Exact limits of the many-body local fields in a two-dimensional electron gas

We have derived the exact limiting behavior of the many-body local fields $G_{\pm }$(q,ω) in a two-dimensional electron gas. At large wave vectors and finite frequencies these functions can be expressed in terms of g(0), the value at the origin of the pair correlation function of the electron gas. We find in particular that in this limit $G_{+}$(∞,ω)=1-g(0), and $G_{\mathrm{-}}$(∞,ω)=g(0). Our results provide new insight into the problem of the many-body vertex corrections in this interacting system.