New mathematical structures in renormalizable quantum field theories

Abstract

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments.

arXiv comment26p,4figs., Invited Contribution to Annals of Physics, minor typos corrected

References in corpus (15)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Hopf algebras, cyclic cohomology and the transverse index theory
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops
• Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality
• Knots and Numbers in ${\varphi }^4$ Theory to 7 Loops and Beyond
• On Overlapping Divergences
• Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs
• Unknotting the polarized vacuum of quenched QED
• Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation
• Combinatorics of (perturbative) quantum field theory
• Flavour Mixing, Gauge Invariance and Wave-function Renormalisation
• Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
• Using the Hopf Algebra Structure of QFT in Calculations

Cited by in corpus (9)

• The massless two-loop two-point function
• Hopf algebra approach to Feynman diagram calculations
• Shuffle relations for regularised integrals of symbols
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• The Structure of the Ladder Insertion-Elimination Lie algebra
• The uses of Connes and Kreimer's algebraic formulation of renormalization theory
• Ward-Schwinger-Dyson equations in ${\varphi }_6^3$ Quantum Field Theory
• What is the trouble with Dyson--Schwinger equations?
• The Epstein-Glaser approach to pQFT: graphs and Hopf algebras