Advanced Electromagnetism: Foundations, Theory and Applications 9810220952, 9789810220952

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ADVANCED ELECTROMAGNETISM Foundations, Theory and Applications

)lo

! Editors

Terence W. Barrett Dale M. Grimes World Scientific

ADVANCED ELECTROMAGNETISM Foundations,

Theory and Applications

Nous ne savons le tout de rien. -

Blaise Pascal (1623-1662)

ADVANCED ELECTROMAGNETISM Foundations, Theory and Applications

Editors

Terence W. Barrett BSE.I, USA

Dale M. Grimes Pennsylvania StateUniversity, USA

\\I, WI

World Scientific Singapore• New Jersey• London• Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite 18, 1060 Main Street, River Edge, NJ 07661 UK office: 51 Shelton Street, Covent Garden, London WC2H 9HE

ADVANCED ELECTROMAGNETISM: FOUNDATIONS, THEORY AND APPLICATIONS Copyright© 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Printed in Singapore.

V

FOREWORD

There can be little doubt that Maxwell's equations constitute one of the great landmarks in physical theory. Their basic accuracy has been confirmed innumerable times, in many different types of experiment. Their invariance properties led Einstein to his special theory of relativity. Moreover, their gauge-theoretic interpretation led to non-Abelian generalizations, fundamental to modem particle physics. Their elegant mathematical form has provided several important influences on the development of mathematics itself. These facts should not, however, deter theoretical or experimental physicists from seeking alternative descriptions, unconventional formulations, surprising electromagnetic effects, or radical generalizations. The various articles in this book provide the reader with a great variety of different kinds of approach to developments of this nature. We have historically motivated accounts, suggestions for new experiments, unconventional viewpoints and attempts at generalizations. We also see novel and ingenious formulations of electromagnetic theory of various different kinds. I am sure that this book will make it clear that electromagnetism is a subject that is in no way closed to stimulating new developments. It is very much alive as a source of fruitful new ideas. Roger Penrose

VII

PREFACE The papers in this book are separated into three categories: Foundations, Theory, and Applications. Tne unifying theme of all chapters in all categories is a broader view of electromagnetism than usually taken. We have deliberately invited some papers that we know challenge the conventional view of electromagnetism. Our justification for this might be described as follows. Before the development of anything begins - whether theorem, thermometer, or theodolite - the design should be frozen; otherwise continuing developments produce continuous design changes, and ultimate frustration. Therefore during the development of a theory, device or system, careful designers must act with incomplete knowledge. A timely example is the design of computers. The field evolves so rapidly that next year's tools, both software and hardware, will be better and less expensive than this year's, yet the product of today must be based upon last year's frozen design. Just as a system design is frozen for progress may be made in system development, theories are frozen so progress may be made in applications. In the 1920s and '30s the founders of quantum theory knew that their understanding was incomplete. Although serious questions regarding interpretations were raised by de Broglie, Schrodinger, Einstein, et al., the descriptive equations formed the basis of contemporary quantum theory, and, in tum, solid state physics and, later, the electronic-based evolution of society. But forging ahead in this way carries an inherent risk: With each success of a working model come additional adherents to the view that the interpretation adopted as correct as a pragmatic measure at the time is correct for all time, rather than a photographic still taken during its evolution. For example a Time magazine quote by Hynek (August 1967) states: ''There is a tendency in the twentieth century to forget that there will be a twenty-first century science, and, indeed, a thirtieth century science from which vantage points our knowledge of the universe may appear quite different." The major point with respect to the present endeavor is that great predictive power without physical insight may be an impediment to future progress. In the case of the theory of electromagnetism, the theory was first simplified before being frozen. Maxwell expressed electromagnetism in the algebra of quaternions and made the electromagnetic potential the centerpiece of his theory. In 1881 Heaviside replaced the electromagnetic potential field by force fields as the centerpiece of electromagnetic theory. According to him, the electromagnetic potential field was arbitrary and needed to be "assassinated" (sic). A few years later there was a great debate between Heaviside and Tate about the relative merits of vector analysis and quaternions. The result was the realization that there was no need for the greater physical insights provided by quaternions if the theory was purely local, and vector analysis became commonplace.

VIII

The vast applications of electromagnetic theory since then were made using vector analysis. Although generations of very effective students were trained using vector analysis, more might be learned physically by returning, if not to quaternions, to other mathematical formulations in certain well-defined circumstances. As examples, since the time when the theoretical design for electromagnetism was frozen, gauge theory has been invented and brought to maturity and topology and geometry have been introduced to field theory. Although most persons view their subject matter through the filter of the mathematical tools in which they are trained, the best mathematical techniques for a specific analysis depend upon the best match between the algebraic logic and the underpinning physical dynamics of a theoretical system. Several chapters in this book challenge the view that the algebraic logic of electromagnetism is constant under all conditions, local and global. Hence the quotation on the frontispiece by the mathematician, physicist, religious philosopher and prose stylist Blaise Pascal (1623-1662):

Nous ne savons le tout de rien. We only know everything about nothing. 1 Since several chapters present experimental and theoretical evidence that we do not know all that might be known about electromagnetism, to think we do is hubris. We hope that the material presented in this book will inspire others to view electromagnetism as a dynamic field worthy of additional research. Each chapter of this book was invited. The chapters are directed towards a new understanding of previously neglected or misunderstood results and experiments, towards a treatment of new experiments, new physical understanding, new mathematical techniques and an extension of electromagnetism when the appropriate boundary conditions require alteration of the foundation algebraic logic. Several chapters in this book use modem analytical tools to examine empirical evidence that was either unknown, too difficult to interpret, or considered unimportant when the old paradigm was frozen. For example, a common viewpoint is that electromagnetic fields are separate from space-time. Chapters of this book that address the issue consider electromagnetic fields to be an integrated part of spacetime. They also stress the importance of the underlying algebraic logic to electromagnetic field theories. This is congruent with the view, beginning with Riemann, that the concept of force is secondary to geometry. Using a group theory description, conventional electromagnetism is a local theory of U(l) symmetry form with electric charge and an absence of magnetic charge. For many, even most, conditions, this description works well. However, electromagnetic solitons are wellknown to be of SU(2) symmetry, indicating a need to extend electromagnetic theory

1We are indebted to 0. Costa de Beauregard for this quotation.

IX

to higher symmetry forms under well-defined topological and boundary conditions. The extension of conventional electromagnetism to higher symmetry forms permits easier and more physically informative descriptions of such entities as solitons, etc. The foundational algebraic logic must reflect the topological and boundary conditions. Armed only with differential calculus there is no awareness that field dynamics is held hostage by the topological restrictions determining the algebraic logic. This view raises a question of importance to those seeking a unification of all forces. Perhaps unification of other forces with electromagnetism needs to be with a higherorder symmetry form of electromagnetism than the U(l) form. Such new approaches have impact not only on the foundations and theory, but also in the applied area. For example, early in this century a dichotomy became apparent between an atomic level application of classical electromagnetism and experimental results. After a score more years, the differences were called irreconcilable; electromagnetic theory was applied to atomic phenomena where results agree and ignored otherwise. One chapter asks if today's technology leads to the same conclusions. Part of the analysis contrasts radiation from antennas and electrons. Although both exchange energy with the fields, descriptive and analysis techniques are disparate: Electron analysis ignores field patterns and emphasizes initial and final states and kinematic radiation properties. Antenna analysis ignores states and kinematics and emphasizes field patterns. Antennas have a lower limit to the diameter-to-wavelength ratio. Electrons do not. The chapter concludes that the imaginary part of the complex Poynting theorem applied to multiple sources has been misinterpreted, and that the working model of an electron needs modification. With these changes, atomic properties are consistent with and derivable from the classical field equations. Two chapters emphasize the importance of exact solutions to electromagnetic field problems. One chapter examines numerical and analytical methods for evaluating field integrals about current-carrying wires and obtains an exact formulation for the vector potential about a current-carrying wire. This chapter is an expansion and compilation of other papers that have received several awards. According to one of the award citations, "Tnis work will significantly impact method of moments wire modeling by eliminating the need to perform numerical integration and extending the range of wire diameters that can be successfully modeled." The other chapter obtains an exact solution to a receiving antenna. Unless users of iterative solution methods are sufficiently imaginative in their choice of starting conditions, solutions will not converge to the complete answer, and the user will not know this lack of convergence. This chapter obtains the electromagnetically complete set of receiving antenna current modes. Among other matters, it is learned that receiving modes are essential for electromagnetic momentum to be conserved during reception. Science and engineering do not march steadily onward, and some of the authors feel that we must return to the literature of the time when the foundations of electromagnetism were being frozen to continue progress in foundations, theory and

X

applications. The feeling is that re-working some of the old problems reveals that the theoretical choices which have worked so well for us in later years are true only conditionally, and, if the conditions are changed, the choices made then are not wrong, but inappropriate under the changed conditions. The intention of these contributors is to place contemporary electromagnetic theory within a larger context of contemporary developments in all of field theory. In the light of the circumstances described above, we can define the field and endeavor: advanced electromagnetism is the study of the wider development of electromagnetism as a field theory, talcing the contemporary formulation as an extremely important special case.

Terence W. Barrett, Vienna, Virginia, U.S.A. Dale M. Grimes, University Park, Pennsylvania, U.S.A.

XI

CONTENTS

Foreword Preface

V

Vil

Foundations 1.

Gauge Theories, and Beyond R. Aldrovandi

2.

Helicity and Electromagnetic Field Topology G. E. Marsh

3.

Electromagnetic Gauge as Integration Condition: Einstein's Mass-Energy Equivalence Law and Action-Reaction Opposition 0. C. de Beauregard

4.

The Symmetry Between Electricity and Magnetism and the Problem of the Existence of a Magnetic Monopole G. Lochak

5. Quantization as a Wave Effect

3

52

77

105

148

P. Comille

6. Twistors in Field Theory J. Frauendiener and S.-T. Tsou 7.

Foundational Electrodynamics and Beltrami Vector Fields D. Reed

8. A Classical Field Theory Explanation of Photons D. M. Grimes and C. A. Grimes 9.

Sagnac Effect: A Consequence of Conservation of Action Due to Gauge Field Global Conformal Invariance in a MultiplyJoined Topology of Coherent Fields T. W. Barrett

182

217

250

278

XII

10. Gravitation as a Fourth Order Electromagnetic Effect A. K. T. Assis

314

1I.

332

Hertzian Invariant Forms of Electromagnetism T. E. Phipps Jr.

Theory 12. Pancharatnam's Phase in Polarization Optics W Dultz and S. Klein 13. Frequency-Dependent Dyadic Green Functions for Bianisotropic Media W S. Weig/ho/er 14. Covariances and Invariances of the Maxwell Postulates A. Lakhtakia

357

376

390

15. Solitons and Chaos in Periodic Nonlinear Optical Media and Lasers J.-H. Feng and F. K. Kneubiihl

411

16. The Balance Equations of Energy and Momentum in Classical Electrodynamics J. L. Jimenez and I. Campos

464

17. Non-Abelian Stokes Theorem B. Broda

496

18. Extension of Ohm's Law to Electric and Magnetic Dipole Currents H. F. Harmuth

506

19. Relativistic Implications in Electromagnetic Field Theory M. Sachs

541

20.

560

Symmetries, Conservation Laws, and Maxwell's Equations J. Pohjanpelto

XIII

Applications 21.

Six Experiments with Magnetic Charge V. F. M ikhailov

22. Ampere Force: Experimental Tests

593 620

R. Saumont

23. The Newtonian Electrodynamics and Its Experimental Foundation P. Graneau

24. Localized Waves and Limited Diffraction Beams

636 667

M. R. Palmer

25. Analytical and Numerical Methods for Evaluating Electromagnetic Field Integrals Associated with Current-Carrying Wire Antennas D. H. Werner

26. Transmission and Reception of Power by Antennas D. M. Grimes and C. A. Grimes

682 763

GAUGE THEORIES, AND BEYOND R. ALDROV ANDI# INSTITlITO DE FISICA TE6RICA UNIVERSIDADE ESTADUAL PAULISTA Rua Pamplona, 145 01405-900 - Sao Paulo SP Brazil

Abstract Gauge theories, which describe the electromagnetic, the weak and the strong interactions, are summarized and translated into differential geometric language. The bundle formalism is shown to allow a very economical presentation of the local aspects of these theories in terms of the Lie algebras of the tangent fields. Bundles provide also the geometrical background for gravitation as described by General Relativity. Gauge field strengths are curvatures on bundles built up with spacetime and the gauge groups, whereas a gravitational field is a curvature on the bundle of the spacetime frames. Difficulties in unifying the gauge field interactions between themselves and with gravitation suggest that we should go beyong the bundle scheme and look for some deformation of it, capable of accomodating the four known basic interactions. An attractive suggestion has been made by Sakharov, that gravitation is a consequence of quantum fluctuations of the vacua of the other fields. It is our aim here to show that gauge theories offer an alternative approach. Deformations are introduced as Lie extensions of the tangent field algebras of gauge theories. They are shown to induce a non-trivial vacuum already at the classical level and to be able to engender effects similar to the gravitational field.

O. Introduction

Of the four interactions of Nature accepted today as fundamental, three are described by gauge theories 1. Electromagnetism, by itself a gauge theory for the group U( 1), is also a partner of the weak interaction in the # With partial support of CNPq, Brasilia and FINEP- Rio de Janeiro.

E-Mail: Raldrovandi @ IFf.UESP.ANSP.BR; Fax:(55)(011)288 8224. 1 The subject is treated in every modem text on Field Theory, from the classical treatise by N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd ed. (J.Wiley, New York, 1980) to the more recent C. ltzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980); there are many specialized texts, such as L.D. Faddeev and A.A. Slavnov, Gauge Fields.Introduction to Quantum Theory (Benjamin/Cummings, Reading, Mass., 1978); for an excellent short introduction covering practically all the main points, see R. Jackiw, Rev.Mod.Phys. 52 (1980) 661.

4

electroweak SU(2)®U(l) Weinberg-Salam theory. The interaction binding the quarks to make up the hadrons is described by chromodynamics, a gauge theory for the "color" group SU(3). The three theories are combined in the "standard model", with the direct product SU(3)@SU(2)®U(l) as the gauge group. Besides an impressive amount of experimental successes, these theories are of sound theoretical health: they are renormalizable, that is, fully compatible with Quantum Mechanics. The phenomenology of weakly interacting elementary particles had, already in the fifties, determined that massive vector bosons should intermediate the interactions, but theories involving vector bosons are in general non-renonnalizable. The problem was solved around 1970, when 't Hooft and Veltman showed that they would be renonnalizable provided the bosons were related to that very special kind of vector fields, gauge fields. Gauge bosons are made massive by the Higgs mechanism, which involves additional scalar fields and a non-invariant ground state (or "vacuum"). The phenomenological evidence for these theories is overwhelming, but some lacunae remain. Direct experimental confirmation is still missing for some predictions, as the pure gluon-gluon interaction and the existence of the Higgs boson. There is the unsettled quark confinement problem and the difficulties with further unification. The Weinberg-Salam model is not a unified theory, as the relation between the electromagnetic and the weak coupling constants involve a third, independent parameter. The grand-unified scheme has been disallowed by the experiments which did not find the predicted proton decay. The problem of unification remains and it is only natural to look for inspiration in the only working theory which is not of the gauge type. We say this because the fourth basic interaction, gravitation as described by General Relativity, stands apart in proud aloofness. Despite a large body of favorable experimental evidence, also here not everything is perfect: there are problems in accounting for spin and a satisfactory solution of the problem of renormalizability is as yet missing. There is a general feeling that, despite the experimental support found for gauge theories and General Relativity, they should be somehow enlarged or deformed, the present dominating trends being those involving strings, supersymmetry and quantum groups. Of all that has been said about the relation of gravitation to the other interactions, perhaps the most enticing suggestion 2 is that by Zeldovich and Sakharov, that gravitation is a manifestation of the vacuum of the other fields. Most of the studies in this line have aimed at showing that "gravitational" characteristics like curvature could be obtained from

2 A.O.

Sakharov, Doklady 12 ( 1968) l 040.

5

quantum fluctuations3. Calculations are extremely difficult and no conclusive results have been obtained. We present here a classical alternative to that proposal, a deformation of the gauge pattern based on the theory of Lie algebra extensions. It is shown that the presence of a noninvariant vacuum can indeed lead to those "gravitational" attributes. The first gauge theory of standing consequence was that of Yang and Mills 4 , published in 1954. At about the same time the mathematicians 5 were giving the final touches to the construct of fiber bundles. It tum~ out that fiber bundles stand in the background6 both of gauge theories and General Relativity, as stage scenes on which dynamics displays its drama. This is not so surprising - the bundle formalism is only pure geometry writ in a highly economical way. Nothing more than a purely descriptive account of each topic will have place in the short space we are allowed here. This is the main reason for the pontifical style adopted. We shall, except in the first chapter, use anti-hermitian group generators and neglect the explicit mention of coupling constants. Section I is an elementary resume of standard classical gauge theory. Some differential geometric language 7 is introduced in Section II, with accent on vector fields and differential forms. Special emphasis is put on the fact that the tangent structure contains all the local information on a manifold. Section III is a portrait, in broad brushstrokes, of the formalism of fiber bundles. Section IV concentrates on the tangent structure of the bundles, which is shown to summarize also the local dynamics of gauge theories. Section V comments on gauge fields and gravitation, their similarities and differences. In Section VI the theory of Lie algebra extensions is sketched, adapted to the bundle tangent structure and used to obtain a "deformed" version of gauge theories. Quantities of gravitational appeal, like curvature and torsion, emerge from the deformation.

3 For a review, see S. Adler, Rev.Mod.Phys. 54 ( 1982) 729. 4 C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191. 5 For general contemporary accounts, see N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, 1970); A. Lichnerowicz, Theorie Globale des Connexions et des Groupes d'Holonomie (Dunod, Paris, 1955). 6 A. Trautman, CzchJ.Phys. B29 (1979) 107 ; M. Daniel and C.M. Viallet, Rev.Mod. Phys. Sl ( 1980) 175. 7 W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry (Academic Press, New York, 1975).

6

1. The Gauge Makeup 1.1 Pure electromagnetism It is well known that the electromagnetic Lagrangian -

-

-

-

£ = ~ {i 'Pyµaµ 'P- i[aµ o/]yµq,} - m q, q, + e Aµ q, yµq, -

1 4

pr1vFµv

(1)

is invariant under the global gauge transformations q, (x)

--+

(2)

'P'(x) = eiaq, (x),

with a a constant phase. This invariance leads to charge conservation. The phase factor eia is an element of the group U(l) of lxl unitary matrices. But £ is also invariant under local transformations, in which the phase is pointdependent, provided the vector potencial changes concomitantly as Aµ(x)

---+

A'µ(x) = Aµ(x) +

le aµa(x).

(3)

The field strength F µv = aµAy - avAµ is invariant. Eq. 1 is the same as £ =~{i'P yµ (dµ- ieAµ)'P- i[(oµ+ ieAµ)

'P ]yµ'P}-m

qi 'II- ~pµvpµv.

(4)

Comparison with the free case leads to the minimal coupling prescription: replace each derivative in the Lagrangian by a "covariant derivative",

(5) In the bundle formalism such derivatives are natural features of the geometric background. The electromagnetic potential Aµ must behave as in Eq. 3 in order to compensate for the derivatives aµa(x) turning up in the point-dependent case. It appears as a "compensating field". The phase factors eia(x) are now point-dependent elements of the abelian gauge group U(l). 1.2 The non-abelian case Gauge theories got really started when Yang and Mills replaced U(l) by the isospin group SU(2) of the 2x2 special unitary matrices, whose non-

7

abelian character makes a great difference. Utiyamas generalized their procedure to any Lie group G generated by matrices la. Each source field will now be a multiplet transforming according to

'II (x)

-->

'1'1(x)

=U(x) 'l'(x) =eia~x)T a 'l'(x).

(6)

Here a, b, c, ... (= 1, 2, 3, ... , dim G) are indices of the group Lie algebra. Ta= p(Ja) is the gauge group generator Ja in the representation p of the multiplet 'II. The aa's are the group parameters and the phase factors U(x), elements of G, are now operators. The derivatives present some new problems. Each group parameter will require a compensating field A aµ(x). The la's satisfy the general commutation relations

(7) where the~ ab's are the structure constants of G, and will transform by la

-->

Ja' = eia~x)Ja la e-iah(x)Jb_

(8)

The vector potential will then be the matrix a

Aµ(x) = J a A µ(x).

(9)

This means that Aµ(x) belongs to the adjoint representation of G, by which the group acts by (Eq. 8) on its own Lie algebra. To sustain the role of compensating field, this matrix potential will have to transform as -1

I

Aµ(x)-+ A µ(x) = UAµ(x)U (x) +

gi UaµU-1.

(10)

In the original Yang-Mills theory, the Ja's are in the SU(2) Lie algebra. In the lowest-dimensional representation, Ti = ~ Oi, the Oi's being the Pauli . o 1 = ( o1 01 ) ; o 2 = ( oi -iO ) ; o 3 = ( 0I _O1 ) . Th e potent1a · I 1s · t hen matnces _

A-

8 R.

Utiyama,Phys. Rev.

1 ( -2

Al µ-I"A2 µ) Al µ+I"A2µ -A3µ ·

101( 1956) 1597.

A3 µ

(11)

The minimal coupling rule accounts for the local symmetry of a field 'I', belonging to the representation p: it is enough the change the derivatives on '11by covariant derivatives, (12)

where Ta= p(Ja)- Singlets are invariant (Ta= 0) and do not "feel" the gauge field. There are two quite distinct kinds of fields, the source fields and the gauge fields, transforming respectively as in Eq. 6 and Eq. 10. The field strength is also a matrix in the adjoint representation, (13) (14)

It is no more invariant as in the abelian case, but only covariant. In detail, (15) Extensive use will be made of the symbol [Aµv], indicating complete antisymmetrization of the bracketed indices. The Bianchi identity (16)

is an automatic consequence of Eq. 15. The operator acting on Faµv is the covariant derivative of the field strength. If we define the dual tensor (17)

the above expression may be written in the compact form

~ Faµv + fi>cAbAFcµv = 0.

(18)

The field equations are the Yang-Mills equations (19)

or equivalently

9

(20) where the Jav's are the source currents. When Jav = 0, the field equations are just the Bianchi identities written for the dual of F. This is the duality symmetry . The Bianchi identities will be seen to have a purely geometrical content, so that this symmetry establishes a relationship between dynamics and the geometric background. In this sense gauge theories are "more geometric" than General Relativity, which shows no such relationship. Eq. 19 can be written directly for any Lie group: it is enough to know the structure constants. It is usually obtained from the Lagrangian £ = ~ trFµvFµv + £source, but comes also by the duality prescription: find the sourceless equation by duality symmetry and add the source current. This rule gives the field equations even when no Lagrangian is available, as in the case of a non-semisimple gauge group9. Dµ, the covariant derivative of the dual, is a covariant coderivative. It follows from the field equations that (21)

because DvDµPµv

=~

[Dv, Dµ] pµv

= ~ ph µv flbcpcµv = 0. The vanishing

of a current divergence leads to a charge conservation by Noether's first theorem, but the vanishing of a covariant divergence, by Noether's second theorem, only implies the invariance and good behaviour of the theory 10. The Lagrangian for a Dirac source field will be £= ½{i'PyJ'(.Jmi= 213. The figure also shows the effect of imposing 81 (mod 2n) and Bi(mod 2n). Notice that the topology of the torus is obtained by identifying opposite sides of a rectangle. If roilmi is a rational number, roi and mi may be chosen so that they do not have a common divisor other than unity, and are therefore relatively prime. For roi ~ 2 and mi> 3, the field line is knotted and corresponds to the torus knot 10 K"'1,"'2• On the other hand, if roi and mi are rationally independent then roilmi is an irrational number and a field line satisfying Eq. (5) is everywhere dense on the torus; i.e., the field line will come arbitrarily close to any point on the surface of the torus. The closed field line given by roi/mi = 213 required five lines for its representation by the map of the torus shown in Fig. 3(d). A field line that is dense on the torus would completely fill such a map.

