In many body quantum theory, Feynman diagrams are commonly used to calculate green's function. My question is: does the diagrammatic method works for all kinds of green's function? For casual green's function, $$ G(x_1, x_2)=-i\langle T{\psi(x_1)\psi^{\dagger}(x_2)}\rangle $$ it can be represented as a line from $x_2$ to $x_1$ and interactions can be represented by some vertices. However, we can have this diagrammatic technique because we have time order product in the bracket and can use Wick's theorems to reduce multi-points functions into products of several two point functions.
Now consider Greater Green's function: $$ G^{>}(x_1, x_2) = -i\langle\psi(x_1)\psi^{\dagger}(x_2)\rangle. $$ In this case, we don't have time ordering and Wick's theorem cannot be applied. Do we still have similar diagrammatic techniques DIRECTLY for calculating $G^{>}(x_1, x_2)$, i.e. we can still represent $G^{>}(x_1, x_2)$ and interaction as some lines and vertices.
