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In this paper we suggested a natural interpretation of the de Broglie-Bohm quantum potential, as the energy due to the oscillating electromagnetic field (virtual photon) coupled with moving charged particle. Generalization of the Schrödinger equation is obtained. The wave function is shown to be the eigenfunction of the Sturm-Liouville problem in which we expand virtual photon to include it implicitly into consideration. It is shown that the non-locality of quantum mechanics is related only with virtual photon. As an example, the zero-energy of harmonic oscillator is obtained from classical equations.

Interpretation of quantum theory (both classical and relativistic) since its birth to the present day is the subject of much debate on the fundaments of quantum physics. At first glance, the situation is reassuring, because now we have dozen different formulations of quantum theory [

It is well known, the Bohmian formulation based on Louis de Broglie’s pilot-wave theory suggests particularly a great conceptual advantage in possible interpretation because it is causal and not local. Finally it leads to the complete Hamilton function that contains the quantum potential, which reflects fundamental properties of nature (see [

The quantum potential plays a central role in the formalism of Bohm and is used in modern physics and theoretical chemistry. It is of great importance because on the one hand the Bohmian formulation and quantum potential allows us to better understand the foundations, and fundamentals of theory. On the other hand quantum potential has multiples practical applications in different fields of knowledge (for example in the solid state physics, in theoretical chemistry etc.) because it gives us an opportunity to simulate different quantum effects without the involvement of the wave functions of system, and without solving the Schrödinger equation. In this case, the Monte Carlo method is applied to the hydrodynamic calculations, which are sometimes the only possibility to get the result, when the Schrödinger equation cannot be solved exactly.

Unfortunately up to now we did not understand clearly the nature of quantum potential and wave functions. This problem on the one hand provokes many speculations and strange hypothesis, but on the other hand this misunderstanding makes it difficult to progress in important areas such as quantum computers, solid state and many others. These fundamental questions were debated by many authors from the beginning of quantum theory. As an example we quote here the paper [

As it was mentioned above, this misunderstanding provokes many strange and exotic ideas and papers, where quantum potential is used for speculations, particularly in cosmology to construct the most intrigue entity- cosmological constant. For this reasons it is of great importance to reveal physical sense and nature of quantum potential, and to determine the specific role of the wave function in the formalism of quantum mechanics.

Recently the paper [

Another paper in which a natural origin of quantization was suggested was published by Garcia-Morales [

These results also suggest that the orthodox formulations of quantum theory [

In present paper the quantum potential is shown to be formed by bounded electromagnetic field-virtual photon (see section 3 for details), which is a principle part and main participant for any bounded quantum system.

Usually quantum potential in the Bohmian formulation of quantum theory is defined this way (we consider here one-particle case because that for many particles is treated the same manner). Schrödinger equation is

Writing the wave function in form

and

The first one is a Hamilton-Jacobi equation written for a modified Hamiltonian:

where

It is clear, our system should contain an oscillating electromagnetic field produced by electron, but we do not see it in Equation (1). At the same time the Bohmian formulation has the following features:

・ there is presence of hidden variables (it should be treated as a hint for presence of bounded electromagnetic field);

・ it is causal (so, it should be a classical field theory);

・ not local (presence of an electromagnetic field in theory).

Taking into account the fact that the only real substance we have in our arsenal to capture experimental events is electromagnetic field (this is an experimental fact), we can conclude that this field we lost in the beginning. This “lost” field was found in [

Recently [

Action of the system under consideration one can write as:

Here

is the interaction between electromagnetic field and charge(s), and

is action for electromagnetic field. It can be seen that both

It should be stressed here, the Bohmian formulation is non-relativistic one by origin, so we may restrict our speculation by non-relativistic case. Let us consider the hydrogen atom as an example, where to simplicity sake we believe

The classical energy equation for our reduced classical system

Here integration is carried out over 4-volume, and Hamiltonian

Here we sum only over n, and index

These expressions actually are complete “quantum” non-local (integrated over the photon volume) equations to describe our system in Minkowsky space, with clearly written non-local hidden variables (beables) of the virtual photon (which are the coefficients

From this relation we immediately obtain the Hamilton-Jacobi Equation (2) written for a complete Hamiltonian of our system (4), where “quantum potential” now has clear sense and must be attributed to presence of the virtual photon (electromagnetic “pilot-wave”). As to the continuity Equation (3) written for

where

To conclude this part it should be useful to make some comments on the Hamilton-Jacobi Equation (2)

Now, when the physical sense of “quantum potential” (as classical potential that corresponds to the oscillating electromagnetic field with energy

is zero). In this case we obtain a classical system with classical Hamilton function

Very fundamental and at the same time useful example suggests harmonic oscillator. We have discussed it from this point of view before in [

Substituting coherent state wave function

which should be recognized as energy of virtual photon in zero-state of harmonic oscillator (remember here the frequency of electron oscillations is the same that has the virtual photon). So, the total Hamilton function for “quantum” (in reality classical) harmonic oscillator in ground state is

with the oscillation frequency of electron (and virtual photon)

One can see again―the total Hamilton function corresponds to the complete mechanical system classical by nature which does not contain the hidden variables. And so called “quantum potential” appears due to the presence of virtual photon with frequency

In light of these results it becomes immediately obvious meaning of Bell’s theorem, as a classical statement about the impossibility of motions with a speed faster than light in the framework of the relativistic theory.

Conclusions of our work can be formulated as follows:

1) The quantum potential is shown to be an additional energy, electromagnetic by origin, which appears due to the presence of coupled harmonic electromagnetic field (virtual photon).

2) The wave functions are shown to be just a complete basis of the Sturm-Liouville problem (written for reduced system with action

3) It is stressed that the non-locality of quantum mechanics is related only with this virtual photon, namely with distribution of harmonic electromagnetic field in the system under consideration.

Anton Lipovka, (2016) Nature of the Quantum Potential. Journal of Applied Mathematics and Physics,04,897-902. doi: 10.4236/jamp.2016.45098