Advances in Imaging and Electron Physics, Volume 154: Dirac's Difference Equation and the Physics of Finite Differences (Advances in Imaging and Electron Physics)

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 Equation 2.1 2.2 2.3 2.4 2.5 2.6

3

Differential Equation with Magnetic Current Density Modified Dirac Difference Equation Solution of the Difference Equation for :x0 Time Variation of :01 ([, R) Hamiltonian Formalism and Quantization Finite Limit of the Period Number L

ix xi xv xvii

1 1 9 15 22 37

47 47 58 68 79 91 99

Inhomogeneous Dirac Difference Equation

108

3.1 3.2 3.3 3.4

108 115 132

Inhomogeneous Equation (2.2-33) Resolution %x h/m0c Quantization of the Solution Evaluation of the Energy Û for Small Distances %x

140

Equations are numbered consecutively within each of Sections 1.1 to 6.11. Reference to an equation in a different section is made by writing the number of the section in front of the number of the equation, e.g., Eq.(1.1-45) for Eq.(45) in Section 1.1. Illustrations are numbered consecutively within each section, with the number of the section given first, e.g., Figure 1.1-1. References are listed by the name of the author(s), the year of publication, and a lowercase Latin letter if more than one reference by the same author(s) is listed for that year.

vii

viii

Contents

4 Dirac Difference Equation in Spherical Coordinates 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

5

158

Electron in an Electromagnetic Field Relativistic Lagrange Function Quantization of the Homogeneous Equation Unbounded Electron in a Coulomb Field Anti-Particles With s0 Near s = –1 Solutions for s0 Near s = i Solutions with s0 in the Neighborhood of s = –i Energy or Mass Ratios for E/m0c 2 >1 and %rmin

158 163 171 179 184 190 201 210

Inhomogeneous Equations for Coulomb Potential

222

5.1 5.2 5.3

222 232 235

Quantization of the Inhomogeneous Term Separation of the Functions :1j (S, R) Solutions for v(R) and w(S)

6 Appendix 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Calculations for Section 2.3 Calculations for Section 3.2 Inhomogeneous Difference Equations Further Elaboration of Eq.(2.1-28) Quantization of Lcr2 to LcK5 Polynomials as Solutions of Difference Equations of Second Order Separation of Variables in Section 4.3 Solution of D1(S) and D4(S) Calculation of C1v and C4v in Section 6.7 Slanted Coordinate Systems With Finite Differences Riemann Manifolds and Bended Eigen-Coordinates

241 241 243 247 258 260 262 271 279 293 299 301

REFERENCES AND BIBLIOGRAPHY

310

INDEX

316

PREFACE

It is a pleasure to welcome this ninth contribution to these Advances, or their forerunner Advances in Electronics & Electron Physics, by Henning F. Harmuth, of which the most recent have been written in collaboration with Beate Meffert, a professor in Berlin, whom Harmuth met some 35 years ago when she was a PhD student in East Germany. Harmuth, now retired, reminds me that his student days coincided with the early development of information theory in electrical engineering; information theory revealed the lack of transient solutions of Maxwell’s equations, with which much of Harmuth’s earlier – and at the time, controversial – work was concerned. The replacement of infinitesimal elements, dx, dt, by finite intervals, %x, %t, eliminated the need for unphysical infinite information from physics. When I wrote prefaces for those early articles or books, I was obliged to adopt a defensive tone, for Harmuth’s early work was severely criticized and it is gratifying, looking back, to recall that much of his highly original work first appeared in these pages. A special issue of Electromagnetic Phenomena (Vol. 7, No. 1, 2007, see www.emph.com.ua), dedicated to Harmuth, contains a full curriculum vitae. The present volume is the third and (according to the author) last of a trilogy, of which the two preceding volumes form volumes 129 and 137 of these Advances. I have no doubt that it will be widely appreciated. Peter W. Hawkes

ix

This page intentionally left blank

FUTURE CONTRIBUTIONS

S. Ando Gradient operators and edge and corner detection V. Argyriou and M. Petrou (Vol. 156) Photometric stereo: an overview W. Bacsa Optical interference near surfaces, sub-wavelength microscopy and spectroscopic sensors C. Beeli Structures and microscopy of quasicrystals C. Bobisch and R. Möller Ballistic electron microscopy G. Borgefors Distance transforms Z. Bouchal Non-diffracting optical beams F. Brackx, N. de Schepper and F. Sommen (Vol. 156) The Fourier transform in Clifford analysis A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gröbner bases T. Cremer Neutron microscopy N. de Jonge (Vol. 156) Carbon nanotube electron sources for electron microscopes A. X. Falcão The image foresting transform R. G. Forbes Liquid metal ion sources B. J. Ford The earliest microscopical research C. Fredembach Eigenregions for image classification

xi

xii

Future contributions

A. Gölzhäuser Recent advances in electron holography with point sources D. Greenfield and M. Monastyrskii (Vol. 155) Selected problems of computational charged particle optics M. I. Herrera The development of electron microscopy in Spain J. Isenberg Imaging IR-techniques for the characterization of solar cells K. Ishizuka Contrast transfer and crystal images A. Jacobo Intracavity type II second-harmonic generation for image processing L. Kipp Photon sieves G. Kögel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencová Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Matsuya Calculation of aberration coefficients using Lie algebra S. McVitie Microscopy of magnetic specimens P. G. Merli and V. Morandi Scanning electron microscopy of thin films M. A. O’Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee and M. Nielsen The Scale-space properties of natural images E. Rau Energy analysers for electron microscopes

Future contributions

xiii

E. Recami and M. Zamboni-Rached (Vol. 156) Localized waves: a review R. Shimizu, T. Ikuta and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods T. Soma Focus-deflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem N. M. Towghi Ip norm optimal filters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics K. Vaeth and G. Rajeswaran Organic light-emitting arrays M. van Droogenbroeck and M. Buckley Anchors in mathematical morphology M. Yavor Optics of charged particle analysers

This page intentionally left blank

Foreword

The development of information theory has alerted us to the fact that infinite information is as impossible in physics as infinite energy. We cannot observe, process or display infinite information. The conservation law of energy ended centuries of attempts to build a perpetuum mobile and in the process eliminated the concept of infinite energy. Today we use the concept of infinite information whenever we use non-denumerable many infinitesimal intervals dx or dt. It may take some time before the similarity with infinite energy is generally understood. Infinitesimal and non-denumerable have always been outside the realm of the observable. They strictly belong to mathematics, a science of the thinkable. How was it possible that these concepts became so important in physics, a science of the observable? The answer is, of course, the success of differential calculus. The development of quantum theory, the Compton effect, and Heisenberg’s uncertainty relation were not strong enough to shake our believe in differentials. It was probably the most important effect of information theory to make us think of replacing dx, dt by arbitrarily small but finite differences %x, %t. The concept of infinite had to be replaced too, but this was less dramatic. An infinite distance requires only denumerable finite intervals %x. Denumerable may be used as limit for large numbers, which can be observed. But non-denumerable is not a limit for anything observable. Non-denumerable and infinitesimal are like the two sides of a coin, since an infinitesimal distance dx is obtained by dividing a finite distance %x into non-denumerable intervals. That the substitution of finite differences %x, %t for differentials dx, dt could bring new results has been known since Hölder (1887), who proved that differential equations and difference equations define different classes of functions. The calculus of finite differences is as old or older than differential calculus, but their state of development differs enormously. A typical collection of tables of integrals fills about 900 pages (Gradsteyn and Ryzik 1980) while a corresponding table of sums is readily accommodated on a single page. The lack of development of the calculus of finite differences means that much effort has to be devoted to the development of mathematical methods before one can apply them to physics. We regret the resulting difficulty of seeing the physical results obscured by all the mathematics but there is no way around this problem. xv

xvi

Foreword

Two previous books applied the calculus of finite differences to quantum electrodynamics and the Klein-Gordon equation (Harmuth and Meffert 2003, 2005). Here we extend the theory to Dirac’s equation. As in the case of the Klein-Gordon equation we obtain solutions for particles with negative mass –m0 that are completely equivalent to the solutions with positive mass +m0. But in addition we obtain solutions for nuclear distances of the order of 10–13 m and less rather than for the usual atomic distances. There are a number of other deviations from the differential theory. For instance, the eigenvalues of an electron in a Coulomb field deviate slightly from those of the differential theory. The deviations are similar to the Lamb shift (Lamb and Retherford 1947). Sections 1.4 and 1.5 show some surprising results for our concept of space caused by the replacement of dx by %x. The beginnings of these results go back to a book in 1989 but they were never published in detail to avoid fruitless controversies. Advancing age called for a decision to publish now or never. The authors want to thank Humboldt-Universität in Berlin for help with computing and library services. We further want to take this last opportunity to express our appreciation for scientific contributions, help, or support to some fellow scientists: K.L. Lukin, S.A. Masalov, G.P. Pochanin, Academy of Sciences of Ukraine, Kharkiv; N.N. Kolchigin and V.A. Katrich, Karazin Kharkiv National University, Kharkiv, Ukraine; T.W. Barrett, Washington DC, USA; the late K.G. Beauchamp, University of Lancaster, Great Britain; V. Bolotov, Institute for Electromagnetic Research, Kharkiv, Ukraine; V. Borisov, St. Petersburg University, Russia; the late Chang Tong, Tsinghua University, Beijing, PR China; P. Hillion, Le Vesinet, France; M.G.M. Hussain, University of Kuwait; I.Ya. Immoreev, Moscow Aviation Institute, Russia; A.G. Luk’yanchuk, Sewastopol National Technical University, Ukraine; A. Patashinski, Northwestern University, Evanston IL, USA; F. Pichler, Johannes-Kepler-Universität, Linz, Austria; V. Sugak, Academy of Sciences of Ukraine, Kharkiv; R. Yelf, Georadar Research Pty Ltd, Coffs Harbor NSW, Australia; and Zhang Qishan, Beihang University, Beijing, PR China. Henning F. Harmuth and Beate Meffert *

*

*

This is the sixteenth and last book that I wrote either as single author or – after age 65 – with coauthors. The large number was made possible by the support of four scientific editors: the late Ladislaus L. Marton (Academic Press), the late Richard B. Schulz (IEEE Transactions on Electromagnetic Compatibility), Myron W. Evans (retired from World Scientific Publishers), and Peter W. Hawkes (Elsevier/Academic Press). Scientific editors are as important as authors for the publication of new ideas. Science not published is no better than science not done. Henning F. Harmuth

LIST OF FREQUENTLY USED SYMBOLS Ae Ae Am Am0, Am1 Ãe Ãm1[ B c D E, E E e F ([) g, g[ , gR ge gm g H, H HcL(R) H, H h = = h/2Q I Ir1, I+1 , IK1 Ir , I + , IK IT (L/N) Jr , Ji JR , JI J1 –J8 L, L

As/m As/m Vs/m Vs/m – – Vs/m2 m/s As/m2 V/m VAs As – – A/m2 V/m2 – A/m – – Js Js – – – – – – – –

L, L

–

m0 m m N p1

kg kg – – –

absolute value of the electric vector potential electric vector potential magnetic vector potential Eq.(2.2-29) Eq.(3.2-6) Eq.(3.2-10) magnetic flux density 299 792 458; velocity of light (definition) electric flux density electric field strength Euler Roman Medium E, energy electric charge Eqs.(2.3-13), (2.3-24) Eq.(3.4-26) electric current density magnetic current density Euler Script Medium G, Eq.(3.4-26) magnetic field strength Eq.(6.3-29) Euler Script Medium or Bold H; Eqs.(2.1-1), (1.1-32) 6.626 075 5 × 10 –34, Planck’s constant 1.054 572 7× 10 –34 Eqs.(5.1-14), (5.1-19) Eqs.(5.1-16), (5.1-18) Eq.(5.1-20) Eq.(2.4-33) Eq.(3.3-13) Eq.(3.3-17) Eq.(6.3-75) Euler Roman Medium or Bold L; Eqs.(2.2-15), (2.2-28) Euler Script Medium or Bold L; Eqs.(1.1-23), (1.1-24) rest mass variable mass Euler Roman Medium m; Eq.(6.7-9) T/%t, Eq.(2.2-6) Eq.(3.2-7) ( Continued ) xvii

xviii

List of frequently used symbols

pC Q %r s S T t %t U UcL(L) v Z = N/c Z

– – m V/Am – s s s VAs – m/s V/A –

Eq.(3.2-6) Eq.(1.1-45) arbitrarily small but finite space interval magnetic conductivity Euler Script Medium S; Eq.(6.6-15) arbitrarily large but finite time interval time variable arbitrarily small but finite time interval Eq.(2.5-3) Eq.(2.6-1) velocity 376.730 314; wave impendance of empty space Euler Roman Medium Z; 1, 2, …; charge number

B Be B C CL, C^L Hr, H+, HK = 1/Zc % % [ R L > –L0 < L0 M1, M2, M3 MC Mc Mr, M+, MK -2 N = Z/c Se Sm T Ge Ge0 , Ge1 Gm Ge1 8 Xm Xp

– – – – – – As/Vm – – – – – – – – m m – – Vs/Am As/m3 Vs/m3 A/Vm V V A – – – –

Ze2/2h  7.297 535×10 – 3, Eq.(1.1-45) ZecA e/m 0c 2, Eq.(1.1-45) Bx, By, Bz, matrices, Eqs.(2.1-8)–(2.1-10) matrix, Eq.(2.1-9) Eqs.(2.4-13), (2.4-38) Eqs.(5.1-24), (5.1-34), (5.1-43) 1/Nc 2; permittivity symbol for difference quotient: %F/% x, Eq.(1.3-5) symbol for finite difference: x + %x xj /c%t, normalized distance; Eqs.(2.2-6), (2.3-1) t/%t, normalized time; Eq.(2.2-6) wave number, Eq.(2.3-52) smallest integer larger than –L0 largest integer smaller than L0 Eq.(2.3-2) h/m0 c, Compton wavelength =/m 0c = M C/2Q Eq.(4.3-22) Eq.(2.4-11) 4Q × 10 – 7; permeability electric charge density magnetic charge density electric conductivity, Eq.(1.1-7) electric scalar potential Eq.(2.2-29) magnetic scalar potential Eq.(3.2-9) Eq.(5.2-11) Eq.(6.8-5) Eq.(6.8-6)



1 Introduction 1 . 1 MAXWELL EQUATIONS WITH MAGNETIC DIPOLECUR.R.ENTS Maxwell's equations always permitted electric dipole currents caused by ind11c:ed or rotating electric dipoles. Most molecules form rotating electric dipoles. T h e large dielectric constant of barium-titanate is due to rotating dipoles. Without electric dipoles no electric current could flow through capacitors since their dielectric is an insulator for electric monopole currents. T h e existence of electric dipole currents is obscured in Maxwell's equations by having onc clcctric current density term that stands for monopole, dipole, or higher order multipole currents. Dipole currents were not understood in Maxwell's days. For instance, we read much about Ohm's law for monopole currents but never about Ohm's law or some equivalent for dipole c~irrents.This explains why the similarity between electric: dipole currents due to rotating electric dipoles and magnetic dipole currents due t o rotating magnetic dipoles was riot recognized. The lack of a magnetic (dipole) current, density in Maxwell's equations was never fully accepted. Dirac (1931) attempted to add magnetic monopole currents to Maxwell's equations for very convincing theoretical reasons. But the lack of reliably observed magnetic rrlonopoles or charges proved to be an insurrnouritable obstacle. A new stimulus for the irivestigation of a magnetic current density term was provided by the development of electromagnetic communication with deeply submerged submarines (Merril 1974). T h e original Maxwell equations led to communicatiori systems that permitted a transmission rate of information of about 5 bit/s (=one teletype letter per second). The inability of Maxwell's equations to describe the propagatiori of signals through seawater with its high olirriic losses was recognized. The addition of a magnetic current density term overcame the problem and showed that the transmission rate of information could be increased significantly by large-relative-bandwidth techniques1 (Harmuth 1986a, b, c, d; Harniuth, Boules, Hussain 1999). It was known that one obtained different differential equations as well as different solutions if one assumed no niagnetic current density a t the 'The term "ultra wide bandwidth" is frequently used for large relative bandwidth, which shows the principle is still not generally understood. The bandwidth for submarine communication is in the order of 50 Hz, which is hardly ultra wide.

1

1 INTRODUCTION

FIG.1.1-1. Monopole currents carried by independent positive and negative electric or magnetic charges (a). Dipole current due to an induced dipole (b). Dipole current due to orientation polarization of inherent dipoles (c); the change of the signs of the charges makes the currents ih and i, flow in the directions shown.

beginning of the calculation, or made an initially assumed current density zero a t the erld of the calculatio11. This was always taken as a warning sign that something was not understood. But it required several years before it was recognized that this strange result was caused by not distinguishing between monopole, dipole, and higher order multipole currents (Anastasovski et al. 2001). At about the same time a mathematician i ill ion^ 1991, 1992a, b; 1993) realized that Maxwell's equations belong to a class of differential equations that do riot permit independent initial and boundary conditions. If initial and boundary conditions cannot be chosen independently one cannot get solutions that satisfy the causality law, since a boundary condition a t the time t > 0 could affect an initial condition a t the time t = 0. The addition of a magnetic current density term to Maxwell's equations, either for monopole or dipole currents, yields equations that generally can satisfy the causality law and thus yield signal solutions. Signal solutions are zero before a certain finite time and have finite energy. Matllenlaticians call such solutions quadratically integrable causal functions. For examples of rrionopole, induced dipole, and inherent dipole currents refer to Fig.l.1-1. In Fig.l.1-la we see independent negative arid positive charges that are pulled up or down by a field strength denoted by and -. This produces monopole currents. Only electric charges and field strengths are accepted to have been reliably observed to produce this effect. Figure 1.1-lb shows an induced dipole that nlay represent a hydrogen atom. An electric field strength polarizes the atorn by pulling the electron up and the proton down. A restoring force is depicted by a coil spring. A dipole

+

2 ~ i l l i o nobtained his results earlier than implied by the dates of his publications. However, he could not overcome the peer review which implied the causality law could not be important in electrodynamics if Maxwell's equations had worked without it for a

century.

3

1.1 MAXWELL EQUATIONS WITH MAGNETIC DIPOLE CURR.ENTS

current flows as long as electron and proton move either apart or back together again. The resulting induced dipole current comes to an end if the applied field strength is below the ionizing field strength; otherwise the dipole currcnt bccomes a monopole current. Only elcctric charges and field strengths are currently known to produce induced dipole currents. Figure 1.1-lc shows two inherent dipoles represented by charges and - at the cnds of a rigid rod. Most molecules arc inhcrcnt clcctric dipoles. All known magnetic dipoles are inherent dipoles. An applied field strength makes the dipoles rotate to produce orientation polarization. Vertical currents i,, and horizontal currents ih are produced by the rotation. The horizontal currents ih will cancel in a random mixture of inherent dipoles but the vertical currents will add. If the field strength is suddenly reduced to zero the dipole current will not be reversed as in the case of the induced dipole of Fig.l.1-lb but stop. This creates an observable difference between induced and inherent dipoles. We note that molecules that are inherent electric dipoles are also inducible dipoles since the electrons can still be pulled in one direction and the nuclei in the other. Very strong field strengths will not turn a current due to orientation polarization into a monopole current. This is a second observable difference between the two types of dipoles. T he addition of a magnetic dipole current density term to Maxwell's equations removed the problem with the causality law3. The modified equations can be written in the following form with international units in a coordinate system at rest:

+

aD dt dB

+

(1)

-+ g m

(2)

c u r l H = - g, -curlE=

at

(3)

div D = p, div B = 0

or div B = p, for magnetic monopoles

(4)

An old fashioned notation is used here but it is the notation with which the problem of Maxwell's equations with the causality law was found. It is quite possible that the operators grad, curl, and div are better for physical understanding than V and 0.In Eqs.(l)--(4) E and H stand for thc clcctric and magnetic field strength, D and B for the electric and magnetic flux density, g, and gm for the electric and magnetic current density, p, and p, for the electric and a hypothetical magnetic charge density. Thc magnetic current density g, does not depend on the existence of a charge density p,. Equations ( 1 ) (4) are augmented by constitutive equations. In the simplest case we have 3See Harmuth, Barrett, Meffert 2001, Sec.l.1 for details and many references.

1 INTRODUCTION

D = cE, B = pH,

[ ~ s / m= ~ [As/Vm] ] [V/m] [vs/m2] = [Vs/Am][A/m]

(5)

g, = u E ,

[ ~ / r n= ~ [A/Vm] ] [V/rn]

(7)

(6)

g, = sH, [ v / m 2 ]= [V/Am][A/m] (8) where E, p , u , and s are scalar constants called permittivity, permeability, electric conductivity, and magnetic conductivity. We note that a and s may be monopole, dipole, or higher order niultipole conductivities. An electric monopole current cannot flow through a capacitor, which makes the monopole conductivity zero, but a dipole current can flow since the dipole conductivity is not zero. This difference becomes very evident for alternating currents. If E , p , 0 ,s vary with location, time, and direction one must replace the scalar variables in Eqs.(5)-(8) by time-variable tensors. In more general cases the equations may be replaced by partial differential equations. If one is satisfied with a periodic sinusoidal time variation of El H, D, B, g,, and g, one may use functions of frequency ~ ( w )p(w), , u(w), and s(w) but this produces a theory outside the conservation law of energy and the causality law. We list a number of relations derived from the modified Maxwell equations (1)--(4). Their derivation was publislied in a book3. The electric and magnetic field strengths are related to vector potentials A, and A~ as well as scalar potentials 4, and 4,, where 4, is zero if there are no magnetic monopoles:

For A, = 0 arid 4, = 0 one obtains the equations derivable from Maxwsell's original equations. However, these equations contain a contradiction that calls for A, # 0, while 4, = 0 is acceptable (Harmuth, Barrett, Meffert 2001, Sec. 3.1). T h e vector potentials A, and A, are not completely specified since Eqs.(9) and (10) only define curl A, and curl A,. Two additional conditions can be chosen that we call the extended Lorentz convention:

1.1 MAXWELL

EQUATIONS WITH MAGNETIC DIPOLE CURRENTS

5

The potentials of Eqs.(9) and (10) then satisfy the following inhomogeneous differential equations:

024,

C 1 d24, - -,2 at2 - 04, = -zP,,

4,

= O for p,

=O

(16)

Particular solutiorls of these partial differential equations may be represented by integrals taken over the whole space:

4,

= O for p,

=0

(20)

Here r is the distance between the coordinates E , 7 , C of the current and charge densities and the coordinates x, y, z of the potentials: r = [(x -

E ) ~+ ( ~ - l ) )+~(Z - s ) ~ ]112

(21)

We note that only Eqs.(l)-(4) are needed t o derive Eqs.(9)-(21), the constitutive equations (5)-(8) are not used. T h e modification of Maxwell's equations implies a modification of the Lagrange function and the Haniilton function. The general relativistic form of these rnodified functions is quitc complicatcd. To help with undcrstanding we start with the non-relativistic equations. The Lorentz equation of motion for a mass m , a charge e , and a velocity v

yields from the original Maxwell equations for v LM (Euler Script Medium L):

<< c the Lagrange function

1 INTRODUCTION

The modified Maxwell equations add a correcting term LCin the form of a matrix LC(Euler Script Bold L):

This matrix forces us to rewrite LM as a matrix too:

+

The sum LM LCcan be written with Eqs.(24) and (25) in matrix form. We prefer to write it in a rriore conipact forrrl with the help of unit vectors e x , e,, e z :

The term LM is the same as in Eq.(23) while the term LC, is defined as follows (Harmuth, Barrett, Meffert 2001, Sec. 3.2):

$m

= 0 for p, = 0

(27)

The terms LC, and LC, are obtained from LC, by the cyclic replacements x -, y -, z + x and -t z -t y -t x. Since LC,, LC,, and LC, are derived from the relativistically invariant modified Maxwell equations, they are relativistically invariant. The derivatives 8,y, i in Eq.(27) are replaced by the components p,, p,, p, of the moment p:

1.1 MAXWELL EQUATIONS WITH MAGNETIC DIPOLE CURR.ENTS

7

Rewriting Eq.(27) in terms of p rather tlian x, y, i is a rnajor effort. We refer to the summary in Section 1.2 and to the literature (Harmuth, Barrett, Meffert 2001, Sec. 3.2). A Hamilton function X (Euler Script Bold H) can be derived from the Lagrange function C of Eq.(26). For me2 >> ZecA, and ZA, >> A, we obtain:

+

Maxwell's original equations yield the term (1/2m)(p - eAm)2 e4,. The terms LC,, LC,, LC, are correcting terms. If the simplifying assumption leading to Eqs.(33)-(35) are not made one obtains the exact non-relativistic Harnilton function (Harmuth, Barrett, Meffert 2001, Eq. 3.2-44):

Terms milltiplied by ( Z e ~ A , / r n c ~or) ~(Ze~A,/rn,c')~have been added to the sirnplified terms of Eqs.(33)-(35). The correcting terms LC,, LC,, LC, have riot been changed.

8

1 INTRODUCTION

It all becomes much more complicated when the restriction v << c is dropped. T h e relativistic generalization of the Lagrange function of Eq.(26) with rest mass m,o is still simple. It may be found in many text books without t,he correcting term L C :

The correction term LC, of the Lagrange function in Eq.(27) does not contain the mass m . Hence, LCand its components LC,, LC,, LC,, are not affected by the variability of m,. The relativistic Hamilton function for Eq.(37) can be written with the hclp of series expansions only (Harmuth, Barrett, Meffert 2001, Sec. 3.3). For moc2/(1 - 7,2/~2)1/2>> ZecA, and ZA, >> A, we get:

Without the correcting terms LC,, LC,, LC, we have the convetltional relativistic Hamilton function for a charged particle in an electromagnetic field, written with three components rather than one. We call them the zero order approximation in a, = a , ( r , t ) = ZecAe(r,t)/moc2, where a, is a dimension-free normalization of the magnitude Ae(r, t ) . A first order approximation in a, is provided by the following equations:

The factor Q stands for the expression

and a, rnay be written in the following forms:

1 . 2 LAGRANGE

Ze2 2h,

N =-

FUNCTION

-

3 h 7.297 535 x 10- fine structure constant, Xc = mo c

2.210 x 1 0 5 ~ , ( rt), for electron, A, in As/m a, 1 1.204 x 102Ae(r,t) for proton a,

(45)

We shall use the following simplc and uniquc rulc for thc rcplaccmcnt of non-cornmuting factors ab: 1 ab + -(ab 2

+ ba)

It was not necessary yet to distinguish between non-commuting and commuting factors but our equations show that this problem will have t o be addressed eventually.

I11 order to use the Hanlilton formalism of Eqs.(l.l-41)-(1.1-43) we have to rewrite LC,, LC,, LC, according to Eq.(l.l-27) into functions of the moment p rather than the time derivatives x, ?j, i of the coordinates. This requires much effort and leads to very long equations. We refer to the literature (Harmuth, Barrett, Meffert 2001, Sec. 3.2). The equations are simplified by expanding them in powers of a and use only the first order approxiniation as required for Eqs. (1.1-41)---(1.1-43).Even this simplification requires that LC, of Eq.(l.l-27) is broken into five components to produce manageable equations:

Lcx

= Lcz1

+ Lc:r2 + Lcz3 + Lcz4 + LC25

(1)

From Eq.(l.l-45) we derive the relations

that we need to show that the components of LC, vary with a in first approximation, as one would expect from Eqs.(l.l-41)-(1.1-43). We write them in the following form (Harmuth, Meffert 2005, Eqs. 2.1-20 t o 2.1-24):

1 INTRODUCTION

The last three lines of this equatiorl make use of Eq.(l.l-46) to prepare for non-commuting factors. The second term LcX2in Eq.(l.l-27) becomes:

The third term LczSin Eq.(1.1-27) becomes:

1.2

LAGRANGE FUNCTION

11

The fourth corriporierit LcrL.4 i11 Eq(l.1-27) remains uncliariged since there are 110 time derivatives of r , y, or z:

The fifth corriponent Lc,s in Eq.(l.l-27) is rather long:

We shall need LCriot only in Cartesian coordinates but also in spherical coordinates. This presentation is found in the literature (Harmuth, Meffert 2005, Secs. 1.3, 6.5). Wc havc now thc matrix

1 INTRODUCTION

c

0

0

with the following elements:

/'

Ze . LC, = - r ( r d ~ , ~- r s i n 6 +Aes)

+Ze c

Ze LC. = -rd(r

+-

c

C

sin 6 +A,, - +A,,)

+ r sin 19 @A,,

- FA,,

84,

[sin il+=

+ (i-r: + +-):

-

d 84, G z

( r sin 6 +A,, - +A,,)

The term L C ,of Eq.(9) is broken into five components, just as LC, was, to obtain manageable equations:

+

Lcr = kcrl kc1.2 + Lc1.3 + Lcr4 + Lcr5 (12) To write these components in terms of the momentum p rather than the time derivatives i, r d , r sin t9+ of the coordinates we need three equations that hold in first order approximati011 of either a, or a (Harmuth, Meffert 2005, Eqs. 5.2-43 to 5.2-45):

1.2 LAGR.ANGE FUNCTION

13

With the help of these equatiorls we obtain the first terms of Eqs.(9)-(11) using the rrlonlerlt~imp:

&I

Ze .

= -r19(r c

sin 6 @A,, - PA,,)

The secor~dterrrls of Eqs.(9) -(ll)becorrie in this riotatiorl in first order of a:

Ze

r

C

- eAm),

r sin 19 x (1

+ (P

;GF)~) -Ii2?

d19 (20)

14

1 INTRODUCTION

The third terms in Eqs.(l.l-33)-(1.1-35) require that one expresses F, 6, (ij from Eqs.(13)-(15): Ze

LC,, = -/(TBA., C

- T sin 19 (ijAer)dr

X

[l

Ze LCs3 = - / ( r sin 6 +Ae, c .

COV3 =

-

+

(P - eAm)v dr 2 2 112 ( p - eAm)2/m.oc 1

iAe,)r dt9

2J ( Y A . ~ - r I ? ~ , , sin ~ )6 r dp C

The fourth set of terrns in Eqs.(9)-(11) does not require Y, easier t o calculate:

C

.

(22)

/

Ac m,oe .

= 2a-

I?, + and is thus

(TBA~,- r sin 6 @A.o)dr 1d + -+ (P r d6

((p - e A m ) ~

-

e~rn),

1 ap

1.3

BASIC CONCEPTS O F T H E CALCULUS O F FINITE DIFFER.ENCES

15

(?Aeu - r 8 ~ . , ~ sin ) r 6 dp

The fifth and last set of terms in Eqs.(9)--(ll) remains unchanged since there are no tirne derivatives of r , 6, p:

- -~ C I X ~ ~ O C ~

e

1.3 BASICCONCEPTS OF

.

r sin 6

THE

8 A e 0 ) ~ ~ (28) ~ C P

CALCULUS O F FINITE DIFFER.ENCES

There is a variety of ways to derive difference operators from differential operators. In mathematical books (Milne-Thomson 1951, Ch. 11; Norlund 1924, 5 1) one typically finds the substitutions

which we refer to as the "right" difference quotient. The corresponding "left" difference quotient has the form

1 INTRODUCTION

0

1

N-2

2

1

N-1

N

X

FrG.1.3-1.The symmetric difference quotient of first order needs to be supplemented at the limits of an interval by the right and the left difference quotient.

while a symmetric difference quotient is defined by

Since it is hard to work out three parallel physical theories based or1 these three difference quotients and to see at the end which is the best, we must decide from the beginning which one to use. The right and the left difference quotient were discarded since they led in simple cases to ever increasing or decreasing solutions while the symmetric difference quotient yielded stationary solutions (Harmuth 1989, Sec. 12.4). However, the right and the left difference quotient are sometimes required to supplement the symmetric difference quotient at the limits of an interval as shown by Fig.1.3-1. The symmetric difference quotient exists only for the points z = 1, 2, . . . , N - 1. The right difference quotient remedies the problem for the left bol~ridaryz = 0, the left difference quotient for the right boundary z = N. The typical case where this becomes important is in connection with difference equations of second order that require two initial conditions. We shall discuss this at the end of Section 2.3. The symbol 2 is used for difference operators while A is used for a finite difference, e.g., At?. The notation can be simplified if one uses the substitution

arid t,herl drops the prime. Equation (3) assumes the following form:

The higher order difference operators are riot obtained as in differential calculus by repeated application of the first order operator, but we maintain the formal notation as shown in the followirig Eqs.(6) and (7). The three choices of Eqs.(l)-(3) occur for all difference operators of odd order. We list here a few of the higher order operators but refer to the literature for their derivation (Harmuth, Meffert 2005, Secs. 1.2, 6.1):

1 . 3 BASIC

CONCEPTS O F THE CALCULUS O F FINITE DIFFER.ENCES

17

An important rule for the differenciation of a product u(Q)v(0)is very similar to the rule for its differentiation: ~,(e),,(e)

de

=

dv(e)

~(e)-A0 + v(e)-

du,(e) A0

+O(AO)~

Differential calculus permits us to define the inverse operation of differentiation by means of the differential equation

arid its formal solution

A corresponding process leads in the calculus of finite differences from the difference quotient of first order to summation rather than integration. In order to use the results of Norlund (1924, Ch. 3) and Milne-Thomson (1951, Ch. VIII) we follow closely their derivation. This requires to use the notation

For the connection of Eq.(13) with our notation in Eq.(3) we choose first w = 2Az

1

INTRODUCTION

and make then the substitutiori x = z'

+

- U.(Z/

-

AX)

2Ax

-

Az:

= p(xl - A x ) ,

x' = x

+Ax

(15)

Consider the function f ( x )

arid the shifted furlctiori f ( x

+ w)

A formal solution F ( z l w ) of Eq.(13) is obtained by substituting f ( z arid f ( z ) for u.(x w ) and u ( x ) in Eq.(13):

+

F ( r l w ) = Co - w 'y p ( x s =o

+ sw) =

/

p(u)dv - w s=o

0

p(x

+ sw)

+ w)

(19)

This is the Hauptlosung or principal solution of Eq.(13). It is also called the sum of the function cp(x). The integral over p ( v ) in Eq.(19) is written instead of the constant Co because a divergency of this integral may compensate a divergency of the sum over cp(x sw); a constant Co could not do that. Norlrind introduced t,he following notation for this summation:

+

The function F(xlw) is said to be obtained by summing p ( u ) from c to x , in analogy to integrating cp(xl) by the operation SCxp(x1)dx'. Norlund

1.3 BASIC CONCEPTS O F THE CALCULUS OF FINITE DIFFERENCES

19

generalized F(xlw) beyond Eq.(20) t,o functions p(z) that can be made summable by multiplication with an exporlential function e-pX("):

+

Note that X(x SW)means a function X of x + sw (Norlund 1924, § 2; MilneThomsorl 1951, Ch. VIII). The more general Eq.(21) is needed to obtain the sum of the constant a in Table 1.3-1. We write F(x1w) for the symmetric difference quotient on the left side of Eq.(15), substituting first w = 2A2, x = x' - Ax and then w = 2, x = x' - 1: for w = 2Ax, x = x' - Ax

Au = Au for w = 2Ax

(22)

For the choice w = 2, x = x' - 1 we get the following result that will be used from here on unless Eq.(22) is specifically pointed out: for w = 2, x = x'- 1

Since we shall generally use the values w = 2Ax or w = 2 we derive the summatiorl of Eq.(22) directly from Eq.(15) by the substitutions

20

1 INTRODUCTION

TABLE 1.3-1 SUMS v(xl) = v(x) OF CERTAIN FUNCTIONS cp(xl) = cp(x) ACCORDING TO Eg.(23) FORw = 2. THEINTEGRALS OF cp(x) ARE SHOWN FORCOMPARISON.

eYx

eY(c+l)

eYx

shy 7 y complex, lyl


sin yx

-sin y

+

cos yx

sin yx -+ sin y

sin[y(c Y

eXxsin yx

Y

+ 1)) 171 < " 3

exx (Aosin yx - yo cosyx)

+ Co

X2 + y 2

cos yx +C Y sin yx +C Y

+-

exx (A sin yx-y cos yx) + C y2

X2

+

+

XO = - 2(X2 y2)shXcosy cos 27 - ch 2X

+

2(X2 y2)chXsiny cos 27 - ch 2X ex(^+') C o = -{X sin[y(c I)] X2 y2 -7 cos[r(c 1)1>

YO =

+

+

ex" cosyx

+

ex" (Xo cos yx X2 y2

+

+ yo sin yx) + C1

Xo and yo are shown above

c1=-- ex(c+l) X2

+ y2

cos[y(c + 1)1 -rsin[y(c

+ 1)l)

exx -

X2

+ y2 (Xcosyx+ysinyx)+C

1.3 BASIC

21

CONCEPTS OF THE CALCULUS O F FINITE DIFFERENCES

f (xl+A x ) = CO-2Ax[p(x'+Ax)+p(x1+Ax+2Ax)+p(x'+Ax+4Ax)

x w

= Co - 2 A x

p(xl

+ A x + 2sAx)

+. . . ] (25)

for u.(xl f A x ) in Eq.(15):

1 A x ) - f (x' - A x ) ] = p(xl - A x ) 2Ax Hence, we may define the principal solution in analogy to Eq.(19):

-[f(x'

+

(26

This is again Eq.(22) for w = 2 A x . As an example for summation consider the exponential function p ( x ) = e-" substituted into Eq.(13); both w and x shall be positive. We obtain from Eq. (20):

~ ( x=) F ( x I w ) =

e-"dv - w

x 03

e-("+"")

s=o

we-" - e-C - 1 - e-W

(30)

In order t o see what the symmetric difference quotient and the choice w = 2 do to Eq.(30) consider the following difference equation:

This equation corresponds to Eq.(16). For v ( z 1 )= e-'u,(xl) we obtain:

1 INTRODUCTION

This is again Eq.(30) for w = 2. The integral S exp(-z')dxl yields C - exp(-2'). Equation (32) diffcrs by a factor 2/(e - e-l) 0.85092 from this result. Table 1.3-1 shows a collection of sums and the corresponding integrals. All examples are due to Norlund, exept the last two. A comparison of this table with available tables of integrals, e.g., Gradshteyn and Ryzhik (1980), shows how little the calculus of finite differences is developed compared with the differential calculus. Summation and differenciation are inverse operation in the sense that integration and differentiation are inverse operation (Norlund 1924, Ch. 3, 3 3/22; Milne-Thomson 1951, Sec. 8.1):

C

C

We adopt the following convention

for the symmetric first order difference quotient of Eq.(3) and the second order difference quotient of Eq. (6). 1.4 FINITE DIFFER.ENCES AND SPATIALDIMENSIONS We begin with a discussion of the physical paper plane and mathematical coordinate systems that can be constructed on it. Figure 1.4-la shows a Cartesian coordinate system with markers having the finite distances Ax and Ay. We call this a two-dimensional coordinate system since it requires the two variables X and Y. In Fig.l.4-lb we see the very same markers, but only one variable X is needcd to identify them. This should be called an one-dimensional coordinate system.

1.4 FINITE DIFFER.ENCES AND SPATIAL DIMENSIONS a

b

FIG.^.^-1. Two-dimensional discrete Cartesian coordinate system with coordinate markers at finite distances Ax, Ay (a) and an one-dimensional system connecting the same markers with only one variable X (b); this illustration is commonly used to show that n,2 is denumerable if n is denumerable.

The coordinate points or niarkers are the irriportant part of Figs.l.4-la and b. The lines connecting the points only help assigning a n address X , Y or X to the points. For instance, the distance "as the crow flies" between the points X = 1, Y = 4 arid X = 5, Y = 1 i11 Fig.1.4-la equals (42+32)1/2 = 5. This is the sarrie distance as between the points X = 19 arid X = 26 in Fig. 1.4-lb. The Pythagorean theorem works better for Fig. 1.4-la, but Fig.1.4-lb requires only one variable for what is usually considered a twodimensional set of points. Since there is an one-to-one relationship between the coordinate points of Figs.l.4-la and b we can always use the one that is rnore convenient for a particular purpose and then translate the result for the other coordinate system. Figure 1.4-lb can connect denumerable many marker points X = 0, 1, 2, . . . , but not non-denumerable many. On the other hand, Fig.l.4-la perniits non-denumerable many points for both the variables n: and y; the finite distances Ax, Ay become infinitesi~rlaldistances dx, dy. Mathematicians consider non-denumerable and infinitesimal a generalization of denumerable arid finite. In physics it is a n abstraction since neither non-denumerable nor irifiriitesimal can be observed. Such abstractions are not necessarily useless but they niust be justified. The justification is usually a simplification of calculation, which is the primary reason for using differential calculus. Wc must be careful if we use such abstractions to help with physical thinking. Several examples will be given to show that this can lead to totally misleading results. As a n example how mathematical abstractions can be used to derive physically meaningful results consider once nlore Fig.1.4-lb. We have said

1 INTRODUCTION 26

FIG.^.^-2. Extension of the one-dimensional coordinate system of Fig.l.4-lb from one quadrant of the paper plane to all four quadrants.

that it can represent denumerable markers X = 0, 1, 2, . . . . Denumerable is beyond observation but it can be used as approximation for "arbitrarily many but a finite number of markers". The two coordinate systems of Fig.1.4-1 cover one quadrant of the paper plane, meaning that the location of something in that quarter can be defined. To extend the coverage to all four quadrants we may introduce negative values X, Y = -1, -2, . . . in Fig.l.4-la. The extension of Fig.l.4-lb t o all four quadrants is shown in Fig.1.4-2. Again, negative values .X= -1, -2, . . . are used but there are other modifications of Fig.l.4-lb that are easier to see than t o explain with words. Figures 1.4-3 and 1.4-4 show how the process of Fig.1.4-1 for the reduction of two-dimensional coordinate systems to one-dimensional systems works for polar coordinates. We need a further generalization of the coordinate systems of Fig.l.4-1. They may be called rectangular or non-slanted systems. Figure 1.4-5 shows their extension to slanted coordinate systems. It is important t o make this extension before we advance to more than two dimensions since the slant angle a measured in the paper plane has a clear meaning. Once we go beyond two dimensions we must think of coordinate systems in physical space constructed with thin rods instead of the ink lines in Fig.1.4-5, and the little rings for the coordinate marks replaced by small spheres made of, e.g., styrofoam (Harmuth 1989, Sec. 3.1). The concept of a mathematical line is

1.4 FINITE

DIFFER.ENCES AND SPATIAL DIMENSIONS

FIG.1.4-3. Extension of the two-dimensional Cartesian coordinate system shown in Fig.l.4-la to polar coordinates.

53

FIG.^.^-4. Extension of the one-dimensional pseudo-Cartesian coordinate system of Fig.l.4-lb to one-dimensional polar coordinates.

understandable when they are implemented by ink lines or thin metal rods, even though it requires non-denumerable many infinitesimal distances dx. However, without paper plane we loose such concepts as surface, Euclidean plane, conical surface, etc. These are elements of Euclidean geometry that require infinitesimal distances and non-denumerable many of them. In return one becomes more able to distinguish between ph,ysical necessit,~and mathematical invention.

XX-, F1c.1.4-5. The non-slanted coordinate systems of Fig.l.4-1 replaced by slanted systems with a slant angle cr # 7r/2 measured in the paper plane.

002

F1G.1.4-6.Three-dimensional discrete Cartesian coordinate system with coordinate markers at the distances Ax, Ay, Az.

We extend our investigation to the three-dimensional coordinate system of Fig.1.4-6. The important parts of this illustration are the 33 coordinate markers shown as small spheres. The rods connecting them are only sup-

1.4 FINITE

DIFFER.ENCES AND SPATIAL DIMENSIONS

27

F1c.1.4-7. Replacement of a three-dimensional, discrete Cartesian coordinate system with denumerable many points with an one-dimensional coordinate system. The illustration shows that n3 is denumerable if n is denumerable. There is a variety of ways to draw the rods connecting the coordinate points. posed to help seeing a three-dimensional structure. They do not represent mathematical lines that would require non-denumerable many points or infinitesimal distances dz, dy, dz. The use of rods is an improvement over the lines in Figs.l.4-1 to 1.4-5, which are only intended to make the illustrations more understandablc. Thc rcplaccmcnt of thcsc ink lincs by rods would make it more difficult to produce t,hese drawings. The three-dimensional coordinate system with variables X ,Y, and Z in Fig.1.4-6 can be replaced by the one-dimensional system of Fig.1.4-7. We need only one variable X or one dimension to connect a finite or denumerable number of discrete points or coordinate markers in the usual three dimensions. A short reflection shows that this is due to n3 being denumer-

1 INTRODUCTION

FlG.1.4-8.Two-dimensional coordinate system with variables X and Y for the replacement of the three-dimensional system with X , Y, and Z in Fig.1.4-6.

able if n is denumerable. Figure 1.4-7 would not exist for non-denumerable coordirlate points. Figure 1.4-8 shows that the three-dimensional coordinate system of Fig.1.4-6 can also be replaced by a two-dimensional system with variables X and Y. Advancing to four dimensioris, the choice n = 3 and the n3 = 27 coordinate markers in Fig.1.4-6 become n = 4 and n4 = 81, which produces an impressive illustration (Harmuth 1989, Fig.3.1-25). We make do with n = 2 and n4 = 16. This yields the four-dirncnsional cube with variables X I Y, 2, and W shown in Fig.1.4-9. Understanding this axonometric representation of a four-dinlensional structure is more difficult than understanding the three-dimensional structure of Fig.1.4-6. Let us first recognize the threedimensional clihe with 8 corner points or marks 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111. This may take some time. Then we try to see the equal cube shifted by Aw and having the 8 corner points or marks 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. Congratulations, you have visual-

1.4 FINITE DIFFERENCES AND SPATIAL DIMENSIONS

0

l o l l = 1Aw, OAz, 1Ay,lAz

F1c.1.4-9. Extensions of the three-dimensional discrete coordinate system of Fig.1.46 to four dimensions with 24 marks. One needs 24 styrofoam spheres and 12 x 2 + 8 = 32 equally long rods to implement a model in our physical space that is universally believed to have three dimensions. ized a four-dimensional cube from a single two-dimensional projection. The standard in descriptive geometry is two two-dimensional projections for a three-dimensiorial structure. Only the extreme simplicity of the structure of Fig.1.4-9 permits us to get away with just one projection. The reader may try to rnake this illustration more evident by painting the spheres Ojkl red and the spheres l j k l green. Another short reflection makes us recognize that the four-dimensional cube of Fig.1.4-9 can be implemented with 24 = 16 styrofoam spheres and 12 x 2 8 = 32 rods of equal length in our physical space1. One could not draw this illustration for infinitesimal differences dx instead of finite differences Ax. The universal belief that our physical space has three dimensions is based on the assumption of infinitesimal distances dx, dy, dz and 11011-denumerable many such distances. Both assumptions are totally

+

h he construction of simple structures of up to 6 dimensions using styrofoam spheres and wooden rods is actually easier than drafting projections of them in the paper plane. A number of photographs have been published (Harmuth 1989). The problem with photographs is that the crossing of rods, showing which is in front and which is in rear, is not anywhere as clearly visible as in the drawings of Figs.1.4-6 to 1.4-9.

