ترکیبیات و فیزیک: کارگاه مینی درباره نرمال‌سازی، ۱۵-۱۶ دسامبر ۲۰۰۶: کنفرانس ترکیبیات و فیزیک، ۱۹-۲۳ مارس ۲۰۰۷، مؤسسه ماکس پلانک برای ریاضیات، بن، آلمان

Provides an overview of applications of combinatorics to various problems, such as applications to Hopf algebras, techniques to renormalization problems in quantum field theory, as well as combinatorial problems appearing in the context of the numerical integration of dynamical systems, in noncommutative geometry and in quantum gravity. Contents 6 Preface 8 List of participants 10 One-particle irreducibility with initial correlations 12 Multiple zeta values and periods: From moduli spaces to Feynman integrals 38 From quantum electrodynamics to posets of planar binary trees 64 Sweedler's duals and Schutzenberger's calculus 78 Primitive elements of the Hopf algebra of free quasi-symmetric functions 90 A Renormalisation Group approach to Stochastic Loewner Evolutions 100 On the causal gauge principle 126 1. Introduction 126 2. Overview of the CGI method 127 3. The abelian model 130 4. Three MVBs 139 5. The Weinberg–Salam model within CGI 140 6. Discussion 142 References 143 Abstract integration, combinatorics of trees and differential equations 146 Rooted trees appearing in products and co-products 164 Magnus expansions and beyond 182 Wilsonian renormalization, differential equations and Hopf algebras 198 1. Introduction 198 2. Basics of wilsonian renormalization 200 3. Rooted trees and power series of non linear operators 213 4. Renormalization, effective actions and Feynman diagrams 224 5. Conclusion and outlook 245 Acknowledgements 246 References 246 Algebraic analysis of non-renormalization theorems in supersymmetric field theories 248 Not so non-renormalizable gravity 258 Renormalised multiple zeta values which respect quasi-shuffle relations 266 Formulas for the Connes–Moscovici Hopf algebra 280 Hopf algebras and the combinatorics of connected graphs in quantum field theory 298 Hopf Algebras of Formal Diffeomorphisms and Numerical Integration on Manifolds 306 A combinatorial and field theoretic path to quantum gravity: The new challenges of group field theory 336 Noncommutative formal Taylor expansions and second quantised regularised traces 360 Motives: An introductory survey for physicists 388 1. Introduction 388 2. The Grothendieck ring 390 3. The Tannakian formalism 395 4. Weil cohomology 401 5. Classical motives 405 6. Mixed motives 409 7. Motivic measures and zeta functions 414 Appendix A. Motivic ideas in physics (by M.Marcolli) 418 References 424 Combinatorics and Feynman graphs for gauge theories 428 Multi-scale Analysis and Non-commutative Field Theory 454 One-particle Irreducibility With Initial Correlations -- Multiple Zeta Values And Periods: From Moduli Spaces To Feynman Integrals -- From Quantum Elctrodynamics To Posets Of Planar Binary Trees -- Sweedler's Duals And Schützenbergers Calculus -- Primitive Elements Of The Hopf Algebra Of Free Quasi-symmetric Functions -- A Renormalisation Group Approach To Stochastic Lœwner Evolutions -- On The Casual Gauge Principle -- Abstract Integration, Combinatorics Of Trees And Differential Equations -- Rooted Trees Appearing In Products And Co-products -- Magnus Expansions And Beyond -- Wilsonian Renormalization, Differential Equations And Hopf Algebras -- Algebraic Analtysis Of Non-renormalization Theorems In Supersymmetric Field Theories -- Not So Non-renormalizable Graviry -- Renormalised Multiple Zeta Values Which Respect Quasi-shuffle Relations -- Formulas For The Connes-moscovici Hopf Algebra -- Hopf Algebras And The Combinatorics Of Connected Graphs In Quantum Field Theory -- Hopf Algebras Of Formal Diffeomorphisms And Numerical Integration On Manifolds -- A Combinatorial And Field Theoretic Path To Quantum Gravity: The New Challenges Of Group Field Theory -- Noncommutative Formal Taylor Expansions And Second Quantised Regularised Traces -- Motives: An Introductory Survey For Physicists -- Combinatorics And Feynman Graphs For Gauge Theories -- Multi-scale Analysis And Non-commutative Field Theory. Kurusch Ebrahimi-fard, Matilde Marcolli, Walter D. Van Suijlekom, Editors. Includes Bibliographical References. This book is based on the mini-workshop Renormalization, held in December 2006, and the conference Combinatorics and Physics, held in March 2007. Both meetings took place at the Max-Planck-Institut für Mathematik in Bonn, Germany. Research papers in the volume provide an overview of applications of combinatorics to various problems, such as applications to Hopf algebras, techniques to renormalization problems in quantum field theory, as well as combinatorial problems appearing in the context of the numerical integration of dynamical systems, in noncommutative geometry and in quantum gravity. In addition, it contains several introductory notes on renormalization Hopf algebras, Wilsonian renormalization and motives.