# Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line.

$$\begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\phi_n(x) \phi_n^* (y) e^{-i E_n (t_x - t_y)}}\\ &=& - i \delta(t_x - t_y) \delta(x - y) \end{eqnarray}$$

I understand that $$\sum_n \phi_n(x) \phi_n^*(y) = \delta(x-y)$$ But I can't figure out where the. $$\sum_n e^{iE_n(t_x-t_y)}$$ goes to.

The trick is: $$\sum_0^{N-1} e^{-iEn(t_x-t_y)} = N\delta(t_x-t_y)$$
• Erm I don't understand, how did you get that relation? Also if you have an $N$, wouldn't the answer be $-iN \delta(t_x-t_y) \delta(x-y)$ ? – chou Apr 1 at 17:08