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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