*

1.2. Topology of Constant a Solutions to the Force-Free Field Equations

Henon 11 has performed a numerical experiment to illustrate the complicated behavior of the field lines for a force-free field with constant a. He used the form of Eqs. (1) with X; identified with B;, to study the system of equations:

t

=Bx= A sinz + Ccosy,

t

=By= B sinx +A cosz,

t

= Bz = C siny + B cosx,

(7)

where A, Band Care constants. By talcing the curl, the field [Bx, By, Bz] is readily verified to be force-free with a = 1. The system is periodic with period 2n. Henon computed the flow given by Eqs. (7) on a 3-dimensional torus, T 3 • The topology of T 3 is obtained by identifying the opposite faces of a cube; i.e., here by using the coordinates x (mod 2n), y (mod 2n), and z (mod 2n).

57 Fig. 4 shows the numerical integration done by . . .. 0.9 Henon for A= fJ, B = fl, and C = 1. The points ..... 0.8 correspond to successive intersections of a field line 0.7 . with the x,y - plane (mod 2n) specified by the section z = 0 0.6 (mod 2n). The points joined ~ N fo.s by a curve correspond to the e>,. same field line, while all the 0.4 isolated points also correspond to a single field 0.3 line that is not constrained to •,c a surface. The latter are 0.2 •..· "semi-ergodic," filling a 0.1 • .. ... region of 3-dimensional .: space. Dombre, et al. 12 have o 0.1 0.2 0.3 0.4 o.s 0.6 0.1 0.8 0.9 1.0 made an extensive study of x mod(2n) this system of equations. Fig. 5 shows the Figure 4. Numerical integration of Equations (4.7). The points behavior of Eqs. (7), with joined by a curve correspond to the same field line, and the isolated Henon's values for A, Band points correspond to a single field line not constrained to a surface. C, in three dimensions. The Adapted from: M. Henon, C.R. Acad. Sci. Paris 262A, 312 (1966). field lines in this figure are constrained to different surfaces depending on the choice of initial conditions. Dombre, et al. showed there are six such surfaces or principal vortices. Fig. 6 shows two views of a far more complicated behavior obtained by yet another set of initial conditions. Here, outside the principal vortices, the field lines show a chaotic behavior. The figures were produced by using the Runge-Kutta method of numerically integrating systems of ordinary, first-order differential equations. 1.0

·-

I

•

•

6

6

6

6 6 6 Figure 5. The behavior of Eqs. (7) depending on different initial conditions. The field lines are constrained to different surfaces. There are six such surfaces or principal vortices.

58

Figure 6. The behavior of Eqs. (7) for another set of initial conditions. Outside the principal vortices, the field lines have a chaotic character.

2.0 Magnetic Field Helicity The helicity of a magnetic field can be expressed as

lf=

L

(8)

A·B dV.

It is a pseudoscalar that is gauge invariant in simply connected domains provided the normal component of B vanishes on the bounding surface of V. In multiply connected domains, such as the volume bounded by a toroidal magnetic surface, the meaning of the integral, and its physical interpretation, is not immediately clear. For the latter case, consider a gauge transformation A ➔ A + Vx. The helicity introduced by the term associated with Vx is

:Jf

= { Vx-B

Jv

dV

= ( V ·(XB)

Jv

dV

=

1av

(XB)-dS +

1

[x]B·dl:

1,

(9)

l:1

where [x] is the jump across Ii, the cut needed to make x single valued (see Fig. 7). Since the bounding surface of the torus is a magnetic surface, the first term on the right hand side vanishes. It can be shown 13 that [x] is constant on I 1• Thus, r can be taken as bounding Li and [x] can be taken outside the integral. Now for any two points p 1 and p 2, the difference in the value of the scalar function XIS

59

XP2- XP1=

1P2 V X·dl.

(10)

Pl

Figure 7. :1;1 is the cut need to make the scalar function X single valued. The jump [X] in the value of x across the surface :1;1 is constant on :1;1, so that r may be taken as bounding I.i-

If the path of integration is closed, but does not cross the cut needed to make x single valued, the integral will vanish by Stokes' theorem. When the path of integration r does cross the cut :1;1, the value of the jump [z] is (11)

where 1:2 is the magnetic flux through the surface I.i- Thus, (12)

The helicity introduced by the gauge transformation is therefore seen to explicitly depend on the topology. Physically, the helicity derived from the gauge term corresponds to the linkage of the flux contained within the torus with that passing through the "hole in the donut." The computation of magnetic energy in such multiply-connected domains has been discussed elsewhere. 14

2.1. Twist, Kink and Link Helicity Using a geometric invariant of a space curve, called the writhing number 15, Berger and Field 16 decomposed the helicity of a magnetic field into the sum of "twist" and "kink" helicities and, based on the work of Fuller, 17 defined the helicity of open field structures. They made use of a theorem 18 from knot theory 19 that states that the linking number of two curves X and Y, without common points, can be written as the sum of the twist number Tw and the writhing number WR, (13)

60

This makes sense, for example, if one considers X and Y to be the edges of a ribbon; the example given by Berger and Field identifies X with the central axis of a flux rope and Y with a field line winding about this axis. The topological interpretation of helicity in terms of the Gauss linking number and its limiting form, the Calugareanu invariant, has been extensively discussed by Moffatt and Ricca. 20 In knot theory, a crossover is defined to be positive or negative by the right hand rule (see Fig. 8). The writhing number, WR,is defined to be the sum (N+ - N-), where N+ and N- are the number Positive Crossover Negative Crossover of positive and negative crossovers. If a a (p) = -I a (p) = +I link is projected onto a plane so that the Figure 8. Positive and negative crossovers as crossovers are at points p, the linking determined by the right hand rule. number is defined as Lxr =

1L

(14)

a(p),

peXnY

where Xn Y denotes the set of crossings of the link X with the link Y, and a(p) corresponds to the sign, defined by the right hand rule, assigned to each crossover . Examples are given in Fig. 9. a(p) "'+I

Lxy= 0

Lxy= +2

(b)

(a)

Lxy=O Lxy=O a(p)

= +I

Lxz= o y

X =-I

~

Lyz= 0

a(p)=+I

~

Figure 9. Examples of linking number as defined by Eq. ( 14). The absence of linkage implies that the linking number vanishes, but the converse is not true as seen in (c).

61

Note that the absence of linkage implies that the linking number vanishes, but the converse is not true as illustrated by the Whitehead link shown in Fig. 9(c). While the Borromean rings shown in Fig. 9(d) are clearly linked, any two components are unlinked. The twist number, Tw, will be limited here to integer values and the total twist will generally be multiples of 21r. Assume that the writhing number is an invariant. As is seen in Fig. 10, the effect of switching a positive crossing to a negative crossing is equivalent to inserting a link on one of the elements of the crossing.

OR

►

WR =O'(p) = +1

WR=

l:

O'(p) = +l

(b)

(a)

WR = l: CT(p) = +l (c)

Figure 10. Switching a positive crossing to a negative crossing is equivalent to inserting a link on one of the elements of the crossing. In (a), pc (XnY); in (b), pc (XnY) u (YnZ); and in (c), p

c (XnY) u (XnZ).

In what follows, uniformly twisted flux tubes will be used for heuristic purposes. It must be understood, however, that such tubes, where the zand ~ components of the field are constant and no cu"ent is present, cannot exist in nature. This is clear from the fact that an axial current must be present since

f

B -di, where the integral is about the tube, does

not vanish. Thus, twisted flux loops will have currents present that will in general give rise to additional magnetic fields that link the twisted loop. For a flux tube of small cross section, the integrand in Eq. (8) can be written as A·BdSdl = A·dlBdS, where di is along the axis of the flux tube. Thus, if an originally untwisted flux tube, assumed to be a torus of circular cross section carrying a flux ,is cut, given a twist of 21r,and reconnected, the helicity introduced by the twist is

(15)

62 Note that this is essentially the same result as obtained in Eq. (12). If the original twist had 2, and in general, a twist of 2nTwcorresponds to a twist been 4n, the result would be 2 2• helicity of Jfr = Tw

=

f

A . di

=-

nh 2e

(27)

and the expression of the current 's intensity in terms of the frequency flow of electron pairs (28) I= -2ev, we derive the self-energy quantization formula (29)

2W=cl>l=nhv.

8. Graneau's and Saumont's the Vector Potential

Experiments

Evidencing

Graneau's 18 "railgun" experiment is an enlarged version of Ampere's "hairpin" one. Feeling the BS force, a sliding bridge connecting two laterally constrained long straight parallel conductors generates by reaction a repulsive tension T along each wire, which can be evidenced at any abscissa x by inserting a mercury-loaded junction. If the emf generator is thought of as infinitely distant, Tis x-independent, and unchanged in virtual displacements of the bridge; its relation to the virtual work is 2T = dW / dx, like in a Swiss clock powered by a descending weight. According to Eq. (23) its expression I A is a corollary to that of the virtual work of the BS force applied to the bridge. Physicality of T then implies selection of a preferred gauge, the form of which can only be A = kl. Existence of the Ampere tension is thus unquestionable. Two more sledgehammer proofs of existence of the repulsive tension T = 2w (w = dW/dl) along a straight current are these: (1) the tangential tension T is radius-independent along a circular current, and (2) it is equivalent to the internal magnetic pressure (1/41r)B 2 stretching a long straight coaxial conductor capped by flat conducting discs; a and b denoting the inner and outer radii, the value 2w = 1 2 ln(b/a) is obtained either via 2w = I(Aa - Ab) with A = l ln r, or by integrating 21rrB 2 dr from r = a tor= b with B = I /21rr. Graneau's 18 "railgun" and "exploding wires" experiments I present by using his own words - with his permission and that of his publisher. 19 Un-indicated cuts are made, but of course the reader can go back to the original. My personal comments are inserted in square brackets.

89 While Graneau's evidencing of the repulsive Ampere tension has led him to adopt an anti-SRT stance, I deem the existence of this force not only quite compatible with relativistic covariance, but indeed a corollary to the Lorentz force applied to the flowing electrons. Let us start with railguns (Ref. 19, pp. 156-8): "The gun consists of a pair of straight and parallel conductors one end [of which] is connected to a source of electric current. This end is the gun breech. The other end is the muzzle through which the projectile leaves. To begin with a short piece of copper bridges the rails near the breech. Called the armature [it is] in sliding contact with the rails. When a heavy current begins to flow down one rail, across the bridge, and back in the other rail, the armature is subjected to a strong electrodynamic force which accelerates it down the rails. This force is transverse to the current [in the bridge) and both Ampere's and Lorentz's laws agree on its magnitude. All guns are subject to recoil forces. The railgun recoil has been shrouded in mystery [with this I frilly agree). Ampere's law claims that the forward force in the armature is balanced by two longitudinal rearward directed forces in the rail close behind the armature [which is quite well formalized by the T =IA concept). [Thus) the rails will be pushed backwards, [deflected laterally, and buckled). On the other hand, if Lorentz's law is correct the rails will not experience a recoil force and not buckle. [Therein lies a great shared misunderstanding: along a straight tensed filament opposite tensions do exist: a stretched string breaks under two repulsive tensions. To actualize at any point, and allow measurement of the tension dormant along a straight current, it suffices to insert a mercury loaded junction.]" From railgun recoil and buckling we now turn to exploding wires (Ref. 19, pp. 148-151): "The response of fuse wires to large current pulses was studied in Warsaw by Jan N asilowski. He noticed that a copper wire was shattered into small pieces by large but short current pulses. The wire showed no sign of melting. Metallurgical examination proved that every wire break was caused by an impulse tension. Not knowing of Ampere tension Nasilowski was baffled. [Long after) he was delighted with the Ampere force explanation."

90

If doubts remain concerning the existence of Ampere's repulsive stress tension along a straight current of axis z and its rendering according to Maxwell's electromagnetism, here is a rejoinder. Maxwell's "etheric" pressure (l/81r)B 2 in the surrounding vacuum, integrated all over a plane orthogonal to z, builds up ( as does any positive energy density) a repulsive tension just equal to I A [the logarithmic divergence in both can be removed by imagining a coaxial return current]. Assuming adherence between the field and its source explains the fact. While Graneau plays in the large, Saumont 20 plays in the small.

M (b )

(a) Fig. 3. (a) Atwood's machine: -y = (M - m)/(M

+ m);

(b) anti-recoil when pushing laterally a

toy aerostat: the system's barycenter moves backward.

His horizontal armature (Fig. 3), moving vertically, is a straight wire some 13 cm long placed on the pan of a high precision mechanical (not electric) balance. Its ends, bent down wards, dip in mercury-loaded cups connected with an electric generator. The large fixed part of the circuit is horizontal and so, referring to formula (24) above, it contributes not to the virtual work - which then equals the self-energy of an ideal circuit defined as the difference between the final and initial positions of the armature. Experimentation evidences, as expected, an upward lift proportional to the square intensity, of value independent of the armature's length. The ends of the severed fix and mobile parts of the circuit are enameled, cut straight, with naked sections facing each other. Thus, the force that is measured is parallel to the current inside the mercury, and repulsive. Counter-tests confirm this. If the armature's ends make U turns, so that the repulsive tension operates downwards, the measured force remains the same. If, inside the mercury, the current runs up at one end and down at the other, a zero force is measured. And if the current runs sideways in the mercury, a zero force is also measured. Other counter-tests eliminate artefacts. Moved up or down, the fixed portion of the circuit regenerates the transverse BS force proportional to the armature's

91

length. Care has been taken of the Archimedian lift on the armature's ends dipping in mercury, and of the aerostatic one due to Joule heating. So illustrating the argument of Section 5, this set of experiments measures the sum and the torque of the mutually opposite wrenches applied to each other by complementary arcs of the circuit, in the form of paired opposite tensions of value ±I A locally tangent to the wire. J. Mourier, 21 expressing the self-induction coefficient k in terms of the ratio of diameter to curvature radius of the armature terminals, and using the local BS forces, gets good agreement with the measurements. 9. From Weber 1848 to Darwin

1920

Valid up to order (v / c) 2 included, Darwin's 22 1920 Galilean-invariant, instantaneous-far-action approximation to electrodynamics is used in Coleman and van Vleck's 6 paper entitled "Hidden momentum in magnets." Let us evidence some far-reaching implications it has. Defined d la Sommerfeld, the effective mass of each point charge is

m =mo+

1 -c 2

-2

(mov2 + QV ) + ... ,

(30)

V denoting the variable electric potential created by the other charges in the Coulomb gauge. The mutual energy contribution to the mass m has consequences we will discuss, We note first that equipartition of the mutual energy of any two charges is implied in formula (30). Also follow6 conservation of the system's total mass

(31)

barycentric moment MR=:Emr,

(32)

and total linear momentum

Weber 24 found in 1848 the electric contribution to inertia and the direct actionreaction term displayed in formula (33); Darwin's more rigorous approach generates QA, not Q V v, in each particle's momentum, thus expressing non-collinearity of velocity and electromagnetic momentum; then it is via summation that I: QA '.:::= I: QVv. Darwin's expression of each particle's effective momentum is the low

92 velocity approximation to Wheeler-Feynman's covariant formula (40) - which is especially relevant for us. Helmholtz objected to Weber's formula that it allows negative masses, which is true of Darwin's formula also and is discussed in the next section: a point charge immersed in a fieldless electric potential, the source of which is not freely falling, is not freely falling either.

10. Coulomb Potential-Induced

Archimedian

Lift - or Rest

The Archimedian lift -mg felt by a body "displacing" a fluid mass m in the presence of a gravity field g mimicks anti-gravity. It also mimicks anti-inertia as, according to Einstein's inertia-gravity equivalence (including his elevator metaphor), a body immersed in a fluid, the container of which is accelerated by g, displays a negative extra inertia -mg. So a body "displacing" a fluid mass m behaves as being endowed with a negative extra mass (not rest mass) -m. For example, inside an accelerating (decelerating) caravan a toy helium balloon is projected forward (backward); and if inside a parked caravan someone displaces laterally a toy balloon resting at the ceiling via barycenter conservation the van anti-recoils, so that relative to the background the person is moved forward (cf. Fig. 3). Darwin's formulas imply the existence of analogous electrodynamic phenomena. Consider, in place of the van, an insulating uniformly charged sphere with inside it a point charge replacing the balloon. In slow relative motion of the sphere, charge uniformity of the enclosed Coulomb potential proper is maintained by the sphere's variable stress tension. As the sum of the stress-energy plus the half-mutual-energy attached to the sphere remains zero, totality of the electrostatic mutual energy VQ is reported on the point charge - a conclusion already drawn in the Introduction from the timelike Aharonov-Bohm effect. If both are at rest, sphere and charge exert no mutual forces. But if the sphere is accelerated from outside, or the charge from inside, formula (32) requires that the point charge of value Q should behave as being endowed with an extra mass c- 2 VQ; Einstein's inertia-gravity equivalence then means that, inside a sphere at V > +511 kV, an electron will levitate and a firing electron gun will anti-recoil. These outrageous claims are derived straightaway from Darwin's semi-relativistic electrodynamics, namely the Coleman-van-Vleck's barycenter formula (32); can we trust them? Let us modify our thought experiment by placing the point charge Q on the axis z of a cylindrical capacitor, thus immersing it in a uniform potential of expression V = q ln(r1/r2). Pictured d la Maxwell, the mutual energy VQ resides in the field between the armatures with its barycenter at Q. Thus if slowly accelerated, or held

93 suspended in a uniform gravity field along z, the point charge, shadowed by this cloud, behaves as being endowed with the extra inertial mass c- 2 v Q.

11. Relativistic Far-Action-Reaction in the Wheeler-Feynman Electrodynamics Explicitly covariant a la Minkowski, the Wheeler-Feynman 17 electrodynamics of point charges is best visualized as a four-dimensional transposition of the statics of filaments. An isomorphism exists between it and the Ampere magnetostatics discussed in Section 6, the translation lexicon being: 3-space --+ 4-space-time; current wire of intensity I --+ timelike trajectory of charge Q; stress tension --+ 4momentum; linear force density --+4-force; torque--+ 6-angular momentum; energy --+ action. Using Fokker's definition of the mutual action of two point charges (i, j, k, l = 1, 2, 3, 4; x 4 = ict; ri = ai - bi) (34)

[compare with (2)) and the source-adhering half-retarded half-advanced LienardWeichert 4-potential [compare with (3))

(35) which obeys the Lorentz condition

(42) WF derived for each point charge of combined 4-momentum

pi= mVi

+ QAi

(36)

and therefore combined action

dL the acceleration equation (ds 2

= Pidxi

,

(37)

= -dxidxi) (38)

evidencing direct action-reaction between any two point charges

(39) [compare with (22)).

94 Integrally equivalent to this "Ampere 4-force," there is the a and b asymmetric Lorentz 6-force Q 0 B(a)[iildai felt by each charge: from (35) we derive

which, subtracted from (39), yields

These WF formulas entail a largely overlooked important consequence: any point charge immersed in a source-adhering 4-potential of expression (35} carries an extra inertia QA i. For example, while the Coulomb interaction force between two heavy point charges at quasi-rest follows from (39), (36) confers on each an extra mass (1/2)c- 2r- 1Q 0 Qb- There also follows from (39) the Weber-Darwin velocitydependent interaction force between moving charges - for instance, between two successive electrons in a cathodic beam. As is well known, 25 non-collinearity of 4-velocity Vi (½Vi = -c 2 ) and 4momentum entails existence of a 6-torque; here, a "potential" torque is defined: (40)

the operational meaning of which is explained below. Opposite potential 6-torques (41) are applied to each other by any two interacting charges [compare with (22)]. It is highly significant that: ( 1) use of the source-adhering Lienard- Wiechert gauge {35} is required for getting the direct action-reaction equation {39}; (2) from it there necessarily f ollows 17 the Lorentz condition (42) From the first remark we derive a far-reaching, relativistically covariant statement: accelerated by any means inside a given electromagnetic field, a point charge Q displays an extra inertial 4-momentum QAi, Ai denoting the source-adhering 4potential. This, together with Einstein's phenomenological inertia-gravity equivalent, validates our frequent recourse to ideally weighing a system {or parts of it) inside a uni/ orm gravity field. We shall now discuss operationality of the "potential" 6-torque (41). Its three [u4] "boost" components were met with in the discussion of equation (33); its three

95 We shall now discuss operationality of the "potential" 6-torque (41). Its three [u4] "boost" components were met with in the discussion of equation (33); its three [uv] components, angular momentum proper, display their magic in the following example. Consider a charge Q flying at a velocity v parallel to a fixed infinitely long and thin straight magnet trapping a flux .The paradox is that, while feeling no force, the moving charge exerts, via its magnetic field B =Ex v, a torque on the magnet, the value of which is [Bx dl] = -(v• dl)E per line element. The integrated torque comes out as -QV /2r, r denoting the constant distance between charge and magnet. As the magnet's fieldless vector potential is of the value A= /21rr,it turns out that just opposite to the torque felt by the magnet there exists a "potential torque" Q[A x v] attached to the flying charge. Thus an angular momentum credit card (similar to the potential energy credit card so widely used) is here accepted. If the magnet of axis z is not infinitely long, but only very long, the charge, before and after running alongside it, passes near a pole, feeling there the Lorentz force generated by a magnetic field coplanar with z; there it initially deposits, and finally cashes, orbital angular momentum in the form discussed above. The preferred gauge here is the one expressing the reaction from the potential 's source.

