Is unphysical states still unphysical in an interaction theory?

Question

Maggiore A modern introduction to quantum field theory Section 4 : In free quantum electromagnetic field theory... since only the two degree of freedom transverse wave, the energy, and the momentum, was concerned, the unphysical states could be ignored.

The book also mentioned that, in interaction picture, one might needed to "check that the interaction between physical states does not produce unphysical states."

However, in an interaction theory, in recent years, experimentally, one could photograph the photon propagate through the media.

This brought up the question weather the treatment/assumption in the free field theory still holds, because, through the imaging, an evidential consideration of geometry came into the play. Therefore, this brought suspicions into the treatment of "unphysical states" in interaction theory.

Is unphysical states still unphysical in an interaction theory? Since geometry, other than the momentum and the energy, were needed to be considered?

Answer 1

Are unphysical states still unphysical in an interacting theory?

Tautology. You just said they are unphysical states, of course they are unphysical.

But I meant, are the unphysical states of the free theory still unphysical in interacting theory?

An ill-posed question. States of the free theory and states of the interacting theory are fundamentally different, the main lesson of Haag's theorem. There is no obvious 1-to-1 mapping. The interaction picture is heuristic, it doesn't mathematically exist.

What about perturbation theory?

If you limit yourself to perturbation theory only, you define asymptotic states as the states of the free theory with renormalized masses / charges / couplings / field normalizations etc. In this case, pretty much by definition, unphysical states are unphysical...

What you need to check is that they decouple under the S-matrix, that is, the S-matrix doesn't map physical states to unphysical states. In QED, this is ensured by the Ward-Takahashi identity, which is a manifestation of gauge invariance.