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?