12. Angular Action-Reaction Opposition, a Feynman Lectures Conundrum, and de Broglie's Photon Spin Density

Going back to Section 3, we replace in equation ( 11) the point charge by a uniformly charged circle coaxial with the dipole of moment M - a "parallel," in geographic jargon. Thus we are left with a purely angular momentum problem. At the latitude a the circles's "potential angular momentum" has the value VM cos a. I will argue that: (1) de Broglie's 26 concept of an electromagnetic spin density of expression 1 u =-{-Ax E+ VH} (43) 41r

must enter angular momentum balance; (2) as a consequence, an opposite angular recoil appears in the magnet. Readers of The Feynman Lectures on Physics are challenged 27 to refute so outrageous a claim, the hint being that not the magnet, but the field, is the place where the missing angular momentum builds up, in the form of orbital angular momentum of the "mutual" Poynting vector Ee x Hm. Existence of a "bootstrap ·merry-go-round" then follows, as no photons are orbiting- and none could, because

96 the sphere's energy quantum is the "anomalously" small one e 2 / R '.::'.(2/137)hv, with v = c/R. Let us radicalize the problem 9 by placing, at the center of a uniformly charged insulating sphere, in its fieldless Coulomb potential V = Q/ R, an initially unmagnetized dipole ferromagnet. If, magnetizing spontaneously later, it generates a magnetic moment M of axis z, the induced electric field confers on the sphere an angular momentum

(44) (2/3 = 1 - 1/3 expressing the difference between spin and precession mentioned in Section 3). If M is positive, a positively charged sphere will thus start rotating eastward, like our Earth. As outside it the magnetic lines go from north to south, and as the electric field points out, the Poynting vector blows eastward, like the tradewind; as its angular momentum has the same sign as the sphere's, it cannot compensate for it. Outside the sphere, at latitude a, the fields and potentials have as non-zero polar components

Er= Qr-

Br= 2Mr-

2,

= Qr-

V

1,

3

Bt

sina,

At

= M r-

2

= -Mr-

3

cosa,

sin a ;

from these we get, outside the sphere, as density contributions vector's orbital angular momentum,

to the Poynting

and to de Broglie's spin,

whence

d3 (Cp Inside the sphere, where E

+ CB)out = -~r-

4 QMsin2a.

=0 and V =const, we get

d 3 (CB)in

= +~r- 4 QM

sin 2a.

Integrating outside the sphere, from R to +oo, we get

(45)

97 a formula saying that the ( algebraic) sum of the Poynting vector orbital angular momentum and of de Broglie's spin density compensates exactly for the sphere's potential angular momentum Cs Physicality of the electric potential and of the photon's spin density could hardly be evidenced better. Inside the sphere, the only density contribution to the volume integrals is VB = Vo x A, which (V being constant) transforms into a surface integral of value 2

CB in= + 3VM,

(46)

just equal to the sphere's Cs. We thus infer that the magnet contains a "hidden angular momentum" -Cs [note that (2/3)VM is the volume integral of -VB). Thus, a zero total angular momentum state of the sphere-and-magnet system is such that both pieces contain opposite electromagnetic angular momenta - in accord with the action-reaction principle. A thought (or why not real?) test could use the Einstein-de-Haas (EdH) effect inside the sphere. At the voltage V such that the potential angular momentum (2/3)VM is just opposite to that -(m/2e)M due to the electron spin, the EdH effect should be inhibited. As, with V0 ~ 511 kV, (1/2)VoM expresses the ferromagnet's spin, this value is V ~ 383 kV (beware: the whole experiment must be conducted at this voltage: uncharging the sphere before making the measurement would intercalate a counter effect). To conclude: de Broglie's 26 concept of an electromagnetic spin density (cf. expression (43)): ( 1) restores the right sign in the Feynman Lectures 27 conundrum, (2) validates action-reaction opposition between the sources of the (combined} field, and (3) confers a testable physicality on the electric potential. 13. De Broglie's

Photon

Energy-Momentum

and Spin Tensors

According to the theory of elasticity, µuv = Euv -Evu (u, v = 1, 2, 3) is the local torque density; from this, one infers 25 that in any medium or field endowed with a spin density u the energy-momentum density Tii (i, j, k, l = 1, 2, 3, 4; x 4 = ict) is asymmetric and obeys the relation (47)

where, inherently antisymmetric in its last two indexes, the spin density often is fully antisymmetric. Using the standard Minkowski-Maxwell equations Lai

Bik =

o,

a,Hk' =

j\

Bij = Hij

+ Mij

,

(48)

[ijk)

(49)

98 we consider the following three asymmetric energy-momentum tensors: (1) MaxwellMinkowski 's (50) M'i = - 4~ { Bik Hi k + ¼Bk1Hkl,5ij} , (2) de Broglie's 26 photon canonical one, where [8i] denotes the Schrodinger or Gordon operator (difference between the partial derivative operators to the right and to the left)

pii

= __!_Ak[8i]Hik,

47r and (3) one implicit all through Sections 2, 4, 5, 7,

Nkl = _ 4~ { Ak/

+ ~A;j',5kl}

(51)

(52)

Using de Broglie's photon spin density a[ijk)

'°'Ai

= __!_

41r ~

Hik ,

(53)

[ijk)

and the magnetic polarization current density z[ijkJ

=

L 8iHik

,

(54)

[ijk)

we derive from the Minkowski-Maxwell equations ( 50) and ( 52) plus the Lorentz condition (40) the spin density conservation equation (55) to be compared with (48). As is well known, the antisymmetric contribution to the Maxwell stress tensor contains the electromagnetic torque and boost densities, E x P + H x M and D x B - E x H. Similarly, the antisymmetric contribution to the A k j' tensor contains the torque and boost densities Ax j and qA- Vj previously encountered. The 4-force densities attached to the three energy-momentum tensors are: (1) the sum of the Lorentz and the Curie or Stern-Gerlach ones, (56)

(2) the sum of the Lorentz one, and of an unfamiliar one to be discussed in Section 15, (57)

99 (3) twice the sum of the two Stern-Gerlach style ones,

akpik

= - ! Bkd8i]Hk'

- Ak [ai]ik . (58) 2 De Broglie's massive photon obeys Minkowski-Maxwell equations with Gordon's operator replacing d 'Alembert's; Proca later proposed equivalent equations for a spin 1 meson. Let us recall briefly how de Broglie's photon spin density shows up in a Maxwellian plane wave. The general plane wave of time frequency v and space frequency k ( k = v / c) can be thought of as a superposition of two waves of opposite circular polarizations. In a pure helicity state each photon has an energy hv and a spin ±h/21r. The mutually orthogonal vectors E and H = 21rvA, of the same magnitude in mixed units, rotate in the wave front, clockwise or anti-clockwise; (1/2)(E 2 + H 2 ) = nhv and Ax H = ±nh/21r respectively measure the energy and spin densities. This holds for a massless photon. A massive photon flies not at the limiting velocity c, but tends to do so as its frequency increases. Spin zero momentum-energy carrying plane waves then exist, propagating a "longitudinal" electric field and vector potential; the probability of their excitation is exceedingly weak. 26 14. Constant-Potential-Dependent

Forces as Source Reactions

Associated with "hidden linear momentum in current loops" of expression (4) is the ponderomotive force V (dl / dt )di; and with the cemf generated along a varied current by a permanent flux from outside, there is the emf I Ad// dt; and with "hidden angular momentum in magnets," there is the ponderomotive torque VdM/dt. All these are reactions from the constant potential 's source thought of as infinitely heavy. 15. On the Akjl

Energy-Momentum

and Aim,ik

Spin Tensors

The space-time scalar Akjk, a four-dimensional action density (and a threedimensional energy density V q - A• j), is a familiar factotum in Lagrangians "where gauge dependence of just a calculation ingredient matters not." The qA and V j components of the A k j', tensor build up the respective "hidden momenta" in formulas (4) and (5). The gauge-dependent 4-force density Ak[8i]jk associated with this tensor in equation (58) is not unfamiliar, as its time component is the difference of known power densities; operationality of the space components then follows via relativistic invariance. Ponderomotive or electromotive force densities such as V8J, A8tq, A8J, have repeatedly shown up in the preceding sections.

100 As said before, this 4-force density appears as relating the tension T = I A and the Biot-Savat force density JB x dl along a current loop. In general we find (u, v, w = I, 2, 3)

d~t

= OwTudlw = I[8wAv -

8vAw]dlw + I8wAu · dlw ;

(59)

that is, the sum of the BS force plus this force. Interlaced with a toroidal magnet (see Section 4) a current loop feels no force. While the prevalent stance is "no Lorentz force, no force," our is: "no BS force, but exact compensation between tension and linear force density." An analogous statement holds along the toroidal magnet of Section 4, where no one disputes the reality of the (unobserved} tension T = ~H and of the Stern-Gerlach force fu = ~8wHu · dlw. The spin density Vm, evidenced in formulas (13) and (46), is a component of the tensor Aim[ik]_ So the Akj' and Aimik tensors are implied in any covariant formulation of the phenomena discussed in this essay. 16. Lorentz Condition

Revisited

The Lorentz condition, a very strong restriction on the gauge, requires that the arbitrary scalar superpotential be a solution of d' Alembert's equation - that is, a sourceless propagating field magnitude. The following statement then holds: the general solution of Maxwell's equations is the sum of physical source-adhering solutions plus the ghostlike general solution of the homogeneous d 'Alembert equation. Plane wave d 'Alembert solutions are "longitudinal waves" propagating a fieldless lightlike 4-potential, Discarding them as "unphysical" is a contrario conceding some physicality to the transverse potential ... Finally, how does the Lorentz condition 8kAk = 0 dovetail with the sourceadhering gauges? Consider permanent regime cases, such as electrons orbiting a heavy nucleus, or Aharonov-Bohm ones bypassing a long solenoid. A permanent regime is one such that in some inertial frame all time derivatives of the field magnitudes are zero; so OtV = 0 and, via the Lorentz condition, 8 · A = 0 Thus the central Coulomb gauge compatible with the Lorentz condition is defined as V = Q / r plus A = O; and the cylindrical Ampere gauge compatible with the Lorentz condition as A= r- 2 ~ x r plus V = 0 (~ denoting the magnetic moment per line element). In either case, we require that the 4-vector Ai be zero at spatial infinity. The Lienard-Wiechert gauge (retarded, advanced, or time-symmetric) - a four-dimensional extension of the Coulomb and Ampere static gauges - satisfies the Lorentz condition.

101

So the Lorentz condition is compatible with any source-adhering gauge defined covariantly. As source-adhering solutions of Maxwell's equations are the zero mass limits of massive photon solutions, their use should be the norm. 17. Lorentz Condition

in the Dirac Electron Theory

All textbooks explain gauge invariance of the Dirac electron theory, none mentioning, however, that the Lorentz condition is inherent- as otherwise a scalar field 8iAi would come up in the second order equation. And none mentions either the following corollary. In 4-frequency terms, any gauge field obeying d'Alembert's equation kiki = 0 and the Lorentz condition kiAi = 0 constrains the gauge to AiAi = 0. Thus a gauge change on the free electron amounts to adding a lightlike vector to its 4-frequency: an extreme-relativistic Lorentz transform implying zero electron mass. The conclusion then is: in the unbounded vacuum the electron's gauge must be Ai = 0. Consequently, a source-adhering gauge should be selected for any electromagnetic potential embedding the electron - including :6.eldlessones. Such a gauge is selected not from within, but from without the Dirac equation. This is how action-reaction with an infinitely heavy source is taken care of: a "macroscopic renormalization" of mass (not rest mass), momentum, or angular momentum. For example, the zero of the energy eigenvalue of an electron guided along the axis of an infinitely long cylindrical capacitor, thus bound to a positive potential V, is at -c- 2 eV. The same is true for a hydrogen atom placed at the center of a large uniformly charged sphere. Of course, part of the electronic wave tunnels outside the container. Anyone familiar with the Dirac equation knows existence of the 10 = 2 x 5 tensorial equations obtained by adding and subtracting the Dirac equation and its adjoint after multiplying the former to the left by '11,and the latter from the right by ,w; these are known 28 as the Franz-Kofi.rueequations. In the absence of a 4-potential this system consists of two uncoupled subsystem of five equations, one of electromagnetic and one of mechanical meaning. A non-zero Ai thus generates an electro- mechanic coupling that is symmetric in the following sense. Ten gauge-invariant space-time tensors exist in Dirac's theory, namely: five Dirac style '11,w and five Gordon or Schrodinger style '11{(ih/41r)[oi] + eAi}w, [oi] denoting the Gordon operator, the difference between the partial derivative operators to the right and to the left; both contributions in the latter five tensors are gauge-dependent. Five of either the Dirac or the Schrodinger ten tensors have an electric and five a mechanic meaning. The symmetric, or crossed, coupling mediated by Ai consists of this: the Dirac

102

style "potential" contribution in respectively an electric (mechanic) Gordon tensor is mechanic (electric). Thus Ai behaves as an electro-mechanic potential. There are three mechanical Schrodinger style tensors, all of interest to us: the canonical rank 2 energy-momentum density Tkl, its trace T, and the canonical rank 3 spin density ai[jk)_ The potential contributions in these, namely Akj', Akjk, and Aimik (jk and mik denoting Dirac's 4-electric current and 6-polarization densities), have repeatedly shown up in the preceding sections: A k j' contains the "hidden momentum densities" Vj and qA and, by contraction, the energy density A· j; A imi k contains the "hidden angular momentum density" V m. So, of the two gauge-dependent contributions in these gauge-invariant tensors, the "potential" one expresses the ponderomotive force or torque, the other one the electromotive force or torque. 18. Conclusion:

Electro-gravific

Interaction

That contact-action and far-action are alternative conceptualizations and formalizations of electrodynamics has been exemplified d la Diogenes by Wheeler and Feynman. 25 In an earlier paper 29 they had shown mathematical equivalence of Maxwellian retarded solutions including the Lorentz damping force, or advanced solutions including a Lorentz anti-damping force, or half-retarded half-advanced solutions with no damping force insofar as nothing other than mechanical inertia and electromagnetic interaction is considered. Electro-gravific interaction, a leitmotiv throughout our essay, includes the statement that QA i with Ai expressed in the source-adhering gauge is an extra inertia to a charge immersed in a given electromagnetic field - one more hint that the 4-potential is a keystone between electricity and gravity. If so, the "de Broglie formula"

pi= mo Vi - eAi

= .!!:._ki 21r

(61)

should be contemplated in its full-fledged expression 30 rather than in the shortened form generally quoted. The presently accepted "Einstein-Maxwell theory" inserts the symmetric free field Maxwell stress tensor in the source of gravity. Our arguments, all expressed in terms "manifestly compatible" with the SRT, converge towards the idea that the symmetric A k j' tensor should be included as gravity source in an "EinsteinCartan"31 formalism. Explanation of effects such as Graneau's and Saumont's, and prediction of others like linear and angular recoil effects, have been proposed. Via asymmetry of the energy-momentum tensors, spin, together with boost its twin, enter the picture - which automatically follows if interacting Dirac electrons and de Broglie massive photons are used as gravity sources.

103

19. Appendix:

Euclidean-Galilean

Rigid Bodies and Wrenches

The general Euclidean displacement of a rigid body consists of a translation and a rotation. At any Galilean time t a rigid body moving in an inertial frame is thus endowed with "instantaneous" linear V and angular A velocities. Between the velocities P~ and P.i of two points Pi and Pi in the solid, there exist the vector relations ~{(Pi - Pj)

= (Pi -

2 }'

pj - p~

=A

X

Pi)· (P~ - P.i) (P j

-

= 0,

pi) .

The condition A x P~ = 0, A '# 0, uniquely defines the axis of the A& V wrench and the "tangent helical motion." The wrench concept also turns up in the statics of solids. A system of forces applied at points of a solid has a "sum" S and, at any point Pi, a "moment" Mi; the following vector relations hold: (Pi - Pi) · (Mi - Mi)

=0 ,

Mi - Mi=

S x (Pi - Pi).

Again, if S -=f;0, there is an "axis" along which S x Mi = 0. Not only in the statics and kinematics, but also in the dynamics of solids, is the wrench concept useful: the power exerted by a wrench of forces moving a solid is S-V+M-A. References 1. E.J. Konopinski, Am. J. Phys. 46 (1978) 499.

2. T.W. Barrett, "Electromagnetic phenomena not explained by Maxwell's equations," in Essays on the Formal Aspects of Electromagnetic Theory, ed. A. Lakhtakia (World Scientific, Singapore, 1993). 3. 0. Costa de Beauregard, Phys. Lett. A24 {1967) 177. 4. W. Shockley and R.P. James, Phys. Rev. Lett. 18 (1967) 876. 5. H. Haus and P. Penfield, Phys. Lett. A26 (1968) 412. 6. S. Coleman and J .H. Van Vleck, Phys. Rev. 171 (1968) 1370. 7. A.S. Goldhaber and W.P. Trower, Magnetic Monopoles (Amer. Soc. Phys. Teachers, 1990), pp. 8-9. 8. J.J. Thomson, Phil. Mag. 8 (1904) 331; see pp. 347-349. 9. 0. Costa de Beauregard, Nuovo Cim. B63 (1969) 611. 10. 0. Costa de Beauregard, Precis of Special Relativity (Academic, New York, 1966), p. 19. 11. P. Penfield and H. Haus, Electrodynamics of Moving Media (M.I.T. Press, Cambridge, Mass. 1967), p. 202 and sq.

104 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

0. Costa de Beauregard, Found. Phys. 22 (1992) 1485. 0. Costa de Beauregard, Phys. Lett. A183 (1993) 41. B. Hoffmann, About Vectors (Prentice Hall, 1966), p. 83. E. Whittaker, A History of the Theories of Aether and Electricity (American Institute of Physics, 1987), pp. 83-88 and 198-206. 0. Heaviside, Electrician (1888) 229. J .A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 21 (1949) 425. P. Graneau, J. Phys. D20 (1987) 391. P. and N. Graneau, Newton versus Einstein (Carlton, New York, 1993). R. Saumont, Phys. Lett. A165 (1991) 389. J. Mourier, preprint. C.G. Darwin, Phil. Mag. 39 (1920) 537. 0. Costa de Beauregard, Phys. Lett. A28 (1968) 365. W.E. Weber, Ann. Phys. 73 (1848) 229. H.C. Corben, Classical and Quantum Theories of Spinning Particles (Holden Day, San Francisco, 1968). L. de Broglie, Mecanique ondulatoire du Photon et Theorie quantique des Champs (Gauthier Villars, Paris, 1957), 2nd ed. p. 67. R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2 (New York, 1963), pp. 17.5, 27.11. 0. Costa de Beauregard, C. R. Acad. Sci. 307 II (1988) 457. J .A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 17 (1945) 157. L. de Broglie, Ann. Phys. {Paris} 3 (1925) 22; see pp. 55-56; reprinted Ann. Fond. Louis de Broglie 17 (1992) 1. F.W. Hehl, P. von der Heyde and G.D. Kerlick, Rev. Mod. Phys. 48 (1976) 393.

THE SYMMETRY BETWEEN ELECTRICITY AND MAGNETISM AND THE PROBLEM OF THE EXISTENCE OF A MAGNETIC MONOPOLE

Georges

Lochak

Fondation Louis de Broglie 23, quai de Conti 75006 Paris, France

1.

Introduction

It seems fair to say that there are about as many physicists who consider the magnetic monopole as a monster hidden in the depths of the Loch Ness, as there are who regard this idea as so necessary for the beauty of nature, that God cannot possibly have failed to think about it. I belong of course to the latter species ! It is well known that the hypothesis of separated magnetic poles is an old one, but the present paper is neither devoted to its history nor to a comprehensive bibliography on the subject : there already exist several papers or books of this kind [I], [2], [3], [4]. Here, we shall quote only those papers that are useful for our purpose, which is to give arguments in favor of the hypothesis of magnetic monopoles, the possibility of their observation and the explanation of the fact that they were not yet observed with certainty. Therefore, we shall not survey all aspects of the problem. In particular, although this is a commonly favoured point of view, there will be no further mention of a possible hyper-heavy monopole. Keeping away from G.U.T., we shall remain in the framework of electrodynamics. On the other hand, we shall not confine ourselves to symmetry arguments, but shall present a wave equation for a magnetic monopole, which parallels the Dirac equation for the electron. This equation describes a monopole quite different from the one which is usually considered, but it satisfies all the electrodynamical, mechanical and gauge properties

106

commonly assumed at present. Needless to say, all these "properties" are conjectural and it is very likely that either there are no monopoles at all (God didn't think about it) or, if there is one, there must be a world of monopoles, just as large and diverse as the world of electrically charged particles.

2.

At

the

beginning

was

symmetry

In 1894, one century ago, Pierre Curie wrote a paper on "Symmetry in Physical Phenomena" [5], where he put forward the idea of a general constructive role of symmetry in physics and emphasized the importance of dissymmetry in the appearance of phenomena. He described the Curie groups : a classification of physical invariance groups of limited objects in tridimensional space, in analogy with crystallographic groups, which are the invariance groups of an unlimited periodic medium. As an example, he described the symmetry of electromagnetic phenomena and therefore of fields - entirely on the basis of experiments, without using Maxwell's equations (as, for instance, in [7]). He added a short paper [6] in which the possibility of ''free magnetic charges" was shown as a consequence of the laws of symmetry of electromagnetic field 1. There is a difference between electric and magnetic charges, which is a consequence of the fact that the electric field is a polar vector and the magnetic field is an axial one 2 : E has the symmetry of a radial vector r, a velocity v, a linear momentum p, a force F, while H has the symmetry of the external product of two polar vectors, like rxr' or rxp. As a consequence, consider the force exerted by each field on the corresponding charge F =eE

I

F =gH

(2 .1)

1 It is said in reference [2] that Curie "suggests out of the blue that magnetic charge might exist". It is no more "out of the blue" than all the predictions made in our century on the basis of symmetry, including the famous paper by Dirac himself on magnetic poles. Moreover, it was the first prediction of this kind. 2 It is worth noticing that, despite the obvious difference between the two fields it is not so easy to prove experimentally which is polar and which is axial [5].

107

If we assume charge must be the image of an a north pole is laws.

that these law of force are P-invariant, the electric a scalar and the magnetic charge a pseudo-scalar : electric charge has the same sign, while the image of a south pole and we find the following symmetry

Electric field E and current J

I

Electric charge e

Magnetic field H and current K

Magnetic charge g

t.. t..

(+)

t.. '-

e '-

E

(+)

(+)

~.. (-)

Fig. 1 Symmetry laws of electric and magnetic quantities

One can see on Fig. I that, while the electric current is a polar vector, like the electric field, the magnetic current must be axial like the magnetic field, in virtue of the definitions : J=ev,

K=gv

(2.2)

It is astonishing to find a pseudoscalar physical constant g, because a physical constant has no tensorial variance : for instance, c does not vary as a velocity and h does not vary as an action or as a kinetic moment. Only physical quantities can have tensorial variances, not constants, and here, there is a confusion between the value of a constant and the variance of the corresponding physical quantity. We shall see that it is not so in quantum mechanics : the elementary magnetic charge will be a scalar, as it must be, but physical properties will be given by a pseudo-scalar charge operator. Magnetic current will be an axial vector different from (2.2).

108

In other words, Fig. I that summanzes the work of Pierre Curie is true, but eq. (2.1) and (2.2) are not, and this is very important because a classical objection against the hypothesis of magnetic poles is that it is purely formal [2], [7], [8]. Actually, let us introduce densities of electric and magnetic currents and charges J, K, p, µ, in Maxwell's equations :

div E = 41tp; div H

=41tµ

This system is invariant

(2 .3)

under the transformation

E ==E' cosy+ H' sin 'Y; H = - E' sin y+ H' cos 'Y

p = p' cosy+µ'

sin 'Y ; µ ==- p' sin y+ µ' cos 'Y

J = J' cosy+ K' sin 'Y; K = -J' sin y+ K' cos 'Y

(2.4)

And the argument is that, by suitably choosing the angle y, one can arbitrarily eliminate magnetic (or electric) quant1t1es. But this is true only if J and K are colinear, and it will not be true in our case, which invalidates the argument.

3.

