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arXiv:1701.06893

Semiclassical Newtonian Field Theories Based On Stochastic Mechanics I

Maaneli Derakhshani

Published Jan 17, 2017• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1701.06893

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Journal of Mathematical Physics•May 1, 2016•1 citation
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 NNN-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.
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.
Semiclassical gravity, in which a classical spacetime is sourced by the quantum expectation value of the stress-energy tensor, is a standard framework for describing the gravitational interaction of quantum matter. In the nonrelativistic limit this approach leads to the Schrödinger-Newton equation, which is often assumed to be consistent at least in the weak-field regime. In this work, we reexamine this assumption for spatial quantum superpositions of massive particles. We show that, when the quantum state is properly normalized, no modification of the Newtonian gravitational potential arises at leading order. However, at first post-Newtonian order the semiclassical coupling generically produces state-dependent contributions involving the mass density and the mass current of the superposition. These terms have a parametric scaling which is different from that of the corresponding relativistic corrections and which does not have Planck mass suppression. Our results therefore impose a strong post-Newtonian consistency constraint on deterministic semiclassical gravity, indicating that sourcing the metric solely by expectation values is insufficient to recover a consistent relativistic weak-field expansion.
We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.
The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.
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.