Do quantum fields in QFT interact with each other constantly and continuously, or only from time to time?
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Quantum fields interact constantly.
However when you consider the perturbation theory for each order of coupling constant your amplitudes look like originating from discrete and finite number of acts of interaction between free particles. These contributions are represented by Feynman diagrams which may be the source of your question. You should understand that this is just an artifact of perturbative description. Full amplitude with continuous interaction is given by sum of infinite number of such diagrams.
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