Equivariant Cohomology and Localization of Path Integrals

ISBN-13: 9783662142844 / Angielski / Miękka / 2014 / 315 str.

Richard J. Szabo

Opis

This book reviews localization for the equivariant evaluation of techniques It Feynmanpathintegrals. methods for developssystematicgeometric study the semi classical of ing for properties phase spacepath integrals dynamical the relationswith and systems, emphasizing integrable topologicalquantum field theories. with a detailed review ofthe relevant Beginning mathemat ical and the Duistermaat Heckman background equivariant cohomology it demonstrates theorem, how the localization ideas are related to classical andhow be can extended to derive local integrability they formally explicit izationformulas for in instances path BRST integrals special using quantiza tion Various localizations techniques. loop are andrelatedto space presented notions in and field quantum Thebookem integrability topological theory. thecommon thatsuchlocalizablemodels phasizes symmetries always possess and uses these to discussthe of ofthe local symmetries range applicability ization formulas.Anumber of and mathematical are physical applications inconnectionwith presented Morse elementaryquantummechanics, theory, index character formulas for Lie theorems, semi simple groups, quantization of inmatrix modular spinsystems, unitaryintegrations models, invariantsof Riemarm field two dimensional surfaces, supersymmetricquantum theories, conformal field field theories and Yang Mills theory, theory, cohomological the in field Somemodern loopexpansion of quantum theory.techniques path such coherent state also integral quantization, as are discussed. methods, The relationsbetween localizationandother ideas in equivariant topological field such asthe and theory, for Batalin Fradkin Vilkovisky Mathai Quillen are and used to discuss the malisms, presented between general relationship field theoriesandmore conventional models. topological physical Copenhagen, 1999 Richard J. Szabo September Iwould like to thank I. G. F. Acknowledgements: Kogan, Landi, Lizzi and G.

Recenzje

"A thorough exposition of the current state of applying equivariant cohomology to quantum field theory. [...] If one takes the attitude that this material may make mathematical sense within the next fifty years, the book can be appreciated as a well-organized exposition of the topological content of quantum field theory from a physics viewpoint." (Mathematical Reviews 2002a)

Spis treści

Equivariant Cohomology and the Localization Principle.- Finite-Dimensional Localization Theory for Dynamical Systems.- Quantum Localization Theory for Phase Space Path Integrals.- Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems.- Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations.- Beyond the Semi-Classical Approximation.- Equivariant Localization in Cohomological Field Theory.- Appendix A: BRST Quantization.- Appendix B: Other Models of Equivariant Cohomology.

Blurb

This book, addressing both researchers and graduate students, reviews equivariant localization techniques for the evaluation of Feynman path integrals. The author gives the relevant mathematical background in some detail, showing at the same time how localization ideas are related to classical integrability. The text explores the symmetries inherent in localizable models for assessing the applicability of localization formulae. Various applications from physics and mathematics are presented.