Completeness

Often one needs to sum over all intermediate states. Starting with equation 4.56 and multipling it by $f_a^{(\pm)}(x)$ and suming over all states, we have

$$i\int d^3x^\prime \sum_a f_a^{(\pm)}(x) {f_a^{(\pm)}}^*(x^\p... ...e) = \pm\sum_a f_a^{(\pm)}(x)\delta_{ab} = \pm f_b^{(\pm)}(x).$$ (4.176)

Comparison with equation 4.170 shows

$$\sum_a f_a^{(\pm)}(x) {f_a^{(\pm)}}^*(x^\prime) = \mp i\triangle_{\pm} (x-x^\prime) .$$ (4.177)