Combinatorics of renormalization as matrix calculus

Abstract

We give a simple presentation of the combinatorics of renormalization in perturbative quantum field theory in terms of triangular matrices. The prescription, that may be of calculational value, is derived from first principles, to wit, the ``Birkhoff decomposition'' in the Hopf-algebraic description of renormalization by Connes and Kreimer.

10 pages, revised version, typos corrected, to appear in Phys. Lett. B

References in corpus (6)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
• Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation
• Euclidean asymptotic expansions of Green functions of quantum fields (II) Combinatorics of the asymptotic operation
• Renormalization without infinities

Cited by in corpus (16)

• Rota-Baxter Algebras and Dendriform Algebras
• A Lie theoretic approach to renormalization
• Free Rota-Baxter algebras and rooted trees
• Hopf algebra approach to Feynman diagram calculations
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Shuffle relations for regularised integrals of symbols
• Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
• Rota-Baxter algebras and new combinatorial identities
• Matrix Representation of Renormalization in Perturbative Quantum Field Theory
• Renormalization of multiple zeta values
• The combinatorics of Bogoliubov's recursion in renormalization
• Representations of Rota-Baxter algebras and regular singular decompositions
• Representations and Modules of Rota-Baxter Algebras
• Renormalised Chen integrals for symbols on Rn and renormalised polyzeta functions
• Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations