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Functional integration : action and symmetries /
Functional integration successfully entered physics as path integrals in the 1942 Ph. D. dissertation of Richard P. Feynman, but it made no sense at all as a mathematical definition. Cartier and DeWitt-Morette have created, in this book, a fresh approach to functional integration. The book is self-c...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2006.
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| Colección: | Cambridge monographs on mathematical physics.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover Half-title Series-title Title Copyright Contents Acknowledgements Cècile thanks her graduate students List of symbols, conventions, and formulary Symbols Conventions Part I The physical and mathematical environment 1 The physical and mathematical environment A: An inheritance from physics B: A toolkit from analysis C: Feynman s integral versus Kac s integral Part II Quantum mechanics 2 First lesson: gaussian integrals 2.1 Gaussians in R 2.2 Gaussians in R 2.3 Gaussians on a Banach space 2.4 Variances and covariances 2.5 Scaling and coarse-graining References 3 Selected examples 3.1 The Wiener measure and brownian paths (discretizing a path integral) 3.2 Canonical gaussians in L and L 3.3 The forced harmonic oscillator 3.4 Phase-space path integrals References 4 Semiclassical expansion; WKB 4.1 Introduction 4.2 The WKB approximation 4.3 An example: the anharmonic oscillator 4.4 Incompatibility with analytic continuation 4.5 Physical interpretation of the WKB approximation References 5 Semiclassical expansion; beyond WKB 5.1 Introduction 5.2 Constants of the motion 5.3 Caustics 5.4 Glory scattering 5.5 Tunneling References 6 Quantum dynamics: path integrals and operator formalism 6.1 Physical dimensions and expansions 6.2 A free particle 6.3 Particles in a scalar potential V 6.4 Particles in a vector potential 6.5 Matrix elements and kernels References Part III Methods from diffierential geometry 7 Symmetries 7.1 Groups of transformations. Dynamical vector fields 7.2 A basic theorem 7.3 The group of transformations on a frame bundle 7.4 Symplectic manifolds References 8 Homotopy 8.1 An example: quantizing a spinning top 8.2 Propagators on SO(3) and SU(2) [4] 8.3 The homotopy theorem for path integration 8.4 Systems of indistinguishable particles. Anyons 8.5 A simple model of the Aharanov Bohm effect References 9 Grassmann analysis: basics 9.1 Introduction 9.2 A compendium of Grassmann analysis Contributed by Maria E. Bell 9.3 Berezin integration 9.4 Forms and densities References 10 Grassmann analysis: applications 10.1 The Euler Poincarè characteristic 10.2 Supersymmetric quantum field theory 10.3 The Dirac operator and Dirac matrices References Other references that have inspired this chapter 11 Volume elements, divergences, gradients 11.1 Introduction. Divergences 11.2 Comparing volume elements 11.3 Integration by parts References Other references that have inspired this chapter Part IV Non-gaussian applications 12 Poisson processes in physics 12.1 The telegraph equation 12.2 Klein Gordon and Dirac equations 12.3 Two-state systems interacting with their environment4 References 13 A mathematical theory of Poisson processes 13.1 Poisson stochastic processes 13.2 Spaces of Poisson paths 13.3 Stochastic solutions of differential equations 13.4 Differential equations: explicit solutions References 14 The first exit time; energy problems 14.1 Introduction: fixed-energy Green s function 14.2 The path integral for a fixed-energy amplitude 14.3 Periodic and quasiperiodic orbits 14.4 Intrinsic and tuned times of a process References Part V Problems in quantum field theory 15 Renormalization 1: an introduction 15.1 Introduction 15.2 From paths to fields 15.3 Green s example 15.4 Dimensional regularization References Other references that inspired this chapter 16 Renormalization 2: scaling 16.1 The renormalization group 16.2 The Lambda Phi system References 17 Renormalization 3: combinatorics 17.1 Introduction 17.2 Background 17.3 Graph summary 17.4 The grafting operator 17.5 Lie algebra 17.6 Other operations 17.7 Renormalization.