Relativistic quantum field theory for condensed systems-(III): An explicit expression of Atomic Schwinger-Dyson method

A new explicit expressions of self-energies III mu nu and Sigma are introduced for photons and electrons based on the particle-hole-antiparticle representation (PHA) of Atomic Schwinger-Dyson formalism (ASD). The PHA representation describes exactly the physical processes such as particle-hole excitations (electron-hole) and particle-antiparticle excitations (electron-positron). The self-energy E includes both the quantum component and the classical component (classical external field and Coulomb's field), which are divided into the scalar part Sigma(s) and 4-dimensional vector parts Sigma(0), Sigma(j). The electron propagators are composed of the particle part, hole part and antiparticle part in PHA representation. The general representation of photon self-energy Pi(mu nu) with 16 elements is expressed in terms of only two components (transverse and longitudinal) Pi(t) and Pi(l). The general form of the photon propagators axe written in terms of free propagator D-0 and two independent propagators D-l and D-t, which include two independent photon self-energies. The tensor part of the electron self-energy does not appear in ASD formalism which makes perfectly the closed self-consistent system, when we take the bare vertex approximation, Gamma(mu) -> gamma(mu).