Geometric Approaches to Quantum Field Theory
Geometric Approaches to Quantum Field Theory

Author: Kieran Finn

Publisher: Springer Nature

Published: 2021-10-07

Total Pages: 212

ISBN-13: 3030852695

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The ancient Greeks believed that everything in the Universe should be describable in terms of geometry. This thesis takes several steps towards realising this goal by introducing geometric descriptions of systems such as quantum gravity, fermionic particles and the origins of the Universe itself. The author extends the applicability of previous work by Vilkovisky, DeWitt and others to include theories with spin 1⁄2 and spin 2 degrees of freedom. In addition, he introduces a geometric description of the potential term in a quantum field theory through a process known as the Eisenhart lift. Finally, the methods are applied to the theory of inflation, where they show how geometry can help answer a long-standing question about the initial conditions of the Universe. This publication is aimed at graduate and advanced undergraduate students and provides a pedagogical introduction to the exciting topic of field space covariance and the complete geometrization of quantum field theory.

Geometric and Topological Methods for Quantum Field Theory
Geometric and Topological Methods for Quantum Field Theory

Author: Hernan Ocampo

Publisher: Cambridge University Press

Published: 2010-04-29

Total Pages: 435

ISBN-13: 113948673X

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Aimed at graduate students in physics and mathematics, this book provides an introduction to recent developments in several active topics at the interface between algebra, geometry, topology and quantum field theory. The first part of the book begins with an account of important results in geometric topology. It investigates the differential equation aspects of quantum cohomology, before moving on to noncommutative geometry. This is followed by a further exploration of quantum field theory and gauge theory, describing AdS/CFT correspondence, and the functional renormalization group approach to quantum gravity. The second part covers a wide spectrum of topics on the borderline of mathematics and physics, ranging from orbifolds to quantum indistinguishability and involving a manifold of mathematical tools borrowed from geometry, algebra and analysis. Each chapter presents introductory material before moving on to more advanced results. The chapters are self-contained and can be read independently of the rest.

Geometric Methods for Quantum Field Theory
Geometric Methods for Quantum Field Theory

Author: Hernan Ocampo

Publisher: World Scientific

Published: 2001

Total Pages: 530

ISBN-13: 9810243510

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Both mathematics and mathematical physics have many active areas of research where the interplay between geometry and quantum field theory has proved extremely fruitful. Duality, gauge field theory, geometric quantization, Seiberg -- Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups offer some striking recent examples of modern topics which stand on the borderline between geometry and analysis on the one hand and quantum field theory on the other, where the physicist's and the mathematician's perspective complement each other, leading to new mathematical and physical concepts and results. This volume introduces the reader to some basic mathematical and physical tools and methods required to follow the recent developments in some active areas of mathematical physics, including duality, gauge field theory, geometric quantization, Seiberg -- Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It comprises seven,self-contained lectures, which should progressively give the reader a precise idea of some of the techniques used in these areas, as well as a few short communications presented by young participants at the school.

Geometric Methods For Quantum Field Theory
Geometric Methods For Quantum Field Theory

Author: Hernan Ocampo

Publisher: World Scientific

Published: 2001-04-30

Total Pages: 530

ISBN-13: 9814492825

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Both mathematics and mathematical physics have many active areas of research where the interplay between geometry and quantum field theory has proved extremely fruitful. Duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups offer some striking recent examples of modern topics which stand on the borderline between geometry and analysis on the one hand and quantum field theory on the other, where the physicist's and the mathematician's perspective complement each other, leading to new mathematical and physical concepts and results.This volume introduces the reader to some basic mathematical and physical tools and methods required to follow the recent developments in some active areas of mathematical physics, including duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It comprises seven self-contained lectures, which should progressively give the reader a precise idea of some of the techniques used in these areas, as well as a few short communications presented by young participants at the school.

