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One possible approach to constructive field theory is to define it on a lattice and take the scaling limit, and there are famous results stating that in $d\geq4$ this cannot lead to a non-trivial theory.

What is the status of approaches using the Gaussian free field?

What I mean is this: let $\Omega\subseteq\mathbb{R}^d$ be a smooth domain and consider the Gaussian free field $\{\langle f,\varphi\rangle\}_{f\in H_0^1(\Omega)}$ in $\Omega$ with zero boundary conditions. Let $F$ be a nice functional of $\varphi$, e.g., a mollified two-point function between $x,y\in\Omega$.

Why doesn't the following definition of $\varphi^4$ theory work?

$$ \langle F \rangle_\lambda := \lim_{\Omega\to\mathbb{R}^d} \frac{\mathbb{E}\left[F(\varphi)\exp(-\lambda\int_\Omega\varphi^4)\right]}{\mathbb{E}\left[\exp(-\lambda\int_\Omega\varphi^4)\right]} \,. $$

I suppose the first question one should ask about this is how to make sense of $\int_\Omega\varphi^4$, and then, if the infinite volume limit exists.

Do people in constructive field theory work on this approach? What are some of the major hurdles in such or similar frameworks?

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  • $\begingroup$ Related: mathoverflow.net/questions/260854/… the part at the end. As Martin said, trying to use the free field as your underlying fixed (anchored) probability space and construct (even by a limiting procedure) $\int \phi^4$ as a functional of that free field, is okay in 2d and finite volume but will not work in 3d. There you need to see the problem as that of weak convergence of a sequence of cut-off measures (fixed measurable space, but ton of probability spaces). $\endgroup$ – Abdelmalek Abdesselam Feb 5 at 18:28
  • $\begingroup$ For a proof of the singularity of measures mentioned by Martin you can see, e.g., arxiv.org/abs/2004.01513 $\endgroup$ – Abdelmalek Abdesselam Feb 5 at 18:28
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    $\begingroup$ A sketch of a slightly different proof of that singularity can be found here (this is the unpublished proof mentioned on p.3 of the above preprint). $\endgroup$ – Martin Hairer Feb 5 at 21:43
  • $\begingroup$ @MartinHairer: I guess that's what you showed us at the Imperial College conference in 2019. Thanks for posting it, and making it available. $\endgroup$ – Abdelmalek Abdesselam Feb 6 at 0:16
  • $\begingroup$ @MartinHairer, thanks for the reference to the proof. Is it obvious from the fact the $d=3$ measure is singular w.r.t. the GFF that the same is true also for $d>3$? If not, are there analogous proofs for $d\geq4$? $\endgroup$ – PPR Feb 6 at 0:34
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When $d=2$, this works fine and this is precisely how Nelson originally constructed the $\Phi^4$ measure (in finite volume). Already for $d = 3$, the $\Phi^4$ measure is singular with respect to the free field, even in finite volume, so this approach is bound to fail. The reason why it is singular is subtle, but you can (kind of) convince yourself that this should be the case from the fact that the correct renormalisation in $d=3$ has an additional logarithmic correction on top of Wick renormalisation.

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