Dyson Schwinger Equations: From Hopf algebras to Number Theory

Abstract

We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive non-perturbative results for the short-distance singular sector of a renormalizable quantum field theory. We focus on the short-distance behaviour and thus discuss renormalized Green functions $G_R(\alpha ,L)$ which depend on a single scale $L=\ln q^2/{\mu }^2$.

arXiv comment23 p, 1 Figure, talk given at the workshop on "Renormalization and Universality in Mathematical Physics", Toronta, Canada, October 18-22, 2005. Typos corrected, references updated

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Cited by in corpus (11)

• A Lie theoretic approach to renormalization
• Combinatorial Dyson-Schwinger equations in noncommutative field theory
• The QED beta-function from global solutions to Dyson-Schwinger equations
• The QCD beta-function from global solutions to Dyson-Schwinger equations
• Perturbative renormalisation for not-quite-connected bialgebras
• Some combinatorial aspects of quantum field theory
• Not so non-renormalizable gravity
• Dynamical Mass Generations and Collective Excitations in the (Supersymmetric-)Nambu$-$Jona-Lasinio Model and a Gauge Theory with Left-Right-Asymmetric Majorana Mass Terms
• Systems of Dyson-Schwinger equations
• Mulitgraded Dyson-Schwinger systems
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory