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Stochastic Theory of Quantum Mechanics

The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.


Introduction
The questions on the underlying reality behind the laws and procedures of quantum mechanics have been a battlefield over the entire period since its initial development, and no truce is yet in sight despite an extensive literature that refers to, discusses and criticizes the orthodox Copenhagen interpretation.
On the one hand its defenders claim that the validity of the theory is justified by the many successes of its predictions and applications. In our opinion this is not at all a convincing argument, because many examples mainly from the past show that a solidly founded theory is not always necessary to arrive at spectacularly useful results, as attested by the fact that empiricism has always been a powerful method ( 1 ).
On the other hand its opponents wonder if it was absolutely necessary to introduce the notions of duality, complementarity and contraction of the wave packet. Moreover the essential role played by the process of measurement is a hotly contested point, because it is in opposition with the idea that a physical theory must provide an objective description of the phenomena observed.
The inspiration for this work came from two important publications. The first of them is a book written by Landé ( 2 ). It is an incisive critique of the Copenhagen interpretation of quantum mechanics, but certainly not of its results. Landé significantly claims that "...if Bohr and Heisenberg are severely criticized in this [book] for their subjectivist approach, their language refinement, and their occasional deviation from logic and consistency, it will hardly be necessary to add that my admiration for their achievements in theoretical physics is as great as ever. But I must confess that, during many years of trying to imbue my students with the Copenhagen Spirit, I felt more and more like the Devil' s Advocate, suffering from an ever-growing intellectual distress".
The book starts with a formidable thunderbolt in a cloudless blue sky. Landé reports the existence of a purely mechanical explanation for the electron diffraction experiment due to Duane ( 3 ). The resulting rule states that the angles of reflection of particles (or photons) arriving with momentum p on a crystal with parallel planes spaced at distance L are selected in agreement with the rule: Duane rule is as legitimate as the two familiar old quantum rules for the energy E and  for the angular momentum M, namely Planck rule  T  h  E     for bodies characterized by  a periodicity in time with period T (harmonic oscillator) and Sommerfeld-Wilson rule        2  h  M for bodies periodic with respect to rotations through  2 (any body). The question thus arises to know whether duality would have been invented if Duane rule had been taken into account in the pioneering years of quantum mechanics. Let us also mention that the results of the famous two slits experiment can be explained mechanically by a further development of Duane ideas applied to the theory of diffraction ( 1 ).
We must however confess that Landé raises many pertinent questions, but does not offer satisfactory answers. Quite fortunately a global solution was suggested in a second publication written by Nelson ( 2 ). who presents a plausible interpretation of quantum theory At the risk of contributing to the confusion reigning supreme in this domain, this paper will defend this interpretation. In our opinion the solution to the problem of interpretation consists in providing some form of plausible picture. This can be achieved by purely mathematical solutions as offered by the stochastic theory of quantum mechanics. As its name suggests it relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function.
The present work can be seen as a presentation to other interested people of attempts to understand the theory of Nelson, to explain it to anybody so to say, and to extend it as far as possible in a coherent view of the quantum world.
This work pertains clearly to the domain of mathematical physics. A compromise inconsistent as most compromises are had to be made because this work on the one hand calls for a rigorous treatment guaranteeing solid foundations, but on the other hand is threatened with the charge of being absurdly overloaded with extraneous higher mathematics. This explains the fact that many proofs are (over-)simplified or even completely left aside. On the contrary others can appear to be too much detailed. Another (personal) reason is that we hate expressions like "it is well-known that" and we tried therefore to make this work self contained as far as possible. The complete text with its 190 pages is much too long to be published here and we must be satisfied by presenting here a detailed table of contents of it.

Probability concepts
This chapter presents the basic concepts about probability, conditional probability, probability densities, conditional probability densities, mathematical expectations and conditional mathematical expectations applied to the random variables of stochastic processes that are absolutely necessary for the reading and understanding of the complete text. The readers who are already experts in those domains can safely go to the next chapter.
3 From the point of view of the expert, probability is simply a branch of measure theory, with its own special emphasis and fields of application. There is a familiar inclination on the part of those who possess unusual and arduously acquired expertise to exaggerate its remoteness from anything the rest of us know, but we must warn you that there is perhaps no mathematical subject that shares with probability the characteristic features that on the one hand many of its most elementary theorems are based on deep mathematics, and on the other hand that many of its advanced theorems are often considered as "intuitively obvious" by many people who are ready to accept them without proofs.
We present the probability concepts first very briefly in the general case where a dis- gives the probability to find the value of the random variable x within the indicated interval, and then with more details in the simpler case where there exists a probability density function   x p such that we can write: The (basic) concept of mathematical expectation is then defined. All the points presented up to here are easily generalized to the case of several random variables.
The (less basic) concepts of conditional probability and conditional mathematical expectation are then presented, first briefly in the general context where a distribution func- gives the probability to find the value of the random variable x within the indicated interval under the additional circumstance that the condition M must be fulfilled, and then with more details in the simpler context where there exists a prob- such that we can write: The main properties of the normal or conditional probability densities and mathematical expectations are proved and generalized to the case of several random variables.
The important special case of so-called independent random variables is finally examined in all details necessary for the future developments.

