Radiation reaction from QED: Lightfront perturbation theory in a plane wave background

We derive dynamical, real time radiation reaction effects from lightfront QED. Combining the Hamiltonian formalism with a plane wave background field, the calculation is performed in the Furry picture for which the background is treated exactly while interactions between quantum fields are treated in perturbation theory as normal. We work to a fixed order in perturbation theory, but no other approximation is made. The literature contains many proposals for the correct classical equation describing a radiating particle; we take the classical limit of our results and identify which equations are consistent with QED.

Figure 1

A pulsed plane wave has finite extent in $x^{+}$, but infinite extent in the remaining directions. All particles, independent of their momentum, enter and leave such a pulse at the same lightfront times, here $x^{+}=0$ and $x^{+}=x_f^{+}$, respectively. There is no such symmetry in the usual time parameter $x^0$.

Figure 2

The Furry-Feynman diagram describing the zeroth order (Lorentz-force) contribution to the average electron momentum, $⟨\mathrm{\text{free}}{⟩}_{\mu }$. Double lines indicate the background-dressed propagator. (For simplicity we draw, here and below, the diagrams of the covariant theory, rather than lightfront time-ordered diagrams).

Figure 3

Furry-Feynman diagrams for two of the $\mathcal{O}(e^2)$ contributions to the average electron momentum, and to radiation reaction. In the first line, the cross term from scattering without emission, and in the second line, nonlinear Compton scattering, at tree level, mod squared.

Figure 4

All possible cuts of these diagrams generate all possible terms to be calculated for $⟨k{P}^e⟩$ to order $e^2$: dashed vertical lines correspond to a cut and the insertion of $k{P}^e=k{P}^{(0)}$.

Figure 5

Order $e^4$ contributions to RR come from all possible cuts of these diagrams. (The required counterterms are not shown.) Vacuum polarization and pair production effects are confined to all possible cuts of the second and third diagrams in the third line, the latter containing the four-scalar interaction.