QFT: Radiative Transitions and the Dirac Sea

[qftinsanity]

I'm completely lost on this question in our QFT course, as is everyone I have asked in the class. The professor is a crazy old man who can do this stuff in his sleep, which must be helpful for him since that's about all he does.

I really have no idea what this question is asking, any guidance at all is very welcome. The question is reprinted below, verbatim.

As mentioned in lecture, using the Dirac equation for a single (positive energy) electron, the `Born approximation' low-energy gamma-electron scattering amplitude is dominated by the intermediate (virtual) state in which the electron has made a radiative transition (either absorbing the initial or emitting the final gamma) to a negative energy state, and so the energy denominator [ real minus virtual (intermediate) energy ] is $$\approx 2m$$. On the other hand, for the (more realistic) many electron state of the positive energy electron plus the Dirac sea of filled negative energy electron modes, the radiative transition is of a sea electron to a positive energy mode and the energy denominator is $$\approx -2m$$. Despite this different sign, the resulting scattering amplitude (Thomson amplitude) is the same as before. Explain this, in many-particle terms.

 

[NateD]

This question appears to be asking for an explanation of the fact that the Born approximation low-energy gamma-electron scattering amplitude is the same in both single- and many-particle states, despite the energy denominator having a different sign in each case. The explanation should be given in terms of many-particle physics. The explanation lies in the fact that the scattering process is essentially the same in both cases. In the single particle case, the electron is in a positive energy state and makes a radiative transition to a negative energy state. In the many-particle case, the electron is in a positive energy state and makes a radiative transition to a sea electron in a positive energy state. In both cases, the same process occurs: the electron is in a positive energy state and makes a radiative transition to a higher energy state. The only difference is the sign of the energy denominator, which is positive in the single-particle case and negative in the many-particle case. However, this does not affect the scattering amplitude, as the sign of the energy denominator only affects the overall magnitude of the amplitude, not its direction. Therefore, the scattering amplitude is the same in both cases.