The

Birkeland--Poincare

effect

In 1896, Birkeland introduced a straight magnet in a Crookes' tube and was puzzled by a convergence of the cathodic beam which does not depend on the orientation of the magnet [9]. Poincare explained the effect by the action of a magnetic pole on the electric charges of the beam ( these charges were only conjectured at that time) ; he showed that it is due to the action of only one pole of the magnet, and that, for symmetry reasons, it must be independent of the sign of the pole [10].

109

c::J

S

s

N

N

Rg. 2 The Birkeland-Poincare effect. When a straight magnet is introduced in a Crookes' tube, the cathodic rays converge whatever the orientation of the magnet. Above : the cases considered by Birkeland ; below : two cases corresponding to the same description given by Poincare.

In order to describe this effect, Poincare wrote down the equation of motion of an electric charge in a coulombian magnetic field created by one end of the magnet. The magnetic field is :

H=g.1.r

r3

( 3 .1)

where g is the magnetic charge, and, from the expression Lorentz force, we find the Poincare equation :

d2r = A .1. dr x r dt2 r3 dt

of the

A=eg

me

(3.2)

where e and m are the electric charge and the mass of the electron. Poincare found the following integrals of motion, where A, B, C, A, are arbirary constants :

f

r2 = C t 2 + 2B t + A · (dr = C

' dt

(3.3)

110

r x dr + A r. = A dt r

(3 .4)

He obtained from eq. (3.4) :

( 3. 5) which says that r describes an axially symmetric cone the Poincare cone - and that the acceleration is perpendicular to its surface, so that r follows a geodesic line . If the cathodic rays are emitted far away from the magnetic pole with a velocity V parallel to the z axis, they will have an asymptote which obeys the equations : X=Xo ;y=yo

(3.6)

And we find from (3.3) and (3.4) (3. 7)

The z axis is thus a generating line of the Poincare cone and the half angle 0' at the vertex is given by

(3.8)

Now, the cathodic ray that becomes, after the ermss1on, a geodesic line rotating along the cone, crosses the z axis at distances from the ong1n given by

✓x~ + fa sin

ct>

,/~+fa sin

2cp

sin

3cp

ct>=

21tsin 0' (3.9)

Therefore, if the emitting cathode is a small disc of radius ,/~ + fa orthogonal to the z axis, and if the position of the magnetic pole is such that one of these points is on the surface of the tube, there will be a concentration of the electrons emitted by the periphery of the

111

cathode and even, approximately, of those coming from the whole disc : this is the focusing effect observed by Birkeland. This is an important result because, although the existence of magnetic monopoles as particles is not yet proved (at least, we are not sure of that), the Poincare equation (3.2) and the integral of motion (3.4) are experimentally verified. In eq. (3 .4 ), the first term is clearly the orbital momentum of the electron with respect to the magnetic pole. The second term was later interpreted by J .J. Thomson (see [7], [11]) who showed that :

eg

c

r.r

=-1-1 x x(E x H) d 41tc 00

3x

-00

(3.10)

Thus, with the value of A given 1n eq. (3.2), the second term of the Poincare integral is equal to the electromagnetic momentum and eq. (3.4) gives the constant total angular momentum J = m A. The presence of a non vanishing electromagnetic angular momentum 1s due to the axial character of the magnetic field created by a magnetic pole and acting on the electric charge. Let us add a remark about symmetry [12] : the Poincare cone is enveloped by a vector r which is the symmetry axis of the system formed by the electric and the magnetic charge, and this axis rotates (with a constant angle 0 ') around the constant angular momentum J=mA. But this is exactly the definition of the Poinsot cone associated to a symmetric top [13]. The Poincare cone is nothing but the Poinsot cone of a symmetrical top, which is not surprising because the system formed by an electric and a magnetic charge is axisymmetric and is rotating around a fixed point with a constant total angular momentum. Such a system must have the angular properties of a top, but with a different radial motion because the it is not rigid (the motion along the geodesic lines of the cone has nothing to do with a top). Introducing the following definition with two obvious properties : L

=r x dr · L . r.r =O ·, A . rr =0 dt ,

all that was said can be summarized 1n the following figure

(3.11)

112

Rg. 3 The generation of the Poincare (or Poinsot) cone and the decomposition of the total momentum.

Of course, all these results are true for a magnetic charge in a coulombian electric field : we shall see that this will be true in our case and that our equation for a magnetic monopole will give, at the classical limit, the Poincare equation.

4.

Forces

and

potentials

for

a magnetic

pole

Owing to the second formula (2.1), we can write the equation of motion of a monopole in a particular system where the external field reduces to its magnetic part :

( 4 .1)

Po, to and Ho are the momentum, time and magnetic field in this system. The Lorentz transformation

of the electromagnetic

field is :

Ea= E + vie H ; Ho= H - 1/c v x E

✓ 1-(vlcf Therefore,

✓ 1-(v/c"f

(4.2)

the general form of (4.1) reads

_d_p = 9 _H_-;::=::=1~=c =v=x=-E_ dt ✓ 1-(v/c'f

where dt =dt ✓ 1- (v/c'f 1s the differential (4,3) can be writen :

( 4 .3) of the proper

time

so that

113

d p = g (H - 1/c v x E) dt

(4.4)

The right-hand side of eq. (4.4) is the Lorentz force acting on a magnetic pole, with a minus sign in front of E instead of the plus sign occuring in front of H in the electric case. Now, we go back to the Maxwell equations (2.3) with magnetic current and charge densities and introduce relativistic coordinates

xa={x1, x2,x2,x4}={x, y, z, ict}

(4.5)

In a covariant form, eq. (2.3) becomes aPFap=~Ja;

Ja =(J,ipc)

aPFap=4c1t Ka; iKa=(K,iµc)

(4.6)

where the i = ✓ -1 in front of Ka is due to the axial character of K ; we have the relation of duality: Fap= j_ Ea~ 2

FY° (Ea~ - antisymmetric)

(4.7)

It is clear that we cannot define the field by a Lorentz polar potential only because (4,8)

Then, we must introduce a new potential Ba such that : (4.9) Both right-hand terms in eq. (4.9) must have the same variance. Hence, Ba is a pseudo-potential, i.e. the dual of an antisymmetric tensor of rank three : (4.10)

114

In terms of ordinary coordinates, we have

Aa=(A, iV); iBa =(B, iW)

(4.11)

where B is an axial vector. The fields are defined as 1 aA E=-VV---+curlB C

dt

H = rot A+ VW+

6a;

(4.12)

Actually, we shall not consider "dyons" with electric and magnetic charges, but "true" magnetic poles with a magnetic charge only, so that formulae (4.9) and (4.12) reduce to :

(4.13) The last formulae were derived by de Broglie from his theory of light [ 14] ; they were related to the magnetic monopole by Cabibbo and Ferrari [15].

5.

Dirac

strings

In a celebrated paper of 1931, Dirac raised a fundamental problem about the interaction between electric and magnetic charges i. e. either the motion of an electric charge around a fixed monopole or conversely the motion of a monopole around a fixed electric charge, [16], [17], [18], [19]. Let us choose, as Dirac did, the motion of an electric charge in the magnetic coulombian field H generated by a fixed monopole with charge g. H is thus defined by a vector potential A such that: curl A= g _r_ r3

(5.1)

115

It is clear that there is no continuous and uniform solution A of this differential equation because if we consider a surface l: bounded by a loop A, we find according to Stokes' theorem :

i =i H.dS

curl A.dS

i

= A.di =91;.dS =Qi dO 3

(5.2)

where dS, di and dn are elements of surface, length and solid angle respectively. Now, if the loop is shrinked to a point, while the pole remains inside the closed surface l:, we get :

1 A.di = g1dn A-Ml

I

= 41tg

(5.3)

This equality is impossible for a continuous potential A because then the first integral vanishes. There must be a singular line somewhere around which the loop shrinks. Now, whatever the wave equation, the minimal coupling is given by covariant derivatives : V-iJLA he

( 5 .4)

Dirac introduced 10 the wave function 'ti a non integrable univalent) phase y, defining a new wave function :

(non

(5.5) If we apply the preceding operator, we know that the introduction of this phase y is equivalent to the introduction of a new potential by a change of electromagnetic gauge (5.6)

We can identify the new potential with the gradient of 'Y, but the phase factor ei'Y is admissible only if the variation of y around a closed loop is equal to a multiple of 21t. Then, we must have :

116

n~1 1

Vy.di = (Ay ) loop= 21tn

A.di =

A-tO

A-tO

(

Comparing eq. (5.3) and (5.7), we find the condition between electric and magnetic charges :

famous

It is interesting to confirm this result on a solution (5.1 ). Dirac chose the following solution :

5 .7 )

Dirac

of the eq.

(5.9) In polar coordinates

:

x = r sin 0 c OS (·I Cot e- a - - a - ---'-i --a )

acp ae sineax

acp

Obviously, the eigenvalues

R20 = j{j + 1) D ; f¼O = m D

(11.8)

are the same as those of Z (11.9)

The R k are well-known : they are the infinitesimal operators of the rotation group written in the fixed referential. 0, cp, x are the nutation, the precession and the proper rotation. The role of the rotation group is not surprising because of the spherical symmetry of the system constituted by a monopole in a central electric field. Our eigenfunction problem is thus trivialy solved : instead of the cumbersome calculations of "monopole harmonics", we see, under

135

the simple assumption of continuity of the wave functions on the rotation group, that the angular functions are the generalized spherical functions, i.e. the matrix elements of the irreducible unitary representations of the rotation group [39], [40], [44], [45]. These functions are also the eigenfunctions of the spherical top. This coincidence was quoted by Tamm [46] without explanation, but here, the explanation is evident because we already know the analogy between a symmetrical top and a monopole in a central field. The eigenstates of R 2 and R 3 are (see any textbook on group theory) : o{11',m(8,)T(t). Substituting this trial solution into the partial differential equation above gives a solution of the form: (r t) =

,

Z

(kr)

n+l/2 (kr)l/2

·

'¥!!1(8,n) eIOlt n

'T

,

(4)

where Y1g are the spherical harmonic functions and Zn+ 112 are the spherical Bessel functions. This solution describes a set of homogeneous waves or Fourier modes characterized by a set of numbers called separation constants, which are the wave number k = ro/c and the integers n and m with O ~ m ~ n. A particular solution of a problem will be obtained through the Fourier method by forming a linear combination of the modes which satisfies the initial conditions and with a selection of the modes which are appropriate for the boundary conditions. It is important to point out that the choice of the coordinate system concerning the method

154

of separation of variables depends physical problem considered.

upon

the

symmetry

of

the

Let us consider now the case of the hydrogen atom where an electron moves in the field of a proton at rest at the origin of the laboratory frame. If there are no other particles present, the hydrogen atom is isolated, which implies that the total energy E-rof the system is constant. The symmetry of the problem implies that the potential energy EP = Er - Ee of the interaction varies inversely with the inter-particle distance r. Therefore the time-independent Schrodinger equation becomes: (5)

with a= 2m 0 /n 2 . If we write 'l'(r) = R(r)Y(8,cp), the equation can be uncoupled in two independent equations:

_!__Q._ (r 2 dR) + [a(Er +-9.:.)_ l(l + 1 )] R = O; 2 r dr dr r r2 1 a . aY 1 a 2Y --(s1n8 -) +----+ l(l + 1) Y = 0. sin8 aa aa sin 2 8 acp2

preceding

(6)

We know that the radial equation admits bound state solutions provided that the total energy takes discrete values given by the relation: (7)

where n is an integer number. It follows that the angular frequencies ropn of the radiation emitted by a hydrogen atom result from a change in energy induced by a transition from level n to level p. They are given by the relation: (8)

where we have used the Einstein relation

nro0

= moc 2 .

As pointed out by Barut 48 and also by Elbaz 49 - 51 with a similar approach, quantum mechanics can be formulated without n, m0 and q as a pure wave theory if the bound state is characterized by a fundamental frequency ro 0 . It is important to note that one

155

measures frequencies energy differences.

ropn associated

with

spectral

lines

and not

The transverse equation admits the spherical harmonic function YT(8,q>) as a solution which depends on the integer m and l. Finally, the general solution of the Helmholtz equation (5) is of the form: (9)

with the definitions p = 21 + 1, b 2 = -aE 0 and knowing that 1rt~ 1(x) are the Laguerre polynomials. We note that the integer numbers m, l, n are separation constants which verify the following inequalities: 0

~

Im I ~ l

a = 2m 0 /n

~

2,

n - 1. These integers do not depend on the parameter but

simply

result

from

symmetries

and

boundaries

conditions when we solve the Helmholtz equation (5). In quantum mechanics, the integer numbers m, l, n are called guantum numbers in spite of the fact that they are perfectly understandable from a classical wave point of view. Here also Planck's constant n in the definition of "a" appears as a dimensional constant. Therefore, there remains a question concerning the meaning of the quantization associated with Planck's constant itself.

2.2.

Calculation

of the

Lamb

shift

If there are space or time symmetries

in a physical problem, the solution of a wave equation or a Klein-Gordon equation will depend on a set of harmonic homogeneous waves characterized by the separation constants. Generally, we are interested in a stationary problem as in the case of the hydrogen atom. However the question arises: what happens during the time we jump from one stationary state to another one ? This problem has been recently addressed by B oude 152, 53 for the Lamb shift calculation, where a time-periodic current is created during the passage from one state to another state. Boudet has developed a different method for explaining the Lamb shift using a finite-electrodynamics model that avoids infinities. He demonstrated how the Lamb shift formula can be obtained without divergent integrals. We know that a general solution of the Maxwell equations can be obtained from the vector potential, in the Lorentz gauge, given by the equation:

156

=:f ~

A(r,t)

J(r',t

- R/c) dr'

( 10)

3 ,

V'

R = Ir - r'I.

with the definition If the density have:

J (r ,t) = qU (r) eiro t 1s time-periodic,

current

f

= ~ eirot

A(r,t)

-ik 0 R e R U (r')

dr'

we

( 11)

3 ,

V'

where k 0 =

I ro/c I.

The time average field is:

of the interaction E= 2

hence:

E = _g_:_

2c 2

ff

V

energy

;cJ

A. J* dr 3

n2

2c 2

we can perform

-ik 0 R

e R

U (r) . U (r') dr 3 dr'

V

1s the average

(14)

of ro, associated

1 R

with

U(r') dr 3 dr'

the Coulomb

3

(15)

V'

- A second term ER dependent stationary current:

V'

integral

decomposition

J JRIU(r).

Es _-9.:_ - 2c2

V

(13)

V'

- A first term E 5 , independent self-energy of the electron:

ff

3

cos(k 0 R) R U (r). U (r') dr 3 dr' 3 ,

the foil owing

~_ -9.:._ .LJR,-2c 2

( 12)

,

V'

ff V

and its

V

By definition the real part of the above energy associated with the current, namely: E =~

of the system

upon ro which 1s associated

with a

[cos(k 0 R) - 1] U(r). U(r') dr 3 dr' 3 . (16)

157

One can then demonstrate

the following

identity:

+oo41t

cos(koR) - 1 = 1 41tR (27t)30

Jdf e-ik. R [k2-ki1

-

_!_]k2dkdn

'

k2 (17)

knowing

that:

k

2

1 _!__51_ 2 - 2 - 2 F(k) ' - kO k

with the definition: ( 18)

The substitution

~ = 81t2c3 roq2

of the identity (17) in Eq.(16) gives:

fff V

e-ik. (r-r') F(k)U(r).

U(r') dr 3 dr' 3 dk 3 ,

V K (19)

and by definition,

we have:

f

U(-k) =

e-ik. r U(r) dr 3 .

(20)

V

The current density qU (r) is a real function of space which implies the condition U (-k) = U *(k), therefore we have:

~ = 81t2c3 roq2

f

F(k) IU (k)l 2 dk 3 .

(21)

K

For a positive frequency,

let us write the following quantity: +oo47t

JJ

1 E(ro) = roq2 IU (k)l 2 dkdn . 41t2c3 o o k - ro / c

(22)

It results from the preceding definition that the self-energy ER is given by the relation: 1 ~ =2 [E(ro) + E(-ro)] . (23)

158

After a similar calculation,

the

Es= q22

41t c

2

J

2I IU (k) 12dk 3 .

current

has stationary states, each state energy level En. We assume that a

with pulsation

during the transition

c.onp = (En - EP)/n

of the sense of the transition we

have

c.onp = -c.opn'

knowing

We note that the self-

2 =

IU(k)l

energy only depends on the quantity

By definition, becomes:

> 0 is generated

from the state En to another state EP'

that we have En > EP for n > p by definition.

is independent

(24)

k

We consider that an atom corresponding to some constant sinusoidal

Coulomb energy has the value:

n

1Unp(k)l 2 , therefore --+

therefore

it

p. equation

(23) (25)

Considering

all the transitions

then the self-energy

n

--+

p and p

--+

n for a given state n,

for all sates n has the value: N

L

N

(!1Enp + !1Epn) =

n is an artifact introduced because one uses the Galilean transformation. Therefore the interpretation of the paradox given by Levy-Leblond does not answer the question raised by Lande; and does not constitute an argument against the view that matter waves might be real physical waves as noted by Wignall. 6. MATTER WAVES AND INHOMOGENEOUS A homogeneous quantum mechanical substituting the Planck-Einstein and de phase of a classical wave as follows:

wave Broglie

"' = ei(Et - p .r)/n .

Levy-Leblond pointed out recapture the classical wave limit zero because the phase change transformation. However the constant

ex = q 2 /n c precludes

WAVES 1s obtained by relations in the

(47)

that this formulation does not as the Planck's constant n goes to tends to infinity under a Galilean existence of the fine structure

such a calculation

independently

of

the other constants q and c. This means that the wave function 'JI' can indeed be considered as a classical wave provided that the Planck's constant n keeps a finite value and Galilean covariance is not used. We can make the same comments concerning the limit for n ➔ 0 of the inhomogeneous wave definition given by Eq.(36). In our critical review 93 of the special theory of relativity we have shown that the Gali)ean and Lorentzian covariance concept is not enforced by nature. This point is fundamental to understanding the

166

difference between the above calculation and the approach followed by de Broglie in his interpretation of matter waves. De Broglie describes the electron as a standing wave in its rest frame associated with a Zitterbewegung frequency ro0 = m 0 c 2 /n, and as a traveling wave in the laboratory frame for a m~ving electron. The famous relation A. = h/p is derived by de Broglie by using a Lorentz transformation between the rest frame and the laboratory frame and stating the phase invariance of the two waves in the transformation. The reader interested by a critical review of de Broglie's work can consult three interesting papers 94 - 96 on this subject. De Broglie considered a moving particle in the laboratory frame as a wave group resulting from the superposition of homogeneous traveling waves, or phase waves, consistent with the fact that the group velocity U (k) is constant for some fixed value of k. Now if we adopt the relativistic view point followed by de Broglie concerning his wave packet approach, we are faced with a contradiction. Since matter waves are dispersive, the wave packet must enlarge during motion. But when one considers the electron in uniform motion from the point of view of special relativity, this electron or wave packet is space contracted when observed in the laboratory frame. However, if the electron is a wave packet, neither space spreading nor space contraction is acceptable. The fact is that space contraction due to relativity has never been observed. Furthermore, Terre11 97 and Weisskopf9 8 have argued against the possibility of experimentally seeing this effect. Therefore, we must give up the idea of covariance associated with the change of an inertial frame, which was criticized with reason by Lande 8 7 when he says: The diffraction pattern ( in a crystal experiment) would depend on the arbitrary choice of the reference system. Instead our calculation has been made in the unique laboratory frame, where no matter wave is observable for an electron at rest in this frame, but becomes observable as soon as an electron is accelerated by an external force transforming the homogeneous waves of the wave packet in inhomogeneous ones in order to preserve the profile of the wave packet intact. Therefore we understand the reason why the so-called length contraction of special relativity is not to be found in the shape of extended particles, but in the waves making the particles and the oscillating vacuum. It follows that the matter wave is an observable consequence of this effect, which can be associated with electron acceleration which creates a perturbation in the waving vacuum. The above interpretation gives some support to the idea that matter

167

waves are physically present 1n vacuum, which 1s a conjecture debated in the literature.

well

7. QUANTIZATION

Ehrenfest 99 , too noted that certain quantities remained during slow change of a system and suggested that these have to take on the fixed values of quantum mechanics. show how Ehrenfest's idea can be formulated in both relativistic and relativistic cases and how quantization occurs.

7.1.

The

non-relativistic

constant quantities We will the nonnaturally

case

Following Boyer 101 , let us review the case of an harmonic oscillator of mass m0 and spring constant k 0 . The equation of motion for the displacement of the mass is:

(48) Since the system 1s closed, the total energy

Er 1s constant:

1 1 E T =-2 m0 U 2 + -2 k 0 r 2 = Ct ·

(49)

If the spnng quantity k(t) changes with time by the action of an

external force, the system is not closed and therefore energy is no longer constant since we have:

the total

(50) where EP = kr 2/2 is the potential energy. Now if the spnng quantity k changes very slowly with time over a period of the harmonic oscillator, then we can replace the potential energy by its time average: T

-I T

f 0

E (t) dt = E_I P 2

(51)

Note that the above equation 1s strictly verified when k 1s a constant. If we do the substitution 1n Eq.(50), we obtain:

168

dE,dt

--=---,

1 d kEi, kdt2

(52)

which can also be written in the form: (53)

The relation ET = Ct k 1l 2 follows, where the constant Ct is independent of k. Now the angular frequency of the oscillator is ro(t) = [k(t)/m 0 ] 1l 2 , therefore the quantity E,-/ro remains a constant during the change of k. We remark that the oscillatory motion ro(t) is frequency modulated. We will also use the concept of a frequency-modulated wave in the relativistic case. We know that the corresponding solution of oscillator in quantum mechanics is obtained from Schrodinger equation:

the the

harmonic following

(54)

where the potential energy is EP = k0 r 2 /2. This equation admits takes well-behaved solutions provided that the total energy discrete values ET = (n + 1/2)n w. We note that the separation of the quantized levels is equal to the angular frequency of the classical oscillator w = (k 0 / m 0 ) 112 . If both approaches

lead to the quantization of energy, they differ essentially by the fact that the spring quantity k depends on time in the classical approach, while it is a constant in quantum mechanics. The two preceding formulations of the harmonic oscillator can be used to explain the quantum Hall effect4 6 , since the allowed energy levels for a free two-dimensional electron system moving in a magnetic field are identical to a fictitious onedimensional harmonic oscillator.

7.2.

The

relativistic

case

It 1s a well-known fact that 1n the equation of motion for an electron in a solid, only external forces are considered as applied forces to the wave packet. The force which originates from the lattice periodic field remains hidden in the so-called electron effective mass dyadic. The electron wave packet obeys a law of motion which is given by the equation:

169

M• dU=F dt

(55)

'

++

where M is the effective mass and F the external force. The wave packet velocity U is equal to: U =!aE

(56)

n ak'

where E(k) is the relativistic energy of the electron in terms of the wave vector k. The consistency of the above equation can be verified by using the relation E(k) = n ro (k) which gives the definition of the group velocity defined in Eq.(62). The components of the mass dyadic are:

a2E 2 ak.ak· r J

M:~ =-1 'J

t.

l

(57)

I

In vacuum, the motion of a massive particle, with a rest mass m0 , submitted to a Lorentz force F is described by the relativistic dynamic equation: d - (m "'U) = F dt OI

with the definition relation:

(58)

'

y = (1 - U 2 /c 2 )- 1l 2 which

dy - y3 _Q__u2 d t - c2 d t ( 2 ) .

gives

the following (59)

By using the preceding relation, we can rewrite the Eq.(58) in the following dyadic form:

(60) with the direct and inverse mass dyadics given by the definitions:

ti- 1 =-

1-(I-..!_UU), mo"f

c2

(61)

170

where

++

I

1s the unit dyadic.

