# How to justify RPA (random phase approximation)?

The Random Phase Approximation (RPA) is a technical method used in field theory to account for interactions when calculating correlation functions. It consists of only keeping a certain class of diagrams when doing a perturbative calculation of a certain function, such as a susceptibility or dielectric function.

Is there a simple mathematical justification to this method, other than "it is simple and it fits experimental data", which is already a good justification? Why don't we include the vertex corrections and other self-energy terms?

• See Altland-Simons page 216. Basically, the RPA Feynman diagrams have the largest contribution in the limit of high electron density. – leongz May 24 '17 at 9:02
• Thanks, exactly what I was looking for ! If you want to elaborate your comment into an answer I will accept it. – Dimitri May 24 '17 at 10:18

RPA is the first order approximation in $1/N$ (which is often called the Large-$N$ expansion), where $N$ is the number of fermions in the system. In the system with spin up and spin down, then $N=2$. Therefore, RPA is exact in the hypothetical system with $N\rightarrow \infty$.
• Thanks. In usual theories $N$ is no larger than $2$ or $3$, so is the limit $N \rightarrow \infty$ really relevant ? – Dimitri Apr 3 '17 at 13:44