ON THE FINITE-TEMPERATURE QUANTUM ELECTRODYNAMICS OF FREE-ELECTRONS AND PHOTONS

Consider a free electron moving through black-body radiation (BBR) maintained at temperature T. Quantum-electrodynamic corrections of order α = e2 ≈ 1 137 to its (quasi) classical mechanics are governed by the shift Δf(T;p) in the Helmholtz free energy of the coupled system electron plus BBR, constrained to a preassigned value p of the electron momentum. We calculate Δf by a method far more elementary than Feynman-diagrammatic approaches, encountering neither infrared nor ultraviolet divergences, and requiring as input only the amplitudes A for the forward scattering, by an electron at rest, of the constitutents of BBR, i.e., of photons (A(γ)) and of electrons and positrons (A(pair) ≡ sum of the Møller and Bhabha amplitudes). Correspondingly, Δf = Δf(γ) + Δf(pair). Most of the final formulae are not new, but the attempt to clarify their derivation and their precise physical significance is. By contrast to expectations on general grounds, Δf exhibits (as already known) a kind of pseudo-Lorentzinvariance (PLI) when kT ≪ m = (electron mass), and again when kT ≫ m; in these regimes Δf(T;p) ≈ Δf(T;0) m p0, as if it arose simply from the shift in p0 = (m2 + p2) 1 2 due to a (p-independent!) shift Δf(T;0) in m. We show that PLI reflects the momentum-independence of the amplitudes A, being exact for Δf(γ) (dominant at low T), and approximate for Δf(pair) at high T. The shift in the free-energy density of BBR itself is rederived as a sum of two simple integrals over Δf(γ) and Δf(pair) respectively. Analogously to Δf we derive also the free-energy shift Δφ of a photon with assigned wave-vector. Δφ coincides with the shift in the photon frequency; identifies, automatically, the transverse plasma frequency of the electron-positron component of BBR; and determines the corrections to the Planck distribution. © 1990.