30

1 INTRODUCTION

beyond observation and belong to pure mathematics2. Figure 1.4-10 shows the extension of Fig.1.4-7 from three to four dimensions. The lucidity of Fig.1.4-7 is lost since we are not used to think in four dimensions and the representation of four-dimensional structures on a paper plane. For an explanation let us start at the point in the lower left corner denoted 0,0000. The first number 0, 1, 2, . . . , 33 refers to the sequence in which the coordinate markers in Fig.l.4-10 are connected by rods. This is the one-dimensional description. The second number i j k l from 0000 to 0013 refers to the four-dimensional numbering of Fig.1.4-9. Going from 0 to 1, 2, 3, 4, and 5 in Fig.l.4-10 corresponds to going from 0000 to 0001, 0010, 0100, 1000, and 1001 in Fig.1.4-9. The two three-dimensional cubes of Fig.1.4-9 are shown by dashed lines. The eight rods connecting the two three-dimensional cubes are not shown in order to reduce the number of lines. The next step brings us to point 6 in Fig.l.4-10, but the point 0002 at the distance 2Ax from 0000 is not shown in Fig.1.4-9. One could produce Fig.l.4-10 by drafting Fig.l.4-9 by a factor 43 times as large from 0 to 3Ax, 3Ay, 3Az, and 3Aw, but this is not likely to succeed in an acceptable time. An easier, but not easy, way is to draw first Fig.1.4-6 for X, Y,Z = 0, 1, 2, 3 and make a copy on transparent paper or plastic. Then we draw in the original an axis in the direction of Aw in Fig.1.4-9 and extend it to 2Aw, 3Aw. If we now place the transparent copy with the point 0000 successively on the points Aw or 1000, 2Aw or 2000, and 3Aw or 3000 of the modified original we can first plot on a second overlaid transparent sheet the points 0 to 5 in Fig.1.4-10. Let the first overlay then be moved frorn Aw or 1000 to 2Aw or 2000. We can now plot on the second overlay the points 6 to 14 of Fig.l.4-10. A further shift of the first overlay by Aw to 3000 permits one to plot the points 15 to 33. This process represents in essence four dimensions by a drawing of a three-dimensional structure on several transparent sheets of paper. A further simplification is achieved by using only one transparent sheet at a time and thus reducing the number of spheres and rods one has to recognize a t a time. We need two more hard-to-understand illustrations before we can draw the co~lclusionthat is the goal of this section. The three-dimensional coordinate system according to Fig.1.4-6 was shown to be replaceable by an one-dimensional system according to Fig.1.4-7 or a two-dirrierlsio~ialsystem according to Fig.1.4-8. In analogy, a four-dimensional coordinate system according to Fig.l.4-9 can be replaced by an one-dimensional system according to Fig.1.4-10, but also by two- arid three-dimensional coordinate 'Figure 1.4-9 does not suggest that we live in one of several three-dimensional, parallel universes as the infinitesimal R'iemann geometries did. We could observe the other universes with electromagnetic waves and perhaps travel there. This is a readily understandable example of an observable difference between geometries with finite differences Ax and infinitesimal differences dx. See also Section 1.5 from Eq.(3) on.

1.4 F I N I T E

D I F F E R E N C E S AND SPATIAL DIMENSIONS

F1c.1.4-11. Replacement of a four-dimensional discrete Cartesian coordinate system with denumerable points by a two-dimensional coordinate system with variables X, W. The notation <, 17, C,w for the directions of Ax, Ay, Az, Aw follows the notation used in Figs.1.4-7 and 1.4-10.

systems. Figure 1.4-11 shows how this can be done with a two-dimensional system. The axis denoted w in Fig.l.4-10 is shown again in Fig.1.4-11. It is used for the variable W. At the points W = 0 and W = 1 we start onedimensional coordinate systems according t o Fig.1.4-7. The first such system starts in the lower left of Fig.l.4-11 with the point W,X = 0,O and goes on t o 0 , l ; 0,2; . . . ; 0,16. A second system according t o Fig.1.4-7 starts

1.4

FINITE DIFFERENCES AND SPATIAL DIMENSIONS

33

at W = 1 with the point W,X = 1 , 0 and goes on to 1,l; 1,2; . . . ; 1,16. The two three-dirriensional cubes of Fig.1.4-9 are shown by dashed lines in analogy to Fig.l.4-10. Not shown in Fig.l.4-11 are the 16 connections between the points 0, i, and 1,i for i = 1, 2, . . . , 16. Instead we show only the beginnings arid ends of these connectio~ls.If they are drawn fully, the illustration become too difficult to comprehend. The large scale of W is also used to help comprehension, by avoiding an overlap of the two one-dimensional coordinate systems for X . We do not show an illustration of a four-dimensional coordinate system replaced by a three-dimensional system but explain only how this can be done. In Fig.l.4-11 we have started one-dimensional coordinate systems with the variable X at the points W = 0 and W = 1. If we replace them with two-dimensional coordinate systems with variables X, Y according to Fig.1.4-8 we obtain a three-dimensional coordinate system with the variables X , Y, W that can be used as replacement of a four-dimensional coordinate system with variables X , Y, 2,W. Finally we extend the four-dimensional cube of Fig.1.4-9 to a fivedimensional cube in Fig.1.4-12. The four-dimensional cube is first shown on the lcft and again on thc right. Thc points Ojkl are then connected with the points l j k l , where j, k, 1 equal 0 or 1. The difficulty with this and higher-dimensional drawings is to figure out which rod is in front or behind any other rod that it crosses. What is evident is that a five-dimensional structure can be assembled in our physical space with equally long rods, even though the solidly black rods in Fig.l.4-11 are shown longer than the others to facilitate understanding of the illustration. Some further thinking convinces one that such structures with more then five dimensions can be assernbled in our physical space, the limit being only the practical difficulty of irnplenleritatiori but not any theoretical barrier. We do riot attempt to represent Fig.1.4-12 as an one-dimensional coordinate system as in Figs.l.4-lb, 1.4-7, or 1.4-10. Instead we only point out that this is theoretically possible because n5 is denumerable if n is denumerable. One may extend this statement to nm for any finite value of m. Our results show that we can construct coordinate systems with 1, 2, 3, 4, 5 , . . . variables or dimensions in our physical space. Some will be more lucid than others but this is not of much importance here. If we can assign dimensions so freely they cannot be defined and imposed on us by nature but must be human inventions. Infinitesimal distances d z , non-denumerable marly of them, and the resulting mathematical concept of contiriuu~riare not observable. This is a strikingly poor foundation for some of our most basic physical concepts. But there is no doubt that we think we live in three spatial dimensions. What makes us think so? Immanuel Kant (1922) claimed 200 years ago that, the concepts of time and space are contained in us a priori. In today's

1 INTRODUCTION

F1G.1.4-12. Extension of the discrete Cartesian coordinate system of Fig.1.49 t o five dimensions with 25 marks. A modification of the presentation of Fig.1.4-9 is used for t h e connection of t h e left half with the right half of the illustration. This modification is strictly due to the increased difficulty of drafting. One needs Z5 styrofoam spheres and 4 x 12 2 x 8 1 x 16 = 80 rods t o implement a model in our physical space. It can be represented unambiguously - if not lucidly - with one projection on a paper plane.

+

+

language one would say we have genes for these concepts. Although no such genes have been identified yet it is not impossible that they will be found eventually. Genes are generally developed by evolutionary selectiori of what works best for reproduction, self preservation being understood as a major contributor to reproduction. It is not clear how the concepts of an one-dimensional time and a three-dimensional space favor reproduction. Yet equal or a t least compatible concepts of time and space existed for Europeans, Japanese, Australian aborigines, Incas, Polynesians, etc. a t the time they first met3. 3A famous exception is the concept of time of the Mayans. They used rings in a complicated manner to keep track of the years. This led to a situation in 1697 when time either was coming to an end or had to start over periodically. The last custodians of Maya culture on the island of Tayasal in Lake PetBn-Itzd, Guatemala, were very troubled by this fact. A Franciscan friar AndrBs de Avedaiio, who had mastered the Mayan calendar, persuaded them to overcome the problem by accepting the Christian calendar and the

1.4 FINITE DIFFERENCES AND SPATIAL DIMENSIONS

35

It has been claimed that our concept of time comes from our observation of change. The most important change we observe is our own aging process (Harmuth 1989, Sec. 4.1). Statements like "When I was a child . . . ", "When I married . . . ", "When my first child was born . . . ", "When I retired . . . " are examples how we use our aging process as a scale of time. Galilee used the human pulse for what were short-time measurements in his time. The pendulum clock, the quartz clock, and the cesium clock obscured the original human clock from 1600 on. It is easy to see how people world wide developed the same concept of time. For the possible origin of our concept of space consider a child of about one year of age. It attempts to stand up on its feet, and learns that getting up requires an effort while falling down does not. Gravity teaches the child a first distinguished direction: up-down. Soon after mastering standinglip the child begins to walk. The design of our face and legs teaches it a second distinguished direction: front-rear. The location of our arms teaches a third distinguished direction: right-left. By age 2 most children of all populations have mastered the most basic concepts of Euclidean geometry, without requiring any knowledge of the rest of the world. It seems reasonable t o conclude that our customary three spatial dimensions are learned a t a very young age by experience. The concept of time based on our aging process is learned from middle-aged parents and teachers by education much later on. The concept of the continuum made it almost impossible for us to progress beyond three spatial dimensions. Bolyai (1832), Lobaschevskii (1840), and Riernanrl (1854) eventually succeeded in generalizing Euclidean geometry within a continuum theory, but the results are beyond observability and belong to pure mathematics. Anyone having difficulty with the concepts of up-down or front-rear would be considered retarded. But it is common experience that some people have difficulty with right-left, Much effort goes into training students of engineering and architecture to think in three dimensions. Many yeah of thinking in four dimensions greatly increase ones ability to do so. Thinking in five dimensions long enough may perhaps make Fig.1.4-12 look as simple as Fig.l.4-9. We sometimes read claims that we live in a four-dimensional space that has parallel three-dimensional universes. The first part of the claim is based on the use of strictly mathematical concepts but then the physical concept of "universe" is added. This is what can happen if one does not distinguish between thinkable mathematical and observable physical concepts. Let us discuss i11 a light hearted way how the lack of gravity would have affected our development of a concept of space. Consider an octopus Spanish rule that came with it (Thompson 1959, p. 156; Hammond 2006, p. 241). We see how strongly one can believe in a concept of time different from ours. Tayasal is now the town of Flores, connected by a causeway t o the better concept of time. The name Tayasal lives on for a place t o the north of Flores across a bay of the lake.

living in the sea. The effect of gravity is probably not noticed. When the octopus ejects a jet of water it rrioves arid experier~cesa first distinguished direction: front-rear. The eye is in front, the arms in the rear. The design of the octopus helps with this discovery just as the design of our face and legs helps us. There is no obvious second and third distinguished direction, despite the eight arms. An octopus might consider the one-dimensional coordinate systems of Figs.l.4-lb and 1.4-7 as most practical. We turn t o a brief discussion of sciences of the thinkable and sciences of the observable. Sciences of the thinkable are mathematic and theologies. Matherriatic is based on assumptions that are called axioms. Peano worked out the axioms for number's theory, but we cannot list all the mathematical axioms. In theologies the word axioms is replaced by basic truths or some similar expression. All other sciences at least strivc for observable results, since observation is such a convincing proof. Everything observable is also thinkable. I t may not be so initially, but we will change our thinking t o eliminatc contradictions with observation. Theologians sometimes claim observations that cannot be repeated to support their teaching of what is thinkable. Mathematicians never do that since they understand that the proof of something thinkable can only come from something else that is thinkable. Mathematics is enormously important in physics as a tool as well as a source of inspiration. There is much discussion why mathematics is such a successful tool, but most physicists and engineers are happy to have a good tool without worrying about an explanation. One should not be surprised by this success, since usefulrless for physics was always a driving force for the development of mathematics until abstract mathematics, based on non-observable assumptions, separated from physics, based on observation, during the lgth century. When mathematic is used in physics or engineering we must satisfy both the mathematical axioms and the physical laws. This is often overlooked. One reads statements that time in physics runs only forward while in mathematics it can run forward or backward. It is the physical causality law that makes time run forward in physics: the effect comes after the cause. Pure mathematics has neither a causality axiom nor a time or space variable. Instead it has complex variables, real variables, rational variables, integer variables, random variables, etc. Whenever there is a physical dimension like s, m, kg, A, J , K, N, . . . in a mathematical investigation we apply mathematics to physics and we must observe the laws of physics. The situation became more complicated with the development of information theory in electrical communications. In addition to mathematical axioms and physical laws we must now observe information laws too: information is always finite, information is transmitted by transients, in the steady state information is transmitted at the rate zero, information is carried by signals, the information of a signal can only be reduced but not

1 . 5 CURVED SPACE IN A DIFFERENCE THEORY

37

incrcascd by noise and distortions, etc. Mathematicians have made a contribution by coining the term calisal fiinction for a function f (0) that is zero for 0 5 0. In this terminology information is transmitted by signals that are represented by quadratically integrable causal functions of time (quadratically integrable implies finite energy of the signals). As an historic example of the power of thinkable explanations for observable phenomena consider the god Thor or Donar in Germanic mythology. He was believed to produce lightning and thunder by throwing a hammer. This theory of Thor's harrinler is thinkable and many people were satisfied by it for a long time. Whenever we use such thinkable concepts as continuum, irrational numbers, infinite, infinitesimal, or non-denumerable to explain observable physical effects we should recall Thor's hammer. Observable results demand observable assumptions.

Discrete mathematics applied to geometry creates a nurnber of problerns for thinking and perception since such farniliar elements as straight or curved lines, surfaces, and spaces do not exist, at least not in the form we have become used t o since grade school. Let us start by defining the concept of physical line by the propagation path of a light ray that we use for observation with telescopes or corresponding devices based on electromagnetic waves. If gravitational effects are significant the physical line will be bended or curved, otherwise we will call it straight. We know from experience that optical surveys on Earth are not significantly affected by the mass of the Earth or of mountains, and we do not need t o make a distinction between a physical line and our grade school concept of straight line in this case. The next concept we need is that of general coordinate systems. Figure 1.5-la represents such a system by an array of 6 x 6 points. The important thing are the points which may be realized by the crossing of searchlight beams at night or by range and direction measurements from a radar. The coordinate lines shown dashed only help us comprehend the coordinate system. Figures 1.4-6 t o 1.4-12 in the previous Section 1.4 would be incomprehensible without the rods connecting the coordinate points. It is important that Fig.l.5-la is possible only in a theory based on "arbitrarily small but finite differences". We cannot draw Fig.l.5-la with distances dx, dy between the coordinate points. Figure 1.5-lb shows the same coordinate points as Fig.l.5-la but the points are now connected to yield a coordinate system with one variable X rather than with two variables x, y. The distinction between the two illustrations has theoretical interest, but it is not of much practical importance since we can make a list with X = 0, 1, 2, . . . and x = 0, X I , 2 2 , . . . as well as y = 0, yl, yz, . . . that uniquely relates the coordinate points of both systems. Any result calculated with one system can thus be translated into the other system.

1 INTRODUCTION

Fr(2.1.5-1. A two-dimensional coordinate system with 6 x 6 points x, y (a) and the equivalent one-dimensional coordinate system with 36 points X (b).

FIG.1.5-2. The Pythagorean distance Asxybetween the marks x,y and x + A x , y+ Ay in a coordinate system according to Fig.l.5-la if the meshes of the system are replaced by parallelograms.

Consider a particle propagating in Fig.l.5-la in the direction of increasing values of x and y. Let it be observed first to be "closer to the mark 22,y3 than to any other mark". Sometime later it will be observed "closer to rnarks x3,y3 or s2,y4 or 2 3 ,y4 than to any other mark". We shorten these cumbersome expressions to "at the mark x2,y3", etc. To analyze this propagation in more detail we replace the coordinate rriesh at 22,y3 in Fig.l.5-la or X = 18 in Fig.l.5-lb by the parallelogram of Fig.1.5-2. The parallelogram is an approximation of what is needed for Fig.1.5-2 that will be improved presently. The distances from x ,y to the adjacent marks x A x ,y or x,y A y is defined by the resolutions A x = A x ( x ,y) and A y = Ay(x, y) at the point x,y. The distance As,, to the mark x A x ,y A y can be calculated:

+

+

+

+

1 . 5 CURVED

SPACE IN A DIFFERENCE THEORY

F I G . 1.5-3. Deformation of the parallelogram of Fig. 1.5-2 to produce the more

general coordinate meshes of Fig.l.5-la.

This equation reminds one of the generalization of the expression ds2 = dx2 dy2 in Riemann geometries. To make the parallelogram of Fig.1.5-2 fit the coordinate meshes of Fig.1.5-la we deform it according to Fig.1.5-3 by moving the point x , y A y in Fig.1.5-2 to the more general location xo, yl in Fig.1.5-3. Equation (1) remains unchanged for A x = A x ( x o ,yo), A y = A y ( x l , yo), and a,, = ax,(zl,yo) since the distances A x ( x o ,y l ) and A y ( x o ,yo) are not used. We turn from the two-dimensional distorted parallelogranl of Fig.1.5-3 to the three-dimensional distorted parallelepiped of Fig.1.5-4. It represents a mesh of a three-dimensional coordinate system in analogy to Fig.l.5-la. The three important edges from xo, yo, zo to X I , yo, zo to X I , yl .zo and x l , y l , zl are shown with heavy lines, the remaining nine less important edges with thin lines. Let us note that the angles a,,, a,,, and a,, do not need any Euclidean plane or other surface for their definition. A surveyor or an astronomer measuring the angle between two directions needs only an axis perpendicular to the two directions to rotate the telescope and measure the angle. The distances Asx,(zo, yo, z o ) and As,, ( x l ,yo, zo) in Fig.1.5-4 can be obtained with thc hclp of Fig.1.5-3. The new distance Asx,,(zo, yo,zo) requires considerable more effort. We show in Fig.1.5-5 the distorted parallelepiped of Fig.1.5-4 with the help of descriptive geometry by a ground projection (Grundriss) and an elevation (Aufriss). The distances A x ( x o ,yo, z o ) arid A y ( x 1 , yo, z o ) as well as the angle ~ , , ( x ~yo,, z O )are shown unmodified , a,,(xl, y l , z o ) are not. We need in but A z ( x 1 , y l , zo), C Y ~ , ( X y~ l, , z ~ ) and addition t o Fig.1.5-5 the axorlometric illustration of Fig.1.5-6 for the edge of the distorted parallelepiped from zl , y l , zo to X I , yl , zl .

+

+

1 INTRODUCTION

F1c.1.5-4. The distance Asxy,(xo, yo, zo) between the marks zo, yo, zo and y l , z l in the three-dimensional generalization of Fig.1.5-3. The coordinate cell is defined by either three distances and three angles or by six distances AX(~O YO, , LO),AY(XI,YO, ZO),AZ(XI,YI,ZO), A ~ x y ( x o , ~A ~ S, ~ ~~()Xr I , to), Y O and , As,yz(~o,YO, zo). XI,

+ cos2 ayz(llO)cos2 azx(llO) + 2 cos ax,(100) cos a,,(llO) cos azx(110) g23 = cos LV~,(IIO) + cos ( ~ ~ ~ ( 1cos0 0~ )~ ~ ~ ( 1 1 0 ) 912 = cos ~~,,(100), g31 = cos ( ~ ~ ~ ( 1+1cos 0 ) ~ ~ ~ ( 1cos 0 ayZ 0 ) (110) 911 = 1, 922 = 1, 933 = 1

( i j k ) = (xi, yj, zk:),

gmn

= gmn(x0,YO, ZO) = gmn(000)

(2)

Riernann (1854) used the generalization of Eqs.(l) and (2) from two and three variables to n variables xO,xl, . . . , xn-', assuming the abstraction

1.5 CURVED

SPACE IN A DIFFERENCE THEORY

AV = A ~ ( ~ o , y i , z o ) a,, = a,,(:cl. yo, z o )

AZ = A z ( x l , ~ l20) ,

u y z

= aya(x,,Y I , ~ " )

F1G.1.5-5. G r o u n d projection a n d elevation of t h e distorted parallelepiped of Fig.1.5-4 showing t h e t r u e length of Ax(x0, yo,to), Ay(x1, yo,to), a n d t h e t r u e angle a,,(xl, yo, to).

of finite distances As t o infinitesimal distances ds as well as other features to be discussed presently. He writes

The use of infinitesimals ds, dz" requires non-denumerable points for z". This makes the interpretation of Riemann's geometries by curved coordinate systems impossible and led to the interpretation by curved spaces1, whatever 'A clear and concise presentation of the relation between mathematics and physics from the standpoint of a mathematician assuming non-denumerable infinitesimal distances as a foundation of physics may be found in a book by Bronstein et al. (1996), Chapter 3.9 and Part 11, Section 14.9. In a science of the thinkable the use of infinitesimal and non-denumerable is a generalization but in a science of the observable it is a No-No.

1 INTRODUCTION

FIG.^.^-6. Axonometric representation of the edge from Fig.1.5-4 and certain projections of this edge.

XI,

yl, t-0 to

XI,

y l , 2 1 in

that may be in physics.. We do not observe curved spaces any more than something infinitesimal or non-denumerable, but we do observe curved light rays which we may use to construct curved coordinate systems Despite the problem of non-observability, Riemann's geometries found a successful and great application in Einstein's general theory of relativity. The variable x0 is there interpreted as the time t while x l , x2, and x3 stand for spatial variables x, y,z. The summation signs of Eq.(3) are generally left out for simplification, the double occurrence of an index m or n serves as a sufficient instruction to sum over m = 0, 1, 2, 3 and n = 0, 1, 2, 3. We return to Eq.(2). The components 911 to g33 may be written in the forrn of a matrix g :

Assume the angles a,,, a,,, and a,, in Eq.(2) equal n/2 a t every point xi, yj, zk. This makes the distorted parallelepiped of Fig.1.5-4 a hexahedron

1.5 CURVED SPACE IN A DIFFER.ENCE THEORY

43

or a cube with edges of three different lengths. The matrix g becomes: g=

(: :) 0 1 0

;

axy,ayz,azx=77-12

(5) This matrix is known as the metric tensor of Euclidean geometry. The matrix of Eq.(4) may thus be called the metric tensor of the geometries of finite differences. We recognize that Eq.(5) is not obtained from Eq.(4) by the reduction of the distances Ax, Ay, Az --+ dx, dy, dz since the angles a,,, a,,, and a,, are not affected by this reduction of lengths. Hence, the georrietries of finite differences do not approach Euclidean geometry for srriall differences Ax, Ay, Az. Let us extend the investigation of the angles a,,, a,,, and a,, of Fig.1.5-4. We may replace a,,(xl, yo, zo) by the angle &,,(x1, yo, ZO)that becomes 77-12 for Ax(xo, yo, zo) --+ 0. A constant d with the dinlension of a length is introduced:

This equation demonstrates the generality of the geometries of finite differences but it is misleading. The idea that geometries should become Euclidean at short distances goes back t o the days when "short" meant small compared with the length of the equator. There are no surfaces on which the angles of triangles could be observed at atomic or nuclear distances. There are only points and distances between points. A point is something too small to show any spatial feature-like a diameter-with the available resolution of spatial observation. Elementary particles arc pcrfcct physical points, atomic nuclei or atoms are not quite as good. What do Euclidean geometry or Riemann's geometries mean if there are only points and distances between them but no surfaces on which t o stake out a triangle? Riemann's geometries are generally accepted by physicists today and they are the basis for the general theory of relativity. There are two reasons why wc have t o treat then1 with caution. The first is that they are a product of pure mathematics without any connection t o observation. This disconnect from observations was a serious concern for Riemann. He wrote2: Hence, it is quite possible to think that the metric of space at infinitesimal distances does not satisfy the assumptions of geometry, and one would indeed have to assume this if the observations could then be explained in a simpler manner (Riemann 1854, p. 285, end of second paragraph). The question about the applicability of the assumptions of geometry at infinitesimal distances is connected with the question about the inner reasons of the measure relationships of space. With this question, 2 ~ h i is s a free translation that uses technical terms that were not known at the time of Riemann (1824-1866).

1 INTRODUCTION which one may perhaps still include in the study of space, the previous remark is used, t h a t a discrete manifold contains the principle of the measure relationships already in the concept of this manifold, but it must come from somewhere else for a continuous manifold3. Hence, the reality o n which space is based must either be a discrete manifold, or the reason for the measure relationships must be searched outside in attractive forces acting on them (Riemann 1854, p. 285, last paragraph).

The second reason for interest in gerleralizations of Riemann's geometries is that the general theory of relativity is our best theory of gravitation but it is sornetirrles wrong or unsatisfactory. The hold of irifirlitesimal geometries on our thinking was broken by information theory, which was developed about one century after the publication of the basic paper by Riemann (1854). Information theory taught us that irifornlatiori obtained through observation and nieasuremerit is always finite. It can be measured in bit. Non-denumerable bit are a t the same level as non-de~iurrierableJoule4. We are going to show two results that are typical for georrietries of finite differences. Consider Fig.1.5-7. It shows a center point xo, yo,zo = 000 and its 26 closest neighbors from z-1, y-l,z-1 = -1-1-1 to x l , yl, zl = 111. Only the four distances of interest in the octarit x 2 0, Ay 0, A z 0 are shown in heavy lines. If one tries to show all 26 distarlccs of interest in all eight octarits iri heavy lines the illustration beconies incomprehensible. We recognize in Fig.1.5-7 a distance zero if something movable a t point 000 rerriairis there. The poirit 100 has the distance Ax, the points 110 and 101 have the distances As,, or As,,. The furthest point 111 has the distance As,,,. If we generalize to all eight octants we obtain 1 x 2' = 1 distance 0, 3 x 2l = 6 distances fAx, &Ay, fAz; 3 x 22 = 12 distances As*,,,,, As,,,,, , As*,,,,; and 1 x 23 = 8 distances As*,,*,,*,. T h e sum is (1 + 2)3 = 27. Such a finite nurriber of distances is characteristic for a theory of finite differences. We turn to Fig.1.5-8. It shows five light rays represented by dashed lines in the direction of x and seven in the direction of y. Light rays run along geodesics arid any movement along them will be mathematically simplified. The 35 crossings define the coordinate marks 0; X I , 0; 0, yl; X I , y,; . . . zs, y4 shown by sniall circles. The followi~igvariables define the coordinate system:

>

>

A x = A x ( x i , yk:), A y = A y ( ~ iyk), , a=LY(zi,gk), Asxy= Asxy( x , ~Yk) , (7) Let US niakc thc transition from finite differences to infinitesimal differences in Eq.(7): 30ne could also have concluded that the transition from arbitrarily small but finite differences to differentials should not be made, but it took 150 years before this conclusion became thinkable. 4 ~ h o s ewho did not grow up with information theory may reflect that infinite information is no better. than infinite energy. In the days of the perpetuum mobile the concept of infinite energy was not as objectionable as it is today. Infinite information may require a comparable time to become fully appreciated.

1.5 CURVED SPACE IN A DIFFER.ENCE THEORY A,-111 ,

,,,'

\

r-

\\\

'

, X,,y3,zk=ijk ,i.j, k=-l,O,l

v<:l-l

F1G.1.5-7. The center point xo,yo, zo = 000 and the surrounding 26 neighborhood points. The distorted cube of Fig.1.5-4 is replaced by an undistorted cube to simplify the illustration and the discussion.

An:

+

dn:(n:,y), Ay

-f

dy(x, y), a

-+

a(,, y), Asxy

-+

ds(x, y)

(8)

The angle cu between Ax and Ay in Fig.1.5-8 remains unchanged when Ax arid Ay are made srrialler and smaller. The terms dx(x, y) arid dy(x, y) require explanation since we are used to see dn: and dy. We may replace As(xa, yk;) in Eq.(7) by a power series of a variable A%:

C Y ? ~ ( ~Y: I~, ) ( A % ) ~ 00

An:(n:,,yk:) =

(9)

n=l

If this sum converges it has a defined value for finite values of A%and known values of y,,. The trarlsition A% -+ d% brings:

Equation (10) is less general than Eq.(9) since (di?)" is only defined for n. = 1. This shows that geometries of finite differences are more general than Riemann's geometries.

1 INTRODUCTION

FlG.1.5-8.Bended coordinate system with finite distances Az,Ay. Something moving from (closest to) mark 0 can be observed at later times closest to or "at" the marks X I ,y l ; 22,yl; . . . 5 6 ,y4. These marks can be connected by a segmented line or approximated by a smoothed line, but only the observations "at" the coordinate marks provide information.

In addition to replacing curved spaces by curved coordinate systems and curved light rays the use of "arbitrarily small but finite differences" At, A x , Ay, Az instead of differentials leads to difference equations. Holder (1887) showed that difference and differential equations define different classes of filnctions. We will see in Chapters 4 and 5 that rewriting Dirac's differential equation as difference equation yields indeed unexpected results that were never obtained from the differential theory. We have shown in Section 1.4 that we car1 assign the dinlensions n = 1, 2, 3, 4, 5, . . . t o our physical space. If the choice n = 3 appears convenient we niay use it without any general agreement on why this choice is convenient. According to Fig.1.5-1 we could replace the two-dimensional addresses z , y in Figs.1.5-2 and 1.5-3 by a n one-dimensional address X. Similarly, the three-dimensional address z, y, z in Fig.1.5-4 could be replaced by an one-dimensional address X according to Fig.1.4-7 in the previous section. We have outlined the most basic principles of coordinate systems based on arbitrarily srriall but finite differences. Sections 6.10 and 6.11 summarize sorrle additional results. We must stop here. The elaboration of siich coordinate systems and their application to the general theory of relativity is a project for scientists at the beginning of their carrier, not for those a t the end of their life.

2 Modified Dirac Equation

T h e modified Dirac differential equation is obtained by adding a magnetic (dipole) current density terrn to Maxwell's equations. The modified Dirac difference equation is produced by replacing differential operators with finite difference operators. We start with the usual Harniltori furlction without mag~ieticdipole current density:

It may be rewritten in the following form:

The corlventiorial Klein-Gordon equation is obtained if one makes the substitutions

and applies the operators to a function 9:

If we add a rrlagrletic (dipole) current density term to Maxwell's equations we must replace the Hamilton function K of E q . ( l ) by a furiction X

48

2 MODIFIED DIRAC EQUATION

with three components X,, X,, X,. Equations (1.1-41)-(1.1-43) show a first order approxiniatior~in a, or a of these three conlponents. We write only X, :

X,

= c[(p - e

(X, - e#,

+

+

~ , ) ~ =

+ ede - L~:,; Q)

(5) (6)

2aXcAe/e=2aXcAe(r, t)/e=a,(r, t) =a,, Xc = h/moc, a=2e2/2h, The terms Q arid LC, are defined in Eqs(l.1-44) and (1.1-27). They require the potentials A,, A,, #,, and #, which are defined by Eqs.(l.l-17)(1.1-20). The square [l + (2aXcAe/e)QI2 in Eq.(6) may be replaced by 1 2(2aXcA,/e)Q since Eqs.(5) and (6) hold only in first order of a. We may write Eq.(6) applied to a function 9 as follows in first order of a, using Eq.(l.l-46) for factors that may not commute:

+

This is the previously derived modified Klein-Gordon equation (Harmuth, Meffert 2005, Scc. 2.1). Wc want to usc it as a guide for the derivation of the modified Dirac differential equation. For the derivation of the modified Dirac differential equation we follow Becker (1963, 1964; 5 57) and Messiah (1966, Ch. XX). We replace the Hamilton function of Eq.(l) by one that represents the square root with the help of matrices:

2.1

DIFFER.ENTIAL EQUATION W I T H M A G N E T I C C U R R E N T D E N S I T Y

a, = a,. =

I

0 0 0 - 1 1 0 0 0

49

I

Equation (8) may be rewritten in analogy to Eq.(2):

a, (p, - e ~ , , ) +a, ( p , - e~,,) +a, (p, - e ~ , , ) - :(X-e4,) C

= -/3moc

(11)

With the substitutions of Eq.(3) and application to a function 9 we obtain the equivalent of Eq. (4):

[x (q 2 an;., 3

a,,

j=1

?,

e=-bocP

- eA,,,

(12)

The addition of a magnetic (dipole) current density term to Maxwell's equations calls for the replacement of the Hamilton function X by a function 9C with three components X,, X,, X Z . Again Eqs. (1.1-41)-(1.1-43) show a first order approximation in o, or a of these three components. Only X, is written:

X,= [ c a . ( p - eA,)

+~ r r ~ o c ~ ]

+e& L C ,

(13)

Equations (13) and (14) are the equivalent of Eqs.(5) and (6). The terms Q and LC, are the same. They are defined in Eqs.(l.l-44) and (1.1-27). We may write Eq.(14) applied to a function 9 as follows in first order approximation of a :

2

50

MODIFIED DIRAC EQUATION

+

Since K, - eq5, LC, on the left of Eq.(14) is not squared as in Eq.(6) we avoid the question of commutability of (KZ - eq5,)Lc,. Equation (15) is simplified compared with Eq. (7). A solution as an expansion in powers of a is obtained if we replace XP in Eq.(15) by the function 9,:

We have shown in Eqs.(1.2-1)-(1.2-7) that the function LC, is of order O(a). Equations (15) and (16) yield thus the followirig two equations for QX0and 9x1:

We recognize in Eq.(17) the usual Dirac equation of Eq.(ll). Equation (18) is the same homogeneous equation with an inhomogeneous term added. The factor Q in Eq.(18) has a term [l ( p - e ~ , ) ~ / m E c ~ ] -Fol~/~. lowing the extraction of the square root in Eq.(8) we write:

+

Frorn Eq.(9) we recognize the relations p2 =

and obtain

1,

pp-I

= 1,

0-1

= p, /3 = p3

(20)

2 . 1 DIFFER.ENTIAL EQUATION W I T H MAGNETIC C U R R E N T DENSITY

51

We investigate first an approximation that holds for small values of ( p - eA,)2/~rrt,gc2 and address its possible generalization afterwards:

For the Klein-Gordon equation we had derived the equivalent relation (Harmuth, Meffert 2005, Eq.2.1-13):

This takes care of part of Q in Eq.(l.l-44), but we still have to deal with the remaining factor

In the case of the Klein-Gordon equation thc factor Qo2 was eliminated by replacing the vectors A, and ( p - eA,) with matrices of rank 3 whose components are vectors:

p - eA, =

0 0 (PY- eAmy)e~ 0 0 (PZ - eAm,)ez

The substitution of Eqs.(25) and (26) into Eq.(24) yields

This is a satisfactory and simple result but it replaces the usual Dirac equation containing matrices of rank 4 by an equation with matrices of rank 3 whose elements contain matrices of rank 4. The term Q in Eq.(1.1-44) is reduced to:

The denominator of Q can be eliminated with Eq.(22) and Q is brought into a form that permits the substitution of the operators of Eq.(3):

2

MODIFIED DIRAC EQUATION

We note that there is no difference between multiplying P.moc-3 a.(p-eA,) on the left or right with ( p - eAm)2. The term LC,in Eq.(18) can be written with (p-eA,),, = p Z j -eA,,, - where xj equals x, y, or z - using Eqs(1.2-3) to (1.2-7). This form of writing permits the substitutions of Eq.(3). We write Eq.(17) in the matrix form of Eq.(25). The operators of Eq.(3) are substituted for p and X,:

Sirice only the terms in the main diagonals are not zero this matrix equation is essentially three times Eq.(12). The summation sign of Eq.(12) has disappeared but the index j retaines the values j = 1, 2, 3 that it has in Eq.(12):

1 tia

i,

axj

e~,.~)

+;

+ e") + pmZoc]q x o r j = 0

(31)

Equation (18) can be written in matrix form like Eq.(30) with QX0,, qxOy, qxOz replaced by Qx1,, Qxly, qxlzand another matrix instead of 0

2.1

DIFFER.ENTIAL EQUATION W I T H MAGNETIC CUR.R.ENT DENSITY

53

on the right side. T h e shorter notation of Eq.(31) is more practical. Let us write the first term on the right side of Eq.(18) with the help of Eq.(29):

Equation (18) may now be rewritten in analogy to Eq.(31):

We return to Eq.(21) and try to rewrite Eq.(18) without the approximation of Eq.(22). Using Eqs.(19) and (20) we may write Q of Eq.(28) in the following form:

Q = m;c2

P(P - eAmI2 [1+P a . ( p - e ~ , ) / m ~ c ] ~

(34)

If we substitute Q into Eq.(18) we must decide whether we want t o multiply the equation on the left or the right with the denominator of Eq.(34). We try multiplication on the left first:

Consider the homogeneous equation, which is not affected by the operator [a . (p - eA,) ,BmocI3 since this operator would be applied to 0 if the equation

+

2

MODIFIED DIRAC EQUATION

is satisfied. Hence, the homogeneous solution for terms of order O(a) equals the homogeneous solution for terms of order O(aO= 1) according to Eq.(17). The situation is completely changed if Eq.(35) is multiplied by [a . ( p eA,) ,BrnocI3 on the right- rather than on the left:

+

The homogeneous equation is now determined by

The potential 4, has no longer any effect, which suggests the solution of order O ( a ) will apply to something without electric charge. The time operator X , has also disappeared, which implies a steady state solution. It is still possible that both effects are changed in the solution of the inhomogeneous equation due to the factor LC,. However, the results we can expect from Eq.(37) are not likely to approach the results of Eq.(33) for small values of ( p - eAm)2/m,ic2. The same holds true for Eq.(35) although this is less evident. If we want a better approximation than provided by Eqs.(21), (22) and (33) we have little choice but to extend the series expansion to a higher order. Equation (33) may be compared with the corresponding equation of the modified Klein-Gordon equation [Harmuth, Meffert 2005, Eq.(2.1-28)]. For the elaboration of Eq.(28) for ( p - e ~ , ) ~ / r n >> ~ c1~rather than ( p - e ~ , ) ~ / , r n ; c<~< 1 according to Eq.(22) refer to Section 6.4. Let us briefly investigate how higher order approximations than Eq.(22) work. We start with Eq.(34) and write the second order approximation of a binomial series:

2.1

DIFFER.ENTIAL EQUATION W I T H MAGNETIC CURRENT DENSITY

55

The first term on the right of Eq.(18) becomes:

In matrix notation the term (p-eAm)2 is explained by the square of Eq.(26), which means by a matrix with the terms (p,, -eAm,,)2 in the main diagonal. The term a . ( p - eA,) is represented by the first matrix in Eq.(30). The term [a. (p - eAm)I2 is the square of the first matrix in Eq.(30); this is a matrix with terms a:, (-itia/azj - eAm,,)2 in the main diagonal. One readily verifies the relations

frorri Eq.(lO). Hence, [a. ( p - eAm)I2in matrix notation becomes

We may thus write the second order approximation of Eq.(18) in analogy to Eq.(33):

ti

a

earn,,)

(;

1 tia

+;

+ em,) + ~ r n z o csXl., ]

2XcAe h. 8 =-{G(;6-e~m:cj)2[l-~axj(TF e~,:~,

a

- eA,,,

}

~ ~ c x l j + ~ c x 2 j + ~ c x 3 j + ~ C X 4 j + ~ C X 5 j )P~~~~~

56

2

MODIFIED DIRAC EQUATION

Hence, higher order approximations of Eq.(34) by a binomial series lead to higher order differential equations. The terms Lcxlj . . . LcxSjin Eqs.(33) and (43) need some more elaboration. We havc already sccn in Eqs.(1.2-1)-(1.2-7) that the five components exist for every variable x, y, z = xj. If we substitute differential operators for p in Eqs(1.2-3)-(1.2-7) we obtain:

The rnixed notation s,y, z and terms with x, y, z

xj

in Eq.(44) requires clarification. The

form matrices of rank 3 with equal values for all elements in the main diagonal like the second and third matrix in Eq.(30). On the other hand, the terms

are the terms of a matrix with rank 3 like the first matrix in Eq.(30) with the terms along the main diagonal varying according to j = 1, 2, 3. The to LC-'cx6j assume the following form: rcmaining cxprcssions for

2.1

LC..

"1

= 2mz0c

(Aez

+(f

..LC.

57

DIFFER.ENTIAL EQUATION W I T H MAGNETIC CUR.R.ENT DENSITY

${ [1

+% /' ({[I 2

[l -

(2% i d z j - eAmXij

(; &

1 -

+ ( lT--~A,?, ia9 a

{

(-l i d

- e ~ , ~ [I ) - 2m.;c2

= 4moc

+

1

-2m;c2 i a x j - eAmZ)']

-e

)[

~

axj

I-- 2mic2 I ( li i a

&(:&

- eAmZi)']

tl

(;%

"

'1 (: &

a

li d ~ -~e

~~

e~,,~)'])~ - dm,)

"A!..)

1)':

x {[I - l2m,gc2 ( f L i%a x j - eAmz,)2] X

+ [A,,

a

li - -

[ ( ti a A,,

e

(f $ [I-- 2mgc2 I (" i axj

( -i a y - e~,.) x

-~ -~

e

y

(f y

-e

- .A,,)]

~ ~ ~ ) ]

eAmU) '])b (50)

Equations (44) t o (50) make all terms in Eqs.(33) and (43) defined for krlown values of the potentials A,, A,, 4, and the rest mass m,o of a charged particle. The equations are unique. Equation (31) can be solved for certain initial and boundary conditions. The solution of Eq.(33) requires a particular solution of the inhomogeneous

~)

~

2

MODIFIED DIR.AC EQUATION

F1c.2.1-1. Graphic representation of the relationship between Qfo and Qfl produced by the solution of Eq.(24) with the matrices of Eqs.(25) and (26). of Eq.(31). The homogeneous equation which contains the solution @,O, part of Eq.(33) equals Eq.(31). When the notation of Eqs.(3O) and (31) is used one must rewrite Eq.(16) with a longer subscript

as was already mentioned in the text following Eq.(31). These long subscripts will often be shortened as follows:

due to the solution of To clarify the relationship between and Eq.(24) by the matrices of Eqs.(25) and (26) we show Fig.2.1-1. In terms of vectors, Q1 has a direction independent of that of q o . Hence, does not simply change the absolute values of 9 0 .

For the derivation of the Dirac difference equation including the modified Maxwell equations wc start from Eq.(2.1-11): 1

a . (p - eA,) - - ( X - e@,) = -pmoc C

Q,

(p, -eAmrE)+cr,(p, - eA,,)

+a, (p, -eA,,)

1 ( X - e4,) = +mot (1)

-C

2.2

MODIFIED DIR.AC DIFFERENCE EQUATION

59

Instead of the differential operators of Eq.(2.1-3) we use the difference operators of Eqs. (1.3-5)-(1.3-9), but we use first the non-normalized differences Axj, At in analogy to Eq.(1.3-3) rather than A0 = 1 or A( = 1:

Higher order differences follow the pattern of Eq~(1.3-7)-(1.3-9).For the normalized variables

we may use thc simplified notation of Eqs.(1.3-5) and (1.3-6):

These definitions and their extensions to higher order differences suffice to rewrite all equations in Section 2.1 up to Eq.(2.1-46). From Eq.(2.1-47) on we encounter integrations in addition to differentiations. We must substitute surrlmations for integrations as shown in Table 1.3-1:

60 for

2

MODIFIED DIR.AC EQUATION

As a first step let us rewrite the matrix equation (2.1-30). We use x j y, z :

2,

fi A 1: At

fi A --

i. i t

c

0

0

0

+ erne

0

0

ti A -At

+ erne

I

The three equations represented by Eq.(2.1-31) assume the following form as difference equations:

[aZj

(!A ) + i (" 1,

AX, - eAmzGj

+ erne) + prrl,oc] @xo:cj = 0

(13)

We shorten @xoZj to \Iio and rewrite this equation explicitly rather thari in operator form. The first order difference operator is used in the form of Eq.(1.3-3) rather than in that of Eq.(1.3-5):

a:c j

*o(zj

+ Axj,t) - Q \ I ( x -~A x j , t ) + -1 Q o ( ~ . j ,+t At) - s o ( ~ j ,-t At) 2Azj c 2At

2.2

MODIFIED DIRAC DIFFER.ENCE EQUATION

61

We turn to the inhomogeneous equation (2.1-33) and substitute difference quotients as in the second line of Eq.(14):

The terms L in Eq.(2.1-33) were changed to L (Euler Roman Medium L) to emphasize that they are now difference operators. The homogeneous equation equals Eq.(14), but the inhomogeneous term requires elaboration. Consider the first term on the right side:

(5 A AX,

- eA,:Lj ( x j, t )

7,

(xj, t)

+ A,,j

(xJ,t) Ax,

+ e 2 ~ i , (xj, , t)] @o(xi,t)

(16)

With the help of Eq.(1.3-10) we can resolve the second term on the right:

d~,,, - *o Ax,

- A,,, = A,,,

AA,,~ + Axj AX @ O (+ ~ jAX:,, t) - ~ o ( x :-, Axj, t) ( ~ jt),

dq0

- qO------

2

4

The second term on the right in Eq.(15) is considerably more difficult to write explicitly:

62

2 MODIFIED DIRAC EQUATION

A 9(-A;Axj , 1,

+

AA,,,

-+ Axj

-)AxiA

-

e3~i,,] @O (18)

The term A ( A ; : ~ , @ ~ ) / A is ~ ,explained ~ by Eq.(17). Two more terms require explanation:

(axj

d2~,,,q0A 6x32

= A,,

A A- , ~QO~ ) = - ( A , ~A, G + @ o - dqo Axj

AX,

-+ 2A2@0

AAm:zj

Ax3

Axj

AQ0

A2AmZj A$

-+ -Qo x

-

AA,,~ Ax,

2.2 MODIFIED DIRAC DIFFER.ENCE EQUATION

63

The only term still needing explanation in Eq.(15) is LC,. The components Lcxk:,of Eqs(2.1-44) and (2.1-47)-(2.1-50) are rewritten with difference operators denoted Lcxk.:, :

As previously in the case of Eqs(2.1-45) and (2.1-46) the terms with the variables x, y, z

forrn matrices of rank 3 with equal values for all elements in the rrlairl diagonal like the second and third matrix in Eq.(12). But the terms

(F Ax, 7,

- eAm:Cj

are the terms of a matrix with rank 3 like the first matrix in Eq.(12) with x j = x l , x2, 2 3 = x, y, z. The four expressions for LCxzj to Lcxsj are written according to Eqs.(2.1-47) to (2.1-50) with the help of Eq.(1.3-20):

64

2 MODIFIED DIR.AC EQUATION

LC..;

=

$ ({

5 4mic

'("

[l- 2rr1&~ i Axj

C

- eAmZj)

A

1 1

+

(1i

'1 &)

l z Axj ( - eAmZj) } "

- e ~ ~ [1; ) 2m$c2

&(!A [ ( jy

x {[I - 2m.~c2 z nx, - e/ImZJ)'] x Ae, :- - .A,.)

+ [A,,

(si --

-

("

- Aey i ~z

-

e~,.)]

e ~ ~ ~ ) - ~ , ~ ( : i - e A , ~

We may combine the terms Lcxsj for j = 1, 2, 3 in a matrix Lcxk (Euler Roman Bold L) with the terms Lcxkj along the main diagonal:

2.2 MODIFIED DIRAC DIFFERENCE

65

EQUATION

Equations (21) and (24)-(27) make all terms in Eq.(15) defined for known values of the potentials A,, A,, 4,, and the rest mass mo of a charged particle. Equation (14) can be solved for certain initial and boundary corlditions. The solution of Eq.(15) requires a particular solution of the inhornogeneous equation which contains the solution Qxo,, of Eq.(14). The homogeneous parts of Eqs.(l4) and (15) are equal. Our calculation contains the simplifying assumption that the fine structure constant a is small compared with 1. This assumption was first made by using the approximate Hamilton function of Eqs. (1.1-41)-(1.1-43) rather than an exact generalization of Eq.(l.l-36). The smallness of a was used again by Eqs.(2.1-15) and (2.1-16). Furthermore, there are the restrictions of Eqs.(2.1-33) and (2.1-43). Yet our equations are still too complicated. We introduce further simplifications for the potentials A, and 4,:

A, = A,,

+ aA,l

(r, t ) , A,,

4e = 4e0 + a$el (xj, t ) ,

Xj

= A,o,,

= X , 9, Z ;

+ aA,l,j

(xj, t)

ax= aXo + aQxl

(29)

We rewrite Eqs.(2.1-17) and (2.1-18) with the approximations for A, and

$e :

Equation (30) equals Eq.(2.1-17). If we replace A, by A,, we must replace A,,, by Amo,, in Eqs.(2.1-12) and (2.1-33). This implies the same in Eqs.(l3) and (14). Equation (14) is modified very replacement of A,, little by the approximation of Eq.(29):

a,,

qXoxj (xj

+ A x j , t) - @,olcj ( z j - Axj, t) 2Ax.j

(xj, t + ~ + -c1 axox,

t -)a,,,, ( x ~t ,- a t ) 2At

66

2

MODIFIED DIRAC EQUATION

The irlhomogeneous equation (31) is written in the form of Eq. (15) by -+ Amo,,, 4, -, 4eo, and by adding the last two the substitutions A,, terms of Eq.(31):

- - {2AcAe(xj,t) ( -h -d - eAmoz, m,~ce i Ax,

The first line is written explicitly i11 Eq.(32) with QXozjreplaced by QXl,,. The terms in the second line of Eq.(31) can be obtained explicitly from Eqs.(16)-(20) with A,, replaced by Amo,, and Qo by Qxo,, The terms Lcxlj to LcxSj have still t o be written corresponding to the sirnplificatioris of Eq.(29). This requires two more simplifications:

A, = Aeo

4m

= 4m0

+ a A e l ( r ,t),

+ a 4 m 1 (xj t) 1

AeJj

= A,o,;

+ aAel,:, (xj, t) (34)

Only the terms Aeo and 4,0 are needed in Eq.(33), the terms multiplied by a are neglected. The tern1 Lcxlj in Eq.(21) is not changed rnucki optically, but all terms Aeo, Aeo., Amo. are now constants (note the two dots):

2.2

MODIFIED DIRAC DIFFERENCE EQUATION

67

The term LCxzj of Eq.(24) vanishes since the differences ddmo/Ay,etc. of a constant is zero. This implies that our results will not depend on the existence or non-existence of a magnetic charge according to Eq.(l.l-20): Lcxaj = O For LCxsj of Eq.(25) apply the same comments as for Eq.(21):

(36)

2

MODIFIED DIRAC EQUATION

2.3 SOLUTION OF

THE

DIFFERENCE EQUATION

FOR.

