Nonrenormalizability 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 nonrenormalizability 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.

Figure 1

The first three terms of the diagrammatic expansion of the external field in terms of the external source, for a field theory with quartic coupling. The red dots represent the source insertions $J_1$, and the lines with an arrow are bare retarded propagators $G_{21}^0$. The quartic vertices are all of the type ${\mathrm{\Gamma }}_{1222}$.

Figure 2

Space-time representation of Eq. (50). The propagators with an arrow are retarded propagators. The orange circles represent the mean value of the initial field. The orange lines represent the link coming from the Gaussian fluctuations of the initial field.

Figure 3

Space-time representation of the NNLO contributions to $G_{22}$ that are not included in the classical statistical approximation. The dotted circles outline the 1112 vertices, that are missing in the CSA.

Figure 4

Space-time representation of the divergent part of the graphs of Fig. 3. As explained in the text, these divergent terms are nonlocal in space-time, with support on the light cone.

Figure 5

Contribution to $G_{22}$ with many ${\mathrm{\Gamma }}_{1122}$ subgraphs.

Figure 6

Effect of the noise term on the topology shown in Fig. 5.