Hopf Algebraic Structures in the Cutting Rules

Since the Connes--Kreimer Hopf algebra was proposed, revisiting present quantum field theory has become meaningful and important from algebraic points. In this paper, the Hopf algebra in the cutting rules is constructed. Its coproduct contains all necessary ingredients for the cutting equation crucial to proving perturbative unitarity of the S-matrix. Its antipode is compatible with the causality principle. It is obtained by reducing the Hopf algebra in the largest time equation which reflects partitions of the vertex set of a given Feynman diagram. First of all, the Connes--Kreimer Hopf algebra in the BPHZ renormalization instead of the dimensional regularization and the minimal subtraction is described so that the strategy of setting up Hopf algebraic structures of Feynman diagrams becomes clear.