Anatomy of a gauge theory

Abstract

We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the Slavnov-Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams.

25p, uses feynmf, typos corrected, references added, to appear in Annals of Physics

References in corpus (4)

• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Nonperturbative renormalization for 2PI effective action techniques
• Hopf algebra approach to Feynman diagram calculations
• Quantum fields and motives

Cited by in corpus (36)

• A Lie theoretic approach to renormalization
• Renormalization of gauge fields: A Hopf algebra approach
• Renormalized Quantum Yang-Mills Fields in Curved Spacetime
• Recursive relations in the core Hopf algebra
• Exponential renormalization
• Algebraic structures in quantum gravity
• The QED beta-function from global solutions to Dyson-Schwinger equations
• Recursion and growth estimates in renormalizable quantum field theory
• Quantization of gauge fields, graph polynomials and graph cohomology
• The QCD beta-function from global solutions to Dyson-Schwinger equations
• The structure of renormalization Hopf algebras for gauge theories I: Representing Feynman graphs on BV-algebras
• Gravity-Matter Feynman Rules for any Valence
• A remark on quantum gravity
• Renormalized asymptotic enumeration of Feynman diagrams
• Diffeomorphisms of quantum fields
• On the self-consistency of off-shell Slavnov-Taylor identities in QCD
• The Hopf algebra structure of the $R^{\ast }$-operation
• Algebraic lattices in QFT renormalization
• Ward-Schwinger-Dyson equations in ${\varphi }_6^3$ Quantum Field Theory
• Not so non-renormalizable gravity
• Gauge Symmetries and Renormalization
• Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
• Log-Expansions from Combinatorial Dyson-Schwinger Equations
• Algebraic Structures in local QFT
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Hopf-algebraic Renormalization of QED in the linear covariant Gauge
• Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
• Next-to${}^k$ leading log expansions by chord diagrams
• Renormalization of gauge fields using Hopf algebras
• Field diffeomorphisms and the algebraic structure of perturbative expansions
• Algebraic Structures in the Coupling of Gravity to Gauge Theories
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Off-shell Diagrammatics for Quantum Gravity
• Ladders and rainbows in Minimal Subtraction
• New Approach to GUTs
• The signed permutation group on Feynman graphs