# What tree-level Feynman diagrams are added to QED if magnetic monopoles exist?

Are the added diagrams the same as for the $e-\gamma$ interaction, but with "$e$" replaced by "monopole"? If so, is the force between two magnetic monopoles described by the same virtual $\gamma$-exchange diagrams? I'm anticipating that the answer is 'no', because otherwise I don't see how one could tell magnetic monopoles and electrons apart (besides their mass and coupling strength to the photon). Obviously magnetic monopoles behave differently from electrons when placed in an $E$ or $B$ field.

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This is a very interesting question, but unfortunately, the electron is perturbative precisely when the monopole is not, since the couplings are inverse to each other. So if you do a photon description of the electron perturbation, the monopole perturbation is strong-coupling. This was a subject of several Schwinger papers in the 1960s, although I am not sure what the conclusion is. –  Ron Maimon Jun 19 '12 at 21:04

In fact, the situation for an abelian $U(1)$ gauge theory—which is the case you asked about—is a bit less clear and less well-defined than the case of a non-abelian gauge theory. Think about the running of the coupling constant, for example.

In a non-abelian theory with a Higgs field, one can have classical solutions which look like monopoles, i.e. they create magnetic flux through a sphere at infinity. Nevertheless, they are perfectly non-singular classical solutions, which almost certainly survive in the quantum theory. In a sense, they are composite, that is they are built out of fundamental fields like the gauge fields and the scalars.

From this, you can conclude that when summing up Feynman diagrams you should not include the monopoles as extra degrees of freedom. Rather, their effect should appear after resuming the entire perturbation series. If you truncate the perturbation series to any finite order, you will not capture the presence of the magnetic monopoles.

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