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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?

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  • $\begingroup$ NB that $t\to\infty$ asymptotic states are non-interacting in the LSZ reduction formula. $\endgroup$ – innisfree Aug 21 '17 at 7:57
  • $\begingroup$ The problem is with relativistic quantum mechanics, where particles (excitations of quantum fields) can be created and destroyed by interaction. In QFT experiments (such as the high-energy experiments in accelerators), it is only possible to measure the asymptotic outcome via particles' detectors, but not what happens during the interaction. $\endgroup$ – yuggib Aug 21 '17 at 8:04
  • $\begingroup$ Also from a theoretical point of view, we are not able yet to describe in a satisfactory way what happens during the interaction, at least in "realistic" models of high-energy physics, but only what happens in the limit of asymptotically free fields (where the concept of particle is satisfactorily defined). $\endgroup$ – yuggib Aug 21 '17 at 8:04
  • $\begingroup$ @yuggib: if that is true, then it is quite remarkable that theoretical predictions made from QFT based standard model agree so amazingly well with high-energy experimental results. $\endgroup$ – flippiefanus Aug 21 '17 at 9:39
  • $\begingroup$ @flippiefanus It is only partially surprising, since the perturbative scattering theory of QFT has been showed to be in accordance with the mathematical setting of interacting field theories. Nonetheless, it is not yet known how to satisfy the axioms of QFT for any interacting theory in 3+1 dimensions. $\endgroup$ – yuggib Aug 22 '17 at 5:30
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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.

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  • $\begingroup$ Thanks John. So in the concrete example I gave, where electrons in my AFM tip interact with a single molecule and tell me how many carbons I have, what's going on phenomenologically? It seems like I have a photon field interacting with whatever field carbon atoms belong to, and there's no doubt at any point in time how many particles exist. Pardon if the example is a bit unconventional for QFT. My background is physical chemistry. $\endgroup$ – Dragonsheep Aug 21 '17 at 8:09
  • $\begingroup$ @Dragonsheep: the interaction is strong when the interaction energy is comparable to the rest masses of the particles involved. For any chemical reaction this in never going to be that case so you're quite safe to consider the electrons and nuclei as well defined particles. $\endgroup$ – John Rennie Aug 21 '17 at 8:15
  • $\begingroup$ That's the first time I've heard this answer, but it certainly answers my questions and settles a lot of concern I've had about the unphysicality of a lot of results that fall out of QFT math. Is there a source you'd suggest where I could read more about this interaction-energy limitation? $\endgroup$ – Dragonsheep Aug 21 '17 at 8:20
  • $\begingroup$ How about strongly-interacting theories with confinement e.g. QCD? $\endgroup$ – innisfree Aug 21 '17 at 9:05
  • $\begingroup$ That is never asymptotically free. $\endgroup$ – innisfree Aug 21 '17 at 9:05

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