This page intentionally left blank FUNCTIONAL INTEGRATION 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 math- ematical definition. Cartier and DeWitt-Morette have created, in this book, a new approach to functional integration. The close collaboration between a math- ematician and a physicist brings a unique perspective to this topic. The book is self-contained: mathematical ideas are introduced, developed, generalized, and applied. In the authors’ hands, functional integration is shown to be a robust, user-friendly, and multi-purpose tool that can be applied to a great variety of situations, for example systems of indistinguishable particles, caustics-analysis, superanalysis, and non-gaussian integrals. Problems in quantum field theory are also considered. In the final part the authors outline topics that can profitably be pursued using material already presented. Pierre Cartier is a mathematician with an extraordinarily wide range of interests and expertise. He has been called “un homme de la Renaissance.” He is Emeritus Director of Research at the Centre National de la Recherche Scientifique, France, and a long-term visitor of the Institut des Hautes Etudes Scientifiques. From 1981 to 1989, he was a senior researcher at the Ecole Poly- technique de Paris, and, between 1988 and 1997, held a professorship at the Ecole Normale Sup´erieure. He is a member of the Soci´et´e Math´ematique de France, the American Mathematical Society, and the Vietnamese Mathematical Society. C ecile´ DeWitt-Morette is the Jane and Roland Blumberg Centen- nial Professor in Physics, Emerita, at the University of Texas at Austin. She is a member of the American and European Physical Societies, and a Mem- bre d’Honneur de la Soci´et´eFran¸caise de Physique. DeWitt-Morette’s interest in functional integration began in 1948. In F. J. Dyson’s words, “she was the first of the younger generation to grasp the full scope and power of the Feynman path integral approach in physics.” She is co-author with Yvonne Choquet-Bruhat of the two-volume book Analysis, Manifolds and Physics, a standard text first published in 1977, which is now in its seventh edition. She is the author of 100 publications in various areas of theoretical physics and has edited 28 books. She has lectured, worldwide, in many institutions and summer schools on topics related to functional integration. CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S. J. Aarseth Gravitational N-Body Simulations J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach† A. M. Anile Relativistic Fluids and Magneto-Fluids† J. A. de Azc´arrage and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O. Babelon, D. Bernard and M. Talon Introduction to Classical Integrable Systems F. Bastianelli and P. van Nieuwenhuizen Path Integrals and Anomalies in Curved Space V. Belinkski and E. Verdaguer Gravitational Solitons† J. Bernstein Kinetic Theory in the Expanding Universe† G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems† N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space† M. Burgess Classical Covariant Fields† S. Carlip Quantum Gravity in 2 + 1 Dimensions† J. C. Collins Renormalization† M. Creutz Quarks, Gluons and Lattices† P. D. D’Eath Supersymmetric Quantum Cosmology† F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds† B. S. deWitt Supermanifolds,2nd edition† P. G. O. Freund Introduction to Supersymmetry† J. Fuchs Affine Lie Algebras and Quantum Groups† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y. Fujii and K. Maeda The Scalar-Tensor Theory of Gravitation A. S. Galperin, E. A. Ivanov, V. I. Orievetsky and E. S. Sokatchev Harmonic Superspace† R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity† M. G¨ockeler and T. Sch¨ucker Differential Geometry, Gauge Theories and Gravity† C. G´omez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-Dimensional Physics† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1: Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop Amplitudes, Anomalies and Phenomenology† V. N. Gribov The Theory of Complex Angular Momenta S. W. Hawking and G. F. R. Ellis The Large Scale Structure of Space-Time† F. Iachello and A. Arima The Interacting Boson Model F. Iachello and P. van Isacker The