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

Sergio Albeverio, Seiichiro Kusuoka

TL;DR

This work delivers a rigorous construction of a non‑Gaussian, rotation‑invariant, reflection‑positive probability measure $\mu$ associated with the $\varphi^4_3$ model by a stochastic quantization framework. Starting from regularized Gibbs measures $\mu_{M,N}$ with spatial and momentum cutoffs, the authors analyze the corresponding stochastic quantization equations to obtain tightness and weak convergence in weighted Besov spaces, yielding a limiting stationary flow $X$ whose invariant measure is $\mu$. They establish rotation/reflection invariance and reflection positivity of the limit, and prove that $\mu$ has compact Besov support and is non‑Gaussian, signaling a robust construction of Euclidean (and via OS/RS program, Minkowski) correlators for the $\varphi^4_3$ theory. The methodology hinges on a careful blend of Besov space techniques, paracontrolled calculus, and renormalization through Wick powers of an infinite‑dimensional Ornstein–Uhlenbeck process, enabling progress beyond torus‑based approaches to the full space ${\mathbb R}^3$. The results provide a solid probabilistic and analytic foundation for a stochastic quantization flow whose invariant measure captures essential non‑Gaussian and Euclidean‑invariant features of the $\varphi^4_3$ quantum field theory, with potential implications for understanding relativistic fields via Euclidean methods.

Abstract

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

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

TL;DR

This work delivers a rigorous construction of a non‑Gaussian, rotation‑invariant, reflection‑positive probability measure associated with the model by a stochastic quantization framework. Starting from regularized Gibbs measures with spatial and momentum cutoffs, the authors analyze the corresponding stochastic quantization equations to obtain tightness and weak convergence in weighted Besov spaces, yielding a limiting stationary flow whose invariant measure is . They establish rotation/reflection invariance and reflection positivity of the limit, and prove that has compact Besov support and is non‑Gaussian, signaling a robust construction of Euclidean (and via OS/RS program, Minkowski) correlators for the theory. The methodology hinges on a careful blend of Besov space techniques, paracontrolled calculus, and renormalization through Wick powers of an infinite‑dimensional Ornstein–Uhlenbeck process, enabling progress beyond torus‑based approaches to the full space . The results provide a solid probabilistic and analytic foundation for a stochastic quantization flow whose invariant measure captures essential non‑Gaussian and Euclidean‑invariant features of the quantum field theory, with potential implications for understanding relativistic fields via Euclidean methods.

Abstract

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures associated with the -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 -model. Our starting point is a suitable approximation of the measure we intend to construct. is parametrized by an -dependent space cut-off function and an -dependent momentum cut-off function , that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions that have 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 . Limit points in the sense of convergence in law exist, when both and diverge to . The limit processes are continuous on the intersection of suitable Besov spaces and any limit point of the is a stationary measure of . is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.

Paper Structure

This paper contains 16 sections, 37 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Setting and Main theorems
  4. Weighted Besov spaces, paraproducts and estimates of functions
  5. Weighted Besov spaces and paraproducts
  6. Approximation operators and commutator estimates
  7. Infinite-dimensional Ornstein-Uhlenbeck process
  8. Approximation equations and their transformation
  9. Bounds for the behaviour of $X^{M,N,(2)}$ in $N$
  10. Tightness of the laws of $\{ X^{M,N}\}$
  11. Properties of the $\Phi ^4_3$-measure obtained in Theorem \ref{['thm:tight2']}
  12. Rotation and reflection invariance
  13. Reflection positivity of the limit measure
  14. Support of the limit measure
  15. Non-Gaussianity of the limit measure
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{ M_N; N\in {\mathbb N}\}$ be a given ${\mathbb N}$-valued sequence such that $\lim _{N\rightarrow \infty} M_N =\infty$ and for all $\delta \in (0,1]$. Then, for $\varepsilon \in (0, 1/16]$, there exists a sequence $\{ X^{M_{N(k)},N(k)}; k\in {\mathbb N}\}$ such that $\lim _{k\rightarrow \infty} N(k) =\infty$ and $\{ X^{M_{N(k)},N(k)}; k\in {\mathbb N}\}$ converges in law on $C([0,\infty );

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 66 more