Path Integrals: Part II

Daniel Zwillinger

86 II Concepts and Definitions

Michel Le Bellac, G. Barton

Abstract This chapter links Parts I and II, which were devoted to statistical mechanics, and Part III, which deals with quantum field theory. In Part III we shall revert to the tools developed for studying critical phenomena, namely functional integrals (often called path integrals in this new connection); Feynman diagrams; renormalization; and the renormalization group. A few adaptations are all that these tools need in their new context, and we shall use some elementary examples to illustrate the links between problems in statistical and in quantum mechanics.

Osvaldo Civitarese, Manuel Gadella

Belal Ehsan Baaquie

Foreword Preface Acknowledgements 1. Synopsis Part I. Fundamental Concepts of Finance: 2. Introduction to finance 3. Derivative securities Part II. Systems with Finite Number of Degrees of Freedom: 4. Hamiltonians and stock options 5. Path integrals and stock options 6. Stochastic interest rates' Hamiltonians and path integrals Part III. Quantum Field Theory of Interest Rates Models: 7. Quantum field theory of forward interest rates 8. Empirical forward interest rates and field theory models 9. Field theory of Treasury Bonds' derivatives and hedging 10. Field theory Hamiltonian of forward interest rates 11. Conclusions Appendix A: mathematical background Brief glossary of financial terms Brief glossary of physics terms List of main symbols References Index.

Pierre Cartier, Cécile DeWitt-Morette

Acknowledgements List symbols, conventions, and formulary Part I. The Physical and Mathematical Environment: 1. The physical and mathematical environment Part II. Quantum Mechanics: 2. First lesson: gaussian integrals 3. Selected examples 4. Semiclassical expansion: WKB 5. Semiclassical expansion: beyond WKB 6. Quantum dynamics: path integrals and operator formalism Part III. Methods from Differential Geometry: 7. Symmetries 8. Homotopy 9. Grassmann analysis: basics 10. Grassmann analysis: applications 11. Volume elements, divergences, gradients Part IV. Non-Gaussian Applications: 12. Poisson processes in physics 13. A mathematical theory of Poisson processes 14. First exit time: energy problems Part V. Problems in Quantum Field Theory: 15. Renormalization 1: an introduction 16. Renormalization 2: scaling 17. Renormalization 3: combinatorics 18. Volume elements in quantum field theory Bryce DeWitt Part VI. Projects: 19. Projects Appendix A. Forward and backward integrals: spaces of pointed paths Appendix B. Product integrals Appendix C. A compendium of gaussian integrals Appendix D. Wick calculus Alexander Wurm Appendix E. The Jacobi operator Appendix F. Change of variables of integration Appendix G. Analytic properties of covariances Appendix H. Feynman's checkerboard Bibliography Index.