Status of particles in interacting QFT From my readings in QFT and answers such as this, I've read that the concept of particles and particle-number in interacting systems becomes ill-defined in QFT.
Of course, in the real world, a number of experiments allow me to observe a countably finite number of particles interacting, with that finite number being well-defined throughout the entire experiment. If I take a laboratory measurement that allows me to observe individual atoms (e.g., for concreteness's sake, single-molecule AFM), I'm interacting with a finite number of particles through different fields (in this case, the EM field) without the number of particles ever being fuzzy
So, when people say that particle number is not well defined for interacting fields, is their claim just that current QFT formalism cannot recover/ calculate a finite interacting particles in the way that classical mechanics and "ordinary" QM can? This seems hardly satisfying.
Or do they really mean to say that an observation of single-atoms such as this are not observations of countably many atoms at all?
 A: If you have a free particle theory then the number of particles is well behaved because they are just the Fock states.
The trouble is that when you turn on interactions between the particles then the states of the interacting field are not the Fock states i.e. they are not eigenstates of the particle number operator. In fact we don't know what the states of the interacting field are.
But even for an interacting field, when the particles are far apart they are effectively non-interacting so once again we have states that are to a good approximation Fock states.
So if you're considering a typical scattering calculation then initially when the particles are far apart we have well defined states with a well defined number of particles. And after the scattering event when the particles head back out to infinity we also have well defined states with a well defined number of particles. The problem is that when the particles are near to each other and interacting strongly with each other. That's when the number of particles isn't well defined.
