Semiclassical Newtonian Field Theories Based On Stochastic Mechanics I

TL;DR

This paper extends stochastic mechanics to derive semiclassical Newtonian gravity and electrodynamics, obtaining many-particle Schr{"o}dinger equations and mean-field limits, and compares these to other quantum-classical approaches.

Contribution

It introduces a stochastic mechanics framework for semiclassical gravity and electrodynamics, deriving key equations and analyzing their relation to existing theories.

Findings

Derived N-particle Schr{"o}dinger equations with classical fields.

Recovered Schr{"o}dinger-Newton and Schr{"o}dinger-Coulomb equations as mean-field limits.

Compared the stochastic mechanics approach to other quantum-classical theories.

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{\"o}dinger equation for many particles interacting through operator-valued gravitational or Coulomb potentials, and (2) recovery of the `single-body' Schr{\"o}dinger-Newton and Schr{\"o}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.

Equations

$$d{\mathbf{q}}_i(t)={\mathbf{b}}_i(q(t),t)dt+d{\mathbf{W}}_i(t).$$

$$E_t\left[d{\mathbf{W}}_{in}(t)d{\mathbf{W}}_{im}(t)\right]=2{\nu }_i{\delta }_{nm}dt,$$

$${\nu }_i:=\frac{\hslash }{2m_i}.$$

$$d{\mathbf{q}}_i(t)={\mathbf{b}}_{i\ast }(q(t),t)+d{\mathbf{W}}_{i\ast }(t),$$

$$\delta {\bar{\theta }}_i^{rest}:={\omega }_{ci}\delta t_0=\frac{m_i{c}^2}{\hslash }\delta t_0,$$

$${\bar{\theta }}_i^{rest}={\omega }_{ci}{t}_0+{\varphi }_i=\frac{m_i{c}^2}{\hslash }{t}_0+{\varphi }_i.$$

$${\bar{\theta }}_{joint}^{rest}:=\sum _{i=1}^N{\bar{\theta }}_i=\sum _{i=1}^N\left({\omega }_{ci}{t}_0+{\varphi }_i\right)=\sum _{i=1}^N\left(\frac{m_i{c}^2}{\hslash }{t}_0+{\varphi }_i\right).$$

$$\delta {\bar{\theta }}_{joint+}^{lab}(q(t),t):=\sum _{i=1}^N\frac{{\omega }_{ci}}{m_i{c}^2}\left[E_{i+}(q(t),t)\delta t-m_i{\mathbf{b}}_i(q(t),t)\cdot \delta {\mathbf{q}}_i(t)\right]=\frac{1}{\hslash }\left[\sum _{i=1}^N{E}_{i+}(q(t),t)\delta t-\sum _{i=1}^N{m}_i{\mathbf{b}}_i(q(t),t)\cdot \delta {\mathbf{q}}_i(t)\right],$$

$$\delta {\bar{\theta }}_{joint-}^{lab}(q(t),t):=\sum _{i=1}^N\frac{{\omega }_{ci}}{m_i{c}^2}\left[E_{i-}(q(t),t)\delta t-m_i{\mathbf{b}}_{i\ast }(q(t),t)\cdot \delta {\mathbf{q}}_i(t)\right]=\frac{1}{\hslash }\left[\sum _{i=1}^N{E}_{i-}(q(t),t)\delta t-\sum _{i=1}^N{m}_i{\mathbf{b}}_{i\ast }(q(t),t)\cdot \delta {\mathbf{q}}_i(t)\right],$$

$$E_{i+}(q(t),t):=m_i{c}^2+\frac{1}{2}{m}_i{b}_i^2,$$

$$E_{i-}(q(t),t):=m_i{c}^2+\frac{1}{2}{m}_i{b}_{i\ast }^2,$$

$$\sum _{i=1}^N{\oint }_L{\delta }_i{\bar{\theta }}_{joint+,-}^{lab}=2\pi n,$$

$$\frac{\partial \rho (q,t)}{\partial t}=-\sum _{i=1}^N{\nabla }_i\cdot \left[{\mathbf{b}}_i(q,t)\rho (q,t)\right]+\sum _{i=1}^N\frac{\hslash }{2m_i}{\nabla }_i^2\rho (q,t),$$

$$\frac{\partial \rho (q,t)}{\partial t}=-\sum _{i=1}^N{\nabla }_i\cdot \left[{\mathbf{b}}_{i\ast }(q,t)\rho (q,t)\right]-\sum _{i=1}^N\frac{\hslash }{2m_i}{\nabla }_i^2\rho (q,t),$$

$${\int }_{{\mathbb{R}}^{3N}}{\rho }_0(q)d^{3N}q=1.$$

$${\mathbf{v}}_i(q,t):=\frac{1}{2}\left[{\mathbf{b}}_i(q,t)+{\mathbf{b}}_{i\ast }(q,t)\right]=\frac{{\nabla }_{i}S(q,t)}{m_i}{|}_{{\mathbf{q}}_j={\mathbf{q}}_j(t)},$$

$${\mathbf{u}}_i(q,t):=\frac{1}{2}\left[{\mathbf{b}}_i(q,t)-{\mathbf{b}}_{i\ast }(q,t)\right]=\frac{\hslash }{2m_i}\frac{{\nabla }_i\rho (q,t)}{\rho (q,t)}{|}_{{\mathbf{q}}_j={\mathbf{q}}_j(t)},$$

$$\frac{\partial \rho (\text{normalcolor}q,t)}{\partial t}=-\sum _{i=1}^N{\nabla }_i\cdot \left[\frac{{\nabla }_{i}S(q,t)}{m_i}\rho (q,t)\right],$$

$${\bar{\theta }}_{joint}^{lab}(q(t),t)$$

$$=\frac{1}{\hslash }\left[\int \sum _{i=1}^N{E}_i(q(t),t)dt-\int \sum _{i=1}^N\frac{m_i}{2}\left({\mathbf{b}}_i(q(t),t)+{\mathbf{b}}_{i\ast }(q(t),t)\right)\cdot d{\mathbf{q}}_i(t)\right]+\sum _{i=1}^N{\varphi }_i,$$

$$E_i(q(t),t):=m_i{c}^2+\frac{1}{2}\left[\frac{1}{2}{m}_i{b}_i^2+\frac{1}{2}{m}_i{b}_{i\ast }^2\right]=m_i{c}^2+\frac{1}{2}{m}_i{v}_i^2+\frac{1}{2}{m}_i{u}_i^2.$$

$${\mathbf{p}}_i(q(t),t)=-\hslash {\nabla }_i{\bar{\theta }}_{joint}^{lab}(q,t){|}_{{\mathbf{q}}_j={\mathbf{q}}_j(t)}={\nabla }_{i}S(q,t){|}_{{\mathbf{q}}_j={\mathbf{q}}_j(t)},$$

$$D{\mathbf{q}}_i(t):=\underset{{}_{\Delta t\to 0^{+}}}{\lim }{E}_t\left[\frac{q_i(t+\Delta t)-q_i(t)}{\Delta t}\right]={\mathbf{b}}_i(q(t),t),$$

$$D_{\ast }{\mathbf{q}}_i(t):=\underset{{}_{\Delta t\to 0^{+}}}{\lim }{E}_t\left[\frac{q_i(t)-q_i(t-\Delta t)}{\Delta t}\right]={\mathbf{b}}_{i\ast }(q(t),t),$$

$$D{\mathbf{b}}_i(q(t),t)=\left[\frac{\partial }{\partial t}+\sum _{i=1}^N{\mathbf{b}}_i(q(t),t)\cdot {\nabla }_i+\sum _{i=1}^N\frac{\hslash }{2m_i}{\nabla }_i^2\right]{\mathbf{b}}_i(q(t),t),$$

$$D_{\ast }{\mathbf{b}}_{i\ast }(q(t),t)=\left[\frac{\partial }{\partial t}+\sum _{i=1}^N{\mathbf{b}}_{i\ast }(q(t),t)\cdot {\nabla }_i-\sum _{i=1}^N\frac{\hslash }{2m_i}{\nabla }_i^2\right]{\mathbf{b}}_{i\ast }(q(t),t).$$

$$J(q,t)={\int }_{{\mathbb{R}}^{3N}}{d}^{3N}\mathbf{q}\rho (q,t){\bar{\theta }}_{joint}^{lab}(q(t),t)$$

$$=\frac{1}{\hslash }\left({\int }_{t_I}^{t_F}<\sum _{i=1}^N{E}_i>dt-{\int }_{q_I}^{q_F}<\sum _{i=1}^N{\mathbf{p}}_i\cdot d{\mathbf{q}}_i(t)>\right)+\sum _{i=1}^N{\varphi }_i,$$

$$={\int }_{{\mathbb{R}}^{3N}}{d}^{3N}\mathbf{q}\rho {\int }_{t_I}^{t_F}\sum _{i=1}^N\left\{m_i{c}^2+\frac{1}{2}{m}_i{v}_i^2+\frac{1}{2}{m}_i{u}_i^2\right\}dt+\sum _{i=1}^N{\varphi }_i,$$

$$={\int }_{{\mathbb{R}}^{3N}}{d}^{3N}\mathbf{q}\rho {\int }_{t_I}^{t_F}\sum _{i=1}^N\left\{m_i{c}^2+\frac{1}{2}\left[\frac{1}{2}{m}_i{\left(D{\mathbf{q}}_i(t)\right)}^2+\frac{1}{2}{m}_i{\left(D_{\ast }{\mathbf{q}}_i(t)\right)}^2\right]\right\}dt+\sum _{i=1}^N{\varphi }_i,$$

$$L_i:=\frac{1}{2}\left[(m{\mathbf{b}}_i)\cdot {\mathbf{b}}_i+(m{\mathbf{b}}_{i\ast })\cdot {\mathbf{b}}_{i\ast }\right]-\frac{1}{2}\left(E_{i+}+E_{i-}\right)=(m{\mathbf{v}}_i)\cdot {\mathbf{v}}_i+(m{\mathbf{u}}_i)\cdot {\mathbf{u}}_i-E_i,$$

$$J(q,t)=extremal,$$

Full text

Institute for History and Foundations of Science & Department of Mathematics, Utrecht University, Utrecht, The Netherlands**111Address when this work was initiated: Department of Physics & Astronomy, University of Nebraska-Lincoln, Lincoln, NE 68588, USA.

1 Introduction

Semiclassical theories 222As in field theories where the matter sector is described within the framework of quantum mechanics or quantum field theory, and the gravity sector is described by classical (c-number) fields. of gravity and electrodynamics, based on the formalism of standard quantum theory, have been thoroughly studied over the past 55 years [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In the past 20 years or so, semiclassical Newtonian gravity based on the Schrödinger-Newton (SN) equation [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 15, 36, 37, 38, 39, 40, 17, 16, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] has become a popular focus of discussions in the foundations of quantum mechanics [24, 25, 30, 32, 38, 16, 55, 17, 43, 46, 53, 54], quantum gravity phenomenology [24, 25, 26, 28, 31, 33, 38, 39, 40, 43, 42, 41, 46, 44, 47, 45, 50, 52, 48, 54], and state-of-the-art AMO experimental physics [26, 56, 40, 45, 46, 50, 52, 51, 57, 48]. Variants of the SN equation, based on alternative formulations of quantum theory, have also been developed [58, 29, 59, 60, 61, 62, 38, 55, 63, 64, 65, 66, 48], mostly in the context of dynamical collapse theories [62, 55, 48, 29, 60, 61, 64, 59, 66]. Less discussion has been given to the possibility of semiclassical theories of gravity/electrodynamics based on ‘hidden-variables’ 333This phrase is somewhat misleading for the theories in question, but we will use it abusingly due to its already widespread use in the literature. theories; the only instances we know of are Struyve [65], Kiessling [67], and Prezhdo-Brooksby [58] in the context of the de Broglie-Bohm (dBB) pilot-wave theory [68, 69, 70, 71]. Until now, no such discussion has been given in the context of stochastic mechanical hidden-variables theories [72, 73, 74, 75, 76, 1, 2].

A central reason for considering formulations of semiclassical gravity based on alternative quantum theories is that the SN equation, whether understood as a mean-field approximation to the standard exact quantum description of matter-gravity coupling [7, 13, 14, 15, 16, 17, 54] or as a fundamental theory describing matter-gravity coupling consistent with standard quantum theory [5, 6, 30, 15, 55, 17, 43, 42, 46, 47, 48], is either very limited in applicability or fatally problematic [6, 13, 12, 30, 14, 32, 17, 16, 55, 44, 53, 54].

Understood as a mean-field theory, the nonlinearity of the SN equation (or the stochastic SN equation discussed by us in [54]) means that its solutions lack consistent Born-rule interpretations [30, 32, 55, 16, 53] (see section 4 and subsection 5.1); instead, the SN solutions must be understood as describing self-gravitating classical matter fields that approximate quantum systems involving large numbers of identical particles that weakly interact 444In the sense of the coupling scaling as $1/N$, where N is the number of particles. quantum-gravitationally [77, 78, 79, 16, 17, 54]. Moreover, only SN solutions with ‘small quantum fluctuations’ (i.e., solutions which don’t correspond to superpositions of effectively orthogonal classical field states) can have this physical interpretation [9, 23, 12, 55, 44, 48, 54], implying that the vast majority of SN solutions are (physically) superfluous.

Understood as a fundamental theory, the nonlinearity of the SN equation is fatal because the consequent lack of consistent Born-rule interpretations for the SN solutions destroys the standard quantum interpretation of the matter sector of fundamentally-semiclassical gravity based the SN equation (see subsection 5.1). A fundamentally-semiclassical description of matter-gravity coupling, based on the SN equation, would actually be a nonlinear classical field theory that makes empirical predictions (such as macroscopic semiclassical gravitational cat states; see section 4 for an example) grossly inconsistent with standard quantum mechanics and the world of lived experience [9, 23, 55, 12, 43, 48]. (Analogous comments apply to semiclassical electrodynamics based on the Schrödinger-Coulomb (SC) equation [18, 19]; see subsection 5.1 for a discussion.)

Another key motivation for considering formulations of semiclassical gravity based on alternative quantum theories is that while the standard exact quantum description of matter-gravity coupling yields semiclassical gravity as a consistent mean-field approximation, the matter sector of the standard exact quantum description is afflicted by the quantum measurement problem [80, 68, 81, 82, 83, 70, 38]. This puts a fundamental limitation on the domain of applicability of the standard exact quantum description (whether at the Newtonian level or the fully relativistic level), hence a fundamental limitation on the domain of applicability of semiclassical gravity (whether at the Newtonian level or the fully relativistic level). Namely, the standard exact quantum description and the mean-field semiclassical-gravitational description are only applicable to laboratory experiments involving the coupling of gravity to quantum matter, since laboratory experiments are the only places where the standard quantum formalism can be sensibly applied.

