The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem

YAO Hongjia; HAO Kun; YANG Zhanying; YANG Wenli; SHI Kangjie

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

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The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem

更新时间：2024-01-03

- The convergence of the BPHZ renormalization and Weinberg’s asymptotic theorem
- Journal of Northwest University (Natural Science Edition) Vol. 53, Issue 6, Pages: 1016-1029(2023)
- 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：
- 基金信息：
- DOI：10.16152/j.cnki.xdxbzr.2023-06-011
  CLC：
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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) 53(6):1016-1029(2023)
DOI：

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) 53(6):1016-1029(2023) 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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