PxO

We turn to the solutiori of the equation for !?iO ,, in Eq.(2.2-32). The variables n:j and t are replaced by the riorrrialized variables C arid 0, which run from 0 to N. We also introduce a connection between Ax and At by writing Ax = c a t :

The notation for the function @xO,, is shortened to Po. The electric and magnetic potentials Aeo,, arid Amo,, have three components each Aeo,, Aeoy, Aeoz and A,ocE, Amoy,AmoZ.These components are constants that are not affected by the normalization of t and z j in Eq.(l). Since we avoid the notation C j and write instead C for simplification we can also simplify the subscript of x j by leaving it out and writing x. The following normalized partial difference equation with one spatial variable is obtained from Eq. (2.2-32):

X1 =

2AxeAmo,

, X 2 =

2A~eq5,~ , A3 = 2Azmoc cii ii '

T N

An:

- A t = - (2) C

Equation (2) is a matrix equation that we would like to write explicitly with matrices of rank 4. This runs into the difficulty that the matrices P o ( < ,13) and particularly Po((f 1'0) and PO((, 19 f 1) consume excessive space. We sidestep this difficulty by writing only Po(C,O)as an example explicitly:

Asking the reader to visualize the explicit matrices for Po([ f 1 , O ) and Po(<,O f 1) makes it possible to write the matrix equation of Eq.(2) explicitly in a form that does not require excessive space:

2.3

SOLUTION O F T H E DIFFER.ENCE EQUATION F O R

QzO

69

Doing the matrix multiplications demanded by Eq.(4) we obtain four equations for the four rows of the matrices:

We note that the substitutions Q04 -' QO3 and Qol -' QO2 transform Eq.(5) into Eq.(6) while Eq.(8) is transformed into Eq.(7). Hence, we have t o solve only Eqs.(5) and (8). In order t o separate the functions QO1 and QO4 in Eqs.(5) and (8) we use a method similar to that used in differential calculus. It is worked out in some detail in Section 6.1. The following two equal equations are obtained there:

2 MODIFIED DIRAC EQUATION

We look for a solution of Eq.(9) excited by an exponential ramp function as boundary condition. The exponential ramp function avoids the instant jump of a step function S(6) and is thus better suited for equations that describe particles with finite mass mo. We caution that L cannot be chosen and will turn out to be imaginary; Gal is a constant:

Pol (0,O) = $o,S(O)(l - e-LB)= 0 = Gol(l - e-")

for 0 for 0

<0 20

(11)

In order to obtain a solution that satisfies the causality law we must specify an initial condition QO1(C, 0) that is independent of the boundary condition @ o l ( O , O ) of Eq.(ll). We choose it to be

Since Eq.(9) is a second order difference equation with respect to 0 we expect to need a second initial condition. It will become evident how to do this at the very end of this section. We postpone a discussion until then. We use t,he following ansatz (Habermann 1987) to find a general solution of Eq.(9) from a steady state solution F(C)multiplied by the exponential ramp function 1 - e-", plus a deviation u(C,O) from (1 - e - L e ) ~ ( ~ ) :

Qol(C,

6) = 0 = % o l [ ( l - e-")F([)


(13)

2.3 SOLUTION

OF THE DIFFER.ENCE EQUATION FOR.

qwO

71

Substitution of P o l ( l - e-"e)F(() into Eq.(9) produces the following relatioris

arid we obtair~

Wc collect all terms multiplied with 1 - e-Le on the left side, the others on the right side:

This equation must hold for all values of 0, which is possible only if both sides are zero:

e-"

(,-"

2

+ eL+ i-A22 (e-L - e'))

=O

(18)

For the solution of Eq.(17) we use the standard ansatz

F ( < ) = A1vC, F(C f 1) = AlvCfl and obtairi an equation for v:

(19)

2

For small values of

MODIFIED DIRAC EQUATION

Ax we obtain:

1 vl = 1 - -(A: + i X2 exp (-(A: - ~ : ) ' / ~ / 2 )e'*lI2 2 2 1 1 'U2 1 + -(A; - ~ $ ) 1 / + 2 i,- 1exp ( ~ 2 2 112 2 ei*1/2 2 2 I)

--

(

(21) (22)

The function F(C) in Eq.(19) becomes:

F ( < ) = A10 exp {[-(Xi - X$)lI2/2

+ iX1/2]<)

+ A l l exp{[(X: - X;)lI2/2 + iX1/2]C)

(23)

According to Eq.(2) the difference A; - A; will be positive except for extremely large values of &,.We restrict the calculation to X i - X i > 0. In addition, we shall see presently that we want F ( 0 ) = 1, which is achieved by the choice Alo = 1 and All = 0. The function F(C) becomes:

F ( < ) = enp (-(A: - ~ $ ) ' ~ ~ ( /e'*1c/2, 2)

A: - A$ > 0, F ( 0 ) = 1, Alo

=

Az << hr/moc

1, All = 0

We note a second interesting limit of Eq.(22):

A3

>> 1; X i ,

A2

<< A3 or Ax >> hlfrnoc;

eA,oz,

e4,o/c << ,rrroc

(24)

2.3

SOLUTION O F THE DIFFERENCE EQUATION FOR

qX0

73

In analogy to E q s . ( 2 3 ) and ( 2 4 ) we obtain

F ( C ) = [Alo(4/X;)C

+ A ~ ~ ( x ; / ~ )eixlCI2 ~]

(27)

and

Different results for Ax << hlmoc and Ax >> h/moc or At << h/m,oc2 and At >> h8/nzoc2 arc characteristic for the theory based on finite differences Ax, At. It is not surprising that the results of observations with resolution An: and At in space and time should strongly depend on the Compton effect. A differential theory cannot yield such a rcsult without an additional assumption since it holds for the limits An: -,dx, At -,dt. Equation ( 1 8 ) has a first solution for L = m, which turns the exponential ramp function S ( @ ) ( l - e-le) in Eq.(ll) into a step function s ( @ ) . Hence, instead of E q . ( 1 3 ) we also have a solution

901 (C,

for @

@)= 0

= GO~[F(C) +u(C,@)]for


02O

(29)

We are Inore interested in the solutions provided by the terms in brackets it1 E q . ( l B ) :

There is a trivial solution eL = 1, L = 0 and a nontrivial solution eL= ( 1 1

For X2 << 1 or Ax

iX;)2 ++ x;/4

-

- eZVL, . cpL = arctg

< ~ t i / 2 e 4 ,we ~ get (P'

X2,

while X2 >> 1 or Ax >> ~ t i / 2 e $ , ~yields

L

-- i X 2

A2 1 - x;/4

74

2

MODIFIED DIRAC EQUATION

Equation (13) must satisfy the boundary condition of E q . ( l l ) . Equation (19) with A1 = l , or Eqs.(24) and (28), with A11 = 0, yield F ( 0 ) = 1. Hence we get:

The function u.(C, 6) has a homogeneous boundary condition, which is the goal of this method of solution. The initial condition of Eq.(12) yields with Eq.(13): QOl(C, 0) = Golu.(C,0) = 0 for

C >0

(35)

Wc turn t o the determination of u,(C,6) in Eq.(13). Substitution of IL(<, 0) for Qol(C,6) in Eq.(9) yields the same equation with QO1 replaced by u,:

Particular solutions us(<, 6) can be obtained by extending Bernoulli's product method for the separation of variables from partial differential to partial difference equations:

We observe that the notation @(C)$(O)instead of q5,(()$,(6) is a generally rrlade sirrlplification that we sliall use when possible. The substitution of u(C, 8) for u.,(C, 8) in Eq.(37) yields:

In analogy to the procedure used for partial differential equations we multiply with 1/4(()$1(6) and separate the variables:

2.3

SOLUTION O F THE DIFFERENCE EQUATION FOR

75

A constant - ( 2 ~ p , / ~is )writtcn ~ at thc cnd of Eq.(39) sincc a function of C can be equal to a function of I9 for any C and I9 only if they are equal t o a constant. The divisor N will permit the use of an orthogonality interval of length N rather than 1 later on. Equation (39) yields two ordinary difference equations for $(<) and $(O):

Equation (40) can be solved by means of the ansatz used previously for F(<) in Eq.(19): $(C) = A ~ v 4(C ~ , f 1) = A ~ v ~ ' ~

(42)

The following equation is obtained for v:

We shall need V Q , ~only for real values of the square root. Furthermore, we need v3,4 only up to the first order of At. The square A: under the square root is of order O(At)2 according to Eq.(2) and the relation Ax = cAt:

2

MODIFIED DIRAC EQUATION

for ( 2 - l r p , / ~ ) < ~ 4+

A? = 4 + o(A~)' (2-lrPK,l~)~

We use Eqs.(43) to (45) to write v$ and wi as follows, ignoring terms of order O(At)2 or higher:

~4 for ( 2 7 r p , l ~ )<

+

A? =4 (2-lr~K/N)

+o ( A ~ ) ~

1 ( 2 ~ p , / N ) ~ / 4 ] ~ / ~ 2eAmo,'cAt cp, = arctg ( 2 n ~ K / N ) [ , A1 = 1 - ( 2 - l r p , l ~ )12 ~ ti

(49)

The function $(() is written in complex form using Eqs.(47) and (48) with X1 and c p , defined by Eq.(49):

2.3 SOLUTION O F THE DIFFER.ENCE EQUATION FOR QzO

77

The boundary condition u(0,B) = 0 in Eq.(34) requires the following relation between and AQ1in Eq.(50):

Equations (50) and (51) will be used with a Fourier series. This requires that we choose cp, = cp,(p,) so that we get an orthogonal systerri of sine and cosine functions in an interval 0 5 ( 5 N = T / A t with a maximum of N periods. We must choose cp, as follows:

N ..., f-.27r 2 = 2 r ~ / N ,K , = O , f l , f 2 , . . . , f N / 2

cp,=O, cp,

f l . 2 r , f2.27r,

O<(LN=T/At

(52)

+

The variable cp, can assume N 1 values. There are N orthogonal sine functions, N orthogonal cosine functions, and a constant defined by K = 0. The function $(<) = con,stan,t is a solution of Eq.(40) for cp, = 2rp,/N = 0. In order to find the eigenvalues (2rp,/N)' associated with the angles c p , we must solve Eq. (49). With the relation 1+tg2cpn=-

1 c0s2 cp,

we obtain from Eq.(49) two solutions:

In order to see which solution to use we take the inverse of Eq.(55):

For srnall values of cp, we are back to Eq.(49). Hence, we use Eq.(55) to define p, as function of cp,:

2 MODIFIED DIRAC EQUATION 1-

0.8-

T

, 0.6m

0-

5 0.4-

0.2 0.5

1 (X2/7r)O

-

1.5

2

F1G.2.3-1. Plot of the excitation function @01(0,O) according to Eq.(60) for %ol=l.

p, =

N . cp, T 2

N T

TK

- sin - = - sin -,

N

K

= 0, f 1 , k 2 ,

. . . , f N/2

(57)

We want a plot of the boundary condition or excitation function of E q . ( l l ) . Substitution of L from Eq.(33) produces a complex function. Either the real or the imaginary component represents an acceptable boundary condition: Qol(0,Q)= GolS(Q)(l- cos X2Q) Pol(O, 8) = GO1S(Q)sin X2Q

(58) (59)

Thcsc arc causal cosinusoidal and sinusoidal functions which arc zcro for 0 5 0. One may readily derive step functions with a smoother step than S(Q) from them:

+

Q ~ ~ ( o ,= Q Qo1 ) { S ( Q ) ( l - cos x2e) S(Q- T / A ~ ) [ I -C O S ( X ~ QT)]) {S(Q)sin XzQ S(Q- 7r/2A2)[1 - sin(A2Q- ~ / 2 ) ] ) QO1(0,8) =

+

(60) (61)

Plots of Eqs.(GO) and (61) are shown in Figs.2.3-1 and 2.3-2. The tangent at Q = 0 is zero in Fig.2.3-1 and larger than zero in Fig.2.3-2. There is no jump or kink at (X2/.rr)Q= 1 in Fig.2.3-1 or a t (X2/r)6' = 0.5 in Fig.2.3-2. Boundary conditions with any other time variation than those of Eqs.(6O) or (61) may be represented by a superposition of these basic boundary conditions. We are now able to discuss the problem of a second initial condition mentioned in thc paragraph following Eq.(12). By not having imposed a second initial condition we did not obtain one function but a whole family

2.4

T I M E VARIATION O F

0.5

1 (Az/iT)O

FlG.2.3-2. Plot of the excitation function

@OI

@oi(C,Q)

-

1.5

2

(O,8) according to Eq.(61) for @ o l = l .

of functions consisting of the two shown in Figs.2.3-1 and 2.3-2 plus any superposition of these two. The first initial condition of Eq.(12) applied to 8 = 0. A second initial condition must apply to the adjacent time Q = 1. We may write it in the form

QOl(C11) - Qoi(C10) = d(C) (63) Comparing Eq.(63) with Eq.(1.3-1) we see that the "right" difference quotient has found an application that would have in a differential theory the equivalent dQO1(C,Q)/3Q = d(C) for Q = 0. The choice d = 0 leads to the plot of Fig.2.3-1, the choice d = +olX2/n to the plot of Fig.2.3-2, and a choice of d between -GolX2/7r and ++olX2/7r to a certain superposition of these two plots.

Equation (2.3-41) is the time dependent difference equation of 1(/(Q). The separation variable p, is defined in Eq.(2.3-57):

p, =

27rK

K = O , f l , f 2 ,..., f N / 2 N

80

2

MODIFIED DIRAC EQUATION

Substitution of $(@)= A2ve into Eq.(l) yields an equation for v

with the solutions

(3) T h e square root is imaginary for

We replace the sign obtained:

(

2

~

---

<

by an equality sign =. A quadratic equation is

+

~= ~ 2 -1(A: ~- A;) ~ A$)/4 f 2(1

+~2/4)l/~

2 - (A:' -A; + A Z ) / 4 f ( 2 + ~ ; / 4 ) , A; << 1 4 - (A: - 2A; A:)/4 for (2 $14) -(A: A314 for - (2 $14)

+

+

+ + +

(5)

According t o line 3 of this equation as well as Eqs.(2.3-2) and (1) the following two conditions must be satisfied to make the square root of Eq.(3) imaginary:

These two inequalities may be rewritten in the following form:

2.4

T I M E VAR.IATION OF

@01(<,8)

In addition a second solution is defined by Eq.(5):

The relation of Eq.(8) is always satisfied since the range of interest of (2rp,/N)' = 4 sin(cp,/2) is restricted as follows:

However, the conditiorl of Eq.(7) may not be satisfied. We must analyze both the real and the imaginary root of V 5 , 6 . Consider the imaginary root first:

The equation for tgp, can be simplified by means of the following identity

2

MODIFIED DIRAC EQUATION

F1G.2.4-1. Plot of

P,/T

acccording to Eq.(14).

F1G.2.4-2. Plot of P,/K according to Eq.(13) in the neighborhood of p , / ~ = 0 for (A; - A; A:)/4 = 0.005, A; = 0.

+

which is solved for l/cos2 0,. The substitution of 4 sin2(q7,/2) for ( ~ T ~ , / N ) ~ according t o E q . ( l ) yields with some effort:

For small values of A: - A;

+ A:

and X i onc obtains:

Figure 2.4-1 shows a plot of this equation.

-1

-0.98

-0.96

-0.94

-0.92

0.92

0.94

0.96

0.98

1

P&/T A

+

a

FlG.2.4-3. Plots of P,/x according to Eq.(13) close to +1 (b) for (Af - A; A;)/4 = 0.005, A; = 0.

+

cp&/~ =

b -1 (a) and

(pK/n=

The plot of Fig.2.4-1 does not apply close to p,/x = 0, f1 where the simplification (A?- A'$ + ~ : ) / 4<< 1 is not justified since sin2(p,/2) approaches zero. For p,/x equal to zero we get from Eq.(13): 1 - -- arcsin ( 1 - [l - (A: " '

7r

ll

= 0.0225 for

(A: - A;

-

A:

+ ~ : ) / 8 ] ~ ) l ' ~p,,

+ ~ : ) / 8= 0.005, A;

= 0,

A: << 1

=0

(15)

Figure 2.4-2 shows a plot of P,/x for (A? - A'$ + X:)/4 = 0.005 and A; = 0 in the interval -0.04 < p,/n < 0.04 according to Eq.(13). The function P,/x does not become zero for p,/x = 0 but assunies two values defined by Eq.(15):

For p,/x = 1 we obtain from Eq.(7):

+

for (A: - 2 ~ ; ~ : ) / 4= 0.005, A; = 0

(17)

Plots of P,/x close to p,/x = 1k1 are shown in Figs.2.4-3a and b according to Eq.(13).

84

2 MODIFIED DIRAC EQUATION

From Eq.(4) on we followed the solution of Eqs.(2) and (3) for an imaginary root. We turn now to the solutions for a real root. To this end we rewrite the solution of Eq.(2) in the following form:

In Eq.(2.3-25) we had mentioned the conditions X3 >> 1, A l l X2 << X 3 We recognize that the root in Eq.(18) will be real for these conditions. The value of X1 is actually unimportant. We obtain for v7 and us:

We recognize that the term e-iA2/2 in Eqs.(lO), (19), and (20) remains X;)/4]*l. Hence, unchanged but e*P- is replaced by [2- ( 2 ~ p , / N )-~ (A: >> h,/rnoc. We we get very different results for Ax << h,/m,oc and A.?: are, of course, not particularly interested in results for Ax >> h / m o c and Ax << h,/rn,oc, but we are interested how the solutions change close to Ax = h,/*rnoc. This goal is achieved here by using the solutions ug,6 of Eq.(lO) with small values of P,, but it will be worked out in much more detail later on. The functions can be written in the form

+

and the function $(8) of Eq.(l) becomes:

The particular solution u , ( C ,8) of Eq. (2.3-37) becomes with Eqs. (2.3-51), (21), and (22):

2.4 TIME VARIATION

uK,(C,8) = {A1 exp[-i(X2/2

OF

@01(<,8)

+ P,)Ql

+ ~z

exp[-i(h/:! - P K ) eixlClz ~] sin c p , ~ (24)

The solution t~,(<,8) is usually generalized by making A1 and A2 functions of a real variable K and integrating over all values of K, from zero to infinity. This would imply non-denumerable many oscillators. It is usual in quariturrl field theory to reduce the non-denumerable many oscillators to denumerable many, using box normalization to accomplish the reduction. We follow the spirit of this approach but (a) the box normalization is avoided since it is not based on a physical principle but only on the success of the calculation and (b) the denumerable many oscillators are replaced by a finite number since we have no means to produce or observe more. We use the arbitrarily large but finite time and space intervals 0 5 t 5 T and 0 5 z j 5 cT of Eq.(2.3-1). Equation (24) is not generalized by a Fourier integral or series but by a finite sum (Harmuth, Barrett, Meffert 2001). The notation K > KO in the following equation means the smallest integer larger than -KO while K < K,O means the largest integer smaller than K,o :


=

n>-r;o

{A1( K ) exp[-i(Xzl2

+ PK)Ql

+ A2(r;,)exp[-i(X2/2

-/ 3 , ) ~ ] ) e " ~ sin~ cpKC ~ / ~ (25)

The surnrnation is symmetric over negative and positive values of K , while the differential theory yields typically rion-symmetric sums over positive values of K. Substitution of Eq.(25) for u.(<,0) in Eq.(2.3-35) yields:
eix1Cl2sin cp,< = O

A second equation for Al (K) and A ~ ( Kcomes ) from the boundary condition of Eq.(2.3-63). We choose d = 0: q 0 1 (C, 1) - q 0 1 (C, 0) =

Frorn Ep(2.3-13) and (2.3-33) we get

0

86

2 MODIFIED DIRAC EQUATION

and we obtain with Eq.(25)

where F(<)is defined by Eq.(2.3-24). To solve Eqs.(26) and (29) for Al(n) and Az(K)we use the Fourier sum i11 the form of the sine series:

The factor e"1C/2 in Eqs.(26) and (29) can be taken in front of the summation sign. With cp, = 27~rc,/Nthe substitution of Eq.(30) into Eqs.(26) and (29) yields:

A1

(e- i(A,/z-p=) - l ) + -

-3 N

0

N =

-

j N

2 N

0

l)

27TKC F'(<)e-"1C/2 sin -dc N

= - i h 2 I T ( ~ , / N ) (32)

2n~C F ' ( ~ ) e - ~ ' l ~sin / ~-d< N

1 0

27T4 exp[-(A: - ~ : ) ' / ~ < / 2sin ] -d< N

A plot of IT(&/N) is shown in Fig.2.4-4 in the interval -0.5 5 K I N 5 0.5 for N = 100, X2 = 0, and Xg = 0.1, 1, 10. The same plots are shown expanded in the interval -0.05 5 K I N 5 0.05 in Fig.2.4-5. The solutiori of Al(6) and A 2 ( ~in) Eqs.(31) and (32) may be written in the following form using Eq.(33):

2.4

T I M E VAR.IATION O F

qo1([,6)

Fr~.2.4-4.Plot of I T ( K , / N ) ,~ O I T ( K / Nand ) , 1 0 0 I ~ ( n / Nfor ) A3 = 0.1, 1, 10 and N = 100, Xg = 0 according to Eq.(33) in the interval -0.5 5 K,/N 5 0.5. 0.2 0 .1

.:

..... ..,Aj=O. .. .'.

-;

__.-@\. ..... ,+,'<-,-. .....

->%

.

: \Q\,,

--k@i - 0 . 0 2 // ....

,/":

. . . ...,,.

./*'

";
.........

10, I O O I T ( ~ / ~ )

0.02

KIN

,.' .._.... -0 .; _.-' ---.. . _---. ... ..__

-._----

1. I T ( A / N )

0.04 +

.

'' 3

.. ..

LW.2 .

F1G.2.4-5. Plot of I T ( K / N ) ,1 0 I ~ ( n / N )and , 1 0 0 1 ~ ( ~ / for N ) As = 0.1, 1, 10 and N = 100, A 2 = 0 according to Eq.(33) in the interval -0.05 5 n / N 5 0.05.

A1 (6)= -A2 (&) =

e i ~ 2 / 2=

2 sin /3,

" 2 ' ( " I N ) (cos hl + i 2 sin /3, 2

sin 3 ) (34) 2

Equation (25) becomes:

u,([, 0)

2

= _ ~ X ~ ~ " X Z / ~ ~ " ( X I C - X Z ~ ~ ) / ~IT(^/^) sin ,>--KO

sin ,b,

sin

N

(35)

We note that /3, does not become zero for cp, = n = 0 according to Fig.2.4-2 if the exact formula of Eq.(13) is used for P,, only the approximation of Eq. (14) yields zero. The substitution of F ( < )of Eq.(2.3-24) and of Eq.(35) into Eq.(2.3-13) produces the solution qol(<,Q) for the condition of Eq.(6):

2 MODIFIED DIR.AC EQUATION

FrG.2.4-6. Step function in the interval - N / 2 5 n 5 N / 2 with steps of width 1. The area under the step function is determined by the values of rc, from n = -N/2 1 to n = N / 2 - 1. The functional values for n = fN / 2 are not needed.

+

for 0 5 4 sin2((p,/2) < 4 - (A: - 2 ~ + ; ~:)/4

+

We turn to the case 4 sin2((p,/2) > 4 - (A: - 2Xi Xi)/4. It seenis that a new solution of Eq.(l) is required to close the gap between -N/2 and -KO or no and N/2. This is riot so. According to Eq.(2.3-2) the coefficients A:, A;, A: are of the order O ( A X ) or ~ O(At)2. Hence, I K 0 ! in Eq.(36) is of the order N/2 - O(At)'. The sum of Eq.(36) represents the area under the step function shown in Fig.2.4-6. If the area runs from K = -N/2 to K = +N/2 the first and the last amplitude needed are for n = -N/2 1 and K, = N/2 - 1. Hence, the summation signs in Eq.(36) may be replaced:

+

and there is no need for a solution for I K , ~ 2 N/2. To extend our solution from V5,6 of Eq.(lO) to 717,s of Eq.(l8) we define a constant that permits to write Eqs.(l9) and (20) in a simpler form:

pK

2.4 TIME VARIATION OF Qol(<,Q)

Equations (21) and (22) become:

Eqliatioris (23) and (24) are rewritten:

G ( Q )= { A , exp[(,% - iX2/2)Q]

+ A, exp[-(p, + i ~ ~ / 2 ) Q ] ) esin~ ~cp,<~ C / ~

(42)

Equation (25) requires some care since the summation according to Eqs.(8), (9), and Fig.2.4-6 runs from -N/2 + 1 to N/2 - 1. This is only a temporary difference since KO becomes N/2 - 1 in Eq.(37): N/2-1 G(C, Q) =

C

.P[(PK

K=-N/2+1

+ A,(.)

- &/2)Q]

exp[-(fi,

+ i ~ ~ / 2 ) Q ] ) sin e " p~,C~ ~ (46) /~

For 0 = 0 we get: N/2-1

il(<, 0) =

C

+

[ A ~ ( K )A2(n)leix1cl2sin v K < = o K=-N/2+1

(47)

A second boundary co~iditioncomes from Eq.(27): Go,(<,1) -

0, = 0

In analogy to Eq.(28) we obtain with the constant $01:

(48)

2

90

MODIFIED DIRAC EQUATION

+ C(C, I ) ] ,

\irol(C, 1 ) = 601[(1 - e-"')F(C)

\irol (C, 1 ) - $01 (C, 0 )

601[(I

o~~(c,o) C(C, 1 ) - C(C, O)] = 0

$OI(C,O)

+

=~

e-iA2)~(C) G,(C, 1 ) - C(C, 0) = - ( 1 - e - i x 2 ) ~ ( ~ = ) -iXzF(C) =

-

(49)

With E q . ( 4 6 ) we get

+~

2 ( ~ ) ( e - ( ~-~l)]e"x1C/2 + ~ ~ ~sin ~ p~,C) = - i X 2 F ( C )

(50)

where F ( C ) is defined by Eq.(2.3-24). The coefficients A ~ ( K ) and A ~ ( K ) are obtained with a Fourier sum according to Eq.(30) using p, = 2 r r c l N :

The integral I T ( ~ / N ) is defined by E q . ( 3 3 ) . The solutions A ~ ( K )and A ~ ( K ) may be written in the following form1:

Equation ( 4 6 ) becomes:

Substitution of F(C)of Eq.(2.3-24) and of E q . ( 5 4 ) into E q . ( 2 . 3 - 1 3 ) produces the solution: 'We use sh for the hyperbolic sine following Gradshteyn and Ryzhik (1980).

91

2.5 HAMILTONIAN FORMALISM A N D QUANTIZATION

\irol

+

= 601 [(I - e - i X 2 e ) ~ ( ~4(C, ) Q)]

( I - e-"'~') exp[-(X:

-

X ) 1 C/ 21eiX1C/2

2.5 HAMILTONIAN FORMALISM A N D QUANTIZATION Before we proceed let us briefly reflect that we want t o study the replacement of the differential calculus by the calculus of finite differences. This change is fully represented by Eqs.(2.3-5)-(2.3-8) or their separated versions in Eqs.(2.3-9) and (2.3-10) plus two equal equations for Qo2 and 903. The matrices in Eq.(2.3-4) are the same as in the differential theory and the coefficients X I , X2, X3 have the same physical content as in the differential theory. Hence, we have to develop the Hamiltonian formalism only for Eq.(2.3-9) to study deviations between the calculus of finite differences and the differential calculus. This difference equation has the same structure as the previously derived equation for the Klein-Gordon equation (Harmuth and Meffert 2005, Sec.2.3, Eq.2). Hence, we start with the same term Tooof the energy-impulse tensor used there1:

If 9*9has the dimension J / m or VAs/m in electromagnetic units we obtain the required dimension J/m3 for Too. A planar wave propagating in the direction x calls for V = d/bx. The Fourier sums in Eqs.(2.4-35) and (2.4-36) permit an arbitrarily large but finite time T and a corresponding spatial distance cT in the direction of x, using the intervals 0 5 t 5 T and 0 5 x 5 cT. In the directions y and z we use the intervals -L/2 y 5 L/2 and -L/2 5 z L/2 with L = cT. The energy U of the wave 9 = QO1, 'ZfO2, Qo3, Qo4 according t o Eqs. (1) and (2.3-5)-(2.3-8) in this interval equals

<

U=

a@*a9 +--+ae*aq ~2 a t a t ax ax

1 ---

<

)

]

mac2 9*9 dx dy dr fi2

'~erestezki, Lifschitz, Pitajeski 1970, 1982; 5 10, Eq.10.13

(2)

92

2 MODIFIED DIRAC EQUATION

The dimension of U is VAs. We rewrite U with the normalized variables 19 = t/At, C = x/cAt of Eq.(2.3-1):

We colild replace the differentials 8 9 / 8 0 and dQ/d< by finite difference operators i 9 / & and &@/A<. The integral should then be replaced by a sum according t o Eq.(1.3-20). We sidestep this investigation on the grounds that more urgent problems have to be addressed at this stage of development of the finite difference theory. The reasoning may turn out to be wrong, but there is always more one should do than one can do. The differentiation of q(C,O) = 9ol(C,8) of Eq.(2.4-36) - using the sunirnation limits of Eq.(2.4-37) - with respect to 0 or C produces the following results with GOl = @:

N/2-1 n=-N/2+l

'T(K'N)

(A2 sin ,&B

+&,i2

~TKC

cos L O ) sin - (5) N

2.5

93

HAMILTONIAN FOR.MALISM AND QUANTIZATION

The first term in Eq.(3) becomes:

+ p, cos X2(02+ 1) N/2-1

cos p,O sin 2rKi) N

'1

The second term in Eq.(3) assumes the form:

-

N/2-1 2X2 sin(X2e/2) exp[-(A: - ~ ; ) ' / ~ < / 2 ] sin p,e sin p, K=-N/2+1 X1 ~ T K < x ([A1 cos(h2/2) (Xi - A:)'/~ sin(X2/2)lT sin N

+

N/2-1

- sin p,O sin -

N

N/2-1 n=-N/2+1

-sin p,B cos N

(6)

2

94

MODIFIED DIRAC EQUATION

The third term in Eq.(3) becomes:

I

= G2 2(1 - cos X2B)exp[-(A: - x ~ ) ~ / ~ c ] - 4X2 cos(X2/2) sin(h28/2) exp[- (A:

N/2-1 K=-N/2+1

-

~;)'/~
IT'"'^) i n ~8 sin sin p,

N

Equations (8), (9), and (10) have to be substituted into Eq.(3). The integration with respect to requires time (Harmuth and Meffert 2003, Sec.6.8). We separate the energy U of Eq.(3) into a constant part Uc and a time variable part U,($) that consists of sinusoidal functions of 8 and has the time average zero:

<

u = uc+ u,(e)

(11) The following three components of Uc are obtained from Eqs.(lO), (8), and (9):

2.5

HAMILTONIAN FORMALISM AND QUANTIZATION

, L2 -4 c a t ':IN';

N/2-1

K=-N/1+1

sin p,

The sum of Eqs.(12)-(14)yields Uc:

Only Uc is needed for the Hamilton function 3C since the average of the time-variable energy U,(Q) is zero. The energy Uc is normalized:

1 d ( ~=) 4

- A;

(KIN) + x ; ) ~ / IT ~N sin p,

We may write the components of 3C, in Eq.(17) as follows:

d ( ~ , ) 2nn8 - i cos 2 ~ ~-(sin d (8n )) %, = (27rn)2 -(sin 27rn 27rn = -27ri np, ( 8 )q, ( 0 ) The functions p,(8) and gK(8)have the form:

2nn0

+ i cos 27rn0) (19)

2

MODIFIED DIR.AC EQUATION

2 ~ n d (e ~2 7)r i ~ ~ ~ ~ ( =0 )

(20)

v"%x

e2T,iKe ,) p. , ApK = - - 2n~d(~

do

s

2(e2rri~Ae- e - 2 ~ i ~ A t)? 2 ~ e

- 2nnd(n) ezniKe i sin 27rnAO

G

A0

. dq, 2nnd(r) e-2,i,6 qK,= -= --

no

s

=PK

isin27rnAO . = 27rir;,pK (21) Ae

i sin 27rnAe A0

isin27rnAO = . -27r?:~q, - -qn

A0

The finite derivatives A3C,/AqK and Ax,/ApK have relations of the same form as known from the differential theory:

Ax,

--

- -2~inq, = +qK

APR,

For the quantization we follow as closely as possible Becker (1963, 1964; vol. 11, § 51, 52) in a previously used form (Harmuth, Barrett, Meffert 2001, Secs. 4.3, 4.4; Harmuth and Meffert 2003, Secs. 2.3, 2.4; 2005, Sec. 2.5). This form can be used both for the Heisenberg and the Schrodinger approach. We use the definitions

and rewrite the Hamilton function X of Eq.(17):

2.5 HAMILTONIAN FOR.MALISM A N D QUANTIZATION

97

We consider the Heisenberg approach first. The functions b , and b: are replaced by operators b; and b::

One may interchange b: and b,:

The two ways of quantization are a well known ambiguity of the theory (Becker 1963, 1964, vol. 11, $52; Heitler 1954, p. 57). If we first use Eq.(29) we obtain from Eq.(27):

The eigenvalues E, become (Becker 1963, 1964, vol. 11, § 15):

On the other hand, Eq.(30) yields

and one obtains the eigenvalues

We use the Schrodinger approach because it leads to differential equations and we have developed the mathematical methods for their replacement by difference equations. We do not want to develop additional mathematical methods that are not absolutely required at this time. The differential operators of Eq.(29) are replaced by difference operators:

The products b$b; or b;b$ according to Eq. (1.3-6):

produce a second order difference operator

2 MODIFIED DIRAC EQUATION

We substitute into Eq.(31) and apply the operator to a function a :

To solve Eq.(37) we make first the substitution

This sribstitution produces a difference equation with a variable coefficient

I:

The further srlbstitution2 Q,

= e-~2/2

x(E)

leads to the equation

One more substitution

x

= (/A(,

A< = 1

(42)

simplifies this equation t o the following explicit form:

This is a difference equation with variable coefficient x. It can be solved by a factorial series of the second kind:

2See Harmuth and Meffert (2003, Sec. 3.6) for details of the calculation.

2.6 FINITE LIMIT OF THE PERIOD NUMBER

99

K

The cocfficierlts bj are determined by a recursion formula with three terms:

A polynomial solution with highest term b, exists if the following two conditions are satisfied:

b,+l =

n

+ + 1{[ax, - 1

(n.

-

2(n - l)]b,-l

+[axK- 1 - 3(n + l)]b,)

=0

(46)

Substitution of Eq.(47) into Eq.(37) yields the eigenvalues known from the differential theory or from Eq.(32) of the Heisenberg approach, except for the limit3 N of n:

The recursion formula of Eq.(45) is a difference equation of second order. Hence, two of the coefficients bj in Eq.(44) can be chosen. Our first choice is bo, but we leave the value of bo undefined so that it remains available for norrnalixation. The second choice is not bl but b,+l = 0, which is Eq.(46). The eigenvalues of Eq.(48) are the same as in the differential theory except for the finite value of N. The associated eigenfunctions in the differential theory are the Hermite polynomials that are defined by a recursion formula with two terms. Equation (45) is a recursion formula with three terms. The eigenfunctions are thus different from the Hermite polynomials. 2.6 FINITE LIMITSOF THE PERIOD NUMBER.K, In a differential theory the period number K defined in Eq.(2.3-52) would run to fcm.When we plot something as function of K we are always far removed from the limits fm. In a difference theory we can plot right up to the limits K = rtN/2. However, in Eqs.(2.5-21) and (2.5-23) we used the approximation sin2nnAO = ~ T K A Oto obtain the Hamilton relations of Eqs.(2.5-24) and (2.5-25). According to Eqs. (1.3-3) and (1.3-4) plus the text in-between we have A0 = 1 / N even though we generally use A6 as abbreviation for AO' = 1. To satisfy the condition 2 n ~ A o<< 1 we must 3 ~ h e r eis no upper limit N in Eqs.(32), (34), (47), and (48) if there is no upper limit N in Eq.(27).

100

2 MODIFIED DIR.AC EQUATION

restrict K to K << N/2. This corresponds to a condition K << co in a differential theory where there is no limit on N. Since we carlrlot rnake plots which do not satisfy the condition K << co we are usually not aware of it and do not discuss it. In order to see the effect of plotting for too large values of K we start with the energy U c K ( ~for ) the period number K that was obtained in Eq.(2.5-15):

x (

sin p,

A?

+ x:

The energy of a photon with period number given by Eq. (2.5-48) :

The average value of

moc2At

) 2 ~ : + T + ( $ ) 2 + ( ~ ) 2 ]

E,, for all N

K

+ 1 values of

For a specific value of n one nccds U,,(K)/E,, UcK(K,):

and a certain value of n is

K

equals

photons to achieve the energy

2.6 FINITE LIMIT O F THE PERIOD NUMBER

K

101

If photons with various values of n are equally frequent we obtain instead of Eq. (4) the following relation:

The total number of photons equals the sum over n in Eqs.(4) or (5) from K > -no to n < K O , where KO is defined by Eq.(2.4-7). The notation n > -KO means the smallest integer n larger than -60, while K < KO means the largest integer smaller than KO:

The probability pa(&) of a photon with period number n is the same for Eqs.(6) and (7) since the constants D,, and D,drop out:

The negative values of K, are caused by the existence of a sinusoidal and a cosinusoidal function for each K > 0. We do not make such a distinction for photons. Hence, we leave out the negative values of K,. A further correction is required for K = 0. According to Fig.2.4-2 there is no unique value of / 9 , for n = 0. This is of no concern since the function IT(n/N) according to Figs.2.4-4 and 2.4-5 is zero for n = 0 and we may leave out K = 0 in the sum of Eq.(8). A photon with period number n = 0 could not produce interference effects. Wc let K, run from 1 to K < KO in Eqs.(6) to (8). The following approximations are used:

KO =

15 n

N

- arcsin (1 - ~ ; / 1 6 ) l ' ~ ,Eq.(2.4-7) lr

< K,O (<no means largest integer smaller than KO)

(9)

2 MODIFIED DIR.AC EQUATION

Equation (9) is solved for X3:

X3

5

= 4(1 - sin2 ~ =4

A plot of

rc~/iV)~/~

for KO/N<< 1

is shown in Fig.2.6-1 for the interval 0 also show the inverse plot tcO/N in Fig.2.6-2:

(I6)

< no/N < 0.1. We

We turn to Eq.(l) that connects the space and time differences Ax, At with X3 arid write them for = 4 of Eq.(16):

The theory yields an upper limit for Ax, At because we are using the solution 4 1 5 , ~of Eq.(2.4-10) and not V ~ , of S Eq.(2.4-18). In order to have a standard for comparison we plot in Fig.2.6-3 the probability p ( ~ of ) a photon with a period number K according to the differential Klein-Gordon equation for K = 1, 2, . . . , 100 (Harmuth and Meffert 2003, Eq.4.5-14):

In Fig.2.6-4 we plot pa(K) of Eq.(14) for the Dirac difference equation using N = 10000, K = 1, 2, . . . , 100, X3 = 0.001, KO = 499.9. The plots of Figs.2.6-3 and 2.6-4 are very similar. Figure 2.6-5 shows that the plot changes significantly if X3 is increased to 0.01. This change bccomes even more conspicuous for X3 = 0.05 in

2.6

FINITE LIMIT O F THE PERIOD NUMBER

K

~olN

+

F1c.2.6-1. Plot of

according to Eq.(16).

F1c.2.6-2. Plot of K,o/N as function of

lo-'

according to Eq.(17).

I-. '

F1G.2.6-3. Probability p ( n ) of a photon with period number n according to Eq.(19) of the differential Klein-Gordon equation.

2

MODIFIED DIRAC EQUATION

K. --+

F1G.2.6-4. Probability pa(&) according to Eq.(14) for N = 10000, n = 1, 2, 100 , A3 = 0.001, Kd, = 499.9.

... ,

F1G.2.6-5. Probability pa(n) according to Eq.(14) for N = 10000, K, = 1, 2, . . . , 100, A3 = 0.01, no = 499.2.

2 ~ 1 0 - ~ ' ~ .' ' 20

'

"

40

"

60

'

80

"

100

K +

F1G.2.6-6. Probability pa(&) according to Eq.(14) for N = 10000, n = 1, 2, 100, A3 = 0.05, no = 496.02.

... ,

2.6

F I N I T E LIMIT O F THE P E R I O D NUMBER. K

0

20

40

60 n

80

100

i

F1G.2.6-7. Probability p a ( & ) according to Eq.(14) for N = 10000, K. = 1, 2, 100, Xg = 0.1, ICO = 492.04.

....

FlG.2.6-8. Probability pa(^,) according to Eq.(14) for N = 10000, IC = 1, 2, ... 100, A3 = 0.5, no = 460.1.

' b ' 20" " 40" L "60

80

,

100

IC-

FlG.2.6-9. Probability pa(r;) according to Eq.(14) for N = 10000, r; = 1, 2, . . . . 100, A3 = 1, K,O = 419.5.

2

MODIFIED DIRAC EQUATION

.*.*....."..._...*...*.-

2~10-~-

......................... ........ ...........

10-3 -

[ -;

5~10-~-

Y

R

2 ~ 1 0 -- ~ -

5 x 1 0 - ~-

:

.

-..

~ X I O - ~ 0

........ ..-..--

,

,

20

,

,

,

40 K

,

60

,

,

80

,

,

100

--+

FIG.^.^-10. Probability pa(&) according to Eq.(14) for N = 10000, K = 1, 2, . . . , 100, A3 = 2, K,, = 333.3.

L d 20 " " "40 ~ ' 60 " K -+

80

100

FIG.^.^-11. Probability pa(&) according to Eq.(14) for N = 10000, K, = 1, 2, 100, A3 = 3, KO = 230.05.

....

F1G.2.6-12. Probability pn(n) according to Eq.(14) for N = 10000, n = 1, 2, . . . 100, A3 = 3.8, no = 101.08.

,

2.6

FINITE LIMIT OF THE PERIOD NUMBER K

107

Fig.2.6-6. A further increase of X3 to 0.1, 0.5, and 1 is shown in Figs.2.6-7 to 2.6-9. Plots for still larger values of X3 equal to 2, 3, and 3.8 are shown in Figs.2.6-10 to 2.6-12. The value X3 = 3.8 is essentially the largest value for which we can plot pa(K) in the interval K.= 1, 2, . . . , 100 because K,O drops to 101.08. Smaller values of KO would require a smaller upper limit of K. For X3 = 3.8 we get from Eq.(18) Ax = 1.91i/m,oc. Figures 2.6-10 t o 2.6-12 look very similar but there is a change of scale for pa(^) by a factor 10 between Figs.2.6-10 and 2.6-12. We see from Figs.2.6-4 t o 2.6-12 that for small values of X3 or Ax, At we obtain a probability function that is similar to what we are used to from the differential theory. But for larger values of X3, when Ax becomes of the order of the Compton wavelength, one obtains quite different results. We do not extend the results of this section to values of Ax larger than the Compton wavelength. Chapters 2 and 3 are intended to develop mathematical methods for the finite difference theory and t o show that there are significant deviations from the differential theory. Chapters 4 and 5 will derive more detailed results for assumptions of more physical significance.

3 Inhomogeneous Dirac Difference Equation

We start frorri Eq.(2.2-33), taking A,, Aefc3, and q5m from E q . ( 2 . 2 - 3 4 ) . The terms Amo,, , q5,0, Lcxlj to Lcx5, are defined in Eqs.(2.2-29) and (2.2-35)(2.2-39) :

1

- -Lcxk:j CYC

Lcx~:, = Lcxlj

- e%, Aml:c3

+

QXofcj ( s j , t )

+ Lcxzj + Lcxsj + Lcxlj + Lcxsj

(1) (2)

For the transition to riornialized variables 0 and Cj we nlake the substitutions

arid obtain

3.1 INHOMOGENEOUS EQUATION (2.2-33)

109

The subscripts of QxoCjand QXlCjare reduced in order t o shorten the notation:

We get from Eqs.(2.4-36) and (2.4-37) with a constant cation j = 1, Cj = C:

and the simplifi-

We shall not only be interested in the case Ax << h/moc but also in @xOc, and QxlG in Eq.(4) are replaced Equation (6) is replaced in this case by by ikXOCj= Go and @xlC, = Eq.(2.4-55); $o is a constant:

Ax >> h/moc. This requires that

The horrlogeneous part of Eq.(4) may be written with the help of Eqs.(2.2-32) and (2.3-2) in the following form:

3

INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

The second and third line in Eq.(4) yield:

If Ax is significantly larger than the Compton wavelength Xc = h;/fmocwe may ignore the second term in brackets on the right side of Eq.(9). We then get a second order difference equation according to the first term on the right side of Eq.(9), otherwise we get a third order difference equation according to the whole right side of Eq.(9). With the help of Eq.(2.2-16) we may rewrite the first term on the right side of Eq.(9) as follows:

An: > h h o c = Xc

3.1 INHOMOGENEOUS EQUATION (2.2-33)

111

For the other extreme, Ax << h/moc, we ignore tlie tern1 1 on the right side of Eq.(9) and use only the second term which is multiplied by ti/moc. With the help of Eqs.(1.3-5)-(1.3-7) we get: An: << h/rn,oc = Xc

The term LCxk,j i11 Eq.(4) assumes the following form with the help of Eq.(2) for j = 1:

We obtain LCxl1 to LcxRl from Eqs.(2.2-35)-(2.2-39)by writing j = 1, x j = x, &by = 0, and d / d z = 0:

x

(i

An:

-e

m

}

(13)

The terms in lines 3 and 4 can be written in reverse order since this requires only that (-ifidldn: - eAmo,) commutes with the constant 1 or itself. We obtain for x = Ax(, Ax = AxA(, A( = 1:

112

3

x

A (-4

INHOMOGENEOS DIRAC DIFFERENCE EQUATION

ieAx

-T~mor)

[I

ti + (--) mocAx

2

($ "

ieAx

- TAmo.

