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

TL;DR

This paper introduces a PDE-based construction of the three-dimensional Euclidean ^4 quantum field theory, establishing tightness of measures, non-Gaussianity, and key properties like reflection positivity, using a novel renormalized energy method.

Contribution

It presents a new PDE approach to construct the ^4 quantum field theory in three dimensions, including a renormalized energy method and analysis of Gibbs measures.

Findings

Proves tightness of Gibbs measures as lattice spacing vanishes.

Shows the limit measures are non-Gaussian and satisfy key axioms.

Establishes Dyson--Schwinger equations for correlation functions.

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 $\varepsilon$ and side length $M$. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. 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.