Articles liés à Feynman Motives

Feynman Motives - Couverture rigide

Marcolli, Matilde

 
9789814271202: Feynman Motives
Couverture rigide
ISBN 10 : 9814271209 ISBN 13 : 9789814271202
Editeur : World Scientific Publishing Co Pte Ltd, 2010

Synopsis

This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer. Two different approaches to the subject are described. The first, a bottom-up approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch Esnault Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, top-down approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry. The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.

Les informations fournies dans la section « Synopsis » peuvent faire référence à une autre édition de ce titre.

Présentation de l'éditeur

This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer. Two different approaches to the subject are described. The first, a bottom-up approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch Esnault Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, top-down approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry. The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.

Revue de presse

This book can serve as an excellent guide for graduate students and researchers to this new area, in particular to the reasons of enigmatic reappearance of Euler's multiple zeta values as basic Feynman periods.-- Yu I Manin, Max-Planck-Institute for Mathematics, Bonn, Chair of the Fields Medal Committee, 1998

Anyone interested in this rapidly expanding field will have to look through this monograph; for now, it is the only coherent source of information about the project, apart from reading scattered papers, written in different mathematical languages.-- Gunther Cornelissen, University of Utrecht, The Netherlands

This book is certainly a positive contribution to the field ... it serves very well as a friendly introduction for those new to the field and as a useful and informative overview of the current state of affairs in this ever-burgeoning subject. In particular, some readers may find this book a useful primer for tackling the more detailed exposition of the top-down approach."-- Mathematical Reviews

"I think its contents are important. It remains to add that, physically, it has been nicely produced by World Scientific. --Contemporary Physics

Anyone interested in this rapidly expanding field will have to look through this monograph; for now, it is the only coherent source of information about the project, apart from reading scattered papers, written in different mathematical languages. --Gunther Cornelissen, University of Utrecht, The Netherlands

This book will appeal to mathematicians who seek an entry point into this area, as well as to physicists who enquire about the most basic mathematical notions involved. --Zentralblatt MATH

This book will appeal to mathematicians who seek an entry point into this area, as well as to physicists who enquire about the most basic mathematical notions involved. --Zentralblatt MATH

Les informations fournies dans la section « A propos du livre » peuvent faire référence à une autre édition de ce titre.

Éditeur
World Scientific Publishing Co Pte Ltd
Date d'édition
2010
Langue
anglais
ISBN 10 
9814271209
ISBN 13 
9789814271202
Reliure
Relié
Nombre de pages
236
Coordonnées du fabricant
non disponible
Personne responsable
GPSR Kontakt
gpsr@libri.de

Europaallee 1
Bad Hersfeld
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