Interacting Boson–Fermion Model† C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† C. Johnson D-Branes J. I. Kapusta and C. Gale Finite Temperature Field Theory,2nd edition V. E. Korepin, N. M. Boguliubov and A. G. Izergin The Quantum Inverse Scattering Method and Correlation Functions† M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory† N. Manton and P. Sutcliffe Topological Solitons N. H. March Liquid Metals: Concepts and Theory† I. M. Montvay and G. M¨unster Quantum Fields on a Lattice† L. O’Raifeartaigh Group Structure of Gauge Theories† T. Ort´ın Gravity and Strings A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† R. Penrose and W. Rindler Spinors and Space-Time, volume 1: Two-Spinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and Space-Time, volume 2: Spinor and Twistor Methods in Space-Time Geometry† S. Pokorski Gauge Field Theories,2nd edition† J. Polchinski String Theory, volume 1: An Introduction to the Bosonic String† J. Polchinski String Theory, volume 2: Superstring Theory and Beyond† V. N. Popov Functional Integrals and Collective Excitations† R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton† C. Roveli Quantum Gravity W. C. Saslaw Gravitational Physics of Stellar Galactic Systems† H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations,2nd edition J. M. Stewart Advanced General Relativity† A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells Jr Twister Geometry and Field Theory† J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics † Issued as a paperback Functional Integration: Action and Symmetries P. CARTIER AND C. DEWITT-MORETTE cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationonthistitle:www.cambridge.org/9780521866965 © P. Cartier and C. DeWitt-Morette 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 isbn-13 978-0-511-26029-2 eBook (EBL) isbn-10 0-511-26029-6 eBook (EBL) isbn-13 978-0-521-86696-5 hardback isbn-10 0-521-86696-0 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Acknowledgements page xi List of symbols, conventions, and formulary xv PART I THE PHYSICAL AND MATHEMATICAL ENVIRONMENT 1 The physical and mathematical environment3 A: An inheritance from physics 3 1.1 The beginning3 1.2 Integrals over function spaces6 1.3 The operator formalism6 1.4 A few titles7 B: A toolkit from analysis 9 1.5 A tutorial in Lebesgue integration9 1.6 Stochastic processes and promeasures 15 1.7 Fourier transformation and prodistributions 19 C: Feynman’s integral versus Kac’s integral 23 1.8 Planck’s blackbody radiation law 23 1.9 Imaginary time and inverse temperature 26 1.10 Feynman’s integral versus Kac’s integral 27 1.11 Hamiltonian versus lagrangian 29 References 31 PART II QUANTUM MECHANICS 2 First lesson: gaussian integrals 35 2.1 Gaussians in R 35 2.2 Gaussians in RD 35 2.3 Gaussians on a Banach space 38 v vi Contents 2.4 Variances and covariances 42 2.5 Scaling and coarse-graining 46 References 55 3 Selected examples 56 3.1 The Wiener measure and brownian paths 57 3.2 Canonical gaussians in L2 and L2,1 59 3.3 The forced harmonic oscillator 63 3.4 Phase-space path integrals 73 References 76 4 Semiclassical expansion; WKB 78 4.1 Introduction 78 4.2 The WKB approximation 80 4.3 An example: the anharmonic oscillator 88 4.4 Incompatibility with analytic continuation 92 4.5 Physical interpretation of the WKB approximation 93 References 94 5 Semiclassical expansion; beyond WKB 96 5.1 Introduction 96 5.2 Constants of the motion 100 5.3 Caustics 101 5.4 Glory scattering 104 5.5 Tunneling 106 References 111 6 Quantum dynamics: path integrals and the operator formalism 114 6.1 Physical dimensions and expansions 114 6.2 A free particle 115 6.3 Particles in a scalar potential V 118 6.4 Particles in a vector potential A 126 6.5 Matrix elements and kernels 129 References 130 PART III METHODS FROM DIFFERENTIAL GEOMETRY 7 Symmetries 135 7.1 Groups of transformations. Dynamical vector fields 135 7.2 A basic theorem 137 Contents vii 7.3 The group of transformations on a frame bundle 139 7.4 Symplectic manifolds 141 References 144 8 Homotopy 146 8.1 An example: quantizing a spinning top 146 8.2 Propagators on SO(3) and SU(2) 147 8.3 The homotopy theorem for path integration 150 8.4 Systems of indistinguishable particles.