  We derive from a microscopic Hamiltonian a set of stochastic equations of
motion for a system of spinless charged particles in an electromagnetic (EM)
field based on a consistent application of a dimensionful 1/c expansion of
quantum electrodynamics (QED). All relativistic corrections up to order 1/c^3
are captured by the dynamics, which includes electrostatic interactions
(Coulomb), magnetostatic backreaction (Biot-Savart), dissipative backreaction
(Abraham-Lorentz) and quantum field fluctuations at zero and finite
temperatures. With self-consistent backreaction of the EM field included we
show that this approach yields causal and runaway-free equations of motion,
provides new insights into charged particle backreaction, and naturally leads
to equations consistent with the (classical) Darwin Hamiltonian and has quantum
operator ordering consistent with the Breit Hamiltonian. To order 1/c^3 the
approach leads to a nonstandard mass renormalization which is associated with
magnetostatic self-interactions, and no cutoff is required to prevent runaways.
Our new results also show that the pathologies of the standard Abraham-Lorentz
equations can be seen as a consequence of applying an inconsistent (i.e.
incomplete, mixed-order) expansion in 1/c, if, from the start, the analysis is
viewed as generating a low-energy effective theory rather than an exact
solution. Finally, we show that the 1/c expansion within a Hamiltonian
framework yields well-behaved noise and dissipation, in addition to the
multiple-particle interactions.
