Classical limit of time-dependent quantum field theory — a Schwinger–Dyson approach

We rewrite the Martin–Siggia–Rose (MSR) formalism for the statistical dynamics of classical fields in a covariant second order form appropriate for the statistical dynamics of relativistic field theory. This second order formalism is related to a rotation of Schwingerʹs closed time path (CTP) formalism for quantum dynamics, with the main difference being that certain vertices are absent in the classical theory. These vertices are higher order in an ℏ expansion. The structure of the second order formulation of the Schwinger–Dyson (S–D) equations is identical to that of the rotated CTP formalism apart from initial conditions on the Greenʹs functions and the absence of these vertices. We then discuss self-consistent truncation schemes based on keeping certain graphs in the two-particle irreducible effective action made up of bare vertices and exact Greenʹs functions.