Non-renormalizability of the classical statistical approximation

In this paper, we discuss questions related to the renormalizability of the classical statistical approximation, an approximation scheme that has been used recently in several studies of out-of-equilibrium problems in Quantum Field Theory. Although the ultraviolet power counting in this approximation scheme is identical to that of the unapproximated quantum field theory, this approximation is not renormalizable. The leading cause of this non-renormalizability is the breakdown of Weinberg's theorem in this approximation. We also discuss some practical implications of this negative result for simulations that employ this approximation scheme, and we speculate about a possible modification of the classical statistical approximation in order to systematically subtract the leading residual divergences.