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?



Mathematical construction of $\phi^4$ Euclidean field theory
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.