Thus it stands to reason that a formulation of quantum theory convincingly free of the measurement problem might, when extended to a semiclassical description of gravity (whether as a fundamental theory or a mean-field theory), yield a superior formulation of semiclassical gravity than the options based on standard quantum theory. Arguably, this suggestion has already been confirmed (at least at the Newtonian level) by dynamical collapse versions of fundamentally-semiclassical Newtonian gravity, insofar as the models of Derakhshani [55, 48] and Tilloy-Diósi [66] seem to have consistent statistical interpretations while adequately suppressing gravitational cat state solutions at the macroscopic scale. In addition, the works of Struyve [65] and Prezhdo-Brooksby [58] suggest that the dBB theory offers a more empirically accurate semiclassical approximation scheme than does standard quantum theory, at least for simple examples considered at the relativistic level [65] and the Newtonian level [58, 65] (see subsection 5.2 for more detail); however, Struyve has pointed out [65] that the dBB theory does not yield a consistent model of fundamentally-semiclassical Einstein gravity 555The reason, basically, is that a classical stress-energy tensor built out of a dBB field beable $\phi_{B}(\mathbf{x},t)$ entails non-conservation of stress-energy, i.e., $\nabla^{\mu}T_{\mu\nu}\left(\phi_{B}\right)\neq 0$, since $\phi_{B}$ does not conserve total energy along its space-time trajectory. Thus $T_{\mu\nu}\left(\phi_{B}\right)$ cannot be used on the right-hand-side of the classical Einstein equations as this would violate the Bianchi identities $\nabla^{\mu}G_{\mu\nu}=0$.. In our assessment (see subsection 5.2) the dBB theory does not allow for a consistent model of fundamentally-semiclassical Newtonian gravity/electrodynamics, either. It would seem, then, that there does not yet exist a compelling and widely applicable model of semiclassical gravity based on a theory of hidden-variables, whether in the form of a fundamental theory of matter-gravity coupling or a mean-field approximation to an exact ‘quantum’ description of matter-gravity coupling.

The primary objectives of this two-part series are: (i) to construct a fundamentally-semiclassical theory of Newtonian gravity/electrodynamics within the framework of stochastic mechanics, in particular a new formulation of stochastic mechanics we have recently proposed [1, 2] to answer the long-standing “Wallstrom criticism”; (ii) to show that fundamentally-semiclassical Newtonian gravity/electrodynamics based on our new formulation of stochastic mechanics - which we call “zitterbewegung stochastic mechanics” (ZSM), hence ‘ZSM-Newton’ and ‘ZSM-Coulomb’ - has a consistent statistical interpretation and recovers the standard exact quantum description of matter-gravity coupling as an approximation (valid for all practical purposes), while also being free of the measurement problem; (iii) to show that the SN/SC equation and the stochastic SN/SC equation can be recovered as mean-field approximations for large numbers of identical ZSM particles that weakly interact 666Also in the sense of coupling scaling as $1/N$. classical-gravitationally/electrostatically; and (iv) to show that ZSM-Newton/Coulomb yields a new ‘large-N’ prescription that makes it possible to: (a) accurately approximate the time-evolution of a large number of identical ZSM particles that strongly interact classical-gravitationally/electrostatically, within a consistent statistical interpretation; (b) avoid macroscopic semiclassical gravitational cat states and recover classical Newtonian gravity/electrodynamics for the center-of-mass descriptions of macroscopic particles; and (c) recover classical Vlasov-Poisson mean-field theory for macroscopic particles that weakly interact gravitationally/electrostatically.

In the present paper, we will carry out objectives (i-iii), leaving (iv) for Part II. We will also compare ZSM-Newton/Coulomb to extant theories of semiclassical Newtonian gravity/electrodynamics that are based on standard quantum theory, dynamical collapse theories, and the de Broglie-Bohm theory.

The paper organization is as follows. Section 2 reviews ZSM for the case of many free particles. Section 3 formulates the basic equations of ZSM-Newton/Coulomb, explicates the physical interpretation of those equations, and shows how the standard exact quantum description of matter-gravity coupling is recovered as a special case valid for all practical purposes. Section 4 shows how to recover the SN/SC equation and the stochastic SN/SC equation as mean-field approximations for large numbers of identical ZSM particles that weakly interact gravitationally/electrostatically. Finally, section 5 compares ZSM-Newton/Coulomb to extant theories of semiclassical Newtonian gravity/electrodynamics, pointing out conceptual and technical advantages entailed by ZSM-Newton/Coulomb, as well as possibilities for experimental discrimination between ZSM-Newton/Coulomb and these other semiclassical theories.

2 Overview of ZSM for many free particles

ZSM was developed in order to answer Wallstrom’s criticism of stochastic mechanical reconstructions of Schrödinger’s equation; namely, that they don’t give a plausible justification for the quantum mechanical requirement that wavefunctions (for spinless particles) must always be single-valued while allowing generally multi-valued phases [84, 85, 1]. In other words, why it should be that the wavefunction phase $S$ (in polar form) must change along a closed loop in configuration space by integer multiples of Planck’s constant. A formulation of stochastic mechanics that plausibly answers this criticism is, in our view, a necessary condition for seriously considering extensions of stochastic mechanics to more general physical situations, hence why we will base our approach on the ZSM formulation.

To prepare for the formulation of ZSM-Newton/Coulomb, it is useful to first review ZSM for $N$ particles that are classically non-interacting [2].

Our starting point is the following four phenomenological hypotheses.

First, we posit a Minkowski space-time that contains, on a $t=const$ hypersurface, N harmonic oscillators centered around 3-space positions $\mathbf{q}_{0i}$ for $i=1,..,N$. As ZSM is a phenomenological framework, we need not specify here the precise physical nature of these harmonic oscillators (this is task is left for future work). However, we assume that these oscillators have, in their respective translational rest frames, natural frequencies $\omega_{ci}\coloneqq\left(1/\hbar\right)m_{i}c^{2}$, where $c$ is the speed of light and the $m_{i}$ are mass parameters that set the scales of the natural frequencies. It is reasonable to call these natural frequencies “Compton” frequencies, hence the label $\omega_{ci}$. We will refer to these oscillators hereafter as* *“zitterbewegung (zbw) particles” [1, 2].

Second, we adapt Nelson’s ether hypothesis [73, 86, 87, 88, 89] by supposing now that the Minkowski space-time is pervaded by a frictionless classical fluid medium (which we will also call an “ether”), with the qualitative properties that (i) it is fluctuating everywhere with the same intensity, and (ii) it is an oscillating medium with a spectrum of modes superposed at each point in 3-space. More precisely, we imagine the ether as a continuous (or effectively continuous) medium composed of a countably infinite number of fluctuating, stationary, spherical waves superposed at each point in 3-space, with each wave having a different fixed angular frequency $\omega_{0}^{i}$ where $i$ denotes the i-th ether mode. The relative phases between the modes are taken to be random so that each mode is effectively uncorrelated with every other mode. Again, since ZSM is a phenomenological framework, specifying the precise physical nature of this ether is left to future work.

Third, we follow Nelson [73, 86, 74] in hypothesizing that each particle’s center of mass, as a result of being immersed in the ether, undergoes an approximately frictionless translational Brownian motion (due to the homogeneous and isotropic ether fluctuations that couple to the particles by possibly electromagnetic, gravitational, or some other means), as modeled by the first-order stochastic differential equations

[TABLE]

Here the index $i=1,...,N$, the particle trajectories $q(t)=\{\mathbf{q}_{1}(t),\mathbf{q}_{2}(t),...,\mathbf{q}_{N}(t)\}$ $\in$ $\mathbb{R}^{3N}$, and $\mathbf{W}_{i}(t)$ are Wiener processes modeling each particle’s interaction with the ether fluctuations. The Wiener increments $d\mathbf{W}_{i}(t)$ are assumed to be Gaussian with zero mean, independent of $d\mathbf{q}_{i}(s)$ for $s\leq t$, and with variance

[TABLE]

where $E_{t}$ denotes the conditional expectation at time t. We then follow Nelson in hypothesizing that the magnitude of the diffusion coefficients $\nu_{i}$ are defined by

[TABLE]

Along with (1), we also have the backward equations

[TABLE]

where $\mathbf{b}_{i*}(q(t),t)$ are the mean backward drift velocities, and $d\mathbf{W}_{i*}(t))$ are the backward Wiener processes. As in the single-particle case, the $d\mathbf{W}_{i*}(t)$ have all the properties of $d\mathbf{W}_{i}(t)$ except that they are independent of the $d\mathbf{q}_{i}(s)$ for $s\geq t$. With these conditions on $d\mathbf{W}_{i}(t)$ and $d\mathbf{W}_{i*}(t)$, equations (1) and (4) respectively define forward and backward Markov processes for $N$ particles on $\mathbb{R}^{3}$ (or, equivalently, for a single particle on $\mathbb{R}^{3N}$). Note that we take the $\mathbf{b}_{i}$ $(\mathbf{b}_{i*})$ to be functions of all the particle positions, $q(t)=\{\mathbf{q}_{1}(t),\mathbf{q}_{2}(t),...,\mathbf{q}_{N}(t)\}$ $\in$ $\mathbb{R}^{3N}$. The reasons are: (i) all the particles are continuously exchanging energy-momentum with a common background medium (the ether) and thus are in general physically connected in their translational motions via $\mathbf{b}_{i}$ $(\mathbf{b}_{i*})$, insofar as the latter are constrained by the physical properties of the ether; and (ii) the dynamical equations and initial conditions for the $\mathbf{b}_{i}$ $(\mathbf{b}_{i*})$ are what will determine the specific situations under which the latter will be effectively factorizable functions of the particle positions and when they cannot be effectively factorized. Hence, at this level, it is only sensible to write $\mathbf{b}_{i}$ $(\mathbf{b}_{i*})$ as functions of all the particle positions at a single time.

Fourth, we suppose that, in their respective (now) instantaneous mean translational rest frames (IMTRFs), i.e., the frames where $\mathbf{b}_{i}=\mathbf{b}_{i*}=0$, the zbw particles undergo driven oscillations about $\mathbf{q}_{0i}$ by coupling to a narrow band of ether modes that resonantly peak around their natural frequencies. However, in order that the oscillation of each zbw particle doesn’t become unbounded in kinetic energy, there must be some mechanism by which the zbw particles dissipate energy back into the ether so that, on the average, a steady-state equilibrium regime is reached for their oscillations. So we posit that on short relaxation time-scales $\tau_{i}$, which are identical for zbw particles of identical rest masses, the mean energy absorbed by the i-th zbw particle, as a result of its driven oscillation by the resonant ether modes, equals the mean energy dissipated back to the ether by the i-th zbw particle. Accordingly, we suppose that, in this steady-state regime, each particle undergoes a constant mean harmonic oscillation about its $\mathbf{q}_{0i}$ in its IMTRF, as characterized by the fluctuation-dissipation relation $<H_{i}>_{steady-state}=\hbar\omega_{ci}=m_{i}c^{2}$ where $<H_{i}>_{steady-state}$ is the conserved mean energy due to the steady-state oscillation of the i-th zbw particle.

Now, as a consequence of this last hypothesis, it follows that in the IMTRF of the i-th particle, the mean zbw phase change is given by

[TABLE]

and the corresponding absolute mean phase is

[TABLE]

Then the joint (mean) phase for all the particles will just be

[TABLE]

Note that we cannot talk of the zbw phase other than in the IMTRFs of the particles, because we cannot transform to a frame in which $d\mathbf{q}_{i}(t)/dt=0$, as this expression is undefined for the Wiener process.

Lorentz transforming back to the mean translational lab frame where $\mathbf{b}_{i}(\mathbf{q}_{i}(t),t)\neq 0$ and $\mathbf{b}_{i*}(\mathbf{q}_{i}(t),t)\neq 0$, and approximating the transformation for non-relativistic velocities so that the gamma factor $\gamma=1/\sqrt{\left(1-b_{i}^{2}/c^{2}\right)}\approx 1+b_{i}^{2}/2c^{2},$ it is readily shown [2] that the forward and backward joint phase changes become

[TABLE]

and

[TABLE]

where the forward and backward total particle energies are

[TABLE]

and

[TABLE]

neglecting the momentum term proportional to $b_{i}^{3}/c^{2}$.

Since each zbw particle is essentially a harmonic oscillator, each has its own well-defined phase along its space-time trajectory. Consistency with this last statement entails that when $\mathbf{b}_{i}(q,t)\neq\sum_{i}^{N}\mathbf{b}_{i}(\mathbf{q}_{i},t)$, the joint phase must be a well-defined function of the space-time trajectories of* *all the zbw particles (since we posit that all particles remain harmonic oscillators despite having their oscillations physically coupled through the common ether medium they interact with). Thus we can see that, for a closed loop L along which each zbw particle can be physically or virtually displaced, we have the constraint

[TABLE]

where $n\in\mathbb{Z}$. Moreover, condition (12) will also hold for a closed loop with time held constant.

Associated to (1) and (4) in the lab frame are the forward and backward Fokker-Planck equations

[TABLE]

and

[TABLE]

where $\rho(q,t)$ is the probability density for the particle trajectories and satisfies the normalization condition

[TABLE]

Imposing Nelson’s time-symmetric kinematic constraints

[TABLE]

and

[TABLE]

where $i,j=1,...,N$, then (13-14) reduce to

[TABLE]

with $\mathbf{b}_{i}=\mathbf{v}_{i}+\mathbf{u}_{i}$ and $\mathbf{b}_{i*}=\mathbf{v}_{i}-\mathbf{u}_{i}$.

To give (17) a coherent physical interpretation, we introduce the presence of an external (to the particle) osmotic potential $U(q,t)$ which couples to the $i$-th particle as $R(q(t),t)\coloneqq\mu U(q(t),t)$ (assuming that the coupling constant $\mu$ is identical for particles of the same species), and imparts a momentum, $\nabla_{i}R(q,t)|_{\mathbf{q}_{j}=\mathbf{q}_{j}(t)}$. This momentum then gets counter-balanced by the ether fluid’s osmotic impulse pressure, $\left(\hbar/2m_{i}\right)\nabla_{i}ln[n(q,t)]|_{\mathbf{q}_{j}=\mathbf{q}_{j}(t)}$, so that the $N$-particle osmotic velocity is the equilibrium velocity acquired by the $i$-th particle when $\nabla_{i}R/m_{i}=\left(\hbar/2m_{i}\right)\nabla_{i}\rho/\rho$ (using $\rho=n/N$), which implies $\rho=e^{2R/\hbar}$ for all times. It is supposed that $R$ generally depends on the coordinates of all the other particles because: (i) if $U$ was an independently existing field on configuration space, rather than sourced by the ether, then the diffusions of the particles through the ether would not be conservative (i.e., energy conserving), in contradiction with Nelson’s hypothesis that the diffusions are conservative, and (ii) since the particles continuously exchange energy-momentum with the ether, the functional dependence of $U$ should be determined by the dynamical coupling of the ether to the particles as well as the magnitude of the inter-particle physical interactions (whether through a classical inter-particle potential or, in the free particle case, just through the ether).

To obtain the time-symmetric mean dynamics for the translational motions of the N particles, i.e., the dynamics for $S$, we integrate the time-asymmetric joint phases, (8) and (9), and then average the two to get

[TABLE]

where, from the kinematic constraints (16-17) we have

[TABLE]

It also follows that

[TABLE]

which establishes the i-th Nelsonian current velocity as the i-th translational mean velocity component of (19), and the velocity potential $S$ as the joint mean phase of the zbw particles undergoing Nelsonian diffusions.

