The Hopf Algebra of Feynman Graphs in Quantum Electrodynamics

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We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward–Takahashi (WT) identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of quantum electrodynamics. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT identities, which leads in particular to the well-known identity Z1 = Z2.

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