A PDE construction of the Euclidean \(\Phi^4_3\) quantum field theory. (English) Zbl 1514.81190
Summary: 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.
MSC:
| 81T08 | Constructive quantum field theory |
| 81S20 | Stochastic quantization |
| 81T25 | Quantum field theory on lattices |
| 81T27 | Continuum limits in quantum field theory |
| 81T70 | Quantization in field theory; cohomological methods |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 82B30 | Statistical thermodynamics |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
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