To construct the dynamics for the zbw particles, we first introduce the mean forward and mean backward derivatives:

[TABLE]

and

[TABLE]

where we used the Gaussianity of $d\mathbf{W}_{i}(t)$ and $d\mathbf{W}_{i*}(t)$ in equations (1) and (4). Finding $D\mathbf{b}_{i}(q(t),t)$ (or $D_{*}\mathbf{b}_{i}(q(t),t)$) is straightforward: expand $\mathbf{b}_{i}$ in a Taylor series up to terms of order two in $d\mathbf{q}_{i}(t)$, replace $dq_{i}(t)$ by $dW_{i}(t)$ in the last term, and replace $d\mathbf{q}_{i}(t)\cdot\nabla_{i}\mathbf{b}_{i}|_{\mathbf{q}_{j}=\mathbf{q}_{j}(t)}$ by $\mathbf{b}_{i}(\mathbf{q}(t),t)\cdot\nabla_{i}\mathbf{b}_{i}|_{\mathbf{q}_{j}=\mathbf{q}_{j}(t)}$ when taking the conditional expectation at time t (since $d\mathbf{W}_{i}(t)$ is independent of $\mathbf{q}_{i}(t)$ and has mean 0). We then have

[TABLE]

and

[TABLE]

With these derivatives operators in hand, we can define the ensemble-averaged, time-symmetric particle phase as

[TABLE]

where we have used the time-symmetric mean Legendre transformation

[TABLE]

and the relation $\bar{\theta}_{joint}^{lab}=-\frac{1}{\hbar}S$. Following Yasue [90, 91], we can apply the stochastic variational principle

[TABLE]

which by straightforward computation [2] yields

[TABLE]

Moreover, since by D’Alembert’s principle the $\delta\mathbf{q}_{i}(t)$ are independent (see Appendix A of [2]), it follows from (29) that we have the individual “mean acceleration” equations of motion

[TABLE]

By applying the mean derivatives in (29), using that $\mathbf{b}_{i}=\mathbf{v}_{i}+\mathbf{u}_{i}$ and $\mathbf{b}_{i*}=\mathbf{v}_{i}-\mathbf{u}_{i}$, and replacing $q(t)$ with $q$ on both sides, straightforward manipulations give

[TABLE]

Computing the derivatives in (31), we obtain

[TABLE]

Integrating both sides of (32), setting the arbitrary integration constants equal to the rest energies, and replacing $q(t)$ with $q$, we then have the N-particle quantum Hamilton-Jacobi equation

[TABLE]

This equation describes the total energy field over N Gibbsian statistical ensembles. That is, each coordinate $\mathbf{q}_{i}$ of $S$ is associated to a fictitious ensemble of identical noninteracting zbw particles, where the members of the ensemble differ from each other only by their initial positions and velocities; the ensemble density is given by $\rho$, and reflects ignorance of the actual position and velocity of the actual $i$-th zbw particle at time $t$.

Upon evaluation at $q=q(t),$ we have the total energy of the actual particles along their actual mean trajectories. We can now see explicitly that the local evolution equation for the time-symmetric phase (19), under the variational constraint (28), will just be (33).

The general solution of (33), i.e., the joint phase field of the zbw particles in the time-symmetric mean lab frame (the frame in which the current velocities of the zbw particles are non-zero), is clearly of the form

[TABLE]

Since each zbw particle is posited to be a harmonic oscillator of identical type, each has its own well-defined phase along its space-time trajectory. Consistency with this last statement implies that when $\mathbf{v}_{i}(q,t)\neq\sum_{i}^{N}\mathbf{v}_{i}(\mathbf{q}_{i},t)$, the joint mean phase must be a well-defined function of the time-symmetric mean trajectories of all zbw particles (since we posit that all the zbw particles remain harmonic oscillators despite having their oscillations physically coupled through the common ether medium they interact with). Then, for a closed loop L along which each zbw particle can be physically or virtually displaced, it follows that

[TABLE]

And for a closed loop $L$ with $\delta t=0$, we have

[TABLE]

If we also consider the joint phase field $S(q,t)$, a field over the N ensembles, then the same physical reasoning applied to each member of the i-th ensemble yields

[TABLE]

In other words, the loop integral is now an integral of the ensemble’s momentum field along any closed mathematical loop in 3-space with time held constant; that is, a closed loop around which the actual (i-th) particle with momentum $\mathbf{p}_{i}$ could potentially be displaced, starting from any possible position $\mathbf{q}_{i}$ it can occupy at fixed time $t$. It also clear that (37) implies phase quantization for each individual particle ensemble, upon keeping all but the i-th coordinate fixed and performing the closed-loop integration.

Applying the Madelung transformation [92, 93, 86, 87, 84, 68, 94, 85] to the combination of (18), (33), and (37), we can construct the N-particle Schrödinger equation

[TABLE]

where the N-particle wavefunction $\psi(q,t)=\sqrt{\rho(q,t)}e^{iS(q,t)/\hbar}$ is single-valued by (37). Of course, in the case of non-interacting particles, the $N$-particle wavefunction is nodeless and thus we will have $n=0$ in (37); only in cases of particles in bound states can we get $n>0$.

We should note that the solutions of (38) are generally non-factorizable fields on 3N-dimensional configuration space, which implies non-separability of $S$ and $R$ (hence non-factorizability of $\rho$) in general. Insofar as ZSM starts with the heuristic hypothesis of an ontic ether that lives in 3-space and couples to ontic zbw particles in 3-space, this would seem prima facie paradoxical, assuming one takes the mathematical representation of $S$ and $R$ as a literal indication of the ontic nature of the hypothesized ether (i.e., that if $S$ and $R$ live in configuration space, then so must the ether). As discussed at length in [2], there are three possible ways to resolve this apparent inconsistency: (i) postulate that the ether lives in configuration space, but, as a matter of physical law, determines the motion of N zbw particles in 3-space; (ii) postulate that the ether lives in configuration space along with a zbw ‘world particle’ (in analogy with Albert’s formulation of the de Broglie-Bohm theory [95]), and employ a philosophical functionalist analysis to deduce the emergence of N zbw particles floating in a common 3-space; and (iii) view the $S$ and $R$ fields on configuration space as convenient mathematical representations of some corresponding ontic fields on 3-space (in analogy with Norsen’s “TELB” approach to the de Broglie-Bohm theory [96, 97]) which couple to N zbw particles in 3-space. As also discussed in [2], we view option (iii) to be the most natural and fruitful one for ZSM, and we will implicitly assume this viewpoint throughout this paper.

With the overview completed, we can now develop ZSM-Newton/Coulomb.

3 ZSM-Newton/Coulomb: Basic equations

ZSM-Newton/Coulomb is just the generalization of $N$-particle ZSM to include classical Newtonian gravitational and Coulomb interactions between the $zbw$ particles.

We suppose again that each particle undergoes a mean zbw oscillation in its IMTRF, and now also that each zbw particle carries charge $e_{i}$, making them classical charged harmonic oscillators of identical type. (We subject these particles to the hypothetical constraint of no electromagnetic radiation emitted when there is no translational motion; or the constraint that the oscillation of the charge is radially symmetric so that there is no net energy radiated; or, if the ether turns out to be electromagnetic in nature as Nelson suggested [87, 88], then that the steady-state zbw oscillations of the particles are due to a balancing between the time-averaged electromagnetic energy absorbed via the driven oscillations of the particle charges, and the time-averaged electromagnetic energy radiated back to the ether by the particles.) So the classical Newtonian gravitational and Coulomb interactions between the particles are defined by the gravitational potential (in CGS units)

[TABLE]

and the Coulomb potential

[TABLE]

respectively, under the point-like interaction assumption $|\mathbf{q}_{i}(t)-\mathbf{q}_{j}(t)|\gg\lambda_{c}$.

Then the joint-phase change of the zbw particles in the mean forward joint lab frame ($b_{i}\ll c$ approximated) is defined by

[TABLE]

where $\delta\theta_{joint-}^{lab}$ differs by the replacement of $\mathbf{b}_{i}$ with $\mathbf{b}_{i*}$. Note that when $|\mathbf{q}_{i}(t)-\mathbf{q}_{j}(t)|$ becomes sufficiently great that $V_{g,c}^{int}$ is negligible, (41) reduces to an effectively factorizable sum of the mean forward phase changes for all the zbw particles. (Effectively, because the ether will of course still physically correlate the phase changes of the particles, even if negligibly.) We can then write

[TABLE]

and

[TABLE]

Because each zbw particle is a harmonic oscillator, each has a well-defined phase along its mean forward/backward space-time trajectory. Consistency with this last statement entails that when $V_{g,c}^{int}>0$, the joint phase must be a well-defined function of the mean forward/backward space-time trajectories of all the zbw particles (since we again posit that all the *zbw *particles remain harmonic oscillators when coupled to each other via $V_{g,c}^{int}$). Then for a closed loop L, along which each zbw particle can be physically or virtually displaced, the mean forward joint phase in the lab frame will satisfy

[TABLE]

and for a loop with time held fixed

[TABLE]

and likewise for the mean backward joint phase. It also follows from (44-45) that

[TABLE]

and

[TABLE]

where the closed-loop integral here keeps the coordinates of all the other particles fixed while particle 1 is displaced along L.

In the lab frame, the forward and backward stochastic differential equations for the translational motion are then

[TABLE]

and

[TABLE]

with corresponding Fokker-Planck equations

[TABLE]

and

[TABLE]

Imposing the time-symmetric kinematic constraints (16-17), (50-51) reduce to

[TABLE]

and we have again $\mathbf{b}_{i}=\mathbf{v}_{i}+\mathbf{u}_{i}$ and $\mathbf{b}{}_{i*}=\mathbf{v}_{i}-\mathbf{u}_{i}$.

Now, integrating $\delta\theta_{joint+}^{lab}$ and $\delta\theta_{joint-}^{lab}$, and then averaging the two, we have

[TABLE]

where

[TABLE]

Then

[TABLE]

which follows from the time-symmetric mean Legendre transformation

[TABLE]

and using $\bar{\theta}_{joint}^{lab}=-\frac{1}{\hbar}S$. Applying

[TABLE]

we have [2]

[TABLE]

for $i,j=1,...,N$ and $i\neq j$. And from the independent $\delta\mathbf{q}_{i}(t)$, we have the individual equations of motion

[TABLE]

Applying the mean derivatives, using that $\mathbf{b}_{i}=\mathbf{v}_{i}+\mathbf{u}_{i}$, $\mathbf{b}_{i*}=\mathbf{v}_{i}-\mathbf{u}_{i}$, and replacing $q(t)$ by $q$ on both sides, (58) becomes

[TABLE]

Identifying

[TABLE]

using (16-17) in (60), integrating both sides, and setting the arbitrary integration constants equal to the particle rest energies, we then get

[TABLE]

with general solution

[TABLE]

Recall we made the plausible assumption that the presence of classical external potentials doesn’t alter the harmonic nature of the mean zbw oscillations of the particles. Hence, each zbw particle has a well-defined mean phase along its time-symmetric mean trajectory. Accordingly, when $\Phi_{g,c}$ is not negligible, the joint phase is a well-defined function of the time-symmetric mean trajectories of* *all the zbw particles.

So for a closed loop L along which each zbw particle can be physically or virtually displaced, it follows that

[TABLE]

and

[TABLE]

the latter for a closed loop $L$ with $\delta t=0$. For the joint mean phase field $S(q,t)$, applying the same physical reasoning to each member of the i-th ensemble yields

[TABLE]

Clearly (66) implies phase-field quantization for each ensemble, upon keeping all but the i-th coordinate fixed and performing the closed-loop integration. Applying the Madelung transformation to the combination of (66), (62), and (52), we can construct the N-particle Schrödinger equation for classically interacting zbw particles in the presence of external fields:

[TABLE]

where $\psi(q,t)=\sqrt{\rho(q,t)}e^{iS(q,t)/\hbar}$ is single-valued via (66).

Note the inclusion of hats on the interaction potentials and their coordinates, in contrast to the quantum Hamilton-Jacobi (QHJ) equation (62). As shown by Holland [68] and Oriols & Mompart [98], there exists a correspondence between quantum operators in the Schrödinger equation, and c-number variables in the QHJ equation. For example, the quantum expectation value of the position operator corresponds to the ensemble averaged position coordinate via $\left\langle\psi\right|\hat{\mathbf{q}}\left|\psi\right\rangle=\int_{\mathbb{R}^{3N}}d^{3N}\mathbf{q}\,\psi^{*}\,\mathbf{\hat{q}\,}\psi=\int_{\mathbb{R}^{3N}}d^{3N}\mathbf{q}\,\rho\,\mathbf{q}=<\mathbf{q}>$. For another example, the quantum expectation value of the Hamiltonian operator is equivalent to the ensemble average of the total energy in the QHJ equation:

[TABLE]

So the classical potentials are, in effect, ‘quantized’ at the level of the Schrödinger equation, insofar as they depend on operator-valued position coordinates and satisfy the Poisson equations 777The use of delta functions in the definitions of the mass and charge densities is justified because we are using the point-like approximation for interactions between the particles. In actuality, the mass and charge densities should presumably depend on some form-factor $f(|\mathbf{x}-\mathbf{q}_{i}|)$ which distributes the mass or charge of the particle on its Compton length-scale $\lambda_{c}$. Additionally, in scattering events where $|\mathbf{q}_{i}(t)-\mathbf{q}_{j}(t)|\sim\lambda_{c}$ the point-like approximation will no longer hold and it will become necessary to include this form-factor in calculating the interactions. The precise expression for this form-factor will depend on the specific physical model used for the zbw particle, which at present we do not have (although see section 5 of [2] for a discussion of possibilities). Nevertheless, as we are only concerned here with the non-relativistic regime, the point-like approximation will suffice.

[TABLE]

[TABLE]

Accordingly, the equation set (67-70) gives a statistical mechanical description of $N$ zbw particles undergoing Nelsonian diffusions, while interacting both gravitationally and electrostatically through the classical potentials (39-40).

Equations (67-70) correspond to the standard quantum mechanical equations for $N$ particles interacting gravitationally or electrostatically in the Newtonian regime [77, 78, 79, 17], and that the standard quantum mechanical equations are the Newtonian limits of the standard theories of perturbatively quantized gravity and perturbative quantum electrodynamics [17, 43]. But because we derived (67-70) within the ZSM framework, we can go further than the standard quantum description. That is, we can use solutions of (67), or the equivalent solutions of the Madelung equations (52) and (62) under the constraint (66), to deduce an ensemble of possible trajectories for the actual (zbw) particles.

In particular, it is readily shown that the i-th mean acceleration

[TABLE]

where $\mathrm{D}/\mathrm{D}t$ is the convective derivative. Integrating this last equation for different possible initial conditions $\mathbf{q}_{j}(0)$ allows us to construct an ensemble of mean trajectories, only one of which is realized by the actual $i$-th* zbw* particle. We can then find the gravitational and Coulomb potentials sourced by the actual zbw particles, along their mean trajectories, as follows:

[TABLE]

[TABLE]

where the superscript “$m.t.$” refers to the interaction potentials sourced by the mean trajectories of the actual zbw particles.

