{"title":"On the Resolvent of H+A\\(^{*}\\)+A","authors":"Andrea Posilicano","doi":"10.1007/s11040-024-09481-0","DOIUrl":null,"url":null,"abstract":"<div><p>We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of <span>\\(H+A^{*}+A\\)</span>. <i>Math. Phys. Anal. Geom.</i> <b>23</b> (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind <span>\\(H+A^{*}+A\\)</span>, where <i>H</i> and <i>A</i> play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind <span>\\(H+A^{*}_{n}+A_{n}-E_{n}\\)</span>, the bounded operator <span>\\(E_{n}\\)</span> playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 3","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-024-09481-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-024-09481-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of \(H+A^{*}+A\). Math. Phys. Anal. Geom.23 (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind \(H+A^{*}+A\), where H and A play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind \(H+A^{*}_{n}+A_{n}-E_{n}\), the bounded operator \(E_{n}\) playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.
我们对先前在 [A.Posilicano：On the Self-Adjointness of \(H+A^{*}+A\).Math.Phys.Geom.23 (2020)] 关于形式 QFT 类哈密顿的自相交实现的 \(H+A^{*}+A\)，其中 H 和 A 分别扮演自由场哈密顿和湮灭算子的角色。我们给出了解析域和自相接域的显式表示；随后的克雷昂式解析式导致了这些自相接实现作为 \(H+A^{*}_{n}+A_{n}-E_{n}\)类型的截止哈密顿的极限（关于规范解析意义上的收敛）的特征，有界算子 \(E_{n}\)扮演了重正化反项的角色。这些抽象结果适用于量子场论的各种具体模型。
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.
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