  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.