Therefore

the relativistic

mass

dyadic

of a moving electron in vacuum indicates that the electron is submitted to an applied force from the oscillating vacuum. This internal force can be understood as a resistive force arising from the vacuum. Our approach to deduce the relativistic de Broglie relation will differ from Crawford's calculation, which is reviewed in the appendix, by treating the wave concept independently of any conservation law of energy and momentum. By definition the group velocity of a wave which satisfies the dispersion law (34) is:

u = aro = c2!.. It follows the identity not justify

immediately

= c 2k/U

that ro

yU = ck/k 0 , therefore

n

that the quantity

(62)

ro

dk

and 'Y = ro/ck

the identification

is a constant,

n

0

which

= m 0 c/k

since it depends

imply 0

does

on two

independent parameters m 0 and k 0 • The preceding calculation cannot be taken as a demonstration of the Planck and de Broglie relations (1) since such a calculation cannot justify the necessity of Planck's

constant

by calculating

n.

Therefore

we proceed

with

our demonstration

the quantity:

aui

c2

-=-(6-k. ro

aJ

By definition,

we have:dUi =

IJ

1 --U-U-)

c2

dkL aui ak. j

hence:

c2

dU = -

++

( I -

(1)

Now using the relation

(63)

I J

J

1 '

1

UU). dk . 2 C

(64)

(59), we get:

d ( yU) = 'Y(I+

~ U U ). d U .

(65)

C

The substitution

of Eq.(64)

in the above equation

gives:

171

yc2 d(yU) =-dk

(66)

.

Cl.)

With the definition given by Eq.(62), preceding equation in the following form:

we

can

yc2 dki d(yU-) = - dk- = yU- . l Cl.) l l kl

rewrite

the

(67)

Therefore we get the differential equation d[Ln(yU /ki)] = 0 whose integration gives the relation yU i = Ctiki. We can use the relation yU = ck/k 0 to prove that the constants Cti are the same. It follows the identity yU = Ctk where the integrating constant Ct does not depend on k 0 and can now be chosen equal to n/m 0 in order to recover de Broglie's formula, which finally gives the quantization relation m 0yU = n k. The above differential equation implies the quantization of velocity during the acceleration process, which is confirmed by Jennison's experiments 102, 103 that will be discussed when we consider the relation between matter waves and inhomogeneous waves. To obtain the second quantization relation, it suffices to calculate the variation of the kinetic energy with respect to ro which is given by the equation:

(68) Therefore

we obtain a first equation: (69)

The relation (59) gives a second equation: (70)

We can differential quantization

subtract the two preceding equations to equation d[Ln(yc 2/ro )] = 0 which implies relation m0 yc 2 = n ro.

obtain the the second

172

We note that the Planck-Einstein and de Broglie relations obtained in the preceding calculation can now be used to recover the well-known formulas of solid state physics. For example, Eq.(57) leads to the inverse mass dyadic of a particle which can be rewritten in the form: (71)

The second quantization relation is consistent with the above formula which allows to obtain the definition (61 ). The analogy between the vacuum and solid-state physics is a fruitful concept which has also been used to explain the Dirac sea of electrons by regarding the vacuum as a close analog of a semi-conductor 104 with two bands separated with a 2m 0 c 2 gap. Now let us postulate that k and ro are implicit functions of time. Then our wave packet is built with inhomogeneous waves and the acceleration of the wave packet implies that the medium is dispersive since we have: (72)

The acceleration of the particle will be different from zero only if the matter waves are dispersive. The quantization relations we have obtained are precisely the Planck-Einstein and de Broglie relations provided we use the well-known formula of ray theory: (73)

This equation shows that the group velocity of matter waves is just equal to the velocity of the particle whose motion they govern. This indicates that de Broglie's postulate is internally consistent. However, it is not well-known that the above equality is not always verified as shown by Molcho 1 . Therefore we have found the Planck-Einstein and de Broglie relations where Planck's constant n is a mere integration constant by considering the acceleration of a material particle with a given rest mass m0 represented by inhomogeneous standing waves. The quantization results from the fact that the deformation of a wave

173

cannot be of any form if we want to preserve the oscillating behavior of the inhomogeneous wave. This point can be better understood by considering the case of the quantization of a progressive wave reviewed in the appendix. In several papers 67 • 83 • 93 , 105 we have given some indications, which will be reviewed hereafter, how to use this approach in a non-dualistic theory where the particle as a wave packet integrates the structure of the quantized oscillating field in vacuum. The concept of the quantization of moving standing inhomogeneous waves was beautifully verified by Jennison 102• 103 with experiments carried out on phase-locked cavities as a model for material particles. For about a decade, a research group at the University of Kent had been trying to understand the electron. They identify the electron with a phase-locked cavity formed entirely from electromagnetic waves. They experimentally showed that a standing electromagnetic wave trapped in a cavity not only had rest mass but also possessed inertia since a moving trapped standing wave can only be stopped by applying a restraining force. They also found that their phase-locked cavity formed a remarkably rigid rod which maintains the same electromagnetic length to a surprisingly high degree of accuracy when it is accelerated. Therefore, the standing waves must deform themselves during the acceleration process in order to keep the proper length of the cavity unaltered. This implies quantization of the motion of the cavity. Jennison clearly indicates that if the pushing force is applied to the cavity for a long time, then the cavity accelerates by progressing up a staircase of velocities in quantum jumps determined by Planck's constant according to Jennison's statement. 8. CONCLUSION

We have been able to deduce the Planck-Einstein and de Broglie relations as the result of a wave effect if we assumed that wave packets are built with inhomogeneous standing waves in vacuum. The similarity of quantization between classical scalar waves and quantum mechanical waves may suggest a new possible interpretation of quantization. We note that the deformation of the waves during the acceleration of a wave packet has been studied with respect to time from a Lagrangian point of view. We know that the quantum n is dimensionally an energy multiplied by time which has been associated with the time deformation of the waves and therefore with motion. Since the quantum q 2 is an energy multiplied by space, there is also a possible quantization effect related to space. In fact inhomogeneous standing waves depend on

174

both time and space when defined with Eulerian variables. It has been recently suggested by Woltjer et al. 106 that the quantum Hall effect in solid state physics might have its origin in the spatial inhomogeneity of the electron density across the sample which supports the above idea. We recall that the quantum Hall effect manifests itself as a series of plateaux in the Hall resistance which occur when the Hall resistance is given by RH = h/nq 2 , where n is an integer or a fractional number. Therefore we can explain the quantization concept in physics by noting that the waving vacuum imposes constraints which imply the existence of operators of the form d[Ln(A•B)] = 0 or d[Ln(A/B)] = 0, which give the quantization of the variable A with respect to B, namely, A•B = Ct or A = Ct B. The approach defined in this paper concerning quantization has also made manifest the concept of inhomogeneous wave, providing us with a continuous wave theory at a microscopic space-time scale, which may appear discontinuous at a macroscopic space-time scale, with the quantum constants n and q as the links between the two scales, in order to allow a minimum space or time for the propagation and deformation of these waves. This is a very important concept in physics since we succeeded in deriving Maxwell's equations 83 , 84 from it and demonstrated the link with matter waves. 9. APPENDIX

9.1.

Case

of a progressive

wave

To understand how quantization comes about we consider the transformation of an homogeneous progressive wave into an inhomogeneous progressive wave with respect to time only. To show this transformation let us examine the example of a plotter with a ruler moving in the same direction as the paper with velocities such that we have UP > Ur > 0 and a pen oscillating along the ruler with a period T as shown in the figure 1. The relative velocity between the ruler and the paper is by definition: (74) If the relative velocity is constant, the drawing on the paper will

be a perfect sinusoid with a wavelength defined by the equation: (75)

175

Up

Ur

►

Figure 1 : The plotter and the Inhomogeneous wave

176

With no loss of generality we can consider the case where the paper velocity UP is constant and the ruler velocity Ur is zero for time t < 0,. Therefore the relation A = UPT follows . After time t = 0 the ruler accelerates during the time T; the relative velocity U is no more constant and therefore an observer will see a deformation of the wave during the time T. The observer can measure on the paper a new wavelength:

In the present case, the wave is shortened as shown in the Fig. I, the ruler being located to the left of the sinusoid drawing for both the paper and the ruler moving to the right. The time taken by the plotter to draw the deformed curve on the paper can be calculated by the observer in the laboratory frame as T O = A.0 /U P instead of T = A/UP. From these relations one can deduce the phase velocities ro = UP k and ro0 = UP k0 , while the group velocity 1s: U =i\ro = o.>o- ro =U g i\k kO - k P•

(77)

The equality of the group and phase velocities 1s a consequence of the linear dispersion law which results from the constancy of UP and therefore we have t\ 2 ro/t\k 2 = 0. Now we can write the following equation: (78)

We notice immediately that the last equality in the above equation just defines the classical Doppler effect ro = ro0 - Ur k0 . We also have:

L\A.__ i\k _ i\k i\ro ~ _ i\ro - -A ko i\ro roo k o roo If we eliminate

equations,

the quantity

i\ro/ro 0 between

(79)

the two preceding

we get: (80)

,n The deformation of the wave unfolds during the elapsed with a non-uniform relative velocity U such that: U=U r -U

p

=-U

p

time T

(81)

and an acceleration:

Therefore

we have:

6U

-=

u

-

Ur

u·p

(83)

If we add the Eqs. (80) and (83), we obtain: (84)

After taking the limit of the above equation we obtain the differential relation d[Log(kU)] = 0 which gives after integration: h

-+mu-~ o - k .

(85)

In the above equation, the mass particle m0 and the constant h0 , not dimensionally the Planck's constant, are integration constants. Finally we obtain the surprising result that the velocity of the wave deformation which slides on a progressive wave is quantized. This simply means that the relative velocity cannot take any value we wish, if the curve plotted on the paper must keep an oscillatory behavior. Indeed quantization can be interpreted in that case as the result of a wave effect. It is important to point out that the quantization law is here inverse of the one obtained by de Broglie because we are working with a progressive wave. We can also make the same kind of calculation with a spring which deforms itself and obtain a quantization of both the rectilinear and rotational velocities concerning the spring. It would be interesting to do the experiment with a plotter in order to verify Eq.(85).

9.2.

Crawford

calculation

In an interesting paper, Crawford 107 derives the de Broglie relation from the Doppler effect by considering the elastic collision between an electron and a massive mirror which both move respectively with velocities Uc and Um before the collision such that U e > Um > 0. If the electron emits a homogeneous

wave with a

178

phase velocity UP towards the mirror, it will receive the reflected wave from the moving mirror with a classical Doppler shift given by the relation: (86) The coefficient -2 is due to the fact that we are working with standing waves calculated in a classical way. In fact this Doppler effect can be calculated by using conservation of energy and momentum in a collision between a point photon and the mirror. However the relation (86) shows that the final result 1s independent of Planck's constant n. We know that matter waves are dispersive

waves which implies: (87)

with:

(88)

By inserting the relations (86) and (87) 1n the above equation, obtain:

we

(89) The Eqs. (86) and (89) show that we have a phase modulation of the quantities k,ro. All our calculation does not depend explicitly on time, therefore the homogeneous waves become inhomogeneous waves with an implicit time dependence. In this approach Crawford brought out the concept of inhomogeneous wave without stating the fact explicitly. During the process of wave reflection from a mirror the electron undergoes a collision with the mirror. The relative velocity of the electron with respect to a massive mirror is U = Ue - Um before the collision and V = Ve - Vm after the collision, with Ve < 0 < Vm. An elementary calculation using conservation of energy and momentum gives the well-known result that the collision does not change the magnitude of the relative velocity but just reverses its sign such that V = -U. From this calculation one also obtains the shift in particle velocity:

179

(90) If we add Eqs. (89) and (90) we have:

(91) After integration of the above equation, we obtain the de Broglie relation ±A.U e = h/m 0 for its non-relativistic form. It is important to point out that the de Broglie relation cannot be obtained as the result of a pure velocity effect since Eq.(90) is calculated from an acceleration of the electron during the collision with a mirror. Therefore the quantization given by Eq.(91) tells us that the deformation of the waves induced by the acceleration of the electron and the mirror during their interaction cannot take any value if the waves must keep their oscillatory behavior. 10. REFERENCES 1. J. Molcho and D. Censor. Am. J. Phys. S4 (1986) 351. 2. E. J. Post, Rev. Mod. Phys. 39 (1967) 475. 3. A. K. T. Assis. Can. J. Phys. 70 (1992) 330. 4. H. Jehle, Phys. Rev. D3 (1971) 306. 5. H. Jehle. Phys. Rev. D11 (1975) 2147. 6. W. H. Bostick, Int. J. Fus. En. 3 (1985) 9. 7. 0. Buneman. Supplemento Al Volume V, Serie X del Nuovo Cimento V (1957) 92. 8. E. J. Post. Phys. Rev. D9 (1974) 3379. 9. E. J. Post. J. Math. Phys. 19 (1978) 347. 10. E. J. Post, Physics Letters 92 (1982) 224. 11. E. J. Post. Phys. Rev. D2S (1982) 3223. 12. E. J. Post, J. Math. Phys. 2S (1984) 612. 13. -R. M. Kiehn, J. Math. Phys. 18 (1977) 614. 14. H. Jehle. Phys. Lett. B104 (1981) 203. 15. H. Jehle. Phys. Lett. B104 (1981) 207. 16. C. V. Westenholz, Differential forms in mathematical physics (NorthHolland Publishing Company, Amsterdam, 1981). 17. 0. Darrigol. Ann. Phys. 9 (1984) 433. 18. I. J. R. Aitchison, Contemp. Phys. 26 (1985) 333. 19. L. de Broglie. Tentative d'interpretation causale et non lineaire de la M ecanique ondulatoire (Gauthier-Villards, Paris. 1956). 20. L. de Broglie. La reinterpretation de la mecanique ondulatoire (Gauthier-Villars, Paris, 1971 ). 21. D. Bohm, Phys. Rev. 8S (1952) 166. 22. D. Bohm, Phys. Rev. 8S (1952) 180. 23. D. Bohm, B. J. Hiley, and P. N. Kaloyerou. Phys. Rep. 144 (1987) 321. 24. R. C. Bourret. fl Nuov. Cim. 18 (1960) 347. 25. T. W. Marshall, Proceedings of the Royal Society of London A276 (1963) 475.

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TWISTORS

IN FIELD THEORY

JORG FRAUENDIENER

AND TSOU SHEUNG TSUN

November 10, 1994

In this paper we wish to show the role of twistor theory in solving various field equations that are of importance in physics. To introduce the ABSTRACT.

language we start with short reviews of spinors and twistors. These are of necessity extremely brief, giving just those concepts which will be used later, and omitting many very important aspects of either. The interplay between Minkowski geometry and twistor geometry is emphasized. We touch upon the Dirac equations for fields of arbitrary spin, and concentrate on the field equations in electromagnetic theory and their generalizations to Yang-Mills theory. The strict minimum on vector bundles and twistor cohomology is presented to give a taste of the deeper understanding which twistor theory provides for the solution of these equations. We end with some remarks on the non-linear graviton construction and on some new results for the spin 3/2 fields.

l.

INTRODUCTION:

THE

WHITTAKER

FORMULA

In 1903 Whittaker [26] wrote down a general solution of Laplace's equation: 2

(1.1)

"v V

a2v a2v a2v

= 8x2

+

8y2

+

8z2

= 0,

in the form of an integral: (1.2)

V

= fo2' W(z + ix cos o + iy sin o, o)do,

where Wis an arbitrary function of two variables. This can easily be generalized, as was done by Bateman [4], to an integral formula for a solution of the wave equation: (1.3)

namely:

/27r u = lo F(0, '2(cos8) Ee Eci,

-

sin8 fE then fM decreases until the two are equal. With unequal frequencies, the sources are independent and the total reactive energy is W =WE+ WM. As fM becomes equal to fE the reactive energy switches to WE+Mwhere WE+Mf(WE+WM)~ l; gain and bandwidth also switch values. Next consider superimposed, spherical, z directed electric and magnetic dipole radiators of radius a. Since all fields are orthogonal the two sources act independently. The gain is G = 1.8 dB and the Q on a surrounding virtual sphere, Q5 , is i/(ka)3. independently of the frequencies. Fig. 3 shows the reactive power and reactive energy per unit length (energy density in one dimension) from a single, spherical magnetic dipole radiator with radius a [4]. Reactive power R(O n

2

- i ro2L - iT) =-roC + -= + T)(ka)2 R ka

(47)

Equation 47 shows that the reactance to resistance ratio, equal to the Q in simple systems, is l/(ka)3. The input impedance of an electrically small electric dipole is a capacitive reactance in series with a small resistance that decreases as the square of the ka ratio. For an ideal, spherical magnetic dipole radiator, the dual of the electric one, in the limit of small values of ka and keeping only the first order reactive and real tenns

Lim Zin(ka) = iroL + ro4L2C2R = i'T)ka+ T)(ka)4

ka=>O

(48)

Equation 48 shows that the reactance to resistance ratio is 1/(ka)3 and that the input impedance of an electrically small magnetic dipole is an inductive reactance in series with a small resistance; the resistance decreases as the fourth power of the ka ratio. Quite differently, the reactive fields are balanced for superimposed, small ka electric and magnetic dipoles radiating with a minimum reactive energy configuration. The interacting circuits are Fig. 2 and its dual. The effect of the reactive elements cancels and the input is affected only be the real power output. In this resonant circuit condition, the effective input impedance per element is T)and the total input impedance is

Lim Zi (ka) = Tl

ka=>O n

2

(49)

Equation 49 shows that the radiation resistance of matched dipole elements is large and independent of frequency.

267

5. Photons 5 .1 General comments Reasons given for the modem physics maxim that classical field equations are not fully applicable on the atomic scale of dimensions are: (1) The linear momentum carried by photons requires full directivity of the radiation field [24-]and absorption or emission of fully directed radiation is inconsistent with wave theory. (2) Stable localized states require accelerating charges, accelerated charges radiate, yet atomic states are stable [25]. (3) Atomic ka ratios are so small and the radiative Q is so large that antenna radiation, using classical theory, is prohibited. These phenomena are explained by emphasizing quantum theory and classical theory is discarded wherever it becomes inconvenient. We ask if the break with classical theory is necessary. To address the issue, it is first necessary to model the electron [26]. To be consistent with others and still meet our requirements, we model the electron as an electrically neutral kernel with a diameter of 10-20 m. or less [27], enmeshed in a cloud of loosely bound charge the undisturbed radius of which is about 10-12 m. The charge distorts in response to gradients in local force fields, such as nuclear electrostatic and radiation reaction fields. Appropriate values of mass, charge, angular momentum, magnetic moment, etc., are assigned. Validation of the model comes in the following ways. Using it, an electron trapped by a nuclear electrostatic field is subject to strong and disruptive tidal forces. As an electron accelerates it radiates and suffers radiation reaction forces [25]. Both forces act to form the charge cloud into a form of closed loop, chaotic motion. Numerous studies have shown that current loops are stable. Energy conservation together with the electron model results in the Schrodinger equation [4,19] and from it comes the wave character of electrons. We find that once radiation with an appropriate modal structure is triggered, regenerative feedback occurs that non-linearly drives the electron until the available energy has been radiated. While a classical antenna responds to a source in a way that minimizes Qs, with electron radiation the small Qs is not dominate. Instead, the limit condition of positive feedback determines the modal ratio to be [4, 19] F(l,m)

2l+l

= -G(l,m) = Ft6(m,l) = l(l+l)

(50)

where 6 represents the Kronecker delta function, andj = + i. Making use of Eq. 50 we now show that the radiation has the kinematic and wave properties of photons. To begin we impose Eq. 50 onto field Eqs. A5 to Al0 and obtain the fields:

iLr 00

E=

l~:~)

{ l(l+I)

ht~CJ)P}(cosO)

p + [ht(cr)

p~~:;8)

l=l

(51)

(52)

268 With P and R respectively equal to the real and reactive powers on the surface of a concentric sphere of radius r ~ a, where, as before, a is the radius of the smallest virtual sphere that circumscribes the radiator, it follows that P

I

dU

(53)

=dt = r21dONo

(54) U(t) is the real energy radiated through the surface and dil indicates differential solid angle. The rates at which z directed linear and angular momentum, respectively G 2 and Lz,leave the region are

dG2 (a) _ r2 I ...1n N n dt - C 1 w,.1, 0 COSc,

(55)

~a) = ~ §dil Re[N~

(56)

sin9

N,

where is the real part of the q,component of the Poynti.ng vector. Expressions for the x and y momenta components follow similarly. Our results follow from Eqs. 53 through Eq. 56; of them, only Eq. 53 is an orthogonal function of the field products and only it is independent of the radius of evaluation. Values of Eqs. 53, 55, and 56 as r => oo with j = ± i are

(57)

dGz-av_ .!.dUav dt

(58)

..!.dUay _ .! dGz.av - a, dt - k dt

(59)

dt

-c

+ d.Lz-av_ -

dt

A ti.me integration over any time intetval and pulse shape shows that

+ coLz-av = Uav = cGz.av

(60)

The result describes the kinematic properties of a circularly polarized. z directed spherical wave: the ratios of energy to momentum and angular momentum are, respectively, c and +(J)[4,12]. When coupled with the source-derived [19] Einstein relation

u = 1i(J)

(61)

269 the result is the measurable kinematic properties of photons. Since the wave character of the radiation is already explicit in the equations, it shows that this electron model and the classical equations are consistent with quantized radiation. We showed earlier [4,12], using Eq. 50, that as a increases without limit Eqs. 51 and 52 go to

t = e-iocosB(l + cos8)(~ + i 9)+ 0(1/a)

(62)

T1R = _i e-iocosB(l + cos8)(~ + i 9)+ 0(1/a)

(63)

where 0(1/a) indicates terms proportional to 1/a. Eqs. 62 and 63 describe fields that extend over upper half space; the radiation pattern is shown elsewhere [4,12], and describes all kinematic, propagation and interference properties of photons.

5.2 Positive feedback, the nonlinearity [4] The incorporation of Eq. 50 into N 1 and N2, see Eqs. 11 and 12, shows that they are equal to zero, and therefore N4 is zero, see Eq. 15; there is no reactive power component to the field for either polarization. Without a reactive component there is no reactive radiation reaction on the source. A radiating atom radiates into a purely resistive load: it feels the affect of the time-average radiated power, but nothing else. Although the result is that the minimum value of Qs is zero, since we are not knowledgeable of transient energy build up and since the Poynting theorem is silent on the interpretation of density at a point (any function with zero divergence may be added), the range of uncertainty about Qs for fully directed radiation is Qs~0

(64)

It was shown elsewhere [4, Table II], [19, Sec. IV] that positive feedback will self drive a radiating charged cloud with sufficient degrees of freedom until the radiation coefficients are limited by Eq. 50. Detailed field analysis [4, 19] shows that the near fields provide the positive feedback and a radial expansion force on the radiator. Therefore the radiating system expands, nova-like, during radiation, driving Eq. 64 towards an equality.

