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.
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.
This is the first in a two-part series in which we extend non-relativistic stochastic mechanics, in the ZSM formulation [1, 2], to semiclassical Newtonian gravity (ZSM-Newton) and semiclassical Newtonian electrodynamics (ZSM-Coulomb), under the assumption that the gravitational and electromagnetic fields are fundamentally classical (i.e., not independently quantized fields). Our key findings are: (1) a derivation of the usual N-particle Schr{ö}dinger equation for many particles interacting through operator-valued gravitational or Coulomb potentials, and (2) recovery of the `single-body' Schr{ö}dinger-Newton and Schr{ö}dinger-Coulomb equations as mean-field equations valid for systems of gravitationally and electrostatically interacting identical particles, respectively, in the weak-coupling large N limit. We also compare ZSM-Newton/Coulomb to semiclassical Newtonian gravity/electrodynamics approaches based on standard quantum theory, dynamical collapse theories, and the de Broglie-Bohm theory.
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.
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 m continuously undergoes a driven steady-state oscillation of `zitterbewegung' (\emph{zbw}) frequency, 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.
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, 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.
Little is known about dark matter particles save that their most important interactions with ordinary matter are gravitational and that, if they exist, they are stable, slow moving and relatively massive. Based on these assumptions, a semiclassical approximation to the Schrödinger equation under the action of a Coulomb potential should be relevant for modelling their behaviour. We investigate the semiclassical limit of the Schrödinger equation for a particle of mass M under a Coulomb potential in the context of Nelson’s stochastic mechanics. This is done using a Freidlin-Wentzell asymptotic series expansion in the parameter ϵ=ħ/M for the Nelson diffusion. It is shown that for wave functions ψ ∼ exp((R + iS)/ϵ2) where R and S are real valued, the ϵ = 0 behaviour is governed by a constrained Hamiltonian system with Hamiltonian Hr and constraint Hi = 0 where the superscripts r and i denote the real and imaginary parts of the Bohr correspondence limit of the quantum mechanical Hamiltonian, independent of Nelson’s ideas. Nelson’s stochastic mechanics is restored in dealing with the nodal surface singularities and by computing (correct to first order in ϵ) the relevant diffusion process in terms of Jacobi fields thereby revealing Kepler’s laws in a new light. The key here is that the constrained Hamiltonian system has just two solutions corresponding to the forward and backward drifts in Nelson’s stochastic mechanics. We discuss the application of this theory to modelling dark matter particles under the influence of a large gravitating point mass.
A consistent theory including gravitational effects in quantum mechanics has not been found. Although many different approaches have been proposed for theory of quantum gravity, several authors have suggested that gravity may be fundamentally classical. As there is no experimental evidence for a fundamentally quantum vs. fundamentally classical theory of gravity, it is hoped that investigating the consequences of semiclassical gravity, one may propose an experiment to distinguish between the quantum or classical nature of gravity. As opto- and electromechanical devices allow quantum limited measurement and control over massive, mechanical degrees of freedom, they are ideal systems to test violations of standard quantum mechanics arising from gravitational effects. After a brief introduction to the background theory and literature review, we begin by introducing a semiclassical model of gravity based on quantum measurement and control. We show under which conditions the theory behaves like a collapse model, and how a Newtonian potential is recovered in the multi particle case.We then go on to analyse our semiclassical model in two scenarios: firstly in a cosmological setting, and secondly by considering quantum clocks on the surface of the earth. In the cosmological description, we show that a violation of energy conservation, resulting from our model, can be reinterpreted as a dark energy fluid in the spacetime, suggesting that perhaps local violations of conservations of energy may be resolved in a cosmological framework. Secondly, we derive a minimum, gravitational dephasing rate for quantum clocks under the semiclassical model, which unsurprisingly, is is well below the resolution of current state of the art clocks. The next chapters diverge from analysing the gravitational model, and instead focus on novel methods of manipulating and measuring micromechanical devices, that may be used to experimentally probe the results of the semi-classical theory. Firstly we show that a sequence of coherently controlled pulses in an optomechanical system can be used to directly probe the momentum of an oscillator - opening up the possibility for direct detection of momentum diffusion, which is a consequence of the semiclassical gravity model. Secondly, we introduce an electro-mechanical device, and sequence of control operations, that may be understood as an interferometer for the wave function of a mechanical oscillator coupled to a qubit. We show how the resulting interference pattern can distinguish between unitary evolution, thermalisation/decoherence, and a purely classical description of the mechanical oscillator. We also show how the qubit-oscillator interaction can be used generate arbitrary wave functions of the mechanical oscillator, using only classical qubit driving.