Stochastic processes.
The subject of the first chapter can seem futile to the experts. We are however convinced that the theory of probability as taught at the University often serves exclusively to introduce statistics with its confidence intervals, Bayesian rules, Gaussian distribution and all the like. The theory of stochastic processes if ever addressed is severely restricted to the domain of the Poisson distributions and of the Markov chains.

4
This chapter defines roughly a stochastic process as being a random variable that depends on a parameter t usually identified with the time. The special case of a Wiener process is examined in details as an exercise and because of its importance for the future.
General Markov processes are defined as those obeying the Chapman-Kolmogorov relation, and the diffusion Markov processes are then introduced as those special cases characterized by the existence of a drift velocity v and a diffusion tensor w. It is proved that the probability density   t , x p satisfies a partial differential equation called the Fokker-Planck equation and that the conditional probability density   0 0 1 1 t , x t , x p verifies two partial differential equations, one for each pair of variables x and t, named after Kolmogorov who was the first to discover them. The natural initial condition: is proposed for it. This is easily extended from the case of a scalar stochastic process to the case of a vector stochastic process.
It is shown that a diffusion Markov process can be equivalently defined as the solution of a particular system of difference or differential stochastic equations. This point of view has its own merit because it sheds light on curious properties of the trajectories and it leads to uncertainty relations. At this point, mean velocities and stochastic derivatives with respect to t of a stochastic process are introduced as a replacement for the classical velocities and derivatives.
A Markov process can be defined otherwise by the fact that its conditional probability densities must satisfy the relations: for any value of n and for any increasing sequence n 2 1 t t t     of values for the parameter t. This condition that makes play a role to the direction of time thus seems to introduce an element of irreversibility but this is far from true because a Markov process also satisfies also the relations: for the same value of n and for the same sequence of values for t. This property is expressed by saying that the process is reversible, and this implies that the definitions of drift velocities, diffusion tensors, and that the partial differential equations of Fokker-Planck and of Kolmogorov can be written twice, in a forward version generally using indices + when the direction of time is from the present to the future and in a backward version generally using indices -when the direction of time is from the present to the past.
So for example we must consider simultaneously the two definitions of the drift velocities: the two first Kolmogorov equations:  Perhaps unexpectedly, the forward and backward versions of the drift velocities and of the diffusion tensors are not completely independent. We have indeed x v where we have denoted by ij w the common value of ij ij w and w  .
It is very easy to show that the auxiliary vector i u defined by: x p and that the auxiliary vector i v defined by: come into play in the conservation equation: x p the probability current density corresponding to the probability density   t , x p .
A so-called H-theorem proves that the solutions of the Kolmogorov equations always converge to the solution of the Fokker-Planck equation that can thus be considered to represent the ground state of the system. Some of the results obtained previously are used in the study of stochastic variational principles.
The special case of the stationary processes is considered at length in view of many future applications.

Non-relativistic stochastic quantum mechanics.
This chapter exhibits the important relations existing between the orthodox quantum theory and the theory of the diffusion Markov processes. We first start from general properties and relations of those processes to deduce the so-called Nelson first equation involving the two vectors u and v and the tensor w. We then apply a stochastic variational principle to a Lagrangian that is the normal counterpart of the classical Lagrangian for a particle with mass m and charge e subjected to an electric potential V and a magnetic potential A and this leads to the so-called Nelson second equation involving also the two vectors u and v and the tensor w. Now Schrödinger equation is transformed in a succession of two changes of variables into a system of two equivalent equations that are identical to the Nelson equations provided that we accept the identity: where ij g is the metric tensor in the space of the vectors x . This procedure unfortunately gives the impression that there exist different pairs of vectors u and v corresponding to different solutions of the Schrödinger equation and particularly to its different particular solutions when the method of separation of variables applies to it (1). This is unpleasant for two reasons at least. First this leads to the objection that the solutions determine the equations, and not the converse, and second it turns out that all pairs of vectors u and v possess poles, except the one corresponding to the ground state. We can however tackle the problem entirely from the side of diffusion Markov processes and Nelson equations. With the proposed form of the tensor w, those latter equations represent a system of two partial differential equations in the two unknown vectors u and v that we can try to solve 7 for a more or less unique solution, by excluding all those that are not sufficiently regular. The selected pair of vectors u and v is then introduced in the Kolmogorov equations and we can try to solve those in their turn, by taking into account the initial condition eventually.
We shall demonstrate the merits of this method by applying it to the harmonic oscillator that is characterized by the potential: or equivalently by the force: for some positive constant value of k. In this case the Nelson equations for the unknown (one-dimensional) vectors u and v are:    The method of separation of the variables shows that the first Kolmogorov equation is equivalent to the system of the two equations: We see also that the second Kolmogorov equation is equivalent to the system of the two equations: where the dot symbol denotes the derivation with respect to the time variable t and where the prime symbol denotes the derivation with respect to the space variable x. We can easily verify that the functions: