A Microlocal Investigation of Stochastic Partial Differential Equations for Spinors with an Application to the Thirring Model

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED Mathematical Physics, Analysis and Geometry Pub Date : 2024-08-28 DOI:10.1007/s11040-024-09488-7

Alberto Bonicelli, Beatrice Costeri, Claudio Dappiaggi, Paolo Rinaldi

引用次数: 0

Abstract

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if \(d\le 2\).

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旋子随机偏微分方程的微局部研究及对瑟林模型的应用

在 d 维黎曼自旋流形 (M, g) 上，我们考虑自旋场的非线性随机偏微分方程，该方程由狄拉克算子驱动，并与加性高斯矢量白噪声耦合。我们将 Dappiaggi 等人（Commun Contemp Math 27(07):2150075, 2022）为标量对应方程引入的程序扩展到本案例中，该程序允许在微扰水平上计算解的期望值以及相关的相关函数，并从本质上考虑潜在的重正化自由。这个框架主要依赖于微局域分析的工具，其灵感来自量子场论的代数方法。作为一个具体的例子，我们把它应用于一个随机版本的瑟林模型，特别证明了如果 \(d\le 2\) ，它就处于亚临界体制。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.