Construction of a Non-Gaussian and Rotation-Invariant Φ⁴-Measure and Associated Flow on ℝ³ Through Stochastic Quantization

Abstract

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures μ \mu associated with the φ 3 4 \varphi ^4_3 -model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the φ 3 4 \varphi ^4_3 -model. Our starting point is a suitable approximation μ M , N \mu _{M,N} of the measure μ \mu , which we intend to construct. μ M , N \mu _{M,N} is parametrized by an M M -dependent space cut-off function ρ M : R 3 → R \rho _M: {\mathbb R}^3\rightarrow {\mathbb R} and an N N -dependent momentum cut-off function ψ N : R ^ 3 ≅ R 3 → R \psi _N: \widehat {\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R} , that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions ( X t M , N , t ≥ 0 ) (X_t^{M,N}, t\geq 0) that have μ M , N \mu _{M,N} as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes ( X t M , N , t ≥ 0 ) M , N (X_t^{M,N},t \geq 0)_{M,N} . Limit points in the sense of convergence in law exist, when both M M and N N diverge to + ∞ +\infty . The limit processes ( X t ; t ≥ 0 ) (X_t; t\geq 0) are continuous on the intersection of suitable Besov spaces and any limit point μ \mu of the μ M , N \mu _{M,N} is a stationary measure of X X . μ \mu is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that μ \mu satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, namely it is reflection positive.