Nonequilibrium Dynamics of Charged Particles in an Electromagnetic Field: Causal and Stable Dynamics from 1/c Expansion of QED

TL;DR

This paper derives causal, stable equations of motion for charged particles interacting with an electromagnetic field using a systematic 1/c expansion of QED, capturing relativistic corrections and avoiding standard pathologies.

Contribution

It introduces a consistent 1/c expansion approach to derive equations of motion that are causal, free of runaways, and include quantum and relativistic effects up to order 1/c^3.

Findings

Equations include Coulomb, Biot-Savart, and Abraham-Lorentz effects.

No cutoff needed to prevent runaways in the derived equations.

Reveals that standard Abraham-Lorentz pathologies result from inconsistent expansions.

Abstract

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.