Partons in Phase Space
  Within QED, we examine several issues related to constructing a parton-model-based QCD transport theory. We rewrite the QED analog of the parton model, the Weizsaecker-Williams Approximation, entirely in terms of phase-space quantities and we study the phase-space photon and electron densities created by a classical point charge. We find that the densities take a distinctive ``source-propagator'' form. This form does not arise in a conventional derivation of the semiclassical transport equations because of the overuse of the gradient approximation. We do not apply the gradient approximation and so derive the phase-space analog of the Generalized Fluctuation-Dissipation Theorem. Together, this theorem and the expression for the phase-space particle self-energies give a set of coupled phase-space evolution equations. We illustrate how these evolution equations can be used perturbatively or to derive semiclassical transport equations. Our work relies on phase-space propagators and sources, so we describe them in detail when calculating the photon and electron phase-space densities. We use these tools to discuss the shape of a nucleon's parton cloud. 
David A. Brown and Pawel Danielewicz