PUB - Publikationen an der Universität Bielefeld

We present a new construction of the Euclidean Phi(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 epsilon and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as epsilon -> 0, M -> infinity. 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.