Representations of the Renormalization Group as Matrix Lie Algebra

M. Berg; P. Cartier

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arXiv · hep-th/0105315

Representations of the Renormalization Group as Matrix Lie Algebra

M. Berg·P. Cartier

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Abstract

Renormalization is cast in the form of a Lie algebra of infinite triangular matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. As representations of an insertion operator, the matrices are related to 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 three-loop example in scalar field theory. Apart from possible applications in high-precision phenomenology, we give a few ideas about possible applications in noncommutative geometry and functional integration.

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M. Berg, P. Cartier. 2004-02-14. Representations of the Renormalization Group as Matrix Lie Algebra. https://arxiv.org/abs/hep-th/0105315

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