A PDE construction of the Euclidean ${\Phi }_3^4$ quantum field theory

Abstract

We present a new construction of the Euclidean ${\Phi }^4$ quantum field theory on ${\mathbb{R}}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on ${\mathbb{R}}^3$ defined on a periodic lattice of mesh size $\epsilon$ and side length $M$. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $\epsilon \to 0$, $M\to \infty$. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder--Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with $O(N)$ symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson--Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a \emph{distribution} on the space of Euclidean fields.