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FROM CLASSICAL MECHANICS to QUANTUM FIELD THEORY, a TUTORIAL Copyright © 2020 by World Scientific Publishing Co FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL 11556_9789811210488_TP.indd 1 29/11/19 2:30 PM This page intentionally left blank FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL Manuel Asorey Universidad de Zaragoza, Spain Elisa Ercolessi University of Bologna & INFN-Sezione di Bologna, Italy Valter Moretti University of Trento & INFN-TIFPA, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 11556_9789811210488_TP.indd 2 29/11/19 2:30 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY, A TUTORIAL Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-121-048-8 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11556#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore Syarfeena - 11556 - From Classical Mechanics.indd 1 02-12-19 3:03:23 PM January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page v Preface This book grew out of the mini courses delivered at the Fall Workshop on Geometry and Physics, in Granada, Zaragoza and Madrid. The Fall Workshop on Geometry and Physics takes place in the Iberian Penin- sula since 1992. The aim of this workshop is to introduce advanced graduate students, PhD students and young researchers, both in mathematics and physics, with current aspects of mathematics and physics. The International Fall Work- shop on Geometry and Physics is normally held over four days around the first week of September and attracts many young participants. Two main speakers each deliver a 4-hour mini course, and the rest of the programme is made up of talks by invited speakers and contributed speakers. The large number of young participants each year, many of them attending year after year, convinced the Scientific Committee that it would be a good idea to or- ganize correlated mini courses over different years. In particular, it was thought that a reasonable exposition to quantum formalism would allow the young partic- ipants to properly benefit from seminars and talks dealing with various aspects of quantum theories along with their impact on modern mathematics and mathemat- ical methods. The first of these courses was given in Granada by Elisa Ercolessi (University of Bologna) on Methods of Quantization and it was followed, in the meeting in Zaragoza, by Valter Moretti (University of Trento) with a course on Advanced Quantum Mechanics, while in the meeting in Madrid Manuel Asorey (University of Zaragoza) conluded the series with an Introduction to Quantum Field Theory. The lecturers did an excellent job, obviously with different degrees of sophistication according to their own taste, to present quantum theory from classical mechanics to quantum field theory. Even though these papers have ap- peared in print, separately in special issues devoted to the proceedings, we felt that collecting them in a single volume would render better the continuity, the spirit and the aim for which they were delivered. v January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page vi vi From Classical Mechanics to Quantum Field Theory. A Tutorial We would like to thank the speakers for undertaking a revision of their pub- lished texts with the aim of constructing cross-referencing and an overall index to improve the unitary character of the book. We hope that other students and readers who did not attend the mini courses will benefit from these presentations. Giuseppe Marmo, on behalf of the Scientific Committee January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page vii Acknowledgments The Authors would like to thank the Organizers and the Scientific Committee of the Workshop for the invitation to give the courses and for financial support during the stay. vii This page intentionally left blank January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page ix Contents Preface v Acknowledgments vii 1. A Short Course on Quantum Mechanics and Methods of Quantization 1 1.1 Introduction.............................. 1 1.2 OverviewofQuantumMechanics.................. 7 1.2.1 Fundamental definitions and examples . 7 1.2.2 Geometricquantummechanics............... 17 1.3 MethodsofQuantization....................... 26 1.3.1 Coherent states and Bargmann-Fock representation . 27 1.3.2 Feynmanpathintegral................... 39 1.3.3 Weylquantization...................... 47 Bibliography................................. 64 2. Mathematical Foundations of Quantum Mechanics: An Advanced Short Course 67 2.1 Introduction: Summary of Elementary Facts of QM . 67 2.1.1 Physical facts about quantum mechanics . 68 2.1.2 Elementary formalism for the finite dimensional case . 71 2.1.3 Timeevolution........................ 76 2.1.4 Compositesystems..................... 77 2.1.5 A first look to the infinite dimensional case, CCR and quantizationprocedures................... 79 2.2 Observables in Infinite Dimensional Hilbert Spaces: SpectralTheory............................ 85 ix January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page x x From Classical Mechanics to Quantum Field Theory. A Tutorial 2.2.1 Hilbertspaces........................ 85 2.2.2 Typesofoperators...................... 88 2.2.3 Criteria for (essential) selfadjointness . 100 2.2.4 Spectrumofanoperator.................. 105 2.2.5 Spectralmeasures...................... 108 2.2.6 Spectral decomposition and representation theorems . 114 2.2.7 Measurablefunctionalcalculus............... 120 2.2.8 Elementary formalism for the infinite dimensional case . 121 2.2.9 Technical interemezzo: three operator topologies . 125 2.3 More Fundamental Quantum Structures . 126 2.3.1 TheBooleanlogicofCM.................. 126 2.3.2 The non-Boolean logic of QM, the reason why observables areselfadjointoperators................... 131 2.3.3 Recovering the Hilbert spacestructure.......... 134 2.3.4 States as measures on L(H): Gleason’s theorem . 137 2.3.5 von Neumann algebra of observables, superselection rules.............................. 147 2.3.6 Quantum symmetries: unitary projective representations........................ 158 2.4 Just Few Words about the Algebraic Approach . 181 2.4.1 Algebraicformulation.................... 182 Bibliography................................. 187 3. A Concise Introduction to Quantum Field Theory 189 3.1 Introduction.............................. 189 3.2 QuantumMechanicsandRelativity................. 193 3.2.1 Quantummechanics..................... 193 3.2.2 Relativity and the Poincar´egroup............. 197 3.3 QuantumFieldTheory........................ 200 3.3.1 Canonicalquantization................... 201 3.4 TheQuantumVacuum........................ 206 3.4.1 Renormalization of vacuum energy . 206 3.4.2 Momentumoperator..................... 208 3.4.3 Casimireffect........................ 209 3.5 FieldsversusParticles........................ 211 3.5.1 Fockspace.......................... 212 3.5.2 Wicktheorem........................ 215 3.6 FieldsinInteraction......................... 215 3.6.1 Renormalization of excited states . 217 January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page xi Contents xi 3.7 CovariantApproach......................... 218 3.7.1 Euclideanapproach..................... 220 3.8 ConformalInvariantTheories.................... 223 3.8.1 Functional integral approach . 225 3.9 WhatisBeyond?........................... 227 3.10Appendix1.Casimireffect..................... 230 3.11Appendix2.Gaussianmeasures................... 231 3.12Appendix3.Peierlsbrackets.................... 234 Bibliography................................. 236 Index 239 This page intentionally left blank January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page 1 Chapter 1 A Short Course on Quantum Mechanics and Methods of Quantization Elisa Ercolessi Department of Physics and Astronomy, Universit`a di Bologna and INFN-Sezione di Bologna, via Irnerio 46, 40127, Bologna, Italy. email: [email protected] The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then at presenting the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we
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