Renormalization Group And Fixed Points

Author: Timothy J Hollowood
Publisher: Springer Science & Business Media
ISBN: 3642363121
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This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories.

Epfl Lectures On Conformal Field Theory In D 3 Dimensions

Author: Slava Rychkov
Publisher: Springer
ISBN: 3319436260
Size: 14.46 MB
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This primer develops Conformal Field Theory (CFT) from scratch, whereby CFT is viewed as any conformally-invariant theory that describes a fixed point of a renormalization group flow in quantum field theory. The book is divided into four lectures: Lecture 1 addresses the physical foundations of conformal invariance, while Lecture 2 examines the constraints imposed by conformal symmetry on the correlation functions of local operators, presented using the so-called projective null cone – a procedure also known as the embedding formalism. In turn, Lecture 3 focuses on the radial quantization and the operator product expansion, while Lecture 4 offers a very brief introduction to the conformal bootstrap. Derived from course-based notes, these lectures are intended as a first point of entry to this topic for Master and PhD students alike.

The C And A Theorems And The Local Renormalisation Group

Author: Graham Shore
Publisher: Springer
ISBN: 3319540009
Size: 67.73 MB
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The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local RG (LRG). The idea of renormalisation with position-dependent couplings, running under local Weyl scaling, is traced from its early realisations to the elegant modern formalism of the LRG. The key rôle of the associated Weyl consistency conditions in establishing RG flow equations for the coefficients of the trace anomaly in curved spacetime, and their relation to the c-theorem and four-dimensional a-theorem, is explained in detail. A number of different derivations of the c-theorem in two dimensions are presented and subsequently generalised to four dimensions. The obstructions to establishing monotonic C-functions related to the trace anomaly coefficients in four dimensions are explained. The possibility of deriving an a-theorem for the coefficient of the Euler-Gauss-Bonnet density is explored, initially by formulating the QFT on maximally symmetric spaces. Then the formulation of the weak a-theorem using a dispersion relation for four-point functions is presented. Finally, the application of the LRG to the issue of limit cycles in theories with a global symmetry is described, shedding new light on the geometry of the space of couplings in QFT.

Wavelets And Renormalization

Author: G Battle
Publisher: World Scientific
ISBN: 9814499129
Size: 18.46 MB
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WAVELETS AND RENORMALIZATION describes the role played by wavelets in Euclidean field theory and classical statistical mechanics. The author begins with a stream-lined introduction to quantum field theory from a rather basic point of view. Functional integrals for imaginary-time-ordered expectations are introduced early and naturally, while the connection with the statistical mechanics of classical spin systems is introduced in a later chapter. A vastly simplified (wavelet) version of the celebrated Glimm-Jaffe construction of the Φ43 quantum field theory is presented. It is due to Battle and Federbush, and it bases an inductively defined cluster expansion on a wavelet decomposition of the Euclidean quantum field. The presentation is reserved for the last chapter, while the more basic aspects of cluster expansions are reviewed in the chapter on classical spin systems. Wavelets themselves are studied from two different points of view arising from two disciplines. The mathematical point of view covers the basic properties of wavelets and methods for constructing well-known wavelets such as Meyer wavelets, Daubechies wavelets, etc. The physical point of view covers the renormalization group formalism, where there is a close connection between wavelets and Gaussian fixed points. The book is heavily mathematical, but avoids the theorem-proof-theorem-proof format in the interests of preserving the flow of the discussion — i.e., it is written in the style of an old-fashioned theoretical physics book, but the major claims are rigorously proven. The minor themes of the book are reflection positivity, the combinatorics of cluster expansions, and the issue of phase transitions — themes which have nothing to do with wavelets, but which provide necessary cultural background for the physical context. Contents:Mathematical Sketches of Quantum PhysicsWavelets — Basic Theory and ConstructionEquilibrium States of Classical CrystalsA Wavelet Introduction to the Renormalization GroupWavelet Analysis of Φ43 Readership: Applied mathematicians. keywords:Cluster Expansions;Condensed Matter;Euclidean Fields;Functional Integrals;Phase Space;Quantum Mechanics;Reflection Positivity;Renormalization Group;Uncertainty Principle;Wavelets “… the author has succeeded in giving a vivid and pedagogical presentation of a monumental work in recent mathematical physics, illustrating the mutual influence between wavelet analysis and renormalization group techniques of Euclidean field theory.” Mathematical Reviews “This book is a great achievement. It is difficult to read, but very rewarding for those who read it in depth.” Monatshefte für Mathematik

