I was reading a book by Franco Strocchi, this one, and in some points the author claims that the case of $d=3+1$ of triviality of $\phi^4$ theory is now proven. As far as I can tell, we have just some evidence from lattice computations. Am I missing any relevant reference about this matter?
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1Can you explain what you mean by triviality? – Prahar Oct 23 '17 at 15:09
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2Can you also add on what page this statement is made? – DanielC Oct 23 '17 at 15:15
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@DanielC You can find the statement on page 38 starting with "The recent proof of triviality...". – Jon Oct 23 '17 at 18:01
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@Prahar A simple presentation of triviality is given at https://en.wikipedia.org/wiki/Quantum_triviality. My preferred one uses the form of the propagator that is Yukawa-like or a sum of Yukawas and all the part of the spectrum with bound states just missing considering a Källén-Lehmann representation. – Jon Oct 23 '17 at 18:04
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@prahar Triviality in this context means that the continuum limit of the lattice theory only exists for the free theory. – user1504 Oct 27 '17 at 17:29
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Michael Aizenman and Hugo Duminil-Copin announced last year a proof that the scaling limit of lattice $\phi^4$ theory in 4d is trivial. It's been known since the 80s that the scaling limit is Gaussian up to logarithmic corrections for small coupling (work of Feldman, Magnen, Rivasseau, & Seneor). Aizenman & Duminil-Copin found a method which works without restriction on the interaction and field size.
Their preprint appeared on the arxiv in December. I don't think it's been formally published yet, but the argument isn't that hard to follow.
user1504
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Thank you for your answer. It is similar to the one seen on Mathoverflow but I am mot able to find it now. Surely, after the publication of Aizenman's and Duminil-Copin's paper, the matter will be settled. Anyway, I think that Strocchi's sentence was relying just on lattice computations but these are not a real mathematical proof. – Jon Apr 28 '20 at 07:32