November 17, 2003
It is common practice to think that the microcosm structure is very complicated, that microparticles and other objects of microcosm have a mysterious quantum nature. Describing motion of microparticles, one uses additonal (with respect to classical physics) hypotheses: quantum principles, wave function, spin and other concepts, which are absent in classical physics. Quantum mechanics, based on the quantum principles, explains very well all nonrelativistic phenomena of microcosm. It explains those quantum phenomena, where relativistic effects may be considered as small corrections. But quantum mechanics failed to explain essentially relativistic physical phenomena of microcosm. Trying to explain relativistic phenomena of microcosm, researchers invented many new hypotheses (additional to quantum principles). Nevertheless transformation of nonrelativistic quantum technique into the relativistic one has failed.
In our opinion this failure is conditioned by the fact that there are delusions and mistakes in application of fundamental principles of geometry and classical physics to description of microcosm. If these mistakes are overcame, additional hypotheses (quantum principles) are not necessary. All microcosm phenomena can be explained on the basis of classical physics. Speaking on classical physics, we mean only that the principles of classical physics are used (but not those of quantum mechanics). Classical physics uses the quantum constanth, but now this constant is an attribute of the space-time geometry, whereas in quantum mechanics it is an attribute of quantum principles.
But how can one deny principles of quantum mechanics, if the quantum mechanics explains all nonrelativistic quantum phenomena? The answer is as follows. If there is a mistake in a physical conception, there exist two ways of this mistake overcoming:
The first way is excellent, but sometimes it is very difficult, because it needs to discover the mistake. Discovery of mistake appears sometimes to be very diffcult.
The second way is not so well, but it is much easier, because it does not need to discover the mistake. One may suggest new hypotheses, knowing nothing on the possible mistake. We shall refer to the physical conception created on the basis of additional hypotheses as a compensating conception. Any compensating conception is created for explanation of some series of physical phenomena. Application of the compensating theory to other series of physical phenomena fails as a rule, because the additional hypotheses used for construction of the compensating conception are rather special, whereas the mistake remains in the basic physical principles (or in their application).
In the history of science there is an example of compensating conception, existing during several centuries. This is the Ptolemaic doctrine in celestial mechanics. Now we know that the Ptolemaic doctrine contained a mistake. It was the statement about the Earth's immobility. The Ptolemaic doctrine explains very well the planet's motion, the lunar and solar eclipses. Possibility of additional hypotheses (number of epicycles, differents) facilitated fitting of the planet's parameters. But capacities of such a fitting were restricted. In the framework of the Ptolemaic doctrine one could not discover the gravitation law, or calculate the trajectory of a rocket flying to the Moon. In other words, although the Ptolemaic doctrine described very well a series of celestial phenomena, it was unable to further development, and this incapacity to further development is a property of any compensating conception.
Copernicus discovered the mistake and eliminated it. As a result the celestial mechanics (and thereafter all physics) became to enable for further development. Transition from the Ptolemaic doctrine to the Copernicus one was very difficult and troublesome. Why? Not because of the activity of Catholic church, as one thinks sometimes, but because of special investigation style, which is yielded in the process of working with the compensating conception. This investigation style (pragmatic style, or P-style) uses a very short logic (supposition -- experimental test) and avoids long deductive reasonings, because any compensating conception is ultimately inconsistent, and this inconsistency may appear in the long reasonings. Using P-style, our work is very close to the simple fitting. Long work with compensating conception produces generations of researchers which do not trust to long reasonings. They take into account only a direct connection: supposition -- experimental test.
Note in this context, that at first the Copernicus doctrine was less exact, than the Ptolemaic one, because Copernicus considered only circular orbits of planets. Nevertheless, the Copernicus doctrine does enable to further development, but the Ptolemaic one does not.
The contemporary quantum theory have many features of a compensting conception. It uses additional hypotheses such as quantum principles. Attempts of obtaining relativistic quantum theory are accompanied by new hypotheses which are additional to quantum principles. Theorists, which deal with relativistic quantum theory (which is unsatisfactory) and with the theory of elementary particles, dream to invent such an idea, which could explain all relativistic quantum phenomena. The theorists do not trust to theoretical reseanings. They do trust only to such suggestions whose corollaries can be directly tested by experiment. All situation looks as if the quantum theory would be a compensating conception.