Actually, (72) doesn’t contain all the terms that contribute to the total mass-densities of the particles. The complete expression is

[TABLE]

But in the $v_{i}\ll c$ limit, the classical kinetic and quantum kinetic 888While the latter terms are referred to in the literature as “quantum potentials” [99, 92, 100, 94, 68, 70, 98], we prefer the term “quantum kinetics” [1, 2] since, in stochastic mechanics, they arise from the kinetic energy contributions of the osmotic velocities of the particles, as seen from the left hand side of (60). energy terms are negligible relative to the rest-energy terms, allowing us to effectively neglect the contributions of the kinetic energies to the total mass-energy density of the particle.

From the solutions of (67), we can also construct an ensemble of possible stochastic trajectories for the $i$-th particle:

[TABLE]

[TABLE]

These stochastic trajectories can also be used in the definition of the mass and charge densities, implying classically fluctuating mass and charge densities, hence classically fluctuating gravitational and Coulomb potentials satisfying the Poisson equations

[TABLE]

[TABLE]

where “$s.t.$” refers to the interaction potentials sourced by the stochastic trajectories of the actual zbw particles.

Thus we see here that there are three ‘levels’ of interaction potentials, with $\Phi_{g,c}^{s.t.}$ being the most fundamental (in the sense of being the potentials sourced by the actual, stochastic trajectories of the actual zbw particles), followed by $\Phi_{g,c}^{m.t.}$ (in the sense of being the potentials sourced by the mean trajectories of the actual zbw particles), and then $\Phi_{g,c}$ or $\hat{\Phi}_{g,c}$ (in the sense of being operator-valued potentials that reflect a statistical ensemble of possible potentials sourced by the possible mean trajectories of the actual zbw particles). Indeed the operator-valued interaction potentials $\hat{\Phi}_{g,c}$ have physical meaning inasmuch as

[TABLE]

and

[TABLE]

and likewise for the Coulomb potentials.

Note the conceptual difference between the expected values of the interaction potentials, equation (79), and the potentials obtained from (72-73) (i.e., the potentials sourced by the mean trajectories of the actual zbw particles). The former are obtained from averaging the interaction potentials over N statistical ensembles of mean trajectories; the latter are obtained from taking the conditional expectation at time *t *of (75-76), considering the time-symmetrized mean (current) velocity $\mathbf{v}=\left(1/2\right)\left(\mathbf{b}+\mathbf{b}_{*}\right)$, and using the corresponding mean trajectories to source the potentials.

It is interesting to compare (80) to the Poisson equation associated with the N-body Schrödinger-Newton (SN) gravitational potential [23, 30, 17, 55, 43, 42, 48]:

[TABLE]

In the SN equations, the solution of (81) describes the net interaction potential sourced by N matter density fields on space-time (each field corresponding to an elementary ‘particle’), and this potential feeds back into the Hamiltonian of the Schrödinger equation to generate a nonlinear Schrödinger evolution. In ZSM-Newton, by contrast, the solution of (80) describes the ensemble-average of the net interaction potential sourced by the N point-like zbw particles, and this potential does not feed back into the (derived) Schrödinger Hamiltonian. Everything said here also holds for the Coulombic analogue of (80) and its comparison to the N-body Schrödinger-Coulomb equations [77, 78, 79, 55, 17]. (See subsection 5.1 for a more detailed comparison of ZSM-Newton/Coulomb to N-body Schrödinger-Newton/Coulomb.)

Earlier we observed that the complete expression for the mass densities of the zbw particles is given by the rhs of (74). While we also noted that the classical and quantum kinetic energy terms can be neglected in the Newtonian regime, let us see what happens if we do use the solution of (74) in the QHJ equation (62) and the Schrödinger equation (67). For maximum clarity, we restrict to the two-particle case $q=\left\{\mathbf{q}_{1},\mathbf{q}_{2}\right\}$ and drop the Coulomb potentials and rest-energy terms:

[TABLE]

where $T_{i}(q,t)\coloneqq\frac{\left[\nabla_{i}S(q,t)\right]^{2}}{2m_{i}}$ and $Q_{i}(q,t)\coloneqq-\frac{\hbar^{2}}{2m_{i}}\frac{\nabla_{i}^{2}\sqrt{\rho(q,t)}}{\sqrt{\rho(q,t)}}$. We can see that the gravitational interaction energy between the two particles depends on their classical kinetic and quantum kinetic energy terms, along with their rest masses. Furthermore, using the Madelung transformation to combine (82) with the continuity equation (52), we obtain the nonlinear two-particle Schrödinger equation

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for $i\neq j$ and $\psi=\sqrt{\rho}e^{iS/\hbar}$.

Because of the nonlinearity of (83), the 3-space coordinates $\mathbf{q}_{1}$ and $\mathbf{q}_{2}$ in the Green’s function of the gravitational potential in (83) can no longer be interpreted as linear operators (hence why we don’t put hats on them) and $\psi$ no longer has a consistent Born-rule interpretation [53]. (That $\psi$ of (83) has no consistent Born-rule interpretation means that Salcedo’s “statistical consistency problem” for quantum-classical hybrid theories [36, 101] is not applicable in the present context, since Salcedo’s problem assumes the validity of the Born rule and standard quantum measurement postulates for hybrid theories.) Nevertheless, $\rho=|\psi^{2}|$ is still (by definition!) the stochastic mechanical position probability density for the two-particle system and still evolves by the continuity equation (52). The important conceptual distinction here is that the Born-rule interpretation of $|\psi|^{2}$ refers to the probability per unit volume of possible outcomes of projective position measurements on the two-particle system, while the stochastic mechanical definition of $|\psi|^{2}$ refers to the probability per unit volume for the particles to be at 3-space positions $\{\mathbf{q}_{1},\mathbf{q}_{2}\}$ at time $t$ as a result of their stochastic evolutions via (77-78). Thus, a break-down of the Born-rule interpretation does not entail a break-down of the stochastic mechanical meaning of $|\psi|^{2}$.

Nonlinear Schrödinger equations, together with entangled states, are often said to imply superluminal signaling [102, 103, 104], due to the well-known theorem of Gisin [102]. However, as Bacciagaluppi has emphasized [104], superluminal signaling only follows if a theory with a nonlinear Schrödinger equation can also reproduce the usual phenomenology of wavefunction collapse with Born-rule probabilities. Since said phenomenology does not apply to solutions of (83), Gisin’s theorem does not seem applicable here. Of course, it may still be the case that the nonlinear Schrödinger equation (83) implies superluminal signaling, but determining this depends on formulating a stochastic mechanical theory of measurement consistent with (83). Since the nonlinearity of (83) makes naive application of the standard stochastic mechanical theory of measurement [105, 106, 107, 108, 109] unreliable, it remains an open question what variant of the stochastic mechanical theory of measurement is consistent with (83). However, it is expected that such a variant will yield empirical predictions in close agreement with the empirical predictions of the standard stochastic mechanical theory of measurement applied to the linear version of (83). The reason is that the nonlinear terms (84-88) are ridiculously tiny in magnitude compared to the leading term proportional to $m_{1}m_{2}$ in (83). So for all practical purposes, we can ignore the nonlinear terms in modeling the Newtonian gravitational interaction between the two zbw particles, leaving us back to the linear Schrödinger equation (67).

What about the ether’s gravitational contribution? The answer to this will depend on the details of a proposed physical model of the ether, which we currently do not have. Nevertheless, our phenomenological hypotheses about the ether say that it is a medium in space-time with superposed oscillations involving a countably infinite number of modes, and that it continuously exchanges energy-momentum with the zbw particles. So it is reasonable to assume that there must be some stress-energy-momentum associated with the ether. How this stress-energy-momentum gravitates is an open question, but a couple possibilities can be noted: (i) it doesn’t gravitate at all, but rather the coupling of the ether to massive zbw particles somehow induces gravity on a Lorentzian manifold, in analogy with Sakharov’s ‘induced gravity’ proposal 999In Sakharov’s approach, quantum vacuum fluctuations from matter fields don’t gravitate through their stress-energy-momentum tensor; rather, one-loop vacuum fluctuations on a Lorentzian manifold (the latter left to ’flap in the breeze’) generate an effective action that contains terms proportional to the Einstein-Hilbert action, the cosmological constant, plus “curvature-squared” terms [11]. [11]; (ii) it gravitates, but its overall contribution to the total system energy density in the non-relativistic limit is negligible compared to the rest-energy of a zbw particle. In our view, if the ether hypothesis of ZSM is correct, one of these two possibilities must be correct, because all mass-energy quantities experimentally measured in high energy scattering experiments and nuclear binding/decay processes seem to come from three sources: (a) the sum of the rest-masses of the particles, (b) the relativistic kinetic energies of the particles, and (c) the mass-energy associated with interactions between particles via the known fundamental forces. Further supporting this view, we will show in a future paper that a natural (semiclassical) general relativistic extension of ZSM suggests a macroscopic model of the ether as a relativistic non-viscous fluid that gravitates in the Einstein equations but gives a negligible contribution to the total rest-energy of a system of zbw particles in the non-relativistic limit. Hence, for this paper, we shall continue with neglecting the ether in gravitational effects (aside, of course, from the ether’s physical influence on the particles through their zbw oscillations and translational motions).

Finally, note that we have ignored the contribution of gravitational and electrodynamical radiation reaction forces. In a separate paper, we will show how these radiation reaction forces can be consistently incorporated into ZSM-Newton/Coulomb through a stochastic generalization of Galley’s variational principle for nonconservative systems [110].

4 Schrödinger-Newton/Coulomb equations as mean-field theories

We show in this section that ZSM-Newton/Coulomb recovers the ‘single-body’ Schrödinger-Newton/Coulomb equations [23, 27, 30, 31, 35, 38, 16, 55, 17, 43, 47, 54, 77, 78, 79] as mean-field approximations when the number of zbw particles is sufficiently large. For clarity, we separate out the gravitational and Coulomb interactions.

The main idea of a ‘mean-field’ (or ‘large N’) theory is to approximate the evolution of many particles interacting (gravitationally and/or electromagnetically), when N is large (i.e., $N\rightarrow\infty$) and the interactions are weak (in the sense that the gravitational coupling between particles scales as $1/N$) [79]. So for example, if a system of identical particles has the mean-field phase-space density $f(\mathbf{q},\mathbf{p},t)$, the mean-field approximation says that the force exerted on a typical particle in the system by the N other particles is approximated by averaging - with respect to the phase-space density - the force exerted on the typical particle at its 3-space location, from each point in the phase space. Mean-field theory can also be used to approximate the net (gravitational and/or electrostatic) force from a cloud of many weakly interacting identical particles, on an external (macroscopic or mesoscopic or microscopic) body such as a force-measurement probe.

It is instructive to first discuss the mean-field approximation scheme for a classical system of weakly interacting particles. Let us consider a slight variation on the example discussed by Golse in [79], namely, a system of N identical classical point particles, weakly interacting gravitationally, with 6N-dimensional Hamiltonian

[TABLE]

where $V_{g}^{int}(\mathbf{q}_{i}(t),\mathbf{q}_{j}(t))=\frac{1}{2}\sum_{i,j=1}^{N(j\neq i)}\frac{m^{2}}{|\mathbf{q}_{i}(t)-\mathbf{q}_{j}(t)|}$ and the $1/N$ factor is the ‘weak-coupling scaling’ 101010Without the scaling, $V_{g}^{int}$ diverge much faster than the total kinetic energy (a sum of N terms) as $N\rightarrow\infty$, since the sum in $V_{g}^{int}$ is composed of $0.5\,N\left(N-1\right)$ terms. With the scaling, however, $N^{-1}V_{g}^{int}$ scales as N in the $N\rightarrow\infty$ limit. Thus the weak coupling scaling ensures that $V_{g}^{int}$ and the total kinetic energy scale in the same way in the $N\rightarrow\infty$ limit. . Physically, the Hamiltonian (89) describes a collisionless dilute gas of gravitationally interacting non-relativistic particles, and is a special case of the Hamiltonian considered by Golse [79] and Bardos et al. [77, 78] (they considered (89) for an arbitrary, symmetric, smooth interaction potential). The dynamics for the point particles is generated by (89) via Hamilton’s equations $\dot{\mathbf{q}}_{i}(t)=m^{-1}\nabla_{\mathbf{p}_{i}}H$ and $\dot{\mathbf{p}}_{i}(t)=-\nabla_{\mathbf{q}_{i}}H$. Consider now the empirical distribution for the N particles: $f_{N}(\mathbf{q},\mathbf{p},t)\coloneqq N^{-1}\sum_{i=1}^{N}\delta^{3}(\mathbf{q}-\mathbf{q}_{i}(t))\delta^{3}(\mathbf{p}-\mathbf{p}_{i}(t))$, which satisfies (in the sense of distributions) the Vlasov equation

[TABLE]

where

[TABLE]

Then, in the limit $N\rightarrow\infty$, the system described by (89-91) is equivalent to a six-dimensional phase-space density $f(\mathbf{q},\mathbf{p},t)$ (representing the phase space density particles of mass $m$ located at position $\mathbf{q}$ with momentum $\mathbf{p}$ at time $t$) evolving by the ‘large N’ Vlasov equation

[TABLE]

where the time-dependent “mean-field” Hamiltonian $H^{m.f.}(\mathbf{q},\mathbf{p},t)$ is given by

[TABLE]

The last term on the rhs of (93) is the “mean-field” potential energy, i.e., the phase-space averaged potential energy of a typical particle of mass $m$ at position $\mathbf{q}$ at time t. It can be rewritten as

[TABLE]

which tells us it should be interpreted, more precisely, as the sum of the elementary potentials created at position $\mathbf{q}$ by one typical particle located at position $\mathbf{q}^{\prime}$ and distributed according to the 3-space particle number density

[TABLE]

with normalization

[TABLE]

Since we are considering a system of identical particles interacting gravitationally in the Newtonian approximation, the elementary potentials are of the form

[TABLE]

So with (94) and (95), we can rewrite (93) as

[TABLE]

where $\rho(\mathbf{q},t)$ is the source in the Poisson equation

[TABLE]

Hence the mean-field Hamiltonian (93) describes, at time $t$, the total energy of a typical particle with momentum $\mathbf{p}$ at position $\mathbf{q}$ and with mean-field gravitational potential energy $\int_{\mathbb{R}^{3}}m\Phi(\mathbf{q},\mathbf{q}^{\prime})\rho(\mathbf{q}^{\prime},t)d^{3}\mathbf{q}^{\prime}$. Correspondingly, the position-space number density (95) can be shown to evolve by the continuity equation

[TABLE]

upon projecting the Liouville equation for $f(\mathbf{q},\mathbf{p},t)$ into position space, where the mean momentum

[TABLE]

It is interesting to consider the special case when

[TABLE]

where $S_{cl}(\mathbf{q},t)$ is a single-valued classical velocity potential associated to a typical particle at position $\mathbf{q}$ at time $t$. We then have

[TABLE]

and

[TABLE]

In this ‘Hamilton-Jacobi’ case, (100) becomes

[TABLE]

and (104) implies the Hamilton-Jacobi equation

[TABLE]

Accordingly, the Madelung transformation on (105-106) yields the nonlinear Schrödinger equation

[TABLE]

with corresponding Poisson equation

[TABLE]

Here, $\chi_{cl}(\mathbf{q},t)=\sqrt{\rho(\mathbf{q},t)}e^{iS_{cl}(\mathbf{q},t)/\hbar}$ is the classical mean-field ‘wavefunction’, a collective variable describing the evolution of a large number of identical particles that weakly interact gravitationally. Note that the set (107-108) looks formally just like the single-body SN equations, but with the addition of an opposite-signed quantum kinetic defined in terms of the classical mean-field wavefunction. Likewise, if we had started with the description of N identical charged particles weakly interacting electrostatically, with Hamiltonian (89) under the replacement $V_{g}^{int}\rightarrow V_{c}^{int}$, then by taking the large N limit and considering the Hamilton-Jacobi case, we would obtain a nonlinear Schrödinger-Coulomb-like system identical to (107-108), with the charge $-e$ replacing the mass $m$.

