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 distribution on the space of Euclidean fields.

我们基于PDE参数提出了基于\（{\ mathbb {R}} ^ 3 \）的欧几里德\（\ Phi ^ 4 \）量子场论的新构造。更精确地说，我们考虑在网格大小为\（\ varepsilon \）和边长为M的周期性晶格上定义的\（{\ mathbb {R}} ^ 3 \）上的随机量化方程的逼近。我们在加权空间中引入了一种新的重新规范化的能量方法，并证明了相应的吉布斯度量的紧度为\（\ varepsilon \ rightarrow 0 \），\（M \ rightarrow \ infty \）。每个极限点都是非高斯的，并且满足反射正性，平移不变性和扩展的指数可积性。这些性质允许验证除了旋转不变性和聚类以外的欧氏QFT的Osterwalder-Schrader公理。我们的论点适用于任意正耦合常数，具有O（N）对称性的多分量模型以及某些远程变量。此外，我们建立了零件积分公式，从而建立了针对欧几里得相关函数的Dyson-Schwinger方程的层次结构。为此，我们将重新归一化的三次项确定为欧几里德场空间上的分布。