Combinatorics of (perturbative) quantum field theory

Abstract

We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann--Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.

47p, to appear in a special volume of Phys. Reports dedicated to the Renormalization Group

References in corpus (11)

• 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
• Problems of the Strategy of Regions
• Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality
• Runge-Kutta methods and renormalization
• Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation
• Action principles, restoration of BRS symmetry and the renormalization group equation for chiral non-Abelian gauge theories in dimensional renormalization with a non-anticommuting ${\gamma }_5$
• Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
• Renormalization in quantum field theory and the Riemann-Hilbert problem
• Wave-Function renormalization and the Hopf algebra of Connes and Kreimer
• Feynman diagrams and polylogarithms: shuffles and pentagons

Cited by in corpus (20)

• Combinatorial Hopf algebras in quantum field theory I
• Two interacting Hopf algebras of trees
• Multi-loop Feynman integrals and conformal quantum mechanics
• Hopf algebra approach to Feynman diagram calculations
• Renormalization as a functor on bialgebras
• Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
• Multifractional spacetimes, asymptotic safety and Hořava-Lifshitz gravity
• New mathematical structures in renormalizable quantum field theories
• Hopf algebra of ribbon graphs and renormalization
• On the Invariance of Residues of Feynman Graphs
• Algorithms to Evaluate Multiple Sums for Loop Computations
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Hopf Algebras of Graphs
• Resurgent transseries $\&$ Dyson-Schwinger equations
• Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
• A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in ${\varphi }^4$ theory
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization
• Hopf Algebra Primitives in Perturbation Quantum Field Theory
• The Epstein-Glaser approach to pQFT: graphs and Hopf algebras