  The one loop effective action in quantum field theory can be expressed as a
quantum mechanical path integral over world lines, with internal symmetries
represented by Grassmanian variables. In this paper, we develop a real time,
many body, world line formalism for the one loop effective action. In
particular, we study hot QCD and obtain the classical transport equations
which, as Litim and Manuel have shown, reduce in the appropriate limit to the
non-Abelian Boltzmann-Langevin equation first obtained by B\"{o}deker. In the
Vlasov limit, the classical kinetic equations are those that correspond to the
hard thermal loop effective action. We also discuss the imaginary time world
line formalism for a hot $\phi^4$ theory, and elucidate its relation to
classical transport theory.
