标题： 霍普夫代数对克雷默玩具模型的重整化 显示英文标题

标题： Hopf algebraic Renormalization of Kreimer's toy model

摘要： 这本硕士论文回顾了由 Dirk Kreimer 开创的利用 Hopf 代数进行重整化的代数表述，并将其应用于通过将单个原始发散反复插入自身的迭代方式给出的量子场论玩具模型。 使用这个例子在减法方案中，我们展示出重整化的费曼规则可以产生到多项式 Hopf 代数的 Hopf 代数同态，并由此研究重整化群在与组合 Dyson-Schwinger 方程相关联时的出现。 特别是，我们将反常维度的微扰展开与指定原始发散的积分核的 Mellin 变换的系数联系起来。 关于根树 Hopf 代数的一个定理通过该 Hopf 代数的自同构关系不同 Mellin 变换。 显示英文摘要

摘要： This masters thesis reviews the algebraic formulation of renormalization using Hopf algebras as pioneered by Dirk Kreimer and applies it to a toy model of quantum field theory given through iterated insertions of a single primitive divergence into itself. Using this example in a subtraction scheme, we exhibit the renormalized Feynman rules to yield Hopf algebra morphisms into the Hopf algebra of polynomials and as a consequence study the emergence of the renormalization group in connection with combinatorial Dyson-Schwinger equations. In particular we relate the perturbative expansion of the anomalous dimension to the coefficients of the Mellin transform of the integral kernel specifying the primitve divergence. A theorem on the Hopf algebra of rooted trees relates different Mellin transforms by automorphisms of this Hopf algebra.