I'm trying to convince myself (mathematically) that energy is conserved at the vertex in [itex]e^-e^+[/itex] annihilation, while solving Exercise 3.5(b) of Halzen and Martin's book (page 83).

I am looking at the time dependent term in the scattering amplitude [itex]T_{fi}[/itex] alone, to recover the delta function term which asserts the energy conservation. I know that I can look at the process in forward time or backward time, with the roles of the e+ and e- reversed.

Suppose the energies of the incoming electron and positron are E and E' respectively, whereas the energy of the outgoing photon is [itex]\omega[/itex] ([itex]\hbar = 1[/itex]).

From what I understand, the transition amplitude is proportional to

The signs in your exponents are wrong. It should be
[tex]e^{i\omega}e^{-i (E+E')}[/tex], corresponding to E+E' in the initial state and omega in the final state.

Thanks clem. I was earlier thinking that the photon contribution will be through the potential term which is sandwiched between the final and initial states, which (I thought) referred to particles alone.