Renormalization and Asymptotic Expansions
Author: Vladimir Alexandrovich Smirnov
Publisher: Birkhauser
Published: 1991
Total Pages: 436
ISBN-13:
DOWNLOAD EBOOK →Author: Vladimir Alexandrovich Smirnov
Publisher: Birkhauser
Published: 1991
Total Pages: 436
ISBN-13:
DOWNLOAD EBOOK →Author: Vladimir A. Smirnov
Publisher: Springer
Published: 2003-07-01
Total Pages: 270
ISBN-13: 3540445749
DOWNLOAD EBOOK →'The sturgeon they sent was second grade fresh,' said the barman. 'Really, what nonsense/' 'Why nonsense?' '"Second grade fresh" that's what I call nonsense/ There's only one degree of freshness the first, and it's the last) (M. A. Bulgakov, The Master and Margarita) The goal of this book is to describe in detail how Feynman integrals can be expanded in suitable parameters, when various momenta or masses are small or large. In a narrow sense, this problem is connected with practical calcula tions. In a situation where a given Feynman integral depends on parameters of very different scales, a natural idea is to replace it by a sufficiently large number of terms of an expansion of it in ratios of small and large scales. It will be explained how this problem of expansion can be systematically solved, by formulating universal prescriptions that express terms of the expansion by using the original Feynman integral with its integrand expanded into a Taylor series in appropriate momenta and masses. It turns out that knowledge of the structure of the asymptotic expansion at the diagrammatic level is a key point in understanding how to perform expansions at the operator level. There are various examples of these ex pansions: the operator product expansion, the large mass expansion, Heavy Quark Effective Theory, Non Relativistic QCD, etc. Each of them serves as a realization of the factorization of contributions of different scales.
Author: Kevin Costello
Publisher: American Mathematical Society
Published: 2022-04-25
Total Pages: 251
ISBN-13: 147047008X
DOWNLOAD EBOOK →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: John Clements Collins
Publisher: Cambridge University Press
Published: 1984
Total Pages: 396
ISBN-13: 9780521311779
DOWNLOAD EBOOK →This book provides a coherent exposition of the techniques underlying these calculations.
Author: E. T. Copson
Publisher: Cambridge University Press
Published: 2004-06-03
Total Pages: 136
ISBN-13: 9780521604826
DOWNLOAD EBOOK →Asymptotic representation of a function os of great importance in many branches of pure and applied mathematics.
Author: Wolfgang Wasow
Publisher: Courier Dover Publications
Published: 2018-03-21
Total Pages: 385
ISBN-13: 0486824586
DOWNLOAD EBOOK →This outstanding text concentrates on the mathematical ideas underlying various asymptotic methods for ordinary differential equations that lead to full, infinite expansions. "A book of great value." — Mathematical Reviews. 1976 revised edition.
Author: A. Erdélyi
Publisher: Courier Corporation
Published: 2012-04-27
Total Pages: 118
ISBN-13: 0486155056
DOWNLOAD EBOOK →Various methods for asymptotic evaluation of integrals containing a large parameter, and solutions of ordinary linear differential equations by means of asymptotic expansion.
Author: Richard Bellman
Publisher:
Published: 1960
Total Pages: 18
ISBN-13:
DOWNLOAD EBOOK →The aim of the work is to present a new formalism for obtaining the perturbation solution of a nonlinear equation. The particular equations written will be used to illustrate the technique which is quite general. Although the method employed appears to be known in the literature of mathematical physics in particular cases, its scope does not appear to have been appreciated, nor does it seem to have been exploited in the mathematical domain. Attention was restricted to obtaining the first term of the perturbation expansion, reserving for a later date the questions of higher order terms, of application to more complex functional equations, of application to specific physical problems, to the derivation of asymptotic expansions, and finally, the problem of validating the technique. (Author).
Author: V. Rivasseau
Publisher: Princeton University Press
Published: 2014-07-14
Total Pages: 347
ISBN-13: 1400862086
DOWNLOAD EBOOK →The last decade has seen striking progress in the subject of renormalization in quantum field theory. The old subject of perturbative renormalization has been revived by the use of powerful methods such as multiscale decompositions; precise estimates have been added to the initial theorems on finiteness of renormalized perturbation theory, with new results on its large order asymptotics. Furthermore, constructive field theory has reached one of its major goals, the mathematically rigorous construction of some renormalizable quantum field theories. For these models one can in particular investigate rigorously the phenomenon of asymptotic freedom, which plays a key role in our current understanding of the interaction among elementary particles. However, until this book, there has been no pedagogical synthesis of these new developments. Vincent Rivasseau, who has been actively involved in them, now describes them for a wider audience. There are, in fact, common concepts at the heart of the progress on perturbative and constructive techniques. Exploiting these similarities, the author uses perturbative renormalization, which is the more widely known and conceptually simpler of the two cases, to explain the less familiar but more mathematically meaningful constructive renormalization. Originally published in 1991. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Author: P.A. Lagerstrom
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 263
ISBN-13: 1475719906
DOWNLOAD EBOOK →Content and Aims of this Book Earlier drafts of the manuscript of this book (James A. Boa was then coau thor) contained discussions of many methods and examples of singular perturba tion problems. The ambitious plans of covering a large number of topics were later abandoned in favor of the present goal: a thorough discussion of selected ideas and techniques used in the method of matched asymptotic expansions. Thus many problems and methods are not covered here: the method of av eraging and the related method of multiple scales are mentioned mainly to give reasons why they are not discussed further. Examples which required too sophis ticated and involved calculations, or advanced knowledge of a special field, are not treated; for instance, to the author's regret some very interesting applications to fluid mechanics had to be omitted for this reason. Artificial mathematical examples introduced to show some exotic or unexpected behavior are omitted, except when they are analytically simple and are needed to illustrate mathematical phenomena important for realistic problems. Problems of numerical analysis are not discussed.