Representations of the Renormalization Group as Matrix Lie Algebra

Renormalization is cast in the form of a Lie algebra of matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. The matrices are triangular, and in general infinite. As representations of an insertion operator, the matrices provide explicit representations of the Connes-Kreimer Lie algebra. In fact, the right-symmetric nonassociative algebra of the Connes-Kreimer insertion product is equivalent to an “Ihara bracket” in the matrix Lie algebra. We check our results in a few threeloop examples in scalar field theory. Apart from the obvious application to high-precision phenomenology, some ideas about possible applications in noncommutative geometry and functional integration are given. This work was partially performed while visiting DMA, Ecole Normale Supérieure. Email contact: [email protected]