# Question about a completeness relation in Peskin Chapter 7: radiative corrections p. 212 equation 7.2

In P&S book, in Charpter 7: radiative corrections p. 212 （attached beneath, the completeness relation: (Equation 7.2) $$\textbf{1}=\left\vert \Omega\right\rangle\left\langle\Omega\right\vert+\sum\limits_{\lambda}\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_\textbf{p}} \left\vert \textbf{p}\right\rangle\left\langle\textbf{p}\right\vert.\tag{7.2}$$ This is in analogy with Equation 7.1: $$(\textbf{1})_{1-particle}=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_\textbf{p}} \left\vert \textbf{p}\right\rangle\left\langle\textbf{p}\right\vert.\tag{7.1}$$ My question is:what exactly $$\sum\limits_{\lambda}$$ mean? P&S say: "the sum runs over all zero-momentum states $$\left\vert \textbf{p}\right\rangle\\$$. I don't really get this, does this mean we use $$\lambda$$ to denote different particles that are present in scattering processes? Such as: $$\lambda=1$$ denotes electron and $$\lambda=2$$ denotes positron?

The original Page 212 of P&S book:

• @Andrew Thanks! I just fix this, sorry about the mess!
– ffz
Nov 28, 2022 at 14:24

$$|\lambda_0\rangle$$ runs over all the single particle states at zero momentum. This will encode spin degrees of freedom -- so for a spin-1/2 particle, you'd sum over $$m=1/2$$ and $$m=-1/2$$ states, where $$m$$ is the spin quantum number (as well as antiparticle states).
• $\lambda_0$ also runs over states with more than two particles. $P$ denotes total momentum of the system. May 14, 2023 at 11:16