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Any Wightman-based approach to Axiomatical Quantum Field Theory states that quantum fields are (operator-valued) distributions. Is there a first rigorous proof of this fact which became trivial as early as 1970? Maybe the name of the person is K. Friedrich or A. Wightman, but I want the exact reference for the proof.

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    $\begingroup$ Proof of what? AQFT simply formalizes quantum fields as operator based distributions. There is no proving that something formal formalizes something informal, one can only prove things about formalizations themselves. $\endgroup$ – Conifold Dec 14 '20 at 23:50
  • $\begingroup$ Then a simple mathematical proof that fields need to be distributions would do. It is heuristically present in various sources, but I need the first proof with mathematical rigor. $\endgroup$ – DanielC Dec 14 '20 at 23:52
  • $\begingroup$ Could you give a specific source and what exactly is being proved from what? I have trouble understanding how heuristic considerations about what formal model to choose can be "proved with mathematical rigor". $\endgroup$ – Conifold Dec 14 '20 at 23:55
  • $\begingroup$ Sure, in section 12 of Geroch Notes on QFT: amazon.com/Quantum-Field-Theory-Lecture-Notes-ebook/dp/…. Here he justifies (without proof) that the sums 141/142/143 do not converge, so he says they need to be smeared (so transformed from functions to distributions). My question: who proved this (not necessarily in the context of a scalar quantum field) transformation from functions to distributions first and where? $\endgroup$ – DanielC Dec 15 '20 at 0:10
  • $\begingroup$ Also the comment by Streater & Wightman in their AQFT book on page 97 is also heuristic (he says that classical fields such as the em field) also need a reinterpretation as distributions, so this is then natural for quantum fields. $\endgroup$ – DanielC Dec 15 '20 at 0:14

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