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

Abstract

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, 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 m continuously undergoes a driven steady-state oscillation of 'zitterbewegung' (zbw ) frequency, ωc = (1/ ) mc 2 , in its instantaneous mean forward (and backward) translational rest frame. With this hypothesis we show that, in the lab frame, S arises from imposing the constraint of conservative diffusions on the time-symmetrized steady-state phase of the 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 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.

Non-Perturbative Guiding Center and Stochastic Gyrocenter Transformations: Gyro-Phase Is the <i>Kaluza-Klein</i> 5<sup>th</sup> Dimension also for Reconciling General Relativity with Quantum Mechanics

“…Several studies concerning the relation of quantum mechanics and stochastic processes are described in books like [34][35][36][37]. Others suggested lectures with many correspondences to the present derivation are in [19] and [38]. However, the present description is novel because it is applied to the gyrocenter, instead of considering the particle motion, when the gyrating particle solution is considered in the presence of e.m. stochastic fluctuations, that has never been studied.…”

“…It is worth noticing that the gyrating particle solution corresponds to the zitter-solution already described in section III (C.1). In fact, it has been recently noticed in [38] that the zittter-solution can overcome the Wallstrom criticism. From (197) they are easily obtained the relations:…”

Stochastic Quantization on Lorentzian Manifolds

“…Such an assumption must be made ad hoc, since the wave function is not a fundamental object in the theory. Several responses against this criticism have been given, such as for example the incorporation of zitterbewegung [47,48], adding a postulate regarding the boundedness of the Laplace operator acting on the probability density [49] or by adding the assumption of unitarity of superpositions of wave functions [34]. It is also worth mentioning that on nodal manifolds winding numbers lead to a periodicity factor in the wave function [11], which could resolve Wallstrom's criticism.…”

Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium

“…Although such (interesting and promising) alternative studies of droplets solve the problem of the appearance of a complex phase in a classical context, it is worth noting that the phenomenological results outlined in section IX, concerning the quantization of droplet orbits in the case of a harmonic potential 9,52 , cannot be explained simply in terms of excited modes of the oil bath, because in these experiments only the droplet undergoes the harmonic potential, the oil bath being 15 See Ref. 67 for an interesting proposal involving a multivalued wave function, also based on Zitterbewegung.…”

“…FIG. 6:Histogram of the positions in x of a single particle, in the case of the first Fock state(67). The full curve (red) corresponds to the quantum probability |Ψ 1 | 2 .…”