Short introduction to my understanding: As far as i understand, virtual particles are usually defined to be the internal lines in Feynman Diagrams. But we know that those are just useful tools to calculate amplitudes in interacting quantum field theories. In a free theory I have no interactions, hence no internal lines and no virtual particles. Virtual particles show up when, in interacting QFTs, we have the so called contact interactions showing up in the perturbative Dyson-Schwinger equations as deltas.
Can we regard virtual particles solely as mathematical tools needed to make predictions in perturbative interacting theories because we don't fully understand how interacting theories work? (I mean how they work in a non-perturbative setting)
If the latter is true, and from my actual understanding I think it is true, what is our "hole" in the understanding of full interacting QFTs?