A PDE Construction of the Euclidean $$\Phi ^4_3$$ Quantum Field Theory

Abstract

AbstractWe present a new construction of the Euclidean $$\Phi ^4$$ Φ 4 quantum field theory on $${\mathbb {R}}^3$$ R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $${\mathbb {R}}^3$$ R 3 defined on a periodic lattice of mesh size $$\varepsilon $$ ε and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $$\varepsilon \rightarrow 0$$ ε → 0 , $$M \rightarrow \infty $$ M → ∞ . 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.