Why do operator valued distributions in QFT refer to free fields only? Operator valued distributions in quantum field theory refer to free fields. The creation operators act in Fock space. They create particle states of various energies and momenta. Every state has a well- defined energy and momentum. Free particles correspond to excitations of fields. One particle can be in a superposition of different energy and momentum states.
Why can't this formalism be applied to interacting fields of particles? Why can't there be fluctuations (virtual particles) in this approach. Virtual particles are necessary for interaction. I don't see why an interaction cannot be viewed in the same approach. Because the operators excite only real, free particles?
 A: Of course there is a more mathematically rigorous way of explaining this, but I'll offer a less rigorous and hopefully more intuitive explanation.
In perturbation theory, when the interactions are weak, the formalism of virtual particles is used and is in fact very useful! This is essentially what Feynman diagrams describe.
You can do this when interactions are weak because you can assume that the particles that appear in the theory are the same particles which appear when the interactions are turned off. So the fields that you used to construct the QFT still correspond to the particles in the theory, and have a standard interpretation in terms of creation and annihilation operators.
When interactions become strong, there is no guarantee that the particle spectrum is the same as when the interactions are turned off. The fields you used to construct the QFT therefore no longer necessarily correspond to particles in the theory. A description of the interaction in terms of the 'free' virtual particles no longer makes much sense.