We shall now develop a similar mean-field approximation scheme for ZSM-Newton.

To model a dilute ‘gas’ of N identical ZSM particles interacting weakly through Newtonian gravitational forces, we introduce the N-particle quantum Hamilton-Jacobi equation (for simplicity we drop the rest-energy terms) with weak-coupling scaling:

[TABLE]

where $S$ satisfies

[TABLE]

for a closed loop $L$ with $\delta t=0$.

Now, it is well-known in classical mechanics [111, 112, 113] that when harmonic oscillators of the same natural frequency are nonlinearly coupled, they eventually synchronize and oscillate in phase with each other. (The relative phase does oscillate, but in the long run those oscillations average out to zero.) Since the zbw particles are essentially harmonic oscillators of identical natural frequencies and are nonlinearly coupled via $V_{g}^{int}$, it is reasonable to expect that, after some time, their oscillations eventually come into phase with each other. When this ‘phase-locking’ occurs between the zbw particles, we can plausibly make the ansatz that

[TABLE]

where all the $S(\mathbf{q}_{i},0)$ are identical.

Furthermore, since the N-particle continuity equation

[TABLE]

has the general solution

[TABLE]

the initial N-particle osmotic potential takes the form

[TABLE]

So it is also plausible to make the ansatz

[TABLE]

where all the $R(\mathbf{q}_{i},0)$ are identical, which implies that the initial N-particle probability density factorizes into a product of identical single-particle densities:

[TABLE]

From (116) it follows that (112) factorizes into N single-particle continuity equations at $t=0$. Physically speaking, we can interpret (115-116) as corresponding to the assumptions that, at $t=0$, the way that the particle-ether coupling happens, in the local neighborhood of each zbw particle, is identical for all zbw particles (hence identical osmotic potentials sourced by the ether regions in the local neighborhood of each zbw particle), and that the particles are interacting so weakly through $V_{g}^{int}$ and the ether that they can be considered (effectively) physically independent of one another.

Now, it is physically plausible to conjecture that, in the limit $N\rightarrow\infty$, 111111Although we will not give a rigorous mathematical proof of this conjecture, we will see later in this section that the conjecture is corroborated by another large N argument that does have a rigorous mathematical justification. Specifically, the large N limit prescription that leads from the quantum N-body problem to the mean-field Schrödinger-Poisson equation that approximates a system of N quantum particles weakly interacting by $1/r$ (e.g., Newtonian or Coulomb) potentials [77, 78, 79, 17, 43, 54]. the generation of correlations between the motions of the particles get suppressed (because of the weak-coupling scaling) so that time-evolution by (112) yields

[TABLE]

and time-evolution by (109) yields

[TABLE]

where $\rho(\mathbf{q},t)$ satisfies

[TABLE]

and $S(\mathbf{q},t)$ satisfies

[TABLE]

along with

[TABLE]

Although $S(\mathbf{q},t)$ and $\rho(\mathbf{q},t)$ look formally like single-particle variables, they are, in fact, collective variables in a mean-field description of the exact many-body description given by (109-110) with (111) and (115). In particular, $\rho(\mathbf{q},t)$ has the physical meaning of the density of zbw particles of mass $m$ occupying position $\mathbf{q}$ at time $t$. Similarly, $S(\mathbf{q},t)$ is the zbw phase of a typical zbw particle at $\mathbf{q}$ at time t. Accordingly, the last term on the right side of (120) is the quantum kinetic energy of the typical zbw particle at $\mathbf{q}$ at $t$, and

[TABLE]

is the mean-field gravitational potential energy of the typical zbw particle at $\mathbf{q}$ at $t$, where $\Phi$ is the elementary potential given by (97) and $\Phi_{g}^{m.f.}$ satisfies the Poisson equation

[TABLE]

It is worth observing that (119) can also be viewed as the position-space projection of the modified Vlasov equation

[TABLE]

where the initial phase-space density is defined by $f_{0}(\mathbf{q},\mathbf{p})\coloneqq\rho_{0}(\mathbf{q})\delta^{3}\left[\mathbf{p}-\nabla S_{0}(\mathbf{q})\right]$ and

[TABLE]

due time-evolution by (119), along with normalization

[TABLE]

From (125) it follows that the position-space projection of a typical zbw particle’s 3-momentum $\mathbf{p}$ at position $\mathbf{q}$ yields

[TABLE]

for all times, where $\rho=\int_{\mathbb{R}^{3}}f\,d^{3}\mathbf{p}$. The force term in (124) is

[TABLE]

and has the physical interpretation of the net force on a typical zbw particle at $\mathbf{q}$ at $t$, due to spatial gradients of the mean-field gravitational potential energy and quantum kinetic energy of the typical zbw particle at $\mathbf{q}$ at $t$. Correspondingly, it can be readily confirmed that the momentum-space projection of (124), in conjunction with $f(\mathbf{q},\mathbf{p},t)=\rho(\mathbf{q},t)\delta^{3}\left[\mathbf{p}-\nabla S(\mathbf{q},t)\right]$, yields 121212It is readily confirmed that the pressure tensor arising from the momentum-space projection of (124) vanishes, because of the delta function distribution in momentum in the definition of $f$.

[TABLE]

Now, applying the Madelung transformation to (119-121) yields the mean-field nonlinear Schrödinger equation

[TABLE]

with corresponding Poisson equation

[TABLE]

where $\chi(\mathbf{q},t)=\sqrt{\rho(\mathbf{q},t)}e^{iS(\mathbf{q},t)/\hbar}$. Here, the mean-field wavefunction is, like the classical mean-field wavefunction, a collective variable describing the evolution of a large number of identical zbw particles that weakly interact gravitationally. We note that, this time, the set (130-131) formally looks exactly like the single-body SN equations, but with the very different physical meaning as a mean-field approximation in the sense just explained. Similarly, if we had started with the description of N identical charged zbw particles interacting electrostatically, with QHJ equation (109) under the replacement $V_{g}^{int}\rightarrow V_{c}^{int}$, then by taking the large N limit as prescribed above, we would get a nonlinear Schrödinger-Coulomb system identical to (130-131) with $-e$ replacing $m$.

Note that when the quantum kinetic and its first $\nabla_{\mathbf{q}}$ are negligible relative to the mean-field gravitational potential energy and mean gravitational force, (130) effectively turns into the classical nonlinear Schrödinger equation (107), since (120) effectively becomes (106). This observation seems to suggest a ‘quantum-classical’ correspondence between the Hamilton-Jacobi case of the classical Vlasov-Poisson mean-field theory for a collisionless gas or plasma of non-relativistic interacting particles, and the mean-field approximation for N-particle ZSM-Newton/Coulomb. However, such a correspondence is only formal; we will later see that the reliability of (130-131) as a mean-field approximation breaks down for macroscopic superposition states.

To confirm the validity of our mean-field approximation proposal for ZSM-Newton/Coulomb, let us reconsider the dilute gas of N identical ZSM particles interacting through Newtonian gravitational forces, but starting our description from the Schrödinger equation (67) (minus the rest-energy terms and the Coulomb potential) with weak-coupling scaling:

[TABLE]

where

[TABLE]

and

[TABLE]

Supposing all the particles are in the same single-particle pure state $\chi(\mathbf{q})$ at $t=0$, we can make the “Hartree ansatz”

[TABLE]

where the $\chi(\mathbf{q}_{i},0)$ are identical. Then, as shown by Golse [79] and Bardos et al. [77, 78], in the limit $N\rightarrow\infty$, the generation of correlations between particles in time indeed gets suppressed (in the quantum BBGKY hierarchy corresponding to (132-135)), and the time-dependent function $\chi(\mathbf{q},t)$ satisfies (130-131). Likewise for the electrostatic analogues of (132-133). Furthermore, we note that (132-135) is equivalent to (109-116) by virtue of the Madelung transformation. 131313Presumably, then, there exists a Madelung BBGKY hierarchy corresponding to (109-116), for which one can rigorously prove that in the limit $N\rightarrow\infty$ the mean-field Madelung equations (119-123) are recovered. We are unaware of such a proof in the mathematical physics literature, however.

Now, using the solution of (130-131), we can calculate the mean trajectory of a typical zbw particle at position $\mathbf{q}$ through the mean equations of motion

[TABLE]

[TABLE]

as well as the forward/backward stochastic trajectory through the stochastic equations of motion

[TABLE]

[TABLE]

Considering that (130-131) is the leading-order large N approximation to (132-135), trajectories calculated from (136-139) are expected to only very roughly agree with the exact trajectories calculated using the solutions of (132-135), whether for a dilute gas or plasma of identical zbw particles. Of course, in practice, it is impossible to show this explicitly as it is a non-trivial problem to numerically solve the system (132-133), even for just two particles.

Nonetheless, we can improve the mean-field approximation to (132-133) by including the next-order terms in the large N limit. This was recently done by us in [54] by (i) taking the Newtonian limit of the Einstein-Langevin equation of semiclassical stochastic gravity [14], and (ii) directly reconstructing the next-order terms for the mean-field SN equations. The resulting ‘mean-field stochastic SN equations’ read

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The bilocal field $N(\mathbf{q}_{A},\mathbf{q}_{B};t_{A},t_{B})$ is known as the “noise kernel”, and essentially serves as a measure of small (i.e., Gaussian) ‘quantum fluctuations’ of the mass-energy density of the N-particle system, as described by (142-143), between two nearby space-time points $\left\{\mathbf{q}_{A},t_{A}\right\}$ and $\left\{\mathbf{q}_{B},t_{B}\right\}$. (Technically, the noise kernel defined by (142-143) is divergent due to the spatial delta function. This can be remedied by replacing the delta function with a smearing function [17, 44], but for our purposes this detail is inessential.) Furthermore, the noise kernel plays the role of the diffusion coefficient for the classical stochastic (colored) noise field $\xi(\mathbf{q},t)$ (where $<...>_{s}$ refers to the statistical average), the latter of which phenomenologically models the back-reaction of the quantum fluctuations on the gravitational field via $\Phi_{g}^{m.f.+}$. 141414The fact that the noise field is colored instead of white implies that $\xi(\mathbf{q},t)$ is a smooth function, which further implies that the solution of (140-141) is a smooth function. In other words, the noise field in (141) reincorporates the quantum coherence of the gravitational potential to first-order in the large N approximation. To see this last point more explicitly, we can observe that the stochastic correction to $\Phi_{g}^{m.f.}$

[TABLE]

is known [14] to formally reproduce the symmetrized two-point correlation function for the quantized gravitational potential: 151515Equation (145) is deduced as follows. Start from the equality $2<h_{ab}(x_{A})h_{cd}(x_{B})>_{s}\begin{gathered}=\left\langle\Psi\right|\left\{\hat{h}_{ab}(x_{A}),\hat{h}_{cd}(x_{B})\right\}\left|\Psi\right\rangle\end{gathered}$ , where $h_{ab}(x_{A})$ is the classical stochastic metric perturbation at spacetime point $x_{A}$ satisfying the regularized Einstein-Langevin equation (see equation (3.14) of [14]), $\hat{h}_{ab}(x_{A})$ is the quantum metric perturbation operator in the theory of perturbatively quantized gravity (which is equivalent to the weak-field limit of covariant path integral quantum gravity), and $\left|\Psi\right\rangle$ is the quantum state for a quantum field $\hat{\phi}(x)$ in the large N expansion of covariant path integral quantum gravity [7, 13, 14]. Implement the Newtonian limit by assuming $v\ll c$, $g_{ab}=\eta_{ab}+\delta\eta_{ab}$, and $1\gg|T_{00}|/|T_{ij}|$; thus $\Phi_{g}^{+}\coloneqq\frac{1}{2}h_{00}$ and $\hat{\Phi}_{g}\coloneqq\frac{1}{2}\hat{h}_{00}$. Finally, project $\left|\Psi\right\rangle$ onto the coherent state with corresponding complex field $\chi$ [54]. (We show in [54] that the coherent state projection of $\left|\Psi\right\rangle$ is equivalent to the large N limit of the many-body wavefunction $\psi$ of the exact Newtonian quantized-gravitational level of description.) The result is (145).

[TABLE]

We say “formally” because the non-linear evolution (140-141) implies failure of the Born-rule interpretation for $\chi$. Thus the ‘expectation value’ of the rhs of (145) cannot be understood as the standard quantum expectation value. However, since $\chi$ does have a consistent stochastic mechanical statistical interpretation (namely, $|\chi|^{2}$ corresponds to the number density of zbw particles at 3-space point $\mathbf{q}$ at time t), we can ascribe a stochastic mechanical statistical interpretation to the rhs of (145), in the sense that it is equivalent (by the Madelung transformation) to the stochastic mechanical correlation function:

[TABLE]

where $\Phi_{g}(\mathbf{q}_{A},t_{A})$ and $\Phi_{g}(\mathbf{q}_{B},t_{B})$ are solutions of the mean-trajectory Poisson equation (72).

Accordingly, if we use the solution of (140) in (136-139), the resulting trajectories should slightly better approximate the exact trajectories obtained from using the solutions of (132) for very large but finite N. Note that with the solution of (140), the trajectories constructed from integrating (136-137) contain classical (non-Markovian) stochastic fluctuations through the stochasticity of the solution of (140). On the other hand, the trajectories constructed from integrating (138-139) contain classical stochastic fluctuations through the solution of (140) and the (Markovian) stochasticity encoded in the Wiener process $d\mathbf{W}$ ($d\mathbf{W}_{*}$). Note, also, that even though (140-143) are formulated for the case of a dilute system of gravitationally interacting particles, they can also be applied to dilute systems of electrostatically interacting particles, simply by replacing $m\Phi_{g}^{m.f.+}\rightarrow e\Phi_{c}^{m.f.+}$ in (140), which implies the replacements $\xi/c^{2}\rightarrow-\xi/c$ in (141) and $m^{2}c^{4}\rightarrow e^{2}c^{2}$ in (143). Then the ‘stochastic mean-field Schrödinger-Coulomb equations’ provide a next-order correction to the large N limit of the electrostatic analogue of (130-131), and thereby partially reincorporate the quantum coherence of the N-particle electrostatic potential operator. This last point can be seen most explicitly by observing that (145) holds in the electrostatic case as well, when we replace $\Phi_{g}^{+}\rightarrow\Phi_{c}^{+}$ and $\hat{\Phi}_{g}\rightarrow\hat{\Phi}_{c}$.

