Knots and Feynman Diagrams

Abstract

This book provides an accessible and up-to-date introduction to how knot theory and Feynman diagrams can be used to illuminate problems in quantum field theory. Beginning with a summary of key ideas from perturbative quantum field theory and an introduction to the Hopf algebra structure of renormalization, early chapters discuss the rationality of ladder diagrams and simple link diagrams. The necessary basics of knot theory are then presented and the number-theoretic relationship between the topology of Feynman diagrams and knot theory is explored. Later chapters discuss four-term relations motivated by the discovery of Vassiliev invariants in knot theory and draw a link to algebraic structures recently observed in noncommutative geometry. Detailed references are included. Dealing with material at perhaps the most productive interface between mathematics and physics, the book will be of interest to theoretical and particle physicists, and mathematicians.

Holonomy of the Ising model form factors

“…This is typically a YangBaxter view point : it is easier to solve an integrable model with a spectral parameter that enables to describe the Yang-Baxter structure, than trying to solve that model for a given value of that parameter (quantum groups, knot theory, etc.). It is easier to solve the anisotropic Ising model than the isotropic one, and, similarly, it is easier to consider multiple integrals that depends on a variable, than evaluating constants [49] (polynomial expressions of ζ(3)), ζ(5), ...) corresponding to these multiple integrals at a given value of that parameter: this way of looking at the problem enables to see the emergence of highly non trivial algebraic structures on linear differential operators, that are a very efficient and powerful tool of experimental mathematics, and other formal calculations, to study factorizations of multiple integrals.…”

Non Local Theories: New Rules for Old Diagrams

“…Moreover, it might be possible to adapt the reformulation of the BPHZ procedure in terms of Hopf algebras, due to Kreimer [17].…”

New mathematical structures in renormalizable quantum field theories

“…(4), see [11]. By all computational experience, graphs which have such a Gauss code deliver a residue ∼ ζ(3).…”

“…Hence we briefly review the role of number theory in connection with residues in quantum field theory. This is certainly one of the most surprising subjects worthy of study in quantum field theory: the intimate connection between transcendence and number theory, the topology of Feynman graphs and gauge symmetries has slipped our attention far too long, but slowly is becoming a prominent theme in physics and mathematics [18,11]. We will review the main results and briefly comment on common structures between generalized polylogs and Feynman graphs.…”