5.3 Transmission and reception characteristics The radiation has symmetry similar to that described in Section 3, and the transmission and reception characteristics described in Sections 4.3 and 4.4 carry over to this radiation.

5.4 Radiation reactionforce Although the radiation reaction force on a free charge is small, often that on a bound charge is not. For example, the radiation reaction force Frr(t) on a bound electron radiating as an electric dipole, Pl (t) = qz(t), from the surface of a virtual sphere of radius a is [4,12], see Appendix C: (65)

270 Similarly, radiation reaction of a linear quadrupole satisfies the equation

where Eqs. 65 and 66 follow from the modal power expressions and the quadrupole moment p2(t) = 2q[z(t)]2. Going briefly to the frequency domain shows that the magnitudes of the terms in Eq. 65 are proportional to (ka)n, where n is the order of the time derivative. Even order time derivative terms originate with reactive power; the first term on the right side is the restoring force due to the field reactive energy and the second is due to the effective mass of the reactive fields. Odd order time derivative terms originate with real power. The third term on the right side of Eq. 65 is both the dominant resistive loss term with dipole radiation and the total radiation reaction force term on a free charge [28]. Similar differential equations describing radiation reaction force on model shows that the first odd power reaction term is proportional to (ka)U+l. For an electric dipolar radiator, Eq. 65 shows that the reactive energy, and therefore the reactive radiation reaction force, has the same order of magnitude as the Coulomb force about a hydrogen atom. Therefore an electron producing electric dipole radiation is subject to a radiation reaction force of a magnitude similar to the Coulomb binding force. The force is present until the charge assumes a non-radiating form, presumably a closed loop. The absence of surface reactance leaves only surface resistance; the effect of even order derivatives has been canceled. Therefore, at resonance, the input impedance felt by the driving circuit is resistive and less by a factor of (ka) 3 than the input reactance of an isolated dipole; it is less by a factor of (ka) 5 than the input reactance of an isolated quadrupole. For a resonant system only real power is present and only it produces a radiation reaction, see Eq. 49. The result, in marked contrast with isolated elements, is that the input impedance of each of the four ensemble elements is resistive and the same order of magnitude. An example of increased real power output when the reactance is canceled in a conventional antenna is shown elsewhere [29, Fig. 4]. We conclude that if a radiating charged cloud of radius a begins to radiate with properly oriented and phased multipolar dipolar and quadrupolar moments that radiation will drive higher order moments that will, in tum, drive the lower order number ones with positive feedback. The reactive reaction force will be zero and the resistive reactive term will be small. Therefore each radiating mode increases in magnitude until limited by Eq. 50. Power output increases until all available energy has been radiated. The radiation satisfies Eqs. 60, 62, and 63, which describe the known kinematic and wave properties of photons.

5 .5 Comments about the solution In his Nobel address. Planck [30] referred to "spherical waves" and implied that transmitted power has a significant angular spread. He also asked what happens to a photon after emission. "Does it pass outward in all directions continually increasing in volume and tending towards infinite dilution? Or does it fly like a projectile in one direction only?" Pais [31] refers to photons as "needle-like" and to "needle rays." Equations 51 and 52 show that power is not fully directed at the radiator yet Eqs. 62 and 63 show that it is fully directed in the distant field. Apparently, far from being needlelike, as the relative magnitudes of the field terms change, real power changes direction, menorah-like. and becomes fully directed in the distant field. This may also be seen from a series solution of Eq. 55, where full directivity occurs only in the limit of cr much

271

greater than l. Were the wave train infinite in length, all upper half space would be occupied, instead it is limited in length to a wave train determined by the transition time. Since radiation onset is dependent upon the simultaneous occurrence of an ensemble of multipolar moments, presumably it is difficult to start. The charge circulates in closed paths with chaotic circulation patterns and, for spontaneous radiation to begin, it is necessary to form simultaneously electric and magnetic dipole pairs that generate the dipole fields of Eqs. 51 and 52. The probability of such an event determines the lifetime of the state. Once the condition is met, the process becomes regenerative and radiation continues until the available energy has been radiated. Equation 57 shows the power carried per mode is proportional to (U+l). Harrington showed [11] that the maximum gain per mode is (U+l). Directedness, by definition the ratio of c times the linear momentum-to-energy ratio, follows from Eqs. 57 and 58 and is equal to l'/(l'+ 1), where the modal series is terminated at l'. The set of fields produced by putting Eq. 50 into Eqs. A5 to Al0 is infinite in time extent and is silent about transient behavior. From the theory of single mode radiation with ka « 1, Qs of each isolated mode of order l approaches Lim - l[(U-1)!!]2 ka=>0 Qs - (ka)U+l

(67)

With the recursion relationship of Eq. 50, if the field modes were orthogonal Qs would be about Lim - [(U-1)!!]2 k:a=>0Qs - (ka)U+l

(68)

According to Eq. 68, the large reactive energy would dominate over the electron energy and Qs estimates based upon electron energy only would be grossly in error, yet they are not. Using results of this paper, however, the field modes are not orthogonal and Eq. 68 does not apply. Instead, Qs is small and satisfies Eq. 64. We anticipate agreement with line width calculations based on electron energy. Therefore it is correct for the QED and semiclassical solutions to avoid the subject of reactive energy.

6. Conclusions We critically examine the form of reactive power on closed, virtual, spherical surfaces containing groups of radiators. We apply the continuity equation to the power expression to find the reactive energy per unit radius. The reactive energy satisfies a longitudinal wave equation. The energy stands within a half wavelength of an initial position and forms traveling, outbound packets whose magnitudes decrease with radius at least as fast as 1/r3. The phase may be either capacitive or inductive. We integrate the peak value of energy density over each phase and put the larger integrated value equal to the total reactive energy; we use that energy to find Qs. This technique permits us to evaluate Qs even where the power functions are non-orthogonal functions of the fields. We use the time-domain Poynting theorem to make a critical examination of reactive power. The reactive part of the complex Poynting vector is equal to the square root of products of sums over certain field squares, or it is zero. The sums are perfect squares and easily calculated in three cases: (1) Either TE or TM modes but not both. Reactive power is an orthogonal function of the fields and orthogonality forms the basis for Qbased size-wavelength limitations on small radiators. (2) Properly matched and phased TE and TM modes with linear polarization for which non-orthogonality leads to strong

272 reactive coupling between modal equivalent circuits and mitigated size-wavelength restrictions. (3) Properly matched and phased TE and TM modes with circular polarization for which there is no reactive power. The third case describes a radiation field that is emitted from a virtual, spherical surface and is photon-like; our analysis is limited since our mathematics only handles fields of infinite duration. The radiation has neither a reactive power density at any spatial angle nor a net reactive power for either polarization; the radiator works into a purely resistive load even as the modal number increases without limit. The only radiation restriction is Qs ~ 0. Once a region containing sources with sufficient degrees of freedom starts to radiate with properly oriented TE and TM fields, that radiation regeneratively drives itself and all other moments to the recursion relationship of Eq. 50 [4,19]. The radiated fields satisfy Eqs. 57, 58, and 59, which, in turn, describe the kinematic and wave properties of photons. If radiation is triggered by an external plane wave, the radiated wave has the same frequency, phase, polarization, and direction of travel as the trigger. In these terms, a photon is fully described by the classical field equations.

7. References 1. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw Hill, 1968) 2. A. 0. Barut, B. Blaive, The Hydrogen Atom as an Antenna, IAEA and UNESCO, International Centre for Theoretical Physics, 1991 Miramare- Trieste. 3. S. A. Schelkunoff, Advanced Antenna Theory, (John Wiley, 1952), Chp. 2. 4. D. M. Grimes, in Essays on the Formal Aspects of Electromagnetic Theory, A. Lakhtakia, ed. (World Scientific Publishing Co., 1993) pp. 310-356. 5. W. Heitler, The Quantum Theory of Radiation, 3rd ed.(Oxford, 1960), Chp.V. 6. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, (Addison-Wesley, 1963) Vol.I, Chp.32. 7. H. C. van der Hulst, Light Scattering by Small Particles, (John Wiley, 1957) 8. D. M. Grimes, J. Math. Phys. 23 (1982) 897-914. 9. L. J. Chu, J. Appl. Phys.19 (1948) 1163-1175. 10. H. L. Anderson, ed. Physics Vade Mecum, (Am. Inst. of Physics, 1981) p.295. 11. R. F. Harrington, J. Res. Nat. Bur. Stds. 64D (1960) 1-12. 12. D. M. Grimes, Physica, Vol. 32D (1988) 1-17. 13. H. A. Wheeler, IEEE Trans. Ant. Prop. 23 (1975) 462. 14. R. C. Hansen, Proc. IEEE 69 (1982) 170-182. 15. R. E. Scott, Linear Circuits (Addison-Wesley, 1960) Chp.19. 16. D. M. Grimes, C. A. Grimes, IEEE Trans. Electromag. Compat., to be published. 17. D. M. Grimes, V. Badii, K. Tomiyama, Radio Science, 26 (1991)101-109. 18. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1942) 19. D. M. Grimes, Physica 20D (1986) 285-302. 20. V. Badii, K. Tomiyama, D. M. Grimes, Appl. Comp. Electromagnetic Soc. J., S ( 1990) 62-93. 21. R. L. Fante, IEEE Trans. Ant. Prop. 17 (1969) 151-155. 22. R. L. Fante, IEEE Trans. Ant. Prop. 40 (1992) 1586-1588. 23. C. A. Grimes, D. M. Grimes, 1991 IEEE Aero. Appl. Conf. Digest, paper #3; see also US patent No.4,809,009. 24. A. Einstein, Phys. Z. 18 (1917) 121. Also World of the Atom (Basic Books, New York, 1966) H.A. Boorse and L. Motz, eds., p.888. 25. G. A. Schott, Electromagnetic Radiation and the Mechanical Reactions arising from it, (Cambridge, 1912) 26. A. Pais, in Aspects of Quantum Theory, A. Salam, E. P. Wigner, eds, (Cambrige, 1972) pp.79-93. 27. H. Dehmelt, Science 247 (1990) 539.

273 28. W. K. H. Panofsky, M. Philips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962) 29. D. M. Grimes, C. A. Grimes, in Advanced Electromagnetism: Foundations, Theory and Applications, T.W. Barrett, D.M. Grimes, eds. (World Scientific, 1995), Fig. 4. 30. M. Planck, in World of the Atom (Basic Books, 1966) H.A. Boorse and L. Motz, eds., p.491. 31. A. Pais, The Science and Life of Albert Einstein (Oxford University Press, 1982) Chp.21.

Appendix A: The multipolar field expansion [28] A convenient way to obtain the general expression for all possible radially directed, vector phasor electromagnetic fields is to begin with a scalar field T(r,8,(/J) that satisfies the scalar Helmholtz equation:

(Al) expressed in spherical coordinates. For fields that vanish as r ⇒ oo, that occupy all space with radius r > a, where a is the non-zero radius of a virtual sphere enclosing an antenna, and that remain non-singular on the axes, a useful solution of Eq. Al is

TT~ht(kr) P'7(cos8)e-im;

(A2)

where ht( a) are spherical Hankel functions of the second kind, P T(cos8) are associated Legendre function of order I. and degree m, and I. and m are integers in the ranges and

(A3)

j = +{:[ and we use ei01 time dependence. Double notation using both j and i is to make simpler separation of polarization from time effects, and to restrict m to positive integers. Circular polarization is withj =±i and linear, x or y directed polarization is with retention, respectively, of the real or imaginary part with respect to j. Although the physics convention is to use e-illJI time dependence, this paper is concerned with both physics and electrical engineering, and that choice makes the reactances of capacitors and inductors, respectively, positive and negative -- the reverse of the electrical engineering convention. If the roles of i and j were reversed, since jf represents spherical Bessel functions, the productjjf would appear, and would surely lead to confusion. For those who prefer e-illJI time dependence, take the complex conjugate with respect to i. Both electric and magnetic vector phasor fields satisfy the vector Helmholtz equation. ff satisfies Eq. A 1, then the vector Vx(f 'tr) satisfies the vector Helmholtz equation. Therefore, putting

rr::

(A4) gives fields that satisfy the appropriate wave equation. To get the full solution, we solve for the TE mode t fields and the TM mode O fields using Eq. A4, then use the Maxwell equations to complete the field sets. The complete solution is a sum over all functions that satisfy the vector Helmholtz equation and are weighted to match radiation and boundary conditions. Straight-forward manipulation

274 shows that the unique and complete expression for outbound fields external to the sphere is the equation set:

L L

h ( ) i-lF(l,m) l(l+ 1) t a P'2(cos9)e-jmq,

l=tm=O

a

oo

Er= i

Ee=

11f4 =

oo

rr

i-l [zF(l,m)h~ (a)dP7(cos8) - G(l,m)ht(O) ~(cos8)]e-jmq, l=tm=O d8 sm8

rr

i-l [F(l,m)ht(O)dP'2(cos8) - iG(l,m)h ~ (a) ~(cos9)]e-jmq, l=tm=O d8 sm8

~ = -j 11H0= j

rr r~

i-l [zF(l,m)h~ (a) ~(cos8) l= I m=O sm8

i-l [F(l,m)ht(a)~(cos8) l= I m=O sm8

- G(l,m)ht(O)dP'2(cos8)]e-jmq,

(A5)

(A7)

(A8)

(A9)

d8

- iG(l,m)h ~ (a) dP'2(cos8)]e-jmq,

(AIO)

d8

where h: (a) are the functions

(Al I) Asymptotic values of the radial functions for a>> l are: (A12) The complete description of a specific field is obtained by evaluating the constant coefficients F(l,m) and G(l,m) on the surface of the generating sphere. Modes with F(l,m) coefficients are TM and those with G(l,m) coefficients are TE. These equations describe the complete set of fields that can exist within the described constraints. Appendix B: The Q-bandwidth product

We seek to evaluate complex power for the cases l = 1 and 2, with m = 1. For this purpose we note that a general form for spherical Hankel functions is: (Bl)

275 where A 1 = 1/a, B 1 = - 1, A2 = 3/a2 - 1, and B2 = - 3/a. The m = 1 associated Legendre polynomials for I.= 1 and 2 are (B2)

Pl= 3 sinlkos8

Pf= sin8

B .1 The resonant ensemble Under the design condition Ft= - Gt, with~= F2/F1, and with N3 as the complex Poynting vector, it follows that P _ C-

~ { [ 4 36~2] _ . 108[ (J:2 5~) 2

2~k

3+ 5

l

5a

5

2(~2 5~ 2-)] ~ - 9 + (J 6 + 9 - 162

}

(B3)

Since the system has zero reactance on a virtual surface with normalized radius ao, it follows from the right side of Eq. B3 that J:2 5~ 2(~2 5~ 2-) -(~ • 9) - 00 6 + 9 - 162

(B4)

Using Eq_41, Qs is found to be

(B5)

To relate Qs to the bandwidth, we note that half power points occur at the frequencies where resistive and reactive components have equal magnitude. Equation B3 shows that this happens when (B6) where a 2 is the normalized radius and roi is the radian frequency at the half power point. Substituting Eq. B6 into Eq. B5 gives the result (B7)

Defining the bandwidth to be flro = 21roi-~I,and requiring that flro be much smaller than either ~ or a>i,Eq. B7 is approximately equal to (B8)

276

Appendix C: Dipolar radiation reaction force The radiation reaction forces at the surface of the virtual sphere of radius a follow from the modal equivalent circuits. Using the (i,m) = (1,0) case as an example, the radial field component is h1(a)

Er= 2--

(Cl)

P1(cos8)

(j

which, in the near field limit, becomes (C2)

The first term on the right follows from the first term in the expansion of the spherical Hankel function and the second from electrostatics where Pl is the magnitude of the electric moment. The relationship between F1 and Pl follows, and is (C3)

The angular field components are (C4)

The radiated power at any radius r ~ a follows, and is (C5) We next express Eq. C5 in terms of a generalized voltage and current, i.e., force and flow. Since the electric and magnetic field intensities are respectively proportional to h a) and h 1(a), we define the equivalent voltage and current, V and I, to be

i(

(C6) We use Eq. C6 to solve for the modal equivalent circuit. Dividing V by I gives

i

i'flh Z=-~ h1

Tl = -+

ia

1 1/[-+-]

1

iT1a Tl

(C7)

Equation C7 is in a form convenient for circuit synthesis and, from it, comes the circuit of Figure 2. For the present purpose we choose to re-write it in series terms as

2n (C8)

We next switch to the time domain by replacing im by d/dt and introduce an equivalent charge Q(t) where, by definition, I = iroQ. The result is that V(t)

= Q(t) £a

+ µa

d 2Q(t) _ µa2 d3Q(t)

dt2

C

dt3 + ...

(C9)

The objective is to re-state Eq. C9 in terms of the force on time varying dipole moment Pl (t). The form is dictated by Eq. C9, and we need only solve for the unknown constant K, where:

To find K, we evaluate Eq. C5 in the far field limit, using Eq. C3 to find

p =-cp12k4

(Cll)

12Jt£ Only the last term in Eq. CIO describes loss. Evaluating it in the frequency domain and combining with Eq. Cl I shows the loss to be (Cl2) Eqs. Cl I and C12 show that K=-

1 6xa2

Substitution of Eq. C13 into Eq. CIO gives Eq. 65.

(C13)

SAGNAC EFFECT: A consequence of conservation of action due to gauge field global conformal invariance in a multiply-joined topology of coherent fields TERENCE W. BARRETT

1453 Beulah Road, Vienna, VA 22182, U.S.A.

Abstract: The Sagnac effect underlying the ring laser gyro is a coherent field effect and described here as a global, not a local, effect in a multiply-joined, not a simply-joined, topology of those fields. Given a Yang-Mills or gauge field formulation of the electromagnetic field (Barrett, 1993), the measured quantity in the Sagnac effect is the phase factor. Gauge field formulation of electromagnetism requires in many cases uncoupling the electromagnetic field from the Lorentz group algebra. As conventionally interpreted, the Lorentz group is the defining algebraic topology for the concepts of inertia and acceleration from an inertialess state. However, those concepts find new definitions here in gauge theory and new group theory descriptions. The explanation for the origins of the Sagnac effect offered here lies in the generation of a constraint or obstruction in the interferometer's field topology under conditions of conserved action, that constraint or obstruction being generated only when the platform of the interferometer is rotated. Whereas previous explanations of the Sagnac effect have left the conventional Maxwell equations inviolate but seen a need to change the constitutive relations, here we see a need to do the opposite. Just as the Lorentz group description appears only as a limiting (zero rotation or stationary) case in this new explanation, so Minkowski space-time is also viewed as a limiting case appropriate for the Sagnac interferometer in the stationary platform situation, with Cartan-Wey! space-time appropriate for rotated platform situations. We attribute the existence of a measurable phase factor in the Sagnac interferometer with rotated platform as due to the conformal invariance of the action in the presence of the creation of a topological obstruction by the rotation.

Contents: l. Sagnac Effect Phenomenology 1. 1. the kinematic description 1.2. the physical optical description 1.3. the dielectric metaphorical description 1.4. the gauge field description 2. The Lorentz group and the Lorenz gauge condition 3. The Phase Factor Concept 3.1. SU(2) group algebra 3.2. Short primer of topological concepts 4. Minkowski Space-Time Versus Cartan-Weyl Form 5. Conclusion 6. References

1. Sagnac Effect Phenomenology The Lorentz group algebra is the defining field algebra for the set of all inertial frames and the space-time symmetry. Any frame of reference that is not an inertial frame is an "accelerated" frame and experienced as a force field. However, we shall attempt to show that for noninertial frames the Lorentz group is not the defining algebra. Such situations are

278

279 measured by rotation sensors. Of these, Sagnac ( 19 l 3a,b, 1914) first demonstrated a ring interferometer which indicates the state of rotation of a frame of reference, i.e., a ring interferometer as a rotation rate sensor. The ring interferometer performs the same function as a mechanical gyroscope. When a laser is used as the source of radiation in the interferometer, it is called a ring laser gyro. Fig. 1 shows the basic Sagnac interferometer. One light beam circulates a loop in a clockwise direction, and another beam circulates a loop in a counterclockwise direction. When the interferometer is set in motion, interference fringes (phase difference) are observed at the overlap area H, i.e., in the heterodyned counterpropagating beams. Details of the interferometer can be found in Post ( 1967) and Chow et al ( 1985) and reviews of the Sagnac effect have been given by von Laue (1920), Zemicke (1947) and Metz (1952). There are two basic kinds of ring interferometers sensing rotation: the passive ring resonator and the active ring laser gyro. The general theory of the ring laser gyro is addressed in Gyorffi & Lamb ( 1965), Aronowitz ( 1971) and Menegozzi & Lamb ( 1973).

0 OUT Fig. 1. The basic Sagnac interferometer. The clockwise and counterclockwise beams are not separated in reality and are only shown so here for purposes of exposition.

Here, we shall quickly cover the main descriptive features and move on to address the central issue of this paper: explanations of the effect. In the case of the passive ring resonator, the interference fringes are described by: (1) where llq,is the phase difference between clockwise and counterclockwise propagating beams, Q is angular rate in rad/sec., A is the vacuum wavelength, Bis area enclosed by the light path, and a velocity field v defines the angular velocity: V xv= 211.

(2)

Rosenthal ( 1962) suggested a self-oscillating version of the Sagnac ring interferometer which was demonstrated by Macek and Davis (1963). In this version, the clockwise and counterclockwise modes occur in the same optical cavity. In the case of the

280 laser version of the self-oscillating version of the Sagnac interferometer, i.e., the ring laser gyroscope, and in contrast to the Sagnac ring interferometer, a comoving optical medium in the laser beam affects the beat frequency, which, rather than the phase difference, is the measured variable. In this version, the frequency difference, ll.m, of the clockwise and counterclockwise propagating beams with respect to the resonant frequency, OJ, is described by: ll.m _

m-

2fn

2 ( 1-

a)v • dr

cfnds

(3)

where n is the index of refraction of the stationary medium~ and a is a coefficient of drag. The same frequency difference with respect to the angular velocity is given by: ll.m =

(ii 20. =

v-;

,l

411n.

