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

Abstract

We present a self-contained construction of the Euclidean Φ 4 quantum field theory on R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on R 3 defined on a periodic lattice of mesh size ε and side length M . We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as ε → 0, 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 nontrivial Euclidean QFT apart from rotation invariance and clustering. 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. Our argument applies to arbitrary positive coupling constant and also to multicomponent models with O(N ) symmetry.

Introduction to SPDEs from Probability and PDE

“…Motivations for studying SPDE come from many directions in both mathematics and the applied sciences. To mention only a few references, they have found applications in: constructive quantum field theory, [DH87, GH18,Hai14b,Kup18]; the study of disordered media, [Lab19,AC15]; non-linear filtering, [Zak69,Par79]; population dynamics, [Daw72,Fle75] and stochastic fluid dynamics including models of turbulence, [Kra68,Fla11]. In particular we refer to the introductions of [DPZ14, Par07, DKM + 09] for more detailed summaries of some of these applications along with others not mentioned here.…”

The Euclidean $ϕ^4_2$ theory as a limit of an interacting Bose gas

“…Hence, a further mass renormalization of V (φ) is required, which results in a measure µ that is mutually singular with respect to the freefield Gaussian measure P. The mathematically rigorous construction of the Euclidean φ 4 3 theory, first achieved in the seminal work of Glimm and Jaffe [38], is one of the major successes of the constructive field theory programme started in the sixties. By now, several different constructions of this theory have been developed, based on, first, phase cell expansions [24,38,39,59], then on renormalization group methods [5,13,34], later on correlation inequalities [14], and, most recently, on paracontrolled calculus [19,40], as well as variational methods [4].…”

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

“…Together with the abstract ergodicity result of [15], this also allows to show an analogous version of Theorem 1.1 for the equation (1.4). We also point out the papers [10,11], which provide a construction of the Φ 4 3 -model on the Euclidean space R 3 . In the above context, the stochastic wave equation…”