1 is a set of such points R, that vectors −−→ P 0 P 1 and − − → P 0 R are collinear. The tubular character of the straight (thick straight) means that there are many directions − − → P 0 R, parallel to the vector −−→ P 0 P 1 . On the other hand, the motion of a free particle in the curved space-time is described by the equation of a geodesic where Γ i kl is the Christoffel symbol. Equations (2.8) describe the parallel transport of the velocity vectorẋ i of the particle along the direction dx i =ẋ i dτ , determined by the velocity vectorẋ i . If there are many vectors parallel to the velocity vectoṙ x i , the parallel transport (2.8) appears to be not single-valued, and the world line becomes to be random.
The flat uniform isotropic space-time is described by the world function [23] where σ M is the world function of the Minkowski space, c is the speed of the light. The distortion function D (σ M ) describes the character of quantum stochasticity. In the space with nonvanishing distortion D (σ M ) the particle mass is geometrized [23]. The quantity b ≤ 10 −17 g/cm is the constant, describing connection between the geometric mass µ and the usual mass m by means of the relation m = bµ. Form of the distortion function D (σ M ) is determined by the demand that the stochasticity generated by distortion is the quantum stochasticity, i.e. the statistical description of the free stochastic particle motion is equivalent to the quantum description in terms of the Schrödinger equation [23].

Statistical description
The only parameter for the free particle is its mass m, and the change (3.2) in the nonrelativistic case has the form where u = u (t, x) is the mean value of the stochastic velocity component. Quantum constant appears here as coupling constant between the regular and stochastic components of the particle velocity. The velocity u is considered to be a new dependent variable, and dynamic equation for u is obtained as a result of the action variation with respect to u [28]. The velocity u is supposed to be small as compared with the speed of the light c.
In the relativistic case the change (3.2) takes the form m 2 → m 2 eff = m 2 1 + u l u l + λ∂ l u l , λ = mc On one hand, the action (3.6) describes a set of deterministic particles interacting between themselves via self-consistent vector field κ l . On the other hand, the action (3.6) describes a quantum fluid. Irrotational flow of this fluid is described by onecomponent wave function ψ, satisfying the Klein-Gordon equation [29,28].
From physical viewpoint the quantization procedure (3.5), when the mass m is replaced by its mean value m eff , looks rather reasonable. Indeed, the value of m eff depends on the state of the stochastic velocity component. Stochastic velocity component has infinite number of the freedom degrees and consideration of influence of the mean value u l on the regular component of the particle velocity appears to be very complicated. It is described by partial differential equations, whereas in absence of this influence the regular particle motion is described by the ordinary differential equations. From physical viewpoint such an interpretation of the quantization looks more reasonable, than conventional interpretation in terms of wave function and operators.
The wave function and spin appear here as a way of description of the ideal fluid [13], i.e. as fluid attributes. In other words, statistical description and hydrodynamic interpretation of the world function are primary, and wave function is secondary. Hierarchy of concepts is described by two schemes in figure 2. Note that transition from ACQP to MCQP became possible only after solution of such a pure mathematical problem as integration of complete system of dynamic equations for ideal fluid. Systematical application of this integration for description of quantum phenomena began in 1995 [14,30].

Methods and capacities of MCQP
Let us formulate the first difference between ACQP and MCQP. As any statistical theory MCQP contains two sorts of investigated objects: stochastic system S st and statistically averaged system S st , whereas ACQP investigates only one sort of objects: so called quantum systems S q . Systems S st and S q are continuous dynamic systems. They have dynamic equations and coincide practically always. The system S st is a discrete stochastic system, for which dynamic equations are absent. Two kinds of measurement in MCQP (S-measurement and M-measurement) is another side of existence of two sorts of objects S st and S st . Single measurement (S-measurement) is produced over single stochastic system S st . Result of S-measurement is random. It means that the result of S-measurement is not reproduced, in general, in other S-measurements on other S st , prepared in the same way. S-measurement always leads to one random but definite value R ′ of the measured quantity R. This result does not influence on the wave function ψ, because wave function describes the state of S st and has no relation to S st .
