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arXiv:0812.5116

Quantization as Asymptotics of Diffusion Processes in the Phase Space

E. M. Beniaminov

Published Dec 30, 2008• 1 citation
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Abstract

This work is an extended version of the paper arXiv:0803.2669v1[math-ph], in which the main results were announced. We consider certain classical diffusion process for a wave function on the phase space. It is shown that at the time of order $10^{-11}$ {\it sec} this process converges to a process considered by quantum mechanics and described by the Schrodinger equation. This model studies the probability distributions in the phase space corresponding to the wave functions of quantum mechanics. We estimate the parameters of the model using the Lamb--Retherford experimental data on shift in the spectrum of hydrogen atom and the assumption on the heat reason of the considered diffusion process. In the paper it is shown that the quantum mechanical description of the processes can arise as an approximate description of more exact models. For the model considered in this paper, this approximation arises when the Hamilton function changes slowly under deviations of coordinates, momenta, and time on intervals whose length is of order determined by the Planck constant and by the diffusion intensities.

DOI
10.48550/arxiv.0812.5116

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This work is an extended version of the paper arXiv:0803.2669v1[math-ph], in which the main results were announced. We consider certain classical diffusion process for a wave function on the phase space. It is shown that at the time of order 10−1110^{-11}10−11 {\it sec} this process converges to a process considered by quantum mechanics and described by the Schrodinger equation. This model studies the probability distributions in the phase space corresponding to the wave functions of quantum mechanics. We estimate the parameters of the model using the Lamb--Retherford experimental data on shift in the spectrum of hydrogen atom and the assumption on the heat reason of the considered diffusion process. In the paper it is shown that the quantum mechanical description of the processes can arise as an approximate description of more exact models. For the model considered in this paper, this approximation arises when the Hamilton function changes slowly under deviations of coordinates, momenta, and time on intervals whose length is of order determined by the Planck constant and by the diffusion intensities.
We consider the process of diffusion scattering of a wave function given on the phase space. In this process the heat diffusion is considered only along momenta. We write down the modified Kramers equation describing this situation. In this model, the usual quantum description arises as asymptotics of this process for large values of resistance of the medium per unit of mass of particle. It is shown that in this case the process passes several stages. During the first short stage, the wave function goes to one of "stationary" values. At the second long stage, the wave function varies in the subspace of "stationary" states according to the Schrodinger equation. Further, dissipation of the process leads to decoherence, and any superposition of states goes to one of eigenstates of the Hamilton operator. At the last stage, the mixed state of heat equilibrium (the Gibbs state) arises due to the heat influence of the medium and the random transitions among the eigenstates of the Hamilton operator. Besides that, it is shown that, on the contrary, if the resistance of the medium per unit of mass of particle is small, then in the considered model, the density of distribution of probability ρ=∣ϕ∣2ρ=|ϕ|^2ρ=∣ϕ∣2 satisfies the standard Liouville equation, as in classical statistical mechanics.
We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker--Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schrödinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schrödinger equation taking into account dissipation of the process in the initial model.
We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker-Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schrodinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schrodinger equation taking into account dissipation of the process in the initial model, and explaining the decoherence of the wave function.
A diffusion process for charge distributions in a phase space is examined. The corresponding charge moves in a force field and under an action of a random field. There are the diffusion motions for coordinates and for momenta. In our model, an inner state of the charge is defined by a complex vector. The vector rotates with a great constant angular velocity with respect to the proper time of the charge. A state of the diffusion process is a (complex-valued) wave function on the phase space. As in quantum mechanics, we assume that, for the wave functions, the superposition principle holds. The diffusion process averages out vectors of inner states from different points of the phase space. A differential equation for this diffusion process is founded and examined. We demonstrate that the motion (described by this process) decomposes into a fast motion and a slow motion. The fast motion reduces an arbitrary wave function to a function from a subspace whose elements are parameterized by complex-valued functions of coordinates. The slow motion occurs in this subspace and it is described by the Schr\''odinger equation. The parameters of the suggested model are estimated. The duration of the fast motion is of order 10−1110^{-11}10−11 s.
In the paper, we discuss the studies of mathematical models of diffusion scattering of waves in the phase space, and relation of these models with quantum mechanics. In the previous works it is shown that in these models of classical scattering process of waves, the quantum mechanical description arises as the asymptotics after a small time. In this respect, the proposed models can be considered as examples in which the quantum descriptions arise as approximate ones for certain hypothetical reality. The deviation between the proposed models and the quantum ones can arise, for example, for processes with rapidly changing potential function. Under its action the diffusion scattering process of waves will go out from the states described by quantum mechanics. In the paper it is shown that the proposed models of diffusion scattering of waves possess the property of gauge invariance. This implies that they are described similarly in all inertial coordinate systems, i.e., they are invariant under the Galileo transformations. We propose a program of further research.
The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.