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arXiv:1607.08838

A Suggested Answer To Wallstrom's Criticism: Zitterbewegung Stochastic Mechanics II

Maaneli Derakhshani

Published Jul 29, 2016• 1 citation
arXivHTMLPDF

Abstract

The "zitterbewegung stochastic mechanics" (ZSM) answer to Wallstrom's criticism, introduced in the companion paper [1], is extended to many particles. We first formulate the many-particle generalization of Nelson-Yasue stochastic mechanics (NYSM), incorporating external and classical interaction potentials. Then we formulate the many-particle generalization of the classical zitterbewegung \emph{zbw} model introduced in Part I, for the cases of free particles, particles interacting with external fields, and classically interacting particles. On the basis of these developments, ZSM is constructed for classically free particles, as well as for particles interacting both with external fields and through inter-particle scalar potentials. Throughout, the beables of ZSM (based on the many-particle formulation) are made explicit. Subsequently, we assess the plausibility and generalizability of the\emph{zbw} hypothesis. We close with an appraisal of other proposed answers, and compare them to ZSM.

DOI
10.48550/arxiv.1607.08838

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The "zitterbewegung stochastic mechanics" (ZSM) answer to Wallstrom's criticism, introduced in the companion paper [1], is extended to many particles. We first formulate the many-particle generalization of Nelson-Yasue stochastic mechanics (NYSM), incorporating external and classical interaction potentials. Then we formulate the many-particle generalization of the classical zitterbewegung \emph{zbw} model introduced in Part I, for the cases of free particles, particles interacting with external fields, and classically interacting particles. On the basis of these developments, ZSM is constructed for classically free particles, as well as for particles interacting both with external fields and through inter-particle scalar potentials. Throughout, the beables of ZSM (based on the many-particle formulation) are made explicit. Subsequently, we assess the plausibility and generalizability of the\emph{zbw} hypothesis. We close with an appraisal of other proposed answers, and compare them to ZSM.
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 mmm continuously undergoes a driven steady-state oscillation of `zitterbewegung' (\emph{zbw}) frequency, ωc=(1/ℏ)mc2\omega_{c}=\left(1/\hbar\right)mc^{2}ωc​=(1/ℏ)mc2, 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.
Interaction is so ubiquitous that imaging a world free from it is a difficult fantasy exercise. At the same time, in understanding any complex physical system, our ability of accounting for the mutual interaction of its constituents is often insufficient when not the restraining factor. Many strategies have been devised to control particle-particle interaction and explore the diverse regimes, from weak to strong interaction. Beautiful examples of these achievements are the experiments on Bose condensates, or the recent experiments on the dynamics of spin chains. Here I introduce another possibility, namely replacing the particle-particle interaction with an external stochastic field, and once again reducing the dynamics of a many-body system to the dynamics of single-particle systems. The theory is exact, in the sense that no approximations are introduced in decoupling the many-body system in its non-interacting sub-parts. Moreover, the equations of motion are linear, and no unknown external potential is inserted.
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 Schrödinger 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 mmm continuously undergoes a driven steady-state oscillation of `zitterbewegung' (\emph{zbw}) frequency, ωc=(1/ℏ)mc2ω_{c}=\left(1/\hbar\right)mc^{2}ωc​=(1/ℏ)mc2, 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 Schrödinger's equation with the Hamilton-Jacobi-Madelung equations plus the quantization condition, Schrödinger 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.
Continuing the development of the ZSM-Newton/Coulomb approach to semiclassical Newtonian gravity/electrodynamics [1], we formulate a ZSM-Newton/Coulomb version of the large N approximation scheme proposed by Oriols et al. [2]. We show that this new large N scheme makes it possible to self-consistently describe the center-of-mass evolution of a large number of gravitationally/electrostatically interacting, identical, \emph{zbw} particles, without assuming that the particles are weakly coupled, and without entailing the problematic macroscopic semiclassical gravitational/electrostatic cat states characteristic of the mean-field Schr{ö}dinger-Newton/Coulomb equations. We also show how to recover N-particle classical Newtonian gravity/electrodynamics for many gravitationally/electrostatically interacting macroscopic particles (composed of many interacting \emph{zbw} particles), as well as classical Vlasov-Poisson mean-field theory for macroscopic particles weakly interacting gravitationally/electrostatically. Finally, we outline an explicit model of environmental decoherence that can be incorporated into Oriols et al.'s scheme as applied to ZSM-Newton/Coulomb.