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Is it known (rigorously) whether or not there exist (1+1)D Wightman QFTs which can (in some reasonable sense) be said to correspond to physicists' unitary minimal models $\mathcal{M}(m+1,m)$, $m\in\mathbb{N}^+$, $m\geq 3$?

For example, the case $m=3$ is supposed to model the continuum 2D Ising model at criticality [1], and it seems that based on some heuristics physicists have conjectured -- or proven, at a level of rigor acceptable to them -- that the Schwinger functions of such a QFT should be $$ S_N(z_1,\cdots,z_N) = \Big[\sum_{\substack{\varsigma_1,\ldots,\varsigma_N \in \{-1,+1\}\\ \varsigma_1+\cdots+\varsigma_N=0}} \prod_{1\leq m<n\leq N} |z_m-z_n|^{\varsigma_m\varsigma_n/2} \Big]^{1/2},$$ $z_1,\ldots,z_N\in \mathbb{R}^2$ pairwise distinct. See Section 12.3.3 of [1]. (Moreover, some aspects of this have been rigorously proven. See the answer on this question.) Since a Wightman QFT is determined by its Wightman functions, and since its Wightman functions are determined via analytic continuation by its Schwinger functions, I would expect that the sequence $\{S_N\}_{N=0}^\infty$ uniquely specifies a Wightman QFT. Since one can analytically continue $S_N$ back to ``real time'' by hand, I would further expect that to construct such a QFT it suffices to check that the $S_N$ satisfy the original Osterwalder-Schrader axioms, of which the trickiest seems to be reflection positivity (/Wightman positivity after analytic continuation). Hence, I would expect that a necessary and sufficient condition for there to exist a Wightman QFT whose Schwinger functions are the $S_N$ is that they satisfy reflection positivity, but this (if true) seems non-obvious to me.

[1] Francesco, Mathieu, and Senechal, Conformal Field Theory, Chapters 7, 12

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  • $\begingroup$ I think I recall that the Wightman axioms are relevant for mass-gapped theories so perhaps it's worth checking the mass assumption in those axioms. If it's nonzero mass then all CFTs are excluded. (It could also be the case that the only models that can be described by the axioms are massive and my recollecition is faulty $\endgroup$
    – AlexArvanitakis
    Oct 25, 2021 at 17:19
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    $\begingroup$ @Alex I believe the Wightman axioms are expected to apply regardless of the presence of a mass gap, though it is certainly the case that the most famous examples of rigorously constructed interacting QFTs have mass gaps. $\endgroup$
    – Ethan Sussman
    Oct 25, 2021 at 17:41
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    $\begingroup$ I believe the only relevance of the masslessness would be to the rate of convergence in Wightman's cluster axiom --- in a theory with a mass gap, the convergence should be exponentially fast, while the $S_N$ only satisfy the Euclidean cluster axiom with a polynomial rate of convergence. $\endgroup$
    – Ethan Sussman
    Oct 25, 2021 at 18:25
  • $\begingroup$ Now cross-posted at physics.stackexchange.com/questions/674282/… $\endgroup$
    – Abdelmalek Abdesselam
    Oct 31, 2021 at 11:14

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Not exactly an answer but too long for a comment.

This is a very good question. You are right that the only nontrivial thing to check here is Osterwalder-Schrader positivity. The only proof I know is through OS positivity of the Ising model on the lattice and passing to the continuum limit which preserves (non strict) inequalities. In fact I essentially asked Slava Rychkov that same question, see the talk https://www.youtube.com/watch?v=c0u55y1sRcc around the 40 min mark. There seems to be a way of proving positivity in the CFT bootstrap approach but I didn't see that written in detail somewhere for the 2d Ising CFT.

A useful reference for this stuff is the recent article by Kravchuk, Qiao and Rychkov in JHEP "Distributions in CFT. Part II. Minkowski space" https://link.springer.com/article/10.1007%2FJHEP08%282021%29094

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  • $\begingroup$ Thanks for the comment-answer! I strongly suspected that one could prove OS positivity via the argument you outlined but wasn't completely sure the details would all align properly (and wasn't sure there's no easier way). But a proof via the boostrap is even better! (Assuming that "seems to be" can be replaced by "is" in your answer.) $\endgroup$
    – Ethan Sussman
    Oct 29, 2021 at 12:51
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    $\begingroup$ Maybe you should cross post (and let it be known that you did) on physics stackexchange with the conformal field theory tag. Petr Kravchuk may give you a better answer. $\endgroup$
    – Abdelmalek Abdesselam
    Oct 29, 2021 at 12:57
  • $\begingroup$ Also: is it possible to see why the analytic continuations of the $S_N$ back to real time satisfy spectral positivity without using reflection positivity? From the domain of analyticity, one should be able to deduce that the Fourier transforms of the Wightman functions (rewritten in terms of coordinate differences) decay superexponentially away from the power of the forward light cone, but how would one conclude that they are actually supported there? (commented continued below) $\endgroup$
    – Ethan Sussman
    Oct 29, 2021 at 13:57
  • $\begingroup$ As far as I can tell, superexponential decay suffices to go through the Osterwalder-Schrader argument of constructing the Hilbert space and Hamiltonian and showing that $e^{-Ht}$ is a contraction for $t>0$ (also in Simon's P(Phi)_2 book) to deduce the support property, but that requires reflection positivity... $\endgroup$
    – Ethan Sussman
    Oct 29, 2021 at 13:57

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