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Of course the discrete particle of classical mechanics that when found in big numbers must be dealt with many-body statistical mechanics has always a certain conceptual and methodological clash with classical continuous fields and with continuum mechanics that has traditionally required convinient mathematical objects like dirac delta generalized functions.

It is IMO a significative parallelism that just like when approximating matter many-body systems thru continuous models like it is done in fluid and solid continuum mechanics one makes the unrealistic assumption that at ech point there are infinite molecules, in QFT there are also the appearance of infinities that have to be treated with regularization-renormalization.

Can we say that QFT solves rigorously the continuous-discrete controversy that seems to always chase physical models?