The massive measurement (M-measurement) is produced over many single dynamic systems S st , constituting S st . M-measurement is a set of many S-measurements, produced over different elements of E [S st ] or of S st . M-measurement of the quantity R leads always to a distribution F (R ′ ) of possible values R ′ , because S-measurements in different elements of S st give different results, in general. Result of M-measurement is always a distribution, even in the case, when all S-measurements, constituting the M-measurement, give the same result. In the last case the distribution is a δ-function. Distribution F (R ′ ) is reproducible. It is reproduced in other M-measurements, carried out in the same state ψ.
Thus, S-measurement is irreproducible, in general, and gives a definite result. The M-measurement is reproducible and, in general, does not give a definite result (but only a distribution of results). Nevertheless, formally the M-measurement of the quantity R may be considered to be a single act of measurement of the quantity R ′ distribution, which is produced over the dynamic system S st .
Theory can predict only results of M-measurement, which leads to a reproducible distribution except for the case, when the measured system is an element of the statistical ensemble, which is found in such a state, where the distribution of the measured quantity is a δ-function. In this case the result of S-measurement can be predicted, because it is determined uniquely by the distribution, which is a result of M-measurement and can be predicted by the theory.
Is it possible that M-measurement of the quantity R in the state ψ lead to a definite result R ′ ? Yes, it is possible, if M-measurement is accompanied by a selection of those S-measurements, constituting the M-measurement, which give the result R ′ . Such a measurement is called selective M-measurement (or SMmeasurement). SM-measurement has properties of S-measurement (definite result R ′ ) and those of M-measurement (it is produced in S st , and influences on the state of S st ). SM-measurement is accompanied by a selection, and the question about its reproducibility is meaningless, because it depends on the kind of selection, which determines the result of the SM-measurement.
In ACQP there is only one type of objects (quantum systems S q ) and only one type of measurement (Q-measurement). The Q-measurement is essentially Mmeasurement, because it always gives distribution of the measured quantity R. Result of Q-measurement is determined by the rule of von Neumann. All predictions in ACQP as well as in MCQP concern only with results of M-measurement. In ACQP there is only one object and only one type of measurement. Nobody distinguishes in ACQP between S-measurement and M-measurement.
In ACQP the statistically averaged system S st is considered to be an individual quantum system S q and a single measurement on S q = S st is considered to be a measurement with properties of both M-measurement and S-measurement (i.e. SM-measurement). In classical physics a single physical system is described as a discrete dynamic system, whereas a statistical ensemble of single physical systems is described as a continuous dynamic system. In ACQP the question, why the continuous dynamic system S q describes a single physical object (for instance, a single particle), is not raised usually. If nevertheless such a question is raised, one answers that quantum system is a special kind of physical system which distinguishes from classical system and whose state is described by such a mysterious quantity as wave function, or something like this. The fact that the wave function is a method of the statistical ensemble description is unknown practically.
In ACQP there is only one object, and for description of a single measurement one uses SM-measurement (but not S-measurement), because S-measurement does not exist in framework of ACQP. The SM-measurement is produced under the dynamic system S q = S st and influences on the wave function, describing the state of S q . The S-measurement is produced under the system S st . It does not influence on the wave function, describing the state of S q = S st . Character of influence on the wave function is the main formal difference between S-measurement and SM-measurement.
