# 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.

## Key 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.