From Classical Mechanics To Quantum Field Theory, A Tutorial PDF
Preview From Classical Mechanics To Quantum Field Theory, A Tutorial
CLASSICAL FROM MECHANICS QUANTUM TO FIELD THEORY A TUTORIAL 11556_9789811210488_TP.indd 1 29/11/19 2:30 PM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk CLASSICAL FROM MECHANICS QUANTUM TO 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 January3,2020 9:10 FromClassicalMechanicstoQuantumFieldTheory 9inx6in b3742-main pagev Preface ThisbookgrewoutoftheminicoursesdeliveredattheFallWorkshoponGeometry and Physics, in Granada, Zaragozaand Madrid. TheFallWorkshoponGeometryandPhysicstakesplaceintheIberianPenin- 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. Thelargenumberofyoungparticipantseachyear,manyofthemattendingyear after year, convincedthe Scientific Committee that it would be a goodidea to or- ganize correlated mini courses over different years. In particular, it was thought that a reasonableexpositionto quantumformalismwouldallow the youngpartic- ipants to properly benefit from seminars and talks dealing with various aspects of quantumtheoriesalongwiththeirimpactonmodernmathematicsandmathemat- 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 January3,2020 9:10 FromClassicalMechanicstoQuantumFieldTheory 9inx6in b3742-main pagevi vi From Classical Mechanics toQuantum 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 readerswho did not attend the mini courseswill benefit fromthese presentations. Giuseppe Marmo, on behalf of the Scientific Committee January3,2020 9:10 FromClassicalMechanicstoQuantumFieldTheory 9inx6in b3742-main pagevii 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 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January3,2020 9:10 FromClassicalMechanicstoQuantumFieldTheory 9inx6in b3742-main pageix Contents Preface v Acknowledgments vii 1. A Short Course on Quantum Mechanics and Methods of Quantization 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of Quantum Mechanics . . . . . . . . . . . . . . . . . . 7 1.2.1 Fundamental definitions and examples . . . . . . . . . . . 7 1.2.2 Geometric quantum mechanics . . . . . . . . . . . . . . . 17 1.3 Methods of Quantization . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.1 Coherent states and Bargmann-Fock representation . . . 27 1.3.2 Feynman path integral . . . . . . . . . . . . . . . . . . . 39 1.3.3 Weyl quantization . . . . . . . . . . . . . . . . . . . . . . 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 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . 76 2.1.4 Composite systems . . . . . . . . . . . . . . . . . . . . . 77 2.1.5 A first look to the infinite dimensional case, CCR and quantization procedures . . . . . . . . . . . . . . . . . . . 79 2.2 Observables in Infinite Dimensional Hilbert Spaces: Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 ix
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