Replacement of S-measurement by SM-measurement in ACQP leads to such paradoxes, as the Schrödinger cat paradox, or the EPR-paradox, which are based on the belief that a single measurement changes the wave function (i.e. that a single measurement is a SM-measurement). Paradoxes are eliminated, if one take into account that a single measurement is a S-measurement, which does not change the wave function). See detail discussion in [27]. Some properties of S st appear as a result of properties of S st . Another ones appear as a result of statistical averaging. The last are interpreted as collective properties. For instance, in statistical physics (MCTP) the temperature is a collective property, because one molecule has no temperature. Only collective of molecules has a temperature in statistical physics. In thermodynamics (ACTP) there is no concept of collective properties, and the temperature is a property of any amount of matter (one molecule, or even a half of molecule). In MCQP the wave function ψ is a collective quantity, which describes the state of S st . If we say that individual stochastic system S st is found in the state ψ, it means only that S st is taken from the ensemble E [S st ], whose state is described by the wave function ψ. In MCQP the spin of a particle may be a property of individual particle S st , and it may be a collective property of S st . In different cases we have different results.
In the case of dynamic system S P described by the Pauli equation the electron spin is the collective property [14]. It means, in particular, that the dynamic system S S , described by the Schrödinger equation, and dynamic system S P consist of similar individual systems S st . The difference between them is described by the type of the fluid flow. In the case of S S the fluid flow is irrotational, whereas in the case of S P the fluid flow is rotational. In other words, in S P the electron spin is conditioned by collective property (vortical flow). In the given case the electron spin is a result of the fluid flow vorticity. (Let us remember that the statistical ensemble is a fluidlike dynamic system.) In the case of dynamic system S D , described by the free Dirac equation, the electron spin is a property of an individual stochastic particle [31]. In this case the dynamic disquatization of the Dirac equation, i.e. determination of classic analog S Dcl of the Dirac particle S D shows that S Dcl is a dynamic system, having ten degrees of freedom. It may be interpreted as two classical particles, rotating around their common center of inertia. Angular momentum of this rotation forms spin of S Dcl . Thus, in the case of the Dirac electron S D the spin is a property of an individual stochastic system.
In ACQP there are no collective properties, because there is only one object: the quantum system S q . The statement of the question whether the spin of the particle is a collective property is meaningless in the framework of ACQP. Is it important for investigations to distinguish between individual and collective properties? Sometimes it is of no importance, but sometimes it is important. We can investigate this question in the example of temperature in the statistical physics. Collective properties of temperature are of no importance, provided we deal with the mean values of it. If we deal with fluctuations of temperature and those of other quantities, the collective properties of temperature become to be important. In any case, method of investigation of MCQP appears to be more subtle and effective, than that of ACQP.
In general, MCQP as any model conception possesses more subtle and flexile methods of investigation. For instance, the change (3.5) carries out essentially the quantization procedure, i.e. transition from classical description to the quantum one. Let us imagine that the quantization (3.5) is not exactly true, and one needs to correct it slightly. In the framework of MCQP one needs only to change the form of the expression (3.5). It is not clear how such a modification can be realized in the framework of ACQP, because ACQP is connected closely with linearity of dynamic equations in terms of the wave function. One cannot imagine quantum mechanics which is nonlinear in terms of wave function, because in this case the quantum principles are violated.
Having a long history [1,2], the hydrodynamic interpretation should rank among new methods of investigation. But in the framework of ACQP the hydrodynamic interpretation is secondary as it follows from the schemes in figure 2. In the framework of MCQP the hydrodynamic interpretation is primary, and possesses more subtle methods of investigation. In the framework of ACQP a transition to semiclassical approximation is carried out by means of transition to the limit → 0 with some additional conditions. In MCQP this transition is realized by means of dynamic disquantization [31], which is a relativistic dynamic procedure. At the dynamic disquantization one removes transversal components ∂ ⊥k = ∂ k − j k j l (j s j s ) −1 ∂ l of derivative ∂ k , which are orthogonal to the flux 4-vector j k . After this transformation the dynamic equations contain derivatives only in direction of the vector j k . Such a system of partial differential equations can be reduced to a system of ordinary dynamic equations. As a result of dynamic disquantization the system of partial differential equations turns to a system of ordinary differential equations. In other words, the statistical ensemble of stochastic systems turns to a statistical ensemble of dynamic systems. As a result the continuous system can be interpreted in terms of a discrete dynamic system (with finite number of the freedom degrees). In the nonrelativistic case the dynamic disquantization is equivalent to → 0. In the relativistic case the quantum constant remains in the discrete dynamic system [31]. It allows one to obtain a more subtle interpretation.
