Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization
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arXiv
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| Format: | Preprint |
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2021
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| author | Albeverio, Sergio Kusuoka, Seiichiro |
| author_facet | Albeverio, Sergio Kusuoka, Seiichiro |
| contents | A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $μ$ associated with the $φ^4_3$-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 $φ^4_3$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_08040 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization Albeverio, Sergio Kusuoka, Seiichiro Probability Mathematical Physics Analysis of PDEs 81S20, 81T08, 60H15, 35Q40, 35R60, 35K58 A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $μ$ associated with the $φ^4_3$-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 $φ^4_3$-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. |
| title | Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization |
| topic | Probability Mathematical Physics Analysis of PDEs 81S20, 81T08, 60H15, 35Q40, 35R60, 35K58 |
| url | https://arxiv.org/abs/2102.08040 |