Introduction To The Functional Renormalization Group

Author: Peter Kopietz
Publisher: Springer
ISBN: 3642050948
Size: 55.47 MB
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The renormalization group (RG) has nowadays achieved the status of a meta-theory, which is a theory about theories. The theory of the RG consists of a set of concepts and methods which can be used to understand phenomena in many different ?elds of physics, ranging from quantum ?eld theory over classical statistical mechanics to nonequilibrium phenomena. RG methods are particularly useful to understand phenomena where ?uctuations involving many different length or time scales lead to the emergence of new collective behavior in complex many-body systems. In view of the diversity of ?elds where RG methods have been successfully applied, it is not surprising that a variety of apparently different implementations of the RG idea have been proposed. Unfortunately, this makes it somewhat dif?cult for beginners to learn this technique. For example, the ?eld-theoretical formulation of the RG idea looks at the ?rst sight rather different from the RG approach pioneered by Wilson, the latter being based on the concept of the effective action which is ite- tively calculated by successive elimination of the high-energy degrees of freedom. Moreover, the Wilsonian RG idea has been implemented in many different ways, depending on the particular problem at hand, and there seems to be no canonical way of setting up the RG procedure for a given problem.

A Primer On Quantum Fluids

Author: Carlo F. Barenghi
Publisher: Springer
ISBN: 3319424769
Size: 76.45 MB
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The aim of this primer is to cover the essential theoretical information, quickly and concisely, in order to enable senior undergraduate and beginning graduate students to tackle projects in topical research areas of quantum fluids, for example, solitons, vortices and collective modes. The selection of the material, both regarding the content and level of presentation, draws on the authors analysis of the success of relevant research projects with newcomers to the field, as well as of the students feedback from many taught and self-study courses on the subject matter. Starting with a brief historical overview, this text covers particle statistics, weakly interacting condensates and their dynamics and finally superfluid helium and quantum turbulence. At the end of each chapter (apart from the first) there will be some exercises. Detailed solutions can be made available to instructors upon request to the authors.

Cosmological Applications Of Algebraic Quantum Field Theory In Curved Spacetimes

Author: Thomas-Paul Hack
Publisher: Springer
ISBN: 3319218948
Size: 66.46 MB
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This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology; and a fundamental study of the perturbations in inflation. The two central sections of the book dealing with these applications are preceded by sections providing a pedagogical introduction to the subject. Introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation is also given. The reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but no background in QFT on curved spacetimes or the algebraic approach to QFT is required.

Quantum Potential Physics Geometry And Algebra

Author: Ignazio Licata
Publisher: Springer Science & Business Media
ISBN: 331900333X
Size: 25.85 MB
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Recently the interest in Bohm realist interpretation of quantum mechanics has grown. The important advantage of this approach lies in the possibility to introduce non-locality ab initio, and not as an “unexpected host”. In this book the authors give a detailed analysis of quantum potential, the non-locality term and its role in quantum cosmology and information. The different approaches to the quantum potential are analysed, starting from the original attempt to introduce a realism of particles trajectories (influenced by de Broglie’s pilot wave) to the recent dynamic interpretation provided by Goldstein, Durr, Tumulka and Zanghì, and the geometrodynamic picture, with suggestion about quantum gravity. Finally we focus on the algebraic reading of Hiley and Birkbeck school, that analyse the meaning of the non-local structure of the world, bringing important consequences for the space, time and information concepts.

Bosonization Of Interacting Fermions In Arbitrary Dimensions

Author: Peter Kopietz
Publisher: Springer Science & Business Media
ISBN: 3540684956
Size: 56.51 MB
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The author presents in detail a new non-perturbative approach to the fermionic many-body problem, improving the bosonization technique and generalizing it to dimensions d1 via functional integration and Hubbard--Stratonovich transformations. In Part I he clearly illustrates the approximations and limitations inherent in higher-dimensional bosonization and derives the precise relation with diagrammatic perturbation theory. He shows how the non-linear terms in the energy dispersion can be systematically included into bosonization in arbitrary d, so that in d1 the curvature of the Fermi surface can be taken into account. Part II gives applications to problems of physical interest. The book addresses researchers and graduate students in theoretical condensed matter physics.

Renormalization Group And Effective Field Theory Approaches To Many Body Systems

Author: Achim Schwenk
Publisher: Springer
ISBN: 3642273203
Size: 21.56 MB
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There have been many recent and important developments based on effective field theory and the renormalization group in atomic, condensed matter, nuclear and high-energy physics. These powerful and versatile methods provide novel approaches to study complex and strongly interacting many-body systems in a controlled manner. The six extensive lectures gathered in this volume combine selected introductory and interdisciplinary presentations focused on recent applications of effective field theory and the renormalization group to many-body problems in such diverse fields as BEC, DFT, extreme matter, Fermi-liquid theory and gauge theories. Primarily aimed at graduate students and junior researchers, they offer an opportunity to explore fundamental physics across subfield boundaries at an early stage in their careers.