Pl '

(4)

where P is the perimeter of the light path. The Sagnac interferometer path length change in terms of phase reversals is independent of the waveguide mode and completely independent of the optical properties of the path (Post, 1972a). There are details of ring laser gyroscope operation, such as lock-in and scale factor variation, which necessitate the amendment of the above descriptions, but these operational details will not be addressed here, and we move directly to consider explanations of the effect. There are three current explanations/descriptions of the Sagnac effect: (a) the Kinematic Description. (b) the Physical-Optical Description. (c) the Dielectric Metaphorical Description. It is the intention of the present essay to introduce a fourth: (d) the Gauge Field Explanation. We shall examine each of these approaches in turn. 1. 1. Kinematic Description

The force field exerted on the fields of the Sagnac interferometer can be either due to gravitational, linear acceleration or rotational velocity (kinematic acceleration) field effects. The kinematic acceleration field is due to Coriolis force contributions, but the gravitational field and linear acceleration field do not have such contributions. There is also a distinction between the different fields with respect to the convergence/divergence of those fields. For example, the lines of gravitational force converge to a nonlocal point, e.g., the center of the earth, in the case of a platform at rest on the earth, but in the case of a linearly accelerated platform the lines of force converge to a nonlocal point at infinity. No Coriolis force is present in either of these cases. In contrast to these, with a platform with rotational velocity, the lines of force diverge from the local axis of rotation and a Coriolis force is present. If the platform with rotational velocity is also located on or near the earth,

281

it will also experience the gravitational force of the earth, besides the Coriolis force. However, only the platform undergoing kinematic acceleration (rotational velocity) is in a state of motion with respect to all inertial frames. The kinematic description thus primarily implicates the Coriolis (acceleration) force and a state of kinematic acceleration is associated with a state of absolute motion with respect to all inertial frames. For example, Konopinski ( 1978) defined the electromagnetic vector potential as field momentum exchanged with the kinetic momenta of charged particles. According to this author most defining relations between potentials and fields, such as, e.g., in equations of motion, are defined in a static condition. In these static conditions, and moreover, local conditions, q(/>can be defined as a "store" of field energy, and qA a "store" of momentum energy. That is the conventional interpretation of the four C

vector potential. However, the subject of Konopinski's paper is a second condition which is global

f

m character. The model he considers is not a point but a volume dV(r) of an electromagnetic field with an energy and vector momentum defined as in the Coulomb gauge:

(5) _(ExB)

g ( r,t ) ----,

(6)

4nc

and with a total mass

(7) which is constant in the absence of fluxes through the surface enclosing the volume. The test condition for this situation consists of a point charge q at a fixed position r q in a static external field E 0 = -V ,B0 = -V x A( r). The fact that the model considered is a volume introduces global (across the volume) and local (within the volume, e.g., pointlike) conditions. The equation of motion for a point charge is given as: d(Mv+ q A) ___ c_=-Vq((/>dt

(

)

~ •A),

(8)

C

which describes changes in conjugate momentum (left side) = interaction energy (right side). With q constant (the left hand side), any variation of v causes A to vary and viceversa. The total field momentum that changes when the position r q is changed is derived

as:

282 ,(

P(rq)=ydV(r)

E (r - r ) x B0 (r) q

q

4,cc

.

(9)

Introducing source terms gives the field momentum P(r q) as: (10)

The result is that under the total momentum conditions expressed by Eq. (8), changes in the total field momentum when the position of the particle at r q is changed, must result in changes in the kinetic momentum of the particle Mv . Considering now the present topic of interest, the Sagnac effect, one may state the equilibrium condition in the reverse causal condition to that considered by Konopinski, namely: changes in the velocity of the system defining the kinetic energy of a "particle", v, result in changes in the vector potential in the total field momentum, A. Moreover, in the Sagnac effect there are two vector potential components with respect to clockwise and counterclockwise beams. The measured quantity, as will be explained more fully below, is then the phase factor or the integral of the potential difference between those beams and related to the angular velocity difference between the two beams. Therefore, as the vector potential measures the momentum gain and the scalar potential measures the kinetic energy gain, the photon will acquire "mass" 1. Konopinski using variational principles formulated a Langrangian for this global field situation with Av as "generalized coordinates" and dµAv as "generalized velocities".

L=-(dµAv)2 +JvAv_ 81C C

(ll)

With gauge invariance (the Lorentz gauge) there are no source terms, iv= 0, so:

(12) Thus, in our adaptation of this argument, Eq. (12) describes the field conditions of the Sagnac interferometer when its platform is stationary, but conveys no more information than the field tensor Fµv = dµAv - dvAµ- On the other hand, (i) the conservation condition expressed by Eq.s (8) and (11) describe the Sagnac interferometer platform in rotation and kinetically; and (ii) it is relevant that Eq. ( 12), but not Eq. ( 11), is determined

1Moyer ( 1987), addressing the combining of electromagnetism and general relativity, identified charge with a Lagrange multiplier and the Hamiltonian with the self-energy mc 2 , using an optimal control argument rather than the calculus of variations. The Lagrangian identified is a function of the electromagnetic scalar potential and the vector potential, i.e., the four potential.

283 by the Lorenz gauge2. Therefore the field algebraic logic underlying the Sagnac effect, i.e., the Sagnac interferometer platform in rotation, is not that of the Lorenz gauge.

1.2. Physical-Optical Description Post's physical optical theory of the Sagnac effect (Post, 1967, 1972b) demands the loosening of the ties of the theory of electromagnetism to a Lorentz3 invariant structure. Post correctly observed that it is customary to assume no distinction in free space between dielectric displacement, D, and the electric field, E, nor between magnetic induction, B, and the magnetic field, H. This identification is called the Gaussian identification and is justified by the supposed absence of polarization mechanisms in free space. The Gaussian identification, together with the Maxwell equations leads to the d'Alembertian equation, which is a Lorentz invariant structure. As the d'Alembertian does not permit mixed spacetime derivatives, it cannot account for the nonreciprocal asymmetry between clockwise and counterclockwise beam rotation in the Sagnac interferometer. Therefore Post suggested that in order to account for this asymmetry, either the Gaussian field identification is incorrect in a rotating frame, or the Maxwell equations are affected by the rotation. However, in offering this choice Post tacitly assumed no linkage between the Gaussian field identification and the Maxwell equations (i.e., their exclusivity was assumed). He also assumed that the solution to the asymmetry effect must be a local effect, because he was convinced that both the Gaussian field identification and the conventional Maxwell equations describe local effects. Below, on the other hand, we argue that the field arrangements in the interferometer should be described as a global situation and, as a result, the occurrence of an asymmetry does not warrant a change in the local Abelian Maxwell equations (also rejected by Post), but to the required use of nonlocal, non-Abelian Maxwell equations in a multiply-connected interferometric situation (neglected by Post); and also does not warrant a change in the local Gaussian field identification (adopted by Post), but warrants the use of nonlocal non-Abelian field-metric interactions (neglected by Post). Both nonlocal non-Abelian equations and nonlocal interactions are required because the asymmetry under discussion arises in the Sagnac interferometer, that interferometer being a global, i.e. nonlocal, situation; and the amendment suggested by Post, whether of the local equations or the local identification, is inappropriately ad hoc in the presence of that global situation. Rather, the field topology of the Sagnac interferometer requires drastic redefinition of those equations and the full interaction logic, rather than a topologically inappropriate amendment of their local form regardless of field topology. Nonetheless, Post understood that there was, and is, a problem in defining fieldmetric relations in empty space and referred to, among others, the Pegram ( 1917) experiment as illustrative. Quickly stated, in this experiment, simultaneously and around the same axis, a coaxial cylindrical condenser and solenoid are rotated. The rotation produces a magnetic field in the solenoid in the axial direction and between the plates of the condensor. The condensor can then be charged by shortening the plates of the condenser. Post observed that the experiment indicates a cross relation between electric and magnetic fields in a vacuum, a relation which is denied by a Lorentz transformation. Post ( 1967) also noticed correctly that Weyl and Cartan were aware of the metric independence of Maxwell's equations. But Post (1974) presented the view that the 2Evidently, this gauge is due to L. Lorenz (1829-189 I) of Copenhagen, not H.A. Lorentz (1853-1928) of Leiden (cf. Whittaker, 1951, vol. l, pp. 267-268, and Penrose & Rindler, 1984, vol. 1, page 321 (footnote)). 3H.A. Lorentz ( 1853-1928).

284 asymmetry of the conventional Maxwell equations (absence of magnetic monopoles) is compatible with a certain topological symmetry, which he then used to suggest that the law of flux quantization,

f F = 0,

is a fundamental

law. This suggestion,

however,

is

contradicted by the law of flux quantization being global, rather than local, in nature, and being based on a multiply-connected symmetry. Post's suggestion was motivated evidently by the assumption - incorrect from our perspective - that the space-time situation of the Sagnac interferometer is simply connected. He also distinguished two points of view: (I) Topology enters physics through the families of integration manifolds that are generated by physical fields and space-time is the arena in which these integration manifolds are embedded. (2) Space-time is endowed with a topological structure relating to its physics. The first point of view brings the field to center stage; the second, the metric. However, from a gauge field perspective these two points of view are neither unconnected nor exclusive and thus do not constitute an exclusive choice. Under a gauge field formulation, there can be interaction between field and metric. The exclusive choice offered by Post is understandable in that Post did not distinguish between force-fields and related gauge-fields. From a gauge theory point of view, however, one is not forced to choose exclusively between these two alternatives. Post also proposed that the global condition for the A potential to exist is that all cyclic integrals of F vanish. However, this statement is again based on consideration of simply-connected domains, for which the local Maxwell theory is, indeed, described by

f F = 0. He stated

that flux quantization is formally incompatible with the magnetic-

monopole hypothesis, because of his belief that a global A exists only if

f F = 0. But as

we shall show below, in a multiply-connected domain in the presence of a topological obstruction, a global phase factor defined over local A fields exists. Therefore in this global, multiply-connected situation,

f F * 0.

Nonetheless, Post's physical optical theory was a major advance in understanding and is based on the following valid observations: (I) The Maxwell equations have no specific constitutive relations to free space. The traditional equalities in free space, E = D and B = H, assume the Gaussian approximation (absence of detected polarization mechanisms in free space) discussed above, and define the properties of free space only as seen from inertial frames. This is because those relations and the Maxwell equations lead to the standard free-space d'Alembertian wave equation, and the d'Alembertian, as Post pointed ou4 is a Lorentz invariant structure. (The Gaussian approximation corresponds to the Minkowskian metric (c 2 , -1, -1, -1), or ( 1,-1, -1, -1) with ( c 2 = 1), defining the Lorentz group as a symmetry property of the space-time continuum.) (2) Only in the case of uniformly translating systems does the mutual motion of observer and platform completely define the physical situation. Post's solution to the problems raised by these observations is, as we have seen, to modify the constitutive relations, but not to modify the field or Maxwell equations. This solution, we have suggested, inappropriately assumes that the Sagnac interferometer is a

285 simply-connected geometry. The Pegram experiment, described above, indicates the presence of cross coupling between electric and magnetic fields. According to Post this cross-coupling is responsible for the Sagnac effect and is due to the constitutive relations on a rotating frame. However, it is ironic that it is not possible to assume the physicality of such cross terms without modification of the Maxwell equations which Post left untouched. Therefore, while we may agree that the cross-coupling is related to the Sagnac effect, below we ascribe its existence to the presence of non-Abelian Maxwell relations (Barrett, 1993), rather than amended constitutive relations. We can also agree that "A nonuniform motion produces a real and intrinsic physical change in the object in motion; the motion of the frame of reference by contrast produces solely a difference in the observational viewpoint". (Post, 1967, page 488). However, we would reword the distinction as follows: whereas the linear motion of the frame of reference of an interferometer incorporating area II produces solely a local difference in the observational viewpoint describable by the Lorentz gauge, nonuniform motion of an interferometer incorporating area II , as on a rotating platform, produces a global difference describable by the Ampere gauge. Despite these, considered here, incorrect theoretical positions, Post ( 1967) greatly improved and advanced the understanding of the Sagnac effect by squarely addressing the physical issues. We shall return to these physical issues later. He was also a leader in realizing the necessity of distinguishing local versus global approaches to physics (Post, 1962, 1971, 1974, 1982). 1.3. Dielectric Metaphor Description Chow et al (1985) commenced their dielectric metaphor for the gravitational field with the Plebanski ( 1960) observation that it is possible to write Maxwell's equations in an arbitrary gravitational field in a form in which they resemble electrodynamic equations in a dielectric medium. Thus the gravitational field is in some sense equivalent to a dielectric medium and represented by the metric gµv = gvµ of the form: 0

0

0

0 -]

0

0

0

0

-]

0

0

0

0

-]

1 gµv

= 71µv + hµv

=

+

0

ho1

h10

0

0

0

h20

0

0

0 '

h30

0

0

0

ho2 ho1 (13)

where 1Jµv is the metric of special relativity and hµv is the effect of the gravitational field. Using the definition (14) Chow et al ( 1985) defined amended constitutive relations. That is, just like Post ( 1967) examined above, these authors chose to introduce any metric influences on the fields into the constitutive relations rather than into amended Maxwell equations. The amended constitutive relations offered by these authors are:

D = E - c( Bx h),

(15)

286 1

B=H+-(Exh).

(16)

C

These authors then proceeded to derive an equation of motion for the electric field. However, as we have seen above, because an equation of motion (d'Alembertian) assumes a Lorenz gauge 4 (i.e., the special theory of relativity), and as hµv is introduced as a correction to the special theory components, T/µv• such a derivation must be at the price of group algebraic inconsistency. Furthermore, if, as is claimed by many (commencing with Heaviside, 1893) there is a formal analogy between the gravitational potential, h, and the vector potential, A, and the field V x h and the magnetic field B, that analogy can nonetheless be introduced into the electromagnetic field in ways other than by the amended constitutive relations, Eq.s (15) and (16). The next section addresses this other way. 1.4. The Gauge Field Explanation

Having found the physical optical and the dielectric metaphorical explanations of the Sagnac effect wanting, we now introduce a gauge theory5 explanation, but before doing so we examine Forder's ( 1984) analysis of ring gyroscopes. A presupposition of gauge theory is constant action and Forder ( 1984) showed that the adiabatic invariance of the action implies the invariance of the flux enclosed by the contour, which is Lenz's law. There is a common framework for treatment of ( 1) the ring laser gyro, (2) conductors, and (3) superconductors. The treatment of ( 1) is according to the adiabatic invariance of the quantum phase:

'1.q,= 0 = Lllk + m'1.T,

(16)

where L is the length of the contour; and the treatments of (2) and (3) is according to the adiabatic invariance of the magnetic flux:

'1.(f)= 0 = L0 '1.i+ E '1.T, q

(17)

where 4, is the inductance of the contour. Forder's thesis implies: 1. treatment of the action as an adiabatic invariant when the gyro is subject to a slow angular acceleration; 2. generalization of the action integral to a non-inertial frame of reference which requires the general theory of relativity; 3. defining particles on the contour (of a platform), which provide angular momentum and the action of their motion in the rotating frame, that angular momentum and action involving not only (A) the particles' (linear) momentum, but also (B) a term proportional to the particles' energy.

4 L. Lorenz ( 1829-1891) of Copenhagen, not H.A. Lorentz ( 1853-1928) of Leiden. 5Toe concept of gauge was originally introduced by Weyl (1918a, 1929) to describe a local scale or metric invariance for a global theory (the general theory of relativity). This use was discontinued. Later, it was applied to describe a local phase invariance in quantum theory.

287 Forder's claim is that (B) distinguishes rotation sensitive from rotation-insensitive devices. We quickly outline the main points of this claim. With the intrinsic angular momentum defined: (18)

(and the action= integral over one complete cycle), Forder's model is one of particles at the periphery of the frame at a distance l. If Hamilton's principal function is S(t, l), the energy is E and momentum is p for each particle, then (19)

and the transit time around the contour is:

T= di_

(20)

dE

Free propagation is assumed for both particles and light propagating between perfectly reflecting surfaces arranged around the contour. Then

S(t,l) =pl-Et,

(21)

and I= pL,

(22)

f

where L = dl is the length of the contour. The period of motion is then: (23)

where v = dE is the velocity of the particle. dp If the contour and observer are accelerated to an angular velocity changes to:

n, the action (24)

( i = 1,2, 3) and where the Pi components are the spatial parts of the four-momentum Pµ

asA

= --µ

ax ·

Forder's argument is that:

(25)

288 (26)

i.e., the action is an adiabatic invariant under the acceleration, and that:

(27) where ll.T is a synchronization discrepancy between clocks at different points in a rotating frame. The action for a ring laser gyro is then simply: (28)

where A dx 0 T =f-=--A C

di

dE

=

LdpA A

dE

L +ll.T=A+ll.T

(29)

V

is the transit time around the contour, and

A dEA V

=-A-

dp

(30)

is the proper velocity of the particle. Forder then claimed that although the proper velocity is c for all observers, in a rotating frame a photon takes a different length of time to traverse the contour. However, it is difficult to understand how this can be claimed. If the clockwise and counterclockwise beams in the Sagnac interferometer traverse different paths, then one might state that, under platform rotation, the physical distance changed to compensate for changes in ll.T, the time taken to traverse the length of the rotating interferometer back to point H (see Fig. 1). But this is not the case. It is the same interferometric path for both beams. So how it can shorten for one beam and lengthen for another is mysterious. Leaving that aside: depending on the direction of propagation, increased or decreased by ll.T defined:

the period is

(31)

where S is the contour area, and a link was established by Forder between the Sagnac effect and action constancy - a constancy which underlies gauge theory to which we now tum. In the case of conventional electromagnetism phase is arbitrary (there is gauge invariance) and fields (of force) are described completely by the electromagnetic field tensor, f µv· This is the case when the theoretical model addresses only local effects. In the case of parallel transport, however, the phase of a wave function v,representing a particle of charge e at point x is parallel to the phase at another point x + dxµ if the local values differ by:

289 A

D

A' B' AB

SU(2) ==S 3

B

SU(2)/Z2 ==S0(3) B

A

E

A'B' AB

SU(2) ==S 3

C

B'

A' B

A

SU(2)/Z2 = S0(3)

SU(2)==S3

Fig. 2. Generation of a Topological Obstruction (source of magnetic flux) in the Sagnac interferometer. A. parallel transport along two or more different paths; B. parallel transport along two different paths with a path reversal; C. parallel transport along the same path; D. parallel transport along the same path with a path reversal around an obstruction; E. parallel transport along the same path with a path reversal around an obstruction and with twist. Paths taken by two counterpropagating beams are separated in C, D and E for purposes of exposition. The counterpropagating beams are superposed in reality with A=B' and B=A' (C) and A=B=A '=B' in (D) and in (E). SU(2) is a 3-sphere 3 in 4 space; Z2 are the integers modulo 2;

s

g.

SU(2)/Z2 = S0(3) is a 3-sphere in 4-space with identity of pairs of opposite signs, e.g., _j+aj = The twist in E corresponds to a ''patch" condition and in the Sagnac interferometer is caused by the presence of angular velocity(= linear acceleration) as shown in E.

290

C.

A. Q

D.

B. Q

Fig. 3. Generation of a Topological Obstruction (source of magnetic flux) in the Sagnac interferometer. A. two beam parallel transport along two different paths from P to Q around an obstruction with progress in the same direction along the paths; B. two beam parallel transport along two different paths around an obstruction with a path reversal; C. two beam parallel transport along one path around an obstruction with P=-Q and Q'=-P; D. two beam parallel transport along one path around an obstruction with P=-Q and Q'=-P and with a twist. Paths taken by two counterpropagating beams are separated in C and D for purposes of exposition. The twist in D corresponds to a "patch" condition and in the Sagnac interferometer is caused by angular velocity(= linear acceleration). The counterpropagating beams are superposed in reality. Fig. JC corresponds to Fig. 2D; and Fig. JD to Fig. 2E. A. DIRECT PRODUCT OF COUN1ERPROP1'GA BEAMS

TING

Fig. 4. Generation of a Topological Obstruction (source of magnetic flux) in the Sagnac interferometer. Commencing with a direct product of two counterpropagating beams, on s3, A,the sphere s3 is cut in B, given a twist in C and joined in D. The twist and join in D corresponds to a "patch" condition and in the Sagnac interferometer is caused by angular velocity(= linear acceleration). Fig. 4C co"esponds to Fig.s 2D and JC; and Fig. 4D to Fig.s 2E and JD. Adapted from Hong-Mo & Tsun, 1993.

291 (32)

where µ = 0, 1,2, 3 and aµ ( x) represents a set of functions. A gauge transformation at x with a phase change ea(x) is written as: 1/f(x) ➔ 1/f'(x) = exp[iea(x)]'lf(x),

at x:

(33A)

but at x + dxµ it is written as: (33B) 1/f(x + dxµ)

➔

1/f'(x + dxµ) = exp[ie{ a( x) + dµa( x)dxµ}

]'If(x).

So in the case of a phase change with phase parallelism (Hong-Mo & Tsun, 1993): (34) where dµ signifies d I dxµ. Fig.s 2, 3 and 4 are examples of representations defining topologically parallel transport for two counterpropagating beams. Referring to Fig. 2, along either of the paths, a change in phase is given by: (35) The difference between the phases at Q along two distinct paths is: Q e

Jaµ(x)dxµ

Q -e

Jaµ(x)dxµ

=e

faµ(x)dxµ

(36)

r, Use of Stokes' theorem obtains the surface integral: (37)

over any surface I: bounded by the closed curve

r 2 - r 1 with (38)

If f µv is nonzero, then parallel-transport of phases is path-dependent. But f µv is gauge invariant (because as a difference, it is independent of any phase rotations at that point). Therefore f µv has the defining characteristics of the electromagnetic field tensor and the a; ( x) of gauge potentials. This is as far as conventional Maxwell theory takes us, leaving

the situation of a rotated platfonn shown in Fig.s 2, 3 and 4 underdescribed. To progress further, requires Yang-Mills theory (Yang and Mills, 1954a,b) to which we now tum.

292 Yang-Mills theory is a generalization of electromagnetism in which VI, a wave function with two components, e.g., VI= VI'( x), i = J,2, is the focus of interest, instead of a complex wave function of a charged particle as in conventional Maxwell theory. In the case of the Sagnac effect, we assign i = ( 1) to clockwise, and i = (2) to counterclockwise propagating beams, but the clockwise and counterclockwise propagating beams in the interferometric situation we are considering can only be distinguished after parallel transport following a patch condition, which is a condition initiated when the Sagnac interferometer platform undergoes an angular rotation. A change in phase then means a change in the orientation in "internal space" under the transformation: VI ➔ SVI, where S is a unitary matrix with a unitary determinant. With the gauge potential now defined as a matrix, Aµ(x), with g as a generalized charge, and with parallel transport along separate arms of the interferometer, we have : beam traveling clockwise and platform rotated (at x ): V1(x) ➔ Vl'(x) = S(x)V1(x),

(39A)

beam traveling counterclockwise and platform rotated (at x + dxµ): 'If(x) ➔ 'If'( x)

= S( x + dxµ )'If( x),

(39B)

or more explicitly: beam traveling clockwise and platform rotated (at x ): Vlp( x)

➔rrlorh·u•

VIQ(x) = exp[igAv(

X

+ dxµ )dx v]exp[igAµ ( x)dxµ] VIp( x),

(40A)

beam traveling counterclockwise and platform rotated (at x + dxµ): VIp( x)

➔r,muu 0. So even the inertia of a body can be seen as a sixth order electromagnetic effect. In order to get the correct precession of the perihelion of the planets using Eqs. (43) and (44) we need to impose a more restrictive constraint. The calculation is a standard one, [34], and the correct result (Lcp 61rGM/(c 2 a(l - t: 2 )), can be obtained through Eq. (43) with r / /3= - 7/3. This is a relevant result as it helps to fix the values of /3and rIn conclusion we can say that with this preliminary model we tried to show the possibility of deriving gravitation from electromagnetism, even with the correct orders of magnitude. This is a constructive model in which we utilized only electromagnetism to show that a neutral dipole can exert an attractive force on another neutral dipole. We restricted ourselves to the sixth order but it is reasonable to suspect that there will be present other effects at the eight and higher orders. The study of these effects requires a much larger analysis which is outside the scope of this work. We hope this model can cast some light on the unification of the forces of nature.

