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I am currently learning quantum field theory and trying to wrap my head around a couple of concepts that are still eluding my understanding:

What is the reason for transitioning from wavefunctions to quantum field operators that act on the vacuum state to give quantum states? I understand that this has something to do with invariance under Poincaré transformation, and I understand that if you want to take a field equation (say the KG equation for example) and quantize it, then you can use the prescribed methods, but why would we do this for particles which can already be described by the Dirac equation? Why should we say that any given particle is a manifestation of some underlying quantum field?

Any help with this would help me greatly.

Qmechanic
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user132849
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    related to https://physics.stackexchange.com/q/332189/36194 – ZeroTheHero Feb 28 '22 at 05:58
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    you can probably find the answer here: https://physics.stackexchange.com/q/122570/226902. This is also relevant https://physics.stackexchange.com/a/128431/226902 . See also this answer: https://www.quora.com/Why-do-we-use-quantum-field-theory-instead-of-relativistic-quantum-mechanics-What-is-wrong-with-relativistic-quantum-mechanics – Quillo Feb 28 '22 at 09:50
  • Great thanks for the resources Quillo and ZeroTheHero! – user132849 Feb 28 '22 at 18:03
  • Okay let me refine my question a little then: What a priori reason do we have for transitioning from particles to fields? Many of the answers cite the non-constant particle number as a motivation for this, and this would make sense to me, but was this known at the time of the development of QFT, or is this a later development that in retrospect justified it? Thanks! @Quillo – user132849 Feb 28 '22 at 18:18
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    Yes it was known and it was one of the main reasons to go from relativistic QM to QFT (plus other technical problems of relativistic QM regarding causality and boundedness of energy). – Quillo Feb 28 '22 at 19:08
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    The disappearance of particles of one type and the appearance of particles of another was central to Fermi's 4-interaction for beta decay, the first "killer app" of QFT (according to CN Yang), in 1933. – Cosmas Zachos Feb 28 '22 at 20:27

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