Dyson–Schwinger Equations, Renormalization Conditions, and the Hopf Algebra of Perturbative Quantum Field Theory
| Author | : Paul-Hermann Balduf |
| Publisher | : Springer Nature |
| Total Pages | : 373 |
| Release | : |
| Genre | : |
| ISBN | : 3031544463 |
Dyson Schwinger Equations Renormalization Conditions And The Hopf Algebra Of Perturbative Quantum Field Theory
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Dyson–Schwinger Equations, Renormalization Conditions, and the Hopf Algebra of Perturbative Quantum Field Theory
| Author | : Paul-Hermann Balduf |
| Publisher | : Springer |
| Total Pages | : 0 |
| Release | : 2024-04-26 |
| Genre | : Science |
| ISBN | : 9783031544453 |
This book offers a systematic introduction to the Hopf algebra of renormalization in quantum field theory, with a special focus on physical motivation, the role of Dyson–Schwinger equations, and the renormalization group. All necessary physical and mathematical constructions are reviewed and motivated in a self-contained introduction. The main part of the book concerns the interplay between Dyson–Schwinger equations (DSEs) and renormalization conditions. The book is explicit and consistent about whether a statement is true in general or only in particular renormalization schemes or approximations and about the dependence of quantities on regularization parameters or coupling constants. With over 600 references, the original literature is cited whenever possible and the book contains numerous references to other works discussing further details, generalizations, or alternative approaches. There are explicit examples and remarks to make the connection from the scalar fields at hand to QED and QCD. The book is primarily targeted at the mathematically oriented physicist who seeks a systematic conceptual overview of renormalization, Hopf algebra, and DSEs. These may be graduate students entering the field as well as practitioners seeking a self-contained account of the Hopf algebra construction. Conversely, the book also benefits the mathematician who is interested in the physical background of the exciting interplay between Hopf algebra, combinatorics and physics that is renormalization theory today.
Quantum Field Theory
| Author | : Bertfried Fauser |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 436 |
| Release | : 2009-06-02 |
| Genre | : Science |
| ISBN | : 376438736X |
The present volume emerged from the 3rd `Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions are committed to the idea of this workshop series: To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics.
Arithmetic and Geometry Around Quantization
| Author | : Özgür Ceyhan |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 295 |
| Release | : 2010-01-12 |
| Genre | : Mathematics |
| ISBN | : 0817648313 |
This volume comprises both research and survey articles originating from the conference on Arithmetic and Geometry around Quantization held in Istanbul in 2006. A wide range of topics related to quantization are covered, thus aiming to give a glimpse of a broad subject in very different perspectives.
Algebraic Renormalization
| Author | : Olivier Piguet |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 142 |
| Release | : 2008-09-11 |
| Genre | : Science |
| ISBN | : 3540491929 |
The idea of this book originated from two series of lectures given by us at the Physics Department of the Catholic University of Petr6polis, in Brazil. Its aim is to present an introduction to the "algebraic" method in the perturbative renormalization of relativistic quantum field theory. Although this approach goes back to the pioneering works of Symanzik in the early 1970s and was systematized by Becchi, Rouet and Stora as early as 1972-1974, its full value has not yet been widely appreciated by the practitioners of quantum field theory. Becchi, Rouet and Stora have, however, shown it to be a powerful tool for proving the renormalizability of theories with (broken) symmetries and of gauge theories. We have thus found it pertinent to collect in a self-contained manner the available information on algebraic renormalization, which was previously scattered in many original papers and in a few older review articles. Although we have taken care to adapt the level of this book to that of a po- graduate (Ph. D. ) course, more advanced researchers will also certainly find it useful. The deeper knowledge of renormalization theory we hope readers will acquire should help them to face the difficult problems of quantum field theory. It should also be very helpful to the more phenomenology oriented readers who want to famili- ize themselves with the formalism of renormalization theory, a necessity in view of the sophisticated perturbative calculations currently being done, in particular in the standard model of particle interactions.
Universality and Renormalization
| Author | : Ilia Binder |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 424 |
| Release | : |
| Genre | : Science |
| ISBN | : 9780821871539 |
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).
A Combinatorial Perspective on Quantum Field Theory
| Author | : Karen Yeats |
| Publisher | : Springer |
| Total Pages | : 120 |
| Release | : 2016-11-23 |
| Genre | : Science |
| ISBN | : 3319475517 |
This book explores combinatorial problems and insights in quantum field theory. It is not comprehensive, but rather takes a tour, shaped by the author’s biases, through some of the important ways that a combinatorial perspective can be brought to bear on quantum field theory. Among the outcomes are both physical insights and interesting mathematics. The book begins by thinking of perturbative expansions as kinds of generating functions and then introduces renormalization Hopf algebras. The remainder is broken into two parts. The first part looks at Dyson-Schwinger equations, stepping gradually from the purely combinatorial to the more physical. The second part looks at Feynman graphs and their periods. The flavour of the book will appeal to mathematicians with a combinatorics background as well as mathematical physicists and other mathematicians.
Feynman Amplitudes, Periods and Motives
| Author | : Luis Álvarez-Cónsul |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 302 |
| Release | : 2015-09-24 |
| Genre | : Mathematics |
| ISBN | : 1470422476 |
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.
Renormalization and Effective Field Theory
| Author | : Kevin Costello |
| Publisher | : American Mathematical Society |
| Total Pages | : 251 |
| Release | : 2022-04-25 |
| Genre | : Mathematics |
| ISBN | : 147047008X |
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.
From Classical Field Theory to Perturbative Quantum Field Theory
| Author | : Michael Dütsch |
| Publisher | : Springer |
| Total Pages | : 536 |
| Release | : 2019-03-18 |
| Genre | : Mathematics |
| ISBN | : 3030047385 |
This book develops a novel approach to perturbative quantum field theory: starting with a perturbative formulation of classical field theory, quantization is achieved by means of deformation quantization of the underlying free theory and by applying the principle that as much of the classical structure as possible should be maintained. The resulting formulation of perturbative quantum field theory is a version of the Epstein-Glaser renormalization that is conceptually clear, mathematically rigorous and pragmatically useful for physicists. The connection to traditional formulations of perturbative quantum field theory is also elaborated on, and the formalism is illustrated in a wealth of examples and exercises.
