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The DC electrical conductivity of an ultrarelativistic QED plasma is studiedin real time by implementing the dynamical renormalization group. Theconductivity is obtained from the realtime dependence of a dissipative kernelrelated to the retarded photon polarization. Pinch singularities in theimaginary part of the polarization are manifest as growing secular terms thatin the perturbative expansion of this kernel. The leading secular terms arestudied explicitly and it is shown that they are insensitive to the anomalousdamping of hard fermions as a result of a cancellation between self-energy andvertex corrections. The resummation of the secular terms via the dynamicalrenormalization group leads directly to a renormalization group equation inreal time, which is the Boltzmann equation for the (gauge invariant) fermiondistribution function. A direct correspondence between the perturbativeexpansion and the linearized Boltzmann equation is established, allowing adirect identification of the self energy and vertex contributions to thecollision term.We obtain a Fokker-Planck equation in momentum space thatdescribes the dynamics of the departure from equilibrium to leading logarithmicorder in the coupling.This determines that the transport time scale is given byt_{tr}=(24 pi)/[e^4 T \ln(1/e)}]. The solution of the Fokker-Planck equationapproaches asymptotically the steady- state solution as sim e^{-t/(4.038t_{tr})}.The steady-state solution leads to the conductivity sigma = 15.698T/[e^2 ln(1/e)] to leading logarithmic order. We discuss the contributionsbeyond leading logarithms as well as beyond the Boltzmann equation. Thedynamical renormalization group provides a link between linear response inquantum field theory and kinetic theory.Comment: LaTex, 48 pages, 14 .ps figures, final version to appear in Phys.  Rev.

Dynamical renormalization group approach to transport in ultrarelativistic plasmas: The electrical conductivity in high temperature QED