BPHZ重整化的收敛与温伯格渐进定理

要宏佳; 郝昆; 杨战营; 杨文力; 石康杰

doi:10.16152/j.cnki.xdxbzr.2023-06-011

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BPHZ重整化的收敛与温伯格渐进定理

西北大学学报110周年刊庆 | 更新时间：2024-01-03

- BPHZ重整化的收敛与温伯格渐进定理
- The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem
- 西北大学学报（自然科学版） 2023年53卷第6期页码：1016-1029
- 作者机构：
  
  1.西北大学　现代物理研究所，陕西　西安　710127
  2.西北大学　物理学院，陕西　西安　710127
- 作者简介：
  
  要宏佳，女，从事量子可积模型研究，yyhongjia@163.com。
  郝昆，男，研究员，博士生导师，从事量子可积模型、量子场论研究，haoke72@163.com；
  [ "杨文力，男，陕西宝鸡人，二级教授、博士生导师，主要从事理论物理的研究和教学工作。近十年来致力于U(1)对称破缺量子可积系统精确解及相关问题的研究，取得了系列重要研究成果，解决了挑战量子可积系统四十多年的典型问题，建立了精确求解可积模型的统一理论框架，促进了该领域研究的发展。在理论物理相关领域具有重要影响的Physical Review Letters, Journal of High Energy Physics, Nuclear Physics B, Physics Letters B, Physical Review D, Communicationsin Mathematical Physics等期刊上发表SCI论文200余篇。2014年获得国家杰出青年基金资助，2015年入选教育部长江特聘教授，2016年入选享受政府特殊津贴专家。国际纯粹与应用物理联合会(IUPAP)数学物理分委员会委员、中国物理学会常务理事、全国凝聚态理论与统计物理专业委员会委员、意大利国际理论物理中心(ICTP)正规协联成员、美国数学学会会员。为国家科学技术奖、陕西省科学技术奖和科技部重大专项、国家自然科学基金委项目、国家留学基金委项目等会评专家。" ]
- 基金信息：
  
  国家自然科学基金(12275214;12247103;12047502;11805152);陕西省自然科学基础研究计划(2021JCW-19;2019JQ-107)
- DOI：10.16152/j.cnki.xdxbzr.2023-06-011
  中图分类号：
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要宏佳, 郝昆, 杨战营, 等. BPHZ重整化的收敛与温伯格渐进定理[J]. 西北大学学报（自然科学版）, 2023,53(6):1016-1029.

YAO Hongjia, HAO Kun, YANG Zhanying, et al. The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem[J]. Journal of Northwest University (Natural Science Edition), 2023,53(6):1016-1029.
要宏佳, 郝昆, 杨战营, 等. BPHZ重整化的收敛与温伯格渐进定理[J]. 西北大学学报（自然科学版）, 2023,53(6):1016-1029. DOI： 10.16152/j.cnki.xdxbzr.2023-06-011.

YAO Hongjia, HAO Kun, YANG Zhanying, et al. The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem[J]. Journal of Northwest University (Natural Science Edition), 2023,53(6):1016-1029. DOI： 10.16152/j.cnki.xdxbzr.2023-06-011.

摘要

BPHZ重整化理论的中心问题是证明重整化后的费曼波函数,R,Γ,在闵氏空间的积分绝对收敛，要证明这一点，只须证明在欧氏空间的对应波函数,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158896&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158901&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158899&type=,在欧氏空间的积分绝对收敛，Hahn和Zimmermann证明了这一结论。该文用温伯格渐近定理也给出了这一结果，由于证明所需要的条件不同，两种方法能涵盖的场论并不完全相同。该文共分4部分：①介绍温伯格渐近定理及,A,n,类函数；②详细推导渐进定理的实质部分；③解释为什么,A,n,类函数在,R,n,绝对可积必须在有界区,σ,绝对可积，证明这个条件由,R,n,向,R,l,（,l,<,n,）的递推关系；④证明Zimmermann给出的,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158903&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158909&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158906&type=,是动量空间,R,n,的,A,n,类函数并且在,R,n,的任何有界区,σ,绝对可积，进一步证明,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158912&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158917&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158914&type=,在,R,n,绝对可积。

Abstract

The central problem of the BPHZ renormalization theory is to prove the absolute convergence of the integral of the renormalized Feynman wave function ,R,Γ, in Minkowski space. To prove this, one only has to prove the absolute convergence of the integral of the corresponding wave function in Euclidean space. Hahn and Zimmermann demonstrated this conclusion. This article also provides this result using Weinberg’s asymptotic theorem. Due to the different conditions required for proof, the field theories covered by the two methods are somewhat different. This paper can be divided into four parts: ① Introduction to Weinberg’s asymptotic theorem and ,A,n, class function. ② Detailed derivation of the substantive part of the theorem. ③ Explain why ,A,n, class functions in ,R,n, must be absolutely integrable in the bounded region ,σ, prove the recursive relation of this condition from ,R,n, to ,R,l,(,l,<,n,). ④ Prove the ,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158919&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158923&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158921&type=, given by Zimmermann is ,A,n, class function in the momentum space ,R,n, and it is absolutely integrable in any bounded region ,σ, of ,R,n,. Further prove that ,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158924&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158930&type=,https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=52158927&type=, in ,R,n, is absolutely integrable.

关键词

BPHZ重整化费曼图对称群绝对可积敛散性

Keywords

BPHZ renormalizationFeynman diagramsymmetry groupabsolutely integrableconvergence and divergence

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