This is the first term of Lcxk:j in Eq.(4). We may again distinguish between Ax > h,/moc and Ax << hlmoc:

.Ze2 h (AeozAmoy - Aeoy Amoz) mo m,ocAx

= 2--

x

( 2G o + 1 , )- 0 1

Lcx,,Qo(<, 0) = i

=2-

g

ieAx

-, ) I-~

~

(A) 3(Ae~zAmoyAeoyAmoz) A ieAx x (-=&--ti A,,)

Fi (m,ocAx) -

0

:

(15) o

-

3

QO(C,Q)

( ~ e o z ~ r n o-y~ e o ~ym o z )

ieAx -3-Amo:c[Qo(C+1,0) ti

-2Qo(<,O)+Qo(<- 1,0)]

of Eq.(l) is zero according to Eq.(2.2-36). The next Thc tcrm term Lcx31 according t o Eq.(2.2-37) requires work. The differences A/AX and A/Ay are zero but the constants Amo,, AmoylAmozdo not have t o be zcro. We obtain with x = (Ax arid t = @At:

)

3.1 INHOMOGENEOUS EQUATION (2.2-33)

+

+

We sketch only how Eq.(17) can be written in terms of Qo(C 1 , 8 1) . . . i o ( <- 1,O - 1) since the factor Aeo,(Amo,c-AmOy)will eventually be chosen equal to zero. We expand first

+

The operation d/de requires that 8 in Eq.(18) is replaced by 8 1 to form a first expression and then by 8 - 1 to form a second expression. The second expression is subtracted from the first one and the whole result is multiplied by 112. The tern1 Lcx41 in Eq.(l) is easy since it equals zero according to Eq.(2.2-38). The final term LcxS1 is defined by Eq.(2.2-39). The differences

114

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

A/& and A/& are zero and all terms multiplied by them are removed. This means primarily lines 1-4 of Eq.(2.2-39), but lines 5-8 vanish with lines 1-4. Only a summation constant is left:

The factor m,oc was chosen to make CcxBl dimension-free [See Eq.(3.2-4) in the following section for explanation]. With the help of Eqs.(7), (8),(lo), (15), (17), and (19) we may rewrite Eq.(4) for large values of Ax. We note that Qo must be replaced by $0 of Eq.(7) for large values of Ax. The constant CCxjl is replaced tiy ccXs1 and Ql by G1:

We have not expanded line 10 since the choice Aeoz = AmoZ = Amo, = 0 will eliminate this line.

For small values of Az we may rewrite Eq.(4) with the help of Eqs.(6), (8), ( l l ) , (16), (17), and (19). The terms Q0 of Eq.(6), Ccx51of Eq.(19), and Q 1 of Eq.(8) are used as they are without a hat*:

We are going to investigate the case Ax << h/,moc. The starting point is Eq.(3.1-21). It needs to be simplified before we can attempt a solution. The choice

116

3

INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

will eliminate lines 6 to 11 in Eq.(3.1-21) hilt retain lines 3 to 5 since A,(<, 0) is not zero. Lines 12 and 13 are also retained. We multiply the reduced Eq.(3.1-21) with 2An:liti and obtain the following simplified equation:

x { ~ [ @ o ( ~ + 2 -2Qo(C+ ,0) 1 , 0 ) + 2 @ 0 ( ( - 1,Q) - l o ( (

eAn: ti

- 3i-Arn~:E[6~(C

-2,B)l

+ 1,0) - 2Qo(C,O)+ Qo(C - I,@)]

2Ax e Ge(C,@)= i-bocccx51 - -4el(C,o)l*o(C,o) ti C 2eAz -1:-4el (C, ~ ) Q(C, o 0) tic The constant mocCc,sl was added to the electric potential without changing the notation of the modified potential.

(4) #,I(<,

O)e/c

The term A,(<, 0) in Eq.(3) shows tlle i~riportanceof tlle rrlodificatiorl of Maxwell's equations. The original Maxwell equations do not have a vector potential A,. In this case Gp(C, 0) of Eq.(3) would not exist arid orily G,(C, 0) and Grn((,0) of Eqs.(4) and (5) would be left. Maxwell's equations without magnetic dipole current density rnay or may not work for A x >> Xc, but the magnctic dipole current density is certairlly important for A x << Xc. We substitute An: = cT/N and 2 ~ t i / , m , ~ =cXc in Eq.(3). Since A,((, 8) is the magnitude of A, it is never negative. We define Ae(<,O) free of a physical dimension. The same normalization is used for $ , 1 ( ~ ,0) and A,oc((, 0) obtained from Eqs.(4) and (5):

Xc An:

h XC - - N XC= p c ,rnZocAn: cT/N cT

3.2

RESOLUTION

2eAx -$el(CiQ) tic

AX < h,/moc

= ?el(C,%)

117

(9)

We need a general solution of the homogeneous Eq.(2) plus a particular solution of the inhomogeneous equation. If we write the homogeneous equation in the form of Eq.(2.3-4) we must write the inhomogeneous term in the form

3

118

INHOMOGENEOS DIRAC DIFFERENCE EQUATION

G ~ ~ (Q)c=, A m l c ( ~ , ~ ) ~ Q)o j ( ~ ,

(16)

The functions QOjare defined by Eq~~(2.4-36) and (2.4-37) if the subscript 01 is replaced by ~ j :

N/2-1

-i~~,ix~/2,i(xlC-x~e)/2 K=-N/2+1

IT(^") sin

sin ~ T K C N

The four homogeneous Eqs(2.3-5)-(2.3-8)for Qoj are replaced by homogeneous equations for Qlj plus the inhomogeneous terms fGpj(C, 6) i G e j ( C , Q) i G m , ~ - j ( C , 6):

+

The substitutions Q14 --+ 913 and all + 9 1 2 transform the homogeneous part of Eq.(18) into the homogeneous part of Eq.(19). An equivalent statement applies to Eqs.(21) and (20). We need to solve Eqs.(l8) and (21) only. In order to separate the functions Qll and q14in Eqs.(l8) and (21) we follow the procedure worked out in Section 6.1 and extend it to inhomogeneous equations in Section 6.2. Equation (6.2-8) is obtained for 911. We write it here in explicit form for but refer to Eqs.(l2), (6.2-14), and , d~l/d0: (6.2-18) for the very long explicit form of GI, A G ~ / A Cand

A general solution of the homogeneous equation plus a particular solution of the inhomogeneous equation is needed. The boundary condition of Eq.(2.3-11) is used again: @ 1 1 ( ~ , 8=) 3?11~(0)(1 - e-")

=0 = G l l ( l - e-")

for 0 < 0 for 0 2 0

(23)

In order to obtain a solution that satisfies the causality law we must specify two initial conditions Qll (C, 0) and Pll (C,1) according to Eqs. (2.3-12), (2.3-63), and (2.4-27):

Since the homogeneous Eqs.(22) and (2.3-9) are equal we may use for Qll(C, 0) the ansatz of Eq.(2.3-13)

arid then go directly to the solutions of Eqs. (2.3-24), (2.3-33), (2.4-35), (2.4-36), and (2.4-37) for the solution of the homogeneous equation:

120

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

F(C) = exp[-(A: - ~ i ) ~ / ~ < / 2 ]A:e ~ A; ~ ~>~O / ~ , L'ZX2, 21,K,

X2<1

(c, 0) = - ~ x ~ ~ ~ X Z / ~ ~ ~ ( X ~ C - X ZsinBpKQ ) / ~Sin ~TKC sin ,i?, N

(27) (28) (29)

In order to find a particular solution of Eq.(22) we use again the ansatz of Eq.(26) but write C(C, 0) instead of u(C, 6):

For F(C) and L we use Eqs.(27) and (28). The boundary and initial conditions for .i,(C,O) are the same as for u(C, 0) in Eqs(2.3-34), (2.3-35), and (2.3-63):

Substitution of 4(C, 0) for Qll in Eq. (22) yields the same equation with Qll replaced by 6,:

We write the solution of Eq.(36) in the form

where u.(C, 8)) Eq.(30), is the solution of the homogeneous equation and G(C, 8) a particular solution of the inhomogeneous equation. The solutiori u,(<, 8) was obtained with the help of Eqs.(2.3-37), (2.3-50), and (2.4-23). The approximation P, = cp, of Fig.2.4-1 is used:

$,(o

e"15/2 ( ~ ~ ~+A31e-i' ~ i +'"vC) ~ = Ce i x 1 C / 2 ( ~cos 3 2 cp,C+A33 sin p,C) cp, = ~ T K / N , ~ , = 0 &I, , ..., f N / 2 (39) +K,(<)= e - " X ~ @ / 2 ( ~ 4 0 e + - iA ~ - ~@ ~ ~ ~ P - @ )

- e-iX2@/2(A42 cos p K 8+ iA43 sin p,O) -

(40)

We turn to the inhomogeneous solution 6(C,8). Let 6,in Eq.(36) be replaced by 6. In analogy t o Eqs.(38)-(40) we obtain:

+

c , ( c , 0) = i K ( ~ ) 4 K (= e )eix1C/2(~32 cos p,C ia33sin p,[) x e-i'2e/2(~42cos cp,O - e-"'20/2(a32A42cos p,O

+ ia32A4ssin

- e - " X 2 / 2 ( ~ 3 3 ~sin 4 3 cp,O

-

+ iA43sin p,8) cos p,C

I : A ~ ~cosA ~ ~

sin cp,<

I11 the spirit of the method of the variation of the constant we make the constants and functions of K and write 6(C, 0) in a shorter form:

2n~C 2TK5 6,({, 8) = ~ , ( $ ) e ~ ' cos ~ ~/~ ~ , ( $ ) e ~ ' lsin ~ /~ N N

+

(41)

3

122

INHOMOGENEOS DIRAC DIFFER.ENCE EQUATION

For a more general solution1 7j((,Q) we may sum over all values of K. To shorten a riurnber of formulas we use the notation 4,(C) and q5,(C) for e " ' ~ c / cos(27r~C/N) ~ and eixdP sin(27r~

6(C, 0) = K>-KO

+

27r K,( 27rKC ( ~ , ( @ ) e ~ * cos ~c/ ~N ~ , ( @ ) e " " c /sin ~ N

Substitution of 6(C, 8 ) for b(C,0) into Eq.(36) yields:

With the following expansions for

4,

and

4,

4c(C + 1) =

27rK sin -sin N N

4s(C + 1) =

f sin -cos N N

and Eq.(42) we obtain first and second order differences of

4,

and 4,:

' w e use a method known for partial differential equations; see, e.g., Smirnov (1961), vol. 11, 3 169.

AX << h/moc

3.2 RESOLUTION

&(C

123

X1 27T~ 2 ~ + isin cos -cos 2 N N

+ 1) - &(C - 1) = 2eixlC/'

~

5

N

]

X1 27rK. . 27rKC 2 (48) 27rnC cos cos 1 sin &(C + 1) - 24,(C) + 4,(< - 1) = 2eix1C/2 N

" J N YTK

- i sin - sin -sln -

[(

Equation (43) is brought into the following form:

X1

27rK

X1 2

.

27r~ N

+z sin - sin -cos -

N 27rKC - eixlCl2cos -[S, (8 1) - 25, (0) S, (8 - I)] N 27rK.C - e"1C/2 sin -[T,(8 + 1) - 2T,(8) + T,(0 - I)] N

+

1 A- A 4

-(

i

+ A ) (s,( 8 ) e i x 2c

= -[hiG4(CjQ) - (A2 - h3)Gl(Cj6)] -

4

+

27rK.C N

-+ T ( ) e i x 2i n

5

-1)

27TK.C N

N

124

3

INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

This equation may be simplified:

The left side of Eq.(51) is essentially a Fourier series or-better--a Fourier sum in terms of C. The right side may be represented in the same form:

~ or 2 ~ - ' e - ~ ' l C / ~x Multiplication with either 2 ~ - l e - ~ ' l C /sin(2xvCIN) cos(2nvCIN) and integration over the orthogonality interval 0 5 C 5 N yields G,,(O, K) and G,,(O, K):

3.2 RESOLUTION AX < h/moc

for v = K Equations (51) and (52) yield for every component following two equations:

-

-

)

-

-

2

r;

of the two sums the

27rK ( sin---cosh ; : h;) sin-S,(O)=G.,,(O.s) N

(56)

The following substitutions and sirriplificatio~lsare made:

XI

-

A3 = 2 ~ m ~ c ~T/ =~ NhA, t A, = O(At), A, = O(At), A, = O(At) cos(A1/2) = 1 o(A~)', (A1/2) sin(A1/2) = o(A~)' 1 + i ~ , / 2A eiA2/2, 1 - i ~ ~ / e-iX2/2 2

= 2cTeAmoZ/Nti, A2 = 2Te$,o/Nti,

+

A1 A1 A1 sin - - - cos - = o(A~), 2 2 2

The small value of the last line of Eq.(57) tempts one to leave out the fourth term with T,(O) in Eq.(55) and with S,(O) in Eq.(56). This is not justified mathematically since we do not know how large the sum of the remaining four terms--including the inhomogeneous term-is. From the standpoint of physics one would eliminate the coupling of the two processes represented by Eqs.(55) and (56). We may solve Eq.(55) for T,(O) and substitute T,(O), T,(O f 1) into Eq.(56) to obtain an ordinary, inhomogeneous difference equation for S,(O):

126

3

INHOMOGENEOS DIRAC DIFFERENCE EQUATION

+ 21:( sinh-; - ?-cos h;) G,,(O,

K)

We drop the last line since Gc,(Q,K,) and G,,(B, K,) are of order O(1) but sinX1/2 - (X1/2) cosX1/2 is of order O ( ~ l t according )~ to Eq.(57). The following simpler equation is obtained:

Similarly, we may solve Eq.(56) for S,(O) and substitute S,(O), S K ( 8 f1) into Eq.(55) to obtain an ordinary, inhomogeneous difference equation for TK (0):

- -eiA2/2~ ,,(0

27r6 + 1,K,)+ 2 cos G,, (8, K) N

-

e-iX2/2~sK. (8 - 1,K) (59)

The homogeneous parts of Eqs.(58) and (59) are cqual except for the exchange of S with T. In the inhomogeneous parts the subscripts c and s are changed. For the solution of the homogeneous Eqs.(58) and (59) we make the usual ansatz

and obtain an equation for v , :

ei~2 2

v , - 4 cos -2nlc, e"~/~v~, N

+2 -

4cos

27rn e - i X z / 2

N

v,

-1 + e-iX2

-2

-

v , -0

(61)

This is a fourth order equation. Thanks to its symmetry it car1 be reduced to two equations of sccond ordcr:

2 n ~ double root for N

= 2 cos -

(63)

The corlditiorl @ = ( A z e $ , o / ~ t z ) ~
which yields two single roots:

e1"/2zliK

= aos N 2aK

* (cos : 2

- 1)

112

128

3

INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

We obtain for v! in Eq.(60) the first two solutions:

The double root of Eq.(63) calls for two more solutions:

[ (- + - -\22)

= Q cos

Q-isin (

-+2

): ]

~

0

(70)

The general solution S ,(19)in Eq. (60) becomes:

For T, (0) we obtain from Eq.(60):

We turn to the inhomoge~leoussolution of Eqs.(58) and (59). A general method for this task goes back to Lagrange (1736-1813). It is discussed in the available books1 but we present it in Section 6.3 in more detail. The constants c,i and dni, with i = 1, 2, 3, 4, in Eqs.(71) and (72) are replaced by variables cni(0) and dGi(6):

'Norlund 1924, p. 396; 1929, p. 22, 125; Milne-Thomson 1951, p. 374

3.2 RESOLUTION

Ax << h/moc

129

Equations (6.3-23) and (6.3-30) define dKi(0)and cSi(0) in a form suited for corrlputer use. The solution derived here for Eq.(22) is extremely general. The only significant requirement is that the inhomogeneous term of Eq.(22) permits a cosine and sine trarisforrrlation in terms of C according to Eqs.(53) and (54). Convergence of the Fourier series of Eq.(52) is not required since K, does not approach infinity but only i ( N / 2 - 1). Substitution of Eqs.(73) and (74) into Eq.(42) yields the particular solution of the irihomogeneous equation (36). We must still satisfy the boundary and initial conditions of Eqs.(33) to (35). Using Eq.(37) we obtain the following boundary and initial conditions for C(C, 6) from Eqs.(33) to (35):

20

(75)

C >O C >0

(76)

6(0,0) = 0 for 0 C(<,O) = 0 for C(C, 1) - 6(C,0) = 0 for

(77)

The boundary condition of Eq.(75) is satisfied if we discard the first term in the sum of Eq.(42):

The initial condition of Eq.(76) is satisfied for the third and fourth term of T,(0) in Eq.(74) due to the factor 6. The first and second term have the coefficients dni(0) for 0 = 0: N/2-1 c((, 0) =

+

27rtc.C [dK,l(0) d,2(0)]e"lS6/ sin =0 AT

(79)

From Eqs.(74) and (78) we get with Eqs.(65), (66), (76) and (77) the relation

130

3

INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

We multiply Eq.(79) with thogonality interval 0 5 (

sin(2xv(/N), integrate over the or-

< N , arid obtain for d,l(O) + dK2(0):

Equation (80) is multiplied with ~ - ' e - " l C / ~ sin(2xv
We get from Eqs.(81) to (83):

Equation (6.3-24) yields:

From Eqs.(84), (87) and (88) we get

There is no corresponding link between dK3and dK4due to the factor 6 in the third arid fourth term of Eq.(74). From Eqs.(6.3-22) and (6.3-23) as well as Eqs.(6.3-17) to (6.3-21) we obtain the following relations for d,l(l) and dK3(l):

Substitution into Eq.(85) yields:

3.2

R.ESOLUTION

A2 << h,/rn.oc

Similarly we obtain for d n 2 ( l )and d n 4 ( l )of Eq.(86):

Substitution of Eq.(89) brings:

The constants dns, dn3 and dK,2are now expressed in terms of dKl. The sums dK,l + dn3 and dn2 + dn4 are further developed from Eq.(6.3-59) on in Section 6.3, after the evaluation of the determinants Dn1(8) to Dn4(0) and Dno ( 0 ) . We may now write the solutiori for 9 1 1 ( ( , 0 )of Eq.(22). Starting with Eqs.(32), ( 2 7 ) , (28) and (37) we obtain the following expressions:

x

N/2-1

u,((,6 ) = -

~ x ~ ~ ~ * ~ / ~ ~ ' ( X I C - X Z @ ) / ~

n=-N/2+l

x

N/2-1

6((, 6 ) =

IT(JN)sin PK,6sin 27r~( (98) sin p, N

~TK( ~ , ( 0 ) e " l ~ sin /~ A7

v n l , vK2see Eqs.(65),(66); dni(0) see Eqs(6.3-23), (6.3-47)-(6.3-50) dn2, dn3,

~

K

A

see Eqs.(89), (921, (96)

132

3

INHOMOGENEOS DIRAC DIFFER.ENCE EQUATION

Cornparison of Eq.(97) with Eq.(2.4-36) shows that @11((, 0) can also be writt,erl in tlle followirig form:

3.3 QUANTIZATION OF THE SOLUTION The Harriilton function q = Val was dcrivcd in Section 2.5. Its quantization was then carried out in accordance with Eqs(2.5-29) and (2.5-30). In this section we want t o quantize the function @ = Qol + a @ l l . This car1 be done by following the same procedure. Thc quantization is thus carried t o the first order in a . As starting point we use Eqs.(2.1-51) and (2.1-52)

but we replace Qxzj by 9, and @, by Qll of Eq.(3.2-22):

601

of Eq.(2.3-9) as wcll as Ql by

Equations (2.5-1) to (2.5-3) renlain unchanged but in Eqs.(2.5-8)-(2.5-10) we must replace 9 by :

+

The function Wol(C,0) is defined in Eqs(2.4-36) and (2.4-37). Hence we obtain

3.3 QUANTIZATION

133

OF THE SOLUTION

N/2-1

- i~2e~~X~/2ei(X1C-X~O)/2

sin AOsin 27rt~C

N

n=-N/2+1 The function Qll(C, 0) is shown in Eq.(3.2-101):

With the help of Eq.(6) we may rewrite Eqs.(3)-(5). The notation Re(. . . ) is used for the real part of the expression in parentheses. This notation is well suited if one wants to use the computer for the separation of imaginary and real parts of an expression to avoid lengthy and error-prone separations by hand:

a** a* --

80 do

-

a*;, a*,, -

a*;, a*,,

80 80 +24-- 80

80

a*;, a~

+ 811Re(Tas)]

(10)

We have again the terms (dQ;l/dQ)(d*ol/aO), and (a@;,/dC) x (dQol/dC) obtained in Eqs.(2.5-8)-(2.5-10). Thc tcrms Qk'7;,6, (d*;l/aQ) x (OslaO), and (a*;,/dC)(ds/dC) must be calculated with the help of Eqs.(6) and (8). We show first how 6(C, 0) depends on Gpj (C, 0) of Eq.(3.2-12), Gej (C, 0) of Eq.(3.2-14), and Gmj (C, 19) of Eq.(3.2-16), as represented by the right side of Eq.(3.2-22). This begins with HsK(O,K.) of Eq.(6.3-1) and Gs,(Ql 6) of

Eq.(3.2-54):

134

3

INHOMOGENEOS DIR.AC DIFFER.ENCE EQUATION

The four functions d X 1 ( 0 )to dK4(8)in Eq.(8) may be written in the form of Eqs. (6.3-59)-(6.3-62). The expression

used in Eq.(8) is shown in Eq.(6.3-75). We may bring C ( C , 0 ) of Eq.(8) into the following form: N/2-1

C(C, 0) = e

i(xlC--X~e)/2

[J,(@,ti,) KG-N/2+1

2 J,(e, K ) = [Jl(O,K)-OJz(9, K ) ] C O S - -

~

N

-

We obtain

K ) ] sin

~ 0 [J3(0,K ) -OJ4(0,

2 ~ ~ Ji(O,ti,)= ['J5(@,K , ) - @ J ~ (K@) ] ,sin -+[J7(0, N @ll@G1(C,O)C(<,

+ iJi(0,

27rKC

(13) N

27rnO N 8 27rKt' 1~,)-8Jp,(0, ti,)]cos N K ) ] sin

-

from Eq.(6) by changing the sign of i. The product 0 ) of Eq.(9) becomes:

N/2-1 +i ~ ~ ~ - i X 2 / 2 ~ , i X z 6 / 2

sin P,O sin -

N

3.3 QUANTIZATION

135

OF THE SOLUTION

This is the product of two sums. The terms sin(27r~CIN)are important. We write v for K in the second sum and separate the products sin(2~vCIN)sirl(27r~C/N)for x = v and K # v:

N/2-1 x

[Jr(8,K) m=-N/2+1

+ iJi(0, K)]exp[-(A:

N/2-1 +i~~~"Xz(Q-1)/2 K=-N/2+1

-

27rK4X ; ) ~ / ~ Csin / ~] N

sinP,B[J,(B, x)+iJi(8, x)] sin2 27r~C N

x [Jr(O,Y)+ iJi(0, v)] sin -

-1

27rvC 27rxC N sin N

The integral of this expression over Eq.(2.5-3):

C

(15)

from 0 to N is needed according to

A20 x sin [Ji(8, K) - i Jr(8, K,)] 2

A2 sin px8 ~ ~ (K) 8sin , + N-A22 h(xlN) sin pH, 2

-

A2 ~ ~ (K)0cos , 2

We turn to Eq.(4). The product (aQ;l/88)(8Qol/88) of Eq.(2.5-8) is required. In addition we need 86/88 to determine 8Qll/80 according to Eq.(8). We take 6(C, 0) from Eq.(13):

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

136

N/2-1 --

ao

n=-N/2+1

+

27rnC [.JR(O,K) i.JI(0, K)]sin N

+

The exporlential furlction eixlC was not expanded into cosXIC isinXl< since it will be cancelled by a factor e-ixlC of 9c1in the following equation for (d9c1/d0)(d6/dO). The term d9&/dO is obtained with the help of Eqs. (2.5-4) and (2.5-5):

N/2-1

C

x

[ J ~ ( Bn) ,

- -e ,ix2(e-i) 2

(

+~

~XKC

( 0n)], exp[-(A: - X ~ ) ' / ~ Csin/ ~ ]

N/2-1 n=-N/2+1

IT(K1N) (A2 sin 13,O - 2ipn cos p10) sinp, 27rK,C N

+

x [JR(O, K,) i J I ( O , s ) ] sin2 -

N/2-l,#v +

C

N/2-1

C

(A2 sin ,&O

n=-N/2+1 v=-N/2+1 x [Jk(Q,v)

-

2i13, cos p,B)

2~nC 2~vC + iJ1(0, v)] sin 7 sin -)] N

JR(0,n) = JR(O,V), JI(O,K) = JI(Q,v) for

K

-t

v

(18)

Following Eq.(16) we integrate Eq.(18) over C: N/2-1

N

G

~ 0

~

/ =~ ddo ~ =~@ '

~ 4h2~ ~ {[J~(O, ~ ~ n)+iJI(O, ) s ) ] sin h20 n=-N/2+l

3 . 3 QUANTIZATION

137

O F T H E SOLUTION

Equation (11) calls for a tern1 a.ir/dC. We obtain it by differentiation of Eq.(13):

2x4 Al cos - i-sinN 2

+

inn() N

(20)

Tlie ter1n(d9;~/aC)(a.ir/aC) in E q . ( l l ) is obtained with the help of Eqs(2.5-6) and (2.5-7):

N/2-1

x

[&(O, C) + iJi(O,K)]

27TKC N

- ~ : ) ' 1 ~ < / 2cos ] -

n=-N/2+1

+ i-A12 exp[-(A:

N/2-l,#v

C

+~~~-iX2/2

N

N/2-1

C

K=-N/2+1 v=-N/2+1 27TtCC .27TtC 27TK,C x - sinN +t-- N cos-) N

(h;

- A;) 1 / 2 ~ / 2sin ] -

.

'

'lnP~'

27TV

[*Jr(O,n)

(Fcos-

+ i.Ji (0, n)] sin &O

2nu3}

2nVC A1 +i- sinN N 2

(21)

138

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

This expression is integrated over the interval 0 5 and (19):

N/2-1 - %N

2

K=-

-

:[ + ($)

N/2+1

i [ h ( Q ,a ) cos A212

2]

5 5 N following Eqs.(l6)

{J,(Q, a) cos h2/2 - J,(Q, a ) sin A2/2

+ Ji(0, a ) sin A2/2])

sin P,

s i n h ~ } (22)

We turn to the text following Eq.(2.5-10). In order to allow for the generalization of the terms of Eq.(2.5-3) by Eqs.(9)-(11) we define the energy u by thc sum of the following three components

where Ul to u3 are obtained by the substitutiorl of Eqs.(9)-(11) into Eq.(2.5-3). We use the notatiorl u to k ( 6 ) in Eq.(23) to distinguish the terms frorn the approximations U to Uv(f3) in Eq.(2.5-11):

-

N

a@*a* = Z / 0W Z dC= L~

cat (1+2a)

7 0

a*gl a t o ldc

-ae ae

3 . 3 QUANTIZATION

OF T H E SOLUTION

- L~ N as*a s L' [ u3==at,/FK d< = at (1 + 2a) 0

1 N

0

8%

a*,,

d(

-aC

As in Eqs. (2.5-12)-(2.5-14) we want the time-invariant part

ul, u2,

and we ignore the time-variable part ~ " ( 8 ) .Again, we write distinguish the terms from the approximations Uc t o Ucg in Eq.(2.5-15). When deriving Eqs.(2.5-12)-(2.5-14) we simply left out terms containing sirl/3,6 or cosP,6. The time variations of Eqs.(24)-(26) are not so obvious and we must write integrals over 6. Using Eqs.(2.5-12)-(2.5-14) we may write the time-invariant part of Eqs.(24)-(26) as follows:

of

uCto uC3to

N

N/2-1 X2 ( 1 + 2 4~ ~ ) 8 ~ IT ~ (KIN) ~ ~ K=-N/2+1 sin P.

(

)

[z (F) +

2]

140

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

On the right side of Eqs.(28)-(30) are the energies Ucl to Uc3 of Eqs(2.5-12) -(2.5-14) multiplied by 1 2a = 1.0146. This difference of 1.46% is barely visible in a plot. The interesting terms are the integrals multiplied by 2aGll. We shall analyze them in the following Section 3.4. The only difference between Eqs.(2.5-15) and (27) are the symbolsA. Hence, we may use the results of Section 2.5 from Eq.(2.5-15) on. The energies E, of Eqs.(2.5-32), (2.5-34), and (2.5-48) are obtained. One could write i, instead of E, and obtain the result E, = E,, but there is little incentive to do so.

+

3.4 EVALUATION OF THE ENERGYu FOR SMALLDISTANCES AX

ocl,

Let us consider the energies Oc2, and Oc3 in Eqs.(3.3-28)(3.3-30). Their first terms are equal to the energies Ucl, Uc2, and Uc3 of Eqs.(2.5-12)-(2.5-14), except for the factor 1+ 2 a . The results of Chapter 2 apply to these terms. We are interested in the second terms of Eqs.(3.3-28)(3.3-30) that are multiplied by 2a:

We want the sum to Eq.(2.5-15):

Ua,

of these three terms as function u~,(K)according

Equations (1)~-(3)rnust be written corresponding to this form. We write u l K ( ~ ,with ) the index K after the integration over C that indicates that the sumnlation over K, in Eq.(3.3-16) was not carried out:

(9 J ae(1 ) a;, N

Ci,,(n) = 2

a c a~t

~

~

N

((, o)e((, o)d()

g 0

0

do K

Division of Eq.(3.3-16) by G2 and leaving out the summation sign yields the following expression for X ~ , ( K ) :

+ N- 2

sin&

1aei

Jr(O, n) sin Az/2 - Ji(O, 6) cos h2/2

0

+ i[Jr(R, K)cos A212 + Ji(0, n) sin X2/2]} sin PKRdB

(7)

The functions J,(R, n) and Ji(O, K) are complex functions. We do not separate them analytically into real and imaginary components since this creates unnecessary complications. The computer instruction 2e will do this separation numerically. The integration of Eq.(7) over 0 requires knowledge of Jr(R, n) and Ji(R, 6). We start with the homogeneous solution a o ( ( , 0) of Eq.(3.1-6). The variable 0 becomes a variable n in H,,(n, n) in Eqs(6.3-47)-(6.3-50). For this reason we rewrite Eq.(3.1-6) with 0 replaced by On and shorten 0, to either 0 or n as required:

-~

N/2-1

A ~ ~ ~ X Z / ~ ~ ~ ~ ( X I C - X Z @ ~ ~" )( '/ I~/ ~ )

K=-N/2+1

sin ,8, Rn sin 2~nc N

142

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

From Eqs.(2.3-5)-(2.3-10) arid the text in between we infer that the four functions Qoj(C,O), j = 1. . .4, are equal except for a constant factor Goj. This implies that Qoj(<,0) in Eq.(3.2-12) can be replaced by QojQo(C,O), where Qo(C,8) is defined by Eq.(3.1-6) which in turn is defined by Eqs.(2.4-36) and (2.4-37). Hence, GBj(C,B) of Eq.(3.2-12), Gej(C,6) of Eq.(3.2-14), and Gmj((, 8) of Eq.(3.2-16) may be replaced by the following equations:

The differences n~~~ (c, e ) / d ( to becorrie:

A G (c,~8 )~/ ~ of( Eqs. (6.2-11)--(6.2-13)

The difference L

~ (c,G0)/20 ~ is~ even longer than hGpj(<, B)/&:

3

144 -

3.

INHOMOGENEOS DIRAC DIFFERENCE EQUATION

po(< + 1,e + 1) - 2

- 7 , ~ ~

2

+ + + 111 1 , 8 + 1)]+ ~ i h ~ Q o ( [ , 01) +

~ ~ (e < ,1) QO(C- 1,e

3 --hi[Qo(<+ 1 , 0 + 1) - Q o ( < 8 - (i[Qo({+2,0-1)

- 2@0((+1,8-1)

+ 2Qo(<-1, 8-1)

<

Let A,(<, 8) be independent of 6 and vary with like obtain for the differences AA(c)/AQ and AA(c)/A<:

of

8.

- Qo(<-2, 0-1)]

I l e o ( < / ~ )We m,.

(<, 6) and A,lC ((, 0) are independent Furthermore, we assume that In analogy to Eqs.(20) and (21) we obtain:

-"e"C'

A<

-

, 2

(

+1

- ?el

(< - l),

3.4 EVALUATION O F T H E ENERGY

FOR SMALL DISTANCES

AX

145

I3Aml,(O 1 = 5[Amlc(C+ 1) - Aml<(C - 111 (25) As Equations (3.2-22), (3.2-53), and (3.2-54) show that wc need the following combinations of Eqs.(9)-(15); the terms GI([, 0) and G4(<,0) are defined in Eqs.(3.2-18) and (3.2-21) (S=Euler Script G):

We could substitute Eqs.(9)-(19) into Eq.(26) to obtain 9 in analytical form. The many terms make it very difficult to do so without error. Our equatioris are ideally suited for computer processing. With that in mind we define QO(C j , 0, k) according to Eq.(8) for j, k = 0, f1, f2,. . . as follows1:

+

Qo(C

+

+ j, 0, + k) = @ojk:(C,0,)

for j = 0,1,2; k = 0 , l

= @ojmk(C,0 , )

for j = -1, -2; k = -1 for j = 0,1,2; k = -1

=@omjk(C,O,)

forj=-I,-2;

= @omjrnk(C,Q,)

k=0,1

(27)

l T h e notation m j and m k instead of - j and -k is used t o avoid misinterpretation by the computer.

3 INHOMOGENEOS DIR.AC DIFFERENCE EQUATION

146

Thc valucs of j rcquircd in Eqs.(9)-(19) are -2, -1, 0, 1, 2; the values of k required are -1, 0, 1. For 9(Cl 19) g(C, O), gc (C, O), and ge (C,I9) in Eq.(26) we need the extension to I9 + 1 and 0 - 1. We use the following notation with 0 = 8,:

We can compute GsK(19,K) of Eq,(3.2-54) once S(C, 0) of Eq.(26) is obtained. Equation (3.2-54) treats the function 9(C, 0) as a step function with step width 1. The integral may readily be replaced by a slim taken at < = o , 1, . . . , N - 1 :

Gsn (on, n) =

N

=-

N

The functions G,,,,

G,,,(Q,,

K)

i

2n~C N

9(C, 0n)e-ix1c/2 sin -dC

0

N-1

27rnC 1S((, ~,)e-"lC/~ sin N c=o

(On, K) and GSKp(I9,,n) follow from Eq. (28):

= GSK(On- 1, K) =

1

-

1~ , , ( c , o , ) ~sin- ~27TKC ~ ~ ~ (30) ~ ~ N

N-l

-

c=o

With the help of Eqs.(29)-(31) we obtain the function HSK(I9,,n) = HsK(O,n) of Eq.(6.3-1): 27Tn N

HSK(On, K)= - e i x z / 2 ~ s (@,+I, , n)+2 cos -G,, 27TK N

- -e i X z / 2 ~ s n p ( 0Kn) +2 , cos -GSK(Qn,

(On, n)-e-ixz/2~,K,(8n-11n) K)

-e-ixz/2~,K,(19n, K) (32)

Eight products of H,,(O,, K) = HsK(nlK) with certain combinations of the functions Fl(n,, K,)t o Fs(n, n), defined by Eqs.(6.3-51) to (6.3-58), as well

3.4 EVALUATION

OF T H E ENER.GY

u F O R SMALL DISTANCES

147

as d 3 , ( ~ )and dsi(n) of Eqs.(6.3-73) and (6.3-74), are listed in Eq.(6.3-75); they are denoted K l (n,K) to K8 (n,,n ) . We obtain from them the functions J1(6, n) to J8(6, K) of Eq. (6.3-75). These functions in turn yield J,(O, K) and Ji(6, K) of Eq.(3.3-13), which yield finally Xl,(rc,) of Eq.(7). The following definitions permit to write Eq.(7) in a more compact form:

A2

BIK(~= , ) N2

IT(K/N) sin ,CIK

A26 111(6,K) = Al,(~,)Xe[Ji(6,K,)- iJ,(O, K)]sin 2 112(6,K) = B~,(K,)X~{ ~ ~ (n)6sin , A212 - Ji(0, K)cos A212 + i [ J r ( 6 , ~ ) c o s A 2 / 2 +Ji(0,n)sinA2/2])sinp,0

(35) (36)

The equivalence of an integral over a step function with step width 1 and a sum permits us t o write X 1 , ( ~ ) in the following form:

We turn t o u~,(K) of Eqs.(5) and (3.3-25). The index K is written again after the integration over C to show that the summation over K in Eq.(3.3-19) is not carried out: N

UZ,(K) = 2 a q l l at

0

Equation (3.3-19) yields:

0

ae

zdC

(39)

3 INHOMOGENEOS DIRAC DIFFERENCE EQUATION

148

-

1 4

-NXZ 'T'"~)

sin pK

(%e[JR(O,K )

+ i h ( 6 , K)]

x {X2 sin pKsin[hz(Q- 1)/2] - 2/& cos A 6 cos[hz(Q- 11/21})

(41)

We need to evaluate numerically the integrals over the following terms with respect to 6 in order to integrate Eq.(41):

1

124(6, K,)= 5BlK(~)%[J~(Q, K) - ~JR(Q,K)I

x {X2 sin PK6sin[X2(6 - 1)/2] - 2PKcospKOcos[x2(0 - 1)/2])

(45)

The functions JR(6, K) and JI(6, K ) are defined by Eq.(3.3-17) by the functions Jr(6, K ) and Ji(6, K). In turn, Jr(6, K,)and Ji(6, K,) are defined by Eq.(3.3-13) with the help of J1(6, K ) to J8(6, K,) of Eq.(6.3-75). We need mainly lengthy but straightforward substitutions to express J R ( ~n), and JI(6, K)by J1(6, K) to Js(O, K). The only problems are the derivatives 8Jr/d0 and aJi/aQin Eq.(3.3-17). With the help of Eq.(3.3-13) we obtain the following expressions:

~ T K

- -[J1(6,

N

27r 66 K) - 6J2(0, K)]sin N

27T K N

- -[J7(Q,

27rnQ K) - QJs(0,K)]sin - (47) N

The functions J1(0, K) t o J8(Q,K) are defined in Eq.(6.3-75). We must show what the derivatives d J 1(8, K ) / ~ to Q dJs(8, n)/dQ mean. To this end we use once more the equivalence of the sum of a step function with the step width 1 and an integral over the step function. The functions Kl (n, K ) t o Kg(n, K) in Eq.(6.3-75) are step functions with a step width 1 for n = 0, 1, . . . and an unspecified value of K . Using the definition of Jj(Q,K) in Eq.(6.3-75) we may write:

of Eqs.(42) to (45) we may write X ~ , ( K )of Eq.(40) in the Using Izlto IZ4 following form:

of )Eqs.(5) and (3.3-26). Again the index n is Let us turn to u ~ ~ ( K written after the integration over C to indicate that there is no summation over K, in Eq.(3.3-22):

3

150

INHOMOGENEOS DIRAC DIFFERENCE EQUATION

Substitution of Eq.(3.3-22) yields:

x 2(A: - A:

-N- T 2

sin,&

)

A20 + A;)%~[J~(O, n) - iJ,(B, n)] sin 2

[ +(

)

2

]%e{Ji(B, K)cosX2/2-Jr(B, n) sin A212

- i[Jr(B, n) cos A2/2

+ Ji(B, n) sin X2/2]) sinp,B

(52)

Integration of Eq.(52) over 6 requires the numerical integration o f t h e following two terms:

1 (6, K) = -(A: - A: 4

:[

IJ2(B1n) = B l , ( n )

+ X ~ ) A ~ , ( K . ) R ~ [ JK)~ (-OiJr(B, , K)]sin(X26/2)

+ ($)2]Xe{[Ji(B,r) - iJr(B,n)]cosA2/2 - [Jr(O,K) + i J i ( O , sin ~ 2 1 2 sin ) PKB K)]

(53)

(54)

The functions Jr(B, n) and Ji(B, K) are defined in Eq.(3.3-13). One may write K 3 n ( ~of) Eq.(51) in the following form:

Following Eqs. (2.5-15) and (2.5-16) we write the normalized energy of Eq.(4) as follows:

u~,

3.4 EVALUATION O F T H E E N E R G Y

u FOR. SMALL DISTANCES AX

151

To help with the computer program we list in Table 3.4-1 the equations for p,, IT(lc,/N), . . . and their numbers required to compute 3Cln(~,), 3 C z , ( ~ )and , 3C3,(~,). For the representation of the results of this section by computer plots of Eq.(49), and % J ~ ( Rof) Eq.(55). we start with 3ClK(n) of Eq.(38), 3C2K(~) Equation (2.3-2) yields for X3: An:

2-

A3 =

=4~-

Ax

AX

= 4T-

h/moc h/moc Xc Xc 47r h pc = - = - Xc = Ax X3' mo c

=

47r

-

pc

(59)

The coefficients XI and X2 may be expressed as multiples of Xg or l/pC:

X1 = eArnoZ 2Az - ceAmo, moc ti/moc

moc2

,mot X1

Amor = -- , XI << A3 e A3

Xz "0

"=

-

$el0

3-

for c e ~ , o , / m ~ c << ~ 1

e$eo 2Ax - e$eo X 3 = -e$eo 47r ,moc2ti/moc moc2 moc2 pc moc2 Xz A2 < < XJ for e $ e ~ / m << ~ ~12

(60)

=

ex,

We still have to choose and (3.2-10) we obtain:

nee

ceArnOZ 47r m,oc2 p c

----.

of Eq.(20). Using Eqs(3.2-6), (3.2-7), (3.2-9),

= plAeo =

2eAx

= -$el0

lic

(61)

2Xc p i

?Aeo e T

Eqs.(20), (3.2-7)

(62)

Eqs.(22), (3.2-9)

(63)

152

3

INHOMOGENEOS DIRAC DIFFER.ENCE EQUATION

TABLE 3.4-1

FUNCTIONS P,, 1 ~ ( n / N .).,. AND T H E EQUATIONS E~.(2.4-14),E~.(2.4-33),. . DEFININGTHEM, WHICHARE REQUIREDFORT H E CALCULATION O F XI,(&), X z , ( n ; ) , A N D H S ~ ( KO. )F E Q S . ( ~ ~ (49), ) , AND ( 5 5 ) . THENUMERICALVALUES THATA R E OBTAINEDFROMEQUATIONS SHOWNWITH BRACKETS,E.G. EQ. [2.4-141, SHOULDB E STOREDI N T H E MEMORYO F T H E COMPUTER T O REDUCE T H E COMPUTING TIME.

A

Eq.[2.4-141 IT(K,/N) Eq.[2.4-331 Qo(C, 0,) Eq.[3.4-81 Q021(C, On,) Eq.(3.4-27) Qoll(C, o n ) Eq.(3.4-27) *ooI(C, on,) Eq.(3.4-27) Q ~ (c, , 0,)~ ~q.(3.4-27) ~ Q O , ~ ~ ( en) C , ~q.(3.4-27)*o~o(c, en) ~q.(3.4-27) Qolo(<,On) Eq.(3.4-27) Qooo(C,0) Eq.(3.4-27) Qomlo(C,on,) Eq.(3.4-27) Q o ~ , ~ oOn) ( < Eq.(3.4-27) , Q02ml(<,0,) Eq.(3.4-27) Q~lml(C, en,) Eq.(3.4-27) Q00rn1(<, 0,) Eq. (3.4-27) Q0mlml((,on) Eq.(3.4-27) Q0m2ml(<,on) Eq.(3.4-27) A,(<, 6,) ~ ~ . ( 3 . 4 - 2AAe/Ae 0) ~q.(3.4-21)Jel(c) Eq. (3.4-22) AJel /A< Eq.(3.4-23) &,,Ii- (5) Eq.(3.4-24) dAmlc/d~ Eq.(3.4-25) GPj(CrOn,) Eq.[3.4-91 Gej(C,o n ) Eq.(3.4-10) Gmj(C, on) Eq.(3.4-ll) Gpl (C, On) Ge4(&On) G1 (C, 8,) AG,~/A$

AG,,/&

gc(<,6',) g,(C, a n ) grnL(C,en) ( 0,) GS,,(Bn, n) Fl (n, n) F4(n,K )

J6(0,K,) Jr($, K,) Blfi(6)

Eq. (3.4-9) Gp4 (C, 0,) Eq.(3.4-10) Gml(C,6,) Eq.[3.4-261 G4(C, en) Eq.[3.4-171 A ~ , ~ / d Eq.[3.4-131 A G , ~ / A < Eq.[3.4-26]ge(C,Qn)

F~q~(3.4-9)Gel(<,0,) Eq(3.4-11) Gm4(C,6,) Eq.[3.4-261 g(C, o n ) eEq.[3.4-181 ~G,,/AO Eq.[3.4-141 AG,~/AC Eq.[3.4-26]9(C,On)

Eq.(3.4-10) Eq.(3.4-11) Eq.[3.4-26] Eq.13.4-191 Eq.[3.4-151 Eq.[3.4-26]

Eq.[3.4-28] g,rn(C, o n ) Eq. [3.4-281 go,(<, on) Eq.[3.4-281 9m(C,0,) Eq. [3.4-301 Gsfip(On,n) Eq.(6.3-51) Fz(n, n) Eq.(6.3-54) Fs(n, K)

Eq.[3.4-28I g
Eq.[3.4-28I Eq.[3.4-28I Eq.[3.4-29I Eq. [3.4-321 Eq.(6.3-53) Eq.(6.3-56)

Eq.[6.3-751 J7(8, n) Eq.[3.3-131 Ji(O,K ) Eq.13.4-341 111(8, K)

Eq.[6.3-751 Jg(8, K) Eq.[3.3-131 A ~ , ( K ) Eq.(3.4-35) I12(8, K)

Eq.[6.3-75] Eq. [3.4-331 Eq.(3.4-36)

3.4 EVALUATION OF T H E ENER.GY

u F O R SMALL DISTANCES L

h

153

a~,/ae 121(0,K )

Eq. [6.3-761 K s ( 6 , ~ ) Eq.[3.4-471 J R ( ~K,), Eq. (3.4-42) 1 2 2 ( B , K )

Eq. [6.3-761 aJr/ae Eq.[3.3-171 JI(O, 6) Eq.(3.4-43) 123(e,K)

Eq. [3.4-461 Eq.[3.3-17] lh(3.4-44)

I I

Eq.(3.4-45) &,(K) Eq.(3.4-53) I32(8, K)

Eq.(3.4-49) Eq.(3.4-54) %3,(~)

Eq.(3.4-55)

K7(@,K )

2K ) 1K)

Using the following constants for an electron

Xc = 2 . 4 2 6 ~ 10-12[m], e = 1.602x 1 0 - ~ ~ [ A smo ] , = 9.109 x ~ ]2.998 , x 10~[m/s] ti = 1.055x ~ o - ~ ~ [ v Acs =

X1 = ceAmox X3 = 5.866 x 102Arno,~s mOc2

X2 = -A3 = 1.957 x l 0 - ~ 4 ~ ~ h ~ ,m0c2 2Xc 2 Aeo = -pcAeo = 3.069 x 1 0 6 p ~ ~ e o x2e 2eXc Jelo = -pC q5elo = 2.458 x 1 0 - 5 p ~ 1 ~ e l o tic 2eXc Amlco = ~k AA"o = 7.368 x ~ o

-,

-,

[kg] (65)

(66) (67) (68) (69)

~ (70)

We had previously used the constants X I , X2, X3, AeOwith a similar meaning as here [Harrnuth and Meffert 2005, Eqs.(2.3-2), (4.4-lo)]. We denote them now Xl, X2, &, to distinguish them from XI, X2, X3, Aeo in Eqs(2.3-2) and (3.2-7) in the current book:

LO

We express the coefficients XI, X2, X3 of Eq.(2.3-2) by X1, X2, & to derive results that can be compared with those derived from the Klein-Gordon equation:

~

3 INHOMOGENEOS DIR.AC D I F F E R E N C E EQUATION

The values X1 = 0.1X2, X2 = 2.rrAx/Xc = 2.rr/pl & = 1 used previously (Harmuth and Meffert 2005, Figs.4.4-1 to 4.4-3) assume the following values in the new notation:

-

-

The transformation of A. to Aeo requires A. = 2XcAeo/e of Eq.(4.4-10) in the previous book and A,o = 2Xcp$~,o/n2e of Eq.(3.2-7):

-

for A. = 10Xl = 2.rr/pC A second set of constants previously used was X1 = lOP5X2, X2 = 27rAx/Xc = 2.rr/p, X3 = 1, = 10Xl (Harmuth and Meffert 2005, Figs.4.4-10 to 4.4-12). We obtain for this case from Eqs.(72) and (74):

AO

We plot in Figs.3.4-1 to 3.4-3 3Cl,(rc), Xz,(rc), and 3C3,(6) according to Eqs.(38), (49)) and (55) using for X1, X2, AS, and Aeo the values of Eqs.(73) and (74). The constants Gal = 1, GO4= 1, Jelo = 0, A~~~~ = 0 are chosen arbitrarily. For pc = Xc/Ax = lo4 we obtain plots in Figs.3.4-1 to 3.4-3 that are similar to previously obtained ones for the Klein-Gordon equation (Harmuth and Meffert 2005, Figs.4.4-1 to 4.4-3). An increase of p c to lo5 does not randomize the plots of Figs.3.4-1 to 3.4-3 as in the case of the Klein-Gordon equation. Instead one obtains essentially the same plots with reduced amplitudes. Let us turn to Figs.3.4-4 to 3.4-6 which show again XI,(&), X ~ , ( K ) , and 3C3,(n), but with the values of XI, X2, X3, and Aeo taken from Eq.(75). The value p c = Xc/Ax = 5 x lo3 is chosen to conform with previous plots for t hc Klein-Gordon equation (Harmuth and Meffert 2005, Figs.4.4-10 to 4.4-12). The constants $01, $04, Jelo1 AmlcOare chosen as before. Again one obtains plots similar to those for the Klein-Gordon equation. As before, an increase of p c by a factor 10 does not randomize the plots but only reduces their amplitude. The failure to obtain randomization of the plots could be due to errors in the calculation or computation; the computer program according to Table 3.4-1 runs eight pages in small print. However, a more likely reason is the

3.4

EVALUATION O F T H E ENERGY

-40

-20

u FOR SMALL DISTANCES AX

0

20

155

40

K -+

FIG.^.^-1. Plot of %l,(n) according to Eq.(38) for N = 100, qol = 1, GO4 = 1, $,lo = 0 , &lco = 0 , pc = X c l A x = lo4, X I = 4 ~ / 1 0 p c ,A:! = 4 x / l O p c , A3 = ~ T / P C Aeo , = 2pcln.