This method was applied for investigation of dynamic system S D , described by the Dirac equation [31]. It appears that the classical analog of the Dirac particle S D is a rotator (but not a single particle), i.e. two particles rotating around their common center of inertia. This explains freely angular and magnetic momenta of the Dirac particle. Dynamic variables describing rotation contain the quantum constant . If → 0, degrees of freedom, connected with rotation are suppressed. Radius of rotator tends to 0, and instead of rotator we obtain pointlike particle with spin and magnetic moment. This result with → 0 agrees with the results, obtained in the framework of ACQP. More soft result of rotator, when = 0, cannot be obtained by methods of ACQP. These methods are too rough. Besides, it appears (quite unexpectedly) that the internal (rotational) degrees of freedom of the dynamic system S D are described in nonrelativistic manner [31,30,32]. Investigation of the dynamic system S D was produced without any additional supposition. It was investigated simply as a dynamic system by means of relativistically covariant methods. These results cannot be obtained in the framework of conventional quantum mechanics (ACQP).
Methods of MCQP are consistent relativistic ones. For description of relativistic quantum phenomena ACQP uses the program of uniting of nonrelativistic quantum mechanics technique with the relativity principles. Unfortunately, this program failed. At any rate, it works unsuccessfully last fifty years, trying to construct relativistic quantum field theory and theory of elementary particles. It seems that uniting of nonrelativistic quantum mechanics technique with the relativity principles is impossible.
As another example of the MCQP methods application, we refer to the problem of the pair production, which is the central problem in the high energy physics. ACQP cannot say anything on the pair production mechanism and on the agents, responsible for this process, whereas MCQP can say something pithy on this problem. MCQP vests responsibility for the pair production on the κ-field (3.7), which is conditioned by the stochastic component of the particle motion. In MCQP the pair production is taken into account on the descriptive (before-dynamic) level, i.e. the pair production is taken into account by consideration of WL as a primary physical object, whereas in ACQP the pair production is taken into account only on the dynamic level, i.e. by means of dynamic equations. In ACQP particles may be produced not only by pairs. The number of particles produced in an elementary act may be arbitrary. The number of produced particles depends on the form of corresponding term in Lagrangian. In MCQP the particles are produced only by pairs particle -antiparticle. This fact is fixed on the descriptive (conceptual) level, and cannot be changed on dynamical level (by a choice of Lagrangian). All this shows that the source of pair production is different in MCQP and ACQP.
Let us imagine that the particle world line turns in the time direction. Depending on situation, such a turn describes either pair production, or pair annihilation. The κ-field creates conditions for such a turn and for the pair production. The fact is that at such a turn in time the world line direction becomes spacelike (p 2 = m 2 < 0) in the vicinity of the turning point. If one forbids the world line to be spacelike, the pair production becomes to be impossible. Such a possibility to change the particle mass and to make it imaginary is rather rare property among the force fields. For instance, the electromagnetic field of any magnitude cannot change the particle mass, and hence, to produce pairs. Pair production is a prerogative of the κ-field. According to relation (3.5) the expression containing κ-field enter in the effective squared mass as a factor. If this expression is negative, the mass becomes imaginary, and the pair production (annihilation) becomes to be possible [28].
