Is there a point interaction model of the electron?

Question

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?

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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

Answer 1

You can integrate out the photon, and leave a contact interaction. This interaction is nonlocal, so that the electron will interact with other electrons far away in space and time (corresponding to emitting a photon and absorbing it much later). This type of description is discussed in Feynman's 1940s papers, and in the 1950s analysis of decoherence in collaboration with Vernon.

When the particle is massive, the effective 4 fermion interaction is local in the long distance approximation. Renormalizability is not an unimportant consideration, the interaction falls to zero at long distances as the ratio of the momentum to the mass of the exchanged particle. This means we don't see such a 4-fermi contact interaction at low energies compared to the GUT or Planck scale, so it is absent in the standard model.