Finally, let us comment on the limitations of the mean-field approximations considered here.

First, the large N limit leading to (119-121) or (130-131) is only applicable when the inter-particle interactions are sufficiently weak that the independent-particle approximation is plausible. Some example applications of (130-131) to self-gravitating N-particle systems that conform reasonably well to the independent-particle approximation, are boson stars [114, 27, 115] and (when one includes short-range interactions between particles) Bose-Einstein condensates [116, 117]; for electrostatically self-interacting N-particle systems, the electrostatic analogue of (130-131) is widely used in condensed matter physics to model ‘jellium’ (i.e., homogeneous electron gas) systems [118, 119]. On the other hand, for strongly interacting N-particle systems such as (say) superconducting microspheres [56, 51, 57, 48], the independent-particle approximation is a poor one and the deterministic or stochastic SN/SC equations cannot be used.

Second, even for dilute N-particle systems, such as considered above, the mean-field approximations provided by (130-131) and (140-143) become empirically inadequate for calculating the gravitational force on an external (macroscopic or mesoscopic or microscopic) probe, when quantum fluctuations of the mass-energy density of the N-particle system become too large. As an example, for the dilute system of N gravitationally interacting ZSM particles, with total mass $M=Nm$, suppose that the solution of (130) or (140) takes the form of a Schrödinger cat state. In particular, an equal-weighted superposition of two identical Gaussians, where one is peaked at $\frac{1}{2}\mathbf{L}$, the other at $-\frac{1}{2}\mathbf{L}$, and both having zero mean momentum:

[TABLE]

Then the Poisson equation for the mass density corresponding to (131) or (141) takes the form

[TABLE]

or

[TABLE]

with

[TABLE]

[TABLE]

If the spatial separation between the two Gaussians is macroscopic, e.g., $\mathbf{L}=1m$, and if $M=1,000kg$, then the classical gravitational field produced by (148) or (149-151) is totally unrealistic. For example, a probe corresponding to a macroscopic test mass passing through the mid-point of the two mass distributions will, according to (148), go undeflected, or, according to (149-151), will oscillate in between the two mass distributions before passing through with no mean deflection (because of the Gaussian property of the noise field). Both predictions are in stark contrast to what the exact N-particle description (132-133) would predict if $\psi(q)$ takes the form of (147) and one applies the textbook quantum measurement postulates [120, 80] or the stochastic mechanical theory of measurement [105, 106, 107, 108, 109]; namely, that the test mass will either deflect towards the left mass distribution or the right mass distribution, with probability $\frac{1}{2}$ each. 161616Of course, the stochastic SN equations (and the Einstein-Langevin equation more generally) are formulated to handle only dilute N-particle systems with small quantum fluctuations in the matter sector. Cat state solutions clearly fall out of this regime, so it is not surprising that the stochastic SN equations make an empirically inadequate prediction in this case. In order to extend the stochastic SN equations to the case of non-Gaussian fluctuations, we would (presumably) need to incorporate into (140-141) the quantum coherence of the full n-point correlation function involving $\hat{\Phi}_{g}$, in terms of some suitable generalization of the noise kernel. This remains an open problem [13, 14, 54]. Furthermore, apart from the fact that the solutions of (148) or (149-151) don’t have consistent Born-rule interpretations [30, 32, 16, 55, 53], the stochastic mechanical statistical interpretation of the solutions of (148) or (149-151) doesn’t predict a probed gravitational field that’s any more consistent with the prediction obtained from (132-133). And, of course, all these issues with cat states apply as well in the electrostatic case.

As we will see in Part II, the limitations of the mean-field approximations considered above can be circumvented by employing a center-of-mass description of a large N system of ZSM-Newton/Coulomb particles. But next let us compare ZSM-Newton/Coulomb, developed thus far, to other semiclassical theories.

5 Comparison to other semiclassical Newtonian field theories

Here we compare ZSM-Newton/Coulomb, developed thus far, to other semiclassical Newtonian field theories proposed in the literature. In particular, we highlight conceptual advantages of the ZSM-Newton/Coulomb approach and possibilities for experimental discrimination.

5.1 Comparison to non-hidden-variable approaches

Anastopoulos and Hu (AH) [17] have shown that the mean-field SN equations (130-131) can be derived from the standard quantum field theoretic description of a scalar matter field interacting with perturbatively quantized gravity (hereafter PQG): simply take the Newtonian limit of PQG to obtain the N-particle Schrödinger equation (67), consider the case of weakly-coupled systems of identical particles, then apply the large N limit (as we did in (132-135)). Complementing their analysis, we have shown [54] that the mean-field SN equations follow from standard semiclassical Einstein gravity (SCEG) [14, 16, 17], under the following prescription: (i) take the Newtonian limit of the semiclassical Einstein equation (see (155) below) to obtain the Poisson equation with the quantum expectation value of the mass density operator as a source; (ii) introduce the Schrödinger equation in Fock space for the state-vector $\left|\psi\right\rangle$, with a gravitational interaction term in the Hamiltonian involving the solution of the Poisson equation; (iii) identify many-body quantum states consistent with the large N regime of Newtonian PQG (i.e., coherent states); (iv) compute the quantum expectation value of the mass density operator with respect to such a state; and (v) project the Fock space Schrödinger equation into the first-quantized position space representation.

Likewise AH have shown [17] that the mean-field SC equations follow from standard relativistic QED: take the non-relativistic limit, consider a weakly-coupled system of identical particles, then take the large N limit. As with the gravitational case, we have also shown [54] that the mean-field SC equations follow from analogously applying steps (i-v) to standard semiclassical relativistic electrodynamics (SCRED). 171717The semiclassical Maxwell equation of SCRED is given by $\nabla_{\mu}F^{\mu\nu}=\left\langle\psi\right|\hat{J}^{\nu}\left|\psi\right\rangle$, where $\hat{J}^{\nu}$ is the charge four-current operator, $\left|\psi\right\rangle$ is some state-vector, and $\nabla_{\mu}$ is the covariant derivative in case the background spacetime is curved. Taking the non-relativistic limit, introducing the Fock space Schrödinger equation for $\left|\psi\right\rangle$, and taking $\psi$ to be a coherent state, one obtains the mean-field SC system [54]. .

Thus, for weakly-coupled systems of identical particles, the large N limit scheme used in ZSM-Newton/Coulomb can also be employed in Newtonian PQG/QED; and in both cases one recovers the mean-field SN/SC equations. These results also agree with the Newtonian limits of SCEG and SCRED, when the latter are interpreted as mean-field theories for weakly-coupled systems of identical particles.

It is remarkable that these correspondences follow despite ZSM-Newton/Coulomb treating the gravitational/Coulomb potentials as fundamentally classical fields sourced by point-like classical particles undergoing non-classical motions in 3-space. In this respect, the ZSM approach is unique among existing formulations of quantum theory that have been extended to fundamentally-semiclassical gravity or electrodynamics.

For example, it is well known [23, 28, 31, 55, 17, 43, 42, 46, 54] that if one formulates fundamentally-semiclassical gravity based on the equations of either standard non-relativistic quantum mechanics [23, 28, 31, 55, 17, 43, 42, 46, 54] or non-relativistic many-worlds interpretations [40, 55], one obtains the N-body SN equations

[TABLE]

and

[TABLE]

where

[TABLE]

It is also well-known [55, 17, 43, 42, 54] that (152-153) can be obtained from the Newtonian limit of the semiclassical Einstein equation

[TABLE]

if one naively assumes that (155) is valid even when $\psi$ is a single-particle wavefunction, whether in a standard quantum theory reading or a many-worlds interpretation (re: the latter context, see [6, 40, 55]). However, like the mean-field SN equations, the solutions of (152-153) lack consistent Born-rule interpretations [30, 32, 16, 55, 53] and include the macroscopic gravitational cat states discussed in section 4. In other words, attempting to formulate fundamentally-semiclassical gravity, based on either standard quantum theory or many-worlds interpretations, results in a nonlinear classical-gravitational field theory that makes absurd empirical predictions.As another example, it was shown in [55, 48] that the N-body SN equations (with stochastic corrections to dynamically induce intermittent wavefunction collapse) arise naturally when one extends the GRW, CSL, and DP theories to fundamentally-semiclassical gravity with a matter density ontology (called GRWmN, CSLmN, and DPmN, respectively). In contrast to SQM-Newton (where SQM = standard quantum mechanics) and MW-Newton (where MW = many worlds), GRWmN/CSLmN/DPmN have been shown to adequately suppress the empirically problematic macroscopic gravitational cat states while also having consistent statistical interpretations [55, 48]. Thus, these dynamical collapse theories of fundamentally-semiclassical Newtonian gravity are empirically viable. At the same time, these dynamical collapse theories also make slightly different empirical predictions from the Newtonian large N limit of PQG and SCEG; and given the empirical equivalence between Newtonian-large-N PQG and SCEG, and N-particle ZSM-Newton (when the nonlinear terms of the latter are neglected), it will also be the case that these dynamical collapse theories make slightly different empirical predictions from N-particle ZSM-Newton. These slight differences in empirical predictions are entailed by the collapse-inducing stochastic correction terms, and the fact that these dynamical collapse theories still allow for stable gravitational cat states in a mesoscopic regime of masses [55, 48]. The slightly different empirical predictions of these collapse theories may be testable by the next (or next-next) generation of state-of-the-art AMO experiments, as argued by us in [57, 48].

As yet another example, the Tilloy-Diósi (TD) model of fundamentally-semiclassical gravity makes use of the flash ontology within CSL or DP dynamics, to describe fundamentally-semiclassical Newtonian gravitational interactions between N particles, with no nonlinear feedback from the wavefunction. (One can also make a GRW analogue of the TD model, as pointed out by us in [55].) TD’s (stochastic) analogue of the SN equations reads

[TABLE]

up to a fixed spatial cut-off $\sigma$. Here the potential $\hat{V}_{G}$ represents the usual Newtonian gravitational potential operator, while the non-Hermitian terms on the right give rise to decoherence and collapse of spatial superpositions of a massive particle. As shown by TD [66], their model adequately suppresses macroscopic gravitational cat states and has a consistent statistical interpretation. By virtue of the non-Hermitian terms in (156), the TD model also makes slightly different predictions from both Newtonian-limited PQG and ZSM-Newton. These differences might also be testable by the next (or next-next) generation of state-of-the-art AMO experiments [48]. A notable conceptual difference between the TD model and ZSM-Newton is that the former predicts point-like mass distributions (which source the classical gravitational field) that discontinuously appear and disappear in space-time, because the flash ontology is used as the means of defining the mass density sources (we have previously made this point in regards to a GRW analogue of the TD model [55]); by contrast, the mass density sources in ZSM-Newton (the zbw particles) involve no such discontinuities.

Concerning theories of fundamentally-semiclassical electrodynamics, perhaps the best-known is Asim Barut’s “self-field QED” [18, 19, 20, 21]. This theory essentially takes the Schrödinger-Coulomb (SC) analogue of (152-153) (or its relativistic generalization, the Dirac-Maxwell system) as its starting point and purports to reproduce the self-energy effects of non-relativistic and relativistic QED to all orders of perturbation linear in alpha. However, there are more basic predictions of the theory that were left (apparently) unaddressed by Barut and his co-workers, and which seem to make the theory empirically inadequate. First, just like the SN equations, the SC analogue of (152-153) does not have a consistent Born-rule interpretation, thereby preventing a naive application of the standard quantum measurement postulates. Second, also just like the SN equations, the SC equations admit macroscopic electrostatic cat states as solutions, and these solutions are clearly not seen in the real world (incidentally, this rules out the possibility of many-worlds interpretations based on the SC equations). Third, even if one attempts to add stochastic corrections to the SC equations in the form of GRW/CSL/DP, numerical simulations of the SC equations indicate that a free particle wavepacket would undergo Coulomb self-repulsion (from the nonlinear electrostatic self-interaction), and this self-repulsion effect would lead to interference maxima in the two-slit experiment much too broad to be in agreement with existing experimental data [121]. As an alternative formulation of fundamentally-semiclassical electrodynamics based on dynamical collapse theories, we might consider a straightforward electrostatic analogue of TD’s equation (156). Presumably such a theory would be free of the problems entailed by the nonlinearity of the SC equations, but this remains to be explored. In any case, it would appear that, in comparison to theories of fundamentally-semiclassical electrodynamics based on standard quantum mechanics, many-worlds interpretations, and dynamical collapse theories (with matter density ontology), ZSM-Coulomb is the only one that’s empirically viable (within its non-relativistic domain of validity) insofar as it’s empirically equivalent to the Newtonian limits of standard QED and SCRED (modulo the tiny empirical differences entailed by the nonlinear correction terms (86-90) discussed in section 3).

5.2 Comparison to alternative hidden-variable approaches

Other formulations of stochastic mechanics exist besides ZSM [122, 123, 90, 84, 85, 1, 2]. Moreover, dBB pilot-wave theory is the most well-developed hidden-variables formulation of quantum theory to date. Do these other hidden-variables theories have consistent and empirically adequate extensions to semiclassical Newtonian field theories, whether in the form of fundamentally-semiclassical theories or semiclassical approximations? How do they compare and contrast to ZSM-Newton/Coulomb?

As mentioned in section 2, all non-ZSM formulations of stochastic mechanics are subject to Wallstrom’s criticism [84, 85, 124, 104, 1, 2]

• they are all empirically inadequate because they either allow for too many solutions or too few solutions, compared to the Schrödinger equation of standard quantum mechanics. For those formulations that allow too many solutions, one can always impose by hand the quantization condition needed in order to make the solution spaces of those formulations isomorphic to the solution space of standard quantum mechanics [84, 85, 124, 104, 1, 2]. This is, of course, an ad hoc move, but one might view it as provisional until such a condition can be justified by some non-ZSM modification of said formulations of stochastic mechanics. In this case, the amended formulations of stochastic mechanics would result in exactly the same mathematical descriptions of Newtonian gravity and electrodynamics as we’ve found for ZSM, both at the exact (i.e., N-particle Schrödinger equation) level and the level of the mean-field approximation schemes. (Differences would arise, however, in physically motivating the mean-field approximation, e.g., ansatz (111) in section 4; since the $S$ function would not be interpretable as the phase of a periodic phenomenon localized to the stochastic mechanical particle, such an ansatz would have to be imposed ad hoc.)

Concerning semiclassical de Broglie-Bohm theories, let us consider the possibilities separately.

