We extend our previous work (see arXiv:quant-ph/0501026), which compared the predictions of quantum electrodynamics concerning radiation reaction with those of the Abraham-LorentzDirac theory for a charged particle in linear motion. Specifically, we calculate the predictions for the change in position of a charged scalar particle, moving in three-dimensional space, due to the effect of radiation reaction in the one-photon-emission process in quantum electrodynamics. The scalar particle is assumed to be accelerated for a finite period of time by a three-dimensional electromagnetic potential dependent only on one of the spacetime coordinates. We perform this calculation in theh → 0 limit and show that the change in position agrees with that obtained in classical electrodynamics with the Lorentz-Dirac force treated as a perturbation. We also show for a time-dependent but space-independent electromagnetic potential that the forward-scattering amplitude at order e 2 does not contribute to the position change in theh → 0 limit after the mass renormalization is taken into account.

We investigate the motion of a wave packet of a charged scalar particle linearly accelerated by a static potential in quantum electrodynamics. We calculate the expectation value of the position of the charged particle after the acceleration to first order in the fine structure constant in the → 0 limit. We find that the change in the expectation value of the position (the position shift) due to radiation reaction agrees exactly with the result obtained using the Lorentz-Dirac force in classical electrodynamics. We also point out that the one-loop correction to the potential may contribute to the position change in this limit.PACS numbers: 03.65.-w, 12.20.-m A charged particle radiates when it is accelerated. The resulting change in its energy and momentum is described by the Lorentz-Dirac (or Abraham-Lorentz-Dirac) force in classical electrodynamics [1,2,3]. (See, e.g., Ref.[4] for a modern review.) Thus, if a charge e with mass m is accelerated by an external 4-force F µ ext , then its coordinates x µ (τ ) at the proper time τ obey the following equation:where the Lorentz-Dirac 4-force F µ LD is given byWe have let c = 1 and defined α c ≡ e 2 /4π. Our metric is g µν = diag (+1, −1, −1, −1). Although there are many ways to derive Eq. (1) in classical electrodynamics for a point charge (see, e.g., Ref.[5]), a natural question one can ask is whether or not this equation can arise in the → 0 limit in QED. It was found in Ref. [6] (after an initial claim to the contrary) that the position of a linearly accelerated charged particle in the Lorentz-Dirac theory is reproduced by the → 0 limit of the one-photon emission process in QED in the non-relativistic approximation. (See, e.g., Refs. [7,8] for other approaches to arrive at the Lorentz-Dirac theory from QED.) In this Letter we describe the generalization of this work to a fully relativistic charged particle. The details will be published elsewhere [9].Consider a charged particle with charge e and mass m moving in the positive z-direction under a potential energy V (z). We assume that V (z) = V 0 = const. for z < −Z 1 and V (z) = 0 for −Z 2 < z for some Z 1 and Z 2 , both positive constants. Thus, there is non-zero acceleration only in the interval (−Z 1 , −Z 2 ). The external 4-force in Eq. (1) generated by this potential is given byThe Lorentz-Dirac 4-force can be givenA dot indicates the derivative with respect to t. We have defined γ ≡ (1 −ż 2 ) −1/2 as usual. Suppose that this particle would be at z = 0 at time t = 0 if the Lorentz-Dirac force was absent (i.e. if e = 0). The true position at t = 0, which we denote δz and call the position shift, can readily be found to lowest nontrivial order in F LD by treating the Lorentz-Dirac force as perturbation. (It was proposed in Ref.[10] that one should treat the Lorentz-Dirac force as perturbation.) The calculation can be facilitated by using the fact that the total energy, m dt/dτ + V (z), changes by the amount of work done by the Lorentz-Dirac force. Thus,Rearranging and integrating, and then changing the order of ...

We investigate the effect of radiation reaction on the motion of a wave packet of a charged scalar particle linearly accelerated in quantum electrodynamics. We give the details of the calculations for the case where the particle is accelerated by a static potential that were outlined in Phys. Rev. D 70 (2004) 081701(R) and present similar results in the case of a time-dependent but space-independent potential. In particular, we calculate the expectation value of the position of the charged particle after the acceleration, to first order in the fine structure constant in the → 0 limit, and find that the change in the expectation value of the position (the position shift) due to radiation reaction agrees exactly with the result obtained using the Lorentz-Dirac force in classical electrodynamics for both potentials. We also point out that the one-loop correction to the potential may contribute to the position change in this limit.

We analyze the change in position (the position shift) of the wave packet of a charged scalar particle due to radiation reaction in the → 0 limit of quantum electrodynamics. In particular, we re-express the formula previously obtained for the position shift in terms of Green's functions for the electromagnetic field, thus clarifying the relation between the quantum and classical derivations of the radiation-reaction force.