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

Martina Hofmanová,Massimiliano Gubinelli

doi:10.1007/s00220-021-04022-0

Martina Hofmanová, Massimiliano Gubinelli

Open Access

https://doi.org/10.1007/s00220-021-04022-0

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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 distribution on the space of Euclidean fields.

Highlights

Let M,ε = ((εZ)/(MZ))[3] be a periodic lattice with mesh size ε and side length M where M/(2ε) ∈ N
We present a new construction of the Euclidean 4 quantum field theory on R3 based on PDE arguments
We consider an approximation of the stochastic quantization equation on R3 defined on a periodic lattice of mesh size ε and side length M

Summary

Introduction

The first work proposing a constructive use of the dynamics is, to our knowledge, the work of Albeverio and Kusuoka [AK17], who proved tightness of certain approximations in a finite volume Inspired by this result, our aim here is to show how these recent ideas connecting probability with PDE theory can be streamlined and extended to recover a complete and independent proof of existence of the full space. We obtain fine estimates on the approximate stochastic quantization equation and construct a coupling of the stationary solution to the continuum This leads to a detailed description of the renormalized cubic term as a genuine random space-time distribution. To this end, we introduce the lattice dynamics together with its decomposition. A we collect a number of technical results needed in the main body of the paper

Notation

Overview of the Strategy

Construction of the Euclidean 4 Field Theory

The Osterwalder–Schrader Axioms and Non-Gaussianity

Integration by Parts Formula and Dyson–Schwinger Equations

Fractional

Findings

Let γ for any λ