[Paper Review] Stochastic Theory of Quantum Mechanics

M. Godart|arXiv (Cornell University)|Jun 13, 2012

Statistical Mechanics and Entropy3 citations

TL;DR

This paper proposes a stochastic foundation for non-relativistic quantum mechanics by modeling particle trajectories as sample paths of a Markov diffusion process, with the conditional probability density serving as a substitute for the wave function. It derives the Schrödinger and Nelson equations to determine drift and diffusion coefficients, and shows that wave function collapse is an artifact of initial condition updating, not a physical process.

ABSTRACT

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.

Motivation & Objective

• To reformulate non-relativistic quantum mechanics using stochastic processes as a foundational framework.
• To replace the wave function with a conditional probability density derived from a diffusion Markov process.
• To derive the Schrödinger and Nelson equations from stochastic dynamics and verify consistency with standard quantum mechanics.
• To reinterpret the measurement process, particularly wave function collapse, as a consequence of updating initial conditions in partial differential equations.
• To extend the stochastic framework to relativistic quantum mechanics, though the extension remains incomplete.

Proposed method

• Model particle trajectories as sample functions of a Markov diffusion process, where the conditional probability density replaces the wave function.
• Use the Schrödinger equation and Nelson's stochastic mechanics to determine the drift vector and diffusion tensor of the process.
• Derive the forward and backward Kolmogorov equations to describe the time evolution of the conditional probability density.
• Employ the Fokker-Planck equation to determine the normal (marginal) probability density of the process.
• Apply the framework to simple quantum systems (e.g., free particle, harmonic oscillator) and compare results with standard quantum theory.
• Attempt to generalize the stochastic formalism to relativistic quantum mechanics, though full solutions remain unsolved except for the free particle case.

Experimental results

Research questions

• RQ1 Can non-relativistic quantum mechanics be consistently derived from a stochastic process framework based on Markov diffusion?
• RQ2 How do the Schrödinger and Nelson equations emerge from the dynamics of a stochastic process?
• RQ3 What is the physical interpretation of wave function collapse within this stochastic model?
• RQ4 Why do trajectories in the stochastic model lack time derivatives, and what are the implications for defining velocity and momentum?
• RQ5 Is it possible to extend this stochastic formulation to relativistic quantum mechanics, and what are the main obstacles?

Key findings

• The conditional probability density of the stochastic process serves as a direct substitute for the wave function in quantum mechanics.
• The Schrödinger equation and Nelson's equations are derived as consequences of the stochastic dynamics, linking quantum behavior to diffusion processes.
• In the stochastic model, any system evolving freely from an arbitrary initial state spontaneously returns to its ground state, a behavior not present in standard quantum mechanics.
• Particle velocities and momenta cannot be defined because sample paths of the diffusion process are almost surely nowhere differentiable.
• The so-called collapse of the wave function is explained not as a physical process but as a mathematical consequence of updating initial conditions in the solution of partial differential equations.
• The extension to relativistic quantum mechanics is theoretically promising but remains incomplete, with no general solution to the Nelson and Kolmogorov equations beyond the free particle case.