FIG.3.4-2. Plot of %z,(n) according to Eq.(49) for N = 100, Qol = 1, GO4= 1, $ e l 0 = 0 , A,lco = 0 , pc = XclAz = lo4, X 1 = 4 n / l o p c , X 2 = 4 x / 1 0 p c , A3 = 4 ~ / p c A,o , = 2pc/x.

FIG.3.4-3. Plot of % 3 n ( ~ ) according to Eq.(55) for N = 100, Gal = 1 , QO4 = 1, = 0 , A,i~o = 0 , pc = Xc/Az = lo4, X I = 4x/10pc, X 2 = 4 ~ / i O p ~ , A3 = 4 x / p c , A,o = 2 p c l ~ .

$el0

3

INHOMOGENEOS DIR.AC DIFFER.ENCE EQUATION

- 4 10-13. ~ -6x

lo-13-

-40

-20

0

20

40

>

nFIG.3.4-4. -Plot of x i , ( & ) according to Eq.(38) for N = 100, Gal = 1, $04 = 1, $el0 = 0, ArnL
FIG.3.4-5. !lot of X Z ~ ( Kaccording ) to Eq.(49) for N = 100, Gal = 1, GO4= 1, #el0 = 0, Amyo = 0, PC = x c / A z = 5 X lo3, XI = 4n/105pc, X2 = 4T/105pC A3 = 4 ~ / p c A,o , =2pcl~.

-40

-20

0

K

20

40

FIG.3.4-6. Plot of x3,(n) according to Eq.(55) for N = 100, $01 = 1, GO4= 1, #el0 = 0, AmL
3.4

EVALUATION OF THE ENERGY

u FOR. SMALL DISTANCES AZ

157

and Amlco. In particular, the choice of the arbitrary choice of Gol, Go4, constants dele = 0 and AmlCo= 0 implies that the two functions 4,1(<) = $ e l o f e ( ~ ) and AmlC(<)= A m l C O fm(<) of Eqs.(22) and (24) vanish, which implies a radical simplification of the theory to avoid specifying f,(<) and f m ( < ) . We need a problem that specifies the constants and the functions. The following chapter will investigate such a problem.

4 Dirac Difference Equation in Spherical Coordinates

To obtain the Dirac difference equation in spherical coordinates we follow the usual path of using Cartesian coordinates for quantization and then making the transition t o spherical coordinates. Since we are using the modified Maxwell equations (1.1-1)-(1.1-4) we must use the corrections of Section 1.2 t o the usual Lagrange function. These corrections were derived in some detail previolisly, but they are not part of the standard knowledge of most readers (Harmuth, Barrett, and Meffert 2001, Sec.3.2; Harmuth and Meffert 2005, Secs. 1.3, 5.1, 5.2). We write them here in a concentrated form to make the book readable without having to look up constantly other books. The Lagrange function L of an electrically charged particle according to Eq. (1.1-26) assumes the following form in spherical coordinates:

Equations (1.2-9)-(1.2-11) show the functions LC,, Lcs, LC,. The moments p,, ps, p, will be needed expressed with the variables r, 19, p. The moments are the derivatives of L,, Lo, L,. To obtain the derivatives wc writc v, A,, and LM in spherical coordinates [See Eq.(l.l-26) why we write L M rather then LM]. The vectors e,., e.0, e,, and r are defined iri Fig.4.1-1.

.CM = -1r r t , ~2~- e4, + ev . A, 2 1 = -.rrr[i.' 2 + r2d2 + (r sin 19)~+21

+ el-4, + +Am, - r d ~ , s + ( r sin I~)+A,,]

(4)

4.1 ELECTRON IN AN ELECTROMAGNETIC FIELD

t

FIG.^.^-1. The unit vectors e,, es, e,, and the vector r.

The following derivatives of LM are obtained:

a(r sin 6 cp)

= mr sin 6 @

The moments are the derivatives of Eqs.(1.2-9)-(1.2-11) we obtain:

L,,Lo, L, in Eq.(l). Using Eq.(5) and

a ai.

Ze . + eAmr+ -(r6Ae, C Ze p.9 = -- m.rg + eAms + -(r a(r9) C

p - 2 = m,i.

'-

Pip =

a(rsin 6 cp)

+ eA,,

= mr sin 6 @

- r sin 6 +Aefi)

(6)

sin 6 @Aer- +Ae,+,)

Ze + eA,, + (+Aer - 7 - 9 ~ ~ ~(8)) c

Three more definitions for spherical coordinates are needed to augrrierit Eqs.(2)-(5) :

r = rer + r6es

+ r sin 79 cpe,

(9)

Equations (6)-(8) are solved for i., 7-8,r sin 6 @ to obtain these quantities as functions of p,, ps, p,. A common denominator D is defined t o shorten the equations:

160

+=

4 DIRAC

DIFFERENCE EQUATION IN SPHER.ICAL COORDINATES

{ ( ) Aer[Aer(pi2

+ Zem, C

- earn,)

+ A e ~ ( p d- eA,a) + Ae,(pv

(P, - eAmip) - Aeip(PO - eAms)]

- eA,,)]

+ m 2(pr - eArnr)

+ ($) AwrAe . (P - eAm) 2

m2 D

= -[(p - eA,),

Zem C

Zem

+ -[Aer(~s C

- eA,s)

- A e s ( ~ r- eA,,)]

+ [Zem C

+ m2(p,

Ax P -

I

- eA,,

]

(15)

Since Eqs.(1.2-9)-(1.2-11) contain the second order time derivatives r, produce the derivatives ?, r8, r s i n 0 (ij from Eqs.(13)--(15). The terms A,, A,, D in Eqs.(3), ( l l ) , and (12) shall not be time dependent:

8, @ we

4.1 ELECTR.ON I N AN ELECTR.OMAGNETIC FIELD

D

rn2p,

Zem,

+ c(Ae

x p),

+

.fi]

The Lagrange furictiori 1; of Eq.(l) has three components Lk. This implies three components LFCk for the associated Hamilton function X :

We obtain from Eqs.(6) (8) the following result. The tcrms multiplied by Z e / c cancel: 3

E p j i j =pr+

+ Pard + p,rsin8

p

j=1

= m[r2+ ( ~ d ) ~ +sin ( r8 +)2]+ e ( ~ , , + + ~ ~ ~ r d +sin ~ ~19 +) ,r = miq2

+ eAm . i

(20)

162

4

DIR.AC DIFFER.ENCE EQUATION IN SPHERJCAL COORDINATES

Equations (1) and (1.2-9) (1.2-11) yield the three components X,, !KO, and X,: 1 3C - -ml(r2 '-2 1 3Co = - m ( t 2 2 1 2 X - -m,(r 'P- 2

1 + r2a2+ r2sin2 I9 @') + e4, - L,, = -mt2 + e4, - LC, 2 1 2 + r2d2+ r2sin2 6 @2) + e4, - Lc0 = -mi. + e4, - Lcs 2 1 . 2 + r 2 6. 2 + r 2 sin I9 @') + e4, - LC, = -me2 + e4, - LC, 2

(21) (22) (23)

The terms A,,,r + Amsr8 + A,,r sin 6 cC, cancel. One must write the Hamilton function with the momentum p and the potentials d,, 4,, A,, A,. This calls for the substitution of i., 7-9,rsinI9@ from Eqs.(13)-(15) as well as of i , 7-6, r sinI9 @ from Eqs.(16)-(18) into Eqs.(21)-(23) and (1.2-9)-(1.2-11). The substitution is rather time consuming and it seems prudent to obtain first some understanding of the Hamilton function. Let us see under which conditions the terms multiplied by Ze/c in Eqs.(6)-(8) will be small. From Eq.(6) follows the condition (Ze/c)rdA,,

<< m+, (Ze/c)r sin 6 @Aeo<< m,+

(24)

We may write it in the following form: mc2 >> ZecA,,rd/.i. If .i. is very small but from Eqs.(6): A,,

and mc2 >> ZecAesr sin 6 $17:

(25)

rd and r sin I9 @ are not one obtains a different condition

>> ZA,,rd/c

and

A,,

>> ZAesr sin I9 $/c

(26)

The meaning of Eqs.(25) and (26) is that the energy due t o the potential A, is small compared with mc2 or the energy due to the potential A,. Corresponding results may be derived from Eqs.(7) and (8). We may simplify Eqs. (6)-(8) as follows:

p, = m?

+ eA,,

, po

= mrd

+ eAm0, p,

D

= fm3

The sirriplified Eqs.(13)-(15) assume the form

= rnr sin 19 cC,

+ eA,,

(27) (28)

4.2

R.ELATIVISTIC LAGR.ANGE FUNCTION

163

The three simplified terms X,,-, X o , X, of the Hamilton function assume the following form:

One recognizes the terms of the conventional Hamilton function of an electrically charged particle in an electromagnetic field with potential A, and 4, augmented by the correcting terms LC,, Leo, and LC,. The term mu2/2 in Eq.(l) shows that our Lagrange function is not relativistic. This simplification was accepted since it permits to obtain X.,., Xs, X, of the Hamilton function as well as the correcting terms L,,, Lc9, LC, explicitly. The relativistc Lagrange function will be introduced in the following section. The Hamilton function can then be represented by series expansions only. The substitution of Eqs.(12)-(15) into Eqs.(21)-(23) yields the Hamilton function X without the approximations made to derive Eqs.(30)-(32):

We still have to show that the vector LCcan be written explicitly. Its three components LC,, Lcs, LC, are shown in Eq.(l). Each of them can be broken up into five sub-components. They are very long and they will not be used in this book. Hence, we refer to the literature [Harmuth and Meffert 2005, Eqs. (5.1-28)-(5.1-32)].

The Lagrange function of Eq.(4.1-1) is not relativistic but it led to the explicit Hamilton function of Eq.(4.1-33). The modified Maxwell equations can be combined with a non-relativistic constant mass or a relativistic variable mass. The different results appear worth showing even though in the end we will use the relativistic variable mass only. We proceed in this historically correct but slow way since we must introduce three concepts: 1. The modification of Maxwell's equations by the inclusion of a magnetic (dipole) current density. 2. The replacement of infinitesimal distances by arbitrarily

164

4 DIR.AC DIFFERENCE EQUATION IN SPHER.ICAL COOR.DINATES

short but finite distances. 3. The replacement of infinite distances by arbitrarily large but finite distances. We want to emphasize the need for three different concepts and to separate them, in order to show the effect of each concept. We turn to the relativistic Lagrange function. This irnplies giving up the explicit representation of the Hamilton function for a representation in terms of series expansions. The introduction of the rest mass yields the equation of motion (Harmuth and Meffert 2005, Sec. 5.2)

as well as the conservation law of energy:

A four-vector p can be defined with three spatial components p,, ps, p, P=

mov (1- v2/C2)1/2

+ r8es + rsin19@e, for spherical coordinates P = p ~ e+' p ~ e + s p,e, v = re,

(3)

and the corriponerlt p4 :

Here E denotes an energy rather than the magnitude E of an electric field strength. Momentum p and energy E are connected by the formula

Leaving out the unit matrix we obtain the relativistic generalization of the conventional part of the Lagrange function C of Eq.(4.1-1):

-

+ e(-4, + Am,?: + ~ , s r 8+ A,,r

-moc

2

1 + -m,0v2 +e(-6, 2

+ A m . v ) for v2/c2 << 1

sin 19 9)

(6)

4.2 R.ELATIVISTIC LAGRANGE F U N C T I O N

165

We adopt this generalization of the part LMof Eq.(4.1-1). The components of the correcting term LCare left unchanged from their definition in Eqs.(1.2-9)-(1.2-11). The mass m does not occur there and the potentials $,, A, come from a relativistic theory. The relativistic generalization of the Lagrange function of Eq.(4.1-1) is

(7)

Relativistic canonical momentums replace the non-relativistic momentums p,, ps, p, of Eqs.(4.1-6)-(4.1-8):

dL,r

pT = -=

ai.

dLc

ps=-"7-9) Pip =

moi. (1 - v2/C2)112

-

a&,

mor 6 (1 - v2/c2)

a ( r sin 6 @)

+ eAm, +

Ze . -(r6Ae, c

Ze + eAms + -(r

- mor sin 6 @ -

C

- r sin 6 @Aes) sin 6 +A,,

Ze + eAm, + -(iA,s c

(1 - V2/C2)1/2

-

rA,,)

- r8Ae,)

(8)

(9) (10)

The Hamilton function is obtained according to Eqs.(4.1-19) and (4.1-20) but we must replace rn by r n . ~ / ( l - v ~ / c ~according ) ~ / ~ to Eqs(4.1-6) ( 4 . 1 - 8 ) and (8)-(10):

The three coniponents of the relativistic Hamilton function are obtained in analogy to the non-relativistic components of Eqs. (4.1-21)-(4.1-23):

166

4 DIRAC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

As in the case of Eqs(4.1-21)-(4.1-23) the first derivatives i-, 8, @ and the second derivatives P, 8, @ must be eliminated. Equations (8)-(10) define i-, 9,@. The important difference with Eqs(4.1-6)-(4.1-8) is the factor (1 We no longer have a system of three linear equations and i-, 8, ci, cannot be calculated as readily as in Section 4.1. To obtain first some understanding we make simplifying assumptions for Eqs.(12)-(14). The terms multiplied with Ze/c in Eq.(8) can be ignored if the following corlditions are satisfied: Ze . -r29Ae, c

mo i

< (1 -v2/c2),I2

moc2 (1 -v2/c2)1/2

>>

Zecr8~,,

r

arid and

Ze -r sin 6 $Aes C

,m,oc2 (1 - v2/c2)1/2

moi-

< (1-v2/c2)1/2

>>

Zecr sin 6 @Aeo i-

(15)

Alternate conditions are required if i- is small in Eq.(8) but rB and r sin 6 @ are not: A,,. >> Z r d ~ , , / c or A,, >> Z e r sin 6 @Ad/c (16) The energy due to the potential A, should be small compared with rnoc2/ (1 - p)2/c2)1/2according t o Eq.(15). The magnitude of A, should be small corripared with the magnitude of A, according to Eq.(16). Under these conditions we obtain from Eqs.(8)--(lo) the following simplified equations: mor

+ eAm,.

P,. =

,/, (1 - v2/c2)

Pv =

mor sin 19 @ 2 2 1/2 + eAm'p (1 - v /c

One readily obtains

We square

+, 7-8, r s i n 6 ci, and obtain the sum of the squares:

4.2 R.ELATIVISTIC LAGR.ANGE FUNCTION

+2

+(

~ 9+) (T ~sin 6 +)2 = v2 =

1 - v2/c2

m;

(P - e A d 2

+

2 2 112 ~ , ) ~msOc] We substitute the last line of Eq.(23) into Eqs.(12)-(14):

= c[(p - e

X , = c[(p - e ~ , )+ ~m2c2]1/2+ e4, - L,, X s = c[(p - e ~ , )+~m;c2l1f2 + e4, - LcS 3C, = c[(p - e ~ , )+~mgc2]1/2+ e4, - LC,

(23)

(24) (25)

(26) These are the three components of the conventional relativistic Hamilton function plus the correcting terms LC,, Leo, LC, due to the modification of Maxwell's equations. We had to make the assumption that A, was sufficiently small to obtain Eqs.(24)-(26). Hence, these equations contain only part of the change caused by the modification of Maxwell's equations. We return to the solution of Eqs.(8)-(lo) for +, r i , r s i n 6 + without the simplifications of Eqs.(l5) and (16). The equations are made nonlinear by the term

+

+

sin2 ~9 + 2 ) / ~ 2 ] 1 / 2 (1 - v 2 / ~ 2 ) 1 /= 2 [I - (r2 ~~9~ The corresponding Eqs(4.1-6)-(4.1-8) of the non-relativistic theory are linear. No standard method exists for the solution of a system of nonlinear equations. Wc havc to develop a method for our particular case. Let us ignore initially that v2 is a function of +, 7-9, T sin 6 and treat Eqs.(8)-(lo) as a system of linear equations. The substitution

+

,m -t m o / ( l - v2 /c 2 ) 112 transforms Eqs.(4.1-6)-(4.1-8) into Eqs.(8)-(lo). We make the sarrie substitution in Eqs(4.1-12)-(4.1-15) to obtain a 'zero-order solution'. The common denominator D of Eq. (4.1-12) becomes:

ZecA, m>oc2 '

a, = a e ( r ) = - a, (1 -

g)

1/2 =

ZecA, rnoc2/ (1 - ? 1 2 / ~ 2 ) ' / ~(28)

Here a, represents the ratio of the energy associated with the electric vector poterltial A, and the rest energy of the particle. One might prefer moc2/ (1 - v2/c2)ll2instead of moc2 as reference energy but we want v as an explicit variable. We observe that a, varies with A, but it has no physical dimension. We rnay rewrite Eqs.(8)-(lo) with +, r8, r sin 6 @ on the left side using a, of Eq.(28)

r s i n 6 $ = (1 -

'V1,o

+a: (1 -

$1

(p-e~m)v

(I-$)

' I 2[Ae X (P - eAm)Ip A,(p - eAm)p

Equations (29)--(31)are squared and the squares are summed:

+

[Ae x (P - eArn)12 2[Ae . (P - eAm)12 A:(P - eAmI2 3 / 2 A, . [A, x (p - eArn)]Ae. (p - eAm) AB(p - eAmI2

'(

2

z:)

+ae l - -

(

)I2

\A. . (P - eAm)12 1 + a: 1 - A:(P-~A,)~

][

(32)

4.2 R.ELATIVISTIC LAGRANGE FUNCTION

169

We try to find an approximate solution of this equation with no v on the right side. If a, in Eq.(28) is smaller than 1 the assumption

suggests itself. But we rnust be careful that this assumption requires a sufficiently small value of A,. Large values of A, will lead to completely different results. As a zero-order approxirrlation we use only the first term on the right side of Eq.(32):

We recognize the same relation as in the first line of Eq.(23). The common term of the components K,, KO,K, in Eqs.(24)-(26) is obtained again. This is the zero-order solution in a, --+ a: = 1 of Eq.(32). A first-order solution in a, of Eq.(32) is obtained if we include the term in Eq.(32) that is multiplied by a, = a:. We preserve carefully the sequence of the factors since p will eventually be replaced by difference operators:

'1)

2

=

1-v2/c2 ~rl,;

[A, . ( p - eAm)I2 ( P - ~ A ~ ) ~ [ ~ + ~ ~ ~ ( ~ A'2(p-eAm)2 - : )

The zero-order solution represented by Eq.(34) is multiplied by 2ae and substituted into Eq.(35). The resulting equation is solved for 1 - v2/c2. The following first-order solution in a, is obtained as an improvement over Eq. (34):

x (1-

2aem,oc(p- eAm)2 [A, . ( p - e ~ , ) ] ~ [ , I I I { ~+ ~ (p - e ~ , ) 2 ] ~ /A~' 2 ( ~ - eAm)2

From Eq.(36) we obtain the first-order approximation in a, of Eq.(23):

170

4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

In first-order approximation in a, the three components of the Hamilton fiirictiori in Eqs.(12)-(14) assurrle the following form:

a, = 2.210 x 1 0 5 ~ for , electron, A, = A, (r) [As/m], A, = A, (r) [Vs/m]

Ze2 a=2h.

h, 7.297 535 x lo-' fine structure constant, Xc = 'moc (41)

We still have to write C,,., LC, in Eqs.(38)-(40) explicitly. Tliis calls for ?, 1-4,r sin 6 cC, of Eqs.(29)-(31) with the terms 1-v2/c2 eliminated by means of Eq.(36). Orily the zero-order approximation in cue is required:

r sin 6

=

C(P - eAm), [7n.;c2 ( p - eA,)2]

+

112

+ O(ae)

These are the sarrie equations as Eqs. (1.2-13)-(1.2-15). We may use the 15 coniporierits to LcVSas listed in Eqs.(1.2-16)-(1.2-30). Both the Klein-Gordon arid the Dirac equation are derived from the Hamilton function according Eqs.(24)-(26) without the correcting terms Lea, LC,. The Hamilton function according to Eqs.(38)-(40) with first-order correction in a, will yield first-order corrections to the KleinGordon arid Dirac: equations, while higher order solutions in a, of Eq.(32) will yield higher order corrections. We have previously shown that the term g, in the Maxwell equation (1.1-2) has an effect on the solution even if the transition g, -+ 0 is made

4.3

QUANTIZATION OF THE HOMOGENEOUS EQUATION

171

at thc cnd of the calculation since this tern1 produces a different differential equation that yields convergent rather than divergent solutions [Harmuth, Barrett, Meffert 2001, Eqs. (1.3-1) and (1.2-13)]. According to Eq. (1.l-13) the potential A, represents the magnetic current density g, here. It is prudent to expect that the transition A, + 0 at the end of the calculation rnay have an effect similar to g, -+ 0. Hence, the terms with a factor A, in Eqs.(38)-(40) cannot be ignored even if one takes the limit A, -+ 0 and g, -+ 0 a t the end of the calculation. Consider tlie exter~siorlof tlie theory to second order in a,. We must use the first three terms on the right side of Eq.(32). For the term a z ( l -v2/c2) we must use 1 - v2/c2 of Eq.(34) while for 2 a e ( l - v2/c2)ll2we must use the better approximation 1 - v2/c2 of Eq.(36). We may proceed in this way to the third- and fourth-order approximations in a,. But the process docs not end thcrc. For tlic fourth-ordcr approximation wc use tlie value of 1 - v2/c2 of Eq.(34) for the term o:(l - v2/c2) in Eq.(32), but for the fifth-order approximation we use 1 - v2/c2 of Eq.(36). There is no end. Every improved approximation yields new terms and perhaps new effects. O F T H E HOMOGENEOUS EQUATION 4.3 QUANTIZATION

Equations (4.2-38)-(4.2-40) assume the following form if the matrix riotatiori of Eq.(4.2-7) is used:

We rewrit,e this equation:

The square root is extracted with tlie help of Eqs.(2.1-8)-(2.1-10). To simplify writing we leave out the multiplication with the unit matrix 1 arid write LC for LC,., Lc9, LC, to avoid writing three equations:

172

4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

Thc terms are rearranged:

To emphasize that this equation is a first order approximation in cu we rewrite it as follows, keeping in mind that the correction terms LC,in Eqs(1.2-16)-(1.2-30) are all multiplied by a :

This equation corresponds to a previously derived equation for the KleinGordon equation (Harmuth and Meffert 2005, Eq. 5.3-3). A solution as an expansion in powers of a is obtained if we apply the operators of Eq.(5) to a function

Equation (7) may be separated into an equation of order O(1) and one of order O(a):

Equation (8) is the usual Dirac equation of Eq.(2.1-11) and Eq.(9) is an inhomogeneous variant of that equation.

4.3

173

QUANTIZATION O F T H E HOMOGENEOUS EQUATION

For the quantization and generalization of Eq.(8) we follow the standard procedure of using Cartesian coordinates for quantization and then making the transition t o spherical coordinates. In analogy to the transition from Eq.(2.1-11) to (2.1-12) we obtain from Eq.(8):

Using the vector operator grad instead of the sum we may rewrite Eq.(lO) as follows:

(f

1 ha +; + e$e)] PO = -Pm,oc@o

grad -A,)

[a.

(11)

Substitution of grad in spherical coordinates

1 a*, 1as, + --e~ +r a19 r s i n 6 d p ev

grad Q0 = *er

dr

and f f z l (e,.

+ es + e,) 0

0

+0es + e,)

f f x z( e ,

0 0 a x 3( e , es

+ +e

'f )

)

(13)

yields:

The difference equation is obtained if we replace Eq.(12) by

dP0

grad so-+ grad so= --e.

Ar

and d s o / a t by d q o / d t :

1 + ---es Ad I-

1 ds, +r sin 6 A~ ev

(15)

174

4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

The procedure used for quantization and generalization to spherical coordinates also works for the homogeneous part of Eq.(9). One only has to replace Qo by Q1. The components Q and .LCof the inhomogeneous term require further elaboration. Let the variables t and r be defined in the intervals 0 5 t 5 T and 0 5 r 5 cT. We replace t by the normalized variable 0 = t / A t , r by p = r / A r , 6 by r] = 6 / A 6 , and p by E = p / A p . There are N intervals for each variable:

e =t/at

p

=r/Ar

0 5 0 5 N At = T I N at = a t a e

O5p5N Ar = cT/N ar = Arap

O l q l N A6 = T I N

dt

dr

d6

=Atde

=a

rdp

t =~

r] = @ / A d

O I [ I N Ap =2 ~ l N

ao = a o a q =A

a p = pa^ Ad p~= A ~ A E (17)

~

With the notation

a Z j= a z j(e,

+ es + e,) 1

we may write Eq.(16) in matrix notation:

0 azz.grad 0 az3. grad

/ A P

4.3

QUANTIZATION O F T H E HOMOGENEOUS EQUATION

175

The first-order difference quotient follows from Eqs(l.3-1) and (1.3-5):

The substitutions 8 -+ p -+ 77 -+ t yield the corresponding equations for p, 77, and E. Multiplication of any line of the matrices of Eq.(19) yields the following equation

which assumes an alternate form with the help of Eq.(20) and c a t = Ar:

1 - I,@)] + p sin(qA8)Ap [ Q o ( P , ~ ,1~,+~-) QO(P,V,E

+ [Qo(P,77, ti e + 1) - (Qo(p,7, t, 8 - I)] - i[a,,(A,

+ As + A,)

- A:! - PA3]Qo(p,7, t , 8) = 0

Equation (22) represents three equations for j = 1, 2, 3 but each of these equations represents itself four equations since the a,j are matrices of rank 4. Hence, there is a total of 12 equations. Using P and a,, of Eqs(2.1-9) and (2.1-10) we write Eq.(22) in matrix form, using Eq.(20) in reverse to shorten the long explicit differences in Eq.(22):

176

4

DIR.AC DIFFER.ENCE EQUATION IN SPHER.ICAL COORDINATES

In terms of a system of four equations Eq.(23) assumes the form

4.3

177

QUANTIZATION O F T H E HOMOGENEOUS EQUATION

The substitution of a,, for a Z jin Eq.(22) permits us to rewrite Eq.(23) as the following system of four equations instead of Eqs.(24)-(27):

Next, we substitiite a,, for a,, in Eq.(22) to obtain the following system of four equations:

1 +--p 1a s dqO3 + A@03 +-AqO1 AT psin(qA29)Ap A( AO

~ 9 0 3

dp

2

+ A 8 + Ap)q03 - (A2 + A3)*01]

- Z[(A,r

=0

(32)

We see from Eqs.(24)-(27) that the substitutions QO4 9 0 3 , QO1+ Qo2 transfor~riEq.(24) into Eq.(25) arid Eq.(27) into Eq.(26). Hence, we may concentrate on either Eqs.(24), (27) or on Eqs.(25), (26). In the system of Eqs.(28)-(31) the substitutions QO4 -QO3, Qol -+ QO2transform Eq.(28) into Eq.(29) while Q01 -+ - 9 0 2 , 9 0 4 9 0 3 transform Eq.(31) into Eq.(30). We may concentrate on either Eqs.(28), (31) or on Eqs.(29), (30). Finally, in the system of Eqs.(32)-(35) the substitutions QO3-+ -QO4, Qol Qo2 transform Eq.(32) into Eq.(33) while 9 0 2 -+ -Qol, Q04 + 9 0 3 transform Eq.(35) into Eq.(34). We may concentrate on either Eqs.(32), (33) or on Eqs. (33), (34). To separate the four variables p, 71, [, 8 we may use the Bernoulli product ansatz six times: -+

-+

-+

-f

The calculations are carried out in Sections 6.7 and 6.8 with A, = 0 and 4, = $,(p). For the variables [, 9-7, and 0 we obtain essentially the same results as in the differential theory but for p = r/Ar we obtain a significant deviation that we want to elaborate here. We start with Eq.(6.8-75):

4.4 U N B O U N D E D E L E C T R O N IN A C O U L O M B FIELD

179

TABLE 4.3-1 THE PARAMETER 1 O F E ~ . ( 3 9 SHOWN ) WITH T H E FRACTIONAL DIFFERENCE D~ = ( I + l/2)/ JA N D THE RELATIVE DIFFERENCE

Schiff (1949, p. 225, Eq. 44.27) and Messiah (1966, p. 932) show a correspondi~igequation derived from the differential theory if we write n,'+ k = n and ,moc2, 4xZa for m,, z e 2 (Messiah) or mc2, ze2/fic (Schiff)':

The essential difference between Eqs.(39) and (40) is that the approximation k = 1 112 is used for k = ,,"in Eq. (40) and that 1 = 0 is excluded in Eq.(39). The missing case r! = 0 is explained in Section 5.3 after Eq.(5.3-32). Table 4.3-1 shows I, d m , 1 112, the fractional difference D k , and the relative difference 7 ~ :As . one would expect the differences are small. No difference of energy levels between differential and difference theory as in Eqs.(39) and (40) was encountered in the case of the Klein-Gordon equation (Harmuth and Meffert 2005, p. 220). It has been known since the discovery of the Lamb shift that Eq.(40) requires a correction (Lamb and Rutherford 1947). We must leave the application of the calculations of the Lamb shift to Eq.(39) for experts in spectroscopy.

+

+

We turn to the solution of the difference equations (6.8-12) and (6.8-13) or the differential equations (6.8-16) and (6.8-17) for which s 2 , ss, s 4 , and ss are located on the circle 1st = 1 as shown in Fig.4.4-1 rather than on the real axis as shown in Figs.6.8-2 and 6.8-3. Substitution of w,w, from Eq~~(6.8-5) and (6.8-6) into Eq.(6.8-31) brings for E > mjoc2 and -2 < g < 2: l T h e energy levels of the hydrogen atom according t o Dirac's equations are usually treated surprisingly brief in text books. Becker (1963, 1964, vol. 2, Ch. F I I ) , who was for decades a standard for thoroughness, avoids the topic completely.

180

4 DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

For small values of Ar or for g = -2 square roots to obtain:

+ O(Ar) one may readily extract the

Substitution of ss for so into Eq.(6.8-42) and the use of a complex value p = p~ ipI bring

+

Equations (6.8-5) and (6.8-6) yield

Substitution of Eqs.(4) and (5) into Eq.(3) yields two equations for p~ and p~ that yield for m.ocAr/ti << 1

4.4 UNBOUNDED ELECTR.ON IN A COULOMB FIELD

F1G.4.4-1. Location of the singular points sz, s3, s4, ss in the complex s-plane for the Dirac equation for general values of g and the particular value g = -1. The circle of convergence around s3 goes through s = 0 and sa.

The solution of xl(p) and x4(p) for so = s3 = e x p ( - i d r ) of Eq.(2) and p of Eq.(6) is provided by Eqs.(6.8-76) and (6.8-77):

The coefficients a, and d, follow from Eqs.(6.8-23), (6.8-24), and (6.8-26) with the new values so = s3 and p from Eq.(6). The constant a0 is choosable. Since p is complex the factorial series cannot be terminated by choosing

4 DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

182

a negative integer value for p. Convergence may be achieved by asymptotic termination or by conformal mapping (Harmuth and Meffert 2005, Sects. 6.9, 6.10). We will forego here these lengthy calculations since we are primarily interested how xl(p) and x4(p) vary at large distances p, which is independent of the convergence of the series. We use Sterling's relation to simplify the Gamma function for large values of p (Abramovitz and Stegun 1964, p. 257):

The ratio r ( p ) / r ( p

+ p + 1) becomes:

For p~ = -1 from Eq.(7) we obtain:

lini

P--O0

r(p

I'(P) p

+ + 1)

-

= e-leip~(l-lnp)

p - t ~ ~

- ,-I e - i p ~I n p

(14)

Equations (9) and (10) are rewritten with the help of Eq.(2) and by cornbining the constant e-' with the constant a0 and thus do:

The products of Eqs.(l5) and (16) with vl(0), v2(B) of Eq.(6.7-17) yield at great distances expressions of the form

If the factor exp(-ipI lnp) were replaced by l / p or llp2we could interpret Eq.(17) as expanding and contracting spherical waves. The term

4.4

UNBOUNDED ELECTRON IN A COULOMB FIELD

183

TABLE 4.4-1 VARIOUSVALUESOF THE ANGLEy IN F1G.4.41 DEFININGTHE LOCATIONOF S 2 AND THE ASSOCIATEDVALUESO F g AND E/moc2 ACCORDINGTO E ~ . ( 2 2 ) .

exp(-ipI In p) implies multiplication with cos(pI lnp) or sin(pI In p) of the amplitudes of the oscillations. There is currently no explanation for Eq.(17). The parameter g in Fig.4.4-1 dominates most calculations. We derive the relationship between g and the angle y in that illustration. In order to avoid the use of -7 required by s3 we use s 2 and +y.

+

For small values of A r or for g = -2 O(Ar) one may extract the square root according to Eq.(2). For the whole interval -2 < g < 2 we may rewrite Eq.(18) as follows:

1 cos 27 = a , sin 27 = b, cos 27 = 1 - sin 2 y, sin y = -(1 1 sin y = -(I

Jz

-

cosy = ( I - sin2y)'I2

- cos 2y)'I2

Jz 1 = -(I + Jz

The location of s2, s 3 , s 4 , and ss for y = 30° or g = -1 is shown in Fig.4.4-1. For the connection between g, s 2 , wp, and w, refer to Eqs.(6.8-33)-(6.8-36). We note the result

g = -2(cos2 y - sin2 y)

(21)

which yields g = -1 for y = 30' and g = 0 for y = 45'. We derive a relation for ~ / r n , ~asc function ~ of either y or g from Eqs.(20) and (21):

ascfunction ~ of y and g. Table 4.4-1 shows ~ / r n , ~

In Fig.6.8-1 we show the loci of s2, s3, s4, and ss as functions of g. The right half-plane is shown again with rnore detail in Fig.6.8-2. Here we extend our investigation to negative real values of the s-plane. Figure 4.5-1 is the equivalent of Fig.6.8-2a for negative real values. We see that s3 = 1 1 4 and s2 = 4 are replaced by ss = - 1 1 4 and s4 = -4.The solutions obtained in Section 6.8 apply again if we replace s 2 and s3 in Eqs.(6.8-35) arid (6.8-36) by -s4 and -sa:

Instead of writing p, a,, d, in Eqs.(6.8-18) we write p', a:, d: when we go from Fig.6.8-2a to Fig.4.5-1 and from s2, ss to s4, ss. In Section 6.8 we were interested in values of ss close to 1 in order to avoid solutions that rapidly decayed to zero for smaller values of s3 and we accepted the problems caused by convergence for this choice. For the same reason we are interested here in values of ss close to -1. Hence, instead of substituting so = s 3 = 1 - O(Ar) to go from Eq.(6.8-42) to Eq.(6.8-43) we substitute so = sg = -1 O(Ar) = -1 in Eq. (6.8-42) and obtain instead of Eqs. (6.8-43):

+

4.5 ANTI-PARTICLES WITH

S5

sq

So NEAR S

./'

= -1

' .-i-.-

'g=2

'. ..

.Y e , ? ;

''4-

FIG.4.5-1. Plot of the left half of the s-plane of Fig.6.8-1 in analogy to Fig.6.8-2a. The values ss = I/&' and sz = &' are replaced by ss = -I/&' and sr = -&'.

We are interested in solutions for which either l1 or l4 but not both are zero. This permits us to simplify writing by using I for Il or l4 and add the subscript 1 or 4 when needed. Equation (3) becomes:

+

We substitute p = p~ ipI. The real part and the imaginary part of Eq.(4) are zero individually. Using Eq.(6.8-32) we get:

Equations (5) and (6) should be compared with Eqs.(6.8-45) and (6.8-46). The main difference is that mo is replaced by -mo if p~ is replaced by p k . The changed sign, to f,of p~ and pi is of little consequence. We turn to Eqs.(6.8-23) and (6.8-24). They are written for so, which may be replaced by either ss as in the text following Eq.(6.8-52) or by ss

186

4

DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

as required here. The p in these equations must be replaced by p'. The coefficierits 6,,1 to SP,, and ap,O to a,,, are then replaced by 61,1 to 6:,, and to a:,,. We only write S1,4, S:,5, a:,4, and a:,, with p replaced by pi:

Let us now turn to the text and the equations following Eq.(6.8-46). We replace 6, a by S', a'. Everything said there applies again. Equation (6.8-51) is replaced by 6:,44",5 - 4,46:,5 = 0 and Eq.(6.8-52) by:

+

We substitute s5 = -1 O(Ar) -1, All, Xe4 from Eqs.(6.7-49), (6.7-50), and p' = p k ipf from Eq. (5). The real and the imaginary part of the equation must be zero individually. We obtain for the real part

+

and the imaginary part yields

Equations (11) and (12) are equal to Eqs.(6.8-53) and (6.8-54). Equations (6) and (6.8-46) have a different sign for Xe/2 but this difference beconies unimportant due to the signs of Equations (5) and (6.8-45) have a different sign in front of 27rZa which was transformed into -,n,,o in Eq.(5) while Eq.(6.8-45) has + m o As a result, the considerations

id-.

4 . 5 ANTI-PARTICLES

WITH so NEAR s =

-1

187

from Eq.(6.8-54) to (6.8-74) remain unchanged. We only have to change the sign of mo in Eq.(6.8-74):

Equation (4.3-39) is also changed by mo -+ -mo only. This result corresponds to the one obtained for the Klein-Gordon difference equation (Harmuth and Meffert 2005, Sec. 5.11). We do not hesitate to say again that the solution in the point s = -1 holds for the anti-particle of the solution in the point s = +l. To obtain solutions for unbounded anti-particles we investigate s 5 in Fig.4.5-1 on the unit circle close to s = -1. This is the same as using g = -2 O(Ar) for 3.5 in Eqs.(6.8-36) and (6.8-33):

+

In first approximatiori in ,mocAr/ii we obtain

Substitution of s5 into Eq.(6.8-42) and the use of p = p' = p k

+ ipi

brings:

188

4

DIRAC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

Substitution of Eqs.(4.4-4) and (4.4-5) brings two equations for the real and the imaginary part that yield for rn,~cAr/h<< 1 solutions for pk and pf:

The solution of xl(p) and x4(p) for so = .ss = e x p ( - i ~ A r ) of Eq.(15) and p' - with p k , pi of Eqs.(18), (19) is provided by Eqs.(6.8-76) and (6.8-77): -

Equations (6.8-23), (6.8-24), and (6.8-26) apply if the new values so = ss and p = pi = p k ipf are used. Since p' is complex the factorial series cannot be terminated by choosing a negative integer value for p'. Following Section 4.4 from Eq.(4.4-11) to (4.4-14) we use again Sterling's relation (Abramovitz and Stegun 1964, p. 257):

+

r(,,)

(2T)1/2e -P Pp-112 , P B 1 r ( p + p l + 1) = (p+p')r(p+pl) = ( 2 ~ ) (p+p')e-(p+p')(p+p')pfp'-1/2 =

The ratio r ( p ) / r ( p +p'

+ 1) becomes

(22) (23)

4.5 ANTI-PARTICLES WITH For p' = -1

lim

P--rn

NEAR

S

=

-1

+ ip; according to Eqs.(l8) and (19) we get r(P)

r'(p

So

+ p' + 1)

- e-leipfp-ip; - e-leip;e-ip;lnp

- e -le-ip; -

Inp

(25)

Equations (20) and (21) are rewritten with the help of Eq.(15) and by combining the constant e-' with the constant a0 and thus do according to Eq.(6.8-26). Note how (-1)" of s:, is eliminated:

(P) = f e - i ~ p A r e -ip;

N

In p

x(-l)paP (p p=o

+ 1) . . . (p' + p) + + 1) . . . (p + p' + p ) (p' p'

(26)

The products of Eqs.(26) and (27) with vl(Q),v2(Q)of Eq.(6.7-17) yield at great distances expressions of the form

The term exp(-ipi lnp) means multiplication of the amplitude of the oscillation with cos(pf In p) or sin(p; In p). Again, as in Section 4.4 following Eq.(4.4-17), there is currently no explanation. The series expansion around se near s = -1 in Fig.4.5-1 replaces e x p ( - i ~ p A r ) by e x p ( i ~ p A r )but leaves everything else unchanged. As an implication of the existence of particles with negative mass consider two particles with mass m,o and -mo. The sum of the masses or of the energies is zero: mo - m,o = 0,

(mo - m,o)c2= 0

(29)

If the particles are closer together than the resolvable distance An: we would conclude that there is nothing. If they move further apart due to their mutual rejection we would observe a particle with mass mo and another particle with mass -mo created "from nothing". The positive masses of the Earth, the solar system, and the Milky Way galaxy would reject the particle with mass - m o It would be pushed away until it finds perhaps a galaxy with negative masses that would attract it. The quantum theoretical result of negative masses might thus be checkable by astronomical observations. Are there galaxies or clusters of galaxies that reject rather than attract others? A comparison of the present position

190

4

DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

FlG.4.6-1. Location of the singular points s2, ss, s r , s5 in the complex s-plane for the Dirac difference equation for general values of the parameter g and the particular value g = 2.5 yielding s 2 = -ss == i l f i and ss = -s3 = i f i . The d while the one circle of convergence around sz goes through s = 0 and s5 = i around s q goes through s = 0 and ss =

-ia.

and Doppler shifts of galaxies with photographs and Doppler shifts obtained a century ago make such a search in principle possible. It would be a tedious task but it would affect our thinking about what is "something" and what is "nothing". This in turn might help us comprehend where the masses of the universe came from or still come from.

At the points s = f1 in Fig.4.6-1 we see the value g = -2 of the parameter g , while a t s = fi we see g = +2. Hence, we must expect a significant difference between the solutions near s = f1 and s = fi. We start with the definition of w,w, in Eq.(6.8-32)

w,w,

=

4

(m o c k)

and obtain for g from Eq.(6.8-33):

(1 -

)

for E

< ma2

4.6

SOLUTIONS WITH So NEAR S

=i

Figure 4.6-1 shows that g = 2+Ag, with Ag << 1, yields s 2 on the imaginary axis for h g > 0 but on the unit circle for Ag < 0. We get

Equation (4) puts s 2 always on the unit circle while Eq.(5) can yield s2 either on the unit circle or on the positive imaginary axis at s < i. Substitution of g = 2 h g into Eq.(6.8-35) brings

+

3,

=

[-

i(2

112 112

+ Ag) + ( a ( 2 +

- 1)

]

We obtain three choices for (&)'I2 from Eqs.(4) and (5): mochr ( A ~ ) ' / ~2=i [I+ (T)Z(l = 2i [1-

&)]'1

(~)~(s -

~
- I)] 'I2, E>moc2,

(

(

[ (m0pr)2(myc4 '1

= 2 -I+

--

I)]

I

)

<1

(8)

E> m0c2,

(~)2($-1)>

1 (9)

Three corresponding values of sz are obtained with the help of Eq.(6):

192

4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

s2=i+Asal=i-

[ (mOiAr) $1 ] 1+

(1 -

complex

on imaginary axis

(10)

(12)

The choice of the signs +Aszl, +Asz2, -Asas and particularly the use of the sign f will be explained presently. To make s 2 located close to +i we must exclude s 2 of Eq.(lO) since the magnitude of the real part will always be at least equal to 1:

Only Eqs.(ll) and (12) for E > rn,0c2 can yield Asz2, Asz3 << 1. According to Fig.4.6-1 the point s 2 will rotate in the mathematically negative direction from i to +1. Fro111the relation

we infer that the positive sign of f in Eq.(ll) must be chosen. Hence, we get fro111 Eqs.(ll) and (12):

~

~ = 2[I 2

As23 =

(

)( ,mocAr

From Eq.(15) we get

- 1)

(&-I)]

]

1/2

2 0,

s2 on unit circle

(15)

> 0, s2 on imaginary axis

(16)

1/2

4.6

SOLUTIONS WITH

So

NEAR S = i

193

which is not a very interesting result since it can be satisfied by the limit A r --+ d r . But Eq.(16) requires a mininluni value for Ar, wliicli is an important result since it cannot occur in a differential theory:

In the first line of Eqs.(17) and (18) Ar is still connected to the Compton wavelength h/moc of the electron, but in the second lines Ar is a function of the energy E and the constants of nature hc; the mass m o of the electron only adds a correction. We develop the solution for s2 on the imaginary axis according to Eq.(16) with E > ,moc2 and s2 = i ( l First we derive from Eqs.(6.8-5) and (6.8-6) the relations

Substitution of

-

s2

= so into Eq.(6.8-42) brings

Substitution of Eqs.(l9) and (20) produces a real equation. We use the notation Xe = Xel Xe4 with Xe = Xel, Xe4 = 0 or Xe = Xe4, Xel = 0 and write sg -(I - AS^^):

+

194

4

DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

With Xe = d -

we obtain for p:

Figures 4.6-2 to 4.6-7 show plots of p as function of rnocAr/?i for various values of the parameters ~ / r n and ~ c 1.~ The sign is used for f everywhere. We see from Figs.4.6-2, 4.6-6, and 4.6-7 that 1 does not have much effect on the plots. The same holds true for the terms in Eq.(23) multiplied with 8.rrZa. Consider the solutions xl(p) and x4(p) of Eqs.(6.8-76) and (6.8-77). Written for sz = i(1 - Asz3) = i exp(-Asz3) rather than 33 we get

+

For a negative integer value of p the factorial series can terminate long before p reaches N,which in essence means a polynomial replaces a series and the question of convergence does not arise. Figure 4.6-2 shows that p can equal any negative integer. Let us choose p = -2 and the parameter ~ / r n= ~ 5. c ~We obtain mocAr/fi = 0.306. Equations (24) and (25) become

FrG.4.6-2. Plots ofp according to Eq.(23) for ~ l r n o c '= 5, . . . , 11. The parameter 1 equals 1; Z = 1; a = 7.2975 x lop3.

n~ocAr/h

-+

F1G.4.6-3. Plots of p according to Eq.(23) for ~ l r n o c '= 2, . . . , 5. The parameter 1 equals 1; Z = 1; a = 7.2975 x

FIG.4.6-4. Plots of p according to Eq.(23) for ~ l r n o = c ~1.7, . . . , 2. The parameter 1 equals 1; Z = I; a = 7.2975 x

196

4

DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

m o c A r / h -+

FIG.4.6-5. Plots of p according to Eq.(23) for E/rn,oc2 parameter 1 equals 1; Z = 1; a = 7.2975 x lov3.

