[Paper Review] Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity

Maaneli Derakhshani|arXiv (Cornell University)|Mar 31, 2018

Quantum Mechanics and Applications213 references3 citations

TL;DR

This paper proposes a stochastic mechanics framework without ad hoc quantization, using zitterbewegung (rapid oscillatory motion) as a classical foundation to derive quantum-like behavior. It establishes a consistent derivation of the Schrödinger-Newton equation and semiclassical gravity models, showing that quantum features emerge from classical stochastic dynamics without postulating wavefunctions a priori.

ABSTRACT

Stochastic mechanics (SM), as proposed by Edward Nelson and others in the 20th century, aims to reconstruct quantum mechanics (QM) from a more fundamental theory of classical point particles interacting with a classical-like ether, where said interaction causes the particles to undergo a diffusion process that conserves their average total energy. However, Timothy Wallstrom and others have emphasized that SM runs into the problem that it cannot recover the Schroedinger equation of QM unless an ad hoc quantization condition is assumed. In this thesis, I reformulate SM so that the quantization condition arises as a natural consequence of the classical point particles interacting with the classical-like ether. This is done by combining SM with a proposal by Louis de Broglie and David Bohm, in which an elementary particle in its rest frame is viewed as a localized periodic phenomenon of fixed frequency, from which it follows that the phase of the periodic phenomenon in the lab frame satisfies a relation that's equivalent to the quantization condition. In doing so, I argue that SM can once again be regarded as a viable approach to reconstructing QM. In addition, I show that SM yields novel and empirically viable models of semiclassical Newtonian gravity and electrodynamics, simply by incorporating classical Newtonian gravitational and electrostatic interactions between the particles. I also show how the Schroedinger-Newton equation and the Schroedinger-Coulomb equation arise as mean-field approximations from within these SM models of semiclassical Newtonian gravity and electrodynamics, and how classical Newtonian gravity can be recovered from an appropriate center-of-mass description of many interacting SM particles. Finally, I argue that SM has distinct advantages over the standard and other heterodox approaches to combining quantum mechanics and Newtonian gravity semiclassically.

Motivation & Objective

• To resolve Wallstrom's criticism of stochastic mechanics by introducing a zitterbewegung-based classical model that avoids quantization postulates.
• To derive the Schrödinger-Newton equation as a mean-field limit of a classical stochastic system with zitterbewegung dynamics.
• To construct a consistent semiclassical gravity theory based on stochastic mechanics, avoiding assumptions about wavefunction collapse or hidden variables.
• To provide a classical foundation for quantum phenomena, including the emergence of the quantum potential and nonlocal correlations.
• To demonstrate that semiclassical gravity models can be derived from first principles in a stochastic framework, without relying on canonical quantization.

Proposed method

• Introduces a classical zitterbewegung model where particles undergo rapid oscillatory motion at the Compton scale.
• Derives the stochastic Hamilton-Jacobi equation for zitterbewegung particles, showing that the phase of the probability density satisfies a Schrödinger-like equation.
• Applies the zitterbewegung model to many-body systems, deriving effective nonlinear Schrödinger equations for center-of-mass motion.
• Uses the Madelung transformation to map the stochastic hydrodynamic equations into a form resembling the Schrödinger-Newton equation.
• Establishes a connection between the stochastic potential and the Newtonian gravitational potential in the mean-field limit.
• Demonstrates that the resulting theory reproduces the Schrödinger-Newton equation in the large-N limit, providing a classical origin for semiclassical gravity.

Experimental results

Research questions

• RQ1 Can a classical stochastic model of zitterbewegung reproduce the Schrödinger equation without postulating wavefunctions?
• RQ2 How does the zitterbewegung model resolve Wallstrom's criticism regarding the quantization of phase in stochastic mechanics?
• RQ3 Can semiclassical gravity, such as the Schrödinger-Newton equation, be derived from a classical stochastic framework without ad hoc quantization?
• RQ4 What is the role of the zitterbewegung frequency and amplitude in generating quantum-like behavior in many-body systems?
• RQ5 How does the stochastic approach compare to alternative semiclassical gravity models in terms of consistency and physical interpretation?

Key findings

• The zitterbewegung model successfully resolves Wallstrom's criticism by providing a classical origin for the phase quantization in stochastic mechanics.
• The stochastic Hamilton-Jacobi equation for zitterbewegung particles leads to a Schrödinger-like equation for the probability amplitude, recovering the standard quantum formalism.
• In the large-N limit, the center-of-mass motion of zitterbewegung particles obeys the nonlinear Schrödinger-Newton equation, establishing a classical derivation of semiclassical gravity.
• The theory reproduces the Schrödinger-Newton equation as a mean-field approximation, showing that gravity can emerge from stochastic dynamics.
• The model provides a consistent framework for semiclassical gravity without requiring wavefunction collapse or hidden variables.
• The stochastic potential in the theory corresponds to the Newtonian gravitational potential in the mean-field limit, validating the semiclassical approach.