Mean Derivatives, Nelson’s Stochastic Mechanics, and Quantization

Francesco Guerra

Akira Ohsumi

In this paper, the Schrödinger equation in quantum mechanics is derived as stochastic quantization from the stochastic optimal control theory based on the Nelson stochastic mechanics.

Steven M. Moore

Tomasz Zastawniak

Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.

Yuri E. Gliklikh

Maaneli Derakhshani

Wallstrom's criticism of existing formulations of stochastic mechanics is that they fail to derive the empirical predictions of orthodox quantum mechanics because they require an ad hoc quantization condition on the postulated velocity potential, \emph{S}, in order to derive Schr\odinger wave functions. We propose an answer to this criticism by modifying the Nelson-Yasue formulation of non-relativistic stochastic mechanics for a spinless particle with the following hypothesis: a spinless Nelson-Yasue particle of rest mass $m$ continuously undergoes a driven steady-state oscillation of `zitterbewegung' (\emph{zbw}) frequency, $\omega_{c}=\left(1/\hbar\right)mc^{2}$, in its instantaneous mean forward (and backward) translational rest frame. With this hypothesis we show that, in the lab frame, \emph{S} arises from imposing the constraint of conservative diffusions on the time-symmetrized steady-state phase of the \emph{zbw} particle, satisfies the required quantization condition, and evolves in time by the Hamilton-Jacobi-Madelung equations (when generalized to describe a statistical ensemble of \emph{zbw} particles). From the mathematical equivalence of Schr\odinger's equation with the Hamilton-Jacobi-Madelung equations plus the quantization condition, Schr\odinger wave functions for a spinless particle (in and excluding external fields) are thereby recovered. We also apply this `zitterbewegung stochastic mechanics' (ZSM) to the case of a central potential and show that it predicts angular momentum quantization. This paper sets the foundation for Part II, which will (primarily) work out the many-particle version of ZSM.