Which classical radiation-reaction equation is the classical limit of QED? · revision history

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Title

Which classical radiation-reaction equation is the classical limit of QED

Plain

The classical Abraham-Lorentz-Dirac equation predicts electrons that run away or accelerate before a force arrives, so physicists use a cleaned-up version, the Landau-Lifshitz equation. The question is which equation quantum theory actually reduces to when quantum effects are switched off.

Precise

The Lorentz-Abraham-Dirac (LAD) equation, third order in time with characteristic time $\tau_0 = 2 e^2/(3 m c^3) \sim 6.3e-24\,\mathrm{s}$, has runaway and preaccelerating solutions; restricting LAD to its non-runaway (critical) manifold gives a second-order equation whose first-order approximation in $\tau_0$ is the Landau-Lifshitz (LL) equation; the two differ from order $\tau_0^2$ on. Determine the classical limit ($\hbar \to 0$ at fixed classical field parameters) of QED for a point charge in a background field to all orders in $\tau_0$, and identify whether it coincides with LL, with LAD on the critical manifold, or with another second-order equation (for example Ford-O'Connell), at orders beyond the leading one in $\alpha$.

Settled by

A derivation from QED in a general (at least plane-wave) background of the classical equation of motion to second and higher order in $\tau_{0}$, compared term by term with LL and with the critical-manifold reduction of LAD.

Kind

well-posed

Answer type

identification

References

(empty)

Literature status

partially-resolved

Status note

The LL equation has been recovered from QED at leading order in $\alpha$ in plane-wave backgrounds; I do not have a verified source for an all-order result as of 2026.

Posed since

(empty)