Furthermore, pair production, obtained in ACQP at canonical quantization of nonlinear relativistic field, does not take place in reality. It was shown at canonical quantization of nonlinear complex scalar field [33], described by the Lagrangian density L =: ϕ * i ϕ i − m 2 ϕ * ϕ + λ 2 ϕ * ϕ * ϕϕ : (4.1) At canonical quantization of (4.1) the W L-scheme of quantization was used, when the object of quantization is WL, i.e. world line considered as the primary physical object. In the W L-scheme the canonical quantization is produced without imposition of additional constraint [u, P 0 ] − = −i ∂u ∂x 0 , E = P 0 = T 00 dx (4.2) where [...] − denotes commutator and T ik is the energy-momentum tensor. The condition (4.2) identifies the energy with the evolution operator (Hamiltonian). This identification is possible in the case, when there are only particles, or only antiparticles. It is possible also in the case, when particles and antiparticles are considered as different physical objects (but not as attributes of WL). In W L-scheme of quantization the number of primary physical objects (WLs) is conserved, and quantization is produced exactly (without the perturbation theory methods). In such a quantization the pair production is absent, that agrees with demands to the field producing pairs. But then the question arises. Why does pair production appear at quantization according to P A-scheme [34,35,36,37], when particle and antiparticle are considered as primary objects? The answer is as follows. Canonical quantization (according to W L-scheme) is possible without imposition of constraint (4.2). It means that the constraint (4.2) is an additional condition, and one should to verify its compatibility with dynamic equations. Unfortunately, nobody had verified this, supposing that (4.2) is a necessary condition of the second quantization and there is no necessity to verify its compatibility with dynamic equations. This test was produced in [33]. It appears that (4.2) is compatible with dynamic equations, provided λ = 0, i.e. the field is linear. In the nonlinear case λ = 0 imposition of the constraint (4.2) leads to overdetermination of the problem. In the overdetermined (and hence, inconsistent) problem one can obtain practically any results, which one wishes. So, authors of [34,35,36,37] wanted to obtain the pair production, and they had obtained it.
These examples show that MCQP and its subtle investigation methods can be useful at investigation of the microcosm phenomena properties.
Thus, MCQP makes the first successes, but not in the sense that it explains some new experiments, which could not be explained before. MCQP uses the more subtle dynamic methods of investigation (consideration of two objects S st and S st , hydrodynamic interpretation of relativistic processes [29,28], dynamic quantization and disquantization [31]), which cannot be used by ACQP because of its axiomatic character. Difference between the methods of MCQP and ACQP is described by the following scheme modified in such a way, to take into account influence of the matter distribution in the space-time (curvature) and existence of new metric fields, generated by the possible asymmetry of the world function [38,39].
Asymmetric world function describes the space-time, where the past and the future are unequal geometrically. One cannot imagine such a thing in the framework of Riemannian geometry. Expansion of the symmetric world function σ (x, x ′ ) over powers of η i = x i − x ′i has the form where metric tensor g ik (x ′ ) describes the gravitational field, and σ ikl (x ′ ) is expressed via derivatives of metric tensor g ik (x ′ ). For asymmetric world function the same expansion has the form [38,39] σ (x, where three coefficients σ i (x ′ ), σ ik (x ′ ) and σ ikl (x ′ ) are independent, and each of them is connected with some metric (geometric) field. Coefficient σ i (x ′ ) describes a "vector field" which is strong and effective at small space-time intervals. Coefficient σ ik (x ′ ) describes the second rank tensor field (gravitational field) which is strong and effective at middle space-time intervals. Finally, σ ikl (x ′ ) is connected with the third rank tensor field, which is strong and effective at large space-time intervals. Maybe, this field is connected with astrophysical problem of dark matter, when one fails to explain observed motion of stars and galaxies by means of only gravitational field. At construction of MCQP one did not use any new hypotheses. On the contrary, flexibility and subtlety of MCQP methods are connected with remove of unwarranted constraints and correction of mistakes in the approach to geometry and to statistical description. In other words, MCQP satisfies the Newton's criterion: "Hypothesis non fingo." Only choice of true space-time geometry is determined properties of physical phenomena in microcosm. This choice must be done in any case. But this choice may be true, or not completely true.