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arXiv:2301.05467

Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion

Folkert Kuipers

Published Jan 13, 2023• 0 citations
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Abstract

We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.

DOI
10.48550/arxiv.2301.05467

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We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.
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We prove a theorem showing that quantum mechanics is not directly a stochastic process characterizing Brownian motion but rather its square root. This implies that a complex-valued stochastic process is involved. Schrödinger equation is immediately derived without further assumptions using Itō integrals that are properly generalized. Fluctuations in space arise from a Brownian motion and the combined effect of a stochastic process with a symmetric Bernoulli distribution typical of tossing a coin.
A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.
where m is the particle mass at rest, c is the speed of light, p is the particle momentum and U is a potential. The root in Eq. (1) complicates substantially the way of quantizing the relativistic motion. In the literature there are two important solutions of this problem: the Klein-Gordon and Dirac equations. Their complex mathematical structures and some problems with definition of the probability density make it difficult to extend these equations for description of relativistic quantum Brownian motion. Moreover, it is not clear if the non-relativistic energy ˆ t E i   and momentum p i   operators, used in quantizing of Eq. (1), are valid in the relativistic quantum mechanics as well. The present paper proposes an alternative approach via a relativistic Wigner-Klein-Kramers equation. The quantum Brownian motion is described by the Madelung quantum hydrodynamics [4] coupled to the Breit-Fermi Hamiltonian. According to the relativistic dynamics the stochastic Langevin equation for a classical relativistic particle moving in a classical non-relativistic environment acquires the form
Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.