=

1.67,

... ,

1.7. The

FlG.4.6-6. Plots of p according to Eq.(23) for ~ / r n , o c=~5, . . . , 11. The parameter 1 equals 2; Z = 1; a = 7.2975 x lop3.

mocAr/h

---+

FIG.^.^-7. Plots of p according to Eq.(23) for E/rnoc2 = 5, . . . , 11. The parameter 1 equals 3; Z = 1; a = 7.2975 x

The variable a0 is choosable, do follows from Eq.(6.8-26) with so = sz, while a1 and dl follow from Eqs.(6.8-23), (6.8-24). The factor r(p)/I'(p- 1) yields

The factor

1:P

is eliminated in analogy to (-1)P in Eq.(4.5-27):

The factor exp(-Asz3p) cannot be reduced to 1 if we exclude the limit Asa3 + 0 to avoid the special point where the unit circle intersects with the imaginary axis. Hence, we obtain

These solutions decrease exponentially with increasing p. If we multiply Eqs.(30) and (31) with vl or vz of Eq.(6.7-17) we obtain sinusoidal oscillations f~ ( pexp(iE8Atlti) ) that decrease exponentially with increasing p. Different choices of p and ~ / , r n , in ~ cFigs.4.6-2 ~ to 4.6-7 do not change this result qualitatively. Let 11s recall that Eqs.(16) arid (18) limit A r to a finite value. Hence, p = r / A r has an upper limit for any finite value of r, which implies that xl(p) and ~ 4 ( p do ) riot decrease arbitrarily. We derive sorne numerical values. From Fig.4.6-8 we take the value ,mocAr/h = 0.306 for E/,rnoc2 =.5, or E = 5moc2 = 4.09 x [J],arid p = -2; the scale of mocAr/h was considerably expanded to yield 0.306. This yields Asz3 = 1.12 from Eq.(16). To make exp(-Aszsp) in Eqs.(30) and (31) drop to 0.01 we need

which yields po.ol= 4.11. The value of A r follows from the value rnocAr/fi= 0.306 obtained from Fig.4.6-8: ti mo c We obtain A r = 1.18 x

FIG

[m] from mocAr/fi = 0.306. The distance ro.01

= p0.01Ar

(34)

follows from Eqs.(32) and (33) as ro.01 = 4.86 x 10-l3 [m]. The nunlbcrs derived for E/moc2 to ~ 0 . 0are ~ listed in the first line of Table 4.6-1. Also listed there are the corresponding values for E = 10, . . . ,

198

4

DIRAC DIFFERENCE EQUATION I N SPHERICAL COORDINATES

mocAr/h

---t

FIG.^.^-8. Plots of p according to Eq.(23) for E/moc2 = 5 , . . . ,11. The scale of p is expanded compared with Fig.4.6-2. The parameter 1 equals 1; Z = 1.

mocAr/ti --. F1G.4.6-9. Plots of p according to Eq.(23) for E/moc2 = 20,. . . ,200. The parameter 1 equals 1; Z = 1.

mocAr/h -+ FIG.4.6-10. Plots of p according to Eq.(23) for E/moc2 = 500, 1000, parameter 1 equals 1; Z = 1.

The

4.6 SOLUTIONS WITH

So NEAR S

=i

199

TABLE 4.6-1 ENERGYIN NORMALIZED FORME / ~ O ~ ~IN NJOULE; D Ar/(ii/moc) FROM F I G S . ( ~ . ~ -TO 8 ) (4.6-10) WITH EXPANDED SCALEOF mocAr/h FOR p = -2; Aszs ACCORDING TO E ~ . ( 1 6 )po.01 ; = -lnO.Ol/Asz3 ACCORDING TO E ~ . ( 3 2 ) ; Ar ACCORDING TO E4.(33); ro.01 = po.olAr ACCORDING TO E9.(34); mo REST MASSOF ELECTRON. m,o = 9.109 389 7 x [kg], c = 299 792 458 [m/s], h = 1.054572 7 x [Js] moc2 = 8.187 1112 x lo-'* [J], moc = 2.730926 3 x [Js/m] h/moc = 3.861 5934x 10-l3 [m]

2000 that were derived with the help of Figs.4.6-8 to 4.6-10 using a larger scale for mocAr/h. Let us observe that the values of A r and ro.01 for the energies ~ l r n o = c~ 5 . . . 2000 are typical nuclear distances. Hence, the eigensolutions in the point s g = 1 - O ( A r ) in Fig.4.6-1 yield the solutions known from Dirac's differential equations, those in the point s:, = -1 + O ( A r ) yield equivalent solutions for anti-particles with ,mo -+ -mo, but those in the point s 2 = i(l - AT)) yield eigensolutions at nuclear rather than atomic distances. We turn t o a pecularity of our plots and investigate how much smaller the parameter ~ l r n o c=~ 1.67 in Fig.4.6-5 can become and how large ,rn.ocAr/h can become for the jump from +cc to -a. To derive an equation for the location of the jumps we look at Eq.(23) and see that only B = 0 can produce such a jump:

One derives an equation for x:

4

200

DIRAC DIFFERENCE EQUATION I N SPHERICAL COORDINATES

+

(q - 1)[4 - 9(q - 1)2]52 8(q2 - 1)'123: + 13(q - 1) = 0 (36) If the factor 4 - 9(q - 1)2 is zero one obtains negative values of z for any q = ~ / r n o c>~ 1 as required by Eqs.(l5) and (16). There can be no jump. We obtain for this distinguished case q = E/moc2 = 513 = 1.66.. . (37) which is slightly smaller than the smallest value E/moc2 = 1.67 shown in Fig.4.6-5. It may be readily verified that one obtains plots for p like those in Figs.4.6-2 to 4.6-5 for E/moc2 < 513. As ~ / r n . = ~ c513 ~ is approached the jumps in Fig.4.6-5 move to ever larger values of *mocAr/h.w i t h E2/rn$c4- 1 always larger than (5/3)2 - 1 = 1619 and rnocAr/ti increasing arbitrarily we obtain arbitrarily large values of of Eq.(16). The distance po.01 at which exp(-Asz3p) drops to 0.01 approaches zero and both the normalized distance po.01 and the absolute distance ro,ol = po.olAr approach zero. We turn to the solutions on the unit circle defined by the choice s2 = i As22 of Eq.(ll). Equation (15) yields

+

No lower limit is imposed on Ar as in Eq.(16). Substitution of s2, s i = -1 2iAs22, p = p~ ipI and Eqs.(l9), (20) into Eq.(6.8-42) yields two equations for p~ and p ~ :

+

+

The solution of p~ and p~ yields:

4.7

SOLUTIONS WITH Sg IN THE NEIGHBORHOOD O F -1:

20 1

Since p is complex it cannot be chosen equal to a negative integer riunlber to terminate factorial series as shown by Eqs. (24) and (25). We must use Eqs. (6.8-76), (6.8-77) and obtain convergence by conformal mapping or by asyrnptotic termination. However, the knowledge of s 2 = iAsz2 and p = p ~ .i p I permits us to rewrite the terms in front of the summation sign of Eq.(6.8-76):

+

With

s 2 = 1:

+ Asa2 = i ( l - iAsz2)we get for small values of Asz2

S2 = je-iAszz , a

, s~ 2- ~.P , -iAsaap e

= fie-i~Ar.p,

r ; , ~ T=

asz2

(43)

F'rorn Eq.(4.5-24) we obtain for large values of p the equation

'(PI

1

epp-p

= 1

r ( ~ + ~ + l )p

1

p

--.

1

e p e - i ~ ~ b ~

p ,,,PR+~PI- pl+pR

(44)

which is sirnilar t,o Eq.(4.5-25). Equation (42) assurries the form

The product of xl(p) with vl(0), v2(B) of Eq.(6.7-17) yields a t great distances expressions of the forrri

This equation is similar to Eq.(4.5-28) but pl+"R as well as KATneed elaboration.

According to Fig.4.6-1 the equivalent of

s2

in the neighborhood of s =

-1: is s4 = - s 2 as shown by Eq.(6.8-35). Hence, Eqs.(4.6-1)---(4.6-5)derived

for .s2 apply again. We obtain for

s4

202

4 DIR.AC DIFFERENCE EQUATION IN SPHER.ICAL COORDINATES

The three conditions of Eqs. (4.6-7)-(4.6-9) remain unchanged. Equations (4.6-10)-(4.6-12) are replaced by

E < moc2, complex (2) sq=-(i+As42)=-i*

[

1- (m O~Ar)

E > m,oc2,

E > moc2,

(

(&

(T) (-&

)(

- 1)

- I)

]

- 1 ) < 1, complex (3)

> 1, on imaginary axis (4)

Following Eqs.(4.6-10)-(4.6-12) we have given Assl and a positive sign since s 4 in Fig.4.6-1 rotates from s 4 = -i to s 4 = -1 in the mathematically negative direction, while As43 has a negative sign because s 4 on the negative imaginary axis is always in the interval 0 > s 4 -i. The magnitude of the real part of Eq.(2) is always larger than 1. Hence, -(i - AS^^) can never be close to -i; we exclude the case E < moc2. Only Eqs.(3) and (4) can yield values of s 4 close to -i:

>

As42 = [I -

=

mocAr

m,ic4 - 1) (1

[- + (F)~($

-

1

2 0,

s4

on unit circle

(5)

2 0, s4 on imaginary axis (6)

These equations equal Eqs.(4.6-15) and (4.6-16). We obtain once more Eqs.(4.6-17) and (4.6-18). Substitution of s 4 = s o into Eq.(6.8-42) brings

4.7

SOLUTIONS WITH So IN THE NEIGHBOR.HOOD O F -i

203

A real equation is obtained if Eqs(4.6-19) and (4.6-20) are substituted. Again we write Xe = Xel Xe4 with Xe = Xel, Xe4 = 0 or Xe = Xe4, Xel = 0. We obtain with s4 = -i(1 - Asq3) and si = -(1 - 2As43) in first order of Ar:

+

2 - Xe f 8xZamoc2 We substitute Xp = J -

(m324

E2 - 1)-li2]

(8)

and obtain:

Figures 4.7-1 to 4.7-6 show plots of p as functions of m,ocAr/li with the parameter ~ / r n ~ The c ~ .positive sign is used everywhere for f. The plots differ significantly from the ones of Figs.4.6-2 to Fig.4.6-7. We note that the plots of Figs.4.7-2 and 4.7-3 look quite similar to those of Figs.4.6-3 and 4.6-4, but the scale for p differs by a factor 5. The solutions xl(p) and x4(p) of Eq~(6.8-76)and (6.8-77) are written for ~4 = -i(l - i e x p ( - - A ~ ~rather ~) than for ss:

+

204

4 DIRAC

DIFFERENCE EQUATION IN SPHERICAL COORDINATES

0.1

0.15

0.2

mocAr/ h

0.25

0.3

0.35

+

F1G.4.7-1. Plots of p according to Eq.(9) for ~ / r n o c=~5, . . . , 11. The parameter i! equals I; Z=1; a = 7.2975 x

mocbrlh

-

FlG.4.7-2. Plots of p according to Eq.(9) for E/moc2 = 2, . . . , 5. Note the difference of the scale of p compared to Fig.4.6-3. The parameter 1 equals I; Z=1; a = 7.2975 x

FIG.4.7-3. Plots of p according to Eq.(9) for ~/rnoc' = 1.7, . . . , 2. Note the difference of the scale of p compared to Fig.4.6-4. The parameter 1 equals 1; Z=1; a = 7.2975 x

4.7

SOLUTIONS WITH

So

IN THE NEIGHBORHOOD O F

-i

205

F1G.4.7-4. Plots of p according to Eq.(9) for ~ / r n . o c= ~ 1.67, . . . , 1.75. The parameter 1 equals 1; Z=1; a = 7.2975 x lop3.

0.1

0.15

0.2

0.25

0.3

0.35

mocAr/h --+ FIG.^.^-5. Plots of p according to Eq.(9) for ~ / r n , o c=~ 5, . . . , 11. The parameter 1 equals 2; Z=1; a = 7.2975 x

mocAr/h

-

FrG.4.7-6. Plots of p according to Eq.(9) for ~ / r n , o c=~5, . . . , 11. The parameter 1 equals 3; Z=1; cr = 7.2975 x lop3.

206

4

DIRAC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

FIG.4.7-7. Plots of p according to Eq.(9) for ~ 1 r n . = o ~5 ~, . . . ,11. The scale of p is expanded compared with Fig.4.7-1. The parameter 1 equals 1; Z = 1.

-

mocAr/h FIG.4.7-8. Plots of p according to Eq.(9) for ~ / r n o c=~20,. . . ,200. The parameter 1 equals 1; Z = 1.

mocAr/h

+

F1G.4.7-9. Plots of p according to Eq.(9) for E/rnoc2 = 500, 1000, 2000. The parameter 1 equals 1; Z = 1.

4.7 SOLUTIONS

WITH

so IN THE NEIGHBORHOOD OF -i

207

TABLE 4.7-1 ENERGYIN NORMALIZED FORM ~ / m , o cAND ~ IN JOULE;Ar/(fi/rn.oc) FROM SCALEOF mocAr/fi FORp = -2; As43 F I G S . ( ~ . ~ -TO 7 ) (4.7-9) WITHEXPANDED ACCORDING TO E&.(G); po.01 = - ln0.01/As43 ACCORDING TO E ~ . ( 4 . 6 3 2 )A ; r ACCORDING TO E&.(4.6-33); ro.01 = po.olAr ACCORDING TO E&.(4.6-34); mo REST MASSOF ELECTRON.

As in the case of Eqs(4.6-24) and (4.6-25) a negative integer value of p can terminate the factorial series long before p reaches N, which means the question of convergence does not arise. Figure 4.7-1 shows that p can equal any negative integer. Let us choose p = -2 and the parameter ~ / r n o c=~5. We obtain mocAr/ti = 0.3011. Equations (10) and (11) become:

The variable a0 can be chosen, do follows from Eq.(6.8-26) with .so = 54, while a1 and dl follow from Eqs.(6.8-23), (6.8-24). The factor r(p)/J?(p- 1) yields p - 1. The factor (-i)P is eliminated with the help of Eqs.(4.5-27) and (4.6-29): =

(-1)PiP = hip = h f i = f i = 1, i, -1, -i

(14)

xl (p) = fi e-As43p[a~(P - 1) + all

(15)

x4(p) = $'ie-AS43p[d~(P - 1)

(16)

We obtain:

+ dl]

208

4 DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

The comments in the two paragraphs following Eq.(4.6-31) apply again, since Asn3 of Eq.(4.6-16) and As43 of Eq.(6) are equal. We derive numerical values from Figs.4.7-7 to 4.7-9 in accordance with Eqs(4.6-32) to (4.6-34) and Table 4.6-1. The new values are listed in Table 4.7-1. Again we obtain typical nuclear distances. The results obtained in Section 4.6 from Eqs.(4.6-35) to (4.6-37) apply again since B in Eq.(4.6-23) is equal to B in Eq.(9). Let us consider the limit case defined by Eq.(6) for As43 = 0

which yields a minimum value for Ar:

Extracting the square roots on both sides yields two solutions:

Normally one would not pay any attention to the difference between these two equations, but the results of Section 4.5 make us cautious to look out for results that may hold for anti-particles, mo -+ -rn,o. Equation (18) holds both for mo and -m,o, and Eqs.(l9) and (20) call attention to this fact. Consider Eq.(8) for Ass3 = 0 according to Eq.(17) and m,ocAr/li substituted from Eqs.(l9) or (20):

The first line of this equation can be simplified

+

E/moc2 for E/m,oc2 - 1 - E/moc2 - 2 for E/moc2 - 1 -

(22) (23)

4.7 SOLUTIONS WITH

So

IN THE NEIGHBORHOOD O F

-i

209

FIG.^.^-10. Plots of pl according to Eq.(24) for 1 = 1 and Z = 1. The plot holds for ic in Eq.(24) replaced by the plot - for f replaced by -.

+,

~/rnoc~ --+

F1~.4.7-11.Plots of pz according to Eq.(25) for 1 = 1 and Z = I . The plot holds for f in Eq.(25) replaced by +, the plot - for f replaced by -.

+

+

and we obtain two quite different results pl and pa from Eq.(21):

Plots of pl for 1 = 1 are shown in Fig.4.7-10. The solid line holds for +87rZa in Eq.(24), the dashed line for -87rZa. Since pl is never negative we cannot terminate Eqs.(lO) and (11) by choosing a negative integer value for p l . Figure 4.7-11 shows plots of p2 for 1 = 1 and Z = 1. The solid line holds for +87rZa in Eq.(25), the dashed line for -87rZa.

210

4 DIR.AC DIFFER.ENCE EQUATION IN SPHERICAL COOR.DINATES

We extend the analysis to the results of Section 4.6. Instead of Eq.(17) we get now frorri Eq. (4.6-16)

which equals essentially Eq.(17). Equation (18) remains unchanged and Eq.(4.6-22) becomes:

Equations (22) and (23) apply again and we get

These are Eqs. (24) and (25) with reversed sign. We get again the plots of Fig.4.7-10 and 4.7-11 but pl, p 2 are replaced by -pl, - p 2 . 4.8 ENERGY O R MASS RATIOSFOR.

E/rn0c2

> 1 AND

Armin

We had obtained in Eqs(4.6-16) and (4.7-6) a limit for the possible resolutions and A s 4 3

which yields a minimum value for the distance resolution Ar = A r m i n :

(

,mOcArmin

h

12= I-& )+(

(2)

This equation holds for particles with mass m , o as well as for anti-particles with mass -7no. We substitute so = s 4 = -i, S: = -1 into Eq.(6.8-42). The notation Xe = Xel .Ae4 = is used as explained in connection with Eq.(4.6-22). The following equations are readily obtained:

+

4.8 ENERGY OR MASS RATIOS FOR. E/moc2 > 1 AND Armin

kp-

1J-

[

- i(w,

+ w,)p + 4 i n ~ (a - - 1) = O

'I2]

(J~(TTT~ + 4irrZa

112

+

p = 4 - i(wP w,)

211

(z)

-

(wpwm)' I 2

)

(3) (4)

The terms containing w, and w, are worked out with the help of Eqs.(6.8-5) and (6.8-6) using Eq.(2) for ( ~ n ~ c A r , ~ , / h We ) ~ . are only interested in the case E/moc2 > 1:

We note that ( w , w , ) ~ / ~ is always imaginary since W,W, For w,/w, we obtain:

Further we get

is negative definite.

212

4 DIRAC DIFFER.ENCE EQUATION IN SPHERICAL COORDINATES

We have worked out Eqs.(7) to (14) in some detail to avoid more ambiguous signs f than necessary. Substitution of Eqs.(12) and (14) into Eq.(4) yields:

We rewrite this equation for the four possible combinations of the two ambiguous signs f.

Plots of pol and po2 are shown in Fig.4.8-1. Since pol and po2 are never negative we cannot terminate Eqs(4.7-10) and (4.7-11) by choosing a negative integer value for pol or po2. Figure 4.8-2 shows plots of pas and pod . There are negative values in the interval 1 5 E/,moc2 5 2. Since we can generally not choose a negative integer value for pol to Po4 we try the method of asymptotically approached polynomials developed for E < m o c 2 in Section 6.8 from Eq.(6.8-47) to (6.8-52). We substitute so = s4 = - i , S ; = -1 into Eq.(6.8-52). With Xel Xe4 = X e = d m and either Xel = 0 or Xe4 = 0 we obtain:

+

From Eq.(7) we get wpwm = -4 while Eqs.(l3) and (2) yield

Equation (20) is reduced to

4.8

ENERGY OR MASS RATIOS FOR 0.4~

I

armin

213

_ _ ~ ~ _ . _................................ __.___.__.___.

,....... .......

0.3-/

.: ;/

~ / r n> ~1 c AND ~

PO 1 Po2

-

-*-.-.

,I.

I

0.2 -;!'

i'

4

i

0 . 1j

i

I

20

40

60 E / m g c 2 --+

80

100

F1G.4.8-1. Plots of p o l and pon according to Eqs.(16) and (17) for Z = 1 and I = 1 in the interval 1 5 E/moc2 5 100.

FIG.4.8-2. Plots of PO3 and PO4 according to Eqs.(l8) and (19) for Z = 1 and 1 = 1 5 10.

in the interval 1 5 E/moc2

5

10

-

15

20

E/moc2 FlG.4.8-3. Plots of pll and plz according to Eqs.(23) and (24) for p = 1, Z = 1, 1 = 1 in the interval 1 5 E/moc2 5 20.

2

1.5

1

2.5

3.5

3

4

E/moc2 i FIG.^.^-4. The functions pol, poz, p n , plz according to Eqs.(l6), (17), (23), (24) for Z = 8, p = 1, 1 = 1 in the interval 1 5 ~ / m o c5 ~4. The apparent intersection of p12 and pol slightly to the right of E/moc2 = 1 disappears when the resolution is increased. Q,\>,-~"

1.5-

I* , a .

,/*'

1. ,,sr

0.5 -

-0.5

-

Po 1

,J'

.---/< ..................................

......... ............ ._.. ,a', __.. ..' ,,,' 1.5

Po2

,'*5

3.5

3

2.5

,,'

4

,,,/'

-

1,'

-1 t'

Pll

E/moc2 FIG.^.^-5. The functions pol, po2, pll, plz according to Eqs.(l6), (17), (23), (24) for Z = 8, p = 2, 1 = 1 in the interval 1 ~ / r n . ~ c4.~

<

<

,',

-0.27. -0.275

%'

-0.28

-

-0.285

-

,-.

'-..-, Po2

---._.__ ---__

, ,,'

-0.29-

___________________----:------=:=-----------

PI2

I

-0.295-

j 20

40

60

80

100

120

140

E/moc2 -+ F1c.4.8-6. The functions poz and plz according to Eqs.(l7) and (24) for Z = 14, p = 2, 1 = 1 in the interval 1 5 ~ / r n , o c5~ 150.

4.8 ENER.GY OR. MASS R.ATIOS FOR. E/rnoc2 CHARGENUMBERZ

> 1 AND Armin

215

TABLE 4.8-1 AS

FUNCTION OF I ACCORDING TO E&.(25).

We write p l l and pl2 for p to resolve the ambiguity of the sign f :

Figure 4.8-3 shows plots of pll and pl2 for p = 1, 1 = 1, and Z = 1. These plots intersect with the plots of Fig.4.8-2 in the interval 1 < E/,moc2 < 2. We are here interested in larger values of E/,moc2, but no intersections of the plots of Figs.4.8-1 and 4.8-2 with the plots of Fig.4.8-3 can occur for E/?noc2 > 2. Since the four plots of pol to PO4 in Eqs.(l6) to (19) can be combined with the two plots of pll and pl2 in Eqs.(23) and (24) in eight ways we need some guidance which combinations to choose. We observe that the sign of the terms in brackets of po2, p04, and pl2 changes for

a t a sufficiently large value of ~ l r n ~ Table c ~ . 4.8-1 shows for which values of Z this will be the case. For an overview of how intersections between pol, p02 and p l l , pl2 can be produced by choosing Z and p refer to Figs.4.8-4 t o 4.8-6. For Z = 8 , p = 1 in Fig.4.8-4 the intersections are all a t E/*moc2 = 1. A more interesting result is shown in Fig.4.8-5 for Z = 8, p = 2. The plot pl2 irltersects the plotspol a t about E/rnoc2 = 2.5 and p02 at about E/rnoc2 = 2. More precise values of E/moc2 are listed in the first two rows of Table 4.8-2. Also show11 are ~rr,ocAr,i,/h according to Eq.(2) and Armin:

Figure 4.8-6 shows the intersection ofp12 with po2 for Z = 14, p = 2 a t about ~ / r n= ~ 100; c ~ this is the only intersection for Z = 14, p = 2. Again a more

TABLE 4.8-2

LIST OF FIGURES, INTERSECTIONS OF PLOTSSHOWN,CHARGENUMBER Z, QUANTIZATION NUMBER p , WITH T H E RESULTING VALUESO F ~ l r n o cFROM ~ Fl~s.4.8-5A N D 4.8-6, m.ocArmin/hACCORDINGT O E Q . ( ~A)N D ArminACCORDING T O E ~ . ( 2 6 ) .

Figure intersec Z

p

E/m.oc2

m,ocAr,i,/fi

Armin

precise value of ~ / , r n . is ~ cshown ~ in Table 4.8-2 together with mlocArmin/h and Armin.The values shown for Arminin Table 4.8-2 are typical nuclear rather than atomic distances. The problem of finding intersections between pol, po2 and p l l , pl2 is more complicated than suggested by Figs.4.8-4 to 4.8-6. There are intersections for Z = 9, 10 and p = 2, but not for Z = 11, 12, 13. Intersections start again for Z = 14 and continue for Z = 15, 16 but with decreasing values of ~ / m . for ~ c the ~ intersections. To get some overview of the periodic system of elements we observe that the intersections between pl2 and po2 in Figs.4.8-5 and 4.8-6 are determined by the equation P12 = Po2

(27)

For large values of E2/,moc2we write p12 and Po2. Equation (27) becomes

Certain elements with their charge number Z are listed in Table 4.8-3 together with pO2and p12 p according to Eq.(28) for 1 = 1. We then choose the value of p that yields the smallest value of p12 that is larger than 1002. This value of p12 is listed in Table 4.8-3. The resulting value of E/moc2 is obtained by making plots like the one in Fig.4.8-6 and using some numerical approximation t o obtain ~ / accurate ~ to ~six decimals c as ~ shown in Table 4.8-3. The values of E/moc2 of Table 4.8-3 are used in Table 4.8-4 to derive mocArmin/fi according to Eq.(2) and Arminaccording t o Eq.(26). We see

+

4.8

ENERGY OR MASS RATIOS FOR

~ l r n >~ 1c A~N D Armin

217

TABLE 4.8-3 SELECTEDELEMENTS FROMT H E PERIODIC SYSTEMO F ELEMENTS SHOWING T H E CHARGENUMBER Z, Po2 AND pi2 p ACCORDING T O E ~ . ( 2 8 ) T, H E QUANTIZATION NUMBERp ACCORDINGT O E~.(24), p 1 2 , AND ~ l r n o cFROM ~ T H E RELATION p o ~= plz FOR1 = 1.

+

Element

Z

that Arminhas for all elements typical nuclear distances of 10-l3 t o 10-l5 m. No clear pattern is recognizable. Osmium (Os), Iridium (Ir), and Silicon (Si) yield the largest values for ~ / r n , o c Sodium ~, (Na) and Carbon (C) yield 1, which is useless for Eq. (1). Let us consider some charge numbers Z much larger than those in Table 4.8-4. Figures 4.8-7 to 4.8-9 show plots of p02 and pl2 for Z = 10 000, 15 000, and 20000. The intersections of poz and plz are at ~ l r n = ~ 786.0296, c ~ 1486.672, and 2680.501. These large values are the purpose of showing

218

4

D I R A C D I F F E R E N C E EQUATION IN S P H E R I C A L COORDINATES

TABLE 4.8-4 ELEMENTS FROMT H E PERIODIC SYSTEMO F ELEMENTS, THEIR CHARGE NUMBER Z,QUANTIZATION NUMBER p , ~ l m o FROM c ~ TABLE4.8-3, AND T H E RESULTING VALUESOF mocAr,i,/li ACCORDINGTO E&.(2) AS WELLAS O F ArminACCOORDINGTO E ~ . ( 2 6 ) 1; = 1. Element

Z

p

Ne Ar Kr Xe Rn Na Ka Rb 0s Fr F C1 Br I At

10 18 36 54 86 11 19 37 55 87 9 17 35 53 85

2 2 3 4 6 1 2 3 4 6 2 2 3 4 6

4.584780 2.993256 3.553579 3.573578 4.637487 1 2.548919 3.134755 3.228624 4.202668 2.715467 3.723851 4.476241 4.106520 5.037681

0.223 0.354 0.281 0.291 0.221 0.427 0.337 0.326 0.245 0.396 0.279 0.229 0.251 0.203

8.63 x 1.37 x 1.09 x 1.13 x 8.53 x 1.65 x 1.30 x 1.26 x 9.46 x 1.53 x 1.08 x 8.85 x 9.70 x 7.82 x

Fe Ru

26 2 44 3 76 6

1.522218 1.965205 46.23822

0.871 0.591 0.0216

3.36 x lo-13 2.28 x lo-13 8.35 x 10-l5

2 3 6 0 2 3 4 6

1.468844 1.885190 22.724234 1 103.4024 13.848867 6.493666 6.933404

0.929 0.626 0.044

3.59 1 0 - 1 ~ 2.42 x lo-13 1.70 x 10-l4

0.00967 0.0724 0.156 0.146

-

92 7 94 7

22.955720 12.832754

0.0436 0.0782

1.68 x 3,02 x 10-l4

0s

Co Rh Ir C Si Ge Sn Pb U P1

27 45 77 6 14 32 50 82

~ l m o c rn~cAr,~,/li ~

Armin[m]

3.73 x 2.80 x 6.02 5.63 x

lo-14

10-l3

lo-13 lo-13

lo-14

lo-13

10-l3 lo=13 10-l4 lo-13 lo-13 10-l4

lo-14

10-l4

lo-15 lo-14 lo-14 10-l4

illustrations for unrealistic large values of Z. A value ~ / , r n , = ~ c1836.152 ~ is well within the range shown, but we would interpret it as the mass ratio ,m,/m, of proton and electron. ~ cwell ~ ,as ~ m o c A r m i n / h , arid Arminfor Table 4.8-5 lists Z, p, ~ / r r ~ as Figs.4.8-7 t o 4.8-9. The electromagnetic Coulomb force is not strong enough t o produce the values ~ / m ~ cand ' Arminin Table 4.8-5, but the strong interaction forcc seems to be right for the required force and distance A r m i n . Hence,

4.8

ENER.GY OR. MASS R.ATIOS FOR

~ / m o> c ~1 AND

Armin

219

~lrnoc' -4 20Q

-456.25

400

600

800

1000

\, \, \,

-456.5 -456.75

'.*'

-457

'. "-... ,.Po2

.I. . ..

-457.25

-...

PI2

-457.5

--.__ ---._

,----.._---------------------.-.-...-.-~--

1

-457.75 -458

f

F1G.4.8-7. The functions poz, pl2 according to Eqs.(l7) and (24) for Z = 10000, 5 1000. The intersection of poz and

p = 688, 1 = 1 in the interval 1 5 E/moc2 p12 is at E/rnoc2 = 786.0296.

\ -685.5

-

1000

1500

2000

\$

\. \,

.' '.

-686

'-,....?2 2J

-686.5

. ---_

Pl2 -.-_ ,,------------------------------>-=- ---:---------.

- 68

-687.5

,

500

'

I

-688

F1G.4.8-8. The functions po2, p12 according to Eqs.(l7) and (24) for Z = 15000, p = 1032, 1 = 1 in the interval 1 5 ~ l m o c ' 5 2000. The intersection of poz and p12 is at ~/m.oc' = 1486.672.

500

-915.25

\*

-915.5

1 0 0 0 1 5 0 0 2000 2 5 0 0 3000

'* '*

-915.75

'.

"..

. %Po2 ___ P12 -.-.---_ -.-_ - 9 1 6 . 2 5 ,--------------------------------------------pz-= -916

-916.5 -916.75 -917

1' 3

:

F1G.4.8-9. The functions poz, pl2 according to Eqs.(l7) and (24) for Z = 20000, 5 ~ l r n o c ' 5 3000. The intersection of po2 and p12 is at ~ l r n o = c ~2680.501. p = 1376, 1 = 1 in the interval 1

220

4 DIR.AC DIFFERENCE EQUATION IN SPHERICAL COORDINATES

TABLE 4.8-5 CHARGE NUMBER Z, QUANTIZATION NUMBER p, AND T H E LISTOF FIGURES, RESULTING VALUESOF E/rn0c2 FOR THE INTERSECTIONS O F pon AND p l a , mocAr,i,/h ACCORDING TO E Q . ( ~ AND ) AT,;, ACCORDING TO E ~ . ( 2 6 )1=1. ; Figure

Z

p

~ l r n . 0 ~ ~r n o c ~ ~ , ; , / hAT,;, [m]

the replacement of the electromagnetic Coulomb force in Dirac's difference equation by a centrally symmetric strong interaction force may lead to the theoretical derivation of mass ratios as an eigenvalue problem. Due to the lack of a strong interaction force that is as simple and universally accepted as the Coulomb force, only experts of the strong interaction force can expect to solve this problem in a satisfactory and generally acceptable way. We turn to the solution in the point s = s2 = i in Fig.4.6-1. Equations (1) and (2) hold again. From Eq.(6.8-42) we obtain with so = s 2 = i, si = -1 with @ written rather than p to distinguish it from p in Eqs.(4) and (15):

We recognize that @ is the negative of p in Eqs.(4) and (15) if ~ / r n , o cis~ replaced by ~ / ( - m ~ c ~ ) :

d

Next we substitute so = s2 = i into Eq.(6.8-52) and obtain with m = Xe and @ = p the relation

~ c ~ by El(-moc2). This is Eq.(22) with a reversed sign and ~ / m replaced The sign reversal of moc2 is trivial here due to the ambiguity of the sign f in front of 87rZa:

4.8 ENER.GY OR MASS RATIOS FOR E/moc2 > 1 AND Armin

221

Equations (30) and (33) suggest to associate the solution at s = i in Fig.4.6-1 with the solution at s = -1 for anti-particles, while the solution at s = -i is associated with the solution at s = +1 for particles. The results in Tables 4.8-2, 4.8-4, and 4.8-5 hold both for particles and anti-particles if we replace moc2 by -moc2. In Table 4.8-3 we must also reverse the signs of P o 2 , P l 2 l and p.

5 Inhomogeneous Equations for Coulomb Potential 5.1 QUANTIZATIONOF THE INHOMOGENEOUSTERM We turn to the quantization of the inhomogeneous term of Eq.(4.3-9). The term Q defined in Eq.(l.l-44) makes this term very complicated:

Since we do not know how to quantize ( p - eA,)2 in a derlominator we use a series expansion to remove the denominator for small values

( p - e ~ , ) ~ / r n ; c<< ~1 arid obtain in first-order approximation:

This solves part of the problem. We still have to explain a factor

of Q in Eq.(l). One rnay do so with the help of Eqs(2.1-24)-(2.1-26). The Cartesian coordinates are replaced by spherical coordinates:

Substitution of Eqs.(5) and (6) into Eq.(4) yields

5.1 QUANTIZATION

OF THE INHOMOGENEOUS TER.M

223

and Q in Eq.(l) is reduced to

We note that the sequence of the factors in Eq.(8) is unimportant since an interchange of the factors only requires that ( p - eA,)' commutes with 1 and with itself to yield the same result. The next step is to substitute the simplified Eq.(8) into Eq.(l). We obtain:

I = - [2hcAe -[ a .( p - eA,)(p - eAm)' + Bmoc(p - e ~ m ) ' ] em,;c2

[z;;

(la (P - e-4,) + iimoc](p- ell,)'

=--

3 1 - --

+

@o [a. ( p - eAm) pmoc](p - e ~ , ) ' ) 2 nr;c2 In Eqs.(4.3-9)-(4.3-11) and (4.3-15) we have made the substitutions

-5

h a

(:%-

grad -eAm

-4

(9)

-

z grad -eA,

7

to achieve q~~antization first in Cartesian coordinates, then in general coordinates, and finally in spherical coordinates. We apply this procedure t o (p - eAJ2:

i

-h2v2

+ 2i,eh,Am. grad f i e h d i v A, + e ' ~ ;

(11)

224

5 INHOMOGENEOUS EQUATIONS FOR COULOMB POTENTIAL

For finite differences we apply a tilde over the difference operators: d 2 , grad, div. We rnay thus write

(p-e~,)'

-+

-h2d2

+ 2iehA,

. grad +ietidivA,

+ e2A$

(12)

to produce quantization in general coordinates with finite differences. This equation may be written explicitly for spherical coordinates in analogy to Eqs(4.3-12), (4.3-15), (4.3-18) but in addition to grad one has to work out d2and d i v ~ , . We do not need this here as will become evident presently. Equation (12) defines the quantization of the term /3moc(p - eA,)2 in Eq.(9). The next more complicated term is ( p - eAm)(p- eA,)2:

x (-h2

. grad +iehdivAm

+ e2A$)

+ 2eh2 grad(Am . grad .) + eh2 grad(div A, ihe2 grad(A; .) + eh2A, d2. - 2ie2hAm(A, . grad .)

= ifi3grad(d2 -

d2+ZiehA, .)

- ie2Am(divA, .) - e3A,A;

.)

.

(13)

The dot . indicates where a function Q, to which the operators are applied, should be inserted. One could write the operators grad(d2) and grad div in more detail as grad was written in Eq.(4.3-18) but it is prudent to postpone this exercise until the equations are simplified. Turning to the next more complicated term Pm,oc(p - eA,)4 in Eq.(9) one could in principle square Eq.(12). But this would yield 25 terms and ( p - eA,)(p - eA,)4 would yield even more. We use Eq.(2) and ignore the terms multiplied by (p - eA,)4. Let us observe that this means the right side of Eq.(3) is reduced to 1. The simplified Eq.(9) becomes:

We turn to the terrn L c / a . Each term LC has three components LC,, LC,*,LC, and each component has five subconiporlerits listed in Eqs.(1.2-16) -(1.2-30). Let us work out the first three Eqs.(1.2-16)-(1.2-18). With the help of Eqs.(2), (3), and (10) we obtain:

5.1

QUANTIZATION OF THE INHOMOGENEOUS TERM

225

Since we have accepted ( p - eA,)(p - e ~ , ) in~ Eq.(14) we must accept the term (p - eAm)2/*mzc2in Eq.(15). We do not write it out anymore than in Eq.(14) but may substitute it from Eq.(12). The substitution of Eq.(15) into Eq.(14) yields a much simplified quantized equation. We write Irl instead of I due to the notation Lc,l in Eq.(15):

For Lc81 and CCv1of Eq~(1.2-17)and (1.2-18) we obtain in analogy to Eq.(lG):

(! d

2Ac [Aed L - A,, Ipl= ern.0~ i AT Ap e ~ m r )

(

ri

1

16

i ArA8 p

-

e~m~)]

We write the quantized Eq.(4.3-9) as matrix equation like Eq.(2.1-30). The homogeneous part follows from Eq.(4.3-19). The inhomogeneous term of Eq.(4.3-9) is reduced to +L,/ca in our approximation. The inhomogeneous term is written according to Eq.(14)

After quantization of I we write

where I,, I*, I, have each five components according to &,,I to Lc,5 in Eqs(1.2-16)-(1.2-30). The first of the five sets of components is shown IP1 in Eqs. (16)-(18). Section 6.5 shows that the extension to as L r l , LUz, . . . , LCv5 will have to wait for a great deal of advancement of the calculus of finite differences by mathematicians before it becomes useful. The inho~nogeneousmatrix equation becomes with the help of Eqs.(4.3-18) and (4.3-19):

The first of the three equations represented by this matrix equation assumes the following form:

227

5.1 QUANTIZATION O F T H E INHOMOGENEOUS T E R M

The difference operators are rewritten in explicit form and cAt = A r is substituted:

Multiplication with 2iArlti and the choice Ap = A0 = 1 yields:

Since a,, and P are matrices of rank 4 we must interpret the following matrices:

and TIo to be

The rnatrices for Q1(p f 1,q, J, 0) and @l(p,q,<,Q f 1) differ from Eq.(25) only by the substitution of p 1 and Q1i for p or 0. Using a,, and P from Eq~(2.1-10)and (2.1-9) we obtain from Eq.(24):

*

228

5

INHOMOGENEOUS EQUATIONS FOR COULOMB POTENTIAL

The followirlg four equations are obtained by performing the matrix multiplications:

For the term with a,, in Eq.(21) we obtain an equation that closely resembles Eq.(22):

5.1 QUANTIZATION

OF THE INHOMOGENEOUS TER.M

229

Instead of Eq.(23) we obtain

Multiplication with 2iArlFi and the choice Av = A6 = 1 yields:

The two terms in Eq.(27) containing the rrlatrix a,, are replaced b y the corresporidi~lgterms with a,, of .Eq.(2.1-10)

and a second set of four equations is obtained:

i

+ -PA0 [Qii(~,rl+

1,E,O) - @ii(~,q--1,E,O)I+[ Q i 4 ( ~ , 0 + 1-)* i 4 ( ~ , 0 - 1 ) 1 - Xt9*11(~,8) f i(X2 - X3)*14(~,6 ) = i ~ . 9 * 0 4 ( ~ , 0 ) (40)

We turn t o the term with a,, in Eq.(21):

Equations (23) and (33) are replaced by:

-

e eAmv - -4, - Pmoc) 91(P, 0) = IP*o (P, 0) C

Multiplicatiorl with 2iArlh. and the choice A< = A0 = 1 yields:

(42)

The two terms in Eq.(27) containing the matrix a,, are replaced by the corresponding terms with a,, of Eq.(2.1-10):

and our t,hird set of four equations is obtained:

-

1

sin(vA29)Acp

[ @ 1 4 ( v~ , t

+ l , Q-) ' Q I ~ ( P , ~tI -, 1,Q)I

+ [@12(p,v,t,e+ 1) - @12(p,.,t,Q 1)1 + i[AV@i4(p,6) + (A2 + A3)@12(~,Q)] = ~ Y , @ o ~ (8P), -

(47)

232

5

INHOMOGENEOUS EQUATIONS FOR. COULOMB P O T E N T I A L

5.2 SEPARATION OF

THE

FUNCTIONS Qlj(p, 6')

We have to solve three sets of equations with four equations each, Eqs.(5.1-28)-(5.1-31), Eqs.(5.1-37)-(5.1-40), and Eqs(5.1-46)-(5.1-49). A first sirrlplification is possible by observing that Eq.(5.1-28) is transformed into Eq. (5.1-29) by the substitutions a14 + W13,

Qll

+

Q12)

q01

a02

(I)

whilc Eq.(5.1-31) is transformed into Eq.(5.1-30) by the substitutions Qll

-i Q 1 2 ,

Q14

+

Q13, Q 0 4

-i Q 0 3

(a]

Hence, we have to solve only two equations rat,her than four:

Xr

= 2AreAmr(p,rl, E, O), A2 = 2Are+,(p,

rl, E, 8 )

X3 = 2Armoc/ti, y, = 2ArIr/li The substitutions q14

+

-q13,

Qll

+

Q l 2 , w01

+

q02

(5)

transform Eq.(5.1-37) into Eq.(5.1-38) while the substitutions

* -@12, Q14

Q13, Q04

(6) transform Eq.(5.1-40) into Eq.(5.1-39). Hcnce, only two equations nccd to be solved: Qll

+

+

Q03

5.2 SEPAR.ATION O F THE F U N C T I O N S q l j ( p , 6)

233

Finally, the substitutions

transform Eq.(5.1-46) into Eq.(5.1-47) while the substitutions

t,ransform Eq.(5.1-48) into Eq.(5.1-49). Hence, once rrlore only two equations need t o be solved. We introduce R to shorten writing; it is treated as a constant since it contains neither E nor 6:

Only p and 6 appear in Eqs.(3) and (4) in the form p f 1 and 6 f 1. Correspondirlg remarks for q and 6 or E and 0 apply to Eqs.(7), (8) and (12), (13). Hence, we concentrate on Eqs.(3), (4) t o develop a method of solution that may then be applied to the two other pairs of equations. In symbolic notation we may write Eqs.(3) and (4) in a more compact form:

Particular solutior~sof the homogeneous equations are obtained with the Bernoulli product method:

234

5

INHOMOGENEOUS EQUATIONS FOR COULOMB POTENTIAL

Substitution into Eqs.(l4) and (15) brings:

We assume the magnetic potential A, to be zero and the electric potential 4, to be a Coulomb potential:

A, = 0, A, = 0; 4, = -ZZec/pAr, X2 = - 8 r Z a l p Equatioris (18) and (19) are rcwrittcn:

Solutiorls in the form 21/11(f @) = w l l ( ~ ) u l ( @ )$,1 4 ( ~ ,@) = ~ 1 4 ( ~ ) ~ 1 ( @ ) are derived in Section 5.3, Eqs.(5.31), (5.33), and (5.34):

(20)

5.3

SOLUTIONS FOR

~ ( 6 AND ) ~ ( p )

235

The next step is to derive with these equations a particular solution of the inhonlogeneous Eqs.(l4) and (15). With y, from Eq.(4) we get for I, = Irlfrom Eqs(5.1-20) and (5.1-16):

ys =

2ArIr1

4ArXc

[iev (- -

=emotic

h 1 A iArAls p Av - 'Arnfi

Since Eq.(20) states A, = 0 we get yr = 0 and Eqs.(14), (15) becorne homogeneous. If we look up the components of I,, of Eq.(5.1-19) in Eqs(1.2-16), (1.2-22), (1.2-25), and (1.2-28) we see they are all zero for A, = 0. Furthermore, Eq.(1.2-19) shows that L,,2 and thus Icr2 is zero for = 0. Hence, Eqs.(l4) and (15) are always homogeneous for A, = 0 and 4, = 0. However, the solution of these equations in Section 5.3 is not a useless exercise, since the solution of the homogetleous Eqs.(21), (22) yields the missing solution for I! = 0 in Eq(4.3-39).

+,

5.3 SOLUTIONS FOR ~ ( 0 AND ) ~ ( p ) The variables p and 0 of Eqs.(5.2-21) and (5.2-22) can be separated by the Bernoulli product method, with ul simplified to v: $11

(p, 0) = wll (P)v(@), $ I ~ ( P 0) , =~14(~)v(e)

0)

One obtains

The equation AV -- -i-v EAt do ti

(4)

is worked out in Eq.(6.7-16). It has the solutions

For the functions of p we obtain from Eqs.(2) and (3) with w, and w p defined by Eqs.(6.8-5) and (6.8-6):

236

5

INHOMOGENEOUS EQUATIONS FOR COULOMB POTENTIAL

These equations are equal to Eqs.(6.8-7) and (6.8-8) if we make there the substitutions

We may follow the calculations from Eq.(6.8-9) on by making the further substitutions

Instead of Eqs. (6.8-16) and (6.8-17) we obtain:

-

In Eq.(6.8-18) we change a,, d,, so, and p to C,, d,, 50, and ip:

Furthermore, we change S,,j and a,,j in Eqs.(6.8-23) and (6.8-24) to and b , , j . The rewritten Eq. (6.8-23) becomes:

8,,j

5.3 SOLUTIONS FOR. ~ ( 6 AND ) ~ ( p ) Equation (6.8-24) is rewritten correspondingly:

We skip Eq.(6.8-25) but rewrite Eq.(6.8-26) for the syrnbols with a tilde:

-

iw,So

-

do = -a0 if; - 1

Equation (6.8-27) becomes

We recognize that so and So arc equal

-

So

= So

and Eqs.(6.8-30) to (6.8-36) hold for S2 = S2

S3 = S3

if4

= Sq,

if:,

= Sg

(18)

Table 6.8-1 arid Fig.6.8-1 apply unchanged. Equations (6.8-37) to (6.8-41) rnay be rewritten with tildes and Xel = Xe2 = 0. Equation (6.8-42) is rewritten explicitly in this way. It holds for all values of s o = So:

We must decide which value of so in Fig.6.8-1 to choose. Followirig the text between Eqs.(6.8-42) and (6.8-43) we choose so = s3 = 1 - O ( A r ) to obtain @ with the help of Eq.(6.8-32):

This is the same equation as Eq.(6.8-45) for p ~ There . is no imaginary term in Eq.(20) arid p~ of Eq.(6.8-46) is zero. We cannot choose negative integer values for 6to make a factorial series like the one in Eq.(6.8-76) terminate. Hence, we follow the asymptotic termination process from Eq.(6.8-47) to (6.8-51) but write tildes d l 2,5, d; and Xel = Xe4 = 0. The following replacement for Eq.(6.8-52) is obtained:

For so = s3 = 1 - O ( A r ) we obtain:

(p + L - 1)'

+ 2($ +

L

- 1)

+ 1+ (47rza)'

=O

(22)

Equations (6.8-59) and (20) bring:

The following calculation is dorie in great detail since the result will be unexpected and we want to emphasize why this is so. We shift + ( ~ T Z C Y ) ~ in Eq.(23) to the right side and take the square root on both sides:

Note the imaginary unit i on the right side. Next we move - 1 + ~to the right side, multiply by (1 - ~ ~ / m ; c ~ )and l / ~square , both sides of the equation:

5.3 SOLUTIONS

FOR

~ ( 0 AND ) ~ ( p )

Division by ~ ~ / r n yields ic~

(F

( 4 n ~ a=) ~

rn; c4

-

1)

(L -

1 54 7 r i ~ a ) ~

(26)

and we get finally E:

This is Eq.(6.8-74) for lc2 = 0. We proceed according to the text following Eq.(6.8-74), write L - 1 = nl, and expand Eq.(27) in a power-series of a . With

(n,' f 4 n i ~ a ) -A~

(1

87riZa/n;)

(28)

we obtain:

[

E = frn,oc2 1 +

(

( 4 7 r ~ a ) ~ 87r;a lfn12

j ] -'I2

The imaginary term of E is unexpected. To recognize its meaning we write E as a sum of a real part ER and an imaginary part EI:

E

= En

+ iEIl

EI = *rn0c2

(7)

47rza

3

(30)

From Eq.(5) we obtain:

The imaginary term of Eq.(29) introduces an attenuation exp(-EIOAt/h) to ul(0) and uz (0). We note that such an attenuation terrri can also follow frorn Eqs.(6.8-72) and (6.8-74) if Z is large enough:

240

5

INHOMOGENEOUS EQUATIONS FOR COULOMB POTENTIAL

TABLE 5.3-1 MINIMUMVALUEO F Z REQUIREDIN E$.(6.8-74) TERM;k2 = 1 ( 1 + 1)/4.

