In classical mechanics, we assumed a particle to have a definite momentum and a definite position. Afterwards, with Quantum mechanics, we gave up the concept of a time-dependend position and momentum, and instead have propability distribution stuff and a Hilbert space containing all the information about the state of the particle. Still, we can "retrieve" the concept of a point particle, by stating that the quantum mechanical state gives a propability distribution for the (point-particle)-properties position and momentum, and moreover, the mean-values for position and momentum follow the classical rules.
What I am seeking for now is basically the same, but not for QM, but for QFT. For an abitrary Quantum-Field-Theory, is there a way to "construct" a particle-concept that gives position and momentum for one particle? I know that in QFT, particles are just excitations of the field, but still, is there a way to assign position and momentum to certain types of excitations?
For example, in Theories of free fields (see my other Question here), one can identify the hilbert-space with a fock-space, and by that one can "construct" some wave-package state, that then (in the context of many-body QM) has a localized position and momentum. Is something like that also possible in an interacting theory (despite the fact that the hilbert spaces of interacting theories are not in correspondence to a fock space)?
My thought is that this should be possible in principle, since the QFT is somewhat a generalization of the QM, that intendes to describe Nature better than QM. There should be something like "backward-compability".