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 not only be of considerable interest theoretical and particle physicists, but also to many mathematicians.
1. Introduction
2. Perturbative quantum field theory
3. The Hopf algebra structure of renormalization
4. Rationality: no knots, no transcendentals
5. The simplest link diagrams
6. Necessary topics from knot theory
7. Knots to numbers: (2,2n - 3) torus knots and (2n - 3)
8. One-loop words
9. Euler-Zagier sums
10. Knots and transcendentals
11. The four-term relation
12. Hopf algebras, non-commutative geometry, and what else?