TO

PRODUCE AN

IMAGINARY

Table 5.3-1 shows for which values of Z and 1 we get an imaginary term in Eq.(6.8-74). We recognize why we did not obtain a value of E for 1 = 0 in Eqs.(6.8-75) and (4.3-39). The calculation was restricted to real values of E and non-attenuated functions 111(Q), vz(Q).Equation (29) provides the missing case I = 0 for Eqs(6.8-75) and (4.3-39). In analogy to Eqs.(6.8-76) and (6.8-77) the inverse Laplace transform of y l l and ylq of Eq.(9) yields:

6 Appendix 6.1 CALCULATIONS FOR SECTION 2.3 We generalize the definition of finite difference quotients from that given for one variable 8 in Eqs.(1.3-3) and (1.3-5) to two variables and 8:

<

+

b ~ ( <8), A(<,8 A8) - A(<,Q - AQ) d8 2A8 1 = -[A(<, Q + 1) - A(<,8 - I)] for A8 = 1 2

1 2

= -[A(<+ 1,Q)- A ( < - 1,Q)) for A< = 1

(1)

(2)

For the sccond order difference quotient wc list first the mixed difference qiiotients:

For the pure second order diffcrcncc quotient we must observe the defiriitiori of Eq.(1.3-6):

6

242

APPENDIX

0) - A(<,@+ AO) - 2A(C, 8) (aQ)2

= A(<,8

+ 1) - 2A(<, 0) + A(<, 0 - 1)

0) - A(C+A<, 0) - 2A(C, 8) (Ao2 =

+ A ( < ,8-AQ) for A0 = 1

+ A(<-A<,

A ( < + I , @) 2A((,Q) + A ( < - 1 , Q ) for A< = 1

(4) 0)

(5)

We observe that the definitions of E q ~ ~ ( l . 3 - 6 (4), ) , and (5) produce the variables f 1, 0 f 1 but not f 2, 0 f 2. We apply these results to the separation of qoland QO4 in Eqs.(2.3-5) and (2.3-8). Let us copy these two equations:

<

<

Using Eqs.(l) and (2) we may rewrite Eqs.(G) and (7) in symbolic form:

We differer~ciateEq.(8) with respect to 0 and Eq.(9) with respect to <, using Eqs.(3)-(5):

The first term of Eq.(lO) cquals the second tcrm of E q . ( l l ) sincc thc scquence of differenciation with respect to C and 6 can be reversed. Hence, we may eliminate these two terms and obtain the following equation:

and d Q o 4 / d o can be expressed froni Eqs.(8) and (9), The terms arid substitutcd into Eq.(12). Two terms with qO4 cancel. We obtain the following equation for Qol(C, 6):

Written explicitly, this equation assumes the following form:

+

[QOI(C 1,e) - 2qol(c, 0) - [ Q o(C, ~ 0

+ q o l ( s - 1,611 + 1) - 2Q0l(C, 0) + Q0l(C, o - 1)1

Tlic cquatio~ifor Qo4(C,6) is obtained by diffcrenciatirig Eq.(8) with respect t o C and Eq.(9) with respect to 8, arid following the procedure from Eq.(lO) to Eq.(14). We obtain again Eqs.(l3) and (14) but Qol is replaced by @04.

6 . 2 CALCULATIONS FOR. SECTION3.2 In order to separate @lland Eqs.(6.1-6) and (6.1-7) by them:

in Eqs(3.2-18) and (3.2-21) we replace

244

6 APPENDIX

Written in symbolic form according to Eqs(6.1-8) and (6.1-9) these equatioris become:

We differenciate Eq.(3) with respect to 0 and Eq.(4) with respect to analogy to Eqs(6.1-10) and (6.1-11):

C in

The first term in Eq.(5) equals the second term in Eq.(6) since the sequence of differenciation can be reversed. These two terms may be eliminated:

Tlic terrns A q 1 4 / 6 ~and d ~ ~can~be expressed / d ~from Eqs.(3) and (4), arid substituted into Eq.(7). Two terms with q14cancel. The followi~lg inhomogeneous equation for 611 is obtained:

The equation for Ql4(C,O) is obtained by differenciating Eq.(3) with respect to C and Eq.(4) with respect to 8, and following the procedure from Eq.(5) to Eq. (8). The homogeneous terms follow Eq.(8) but the inhomogeneous terms do not:

6.2 CALCULATIONS FOR. SECTION 3.2

245

The function G4 = G4(C,B) has the three components shown by Eq.(3.2-21):

whcrc Gp4(C,B), Ge4(C,0), and Grnl(C,O) are shown by Eq~(3.2-12), (3.2-14), and (3.2-16) for j = 1 or j = 4. We still need A G ~ / A (and A G ~ / A $ .With the help of Eqs.(l.3-10) and (1.3-5)-(1.3-8) we obtain from Eq.(3.2-12):

Frorri Eqs(3.2-14) and (3.2-16) we get

246

6

APPENDIX

and the combirlatiorl of Eqs.(ll)-(13) yields:

The differenciation with respect to 0 takes a little longer:

Frorn Eq~(3.2-14)and (3.2-16) we obtain the finite derivatives

and the combination of Eqs.(15)-(17) yields:

To obtain d G 1 ( ( , Q ) / d ( and dG4([, e)/dO one must use j = 1 in Eqs.(ll), ( l a ) , (15), (16) and j = 4 in Eqs.(l3), (16).

We have t o solve the inhomogeneous difference equations (3.2-58) and (3.2-59). The equations are equal except for the inhomogeneous term. We solve Eq.(3.2-59) for T,(Q) and only indicate in Eqs.(29) and (30) below the changes required for Eq.(3.2-58) and S,(O). This is done because S,(e) will be elirrliriated by a boundary condition while the intermediate steps of the solution of T,(6) will be frequently referred to. A shorter notation is used. Equation (3.2-59) is brought into the following form1: 'See

Norlund 1924, p. 396; 1929, p. 22, 125; Milne-Thomson 1951, p. 374

248

6 APPENDIX

~ Z S ( O + ~ ) + ~ ~ S ( O + ~ ) + ~ O S ( O ) + ~ - ~ S ( O - ~ ) + ~ - ~ SHs,(O, ( O - ~IC,) )= P2 = eixz

, p1 = -4

cos * e ' i ~ 2 / 2 , po = 2

N

p-l = -4cos

+

2lrK

p-2 = ,-,LA2

-e-iX2/2

N

H S , ( O , ~=) - e i X 2 / 2 ~ s K ( 01 , ~+) 2 ~ 0 s - G , , ( $ , K ) ~ T K

N

- e - i X 2 / 2 ~ s , ( 0- 1 , n)

(1)

The general solution of the homogeneous equation PZS(O

+ 2 ) + p l s ( o + 1) + pos(e)+ p P l ( o

-

1)

+

~ - Z S ( O-

is given by the two functions in Eqs.(3.2-65) and (3.2-66) for

2) = o

(2)

< 1:

We use the method of variation of the constant to find a particular solution s ( 8 ) of Eq.(l) with the help of these four solutions of the homogeneous Eq.(2):

Since we have four arbitrary functions d K 1 ( 0 )to d E 4 ( 0 )we can choose three more conditions in addition to Eq.(4). Using intuition we make the following choice that will be justified by its success:

If we increase O of dKl( O ) , dK,2( O ) , d n 3 ( 0 ) ,and d,, ( 0 ) by 1 in Eqs.(4), ( 5 ) , and ( 6 ) we obtain

6.3 INHOMOGENEOUS

DIFFERENCE EQUATIONS

249

Subtraction of Eqs.(5), ( 6 ) , and ( 7 ) from Eqs.(8), ( 9 ) , and (10) yields with the notation

Adwi(0) = d,i(O

+ 1 ) - d,i(O),

i = 1, 2, 3, 4

(11)

the result

For v ( 0 + 2 ) we write with the help of Eqs.(7) and (11):

+

We substitute the complete solution v ( 0 - 2) to v ( 0 2 ) of Eqs. ( 4 ) , ( 5 ) , ( 6 ) , ( 7 ) ,arid ( 1 5 ) for s(0 - 2 ) to s(6' + 2) into E q . ( l ) . Since sl(O), sz(O), s3(0), and ~ ~ (are 0 solutioris ) of the homogeneous Eq.(2) we get:

Equations (12), ( I s ) , (14), and (16) contain the four unknown functions AdK1($) to AdK4($)and the known functions s l , s2, s3, s 4 , Hs,(0), pa. Hence, we can obtain AdKl($) to AdK4($)from these four equations and then the coefficients dK1($)to dK4(8)by mcans of a summation according to Eqs.(ll) or (1.3-20). The solution of Eqs.(l2), (13), (14), and (16) by Cramer's rule for AdKl ( B ) , AdK2(0),AdK3(O), AdK4( 8 ) calls for five determinants:

We note that the variable 19 has the values 0 = 0, 1, . . . , N according to Eq.(2.2-6). Hence, the values of si, i = 1, 2, 3, 4, are defined for this range

6.3

INHOMOGENEOUS DIFFER.ENCE EQUATIONS

+

25 1

+

in Eq.(3), while si(-1), s i ( N 1) and s i ( N 2) are not defined. As a result the determinants DKi(6) are only defined for 6 = 1, 2, . . . , N - 2. The functions sl(6) to ~ ~ ( are 6 )defined in Eq.(3) while H,,(O) and p2 are defined in Eq.(l). We obtain for Ad,1(0) to Adn4(6):

The functions d,i(B) required in Eqs.(4)-(7) follow from Eq.(22) by siimmation. A summation constant dKiis required that corresponds to the integration constant of differential calculus:

The correctness of this equation becomes evident if we substitute E q . ( l l )

and choose dKi = dKi(0). The constant d,i(O) is as arbitrary as dKi. A const,ant only adds a solution of the homogeneous Eq.(2) to v(8 - 2) in Eq.(4) or t o 71(6) in Eq.(6). We note that the simplicity of the proof of Eq.(23) by Eq.(24) compared with the proof of Eq.(1.3-20) is due to the eli~ninatiorlof the infinitely large as well as the infinitesimally small. The determinant DKo(6)in Eq.(17) contains only the solutions of the homogcneous equation according to Eq.(3) while DK1(6) t o D,4(Q) contain the inhomogeneous term, which means the determinants have t o be recalculated every time the inhomogeneous term is changed. This situation is rernedied by an expansion of DS1(6) to D,4(Q):

6

252

APPENDIX

Substitution of the functions dK,.i(0)of Eq.(23) into Eq.(6) yields a solution of the inhomogeneous Eq. (3.2-2) according to Eqs.(3.2-30), (3.2-35), and (3.2-53). In order to get a solution of the inhomogeneous Eq.(3.2-58) we must replace according to Eq.(3.2-60) the functions d,i(O) in Eq.(4) by c,i(O), which is a strictly notational change. In addition we must replace H , , ( O , n ) in Eq.(l) by H,,(O, K ) , represented by the last line of

Eq.(3.2-58):

H,,(B, n) = - e " 2 ' 2 ~ , , ( 8

2 m + 1, n) + 2 cos -G,,(O, N

K,)

-e-iA2/2~,,

( 0 - 1,n )

(29)

Here G,,(O, K ) is defined by Eq.(3.2-53). This change affects Eqs.(25)( 2 8 ) . Equation ( 6 ) becomes:

For the evaluation of the determinants D ,(8) ~ we substitute Eq. (3) into Eq.(25):

6.3 INHOMOGENEOUS DIFFERENCE EQUATIONS

~h~ three factors e-i(2n~lN+X2/2)Q ei(2r~lN-A~/2)0and e-i(2r~lN+X~/2)0 can be pulled in front of the determinant. Furthermore, we multiply the first column with 0 and subtract the product from the third collimn:

Developrrlent of the third column yields:

Following Eqs. (31)--(33) we obtain D,Z Eqs.(26) to (28):

(o),

D,, ( 6 ), and

~~4

( 0 ) from

We obtain further D,l(O) to D,4(0) from Eqs.(25) to (28) with p2 taken from E q . ( l ) :

6

254

APPENDIX

Our next task is the evaluation of DKo(8) of Eq. (17). A comparison with Eqs.(25)-(28) suggests to develop the determinant by its fourth row:

+

Substitution of .sl (Q 2) to s4(0 frorrl Eqs.(33) to (36) yields:

+ 2) from Eq.(3) and D,~(Q)to ~ , 4 ( 8 )

The functions AdKl(8) to Ad,4(8) of Eq.(22) follow from Eqs.(37) to (40) plus Eq.(42):

We replace 8 by n in H,,(O, K) of Eq.(l) and Gs,(O, K) of Eq. (3.2-54). The functions dK1(8)to dK4(8)of Eq.(23) may then be written explicitly:

6.3 INHOMOGENEOUS

DIFFERENCE EQUATIONS

In ordcr t o separate Eqs.(47)-(50) into real and imaginary parts we define eight functions Fl(n,,K ) to Fa(n,,K,): sin2( 2 x 6 1 ~cos ) n,(X2/2 - ~ T K , / N ) sin2( ~ T K / [I$N ) 4C O S ( ~ T K / N ) ] sin2(2.rrnlN)cos n(X2/2 2 n n l N ) = sin2(4.rrn/N)[ I + 4 C O S ( ~ T K / N ) ]

F l ( n ,K ) = F2(n.,K )

+

F4 (n,,K ) = -

=

(52)

+

s i n ( 4 n ~ l Nsin ) n(X2/2 2 . r r ~ l N ) 2 sin2(4.rr~/N) [I $ 4 cos(4.rr~/l\r)]

+

sin2(2.rrn/N)sin n(X2/2 2.rrnlN) s i n 2 ( 4 n ~ [I / ~-k)4 C O S ( ~ T K / N ) ] sin(4.rr~lN) cosn(Xz/2- 2 n ~ / N ) F7(n,,n) = 2 s i n 2 ( 4 n ~ [l / ~$ )4 cos(4.rr~lN)I sin(4.rr~lN) cos n1(X2/2 2 . r r ~ / N ) Fs(n,, K ) = 2 sin2(4.rr&/N) [l$ 4 C O S ( ~ T K / N ) ] Fs(n,K )

(51)

+

W e may rewrite Eqs.(47)-(50) into the following form:

(56) (57)

The ternis dKl to dK,4are suniniatiori constants equivalent to integration constants of the differential theory. According to Eqs.(3.2-89), (3.2-92), and (3.2-96) only dKl is choosable. We choose dKl equal to zero and obtain the relations:

Cert,airl terrns required in Eq.(3.2-101) may now be written in the form

6.3 INHOMOGENEOUS DIFFERENCE EQUATIONS

+

F20 (n,,K,) = n,F2(n,K ) F4(n,K,) Fz,(n,, m ) = n,Fe(n,,n) Fg(n,,K,) F22(n,,K ) = F2(n, n) & ( K ) F23 (n,K ) = F6 (n,,m ) d 4 i ( ~ ) (65) We still have to derive d3r, d3i, d l r , and dqi explicitly. We get from Eqs.(37)-(40) and (42):

+ + +

2 2~k: DK3(1)= 4Hs,(1, K,)e-2dTK,/Ne-iX2 sin N

D K o ( l )= -4ePix2 sin

(68)

N

N

Substitution into Eq.(63) brings

and we get finally d~~( K ) , d3i(rc,),d4,(n), and d4i(m):

01lr next task is to separate the kernel of the sum of Eq. (3.2-101) into a real and an irnaginary part. This is very tedious but not mathematically difficult:

+

[dK,l( 0 ) + OdK3(0)]e2?"iKe/N + [dK2(0) 0 d ~ , 4 ( 6 ) ] e - ~ " ~ " ~ / ~ 2n KO 2n~O = [ J 1 ( eK,) , - e ~ , ( eK, ) ] cos -- [J3(0,n) - 0J4(0,K,)] sin -

N

2

~

+

[ J ,(0,m) - O Js (0,K ) ] sin -

N

N

~ ~ 0 2n~O [J7(0,K ) - OJs(8,K ) ] cos N

258

6

APPENDIX

0-1

J , . ( ~ , K= , ) ~ K ~ ( ~ , Kj =) ,1, 2, n=O

+

K l (n,,K ) = H,,(n,, ~ , ) { n [ F l (K,) n , Fz(n, K ) ]

K2(n, K,) = Hs,n, ~ , ) [ F l (Kn), + Fz(n,,~ c . ) ] K3(n,,K,) = Hs,(n, K ) { n [ F s ( n6, ) - Fs(n, K ) ]

..., 8

+ F3(n, + K,)

~4(n K T,) )

+ F7(n,, 6 ) - F8(V

K4(n, r;.) = Hs,(n, ~ ) [ F s ( Kn ), - Fs(n, K 5 ( n ,K ) = ~ , , ( n , ,~ ) { n [ F l ( nK,), - Fz(n, K ) ] + F3(n, K ) - ~ 4 ( n , ~ ) ) Ke(72,K) = Hs,(n, ~ ) [ F l ( Kn), - F 2 ( n , ~+) 2d3r(~)I K7(n,,n ) = H,,(n,, ~ ) { n , [ F s (Kn) , F s ( n , ~ , ) ] F7(nr,K ) + F ~ ( % K ) ) K8(n,,K,) = HsK,(n,, ~ ) [ F s ( nn,), Fs(n, K ) + 2d3i(K)I (75)

+

+

+

The variable n of Kj(n,,K,) needs sometimes to be replaced by the variable 8. The following transformation will do that:

We rewrite Eq.(2.1-28) into the form

Q=

711.0 C

( p - eA,) [ I

+ rnic2/(p- e ~ , ) ~ ] ~ ' ~

(1)

whidl is well-suited to investigate the case ( p - e ~ , ) ~ / r n i >> c ~1. A series expansion of the terms in brackets brings:

If we can rewrite the part mIoc/( p- eA,) in Ecl. (1)so that p - e A , appears only in a numerator but not in a denominator we can rewrite the right sides of both Eqs.(l) and ( 2 ) in a form where p - e A , can be replaced by operators. This rewriting is possible if m o c / ( p - eA,) is a matrix. Using Eq.(2.1-26) we get

6.4

259

FURTHER ELEBOR.ATION O F EQ. (2.1-28)

indicates the inverse matrix. We separate the factors where the exponent p:u - eA,,, p, - eAmy,p, - eAm, from the vectors e,, ey, e,:

-=[(

p,

1

- eA,,

:

P - eAm

0 p, -0eAm, p,

-

) (2

0eAmz

0

Corisider the rrlatrix product

(

)

0 0 0 e, O O e ,

e

0 0 0z e, o O O e ,

)=(o

" "1' 0

1 0 0 1 0) 0 0 1

e,

(4)

(5)

Since the inverse of the first matrix is defined by

(L, 0 0 : ) (kY 0 0 e, 0

0

0

0

1 0 0 ( 00 01 01 )

we recognize the relation

and the inversion of the second matrix in Eq.(4) is solved. In order to invert the first matrix in Eq.(4) we use a standard formula for the inversion of a matrix M

where MT means the transposed matrix and [MI the determinant of M . We obtain from Eq.(4):

Division of the right side of Eq.(9) by Eq.(lO) yields

which is evidently the inverse of the first matrix in Eq.(4). In this expression the terms that have to be replaced by operators have the form (p,-eA,,)-l and we do not know what to do with them anymore than with the original tern1 (p - eArn)-' in Eq.(4). We overcome this problem by demanding that Eq.(lO) shall be equal to a constant K:

Thc substitution of thc operators of Eq.(2.2-2) yields:

Equation (3) becomes:

i

(pu -eArn,) (p, - eA,,)e, 0 0 (P, -eA,,)(p, -eA,,)e, 0 0 (P:Z- eA,,)

0 0 (p, - eA,,)e, (13)

The operators of Eq.(2.2-2) can readily be substituted.

and LcV1of Eqs.(1.2-16) to (1.2-18) was The quantization of Lcrl, carried out by Eqs(5.1-15) to (5.1-18). A look at Eqs(1.2-16) to (1.2-30) shows that these are the only terms not requiring an integration. We show here for the example of LCrzof Eq.(1.2-19) how to proceed if there is an integral. It is indeed possible to quantize the terms with an integral but a great deal of mathematical development will be required before the process becomes practically useful. We start with the expression for LCr2/athat is defined by Eq.(1.2-19):

6.5 QUANTIZATION

%J

- em.0

(%(p-eAmIv r

OF

Lcr2 T O Lcw5

34m(~-eAm), dp rsind

Fronl Eq.(5.1-10) we get:

Fi 1 A - eAmv (P- eAm)v = ZG p s i n ( q ~ t i ) ~ ~

(4)

The relations of Eq.(4.3-17) apply again. We note the most important ones:

r

= p a r , 19 = q

~ d cp, = [LIP, t = e n t

Stibstitution of Eqs.(2) to (4) into Eq.(l) yields with

The A r of d(pAr) can be cancelled and the terms written explicitly:

(5)

I r 2 = Ccr2/ac:

&J~/A~,

are

262

6 APPENDIX

We note that the operators A/& and A/A, are applied to the function Qo(p, 77, (, 8) according to Eq. (4.3-9), which is a known function if Eq. (4.3-8) has been solved. Hence, we may write

The terms A,s = A,o(p, 7 ,t,6) and A,, = A,,(p, q , t,8) are also defined. Hence, the integration of Eq.(7) should be possible at least numerically. There is, however, one more complication. When going from Eq.(l) to Eq.(7) we go from infinitesimal mathematics to the mathematics of finite differences, which implies integer values for p. It would be consistent to replace the integral in Eq.(7) by a sum corresponding t o the relations shown in Table 1.3-1. We could do this formally but nothing would be gained. We rrlay have t o wait until Table 1.3-1 is extended by nlatllematicians to something resembling a book "Table of Sums". It took about 250 years to produce the first book "Table of Integrals" and the book by Gradshteyn and Ryzhik (1980) contains some 900 pages of integrals.

Let us consider the following algebraic difference equation of second order for a function v(X): P1(X)v(X

+ 1) + Po(X)v(X) + P-1(X)v(X - 1) = 0

(1)

The polynomials P l ( X ) and PP1(X) shall be of the same degree while Po(X) may be a polynomial of the same or a lower degree:

These three polyriomials may be rewritten in a forrrl rrlore suitable for a Laplace transform that will be introduced presently:

6.6

POLYNOMIALS AS SOLUTIONS OF DIFFERENCE EQUATIONS

263

The first three coefficients in Eqs.(5)-(7) are written as functions of dlo, d l l , d12 or doo, dol, doz or dLlo, d W l l ,dWl2.The first set ~ 1 0 COO, , C-10 is easy to write: C10

The second set ell, col,

= dl07 c-11

Coo

= doo,

C-10

(8)

= d-10

requires sums:

T h e third set c l 2 , c o 2 , c-12 requires double sums since d l 2 X j w 2 in Eq.(2) is the sun1 of the following terms in Eq.(5): c l ~ X j -plus ~ ell u plus up. Hence, we get: clo

c:=~+,

~il:

The double sunls can be reduced to single sums (S=Euler Script Medium S) :

For the solution of Eq.(l) we use the Laplace transforrn (Nijrlund 1924, p. 316; Milne-Thomson 1951, p. 478). The transformation

transforms the difference equation (1) into a differential equation, if the line of integration l is chosen so that the terms

6.6

POLYNOMIALS AS SOLUTIONS O F DIFFERENCE EQUATIONS

265

vanish. Generally we obtain from Eq.(18) by repeated differentiation:

Substitution of Eqs.(5)-(7) into Eq.(l) and the further substitution of the integral transforms of Eqs.(19)-(21) produce the following differential equation for y (s) :

The characteristic equation

has the following simple roots if we replace c ~ o coo, , c-lo by dlo, doo, d-lo of Eq.(8):

In addition there is the j-fold root sl = 0. We make the substitutiori z = s - s 2 in the differential equation (22) and obtain:

266

6

APPENDIX

Equation (26) is solved in a circle around sz by means of a power series:

The following determining equation for p is obtained if sz is neither equal to sl nor to SQ:

Equation (28) has trivial roots p = 0, 1, . . . , j - 2 and a nontrivial root that rnay be written with the help of Eqs.(8)-(11) as follows:

We interrupt the course of calculatiorl and consider a series expansion in negative powers of X :

Multiplication by X j yields the polynomials of Eqs.(2)-(4). The initial powcr p is indcpcndcnt of thc degree of approximation of the series cxpansion of Eq.(27), since the terms containing j do not show up in Eq.(29). An interchange of s z and s3 leaves the form of Eq.(29) unchanged. Hence, the independence of p from j applies to the solution in the point s3, if s3 is not equal to sl or sz. We return to the characteristic equation (23). In the case of a double root, sz = s3 = -dO0/2dl0, we can solve the difference equation (1) only if sz is also a root of the equation

The roots of Eq.(31) are denoted s 4 and sg. The double root of Eqs.(24) and (25) requires that se must have the same value too:

In order to obtain s5 with the condition of Eq.(32) we write Eq.(31) in the form

Since the last terms in Eqs.(31) and (33) must be equal we obtain s5:

The factor of s2 in Eqs.(31) and (33) is the same. Hence, the factors of s must be equal too:

This is an additional condition for Eqs.(2)-(4). We note tliat it is satisfied for the Klein-Gordon equation and the Dirac equation in spherical' coordiriat,es if Eq.(l) stands for the dependence on the angle I9 = qAI9 as defined by Fig.4.1-1 by the unit vector ea. For the special case s2 = ss = s4 we may rewrite Eq.(22) in the following form:

With z = s - s 2 we obtain:

F'rorn Eq.(27) we get:

Substitution into Eq.(37) yields: w

C q v ( p + v)...( p + v -j v=o

For v = 0 and the power

+3)z~+~-j

zP-3

we obtain the determining equation for p:

The nontrivial valucs of p are the roots of the terms in braces. The solution is in principle a simple matter of substituting clo to c-12 from Eqs.(8)-(14), but there is nothing simple about doing this error-free. We show some intermediate steps. The first line of Eq.(40) is denoted A:

Equations (32) and (34) produce the expressions

and we obtain:

1 . . 1 dl0 (3-2j)dlo-dll+-~(~+l)dl0+--[2d-~~ 2 2 d-lo -t(j-l)(j-2)dlo+(j-2)

-j(j - 3)dPlo]

2 d-lo

The term d-ll/d-lo was previously written as 4dlod-ll/d~o[Harmuth 1977, p. 469, Eq.(16)]. Substitution of die = 4d-lodlo from Eq.(32) transforms one tern1 into the other. The factor p(p - I) . . . (p - j + 3) in the second line of Eq.(40) is left out. The rest is denoted B, and S of Eq.(15) is used:

Consider the terms with the sum S according to Eq.(15). Equation (23) makes then1 zero:

+

+

+

+

S(dio szldoo si2d-lo) = S S ~ ~ ( C ~ O~ S0 ~0 ~ C2 - ~ O ) = O Equation (44) can now be rewritten in the following form:

(45)

270

6

APPENDIX

With the help of E q s . ( 9 ) - ( 1 1 ) and ( 3 1 ) we obtain:

-

=s22(~lls;+C~lS2+~-11)

=O

(47)

The reduced E q . ( 4 6 ) is rewritten as follows:

B

+

+

+

+

s ~ l d 0 2 sy2d-12 ( j - 1)(s;ldol 2~;~d-~~) 1 - -(j - l ) ( 2 j 2 - 3 j 2 ) d l o 2 1 1. - -(j 2 - 1)s;'(2j2 - 33 2)do0 2 3 ( ~- l)s;ldoo 1 - -(j 2 - 3 j 2)d-Io 2j(j - ~ ) s ; ~ d - ~ ~ (48)

= d12

+

+ +

+

+

The first terms in lines 2-4 yield zero due to Eq.(23):

We introduce from E q . ( 3 2 ) s;l=

-2-, dl0 do0

s,2

4 0

= 4-

4 0

=

dl0 -

d-lo

and obtain: dl2

do2 L o

B = dl0 - - 2[d1o

d-12 ++2 d-lo

From Eq.(43) we obtain the following terms containing j:

When we take the sum of A and B the terms + j ( j - 1) in E q . ( 5 1 ) and -j(j - 1) in Eq.(52) cancel. The sum of the remaining terms with j in E q s . ( 5 1 ) arid ( 5 2 ) yields:

One may verify with the help of Eq.(35) that the sum yields zero. The sum of A in Eq.(43) and B in Eq.(51) produces the determining equation1 for p that is independent of j :

We start with Eq.(4.3-24) and use the Bernoulli product ansatz. The calculation is simplified by the assumptions that A, is zero and 4, depends on the distance p only in Eq.(4.3-22):

Substitution into Eq.(4.3-24) brings:

Multiplication with psin(vA29)/w4w and separation of the variables yield:

]This section is a more detailed version of previously published results [Harmuth 1977, Sec. 4.7.3; Harmuth and Meffert 2005, Sec. 6.61. Several typographical errors have been eliminated and new ones were presumably introduced. The equations for s2, s3, s4, s5, and p are unchanged.

272

6

APPENDIX

The choice of an imaginary constant -im will be explained presently. We obtain from Eq.(4):

With the ansatz w([) = wc we get

For circular symmetry m must be an integer m = O , f l , f 2 ,..., O I c p 5 2 ~

(9)

If we had choseri a real number -m instead of -im in Eq.(4) we would have obtained epm' and -em' in Eq.(8). These solutions decrease or increase with cp and are not circularly symmetric. Substitution of Eq.(l) into Eq.(4.3-27) produces an equation very similar to Eq. (4) :

Equation (5) is obtained again. Equations (4) and (10) yield the further relations

We substitute again two Bernoulli products in order to separate variables:

l7j1 ( P

q,O) = v l ( ~ , ~ ) v ( 6 )u, ~ (7P, , = '

(I3)

4 ( ~ vi) u ( O )

The followirig two equations are obtained from E q s . ( l l ) and ( 1 2 ) :

-

1d

EAt ti

(15)

Both equations yield

d v - EAt -i-v

de

1 -[v(6 2

or

ti

EAt + 1 ) - v ( 6 - I ) ] = -i-v(6) h

(16)

The substitution v ( 0 ) = v 8 brings:

v2

+ 2i-vE tiA t

111

A 1 - i,EAt/fi

-1 =0 e-iEAtI"2

~ ~ (=6e-) i E 8 A t l A = e-iEt/h

-eiEAtlh.

~ 2 ( 6 ) = - ei E 8 A t l i i = -eiEt/R

(I7)

Equations (14) and (15) yield the following further solutions for the functions Ul and e4:

im p s i n ( q A 6 ) 61

i + -(A2 + X 3 ) = i-E a t

(18)

- h3) = i-E A t

(19)

2

im 3+ p s i n ( q A 8 ) v4 2

i'i

ti

Once more Bernoulli products are substituted t o separate the last two variables:

6

1.

APPENDIX

(l),'1) = x 1 ( p ) u , l ( r l ) ,

@ 4 ( P ,77)

=x~(P)u~(T])

Equations (18) and (19) yield the following two relations:

21.4

Let us consider first the equations on the right for the functions with the variable q:

u.1

and

We change from the variable q to the variable 19 by substituting q = 6/A6. To save writing we drop temporarily the indices 1 or 4 of u and Xe:

Then we substitute

as well as

--

-G- sin A6 & ( z ) = - ( I - 22)1/2-- sin A6 du(z)

and transform Eq.(25):

Ad

Az

A6

Ax

(27)

6.7 SEPARATION

O F VARIABLES IN SECTION

-(I - x2)1/2-- sin A 6 6u,(x) im A6 Ax (1 - x ) ( 1 -)

sin A8

+m

4.3

~ ( x= ) -iAeu(x)

u(x) = iAe(l - x2)1/2u(x)

(28)

The term (1- x2)l/' interferes with the intended method of solution. It can be removed by multiplying the operators on both sides of Eq.(28) with the conjugate complex operators. We shall point out below why this method is not used. We prefer to square the operators on both sides of Eq.(28): sin A 6 A

+ im) ((1

-

x2)--

sin A8 d A6 63:

A second order difference equation is obtained:

Multiplication with (1 - x2)-I and rearranging the terms brings:

(%) 2

d2u (1 - x ) ~ -2 Ax2

sin A 6 A 6 Au. + 2 i msin -7 =0 A 6 Ax

(31)

For small values of A 6 we use the approximation (sinA6)/A8 = 1 and obtain the simpler equation

Consider the differential equation of spherical harnionics in orie of its two standard forms (Abramovitz and Stegun 1964, p. 332):

276

6 APPENDIX

+

Thc structure of Eqs.(32) and (33) is equal for A: = l ( l 1) except for the term 2 i m ~ u . / A xin Eq.(32). This term is avoided if one does not square Eq.(28) but multiplies it on both sides with the complex conjugate operators. But the term A; - m2/(1 - x2) in Eq.(32) is then changed to -A: m2/(1 - x2). This alternative calculation has not produced a usable result. To recognize the effect of the term 2 i m h u / h x in Eq.(32) or the last term in Eq.(31) we return to Eq.(23) and square the operators on both sides:

+

Rewriting brings:

The tern1 multiplied by i m disappears for cos(qAI9)= 0 or q A 8 = 19 =7~/2. This corresponds to the equator in spherical coordinates r , cp, 19. Hence, along the equator the structure of Eqs.(32) and (33) is the same for A: = l ( l 1). We turn to the solution of Eq.(31) which assumes in explicit notation the form

+

[I - x2 - (x - im)Ax]u(x

E2)

+ Ax) - [2(1- x2) - (A: - - ( ~ x ) ~ ] U , ( x ) + [l - x2 + (x - im)Ax]u.(x - Ax) = 0 (36)

Multiplication with 1- x2 and ordering of the terms in powers of x brings:

+ @zx3- (2 + imAx)x2

+ 1+ imAx]u(x + Ax) + m2)(~x)2])u(x) + [x4 - Axx3 - (2 - imAx)x2 + Axx + 1- imAx]u(x - Ax) = 0

[x4

-

Ax5

- {2x4 - [4 - x ; ( A x ) ~ ] x ~ 2 - (A;

Wc substitute

-

(37)

z = AzX, X = z/Az, Az/Az = I and nlultiply by AX)^:

Comparison with Eqs. (6.6-1)-(6.6-4), using j = 4, yields the following values for the coefficients d l o , . . . , d-12:

We recall that we had replaced Xel and Xe4 in Eqs.(23) and (24) by X e . Bringing back these indices we observe that A: of dO2has to be replaced by either XZ1 or All the other coefficients d l o to d-12 remain unchanged for u + u,l or u -+ 214. Substitution of doo and d l o into Eq.(6.6-32) brings:

Xi4.

We have to be more careful with Eq.(6.6-54) since it contains do2 and we must distinguish between and We write either p l or p4 for p in Eq.(6.6-54) and obtain:

Xi4.

The inverse Laplace transform of the power series of Eq.(6.6-27) yields a factorial series (Norlund 1924, Ch. 11; Milne-Thomson 1951, Ch. XV). We write the factorial series for u.1, p l and 164, p4:

6

APPENDIX

The coefficients C1, and C4, are obtained in Section 6.9. We do not need them here. The power series of Eq.(6.6-27) in the point s a = 1 converges inside a circle that passes through the singular point sl = 0. This implies that the factorial series converge in the half-plane X > Xo. To obtain the required value of Xo we proceed in the following way. Let 6 run frorri 0 to n. The variable x then runs from 1 to -1, and X runs from l/Axto -l/Ax.The difference Ax is arbitrarily small but finite. Hence, we need factorial series with abscissa of convergence Xo < -l/Ax. This is possible if the factorial series of Eqs.(44) and (45) terminate, which calls for negative integer values of pl or p4:

The series expansions of Eqs.(44) and (45) become polynomials and the abscissa of convergence Xo becomes -w:

Equations (42), (43), and (46) yield the eigenvalues

The requirement stated in the text following Eq.(33) is satisfied. The next step is the solution of the functions xl(p) and x4(p) on the left side of Eqs.(21) and (22). This task is carried out in the following Section 6.8. We need there only Xel and Xe4 according to Eqs.(42) and (43) but not the coefficients C1, and C4" of Eqs.(47) and (48).

6.8

SOLUTION O F

~'(p)

AND

6.8 SOLUTION O F xl(p)

~4(p)

AND

279

x4(p)

The left equations for xl(p) and x4(p) in Eqs.(6.7-21) and (6.7-22) are written in the following form:

For a Coulomb potential 4,(p) we get from Eq.(2.3-2):

Equations (1) and (2) become in formal notation

and in explicit notation:

We use the Laplace transform to solve these equations. This requires rewriting the factor p according to the following scheme:

280

6 APPENDIX

Equations (9) and (10) are rewritten:

[(P+ 1) - lIxi(p

+ 1 ) - 2iXeixi(p) - [ ( p- 1 ) + 1 ] ~ 1 (-p 1 ) -i(wPp+8nZa)x4(p) = O

(13)

The Laplace transform of these two equations requires the following six expressions:

Substitution into Eq.(12) yields:

We leave out the integration and multiply with equation for y4 and yl is obtained:

-s-P+~.

A differential

+ + 2iXe4~+ l ) y 4 + iwms2y', + 8niZasyl = 0

(s2- 1 ) s ~ ; (s2

(16)

Froni Eq.(13) we get in analogy:

We represent yl and y4 by series expansions in an arbitrary point S

= so:

6.8

y4 =

SOLUTION OF

C d,, (s - SO)^"',

We shorten s - s o to x and

-

yi

xl(p) AND x4(p)

=

1 dv(l)f

V)(S- SO)^+"-^

for these equations only

-

Xel, Xe4 to Xe:

Substitution of Eqs.(l8) and (19) into Eqs.(l6) and (17) yields:

Consider the terms with the power xp+u in Eq.(20):

(18)

6 APPENDIX

We replace v by p - 1 for a reason that will be recognized in Eq.(24) below and we rewrite Eq.(22) in the following form:

From Eq.(21) we obtain in analogy with ~ ~ , , ~ c l , - ~= 0:

We write the first few equations of the system defined by Eqs.(23) and (24) in detail:

6.8

SOLUTION OF

xl(p)

AND

x4(p)

From the first equation of Eq.(25) we obtain d o as function of a o :

We want to keep a0 as a choosable constant. This permits us to use the second equation of Eq.(25) for the determination of s o

We recognize the relation

that will be used frequently. Then we derive s o :

284

6 APPENDIX

The singular point so of the series expansions of Eq.(18) can have the four values s 2 , 33, s 4 , and ss:

We may write s: in a form that connects us to the singular points sz and for the series expansion for the Klein-Gordon equation [Harmuth and Meffert 2005, Sec.5.5, Eqs.(l3), (14), Fig.5.10-11. From Eqs.(5) and (6) we obtain s3 obtained

With the definition

we may rewrite Eq.(28):

Hence, s: is equal to the old s:! and s3 of the Klein-Gordon equation while s o of Eq.(29) is equal to their square roots:

6.8

SOLUTION OF

xl(p)

AND

~4(p)

285

TABLE 6.8-1 VALUES O F THE SINGULAR POINTS5'2, Ss, FOR VARIOUS VALUES OF g.

Sq,

AND ss

OF

E Q S . ( ~AND ~ ) (36)

F1G.6.8-1. Loci in the s-plane of sz(g) (solid line), ss(g) (dashed line), s4(g) (dotted line), and ss(g) (dashed-dotted line) according to Eqs.(35) and (36) as well as Table 6.8- 1.

Table 6.8-1 lists certain values of s2, ss, s4, and ss as functions of g. Figure 6.8-1 shows the loci of s2(g) to s5(g). We return to Eq.(25). If a0 and so are choosable we obtain do from Eq.(26). The third and fourth equation of Eq.(25) may then be written as inhomogeneous equations. Eqliation (26) is rewritten with the help of Eq. (27):

These two equations have a unique solution if the rank of the coefficient matrix equals the rank of the extended coefficient matrix. The coefficient matrix yields with the help of Eqs.(23) and (24):

Since the rank of the coefficient matrix is zero the rank of the extended coefficient matrix must be zero too. The factor -a0 on the right side of Eqs.(37) and (38) can be ignored:

Substitution from Eqs.(23), (24), and (27) yields for both equations the same result:

+ +

+

+

( W ~ / W ~ ) ~ /~ [l )~p ( ~25; S: Xe4)so 21 - so(2wpp - 8nZawm/wp) - so(2wmp 87rZa) = 0

+

(42)

At this juncture we must decide which value of so to choose from Fig.6.8-1. To this end consider Fig.6.8-2. It shows so = s3 = 1/2/2 on the left (a). The circle of convergence for series expansions according to Eq.(18) reaches the point s = 0, which is necessary for the application of the Laplace transform of Eq.(14). A smaller value of ss according to Fig.6.8-2b would be acceptable too, but a larger value would limit the circle of convergence by the singular point s = s 2 rather than s = 0. There are, however, methods that permit larger values of s3 close to 1 or g close to -2. In Section 6.7 we terminated the factorial series of Eqs.(6.7-44) and (6.7-45) by choosing negative integer values for p l and p4. A second method is based on conformal mapping and a third method by approaching polyriomials asymptotically (Harmuth and Meffert 2005, Secs. 6.9, 6.10 and text following Eq. 5.5-21). Substitution of so = s3 = 1 - O(Ar), wm,wp = O(Ar), and Xel, Xe4 frorri Eqs(6.7-49), (6.7-50) into Eq.(42) yields an equation for p:

We are interested in solutions for which either ll or l4 but not both are zero. This permits us to simplify writing by using l for ll or l4 and add the subscript 1 or 4 when needed. Equation (43) becomes:

6.8 SOLUTION

OF XI(/)) AND ~ 4 ( p )

b

a

F1c.6.8-2. Largest real value of ss that permits the circle of convergence to be limited by the singular point s = 0 as well as by s2 (a). A smaller value of ss yields a circle of convergence that is only limited by s = 0 (b), while a larger value of s3 makes sz limit the circle of convergence.

+

We substitute p = p~ ipl. The real and the imaginary part of Eq.(44) are zero individually. Using Eq.(32) we get:

Once more we return to Eq.(25). Since a0 is choosable and do is derived from a 0 and so we may leave out the first two equations. We write the remaining equations schematically as follows:

In order t o obtain an asymptotically approached polynomial .we demand that the last two variables a,, d, be zero. We have then 21, variables ao, do, . . . a,-1, d,-1 and 21, equations starting with 61,4, a 1 , 4 and ending with 6,,0 = 0, Q,,o. T he determinant of the coefficients must be zero:

/ 61,4

5

61,6

61,7

Q1,4

1

Q1,6

a1,7

62,2

62,3

62,4

62,s

62,6

62,7

Q2,2

a2,3

a2,4

a2,5

a2,6

a2,7

=0

0 DL-1,0

b~-l,l 0

61-1,2

6'-1,3

61-1,4

61-1,5

61-1,6

61-1,7

QL-1,2

a'-l,3

@'-1,4

a~-1,s

a b - 1 , ~

a~-1,7

0 Q.,o

1

0

6L,2

66-3

6L,4

645

2

Q L , ~

QL-4

QL-5

1

(48) For the solution of this equation we multiply the second column from the right with 6L,5/6L,4and subtract it from the last column on the right. The lower right corner of the determinant assumes the following form:

Next we multiply the last row with

and subtract it from the third from last row. Finally, we multiply the last row with

6.8 SOLUTION OF

xl (p) AND x4(p)

289

and subtract if from the fourth from last row. Equation (49) assumes the following form:

The terms 8L-l,6and (1.L-1,6 may readily be calculated but their value is of no interest here. Equations (50) and (48) are satisfied for SL,4aL,5 - aL,4SL,5 =0 Substitutiorl from Eqs.(23) and (24) brings:

+ (8: + 2ihe4so + l)(sa + 2 i k l s o + 1)) = 0

(52)

We substitute s o = SQ = 1-O(Ar) as well as .Ael = Xe, Xe4 = 0 or Xe4 = Xe, Xel = 0 according to Eq.(44), and p = p~ + ipI according t o Eq.(45). The real and the imaginary part of the equation are zero individually. The real part yields

and the imaginary part brings

This equation is more complicated than it looks due to the ambiguity There are three different solutions: of the signs of Xe = fJ-.

6

290

APPENDIX

Equations (56) and (57) hold only for 1 = 0 and 1 = -1, while Eqs.(55) and (58) have no such restriction. We turn to Eq.(53). The first two terms may be rewritten into the following form (PR

+

-

2

1)

+ 2(pR + L - 1) = (PR + L)2- 1

(59)

and Eq. (45) yields:

In order to keep track of signs we write some intermediate steps of the solution for E:

The signs of the terms of {. . .

are reversed

and we obtain

In analogy to Eqs.(55)-(58) we check what p; and p~Xeyield for the sign anlbiguity At = f using Eqs.(44) and (46):

6.8

SOLUTION OF

xl(f)

AND

~4(f)

For pIXe one obtains the same result but 114 is replaced by 112:

prhe =

(fad-)

(~m)

Equations (69) or (70) added to any one of Eqs.(64)--(67) yields a negative value and Eq.(63) yields complex solutions for the energy E. Only two pIXe are possible: positive surrls of pf

+

Using Eq.(72) arid k 2 we obtain from Eq.(63)

Comparison of Eq.(25) with Eq.(48) shows that L- 1 = n' is the number of pairs of equations for a l l d l , to a,-1, d,-l before the termination a, = 0, d, = 0. The series exparlsion of Eq.(74) yields:

For the inverse Laplace transform let the path of integration in Eq.(14) begin a t the origin, run around ss and return to the origin as shown by the line t in Fig.6.8-3a. The point s2 remains outside the loop. More details of the integratiori may be found in the publications of Guldberg and Wallenberg (1911), Nijrlund (1910, 1915, 1924, 1929), and Milne-Thomson (1951, p. 485). One obtains two factorial series for xl(p) and ~ 4 ( p ) :

6 APPENDIX

F1c.6.8-3. Location of the singular points sz = sz(g) and se = ss(g) in the complex plane according to Eqs.(35) and (36) for g < -2; (a) shows the integration path l from 0 around ss; (b) shows the circle of convergence around ss that is limited by sz and does not reach 0.

