A First Principles Derivation of Classical and Quantum Mechanics as the Natural Theories for Smooth Stochastic Paths

TL;DR

This paper derives classical and quantum mechanics from first principles using smooth stochastic processes, offering a geometric interpretation and proposing that quantum phenomena can be modeled with mesoscopic stochastic systems.

Contribution

It provides a first principles derivation of quantum mechanics as a locally realistic hidden variable theory incorporating smooth stochastic fluctuations.

Findings

Classical Hamilton-Jacobi equation derived from stochastic processes.

Quantum mechanics emerges when stochastic fluctuations are included.

Any linear evolution of a density matrix corresponds to a stochastic process.

Abstract

We derive the classical Hamilton-Jacobi equation from first principles as the natural description for smooth stochastic processes when one neglects stochastic velocity fluctuations. The Schr\"{o}dinger equation is shown to be the natural exact equation for describing smooth stochastic processes. In particular, processes with up to quadratic stochastic fluctuations are electromagnetically coupled quantum point particles. The stochastic derivation offers a clear geometric picture for Quantum Mechanics as a locally realistic hidden variable theory. While that sounds paradoxical, we show that Bell's formula for local realism is incomplete. If one includes smooth stochastic fluctuations for the hidden variables, local realism is preserved and quantum mechanics is obtained. Quantum mechanics should therefore be viewed as a "nondeterministic, non-Bell locally realistic hidden variable theory". Since the description is simply a stochastic process, it should be relatively straightforward to create mesoscopic analogue systems that show all the hallmarks of Quantum Mechanics, including super-Bell correlations. In fact, any system that can be described by the linear time evolution of a density matrix is both a stochastic and a Quantum system from our point of view, since we show that the existence of a transition probability density directly implies the existence of a density matrix. Systems for which the stochastic degrees of freedom vary smoothly over time have quantum Hamiltonians with standard kinetic terms.