Two-point Green's functions in quantum electrodynamics at finite temperature and density

Abstract

One-particle propagators of the relativistic electron-positron gas are systematically investigated. With the nonvanishing chemical potential the neutrality of the whole system is secured by a uniformly charged classical background described by a classical current J μ . Due to the translational invariance of this model it is natural to investigate the properties of the propagators in the momentum space. The Fourier-transforms of the Green's functions have been expressed in terms of the generalized spectral Lehmann representation and the second-order spectral functions of the photon and electron propagators have been found. The matter-dependent part of the propagator is finite and only the vacuum part has to be renormalized with the use of standard renormalization counterterms. The singularities of the gauge-independent photon propagator have been further investigated with the use of the spectal representation and nonperturbative expressions for the spectrum of collective excitations have been obtained. In the second order of perturbation they reproduce the asymptotic formulas at T → 0 and T → ∞ cited previously in the literature. In particular, the relativistic plasma frequency (photon effective mass) has been expressed in a simple form in terms of the integrals over the spectral functions. Our formulas for the relativistic plasmon mass squared Ω 2 exhibit an interesting property that at some temperature and density Ω 2 should become negative. However, simple estimates show that this phenomenon occurs at highly nonrealistic temperatures of the order of e 137, i.e., in the region where the perturbation theory fails. The damping of the collective excitations is also considered.