Quantum electrodynamical theory of infrared effects in condensed matter

In quantum electrodynamics electron-photon interaction, caused by p·A coupling with the EM field, is usually treated for a single charge, scattered by a potential μυ and in interaction with all photon modes. In condensed matter, however, correlation effects due to degenerate quantum statistics must be taken into account. In this series of papers, we consider a Fock space containing the occupancy states {|γ〉} involving all photon modes |qλ) and all one electron states |ζ). The radiative damping effects (first order in the fine structure constant α) and the further radiative corrections (all orders of α⩾2) of the cross section σ(Φ) and scattering rates wkk′ are computed using the Van Hove limit method for the Heisenberg operators in the interaction picture, as elaborated by the author and coworkers in several papers on linear response theory in recent years. A straightforward computation of the radiative current −Σζζ′Λc†ζcζ′(ζ|r−r0|ζ′), where Λ is the master super-operator describing transitions under photon emission or absorption, leads to the infrared divergent correction αAησ0∫ω0ω1(dωω)(1+2〈N′ω〉), where α A is the usual non-relativistic vacuum infrared exponent and σ0 the elastic cross section. The quantity η is a ratio of Fermi integrals, which for total degeneracy approaches 2kT3εF causing a considerable reduction in metals. The effect of radiative corrections, which remove this divergence, as carried out in detail in a companion paper is discussed, as well as a comparison with diagrammatic methods.