Nonlinearlity of QFT produces interactions, or so I was told today. Maxwell's eqns, though, are perfectly linear. Does that mean that Maxwell's eqns don't predict interactions? Thanks
Free field theories in which no interaction is present(of course) are linear,you add a nonlinear term to account for interaction like in [itex]\phi^4[/itex] theory in which a self interacting term is present.But that does not mean that interaction term is always nonlinear but nature does not chose one.Maxwell's eqns are linear but they still have interaction.
The Maxwell equations don't lead to self-interaction. That is perfectly fine, as there is no direct (without other fields involved) self-interaction of electromagnetic fields. In addition, "Nonlinearlity of QFT produces interactions" does not mean the reverse ("Interactions in QFT have to come from nonlinearity") has to be true.
I cannot, and I don't think there is one, but the statement "A=>B does not imply B=>A" is a mathematical fact.
Correct. Two classical electromagnetic waves will simply pass through each other without affecting each other at all. This is what we mean by a "noninteracting" or "free" field theory.
Maxwell's equations with no source have no interactions. In the classical theory we have waves which pass straight through each other and in the quantum theory we have photons passing right through each other.