Is the quantum theory a kind of compensation conception? To answer this question distinctly, it is necessary to discover mistakes in fundamental principles of physics and geometry, or in their in applications.
We succeeded to discover a mistake infoundation of geometry, which does not allow one to construct the true space-time geometry, because this geometry is not contained in the list of possible geometries. After revision of the geometry foundations and formulation of the true conception of geometry, a new class of space-time geometries appears. In the space-time geometries of this class the particle motion is primordially stochastic, although the geometry in itself is deterministic.
Such a situation seems to be impossible. Appearance of stochastically moving particles is desirable. To obtain the stochastic particle motion some researchers consider stochastic geometries, or noncommutative geometries. In reality, the deterministic space-time geometry with primordially stochastic particles is a general case of geometry, whereas the deterministic space-time geometry with primordially deterministic particles is a very special (degenerate) case of geometry. All this takes place, provided we formulate foundations of geometry correctly.
Besides, in the space-time geometry with stochastic motion of particles the mass of a particle is geometrized in the sense that the stochasticity of motion is different for particles with different masses. Then the space-time geometry can be chosen in such a way that the statistical description of the stochastically moving particles agrees with experimental data.
Unfortunately, incorrect foundation of geometry is not an unique mistake. There is another mistake in the foundation of the relativistic statistical description. It is common practice to think that consideration of relativistically invariant dynamic equations is sufficient for taking into account the principles of relativity. Maybe, it is sufficient in dynamics, but it is not sufficient at statistical description of relativistic physical objects.
Object of statistical description is to be defined in accordance with the relativity principles. In the nonrelativistic physics the particle (a point in three-dimensional space) is the primary physical object. World line of the particle is a secondary physical object (history of the particle). In the relativistical physics the world line of a particle is a primary physical object, whereas the particle is an attribute of the world line (its section by the hyperplane t= const). Indeed, a world line is a geometrical object, which can be defined geometrically as a set of points in the space-time. The particle can be also defined geometrically as intersection of two geometrical objects: world line and the plane t = const. The reference to the plane t = const means a synchronization (especially, if we consider several particles) and a nonrelativistic character of description. This situation may be described by the sentence: concept of the ''particle state'' is different in the relativistic physics and in the nonrelativistic one. Classics of relativity (for instance, V.A. Fock, "Theory of space, time and gravitation", in Russian, GITTL, 1955, sec. 29) understood and underlined the difference between the relativistic and nonrelativistic concepts of the state.
Statistical description is a calculation of states, and it is very important, what we understand under state. In the nonrelativistic physics the state of several particles is determined by their coordinates x and momenta p at some time moment, i.e. on some planeSt in the space-time. In the nonrelativistic physics a transformation of the statistical description to the moving coordinate system does not generates problems, because at the Galilean transformation the surface St transforms to S't' (t' = const), and St = S't'. In the nonrelativistic case the state x,p on St can be easily recalculated to the state x',p' in other coordinate system on S't'.
In the relativistic physics we can also use nonrelativistic concept of state and describe the state of several particles by x,p onSt. But in the relativistic case at transition to other coordinate system we should use the Lorentz transformation, where, in general, St does not coincide with S't'. If the particles are deterministic, and there are dynamic equations for them, we can recalculate the x,p on St to the x',p' on S't'. But in this case we must use dynamic equations essentially. If the particles are stochastic, and there are no dynamic equations for them, the recalculation of the state x,p on St to the x',p' on S't' becomes impossible. It means that in the relativistic case we cannot use nonrelativistic concept of state and describe the state of particles by their coordinates and momenta x,p on St. In other words, concept of the phase space cannot be applied for description of stochastic relativistic particles. We are forced to use the relativistic (geometric) concept of the particle state, i.e. we must consider a statistical description of world lines, but not that of points in the nonrelativistically defined phase space.
To formulate the relativistical statistical description and consider the world line of a particle as a primary physical object, we should improve our terminology, which contains survivals of nonreltivistic theory. Indeed, the term ''world line of a particle'' means automatically that the particle is a primary physical object and the world line is an attribute of the particle (its history). Considering the world line as a primary physical object, we shall use the term 'WL'', which is an abbreviation of the term ''world line''. As far as application of such abbreviation is sometimes inconvenient, we shall use also the term 'emlon' which is a perusal of Russian abbreviation 'ML' for the world line.