Geometric Analysis and Applications to Quantum Field Theory
Geometric Analysis and Applications to Quantum Field Theory

Author: Peter Bouwknegt

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 213

ISBN-13: 1461200679

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In the last decade there has been an extraordinary confluence of ideas in mathematics and theoretical physics brought about by pioneering discoveries in geometry and analysis. The various chapters in this volume, treating the interface of geometric analysis and mathematical physics, represent current research interests. No suitable succinct account of the material is available elsewhere. Key topics include: * A self-contained derivation of the partition function of Chern- Simons gauge theory in the semiclassical approximation (D.H. Adams) * Algebraic and geometric aspects of the Knizhnik-Zamolodchikov equations in conformal field theory (P. Bouwknegt) * Application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems (A.L. Carey and E. Langmann) * A study of variational methods in Hermitian geometry from the viewpoint of the critical points of action functionals together with physical backgrounds (A. Harris) * A review of monopoles in nonabelian gauge theories (M.K. Murray) * Exciting developments in quantum cohomology (Y. Ruan) * The physics origin of Seiberg-Witten equations in 4-manifold theory (S. Wu) Graduate students, mathematicians and mathematical physicists in the above-mentioned areas will benefit from the user-friendly introductory style of each chapter as well as the comprehensive bibliographies provided for each topic. Prerequisite knowledge is minimal since sufficient background material motivates each chapter.

Geometric And Topological Methods For Quantum Field Theory - Proceedings Of The Summer School
Geometric And Topological Methods For Quantum Field Theory - Proceedings Of The Summer School

Author: Alexander Cardona

Publisher: World Scientific

Published: 2003-03-21

Total Pages: 495

ISBN-13: 9814487678

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This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school.

Structural Aspects Of Quantum Field Theory And Noncommutative Geometry (Second Edition) (In 2 Volumes)
Structural Aspects Of Quantum Field Theory And Noncommutative Geometry (Second Edition) (In 2 Volumes)

Author: Gerhard Grensing

Publisher: World Scientific

Published: 2021-07-15

Total Pages: 1656

ISBN-13: 9811237093

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The book is devoted to the subject of quantum field theory. It is divided into two volumes. The first volume can serve as a textbook on main techniques and results of quantum field theory, while the second treats more recent developments, in particular the subject of quantum groups and noncommutative geometry, and their interrelation.The second edition is extended by additional material, mostly concerning the impact of noncommutative geometry on theories beyond the standard model of particle physics, especially the possible role of torsion in the context of the dark matter problem. Furthermore, the text includes a discussion of the Randall-Sundrum model and the Seiberg-Witten equations.

Geometric, Algebraic And Topological Methods For Quantum Field Theory - Proceedings Of The 2013 Villa De Leyva Summer School
Geometric, Algebraic And Topological Methods For Quantum Field Theory - Proceedings Of The 2013 Villa De Leyva Summer School

Author: Alexander Cardona

Publisher: World Scientific

Published: 2016-09-06

Total Pages: 385

ISBN-13: 9814730890

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Based on lectures held at the 8th edition of the series of summer schools in Villa de Leyva since 1999, this book presents an introduction to topics of current interest at the interface of geometry, algebra, analysis, topology and theoretical physics. It is aimed at graduate students and researchers in physics or mathematics, and offers an introduction to the topics discussed in the two weeks of the summer school: operator algebras, conformal field theory, black holes, relativistic fluids, Lie groupoids and Lie algebroids, renormalization methods, spectral geometry and index theory for pseudo-differential operators.

Operators, Geometry and Quanta
Operators, Geometry and Quanta

Author: Dmitri Fursaev

Publisher: Springer Science & Business Media

Published: 2011-06-25

Total Pages: 294

ISBN-13: 9400702051

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This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.

Topology, Geometry and Quantum Field Theory
Topology, Geometry and Quantum Field Theory

Author: Ulrike Luise Tillmann

Publisher: Cambridge University Press

Published: 2004-06-28

Total Pages: 596

ISBN-13: 9780521540490

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The symposium held in honour of the 60th birthday of Graeme Segal brought together leading physicists and mathematicians. Its topics were centred around string theory, M-theory, and quantum gravity on the one hand, and K-theory, elliptic cohomology, quantum cohomology and string topology on the other. Geometry and quantum physics developed in parallel since the recognition of the central role of non-abelian gauge theory in elementary particle physics in the late seventies and the emerging study of super-symmetry and string theory. With its selection of survey and research articles these proceedings fulfil the dual role of reporting on developments in the field and defining directions for future research. For the first time Graeme Segal's manuscript 'The definition of Conformal Field Theory' is published, which has been greatly influential over more than ten years. An introduction by the author puts it into the present context.