We present a new construction of the Euclidean $Φ^4$ 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.
Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions
University of Michigan-Shanghai Jiao Tong University Joint Institute（密歇根大学-上海交通大学联合学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；Institute of Natural Sciences, School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院自然科学高等研究院）；Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University（上海交通大学教育部科学和工程计算重点实验室）