In the relativistic physics there are two kinds of intersection of WL with the plane t= const: particle and antiparticle. We shall use the term 'SWL' (abbreviation of the term 'section of world line') as a collective concept with respect to concepts of 'particle' and 'antiparticle'. Such a concept should be introduced, because in the relativistic physics the particle and the antiparticle are two different states of WL, whereas in the nonrelativistic physics the particle and the antiparticle are two different physical objects. This difference is very important in the statistical description and not only in it. When in the relativistic physics we consider particle and antiparticle to be two different physical objects, we use inconsistent relativistic approach. Instead of SWL we shall use also the term 'esemlon' which is a perusal of Russian abbreviation 'SML' for the section of world line.
Thus, if we want to construct a consistent statistical description of stochastic relativistic particles, we should construct a statistical description of WLs (emlons). What is the difference between the relativistic and nonrelativistic statistical descriptions? The main difference is as follows. We can use the probability theory in the nonrelativistic statistical description, and nonrelativistic statistical description is a probabilistic description. We cannot use the probability theory in the relativistic statistical description, and the relativistic statistical description is not a probabilistic description.
Why do we unable to use the probability theory in the statistical description of relativistic particles? The answer is as follows. In the nonrelativistic case the density of states is determined by the relationdN=r (x)dV, where dN is the number of states in the volume dV and r (x) is the state density in the vicinity of the point x. The quantity r (x) is nonnegative, and under a proper normalization it can be interpreted as a probability density, which can serve as a foundation for introduction of the probability theory in the statistical description. In the relativistic case the state density (density of WLs) in vicinity of the point x is defined by the relation
dN=jk(x) dSk ,
wheredN is the flux of WLs through the 3-dimensional area dSk, and jk is the state density (density of WLs in vicinity of the point x). The flux jk is 4-vector, which cannot be used for introduction of the probability density, (in particular, because j0 may be negative).
Note that there is no contradiction in the statistical description without probability, because the statistical description is by definition a consideration of many similar or almost similar objects. A set of many independent similar objects S in different states forms a statistical ensemble E[S]. Carrying out a statistical description of objects S, we investigate properties of the statistical ensemble E[S], and make some conclusions about S on the basis of investigations of E[S]. In the statistical ensemble E[S] the stochastic properties of objects S states are compensated, whereas the regular ones are accumulated. Thus, investigating the statistical ensemble E[S], we study regular features of states of the objects S. It depends on situation, whether or not we use the probability theory in our investigation. In the nonrelativistic case an application of the probability theory is possible, whereas in the relativistic case it is impossible. Nevertheless, a statistical description in the relativistic case is possible, although it is carried out by other methods, and it is not so informative as it is in the nonrelativistic case.
The statistical ensemble has two important properties:
1. Statistical ensemble E[S] is a continuous dynamic system independently of whether its elements S are dynamic systems, or stochastic systems.
2. In the nonrelativistic case the statistical ensemble can be used for calculations of statistical average of dynamic variables and functions of them.
In the statistical physics the statistical ensemble possesses the second property, which is connected with application of the probability theory. It is common practice to consider the second property of the statistical ensemble to be its main property. In reality, the first property, which takes place always, is the main property of the statistical ensemble, and the statistical description is based on the first property. A use of the second property, when it takes place, makes statistical description to be more informative, because it adds some details to the statistical description. A use of the first property allows one to construct dynamical conception of the statistical description (DCSD), which can be used always (in both relativistic and nonrelativistical cases).
Thus, revising the foundations of geometry, we obtain a class of space-time geometries with stochastically moving particles. Revising conception of statistical description and making it relativistic, we obtain the statistical description of a stochastic particle in the form of a fluidlike dynamic system E[S]. Connection between the fluid E[S] and stochastic particle S is carried out by means of hydrodynamic interpretation. The world lines of the fluid are interpreted as mean world lines of the stochastic particles. The fluid flux jk and the energy-momentum tensor are regarded respectively as the mean flux of stochastic particles and the mean energy-momentum 4-flux density of these stochastic particles. Such an interpretation is more reasonable, than the conventional interpretation in terms of the wave function.