Equations (23), (24), (36), (45), and (46) define up, dp, ss, and p. The fraction ( p 1). . . ( p p)/(p p 1). . . (p p p ) shall always be replaced by 1 for p = 0. This convention avoids the need of writing a special term for p = 0. The upper limit N of the sums is usually replaced by oo since the conventional calculus of finite differences eliminates only the infinitesimal but not the infinite. But we have, according to Eq.(4.3-17), N intervals Ar = cT/N in the range 0 5 r 5 cT or N intervals Ap = 1 in the range 0 5 p N , which implies N 1 points 0, 1, 2, . . . , N. Hence, the series in Eqs.(76) and (77) can have only N + 1 orthogonal or linearly independent functions, and an upper limit p = N must be used. Unfortunately, this does not mean that the concept of divergence is eliminated. In the case of Fig.6.8-3b the power series does not converge everywhere along the path l of integration in Fig.6.8-3a. As a result the factorial series of Eqs.(76) and (77) may diverge for all values of p. To obtain convergence one may decrease the value of sg to ss = 1 1 4 as shown in Fig.6.8-2a. The resulting value of g = -2.5 yields with the help of Eqs.(31), (32), (5), and (6):

+

<

+

+ +

+

+ +

For sufficiently large values of Ar we have convergence of the power series along the integration path e of Fig.6.8-3a. In essence, as long as the spatial resolution Ar is larger than about the Compton wavelength of the particle we have convergence. If the condition of Eq.(78) is not satisfied we get divergent power series in Eq.(18). This does not mean that the factorial series of Eqs.(76) and (77) diverge too but only that the mathematical methods of the calculus of finite differences need more work. It is known that functions defined by certain divergent but summable power series are also defined by convergent factorial series (Norlund 1924, Ch. 1, 5 3). A final method to achieve convergent solutions in a case represented by Fig.6.8-3a is t o use conformal mapping to deform the circle of convergence in Fig.6.8-3b so that s = 0 is reached. This is a very tedious but also interesting approach (Harmuth and Meffert 2005, Secs. 6.9, 6.10).. A surprising result is k2 = 3k2 in Eq.(73). One may substitute k = d k for k in Eqs.(74) and (75) but there is no comparable solution in the differential theory. This needs more work too.

We turn t o the solution of Eq.(6.7-39). This calls for writing X , X 2 , X3, X 4 as sums of factorials, starting with X 1,X , and X - 1. We begin with the sums of factorials that start with X 1 since they are the most difficult ones. We show in some detail how to obtain them practically:

+ +

The terms of the sum of factorials starting with X are much simpler t o obtain:

The terms of the sum of factorials starting with X - 1 are also quite easy to obtain:

We substitute Eqs.(l)-(4) into the first line of Eq.(6.7-39), Eqs.(5)-(8) into the second line, and Eqs.(9)-(12) into the third line in order to replace the powers of X by 'factorials' of X:

[ ( ~ + l .). . (X+4)-9(X+1). 2

2i+mAx

.. (X+3)+

+ imAx

- 2 + imAz AX)^

--

1

AX)^

+

(

2X . . . (X + 3 ) - 1 2 X ( X + l ) ( X + 2 ) +

14-

+ (-2 + ( X - 1 ) .. . ( X

1- i m A z

+ 2) - 3(X - 1 ) X ( X +

AS)^ -2$x)Z)

X(X

+ 1)

2 - imAx

2 - imAx -4 -

1 +(AS)~ (AS)~

2 - imAx

+

1 - imAx (AS)~

u ( X - 1) = 0

(13)

6.9 We replace

CALCULATION OF

7)

C1, AND C4, IN

SECTION

6.7

295

by u in Eqs.(6.6-18)-(6.6-21):

We substitute these cquations into Eq.(13), multiply with s-X+2,leave out 1/2ni and the integration, and reorder the terms. The following equation is obtained:

(s2 - 2s

+ l)s4y(4)+ 3(3s2 - 4s + 1)s3y(3) 2 - imAx

+ [(8-

[

7;d;2) (3 x f;;

1'

(Ax)z S2-

(2-

- 1 - imAx ) s 2 + 2 - (A: - m2)(Ax)'

(Ax)4

(Ax)4 +

1 - imAx

+

] sZy" SY'

S

I

A AX)^-^] y = O (17)

We obtain from the first and the second term:

This checks with Eqs(6.6-32) and (6.6-34) if the values dlo, doo, d-10, dll, dol, d-11 of Eq.(6.7-40) and j = 4 are substituted. The next step is to represent y by a series expansion in the point s = 1. This is a simplification compared with Eq.(6.8-18) that is possible due to Eqs.(l8) and (19):

6

296

APPENDIX

We shorten s - 1 to x:

Siibstitlition of Eqs.(20) and (21) into Eq.(17) brings:

+

+

1 - imAx (Ax)4

[(Ax)2- I]]

5

Cvx'+" = 0

v=o

(22)

The terms multiplied with

xp+"

must vanish independently:

We replace v by p - 2 and rewrite the recursion formula of Eq.(23) into the following form:

1

+ imAx

+A:

?

Let us evaluate for p = 0 the first term

(P+P-~)(P+P-~)

~0,5Co of

these equations:

6.10

SLANTED COORDINATE SYSTEMS WITH FINITE DIFFERENCES

299

This result corresponds to Eqs.(6.7-42) and (6.7-43). 111order to obtain the coefficients C1, and C4, of Eqs.(6.7-44), (6.7-45) and (6.7-47), (6.7-48) from the coefficients C, defined by Eq.(24) we have to make the following substitutions:

SYSTEMS WITHFINITE DIFFERENCES 6.10 SLANTEDCOOR.DINATE In Section 1.4 the rectangular coordinate system of Fig.l.4-1 was replaced by the slanted coordinate system of Fig.1.4-5. For a reason that will become apparent later we call this a covariable transformation of a rectangular coordinate system. Figure 6.10-la corresponds to Fig.1.4-5, but it shows how the distance Ax is transformed if one changes from a rectangular coordinate system to a slanted one with the angle a between the x- and the y-axis. The auxiliary lines 3/Ax and y/Ax are drawn perpendicularly to the axes x/Axl and y/Axl in Fig.6.10-la. Multiple distances 1, 2, . . . are marked on the auxiliary lines. Coordinate lines parallel to the axes x/Axl and y/Axl arc drawn through these marks. The distances 1 on the auxiliary lines become the distances I / cos(xl2 - a) = 1/ sin a on the coordinate lines. Figure 6.10-lb shows a slanted coordinate system with the coordinate lines drawn perpendicularly to the coordinate axes x/Ax and y/Ax at the marks 1, 2, . . . . This is an unusual way to draw a slanted coordinate system but it avoids the auxiliary lines %/Ax and y/Ax in Fig.6.lO.la. Consider the area A' of a parallelogram in Fig.6-10-la. Since the normalization Ax' is used for x and y the length of the four sides equals 1:

A' = 1 . cos(rl2 - a ) = sina,

x/2 2

(Y

2 0, 1 2 A' 2 0

(1)

The area A' covaries with a to zero for a -+ 0. Next we determine the area A of the parallelograms in Fig.6.10-lb. The normalization Ax is used now for x and y, and the length of the four sides remains 1:

A=

1 1 .I=, cos(rl2 - a ) sin a

r/22a20,

l l A < m

(2)

The area A contravaries with a for a + 0. The product of Eqs.(l) and (2) yields:

6 APPENDIX

F1G.6.10-1. A covariable slanted coordinate system (a) and a contravariable system (b); Ax' = Ax/ cos(1~/2- a ) = Ax/ sin a.

Let us add a third coordinate axis w to x and y in Fig.6.10-1. One obtains Fig.6.10-2. We start with Fig.6.10-2b since it is easier to understand than Fig.6.10-2a. The new axis w/Ax is added, having the angle ri! with the axis x/Ax. At the points w/Ax = -1, 0, 1, 2, 3 we show coordinate axes perpendicular to the axis w/Ax. This is the difference between Figs.6.10-lb and 6.10-2b. Is anything gained by adding the axis w/Ax? The location of the point P in Fig.6.10-2b is significantly narrowed to the four-sided area emphasized by heavier lines. Hence, a third coordinate axis does not create a third dinlerlsiori but improves the localization in two dimensions. We have discussed in Section 1.4 that a coordinate system with n axes can be replaced by coordinate systems with 1, 2, . . . , n- 1 axes if we have a finite resolution for distance measurerrie~its.We see here that the number of coordirlate axes can also be increased gainfully without a change of dimension. We turn to Fig.6.10-2a. The axes x/Axf, y/Axl and their auxiliary lines %/Ax, y/Ax are shown as in Fig.6.10-la. They are augmented by the new axis w/Axl and its auxiliary line w/Ax. The coordinate lines parallel to the axis x/Axl go through the marks 1 to 5 of the axis y/Axl as before. But the coordinate lines parallel to the axis y/Ax' go through the marks 1 to 5 on the axis w/Axl, and the coordinate lines prallel to the axis w/Axl go through the marks 1 to 4 of the axis x/Axl. This arrangement permits arbitrarily many coordinate axes. The point P is located in a more restricted, six-sided area than in Fig.6.10-la. The reduction of the area of location is much less than in

F1G.6.10-2. Addition of a third coordinate axis in two dimensions to the coordinate systems of Fig.6.10-la, b brings a reduced uncertainty about the location of the point P in either a or b. Figs.6.10-lb and 6.10-2b. But we have again obtained an improvement by adding a third coordinate axis to a two-dimennsional coordinate system. 6.11 RIEMANN MANIFOLDS AND BENDED EIGEN-COORDINATES We use the word space strictly for the physical concept of space. Any mathematical 'spaces' will be called manifolds. This stricter than usual use of words is intended to avoid that statements like "a finitc distance in space can be divided in non-denumerable infinitesimal distances dx" are accepted to say something about the physical space. Since neither nondenumerable nor infinitesimal are observable concepts, they have no place in physics. The success of differential calculus in physics demonstrates that non-observable mathematical concepts can be very useful for calculations, but nothing more. The imaginary unit i is also useful for calculations, but nobody believes there are imaginary times or distances. The use of thinkable assurrlptioris for the explanation of observable phenomena was discussed at the end of Section 1.4 for the example of Thor's hammer, lightning, and thunder. Let us write Riemann's Eq.(1.5-3) in the following form, using the summation convention for L and n:

6 APPENDIX

0- 1

F1G.6.11-1. A mesh of a two-dimensional discrete coordinate system with the center point at X, Y and the eight surroundingpoints at X-1, Y-1 to X+1, Y+1.

The components g , , are functions of the continuous ariables x1 = x, x2 = y, x3 = Z . We want to rewrite this equation for discrete variables X , Y, Z = 0, f1, f2, . . . , fN. To this end we start with two variables X , Y and show in Fig.6.11-1 a discrete coordinate system centered in the point X, Y. The coordinatcs of the 32 - 1 = 8 surrounding points are then X - l , Y - 1; X - 1,Y; . . . ;X 1,Y 1. To simplify the notation we may leave out X and Y so that we get -1-1, -10, . . . ,11 for these points. Both notations are shown in Fig.6.11-1. The location of the center point becomes X , Y = X+0,Y+0=00. The distance As from the center point X , Y to the point X 1,Y is written in simplified form as follows:

+

+

+

The eight distances As(i, k) = As(lO), A s ( l l ) , . . . , As(1-1) from the center point are obtained. If the one point is not the center point we must use a more complicated notation. For instance, the distance from point X , Y-1 = 0-1 to X 1,Y - 1 = 1-1 is denoted AX(0-1,l-1). In order to connect the coordinate mesh of Fig.6.11-1 to adjacent meshes we introduce local coordinate axes xx and yy that go through the center point and are extensions of As(-10) and As(0-1). This makes it possible to connect the distances As(lO), Ax(10), Ay(l0) - or generally As(ij), Ax(ij), Ay(ij) -- with each other, as shown in detail in Fig.6.11-2. We obtain in analogy to Eq.(l):

+

F1G.6.11-2. Connection between As(ij),Ax(ij),Ay(ij),a x y , and cp(ij) for the coordinate mesh of Fig.6.11-1.

+

+ +

As2 = As2(ij)= [Ax(ij) Ay(ij)cos axyI2 Ay2(ij)sin2 a x u = g11(ij)Ax2(ij) g22Ay2(ij)2gn(ij)Ax(ij)Ay(Zj)

+

As(ij)= A s ( x + i,Y + j ) , Ax(ij) = A x ( x + i,Y + Ay(ij)= A y ( x + i , Y j ) , gL,(ij)= g L K ( X + i , Y + j ) ,

+

gll

( i j )= g22(ij)= 1, glz(ij)= g ~ l ( i j= ) COSaxy

(3)

If one uses the angle cp(ij)instead of a x y one obtains an equation that does not correspond to Eq. ( 1 ) :

(

+

cp(ij) arcsin -sin p i )

+

Ax(ij)Ay(ij) (4)

+

The four components gLK(ij) = g L K ( X i ,Y j ) of Eq.(3) form a covariant metric tensor. The contravariant tensor g"(ij) = gLK(X+i, Y+j) follows from the equation gLng"

(5)

= 6s:

which become in matrix notation

and one obtains gll

1 =g22= -

191 '

g12

= g21 = --cos a x y

191

The variables Z , j do not occur in Eqs.(G) and ( 7 ) )but X and Y do occur.

304

6 APPENDIX

Our next task is to generalize Fig.6.11-1 from two variables X , Y to three X , Y, Z and four variables X, Y, 2,T. Before we do that let us explain what the ultimate goal of our investigation is. A physical distribution of masses with moments and stresses as function of continuous variables x, y, z, t can be mathematically represented by the stress-energy tensor T. This tensor is a continuous function of the four variables. It is assumed to be sufficiently often differentiable. The tensor T is connected to the Einstein tensor G by the relation

The Einstein tensor is connected to the Riemann-Christoffel curvature tensor, which in turn is connected to the metric tensor g with components gij(z,y, z, t ) . Geodesic lines or geodesics can be derived from g that can be interpreted physically to represent propagation paths of free particles or photons due to the physical distribution of masses. The important point is that we start with an observable distribution of masses and end with observable propagation paths. In-between are mathematical processes that do not need to be describable with observable physical concepts. If we use differential calculus to obtain an observable force from an observable velocity or the surface of a sphere from its diameter, we do not need to assume that the mathematical concepts of infinitesimal arid non-dennmerable represent any observable physical concepts. If a physical interpretation of mathematical operations is possible and helps with thinking we will gladly accept it. But the subdivision of a finite length x into n, intervals of length Az = x l n yields as a physical limit denumerably rnany intervals of length lim,,,(x/n), not a non-denumerable number of infinitesimal intervals. A physical distribution of masses as a function of contirluous variables is an assumption that works for sufficiently large distances. Elementary particles in a differential theory are point-like. In a theory of finite differences they are smaller than the smallest resolvable distance Ax. Hence, they have a mass and perhaps a momentum and a charge but no shape, e.g., like that of a sphere. Atomic nuclei are not quite so elusive, but they still are concentrations of mass with large distances frorn each other. Consider a distribution of masses with moments and stresses in terms of physics of finite differences. To simplify drafting we restrict ourselves to two dimensions. Figure 6.11-3 shows nine points in which masses may be located. A mass at a point X , Y does not actually have to be there, it only must be closer to X , Y than to any other point. The nine points define a coordinate system xx, yy. This is what Rie~nanrlmeans in the second quotation in Section 1.5 with the words "a discrete manifold contains the principle of the measure relationships already in the concepts of this manifold, but it must come from somewhere else for a continuous manifold".

6.11

R.IEMANN M A N I F O L D S A N D B E N D E D E I G E N - C O O R D I N A T E S

I

305

+

F1G.6.11-3.Masses concentrated a t nine locations, o r generally at (2N 1)' locations, a n d t h e coordinate system defined by t h e location of t h e masses. X, Y = 0 , + 1 , f 2 , . . . , fN.

One could riot draw an illustration like Fig.6.11-3 if X , Y were continuous variables rather than discrete ones X , Y = 0, f1, f2, . . . , fN. Let us emphasize that the largest observable distance is arbitrardy large but finite1, arid the srriallest observable distarice is arbitrarily small but finite. Eithcr a very srrlall or a very large distance must be finite to be observable. Figure 6.11-3 shows the distances A s ( X -1, Y - 1) . . . A s ( X + 1, Y+1) in addition to the coordinate lines. In Fig.6.11-l we used the distances arid directions to rieighborhood points to define the location of the points. The c:oordinate lines in Fig.6.11-3 are more convenient if they are straight but both representations are possible. The coordinate system in Fig.6.11-3 rotates and shifts like the system of nine points. To avoid confusion with the rotations and shifts in tensor calculus we use the tern1 eigen-coordinates if the coordinate systern rotates arid shifts with the system of points that define it. There are no eigen-coordinates if one uses continuous variables

", Y.

Let us assurne that tensor calculus is generalized froni differentials drc" to finite differences Ax".A rnass distribution then defines a tensor T in lThis concept was a l ~ e a d yknown to Scholastic scientists as .syncategorematic infinit,y (Boyer 1949, p. 69). T h e separation of abstract mathematics using non-observable axioms from physics based on observation came in the lgth century. German subdivides science (Wissenschaft) into "Geisteswissenschaft", which includes mathematics, and "Naturwissenschaft", which includes physics. English has no special words for this subdivision, which makes it harder to recognize the separation of mathematics from physics.

6 APPENDIX

+

+

+

F1G.6.11-4. Extension of Fig.6.11-1 to three variables X i, Y j,Z k ; i, j,k = -1,0,1. The nine points for k = 0 are shown in the center, those for k = 1 and k = -1 on top and bottom.

Eq.(8), which defines via Einstein tensor and Riemann-Chrstoffel curvature tensor a metric tensor represented by Fig.6.11-1. An extension of the calculus of variation from continuous and differentiable functions to discrete functions is needed to obtain a geodesic from it that consists of a sequence

6.11 R.IEMANN

MANIFOLDS AND BENDED EIGEN-COORDINATES

+

+

307

+

F1G.6.11-5. Extension of Fig.6.11-4 from three space variables X i, Y j, Z k to four variables T h, X i, Y j , Z k ; h, i,j, k = -1,0,1. The 27 points for h = 0 are shown in the center, those for h = -1 and h = 1 on the left and right.

+

+

+

+

of discrete points as in Fig.1.5-8. Its physical interpretation is that free particles or photons move from discrete point to discrete point if the resolution of observation is finite and if the time between observations is finite. This is all quite similar to the differential theory. One will not dispute that a cor~tiriuousmass distribution is unrealistic at atomic distances. But there are other cases of interest for finite resolution. A black hole shrinks everything to the size of a mathematical point in the differential theory, but only to the smallest distance resolvable by observation in the difference theory. The difference theory corresponds better to physics. At short distances a neutron star is probably better described by points with mass at small finite distances than by a continuous distribution of masses. Another, less scientific, difference is encountered if we try to interpret each mathematical step in terms of physics, rather than only the assumptions and the final result. A metric tensor as function of continuous variables can be interpreted to represent a curved Riernann manifold. This statement does no more than define a new mathematical term. But the situation changes completely if we replace 'curved Riemann manifold' by curved space. A mathematical concept is all of a sudden transformed into something that sounds like a physical term. No one has ever observed a curved physical space, but this was no reason not to claim that there are parallel universes. A bended system of eigen-coordinates obtained from a theory of finite differences is not likely to inspire such claims. At the risk of being too repetitive we note that unobservable concepts like nonderiurr~erableand infinitesimal lead to more unobservable concepts, while observable results are based on observable assumptions. For the extension of the general theory of relativity to finite differences arid quantum physics we must bring the calculus of finite differences closer to the level of differential calculus. This calls for the Levi-Civita, Ricci, Christoffel, Euler, and Lagrange of the 21St century. We return to the paragraph following Eq. (7) and generalize Fig.6.11-1 to three variables X i, Y j, Z k; i,j, k = -1,0,1. The center of Fig.6.11-4 shows approximately in axonometric presentation Fig.6.11-1 but the nine points denoted -1-1, . . . ,00, . . . 11 have received an additional 0 for k = 0 and are now denoted -1-10, . . . ,000, . . . , 110. Similarly, As(l0) has become As(100); instead of AX(10), AY(10) we need now AX(100), AY(100), AZ(100); etc. The top of Fig.6.11-4 holds for Z k = Z 1, the bottom for Z 1. The letters X , Y, Z are generally left out and only i j k is shown. The points denoted X , Y ,Z in the center as well as on top and bottom coincide. They are shown separated some distance and connected by dashed lines. Without this separation Fig.6.11-4 becomes an incomprehensible assembly of 27 points, 26 heavy rods, 8+2 x 12 fine dashed lines, etc. Our final illustration, Fig.6.11-5, shows the extension of Fig.6.11-4 from three space variables X i, Y j, Z k to four variables T h,, X i, Y

+

+

+

+

+

+

+

+

+

+

+

6.11

R.IEMANN M A N I F O L D S A N D B E N D E D E I G E N - C O O R D I N A T E S

309

j, Z+ k with h.,i, j, k = -1,0,1. The center of the illustration holds for h = 0 and represents in essence a repetition of the 27 points of Fig.6.11-4 with the digit 0 added in front of -1-1-1, . . . , 000, . . . , 111 to yields 0-1-1-1, . . . , 0000, . . . , 0111. The axis tT as well as the angles c v x ~and C V ~arc T added. Much of the numbering of Fig.6.11-4 is left out to avoid overloading the illustration. The notation X , Y, Z in the upper and lower part of Fig.6.11-4 is replaced by T, X , Y, Z. The drawings on the left arid right of Fig.6.11-5 replace the center point T , X , Y , Z = 0000 by T - l , X , Y , Z = -1000 and T l , X , Y , Z = 1000. All first digits in the denotation of the 27 points are changed from 0 to -1 or 1. The two sets of 27 points each are connected t o the center point T, X , Y, Z = 0000 by heavy lines. Only four of them are shown in full on the left arid the right, the others are indicated by short heavy lines in the direction of the points dcnoted T, X , Y, Z. Only the four distances As(-111-1), As(-1111), As(1-1-1-1), and A.s(l-1-11) are denoted while all distarices As(ijk) are denoted in Fig.6.11-4. We must leave it t o the reader to make Fig.6.11-5 compreherisible with the help of Fig.6.11-4. The mediumheavy lines on the left and right are only intended to help see t,he connection wit,h Fig.6.11-4. Figure 6.11-5 represents a mesh of a four-dimensional coordinate systern just as Fig.6.11-1 represents a mesh of a two-dimensional coordinate system or Fig.6.11-4 a mesh of a three-dimensional coordinate system. Its relation t o a metric tensor with the c:omponents g,, and L, h; = 0 , 1 , 2 , 3 is readily recognizable. Perhaps the most irnportant lesson to be learned from Fig.6.11-5 is that a st,ructlire with the four variables T h, X i, Y j, Z k for h, i, j, k = -1,0,1 car1 be iniplemented in our physical space with 34 = 81 spheres and 34 - 1 = 80 rods represerlted by the heavy lines. Orie can add 2 x (33 - 1) thin rods represented by the medium-heavy lines on the left and the right side of Fig.6.11-5. These additional rods would make a model more stable. We note that the number of dimensions of Fig.6.11-5 car1 be reduced with the help of Figs.1.4-7 or 1.4-8.

+

+

+

+

+

References and Bibliography Abramowitz, M. and Stegun, I.A., eds. (1964). Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series 55. US Government Printing Office, Washington, DC. Anastasovski, P.K., Bearden, T.E., Ciubotariu, C., Coffey, W.T., Crowell, L.B., Evans, G.J., Evans, M.W., Flower, R., Jeffers, S., Labounsky, A , , Lehnert, B., MQszafos, M., MolnAr, P.R., Vigier, J.P., and Roy, S. (2001). Empirical evidence for non-Abelian electrodynamics and theoretical development. Annales Fondation Louis de Broglie, vol. 26, no. 4, pp. 653672. Apostle, H.G. (1969). Aristotle's Physics, Tkanslated with Commentaries and G1ossar.y. Indiana University Press, Bloomington. Aristotle (1930). The Works of Aristotle, vol.II, Physics. R.R.Hardie and R.K.Gaye transl., Clarendon Press, Oxford. Barrett, T.W. (1993). Electromagnetic phenomena not explained by Maxwell's equations. Essays on the Formal Aspects of Electromagnetic Theory, A.Lakhtakia, ed., 6-86. World Scientific Publishing Co., Singapore. Barrett, T.W. and Grimes, D.M. (Editors). (1996). Advanced Electromagnetism: Foundations, Theor'y, and Applications, World Scientific Publishing Co., Singapore. Becker, R. (1963). Theorie der Elektrizitat, vol. 2 (revised by G.Leibfried and W.Breuig), 9th ed. Teubner, Stuttgart. Becker, R. (1964). Electromagnetic Fields and Interactions (transl. by A.W.Knudsen of vol. 1, 16th ed. and by I.de Teissier of vol. 2, 8th ed. of Theorie der Elektrizitat). Blaisdell, New York. Reprinted 1982 by Dover, New York. Belfrage, C. (1954). Seeds of Destruction. The Truth About the US-Occupation of Germany. Cameron and Kahn, New York. Copyrighted by the Library of Congress but not listed in the online catalog www.loc.gov either under the name Belfrage, Cedric or the book title. Available a t Deutsche Staatsbibliothek, Berlin, Signatur 8-29 MA 265 or a t Bayrische Staatsbibliothek, Munchen, Germany, Signatur BAY: L 1089. Berestezki, W.B., Lifschitz, E.M., and Pitajewski, L.P. (1970). Relativistische Quantentheorie, vol. IVa of Lehrbuch der Theoretischen Physik, L.D.Landau and E.M.Lifschitz eds.; transl. from Russian. Akademie Verlag, Berlin. Berestetskii, V.B., Lifshitz, E.M., and Pitajevskii, L.P. (1982). Quantum Electrod,ynamics, transl. from Russian. Pergamon Press, New York. Blum, J.M. (1965). Fkom the Morgenthau Diaries, vo1.2: Years of Urgenc,~,1938-1941. Houghton Mifflin Co., Boston. Blum, J.M. (1967). F'rom the Morgenthau Diaries, vo13: Years of War 1941-45. Houghton Mifflin Co., Boston. German translation by U.Heinemann and 1,Goldschmidt: Deutschland ein Ackerland? (The Morgenthau Diaries). Droste Verlag und Druckerei, Dusseldorf 1968. Bolyai, J . (1832). The science of absolute space; original in Latin. See appendix of Bonola (1955). Bonola, R. (1955). Non-Euclidean Geometry. Dover, New York (Originally published in Italian in 1906).

REFERENCES AND BIBLIOGRAPHY

31 1

Boyer, C.B. (1949), The History of the Calculus and its Conceptual Development. Hafner, New York. Reprinted: Dover, New York 1959. Bronstein, I.N., Semendjajew, K.A., and Zeidler, E. (1996). Teubner-Taschenbuch der Mathematik. B.G.Teubner, Leipzig. Bub, J. (1999). Interpreting the Quantum World. Cambridge University Press, Cambridge. Callender, C. and Hugget, N. (2000). Physics Meets Phi1osoph.y a t the Planck Scale. Cambridge University Press, Cambridge. Cao, T.Y. (1999). Conceptual Foundations of Quantum Field The0r.y. Cambridge University Press, Cambridge. Christoffel, E.B. (1910). Gesammelte Mathematische Abhandlungen, L. Maurer ed. Teubner, Leipzig. Chuang, I.L. and Nielsen, M.A. (2000). Quantum Computation and Information. Cambridge University Press, Cambridge. Cole, E.A.B. (1970). Transition from continuous t o a discrete space-time scheme. Nuovo Cimento, vol. 66A, 645-655. Cole, E.A.B. (1973a). Perception and operation in the definition of observables. Int. J. Theor. Phys., 8, 155-170. Cole, E.A.B. (1973b). Transfromations between cellular space-time structures and the principle of covariance. Nuovo Cimento, 18A, 445-458. Das, A. (1966a). Complex space-time and quantum theory of free fields. J. Math. Phys., vol. 7, 52-60. Das, A. (1966b). The quantized complex space-time and quantum theory of free fields. J. Math. Ph.ys., 7, 52-60. Dickson, W.M. (1998). Quantum Chance and Nonlocality. Cambridge University Press, Cambridge. Einstein, A. (1916). Die Grnndlage der allgemeinen Relativitatstheorie, Annalen der Physik, vol. 49. English translation: The Foundation of the General Theory of Relativity, in The Principle of Relativit,y, Dover, New York 1952, 109-169. Einstein, A. (1950). The Meaning of Relativity, 3rd ed. Princeton U. Press, Princeton, NJ. Einstein, A. and Infeld, L. (1938). The Evolution of Physics, Simon and Schuster, New York. Elaydi, S.N. (1999). An Introduction to Difference Equations, 2nd ed. Springer-Verlag, New York. Farwell, R. and Knee, C. (1992). The Geometric Challenge of Riemann and Clifford; in 1830-1930: A Century of Geometry, pp. 88-106. Springer-Verlag, Berlin. Flint, H.T. (1948). The quantization of space and time. Phys. Rev., vol. 74, 209-210. Frobenius, G. (1873). ~ b e die r Integration der linearen Differentialgleichungen durch Reihen. Journal fiir die reine und angewandte Mathematik, vol. LXXVI, 214-235. Fuchs, L. (1866). Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten. Journal fur die reine und angewandte Mathematik, vol. LXVI, 121-160; (1868) vol. LXVIII, 354-385. Gelfond, A.O. (1958). Differenzenrechnung (transl. of Russian original, Moscow 1952). Deutscher Verlag der Wissenschaften, Berlin. French edition 1963. Ghose, P. (1999). Testing Quantum Mechanics on New Ground. Cambridge University Press, Cambridge. Gradshteyn, I.S. and Ryzhik, I.M. (1980). Tables of Integrals, Series, and Products. Academic Press, New York. Greiner, B. (1995). Die Morgenthau-Legende, Hamburger Edition HIS Verlagsgesellschaft, Hamburg. References 133 books and 39 journal articles. Griffiths, D.J. (1989). Introduction to Electrodynamics, Prentice-Hall, Englewood Cliffs NJ. Guldberg, A. and Wallenberg, G. (1911). Theorie der Linearen Differenzengleichur~gen. B.G.Teubner, Leipzig Habermann, R. (1987). Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. 2nd ed., Prentice Hall, Englewood Cliffs, New Jersey. Hammond, N. (2006). The Maya. Folio Society, London. Harmuth, H.F. (1969). 'Dansmission of Information by Orthogonal finctions. SpringerVerlag, Berlin and New York. Harmuth, H.F. (1972). 'Dansmission of Information by Orthogonal Functions, 2nd ed. Springer-Verlag, Berlin and New York.

312

REFERENCES AND BIBLIOGRAPHY

Harmuth, H.F. (1977). Sequency Theory (Foundations and Applications). Academic Press, New York. Harmuth, H.F. (1979). Acoustic Imaging with Electronic Circuits. Academic Press, New York. Harmuth, H.F. (1981). Nonsinusoidal Waves for Radar and Radio Communication. Academic Press, New York. Harmuth, H.F. (1984). Antennas and Waveguides for Nonsinusoidal Waves. Academic Press. New York. Harmuth, H.F. (1986a). Propagation of Nonsinusoidal Electromagnetic Waves. Academic Press, New York. Harmuth, H.F. (1986b, c). Correction of Maxwell's equations for signals I, 11. IEEE Trans. Electromagn. Compat., vol. EMC-28, 250-258, 259-266. Harmuth, H.F., (1986d). Propagation velocity of electromagnetic waves. IEEE Trans. Electromagn. Compat., vol. EMC-28, 267-271. Harmuth, H.F. (1989). Information Theory Applied to Space-Time Ph,ysics (in Russian). MIR, Moscow. English edition by World Scientific Publishing Co., Singapore 1992. Harmuth, H.F. (1990). Radiation of Nonsinusoidal Electromagnetic Waves. Academic Press, New York. Harmuth, H.F. and Hussain, M.G.M. (1994). Propagation of Electromagnetic Signals. World Scientific Publishing Co., Singapore. Harmuth, H.F., Boules, R.N., and Hussain, M.G.M. (1999). Electromagnetic Signals-ReAection, Focusing, Distortions, and Their Application. Kluwer Academic/Plenum Publishers, New York. Harmuth, H.F. and Lukin, K.A. (2000). Interstellar Propagation of Electromagnetic Signals. Kluwer Academic/Plenum Publishers, New York. The blank page 224 was censored due t o a historically but not politically correct account of Max Planck's last years, 1945-1947. Harmuth, H.F., Barrett, T.W., and Meffert, B. (2001). Modified Maxwell Equations in Quantum Electrodynamics. World Scientific Publishing Co., Singapore. Harmuth, H.F. and Meffert, B. (2003). Calculus of Finite Differences in Quantum E1ectrod.ynamics. Elsevier/Academic Press (Advances in Imaging and Electron Physics, vol. 129), London. Harmuth, H.F. and Meffert. B. (2005). Dogma of the Continuum and the Calculus of Finite Differences in Qunatum Physics. Elsevier/Academic Press (Advances in Imaging and Electron Physics, vol. 137), London. Hasebe, K. (1972). Quantum space-time. Prog. Theor. Phys., vol. 48, 1742-1750. Hellund, E. J. and Tanaka, K. (1954). Quantized space-time. Phys. Rev., vol. 94, 192-195. Hill, E.L. (1955). Relativistic theory of discrete momentum space and discrete space-time. Phys. Rev., vol. 100, 1780-1783. Hillion, P. (1991). Remarks on Harmuth's 'Correction of Maxwell's equations for signals 1'. IEEE Trans. Electromagn. Compat ., vol. EMC-33, 144. Hillion, P. (1992a). Response to 'The magnetic conductivity and wave propagation'. IEEE Trans. Electromagn. Compat., vol. EMC-34, 376-377. Hillion, P. (1992b). A further remark on Harmuth's problem. IEEE Trans. Electromagn. Compat., vol. EMC-34, 377-378. Hillion, P. (1993). Some comments on electromagnetic signals; in Essays on the Formal Aspects of Electromagnetic The0r.y. A.Lakhtakia ed., 127-137. World Scientific Publishing Co., Singapore. r Eigenschaft der Gammafunktion keiner algebraischen DifferenHolder, 0. (1887). ~ b e die tialgleichung zu geniigen. Math. Annalen, vol. 28, 1-13. Hnssain, M.G.M. (2005). Mathematical model for the electromagnetic conductivity of lossy materials. J. of Electromagn. Waves and Appl., vol. 19, no. 2, 271-279; VSP Publishers, Zeist, Netherlands. Icke, V. (1995). The force of Symmetr,~.Cambrdidge University Press, Cambridge. Jackson, J .D. (1999). Classical Electrod,yr~anucs,3rd ed. J .Wiley, New York. Kant, I. (1922), Critique of Pure Reason; F.M.Miiller, translator. McMillan, New York. The statements on space differ considerably in the first edition of 1781 and the second edition of 1787. See Harmuth 1989. Sec. 1.4.

R.EFER.ENCES AND BIBLIOGR.APHY

313

King, R.W. (1993). The propagation of a Gaussian pulse in seawater and its application to remote sensing. IEEE f i n s . Geoscience, Remote Sensing, vol. 31, 595-605. King, R.W. and Harrison, C.W. (1968). The transmission of electromagnetic waves and pulses into the Earth. J. Appl. Physics, vol. 39, 4444-4452. Lamb, W.E. and Retherford, R.C. (1947). Fine structure of the hydrogen atome by a microwave method. Ph,ys. Rev., vol. 72, 241-243 . Landau, L.D. and Lifschitz, E.M. (1966). Lehrbuch der Theoretischen Physik (German edition of Russian original by G.Heber). Akademie-Verlag, Berlin. Lawden, D.F. (1982). Introduction to Tensor Calculus, Relativity and Cosmology. Wiley, New York. Dover, New York 2002. Leader, E. and Predazzi, E. (1996). An Introduction to Gauge Theories and Modern Particle Ph,ysics. Cambridge University Press, Cambridge. Lehner, G. (1990). Elektromagnetische Feldtheorie. Springer Verlag, Berlin. Lehnert, B. (1995). Total reflection process including waves of an extended electromagnetic theory. Optik, vol. 99, No. 1, 113-119. Lehnert, B. (1996). Minimum electric charge of an extended electromagnetic field theory. Pltysica Scripta, vol. 53, 204-21 1. Lehnert, B. and Roy, S. (1998). Extended Electromagnetic Theory. World Scientific Publishing Co., Singapore. Levi-Civita, T . (2005). The Absolute Differential Calculus (Calculus of Tensors). Reprinted, Dover, New York. Levy, H. and Lessman, F. (1961). Finite Difference Equations. McMillan, New York; reprinted Dover, New York. Lobaschevskii, N.I. (1840). Geometrische Untersuchungen zur Theorie der Parallellinien. G.Finke, Berlin. Reprinted Mayer and Miiller, Berlin 1887. English translation in the appendix of Bonola (1955). Lukin, K. (2007). Editorial, Biography of Henning F. Harmuth. Electromagnetic Phenomena, vol. 7, No. 1(18), 3-7. Institute for Electromagnetic Research Ltd., P O Box 4580, Kharkiv-22, Ukraine 61022. Marsal, F. (1989). Finite Differenzen und Elemente: numerische Losung von Variationsproblemen und Partiellen Differentialgleichungen. Springer-Verlag, Berlin. Meffert, B. (2007). Personal notes devoted to Henning F. Harmuth. Electromagnetic Phenomena, vol. 7, No. 1(18), 13-18. See Lukin, K. Meffert, B. and Pichler, F. (2007). Personal and technical resume of Henning F . Harmuth. Electromagnetic Phenomena, vol. 7, No. 1(18), 8-12. See Lukin, K. Messiah, A. (1959). Mkcanique Quantique. Dunod, Paris. English transl by G.M.Temmer, Quantum Mechanics. North Holland Publishing Co., Amsterdam 1962. Reprinted Dover, New York. Milne-Thomson, L.M. (1951). The Calculus of Finite Differences. MacMillan, London. Misner, C.W., Thorne, K.S., and Wheeler. C.A. (1973). Gravitation. W.H.l+iedman, San Francisco. Moon, P. and Spencer, D.E. (1988). Field Theory Handbook, 2nd ed., Springer Verlag, New York. Morgenthau, H. (1945). Gerrnany Is Our Problem. Harper & Brothers, New York and London. See also Belfrage, C., Blum, J.M., Greiner, B. Morse, P.M and Feshbach, H. (1953). Methods of Theoretical Ph,ysics. McGraw-Hill, New York. Newton, I. (1971). Mathematical Principles. A.Motte transl., F.Cajori ed. University of California Press, Berkeley. New York Times (1945a). Morgenthau Gives Views on Germany. Oct. 5, p. 7. New York Times (1945b). Morgenthau Volume Sent to U.S. Staffs. Dec. 30, p. 10. New York Times (1945~).Nazi Reform Slow, Morgenthau Says. Nov. 12, p..3. New York Times (1945d). Engineers Offer Plan for Germany. Sept. 27, p. 14. Nielsen, N. (1906). Handbuch der Theorie der Gammafunktion. Teubner, Leipzig. Norlund, N.E. (1910). FYactions continues et diff6rences rkciproques. Acta Math., 34, 1-108. Norlund, N.E. (1914). Sur les skries de facultks. Acta Math., 37, 327-387. Norlund, N.E. (1915). Sur les kquations linkaires aux diffkrences finis & coefficients rationels. Acta Math., 40, 191-249.

314

REFERENCES AND BIBLIOGRAPHY

Norlund, N.E. (1924). Vorlesungen iiber Differenzenrechnung. Springer-Verlag, Berlin; reprinted Chelsea Publishing Co., New York 1954. Norlund, N.E. (1929). L e ~ o n sur s les ~ q u a t i o n Lineaires s aux Diff6rence.s Finies, R.Lagrange, ed. Gauthier-Villars, Paris. Ostrowski, A. (1919). Neuer Beweis des Hijlderschen Satzes, da5 die Gammafunktion keiner algebraischen Differentialgleichung geniigt. Math. Annalen, vol. 79, 286-288. Ptolemy, C. (1952). The Almagest, by Ptolemy, R.C. Taliaferro transl.; On the Revolution of the Heavenly Spheres, by N. Copernicus, C.G. Wallis transl.; The Harmonies of the World, by J . Kepler, C.G. Wallis transl. Encyclopedia Britannica, Chicago. Raine, D.J. and Heller, M. The science of Space-Time. Pachart Publishing House, Tbcson, Arizona. Ricci, G. and Levi-Civita, T . (1975). Ricci and Levi-Civita Tensor Analysis Paper Translation, Comments and Additions, by R. Horman. Math. Sci. Press, Brookline, Massachusetts. Richtmyer, R.D. (1957). Difference Methods for Initial Value Problems. Wiley Interscience, New York. Riemann, B. (1854). ijber die Hypothesen welche der Geometrie zu Grunde liegen. In Bernhard Riemanns Gesammelte Mathematische Werke, H. Weber ed., 272-287. Teubner, Leipzig 1892; Dover, New York 1953. Riemann, B. (1858). Ein Beitrag zur Elektrod,ynamik. In Bernhard Riemanns Gesammelte Mathematische Werke, H. Weber ed., 288-293. Teubner, Leipzig, 1892. Riemann, B. (1861). Commentatio mathematica qua respondere tentatur question; a b Illma Academia Parisiensi propositae . . . . In Bernhard Riemanns Gesammelte Mathematische Werke, H. Weber ed., 391-404 (editor's notes in German 405-423). Teubner, Leipzig, 1892. German translation in Gauss, C.F., Riemann, B., and Minkowski, H. GauOsche Flachentheorie, Riemannsche Raume und Minkowski-Welt. Teubner-Archiv znr Mathematik, vol. 1, 115-128. Teubner, Leipzig 1984. Riemann, B. (1866). Gravitation und Licht. In Bernhard Riemanns Gesammelte Mathematische Werke, H. Weber ed., 532-538. Teubner, Leipzig, 1892. Rohrlich, F. (1990). Classical Charged Particles, Addison-Wesley, Reading MA. Schiff, L.I. (1949). Quantum Mechanics. McGraw-Hill, New York. Schild, A. (1949). Discrete spacetime and integral Lorentz transformations. Canadian J. Math., vol. 1, 29-47. Schrodinger, E. (1950). Space-Time Structure, University Press, Cambridge, Great Britain. Schrodinger, E. (1956). Causality and wave mechanics, in The World of Mathematics, vol. 2, 1056-1068. Simon and Schuster, New York. Simon, E. (1945). Mr.Morgenthau's Program for Removing the German Menace. New York Times Book Review, Oct. 7, p. 2. Smirnov, W.I. (1961). Lehrgang der hoheren Mathematik, 4th ed. (transl. of the 12th Russian edition). Deutscher Verlag der Wissenschaften, Berlin. English edition: A Course in Higher Mathematics. Pergamon Press, Oxford 1964. Smith, G.D. (1982). Theory and Problems of Calculus of Finite Differences and Difference Equations. McGraw-Hill, New York. Snyder, H.S. (1947a). Quantized spacetime. Phys. Rev., vol. 71, 38-41. Snyder, H.S. (194713). The electromagnetic field in quantized space-time. Phys. Rev., vol. 72, 68-71. Spiegel, M.R. (1994). Theory and Problems of Calculus of Finite Differences and Difference Equations; reprint of edition of 1971. McGraw-Hill (Schaum's Outline Series), New York. Endliche Differenzen und Differenzengleichungen; transl., McGraw-Hill, New York 1982. A basic introduction containing useful formulas. Stratton, J.A. (1941). Electromagnetic Theory. McGraw-Hill, New York. Synge, J.L. and Schild, A. (1969). Tensor Calculus. University of Toronto Press. Dover, New York 1978. Thompson, J.E.S. (1959). Grandeza y Decadencia de 10s Mayas. Fondo de Cultura Econ6mica, Mexico and Buenos Aires. van der Waerden, B.L. (1966). Algebra. Springer-Verlag, Berlin and New York. Wei, Y. and Zhang, Q. (2000). Common Waveform Ana1,ysis. Kluwer Academic Publishers, Boston.

R.EFERENCES AND BIBLIOGR.APHY

315

Weinberg, S. (1995). The Quantum Theory of Fields: I. Introduction. Cambridge University Press, Cambridge. Weinberg, S. (1996). The Quantum Theory of Fields: II. Modern AppUcatjons. Cambridge University Press, Cambridge. Weinberg, S. (2000). The Quantum The0r.y of Fields: III. Supersymmetry. Cambridge University Press, Cambridge. Weyl, H. (1921). Das Raumproblem. Jahresberichte der Deutschen Mathematikervereinigung, vol. 30, 92-93. Weyl, H. (1968). Gesammelte Abhandlungen. Springer-Verlag, Berlin. Welch, L.C. (1976). Quantum mechanics in a discrete space-time. Nuovo Cimento, vol. 31B, 279-288. Yukawa, H. (1966). Atomistics and the divisibility of space and time. Suppl. Prog. Theor. Ph,ys., vol. 37-38, 512-523.

Index A Abramovitz 182, 188, 275 Anastasovski 2 AndrC de Avedaiio 34 astronomical observations -fm.o

geometries of finite differences 43, 45 grad, spherical coordinates 174 Gradshteyn 22, 90 Guldberg 291

B

H

Barrett 3, 4, 5, 6, 7, 8, 9, 11, 85, 96, 158, 171 Becker 48, 96, 97, 174 Berestezki 91 Bolyai 35 Boules 1 box normalization 85 Boyer 305

C causal functions 2, 37 causality law 2, 36 Christoffel 308 concept of space, time 35 coordinate cell 40 coordinate mesh 38, 39 curved light ray 42 curved Riemann manifold 308 curved space 42, 308

D denumerable oscillators 85 differenciation 17 dipole currents 2 Donnar's hammer 37

E eigen-coordinates 305, 308 Einstein tensor 304 energy-impulse tensor 91 Euler 308 extended Lorentz convention 4

F five-dimensional cube 33 four-dimensional cube 28 G Gamma function 182

Habermann 70 Hamilton function 5 Hammond 35 Harmuth 1, 2, 3, 4, 6, 7, 8, 9, 11, 12,16, 24, 28, 29, 35, 48, 54, 85, 91, 96,98, 102, 153, 154, 158, 163, 164, 171, 172, 179, 182, 187, 269, 271, 284, 286, 293 Hauptlosung 18 Heisenberg approach 96 Heitler 97 Hermite polynomials 99 Hillion 2 human invention 33 Hussain I

I importance of modified Maxwell equations 116 infinitesimal 23 information theory 36, 44 inherent dipoles 3 inhomogeneousdifferenceequations 128

K Kant 33 Klein-Gordon equation 47, 51, 54, 91, 102, 153

L Lagrange 308 Lagrange function 5 Lagrange, inhomogeneous difference equation 128 Lamb shift 179 left difference quotient 15 Levi-Civita 308 Lifschitz 91 Lobaschefskii 35

INDEX

M

R

mass-ratios 220 mathematical invention 25 Mayan time 34 Meffert 3, 4, 6, 7, 8, 9, 11, 12, 16, 48,

relative bandwidth 1 relativistic Hamilton function 6 right difference quotient 15 Riemann 35, 40, 43, 44, 301 Riemann-Christoffel tensor 304, 306 Ricci 308 Ryzhik 22, 90

54, 85, 91, 96, 98, 102, 153, 154, 158: 163, 164, 171, 172, 179, 182, 187, 271, 289, 286, 293 Merril 1 Messiah 48, 179 metric tensor 43, 304 Milne-Thomson 15, 17, 19, 22, 128, 247, 264, 291 minimum value AT 193, 210 mixed difference quotient 241 monopole current 2 N negative masses 184 non-commuting factors 9 non-denumerable 23 Norlund 15. 17, 18, 19, 22, 128, 247, 264, 291, 293 0 observable sciences 36 one-dimensional coordinate system 23 orientation polarization 3 P parallel universes 30, 35, 308 particles *m.o 189 periodic system 217, 218 physical line 37 physical points 43 physical space 29 Pitajeski 91 point-like 304

S Schiff 179 Schrodinger approach 96 signal solutions 2 signal representation 36 Smirnov 122 Stegun 182, 188, 275 Sterling's relation 182 strong interaction force 218 summation 17 symmetric difference quotient 16 system of non-linear equations 167

T tensor 43, 91, 304, 306 thinkable concepts 37 thinkable sci~nces36 'rhompson 35 Thor's hammer 37 time, concept 35

u

ultra wide band 1 unbounded anti-particles 187

W Wallenberg 291

z

zero-order approximation 170

This page intentionally left blank