Construction of a non-Gaussian and rotation-invariant ${\Phi }^4$-measure and associated flow on ${\mathbb{R}}^3$ through stochastic quantization

Abstract

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $\mu$ associated with the ${\phi }_3^4$-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 ${\phi }_3^4$-model. Our starting point is a suitable approximation ${\mu }_{M,N}$ of the measure $\mu$ we intend to construct. ${\mu }_{M,N}$ is parametrized by an $M$-dependent space cut-off function ${\rho }_M:{\mathbb{R}}^3\to \mathbb{R}$ and an $N$-dependent momentum cut-off function ${\psi }_N:{\hat{\mathbb{R}}}^3\cong {\mathbb{R}}^3\to \mathbb{R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N},t\ge 0)$ that have ${\mu }_{M,N}$ 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 $(X_t^{M,N},t\ge 0{)}_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+\infty$. The limit processes $(X_t;t\ge 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $\mu$ of the ${\mu }_{M,N}$ is a stationary measure of $X$. $\mu$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $\mu$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.

arXiv comment103 pages. We corrected the second renormalization constant, the propositions in Section 3 and their proofs

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Cited by in corpus (1)

• Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime