Two-electrons Lagrangian in QED

[Fefetltl]

In this thread Ifqm has asked what was the Lagrangian of an electron and a proton interacting together. He proposed a Lagrangian containing electron-proton interaction and the tiny QED field correction (electromagnetic tensor part). His first Lagrangian seems intuitive as a sum of 2 Dirac Lagrangian. The second far less because of the exchange-like terms between protons and electrons.

No one was able to give a Lagrangian for this 2 particles case because the proton is composed of quarks in QFT. But that should be easy for this one because electrons are elementary. So what is the Lagrangian for 2 interacting electrons (in 3D+time space) ?

 

[dextercioby]

Specially relativistic quantum field theory works with one Lagrangian per field type, or per particle type. It also assumes the fields are elementary (the "particle" we name its occupation number one is not composed of other "particles"). So a "Lagrangian for two interacting electrons" is still the QED Lagrangian (in the low energy limit).

 

[Fefetltl]

Thanks dextercioby.

So in this case, what is the Lagrangian for each electron ? The question behind it is what is the interaction term (derivative, mass and QED correction are just Dirac Lagrangian I assume)?

 

[dextercioby]

They are identical particles, so you only need a Lagrangian (which we call "free") and a mean to make them interact. With different type of Fermions, you either have the Fermi theory of weak interaction, or the full electroweak or QCD.

 

[Fefetltl]

What is the mean (the Lagrangian part...) to make them interact ? They are identical, these are electrons. No the interaction is only electromagnetic as far as I know (to roughly $10^{-6}$ atomic unit it is only electromagnetic).