Stochastic Theory of Relativistic Particles Moving in a Quantum Field:   I. Influence Functional and Langevin Equation
  We treat a relativistically moving particle interacting with a quantum field from an open system viewpoint of quantum field theory by the method of influence functionals or closed-time-path coarse-grained effective actions. The particle trajectory is not prescribed but is determined by the backreaction of the quantum field in a self-consistent way. Coarse-graining the quantum field imparts stochastic behavior in the particle trajectory. The formalism is set up here as a precursor to a first principles derivation of the Abraham-Lorentz-Dirac (ALD) equation from quantum field theory as the correct equation of motion valid in the semiclassical limit. This approach also discerns classical radiation reaction from quantum dissipation in the motion of a charged particle; only the latter is related to vacuum fluctuations in the quantum field by a fluctuation-dissipation relation, which we show to exist for nonequilibrim processes under this type of nonlinear coupling. This formalism leads naturally to a set of Langevin equations associated with a generalized ALD equation. These multiparticle stochastic differential equations feature local dissipation (for massless quantum fields), nonlocal particle-particle interactions, multiplicative noise, and nonlocal particle-particle correlations, interrelated in ways characteristic of nonlinear theories, through generalized fluctuation-dissipation relations. 
Philip R. Johnson and B.L. Hu