RELATIVISTIC QUANTUM MECHANICS AND RELATED TOPICS 1122559999__99778899881111224488775577__TTPP..iinndddd 11 2211//22//2222 88::2244 PPMM January19,2018 9:17 ws-book961x669 BeyondtheTriangle: BrownianMotion...PlanckEquation-10734 HKU˙book pagevi TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk RELATIVISTIC QUANTUM MECHANICS AND RELATED TOPICS Eduardo Guendelman David Owen Ben Gurion University, Israel World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 1122559999__99778899881111224488775577__TTPP..iinndddd 22 2211//22//2222 88::2244 PPMM 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. RELATIVISTIC QUANTUM MECHANICS AND RELATED TOPICS Copyright © 2022 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 978-981-124-875-7 (hardcover) ISBN 978-981-124-876-4 (ebook for institutions) ISBN 978-981-124-877-1 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12599#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore SSyyaarrffeeeennaa -- 1122559999 -- RReellaattiivviissttiicc QQuuaannttuumm MMeecchhaanniiccss aanndd RReellaatteedd TTooppiiccss..iinndddd 11 2222//22//22002222 88::4466::0000 aamm February24,2022 11:10 RelativisticQuantumMechanicsandRelatedTopics-9.75inx6.5in b4633-fm pagev Preface The aim of this book is to study the development of relativistic quan- tum mechanics, developing first the free-field equations for spin-0 parti- cles and for spin-1/2 particles, leading to the Klein–Gordon equation and Dirac equations. Interactions of these particles with the electromagnetic field through minimal coupling and other interactions between particles are introduced. Calculationofthe fundamental processesofquantum elec- trodynamics by means of Feynman’s propagatortheoryallowsfor a proper treatmentofdiversescatteringandparticlecreationprocesses. Inaddition, a number of special topics are discussed such as spontaneous symmetry breaking, the global and local cases, the Higgs mechanism, axion–photon interactions using techniques borrowedfrom the scalar QED, pair creation inastrongexternalelectricfield, thetwo-dimensionalrepresentationofthe Klein–Gordon propagator,bound states in the Greens functions approach, and the Breit equation for bound states. Then, the photon–electron inter- actions are treated in the context of a symmetric treatment within elec- trons and photons: Compton scattering, pair creation, and pair annihila- tion. Moreover,non-abeliangaugetheories, the Glashow–Weinberg–Salam model and some electroweak processes are discussed. Finally, we end up with discussing Feynman diagrams. v January19,2018 9:17 ws-book961x669 BeyondtheTriangle: BrownianMotion...PlanckEquation-10734 HKU˙book pagevi TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk February24,2022 11:10 RelativisticQuantumMechanicsandRelatedTopics-9.75inx6.5in b4633-fm pagevii Contents Preface v Chapter 1. Preliminaries 1 Chapter 2. The Relativistic Principle 3 Chapter 3. Klein–Gordon Equation and Its Physical Applications 13 Chapter 4. The Dirac Equation 63 Chapter 5. Photon–ElectronInteractions 139 Chapter 6. Feynman Diagrams 161 Chapter 7. Non-abelian Gauge Theories 167 Chapter 8. Weak Interactions 177 Bibliography 201 vii January19,2018 9:17 ws-book961x669 BeyondtheTriangle: BrownianMotion...PlanckEquation-10734 HKU˙book pagevi TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk February24,2022 11:10 RelativisticQuantumMechanicsandRelatedTopics-9.75inx6.5in b4633-ch01 page1 Chapter 1 Preliminaries Before we begin with our study into the world of relativistic quantum mechanics, it is perhaps useful to place this subject into perspective way. Classicalmechanicsisthestudyofthemotion,i.e.,aparticleinanexternal field, or the motion of several particles interaction either by their mutual gravitational or electromagnetic interaction or rigid bodies with various constraints interacting under these interactions. The laws are governedby Newton’s three laws which can be cast almost equivalently in the eloquent Lagrangian (or Hamiltonian) formulation. For this case, the position of each particle is described in terms of its generalized coordinate qi, and its momentum is given by pi = ∂∂Lq˙ where the LagrangianL equals the kinetic energy minus the potential energy and q˙ ≡ dq. Thus, for a particle of dt mass m in an external field described by a potential V(x) where the ith component of x is xi, the Lagrangian L equals L = m2x˙2−V(x) and the corresponding momentum is given by pi =mx˙i. While these laws are an excellent description of nature when the veloc- ities in the system are much less than the speed of light and the size of the system large, compared with atomic dimensions, they breakdown for speed ≈ c where c is the speed of light or the dimensions are comparable to those of an atom. Relativistic quantum mechanics tries to go beyond this classical description by incorporating special relativity in its founda- tions and quantum mechanics by incorporating the wave-particle duality which is fundamental formulation in the quantum mechanics. These are givenbytheEinsteinandDeBroglierelationships,whichareEph =hν and p = h, where ν and λ, respectively, are the corresponding frequency and λ wavelength associated with a particle. 1