5.2.1 Comparison to fundamentally-semiclassical de Broglie-Bohm theories

There is some ambiguity in how to construct a dBB-based theory of fundamentally-semiclassical Newtonian gravity (or electrodynamics). First, one has to make a choice about which version of dBB dynamics to consider (i.e., the ‘first-order’ or ‘second-order’ version [92, 100, 94, 68, 70, 71, 98, 69]). Second, given a version, one has to make a choice about how to interpret its ontology (e.g., is the wavefunction part of the ontology or does it merely play a ‘nomological’ role in the theory?). Third, one has to make a choice about which part of the dBB ontology - the wavefunction or the particles - plays the role of the mass (or charge) density that sources the classical gravitational (or electromagnetic) field; as it turns out, for versions of dBB in which the particles don’t constitute the only ontic variables, there is no compelling reason why the particles (as opposed to the wavefunction) should be the mass (charge) density source for the classical gravitational (electromagnetic) field, even though that might seem like a prima facie natural choice.

Let us consider this last point in more detail for the gravitational case first, under the first-order ‘dual space’ version of non-relativistic dBB [92, 100, 125, 68, 94, 69]. In other words, the version of dBB theory that posits an ontic 3N-dimensional configuration space, occupied by an ontic ‘universal wavefunction’ $\psi(\mathbf{q}_{1},,,\mathbf{q}_{N},t)$, and an ontic 3-dimensional space (which exists completely independently of the configuration space) occupied by N (spinless) particles with configuration $q(t)=\left\{\mathbf{q}_{1}(t),...,\mathbf{q}_{N}(t)\right\}$. The universal wavefunction 181818The universal wavefunction is required to satisfy the usual boundary conditions of single-valuedness, smoothness, and finiteness. evolves by the Schrödinger equation

[TABLE]

where $V^{int}$ is some scalar interaction potential to be specified and we assume the normalization $\int_{\mathbb{R}^{3N}}|\psi|^{2}d^{3N}q=1$. The particles evolve by the guiding equation

[TABLE]

for all $i=1,..,N$, where the $\nabla S$ form follows if we write $\psi=|\psi|e^{iS/\hbar}$. In addition, we have “equivariance” [70, 98, 69], i.e., the statement that if the initial particle configuration of the dBB system is distributed as $\rho_{0}=|\psi_{0}|^{2}$, then this “quantum equilibrium distribution” [70, 98, 69] is preserved under time-evolution by the quantum continuity equation implicit in (157). In other words, the quantum continuity equation implicit in (157) entails the map $|\psi_{0}|^{2}\rightarrow|\psi_{t}|^{2}$.

Notice that both the wavefunction and the particles ‘feel’ the mass parameters $\left\{m_{1},...,m_{N}\right\}$. More specifically, the time-evolution of $\psi$ (at every point in configuration space) through (157) explicitly depends on all the mass parameters via the kinetic energy operators, while the evolution of $\mathbf{q}_{i}(t)$ depends explicitly on only $m_{i}$ but implicitly on all the other mass parameters through the positions of all the other particles. The dependence of the evolution of $\psi$ on the mass parameters is made even more manifest by starting from the N-particle Bohm-Dirac theory [126, 68], i.e., the most straightforward relativistic N-particle extension of (157-158), and then taking the non-relativistic limit; we would find that the positive-energy components of the Dirac spinor in the Bohm-Dirac theory evolve by a corrected version of (157), where the correction terms are rest-energy terms $\sum_{i=1}^{N}m_{i}c^{2}$ in the Hamiltonian operator.

One might think that since a classical gravitational field lives (by definition!) in 3-space, and since only the particles live in 3-space, this is why the particles should be the (point) sources for the gravitational field. However, recall that the rhs of (153) gives a natural definition of a 3-space mass density in terms of $\psi$ in configuration space.

Consequently, it would seem that inertial mass is a property of both the wavefunction and the particles, and there seems to be no justification for assuming that the particles must be used solely as the mass density sources for a classical gravitational field, if one wants to make a fundamentally-semiclassical Newtonian gravitational theory out of the present version of dBB. Not only that, if one allows $\psi$ to have properties such as energy density, momentum density, etc., one can define the Hamiltonian density

[TABLE]

This Hamiltonian density has the physical interpretation of the energy density stored in the ontic wavefunction, and indicates that the rest-energy terms, hence the $m_{i}$, compose the total mass-energy density of the wavefunction in configuration space. To be sure, nothing in the first-order version of dBB or the dual space version thereof requires that $\psi$ have additional properties like energy density; but nothing excludes these additional properties either. In any case, if one allows $\psi$ to have properties like energy density, then the present version of dBB theory seems to make a compelling case for (at least) taking $\psi$ to be the mass density source for the classical gravitational field.

Given that the dBB theory under consideration is ambiguous about which part of its ontology should be used (or is most natural to use) as the mass density source for a classical gravitational field, let us consider the empirical consequences of using either the wavefunction or the particles or both.

If $\psi$ is used as a source, then the Poisson equation for the classical gravitational field takes the SN form (153), and the Schrödinger equation (157) takes the SN form (152). Because of the conceptual and technical problems with the SN system (152-153), i.e., inconsistency with the Born-rule interpretation and prediction of macroscopic semiclassical gravitational cat states, we must conclude that this version of fundamentally-semiclassical dBB Newtonian gravity (hereafter, dBBfsc-Newton1 where “fsc” = “fundamentally-semiclassical”) is not empirically viable.

If the particles are used as point sources, then the Poisson equation takes the form

[TABLE]

where the $\mathbf{q}_{i}(t)$ are solutions of the guiding equation (158) for all $i=1,..,N$. The solution of (160) then yields the inter-particle gravitational potential energy, which depends on the actual positions of all the dBB particles at a single time, and feeds back into the Schrödinger equation (157), giving

[TABLE]

where

[TABLE]

This version of dBBfsc-Newton (dBBfsc-Newton2) was also considered by Struyve [65], who suggested that it might constitute a viable alternative to the SN equations. (Kiessling considered the electrostatic analogue in [67].) However, dBBfsc-Newton2 appears to be empirically inadequate since $|\psi_{t}|^{2}$ depends now on the actual positions of all the particles at each time, implying that the equivariance property breaks down (i.e., $|\psi(q(t),t)|^{2}$ does not satisfy the quantum continuity equation and can no longer be interpreted as a probability density). The break-down of equivariance means that dBBfsc-Newton2 makes no statistical predictions, whether for gravitational or non-gravitational interactions at the Newtonian level. In other words, even for, say, a non-gravitational position measurement of the dBB particle, the density $|\psi_{t}|^{2}$ corresponding to (161-162) doesn’t have a consistent probabilistic interpretation, whether as a distribution over an ensemble of identical fictitious dBB particles [68, 98] or as a typicality measure [127, 70, 71, 69].

Using both the wavefunction and the particles as mass density sources for classical gravitational fields, doesn’t yield an empirical inadequate theory either; such a theory (dBBfsc-Newton3) lacks the equivariance property (hence making no statistical predictions), and the double counting of the Newtonian gravitational field would lead to gross inconsistency with the gravitational field predicted by classical Newtonian gravity for macroscopic mass distributions.

We must therefore conclude that there does not appear to be an empirically viable formulation of dBBfsc-Newton that’s based on the first-order dual-space version of dBB. Moreover, we do not see how to obtain an empirically viable formulation of dBBfsc-Newton using other versions of first-order dBB theory, whether Albert’s ‘world particle’ formulation 191919In fact, it does not even seem possible to define a fundamentally-semiclassical gravity theory using Albert’s formulation. Albert’s formulation takes as fundamental ontological postulates (i) configuration space $\mathbb{R}^{3N}$, (ii) $\psi$ in configuration space evolving by the N-particle Schrödinger equation, the latter defined in terms of a Hamiltonian that includes an N-particle interaction potential $\hat{V}^{int}(\hat{\mathbf{q}}_{i},\mathbf{\hat{q}}_{j})$ that’s written in a preferred coordinate system, and (iii) a single configuration point (the world particle) in $\mathbb{R}^{3N}$, evolving by the guidance equation. 3-space, and a configuration of particles in 3-space, are claimed to be emergent ontologies in the sense that they are claimed to arise from a philosophical-functionalist analysis of $\hat{V}^{int}$ and the latter’s influence on the motion of the world particle through $\psi$. But a classical gravitational field in 3-space is not part of the emergent ontology. And, of course, presupposing a classical gravitational field living in 3-space is not allowed, as that would contradict the entire purpose of Albert’s formulation (which is to regard configuration space, and the ontological variables living in it, as the fundamental ontologies). [95], Norsen’s TELB formulation 202020The TELB (Theory with Exclusively Local Beables) formulation differs from the dual space formulation in that 3-space is the only ontic space. This approach is (mathematically) motivated by Taylor-expanding $\psi$ in configuration space into an infinite hierarchy of nonlocally coupled fields in 3-space; more precisely, each particle has a single-particle wavefunction pushing it around via the guidance equation, but the single-particle wavefunction is coupled to an infinite hierarchy of 3-space “entanglement fields”, which are themselves nonlocally coupled to the entanglement fields of every other particle (hence why they are called “entanglement” fields). The postulate $\rho_{0}=|\psi_{0}|^{2}$ is still imposed on the single-particle wavefunctions, and equivariance still holds. One could then define classical gravitational fields directly in terms of mass density sources built out of the single-particle wavefunctions, but this would just lead to a TELB version of the SN equation, which would entail all the empirically problematic predictions of the SN equation (e.g., macroscopic gravitational cat states). Furthermore, the break-down of equivariance from using the particles as point sources would still remain. [96, 97], or Dürr-Goldstein-Zanghì’s nomological formulation 212121The nomological formulation is still conjectural, but the basic idea is that the ‘fundamental’ wavefunction is the time-independent Wheeler-DeWitt wavefunctional $\Psi\left(h,\phi\right)$, interpreted as part of physical law rather than physical ontology. Time-dependent wavefunctions are suggested to be derived, effective descriptions for ‘subsystems’ of the universe, and not part of physical ontology either. Only 3-space and particles living in 3-space constitute physical ontology. Accordingly, one cannot not use time-dependent wavefunctions in the definition of an SN-type classical mass-density source in 3-space, as this would be inconsistent with the expected Newtonian limit of the Wheeler-DeWitt equation (i.e., the usual linear Schrödinger equation involving an operator-valued gravitational interaction potential) [128, 129]. Nor could one use the dBB particles as point sources for a classical gravitational field coupling back to the time-dependent wavefunction, as this is inconsistent with the expected Newtonian limits of the Wheeler-DeWitt equation and the guiding equations for $h$ and $\phi$ (i.e., the equations of dBB-Newton). So the nomological formulation of dBB also doesn’t allow for a formulation of fundamentally-semiclassical Newtonian gravity. [128, 70, 129, 71, 69]. Second-order formulations of dBB, namely the “ontological interpretation” advocated by Bohm-Hiley [94] and Holland [68], don’t seem to change the situation either: their only difference from first-order formulations of dBB is that the Schrödinger equation and wavefunction are replaced by the Madelung equations for $|\psi|$ and $S$, with the quantization condition imposed on the latter.

By comparison, while the ontology of ZSM-Newton involves more than just particles, it is clear from the very formulation of ZSM-Newton that the particles must be understood as possessors of inertial mass. This is manifest from (i) the definition of the rest-mass of a zbw particle as corresponding to the energy associated with the Compton frequency oscillation of the zbw particle in its rest frame, and (ii) the definition of the i-th Wiener process, which describes the stochastic evolution of the i-th particle position and depends on the i-th mass parameter through the diffusion coefficient $\hbar/m_{i}$. Furthermore, as we argued in section 3, while the ether of ZSM is expected to carry stress-energy, it is expected to be negligible in the Newtonian regime. Thus, in contrast to dBB, ZSM seems to make the choice of the particles as mass density sources for a classical gravitational field, inevitable. Another difference from dBB is the following: recall from section 3 that, because the Schrödinger equation and wavefunction are derived in ZSM, the use of the particles as sources for a classical gravitational field doesn’t entail the nonlinear coupling in (161-162); rather, as we saw in section 3, the gravitational field that does couple to the Schrödinger equation/wavefunction corresponds (to leading order) to $\hat{V}_{g}^{int}(\mathbf{\hat{q}}_{i},\mathbf{\hat{q}}_{j})$. This is why ZSM-Newton avoids a break-down of the equivariance property. So despite ZSM and dBB sharing many equations in common - the Schrödinger equation (157), the guiding equation (158), and equivariance of $\rho_{0}=|\psi_{0}|^{2}$

• and despite both theories sharing in common a “primitive ontology” 222222Primitive ontology is defined by Allori et al. [130] as “variables describing the distribution of matter in 4-dimensional space-time”. involving particles with definite 3-space trajectories, the different axioms on which ZSM and dBB are based lead to significantly different conclusions about how to formulate a theory of fundamentally-semiclassical Newtonian gravity, and the empirical viability thereof.

It is a straightforward exercise to demonstrate that analogous conclusions follow from consideration of the electrodynamical case, i.e., dBBfsc-Coulomb theories vs. ZSM-Coulomb. This being said, we wish to evaluate a well-known peculiarity of standard dBB theory involving charge-field coupling (i.e., dBB-Coulomb), from the viewpoint of ZSM-Coulomb.

For a single-particle dBB system, in the presence of an external magnetic vector potential $\mathbf{A}_{ext}(\mathbf{q},t)$, the momentum operator in the Schrödinger equation gets a correction $\mathbf{\hat{p}}\rightarrow\mathbf{\hat{p}}-e\mathbf{A}_{ext}$. Now, consider the magnetic Aharonov-Bohm (AB) effect in dBB [131, 94, 68], where $\mathbf{A}_{sol}=\left(\Phi/2\pi r\right)\hat{\theta}$ is the magnetic vector potential sourced by an infinitely long cylindrical solenoid with flux $\Phi$. For an electron wavepacket split into two partial packets passing on either side of the solenoid, where the paths $P_{1}$ and $P_{2}$ traversed by the packets form a loop $C$ encircling the solenoid, the correction to the momentum operator entails a phase shift $\psi\rightarrow\psi^{\prime}=N^{\prime}\left[\psi_{1}+\psi_{2}e^{ie\Phi/\hbar}\right]e^{\left(ie/\hbar\right)\int_{P_{1}}\mathbf{A}_{sol}\cdot d\mathbf{q}}$, when the packets are recombined to form an interference pattern ($N^{\prime}$ is a normalization constant). Correspondingly, the position probability density associated to the interference pattern gets shifted as $|\psi^{\prime}|^{2}=\rho^{\prime}=N^{\prime 2}\left\{\rho_{1}+\rho_{2}+2\sqrt{\rho_{1}}\sqrt{\rho_{2}}cos\left[(S_{1}-S_{2})/\hbar-\delta\right]\right\}$, where $\delta=e\Phi/\hbar$. Note that while the dBB particle moves along with only one of the packets around the solenoid, say the packet traversing path $P_{1}$, with modified momentum $\mathbf{p}=\nabla S_{1}-e\mathbf{A}_{sol}$, both packets ‘feel’ $\mathbf{A}_{sol}$ since each picks up a phase factor $\psi_{a}\rightarrow\psi_{a}e^{(ie/\hbar)\int_{P_{a}}\mathbf{A}_{sol}\cdot d\mathbf{q}}$ such that $\Phi=\oint_{C}\mathbf{A}_{sol}\cdot d\mathbf{q}=\int_{P_{1}}\mathbf{A}_{sol}\cdot d\mathbf{q}-\int_{P_{2}}\mathbf{A}_{sol}\cdot d\mathbf{q}$ and $\oint_{C}\mathbf{p}\cdot d\mathbf{q}=nh-e\Phi$. In other words, even though the motion of the dBB particle is altered by the presence of the vector potential, suggesting (seemingly) that the charge $e$ is a property localized to the dBB particle (like in classical electrodynamics), the fact that the ‘empty’ packet (i.e., the packet moving along $P_{2}$) also picks up a phase factor, and that this phase factor contributes to the shift in the interference pattern of the recombined packets, suggests that charge is also a property carried by the (spatially delocalized) wavefunction [131, 94, 68, 132]. A completely analogous situation arises for the gravitational analogue of the magnetic AB effect, where $\mathbf{A}_{sol}$ is the gravitomagnetic vector potential sourced by a solenoid carrying a mass (instead of charge) current, and all other expressions are identical except for the replacement $e\rightarrow m$ [133]. Analogous considerations apply to the case of the electric/gravitoelectric AB effect.

