# Is there a point interaction model of the electron?

Is there a point interaction model of the electron? Is there a point interaction model of the electron? I imagine something like $\propto(\bar \psi\psi)^2$ (edited). Is such a thing in use?

Since I expect some but that's not renormalizable!-arguments: I think there are theories with scalar photons. Can you integrate out the massless photon completely? And that raises the question more generally: Does going from a renormalizable to an effective description theory change that property?

Edit: I just found the Four-fermion interactions. I guess that leaves the question if, or better how, these can result from swallowing the photon. Here is the question in that light:

Coming from QED, is there a point interaction model of the electron?

PS: I like the last line of the article. Who would have seen that one coming :P

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Maybe I'm misunderstanding, but wouldn't $(\bar\psi\psi)^2$ be a simpler instance of what you're asking? There are actually searches for such things (with no results, of course). –  David Z Apr 19 '12 at 20:30
@DavidZaslavsky: Yes of course, a sqare will suffice here. I was dazed by habits. I'm actually coming from the idea of considering the SO(N) $\phi^4$ variant, but not with global SO(N). One step back, the question really came from someone elses question if there ever are free electrons. –  NikolajK Apr 19 '12 at 20:36