Advances in IMAGING AND ELECTRON PHYSICS VOLUME 154 DIRAC ’S DIFFERENCE EQUATION AND THE PHYSICS OF FINITE DIFFERENCES EDITOR-IN-CHIEF PETER W. HAWKES CEMES-CNRS Toulouse, France HONORARY ASSOCIATE EDITORS TOM MULVEY BENJAMIN KAZAN Advances in IMAGING AND ELECTRON PHYSICS VOLUME 154 DIRAC ’S DIFFERENCE EQUATION AND THE PHYSICS OF FINITE DIFFERENCES H ENNING F. HARMUTH Retired, The Catholic University of America Washington DC, USA BEATE MEFFERT Humboldt-Universität Berlin, Germany AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 84 Theobald’s Road, London WC1X 8RR, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2008 Copyright © 2008 Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
[email protected] Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374221-6 For information on all Academic Press publications visit our web site at books.elsevier.com Printed and bound in The USA 08 09 10 10 9 8 7 6 5 4 3 2 1 To the memory of Max Planck (1858–1947) Founder of quantum physics and distinguished participant of the Morgenthau Plan, 1945–1948. This page intentionally left blank CONTENTS Preface Future contributions Foreword List of frequently used symbols 1 Introduction 1.1 1.2 1.3 1.4 1.5 Maxwell’s Equations with Magnetic Dipole Currents Lagrange Function Basic Concepts of the Calculus of Finite Differences Finite Differences and Spatial Dimensions Curved Space in a Difference Theory 2 Modified Dirac Equat