Since the dBB treatment of the AB effect is formally the same as the ZSM-Coulomb/Newton treatment of the AB effect, this might seem to conflict with the ZSM-Coulomb/Newton hypothesis that the charge (rest mass) of a system is a property localized to zbw particles. However, there is no inconsistency. In ZSM-Coulomb/Newton, the finding that the empty packet in the AB effect picks up a phase factor that contributes to the shift in the interference pattern intensity is a consequence of the following set of postulates: (i) rest-mass and charge are intrinsic properties of zbw particles; (ii) the zbw particles, whose oscillations are dynamically driven by the ether medium, always have well-defined mean phases along their 3-space trajectories; and (iii) the diffusion process for the zbw particles in the ether satisfies the global constraint of being conservative. It might then be asked if ZSM-Coulomb/Newton gives physical insight into what it means, in terms of its proposed underlying ontological picture of the world, for empty packets to electromagnetically (or gravitationally) couple to external fields, even though it is the zbw particles that carry the rest-mass and charge of a system. We can sketch an answer as follows.

As discussed in [134, 120, 1], the superposition principle for wavefunctions is a consequence of the single-valuedness condition, and the single-valuedness condition on wavefunctions in ZSM follows from the union of postulates (ii) and (iii). And as we’ve discussed in [2], an empty packet describes possible alternative histories of a Nelsonian/zbw particle through a different region of the ether (the different region corresponding to the spatial support of the empty wavepacket in 3-space), while also indirectly reflecting spatio-temporal variations in that different region of the ether (because the ether-sourced osmotic potential $U(q,t)$ changes as a function of space and time via the continuity equation and is constrained by boundary conditions in the environment). Thus the empty packet traversing path $P_{2}$ reflects (indirectly) a region of the ether that’s (spatio-temporally) varying along $P_{2}$, and the interference of the recombined packets reflects (indirectly) two regions of ether recombining and interfering while satisfying postulates (ii) and (iii). Since the ether medium is presumed to pervade all of 3-space, and since all components of the ether are presumed to be nonlocally connected to each other, the ether region corresponding to the empty packet is actually not physically independent of the ether region corresponding to the occupied packet. In other words, for the ether to maintain the quantization condition $\oint_{C}\nabla S\cdot d\mathbf{q}=nh$ on the zbw particle, while maintaining that the diffusion of the zbw particle through the ether is conservative, it must know to compensate for the phase shift experienced by the zbw particle passing around the solenoid along $P_{1}$, by correspondingly shifting phase in the region that’s spatio-temporally varying along $P_{2}$. How exactly this works (assuming the ZSM framework is correct) will presumably require developing an explicit physical model of the ether, the zbw particle, and the dynamical coupling of the two, in accord with postulates (i-iii). This is left for future work.

5.2.2 Comparison to semiclassical approximations in de Broglie-Bohm theory

As we’ve seen, there does not appear to exist an empirically viable formulation of dBBfsc-Newton/Coulomb. Nevertheless, it is possible to formulate semiclassical approximation schemes for the ‘fully quantum’ formulation of dBB Newtonian gravity/electrodynamics (hereafter, dBB-Newton/Coulomb).

The dBB-Newton/Coulomb theory corresponds to (157-158) with $V^{int}=\hat{V}_{g,e}^{int}(\mathbf{\hat{q}}_{i},\mathbf{\hat{q}}_{j})$ (for simplicity, we neglect vector potentials). In other words the N-particle Schrödinger equation of dBB-Newton/Coulomb is identical to the N-particle Schrödinger equation of ZSM-Newton/Coulomb, when the nonlinear correction terms predicted by the latter are neglected. The physical interpretation, however, is different.

In ZSM-Newton/Coulomb, the zbw particles carry rest-mass/charge and interact with one another through the classical gravitational/electrostatic fields they source. In dBB-Newton/Coulomb the particles are just points at definite locations, and $\hat{V}_{g,e}^{int}(\mathbf{\hat{q}}_{i},\mathbf{\hat{q}}_{j})$ is a potential energy function on configuration space that influences the evolution of $\psi$ in configuration space; so, to the extent that the particles ‘interact’ gravitationally or electrostatically, they only do so indirectly via the influence of $\hat{V}_{g,e}^{int}(\mathbf{\hat{q}}_{i},\mathbf{\hat{q}}_{j})$ on $\psi$ through the Schrödinger equation (157), and the influence of $\psi$ on the evolution of the particles through the guiding equation (158). Thus the mean-field approximation scheme discussed in section 4 applies just as well to dBB-Newton/Coulomb.

Another dBB-based semiclassical approximation scheme has been suggested by Prezhdo-Brooksby [58] and elaborated on by Struyve [65]. Consider, for simplicity, the dBB theory with two-particle Schrödinger equation

[TABLE]

The guiding equations for each particle are again given by (158), and the 2nd-order equations of motion are

[TABLE]

where $Q\left(\mathbf{q}_{1},\mathbf{q}_{2}\right)$ is the total quantum potential of the two-particle system.

Now, the conditional wavefunction for particle 1, defined as $\psi_{1}(\mathbf{q}_{1},t)=\psi(\mathbf{q}_{1},\mathbf{q}_{2}(t),t)$, satisfies the conditional Schrödinger equation

[TABLE]

where

[TABLE]

Correspondingly, the conditional guiding equation for particle 1 is

[TABLE]

where $S_{1}=S_{1}(\mathbf{q}_{1},t)$. The Newtonian equation of motion for particle 2 is then

[TABLE]

The semiclassical approximation is when $m_{2}\gg m_{1}$ and $\psi$ varies slowly in $\mathbf{q}_{2}$ (compared to $\mathbf{q}_{1}$). Then $K\approx 0$ and $-\nabla_{2}Q\approx 0$. In other words the time-evolution of particle 2 depends (approximately) only on the classical interaction potential $V_{g,e}^{int}$, evaluated at the actual position of particle 1. And the time-evolution of particle 1 depends on $\psi_{1}$ satisfying (approximately) (165) with $K\approx 0$, i.e., particle 1’s effective Schrödinger equation that takes into account the back-reaction of particle 2 through $V_{g,e}^{int}\left(\mathbf{q}_{1},\mathbf{q}_{2}(t)\right)$. Note that, unlike models of dBBfsc-Newton/Coulomb, this semiclassical approximation scheme defines a consistent back-reaction between the two particles in the following sense: the conditional wavefunction of particle 1, in the semiclassical approximation, just corresponds to the effective wavefunction of particle 1, for which $|\psi_{1}|^{2}$ satisfies an equivariance-like property (through the conditional quantum continuity equation implicit in (165)), even though the (semiclassically approximated) evolution of $\psi_{1}$ still depends on the actual position of particle 2 through $V_{g,e}^{int}$.

By contrast, the standard QM semiclassical approximation scheme for two interacting particles [58, 65] is defined by

[TABLE]

[TABLE]

where $\psi(\mathbf{q}_{1},t)$ is a single-particle wavefunction (as opposed to a conditional or effective wavefunction), and the back-reaction from particle 2 on particle 1 is via the average trajectory $\mathbf{\overline{q}}_{2}(t)$ inserted into $V_{g,e}^{int}$ in (169).

Prezhdo and Brooksby [58] have compared the dBB-based semiclassical approximation scheme to this standard QM scheme, for the case of a light particle scattering off a heavy particle, where the heavy particle is bound to a fixed surface. They found that the dBB scheme is superior at tracking the scattering probability as a function of time (when compared to the exact quantum dynamics description), in addition to being computationally simpler to implement than the standard QM scheme.

Struyve [65] has applied the dBB-based scheme to a dBB version of scalar electrodynamics, as well as to a dBB version of canonical quantum gravity under the minisuperspace approximation 232323Canonical quantum gravity under the minisuperspace approximation refers to the Wheeler-DeWitt equation $\mathcal{H}\Psi\left(h,\phi\right)=0$ (and momentum constraint $\mathcal{H}_{i}\Psi\left(h,\phi\right)=0$), under the restriction that the 3-metric $h$ and matter field $\phi$ are homogeneous and isotropic [68, 15, 65]. This corresponds to a time-dependent homogeneous matter field $\phi(t)$ in an FLRW metric with homogeneous scale factor $a(t)$. The Wheeler-DeWitt equation then takes the form $\left(H_{metric}+H_{matter}\right)\psi\left(a,\phi\right)=0$. In the dBB version [65], this latter form of the Wheeler-DeWitt equation is accompanied by guidance equations for the field beables $a(t)$ and $\phi(t)$, which turn out to be coupled to each other via the phase $S$ of $\psi$. In this way, the metric and matter field beables back-react on each other. It is worth mentioning that the minisuperspace approximation is also referred to in the literature as a ‘semiclassical’ approximation; it should not be confused with the dBB-based semiclassical approximation scheme, the latter of which is applied by Struyve on top of the minisuperspace approximation. . In the latter case, he has compared the dBB-based scheme to the standard scheme (applied to standard canonical quantum gravity under the minisuperspace approximation) for cases involving macroscopic superpositions of two Gaussians wavepackets. As it turns out, the dBB-based scheme yields better agreement with the exact dBB version of canonical quantum gravity under the minisuperspace approximation, than does the standard scheme 242424Struyve did not compare the standard scheme to the standard quantum interpretation of the Wheeler-DeWitt equation, the reasoning being that the “problem of time” makes the standard quantum interpretation of the Wheeler-DeWitt equation incoherent. Nevertheless, Struyve pointed out that for approaches to quantum theory that associate approximately classical dynamics to macroscopic superpositions of Gaussian states (such as many-worlds interpretations [15]), the standard scheme is expected to do worse than the dBB scheme in approximating exact solutions of the Wheeler-DeWitt equation (assuming those non-dBB approaches to quantum theory yield consistent quantum interpretations of the Wheeler-DeWitt solutions in the first place)..

The dBB semiclassical approximation scheme for two interacting particles can, of course, be imported into ZSM-Newton/Coulomb. In this sense, the results obtained by Prezhdo-Brooksby are also results that follow from ZSM-Newton/Coulomb. However, in ZSM-Newton/Coulomb, we also have the option of implementing the back-reaction from particle 1 onto particle 2 via the conditional stochastic differential equation for particle 1: $d\mathbf{q}_{1}(t)=\left(\mathrm{Im}+\mathrm{Re}\right)m_{1}^{-1}\hbar\nabla_{1}\ln\psi_{1}|_{\mathbf{q}_{1}=\mathbf{q}_{1}(t)}dt+d\mathbf{W}(t)$. Since the trajectories predicted by this stochastic differential equation differ from the trajectories predicted by the conditional guidance equation (167), we would expect differences in the predictions of the ZSM-Newton/Coulomb version as compared to the dBB version. Although, considering that the semiclassical approximation requires the mass of particle 2 to be much greater than particle 1, we would expect any differences to be very slight. Nevertheless, it would be interesting to revisit the cases studied by Prezhdo-Brooksby and Struyve, to see if the differences might be amenable to experimental/observational discrimination. (Revisiting Struyve’s analyses from the viewpoint of ZSM will of course require extending ZSM to relativistic field theories in flat and curved spacetimes, and to the spacetime metric itself. Future work will show how this can be done.)

6 Conclusion

We have shown how to formulate fundamentally-semiclassical Newtonian gravity/electrodynamics based on stochastic mechanics in the ZSM formulation. In addition, we have shown that ZSM-Newton/Coulomb has a consistent statistical interpretation, recovers the standard exact quantum description of matter-gravity coupling as a special case valid for all practical purposes (even though gravity remains fundamentally classical in the ZSM approach), and recovers the SN/SC and stochastic SN/SC equations as mean-field approximations. We have also compared ZSM-Newton/Coulomb to theories of semiclassical Newtonian gravity based on standard quantum theory, dynamical collapse theories, other possible formulations of stochastic mechanics, and the dBB pilot-wave theory. In doing so, we have highlighted conceptual and technical advantages entailed by ZSM-Newton/Coulomb, and indicated possibilities for experimentally testable differences.

In Part II, we will use ZSM-Newton/Coulomb to formulate a new ‘large-N’ prescription that makes it possible to consistently describe large numbers of identical (ZSM) particles strongly interacting classical-gravitationally/electrostatically. This new large-N prescription will also make it possible to recover classical Newtonian gravity/electrodynamics for macroscopic particles, as well as classical Vlasov-Poisson mean-field theory for macroscopic particles weakly interacting gravitationally/electrostatically.

We wish to emphasize once more the two key results of the present paper: (i) while ZSM-Newton and ZSM-Coulomb treat the gravitational and Coulomb potentials, respectively, as fundamentally classical fields sourced by point-like classical particles undergoing non-classical (stochastic mechanical) motions in 3-space, these semiclassical theories nevertheless recover the standard quantum descriptions of Newtonian/non-relativistic gravitational/Coulombic interactions between particles; and (ii) the large N limit scheme of Golse [79] and Bardos et al. [77, 78], applied to ZSM-Newton/Coulomb, makes it possible to recover the same mean-field approximations as obtained from standard Newtonian PQG/SCEG and standard non-relativistic QED/SCRED (the SN/SC and stochastic SN/SC equations).

In a forthcoming standalone paper, we will elaborate on one of the possibilities for experimentally testing ZSM-Newton vs. other theories of semiclassical Newtonian gravity, namely, the “grav-cat” setup proposed by Derakhshani, Anastopoulos, and Hu [57, 48]. In another standalone paper, we will show how to consistently incorporate gravitational and electrodynamical radiation reaction effects within ZSM-Newton and ZSM-Coulomb, respectively, through a stochastic mechanical generalization of Galley’s variational principle for nonconservative systems [110]. Further down the road, we will show how to extend ZSM to particles and fields in relativistic spacetimes, and then use that framework to formulate consistent hidden-variables theories of semiclassical Einstein gravity and semiclassical relativistic electrodynamics; we will then show that the Newtonian limits of these two theories yield ZSM-Newton and ZSM-Coulomb, respectively.

7 Acknowledgments

I thank Guido Bacciagaluppi, Roderich Tumulka, Bei-Lok Hu, and Dieter Hartmann for helpful discussions.