论文标题
仪表对称性和重新归一化
Gauge Symmetries and Renormalization
论文作者
论文摘要
我们研究了Connes和Kreimer的HOPF代数设置中量子规理论的扰动重新归一化。范·苏里科姆（Van Suijlekom，2007年）表明，量规对称的量子对应物 - 所谓的沃德 - 塔卡哈西和斯拉夫诺夫 - 泰勒的身份 - 对应于各个重量化的HOPF HOPF HOPF ELGEBRA中的HOPF理想。我们将此对应关系概括为超级和非十字级量子场理论，将其扩展到具有多个耦合常数的理论，并添加了有关横向性的讨论。特别是，这使我们能够将结果应用于（有效的）量子相对论，这可能与标准模型相关，正如Kreimer（2008）所建议的那样。为此，我们介绍了关于脱发的浅表性差异代数的不同等级的差异。然后，我们将已知的共同体和反座状身份推广到可超级和不可降低的病例，以及具有多个顶点残基的理论。在我们的主要结果的基础上，我们为这些HOPF理想与相应的重新归一化Feynman规则的兼容性提供了标准。我们发现的直接后果是花冠多项式用于量子阳理论的明确定义，而无需参考特定的重新规定方案。
We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (2007) that the quantum counterparts of gauge symmetries -- the so-called Ward--Takahashi and Slavnov--Taylor identities -- correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang--Mills theory without reference to a particular renormalization scheme.
