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Author : Alain Connes
Release : 2004
Genre :
Kind : eBook
Book Rating : /5 ( reviews)
Book Synopsis Renormalization and Motivic Galois Theory by : Alain Connes
Download or read book Renormalization and Motivic Galois Theory written by Alain Connes. This book was released on 2004. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Frédéric Fauvet
Release : 2009
Genre : Mathematics
Kind : eBook
Book Rating : 739/5 ( reviews)
Book Synopsis Renormalization and Galois Theories by : Frédéric Fauvet
Download or read book Renormalization and Galois Theories written by Frédéric Fauvet. This book was released on 2009. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the outcome of a CIRM Workshop on Renormalization and Galois Theories held in Luminy, France, in March 2006. The subject of this workshop was the interaction and relationship between four currently very active areas: renormalization in quantum field theory (QFT), differential Galois theory, noncommutative geometry, motives and Galois theory. The last decade has seen a burst of new techniques to cope with the various mathematical questions involved in QFT, with notably the development of a Hopf-algebraic approach and insights into the classes of numbers and special functions that systematically appear in the calculations of perturbative QFT (pQFT). The analysis of the ambiguities of resummation of the divergent series of pQFT, an old problem, has been renewed, using recent results on Gevrey asymptotics, generalized Borel summation, Stokes phenomenon and resurgent functions. The purpose of the present book is to highlight, in the context of renormalization, the convergence of these various themes, orchestrated by diverse Galois theories. It contains three lecture courses together with five research articles and will be useful to both researchers and graduate students in mathematics and physics.
Author : Ilia Binder
Release : 2007
Genre : Mathematics
Kind : eBook
Book Rating : 730/5 ( reviews)
Book Synopsis Universality and Renormalization by : Ilia Binder
Download or read book Universality and Renormalization written by Ilia Binder. This book was released on 2007. Available in PDF, EPUB and Kindle. Book excerpt: This book covers a wide range of phenomena in the natural sciences dominated by notions of universality and renormalization. The contributions in this volume are equally broad in their approach to these phenomena, offering the mathematical as well as the perspective of the applied sciences. They explore renormalization theory in quantum field theory and statistical physics, and its connections to modern mathematics as well as physics on scales from the microscopic to the macroscopic. Information for our distributors: Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Author : Kevin Costello
Release : 2022-04-25
Genre : Mathematics
Kind : eBook
Book Rating : 08X/5 ( reviews)
Book Synopsis Renormalization and Effective Field Theory by : Kevin Costello
Download or read book Renormalization and Effective Field Theory written by Kevin Costello. This book was released on 2022-04-25. Available in PDF, EPUB and Kindle. Book excerpt: This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in “mathematics” itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force. —Dennis Sullivan This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding. —Edward Witten Quantum field theory has had a profound influence on mathematics, and on geometry in particular. However, the notorious difficulties of renormalization have made quantum field theory very inaccessible for mathematicians. This book provides complete mathematical foundations for the theory of perturbative quantum field theory, based on Wilson's ideas of low-energy effective field theory and on the Batalin–Vilkovisky formalism. As an example, a cohomological proof of perturbative renormalizability of Yang–Mills theory is presented. An effort has been made to make the book accessible to mathematicians who have had no prior exposure to quantum field theory. Graduate students who have taken classes in basic functional analysis and homological algebra should be able to read this book.
Author : Matilde Marcolli
Release : 2010
Genre : Science
Kind : eBook
Book Rating : 217/5 ( reviews)
Book Synopsis Feynman Motives by : Matilde Marcolli
Download or read book Feynman Motives written by Matilde Marcolli. This book was released on 2010. Available in PDF, EPUB and Kindle. Book excerpt: 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 OC bottom-upOCO approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of BlochOCoEsnaultOCoKreimer 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, OC top-downOCO 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 RiemannOCoHilbert 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. Sample Chapter(s). Chapter 1: Perturbative quantum field theory and Feynman diagrams (350 KB). Contents: Perturbative Quantum Field Theory and Feynman Diagrams; Motives and Periods; Feynman Integrals and Algebraic Varieties; Feynman Integrals and GelfandOCoLeray Forms; ConnesOCoKreimer Theory in a Nutshell; The RiemannOCoHilbert Correspondence; The Geometry of DimReg; Renormalization, Singularities, and Hodge Structures; Beyond Scalar Theories. Readership: Graduate students and researchers in mathematical physics and theoretical physics.
