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On matrix differential equations in the Hopf algebra of renormalization. (English) Zbl 1141.81026

Renormalization is a fundamental technique in Theoretical Physics for the formulation of Quantum Field Theory. It was realized by Kreimer that there is an underlying mathematical structure of renormalization which is related to some Hopf algebra. This was extended and studied in more detail by Connes and Kreimer in a series of articles giving an algebraic approach to perturbative renormalization by introducing the Hopf algebra of Feynman graphs, including the concept of renormalization group.
From the introduction of the article under review: “In this work, we would like to further develop the matrix calculus approach to perturbative renormalization in the abstract context of connected graded Hopf algebras. Any left coideal gives rise to a representation of the group of characters of the Hopf algebra by lower triangular unipotent matrices, the size of which being given by the dimension of the coideal. We investigate the matrix representation of two fundamental concepts which can be defined in this purely algebraic framework: the renormalization group and the beta-function. We retrieve then M. Sakakibara’s differential equations involving the beta-function, giving to his approach the firm ground of triangular matrix calculus.”
The article is pedagogically written and gives a useful list of introductory papers and books on the subject.

MSC:

81T17 Renormalization group methods applied to problems in quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
33B15 Gamma, beta and polygamma functions

Citations:

Zbl 1076.81550