Feynman Amplitudes, Periods and Motives
About this Title
Luis Álvarez-Cónsul, Instituto de Ciencias Matemátics, Madrid, Spain, José Ignacio Burgos-Gil, Instituto de Ciencias Matemátics, Madrid, Spain and Kurusch Ebrahimi-Fard, Instituto de Ciencias Matemátics, Madrid, Spain, Editors
Publication: Contemporary Mathematics
Publication Year 2015: Volume 648
ISBNs: 978-1-4704-2247-9 (print); 978-1-4704-2727-6 (online)
This volume contains the proceedings of the International Research Workshop on Periods and Motives—A Modern Perspective on Renormalization, held from July 2–6, 2012, at the Instituto de Ciencias Matemáticas, Madrid, Spain.
Feynman amplitudes are integrals attached to Feynman diagrams by means of Feynman rules. They form a central part of perturbative quantum field theory, where they appear as coefficients of power series expansions of probability amplitudes for physical processes. The efficient computation of Feynman amplitudes is pivotal for theoretical predictions in particle physics.
Periods are numbers computed as integrals of algebraic differential forms over topological cycles on algebraic varieties. The term originated from the period of a periodic elliptic function, which can be computed as an elliptic integral.
Motives emerged from Grothendieck's “universal cohomology theory”, where they describe an intermediate step between algebraic varieties and their linear invariants (cohomology). The theory of motives provides a conceptual framework for the study of periods. In recent work, a beautiful relation between Feynman amplitudes, motives and periods has emerged.
The articles provide an exciting panoramic view on recent developments in this fascinating and fruitful interaction between pure mathematics and modern theoretical physics.
Graduate students and research mathematicians interested in modern theoretical physics and algebraic geometry.
Table of Contents
- Spencer Bloch – A note on twistor integrals
- Christian Bogner and Martin Lüders – Multiple polylogarithms and linearly reducible Feynman graphs
- Patrick Brosnan and Roy Joshua – Comparison of motivic and simplicial operations in mod-$l$-motivic and étale cohomology
- Sarah Carr, Herbert Gangl and Leila Schneps – On the Broadhurst-Kreimer generating series for multiple zeta values
- Colleen Delaney and Matilde Marcolli – Dyson–Schwinger equations in the theory of computation
- Claude Duhr – Scattering amplitudes, Feynman integrals and multiple polylogarithms
- Vasily Golyshev and Masha Vlasenko – Equations D3 and spectral elliptic curves
- Dirk Kreimer – Quantum fields, periods and algebraic geometry
- Erik Panzer – Renormalization, Hopf algebras and Mellin transforms
- Ismaël Soudères – Multiple zeta value cycles in low weight
- Stefan Weinzierl – Periods and Hodge structures in perturbative quantum field theory
- Karen Yeats – Some combinatorial interpretations in perturbative quantum field theory