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Stochastic representation of quantum mechanics and entangled solitons

Yuri P. Rybakov

Published Jan 1, 2026• 0 citations
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Abstract

Background Using the Einstein's idea on particles-solitons, the stochastic representation of the wave function as a large sum of solitons with random phases is suggested.Purpose To illustrate the main idea of the stochastic representation, let us recall the lectures by N. Wiener on nonlinear problems in random theory.Method In these lectures Wiener used the so-called 𝛼-representation of the wave function for a nonrelativistic particle in three-dimensional space, the wave function being considered as an element of the random Hilbert space with the Gaussian dispersion.Results This representation appears to be equivalent, in accordance with the central limiting theorem, to the sum of many complex solitons with random phases.Conclusions On the basis of this representation it is possible to substantiate the Born's rule for measuring physical observables and also to explain the "spin-statistics" correlation in quantum mechanics.

DOI
10.22363/2658-4670-2026-34-2-253-259

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Background Using the Einstein's idea on particles-solitons, the stochastic representation of the wave function as a large sum of solitons with random phases is suggested.Purpose To illustrate the main idea of the stochastic representation, let us recall the lectures by N. Wiener on nonlinear problems in random theory.Method In these lectures Wiener used the so-called 𝛼-representation of the wave function for a nonrelativistic particle in three-dimensional space, the wave function being considered as an element of the random Hilbert space with the Gaussian dispersion.Results This representation appears to be equivalent, in accordance with the central limiting theorem, to the sum of many complex solitons with random phases.Conclusions On the basis of this representation it is possible to substantiate the Born's rule for measuring physical observables and also to explain the "spin-statistics" correlation in quantum mechanics.
Starting with the projective interpretation of the Hilbert space special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is considered with the aim to model quantum states via classical computer. Entangled solitons construction having been earlier introduced in the nonlinear spinor field model for the calculation of the Einstein-Podolsky-Rosen (EPR) spin correlation for the spin-1/2 particles in the singlet state, the other example is now studied. The latter concerns the entangled envelope solitons in Kerr dielectric with cubic nonlinearity, where we use two-solitons configurations for modeling the entangled states of photons. Finally, the concept of stochastic qubits is used for quantum computing modeling.
Special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is suggested with the aim to model quantum states via classical computer. Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein, Podolsky, Rosen (EPR) spin correlation is calculated and shown to coincide with the quantum mechanical one for the spin-1/2 particles in the singlet state. The concept of stochastic qubits is used for quantum computing modelling.
Stochastic realization of the wave function in quantum mechanics with the inclusion of soliton representation of extended particles is discussed. Two-soliton configurations are used for constructing entangled states in generalized quantum mechanics dealing with extended particles, endowed with nontrivial spin S. Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein-Podolsky-Rosen (EPR) correlation is calculated and shown to coincide with the quantum mechanical one for the 1/2-spin particles. The concept of stochastic qubits is used for quantum computing modeling.
We discuss the statistical origin of quantum theoretical description and show that within the scope of the Mie-Einstein field approach, with particles being considered as lumps of some material field satisfying nonlinear equations, such a description appears to be natural and inevitable since it is impossible to specify initial conditions for the extended particle-soliton. Beside this, it is shown that on solitons’ basis, the special stochastic representation of quantum mechanics can be constructed, the latter one being equivalent to the theory of nonlocal hidden variables.
This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.