Properties of this fluid E[S] depend on the choice of the space-time geometry, and the space-time geometry must be chosen in such a way, that the hydrodynamic interpretation of the fluid E[S] motion agrees with the experimental data. Thus, the space-time geometry should be chosen on the basis of experimental data. On the other hand, the nonrelativistic quantum phenomena are described very well by the conventional quantum mechanics, i.e. in terms of the wave function. To obtain the true space-time geometry and the true statistical description, it is sufficient that the statistical description would agree with quantum mechanics.
Connection between the hydrodynamics and the Schroedinger equation is well known (Madelung, 1926). Unfortunately, it is a one-way connection. Differentiating the Schroedinger equation, one can obtain hydrodynamic equations for irrotational flow of some quantum fluid. To obtain description of the fluid in terms of the wave function, one needs to integrate hydrodynamic equations. This problem of integration is rather simple, provided the fluid flow is irrotational, but this integration is rather complicated mathematical problem in the general case. We succeeded to solve this problem only in the end of eighties. After solution of the integration problem one can transform description of E[S] in hydrodynamic variables (density, velocity) to description in terms of the wave function and vice versa. Connection between statistical description of stochastic particle and the conventional quantum mechanics has been established. The true space-time geometry has been obtained. The model conception of the quantum phenomena (MCQP) has been constructed in the first approximation.
Why only in the first approximation? Because for determination of the space-time geometry only nonrelativistic experimental data were used. These data influence the distance of the space-time geometry only for rather large distances. It is possible that taking into account relativistic experimental data, we obtain some correction to the space-time geometry, which is effective for the smaller distances and essential only in the relativistical case. One should expect this, because the space-time geometry is the main factor which determines physical phenomena in microcosm.
Obtaining MCQP, we do not use any additional hypotheses. On the contrary, we eliminate quantum principles and other unwarranted hypotheses concerning the space-time geometry. Model conception of quantum phenomena (MCQP) is the second stage in the microcosm investigation. The first stage (the conventional quantum theory) may be qualified as the axiomatic conception of quantum phenomena (ACQP). Two stage we can observe in the investigation of thermal phenomena: the first stage -- axiomatic conception of thermal phenomena (ACTP), known as thermodynamics and the second stage -- model conception of thermal phenomena (MCTP), known as statistical physics.
From pragmatic viewpoint the axiomatic conception and the model conception are distinguished by their investigation methods. (For details see thepaper.) By means of methods of statistical physics one can obtain such results, which cannot be obtained in the framework of thermodynamics. The same situation we have comparing MCQP and ACQP.
Let us forget that MCQP is well founded conception, whereas ACQP is only a compensating conception. Compare MCQP and ACQP from viewpoint of their investigation methods.
Investigation of the dynamic systems SD described by the free Dirac equation by methods of MCQPshows, that classical analog SDcl of the " Dirac particle" SD is a classical dynamic system, having ten degrees of freedom. Six degrees of freedom describe SDcl as a free relativistic particle, whereas four remaining degrees of freedom are internal. Dynamic equations describing these internal degrees of freedom contain the quantum constant h. They can be interpreted as a relativistic rotator, i.e. two particles rotating around their common center of inertia. Radius a of the rotator depends on the quantum constant h. If h ® 0, a ® 0 and the rotator reduces to the pointlike particle with spin and magnetic moment.
Investigation of the same dynamic systems SD by conventional methods of ACQP shows that the classical analog SDcl has six degrees of freedom. It means that SDcl is a pointlike relativistic particle. It is of no importance in this context, what the classical system SDcl is in reality (rotator or pointlike particle). It is important that the investigation methods of MCQP allow one to obtain more subtle and more detailed description, than conventional methods of ACQP.
Another example. Investigation of the dynamic systems SKG described by the free Klein-Gordon equation by methods of MCQP shows that this dynamic system describes relativistic esemlon (collective concept with respect to concepts of 'particle' and "antiparticle), moving in some relativistic force field k . The k-field is connected with the mean value of the stochastic component of the esemlon velocity. The k-field is responsible for the effect of pair production. Methods of MCQP allow one to investigate the properties of the pair production mechanism. Investigation of the dynamic systems SKG by conventional methods of ACQP shows that dynamic system SKG describes a relativistic particle and nothing besides this. Again we see that investigation methods of MCQP are more effective and subtle than those of ACQP.