=

=

=

Acknowledgements The author wishes to thank CNPq, FAPESP and FAEP-UNICAMP (Brazil) for financial support during the past few years.

329

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Kluwer Academic Publishers,

[3] P. Heering. On Coulomb's inverse square law. American Journal of Physics, 60:988-994, 1992. [4] P. Graneau. Ampere-Neumann Electrodynamics Palm Harbor, 2nd edition, 1994.

of Metals. Hadronic Press,

[5] P. Graneau and P. N. Graneau. Newton Versus Einstein Interacts with Matter. Carlton Press, New York, 1993.

How Matter

[6] W. Weber. On the measurement of electro-dynamic forces. In R. Taylor, editor, Scientific Memoirs, Vol. 5, pages 489-529, New York, 1966. Johnson Reprint Corporation. [7] W. Weber. On the measurement of electric resistance according to an absolute standard. Philosophical Magazine, 22:226-240 and 261-269, 1861. [8] W. Weber. Electrodynamic measurements - Sixth memoir, relating specially to the principle of the conservation of energy. Philosophical Magazine, 43:1-20 and 119-149, 1872. [9] W. Weber. Wilhelm Weber's Werke, W. Voigt, (ed.), volume 1, Akustik, Mechanik, Optik und Wiirmelehre. Springer, Berlin, 1892. [10] W. Weber. Wilhelm Weber's Werke, E. Riecke (ed.), volume 2, Magnetismus. Springer, Berlin, 1892. [11] W. Weber. Wilhelm Weber's Werke, H. Weber (ed.), volume 3, Galvanismus und Elektrodynamik, first part. Springer, Berlin, 1893. [12] W. Weber. Wilhelm Weber's Werke, H. Weber, (ed.), volume 4, Galvanismus und Elektrodynamik, second part. Springer, Berlin, 1894. [13] W. Weber and E. H. Weber. Wilhelm Weber's Werke, E. Riecke (ed.), volume 5, Wellenlehre auf Experimente gegriindet oder iiber die Wellen tropfbarer Fliissigkeiten

mit Anwendung

auf die Schall- und Lichtwellen.

Springer, Berlin, 1893. [14] W. Weber. Wilhelm Weber's Werke, F. Merkel and 0. Fischer (eds.), volume 6, Mechanik der Menschlichen Gehwerkzeuge. Springer, Berlin, 1 ~QA

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[15] F. Kirchner. Determination of the velocity of light from electromagnetic measurements according to W. Weber and R. Kohlrausch. American Journal of Physics, 25:623-629, 1957. [16] L. Rosenfeld. The velocity of light and the evolution of electrodynamics. Nuovo Cimento, Supplement to Vol. 4:1630-1669, 1957.

II

[17] L. Rosenfeld. Kirchhoff, Gustav Robert. In C. C. Gillispie, editor, Dictionary of Scientific Biography, Vol. 7, pages 379-383, New York, 1973. Scribner. [18] C. Jungnickel and R. McCormmach. Intellectual Mastery of Nature - Theoretical Physics from Ohm to Einstein, volume 1-2. University of Chicago Press, Chicago, 1986. [19] G. Kirchhoff. On a deduction of Ohm's law in connexion with the theory of electrostatics. Philosophical Magazine, 37:463-468, 1850. [20] G. Kirchhoff. On the motion of electricity in wires. Philosophical Magazine, 13:393-412, 1857. [21] P. Graneau and A. K. T. Assis. Kirchhoff on the motion of electricity in conductors. Apeiron, 19:19-25, 1994. [22] H. von Helmholtz. On the conservation of force; a physical memoir. In J. Tyndall and W. Francis, editors, Scientific Memoirs, Vol. 7, pages 114162, New York, 1966. Johnson Reprint Corporation. [23] J. C. Maxwell. On Faraday's lines of force. In W. D. Niven, editor, The Scientific Papers of James Clerk Maxwell, pages 155-229 (Vol. 1), New York, 1965. Dover. Article originally published in 1855. [24] J. C. Maxwell. A dynamical theory of the electromagnetic field. In W. D. Niven, editor, The Scientific Papers of James Clerk Maxwell, pages 526-597 (Vol. 1), New York, 1965. Dover. Article originally published in 1864. [25] J. P. Wesley. Weber electrodynamics extended to include radiation. ulations in Science and Technology, 10:47-61, 1987.

Spec-

[26] J. P. Wesley. Weber electrodynamics, Part I. General theory, steady current effects. Foundations of Physics Letters, 3:443-469, 1990. [27] J. P. Wesley. Selected Topics in Advanced Fundamental Wesley Publisher, Blumberg, 1991.

Physics. Benjamin

[28] H. von Helmholtz. On the theory of electrodynamics. Philosophical zine, 44:530-537, 1872.

Maga-

331

[29] J. C. Maxwell. A Treatise on Electricity Dover, New York, 1954. [30] T. E. Phipps Jr. Toward modernization Essays, 3:414-420, 1990.

and Magnetism,

volume 1-2.

of Weber's force law. Physics

[31] T. E. Phipps Jr. Weber-type laws of action-at-a-distance Apeiron, 8:8-14, 1990.

in modern physics.

[32] T. E. Phipps Jr. Derivation of a modernized Weber force law. Physics Essays, 5:425-428, 1992. [33] A. K. T. Assis and J. J. Caluzi. Letters A, 160:25-30, 1991.

A limitation of Weber's law. Physics

[34] A. K. T. Assis. On Mach's principle. 2:301-318, 1989.

Foundations

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Letters,

[35] A. K. T. Assis. On the absorption of gravity. Apeiron, 13:3-11, 1992. [36] A. K. T. Assis. Deriving gravitation from electromagnetism. Journal of Physics, 70:330-340, 1992.

Canadian

[37] A. K. T. Assis. Compliance of a Weber's force law for gravitation with Mach's principle. In P. N. Kropotkin et al., editor, Space and Time Problems in Modern Natural Science, Part II, pages 263-270, St.-Petersburg, 1993. Tomsk Scientific Center of the Russian Academy of Sciences. Series: "The Universe Investigation Problems," Issue 16. [38] A. K. T. Assis. A steady-state cosmology. In H. C. Arp, C. R. Keys, and K. Rudnicki, editors, Progress in New Cosmologies: Beyond the Big Bang, pages 153-167, New York, 1993. Plenum Press.

Hertzian Invariant Forms of Electromagnetism Thomas E. Phipps, Jr. 908 South Busey Avenue Urbana, Illinois 6180 I

Abstract. Recent experimental evidence against the Lorentz force law and for the original Ampere law of current-on-current force action is cited. If valid, this motivates reexamination of Wilhelm Weber's electrodynamics, which was designed to accord with Ampere's law, with instant action-at-a-distance, and with Newton's third law. Discussion aimed at better understanding of the latter is given. Unorthodox field theories, termed Hertzian and neo-Hertzian, offer an alternative route to compatibility with the new observations and with Newton's third law. These lack compatibility with spacetime symmetry, but constitute invariant covering theories of Maxwell's equations, so they reproduce the Maxwellian "physics of one laboratory." Hertzian theory provides manifest Galilean invariance and thus expresses a relativity principle at first order in velocity. Neo-Hertzian theory employs Einstein's proper-time invariant, interpreted as being the time shown by a clock comoving with the field detector or absorber, as a plausible method of achieving higher-order invariance. The free-space neo-Hertzian wave equation is solved, physical interpretation is discussed, and an application is made to the description of stellar aberration. This example is chosen because it is one that poses unrecognized difficulties for special relativity theory and that cannot be treated at all by Hertzian first-order theory.

1. Two Systems of the World Today, much as in the time of Galileo, there exist two grand over-arching scientific systems of the world, at swords' points with each other -- that is, two theoretical descriptive structures, both couched in terms of universals, both mutually at odds as to basic conceptions and styles of thought, and neither recognizably falsified by empiricism. And, just as in Galileo's time, only one of these systems finds advocates in academia.

In essence the two forms of description are the field (progressive contact action) mode and the action-at-a-distance mode. They are typified, respectively, by (I) universal covariance, (2) Newton's third law. These latter precepts make the most sweeping possible mutually contradictory assertions about nature or human attempts to describe it. The first claims that all laws of nature, when referred to inertial systems, can be expressed in Lorentz covariant four-vector form. (We do not discuss general covariance, but restrict attention here to the "special" variety.) The second is addressed to all forces in nature and claims that every force discoverable possesses or engenders its equal, opposite, collinear, and simultaneous reaction force. The first, incorporated in Einstein's special relativity theory, denies the premises of the second, such as the concept of distant "simultaneity." The second, incorporated in Mach's principle, denies the premises of the first, such as universal action-retardation.

332

333 In the physics of the late twentieth century a general impression exists that "action-at-a-distance" is played-out -- despite overwhelming evidence for quantum nonlocality of action at arbitrary distances -- and Newton's third law discredited. Research interest (for which read government funding) is indeed played-out, but this is a matter of fadism, not physics. No empirical evidence has ever been adduced against Newton's third law -- which was once so firmly established that patent offices worldwide used its violation-in-principle as sole justification for rejecting inventions of perpetual motion machines. [In consequence of special relativity's having "discredited" Newton's third law, presumably such patents are granted without challenge nowadays ... ?] Nor has any empirical failure of universal covariance ever been acknowledged ... although we shall have occasion to raise doubts on that subject.

The two world-schemes come directly into irreconcilable conflict wherever the physicist's natural laws address natural forces. In electromagnetism, which serves as a fair testing ground, universal covariance selects the Lorentz (or Grassmann, Biot-Savart, Laplace, etc.) law as the only admissible candidate. Yet at the time Maxwell wrote his treatise [I] the choice among numerous experimentally supported candidate laws describing the force exerted by one current element on another was entirely open. Maxwell in fact stated his own preference for the original force law proposed by Ampere, which had been contrived to obey Newton's third law. It this respect Ampere's law was unique and retains its unique position to this day. Ampere's law and Newton's third law are so closely bound-up that any test of one must test the other. Of course Ampere's law did not honor covariance, spacetime symmetry, or retarded action-at-a-distance. Any allegation of instant action-at-a-distance has been viewed by Einstein and his followers as "spooky." That seems to be the prevailing judgment of modem academia. Yet the same savants, when wearing their quantum mechanical hats, acknowledge -- with reference to "quantum nonlocality" -- that there are things in heaven and earth not known in Einstein's philosophy. What they do not acknowledge is that these are precisely the things that were known in Newton's philosophy -- viz., in that part of it built upon the feigning of no hypotheses. Incidentally, as a matter of history, when Newton embarked upon theology -- thereby feigning hypotheses -- the latter led him to express total repugnance to his own instant action-at-a-distance formulation of nature's laws. But let us not dwell on one of the more depressing chapters in the history of scientific schizophrenia. 2. Observational Evidence Among empiricists (nowadays generally languishing outside the groves of higher learning, wherein universal covariance represents not a scientific hypothesis but an assimilated truth) renewed interest in Ampere's law has lately been stimulated by evermounting experimental evidence in its favor, patiently amassed by investigators such as Graneau [2], Saumont [3], Pappas [4], etc. Other chapters in this book discuss that evidence in greater detail. The experiments, taken collectively, become steadily more difficult to ignore. Of particular note is a 1994 experiment conducted at Oxford by Neal Graneau, which seems crucial against the Lorentz force law. The apparent defeat of that

334 law creates a vacuum at the level of fundamental presuppositions in physics into which something must fly. This circumstance could conceivably open a crack in previously closed minds through which Newton's third law (as a pragmatism presumably hiding quantum-level mechanisms of truly universal purview) might enter and ultimately regain its eminence. In short we surmise that Newton's third law may recover its credit among physicists not despite but because of its spookiness. To summarize the experiment of Neal Graneau: it was a slightly but crucially altered repetition of the Robson-Sethian (R&S) experiment [5]. Those investigators had the ingenious idea of settling the 150-year disagreement between Ampere and Grassmann-Lorentz by using a high-voltage circuit wherein a straight, mobile conductor, termed the "armature," was electrically coupled to, but mechanically decoupled from, a pulsed high-current source by means of arc gaps, upper and lower. The armature was suspended in such a way that Ampere longitudinal forces, produced by current in the external partial circuit, if existent, would cause it (the armature) to jump up in a clearly visible way. In order to prevent force cancellation, R&S were careful to arrange a vertical shape asymmetry of their external partial circuit ... but their arc gaps were symmetrical, i.e., of equal width at top and bottom. (Cylindrical symmetry about the vertical armature canceled any horizontal impulses.) They reported a null result [5] -- no motion of the armature -- thus apparently establishing the nonexistence of Ampere forces as distinct from Lorentz forces. A subsequent inquiry [6] into the theory of the R&S experiment showed that in any experiment using a straight, mobile force-sensing element the longitudinal motion of that element could not be affected (according to a wide variety of force laws, including all the likely candidates such as those of Lorentz, Ampere, Gauss-Riemann, etc.) by the shape of the external partial circuit, but only by the geometry of the arc gaps. This "shape-independence theorem" was a purely theoretical result, logically demonstrated. Thus the care taken by R&S to ensure external partial circuit shape asymmetry was misdirected. To render the experiment crucial by preventing force cancellation it was necessary instead to provide gap asymmetry. According to the Lorentz force law, neither gap nor shape asymmetry could cause the armature to jump up against gravity, because that law prescribes a cross-product force action strictly transverse to current flow; i.e., transverse to the vertical armature and thus restricted to the horizontal plane.

In Neal Graneau' s variant of the R&S experiment the lower arc gap was made smaller than the upper one. In such geometry Ampere's law predicts a net upward component of longitudinal force on the armature, because of stronger repulsive "Ampere tension" at the narrower gap. When the current flowed briefly in the arcs, the armature was observed to jump up by about the amount of impulse predicted by the Ampere law. This outcome appears inexplicable by the Lorentz force law. That law's adherents must attribute it to apparatus effects and, as far as its agreement with Ampere is concerned, to coincidence. When the experiment has been reported in the literature, it will merit the close attention of any physicists willing to consider departures from the status quo.

335

3. Weber's Electrodynamics

Ampere's law (see Chapters 22 and 23) is somewhat complicated. For all its empirical agreements, it lacks the elemental formal simplicity one would like to see in basic physical propositions. Its complexity probably reflects that of its structural unit, the "current element." Such considerations may have induced Wilhelm Weber ( 1804-18 91) to seek a more fundamental formulation in terms of actions-at-a-distance between pairs of point charges. By introducing a new concept into physics -- that of the velocity-dependent potential -- he was able to discover a truly "relativistic" force law between point charges that depended only on the instantaneous value of their separation r and on the time derivatives r, r of that separation, such that this force law reproduced Ampere's law for ponderomotive actions between neutral current-carrying conductors, and such that Newton's third law was obeyed between the point particles constituting the currents. Moreover, he was able to derive his force law,

(1-~2 + rr) 2 '

F = q1q2 r2

2c

c

by differentiation, -dV/dr, of an extremely simple potential energy function,

(1- 2c;-22) '

V = q1q2 r

wherein q 1_ q2 denote electric charges and c is a units ratio found to be numerically equal to the speed of light. [Graneau [2] remarks that, "Weber attributed no particular importance to c. Today it appears truly astonishing that the velocity of light should have revealed itself in a simultaneous action-at-a-distance theory such as Weber's."] The Weber derivation of Ampere's law depended on a particular model of "current" as equal counterflows of plus and minus charge. Thjs betokened the primitive knowledge in his day of the physical nature of electric current. But recent independent investigations by Wesley and by Assis [7] have shown the equivalence of Weber's and Ampere's laws to be a model-insensitive feature, so that it holds for a more realistic model of current as negative electron drift past fixed positive lattice ions. The new book by Assis [7] may be considered essential reading for anyone who wishes to pursue Weber's theory in depth. We shall not attempt this here, but merely remark that every evidence supports a view of the original Weber theory as valid only through order O(c- 2 ), and that to avoid paradoxical (if not pathological) behaviors it will be necessary to modify his formulation at higher orders. One possibility [8] is to substitute for the above potential energy a modified form V= q,q, r

✓,_ ;-' C

2

'

which has the advantage of enforcing a "speed limit" on relative particle velocity. However, it is at once apparent that any attempt to modernize Weber's approach must encounter difficulty in the description of radiation, which is known to propagate retardedly at speed c ( or at least to behave as if it did).

336 Is it out of the question to describe radiative actions by means of some kind of instant-action force law? This is one of those research topics that must be left as a challenge to the ingenuity of future mathematicians. One can say at present only that radiation obviously cannot occur without radiation reaction (vide Newton's third law!), and radiation reaction is traditionally described in terms of r' ... so it would appear that any force law suited to treat radiative action-reactions must include higher-order time derivatives of r than does Weber's original law or its above-suggested modification. Much -indeed, practically everything -- remains to be learned in this subject area. Field theory, as it relates to force action-reaction, can rather obviously be replaced by a Weber-type instant-action "true relativity" theory (which avoids reference to frames and uses particle-relative coordinates, just as Weber did), with little or no loss of empirical agreement. But to replace the radiation-descriptive side of field theory by such an alternative will require new understandings and inspirations. A clue may be found in the observations by Gray [9] (a) that Kirchoff was able to derive transmission-line equations, predicting retarded propagation, using only Weber's theory and (b) that the vacuum can be modeled as a transmission line. This appears to establish the essential principle that speed-c retardation can emerge from instant-action mathematics. The fact that this seems to lie beyond intuitive understanding may reflect some inherent difficulty or, more likely, shows the penalty to science of a century's neglect of important descriptive alternatives. A lover of science can only deplore the manifestly bad strategy that dictates sustained neglect of one of the "two methods" to which Maxwell referred in the Preface to the First Edition of his treatise [1]: "In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have succeeded in explaining the principal electromagnetic phenomena ... while at the same time the fundamental conceptions of what actually takes place ... are radically different." It is the radical difference of research paths in the plural that provides science's only protection or insurance against systematic radical departures of a chosen path into idiocy or madness. The democracy of science -- mutual admiration, the rule of consensus, etc. -has shown itself no protection at all. History did not -- or at least has not yet seen fit to -- validate the physical path (an hypothesized ether) chosen by Maxwell himself. So, at the end of the twentieth century, physicists find themselves with neither physical nor mathematical ammunition to shoot at their manifold problems, except the paradigm of mathematical field theory with which Maxwell and Einstein, the master hypothesizers, provided them. Now it is four-vectors that fly straight out of our minds, into our published papers, and on through the air to deliver physical effects of the sort that Dr. Johnson stubbed his toe on. One surmises that Faraday (whom Maxwell mathematized and attempted to apotheosize, even as his intellectual heirs have apotheosized Maxwell) would not be happy with all that. Might not simple prudence, the "common sense" of the layman, counsel avoidance of a lemmingrush into one method or the other -- given the track records of both? Killing off deviant or unfit thinking is great Darwinism and first-class political strategy. It pays the air fare

337 to the AAAS meeting, shoes the baby, and pleases the government committee. But as a strategy for the advancement of human science it carries the dominant gene of death.

4. Force and Inertia Electromagnetic fields are defined in terms of forces -- e.g., force on a unit charge. (Curiously, though, knowledge of the fields provides no information on the force they exert on a moving test charge ... that being the province of a separate "force law," usually taken to be that of Lorentz.) Force originates as a strictly mechanical concept linked to mass. Yet all (nongravitational)force laws, in particular electromagnetic ones, make no explicit reference to mass and seem to suggest that force in all its physical effects is granted an uncontested divorce from mass. Very elementary considerations based on Newton's second law, however, make it clear that in all physical situations there is a hidden connection between force and inertial mass (besides the equation of motion, F = ma) such that the divorce is never absolute. To grasp this, it is necessary to recognize a distinction between (a) "physical force," which by integration with respect to distance is what yields change of kinetic energy, and (b) "formula force," which is what one finds in textbooks (or what is given by "force laws"). The relationship between the two is that physical force is always less than or equal to formula force. This simple fact, which ought to be known to freshmen, is demonstrated as follows: Let two point masses M and m occupy positions x M, xm on a horizontal x-axis in an inertial "laboratory" frame. They attract or repel each other with a "formula force" or force law F,aw(r), where r = lxM - xml, due to electrostatics or any other physical cause. We confine attention to low speeds, for which reliance can be placed on Newton's second law, which states that d -(MV) dt

d

= --(mv)

= F,aw,

dt

where V = Ix MI,etc. Note that it is the formula force that appears in the equation of motion. Both bodies are initially held at rest in the laboratory. We consider two cases:

Case A. Mass m is held fixed while mass M is released and allowed to move freely along the x-axis. Since xm = const, we have

u=~r=lxM-Ol=lxMl=V. dt

For constant mass M the equation of motion yields d

-(MV)

dt

d

dr d

dt

dt dr

= M-V = M--V

d

=MV-V

dr

which integrates to I -MV 2

2

=(KE)M=

JF,aw(r)dr.

=F,aw(r),

338

In this case in which the force exerter

m is held fixed, which is the one usually consid-

ered, the full formula force F,aw(r) is effective in imparting kinetic energy (KE) to the test mass M. This is exactly what one would expect. Case B. The masses are released simultaneously and are both free to move along the x-axis after release. Before release the total momentum of the bodies is zero. After release it is also zero: MiM +mxm = O or xm =-(MI m)xM. As before we have V = and find that

IxMl

or

V=(M+m m )u. Thus !!_(MV) dt

= M!!_V = Aj dt

m

lM +m

iv-'

)!!_u dt

= µ!!_u, dt

where µ is reduced mass. The equation of motion then yields d -(MV) dt

d

= µ-u

dt

dr d

d

= µ--u

dt dr

= µu-u

dr

= F,aw(r) ,

which integrates to I -µu 2

2

= JF,aw(r)dr .

It follows that _!_ µu2 2

= _!_( Mm )( M + 2 M+m

m

m)2vi = _!_MV2( M + m) 2

m

=(KE)M( M;m)= ffi-(r)dr, This can alternatively be written in terms of a physically effective "reduced force," (KE)

M

= JF,ed(r)dr

,

where Fred(r) = (

m )F,aw(r)•

M+m

Note that Case A is the special limiting situation, Fred= F,aw, of Case B in which m ➔ oo; that is, the force exerter m has infinite inertial (not gravitational) mass -- which is equivalent to supposing the mass m to be "anchored" in the initial rest inertial system, so that (for whatever physical reason) it cannot move with respect to that system. This is another way of saying that the anchored m can sustain the full reaction force of its action on M. We see that in general for m < oo we have Fred< F,aw. This means that any force exerter of relatively low mass, imperfect "anchoring," or poor "footing," cannot impart as much kinetic energy to the object on which it acts as the force formula (or force law employed in the equation of motion) would lead us to believe. This shortfall of motional energy is quantified by the "force reduction factor" [ml(M+m)], where these masses are inertial ones associated with the force exerter (m) and exertee (M).

339 Another way to put this proposition is that in stating or defining force laws it is universal practice to make the tacit assumption that the force exerter whose action is quantified by F,awdoes not recoil under reaction force. But of course it does recoil physically -- that is made explicit by Newton's third law -- so "physical force," here quantified by Fred, is always less than or equal to "formula force," F,aw. We thus acquire a new insight into the third law: Its basic meaning is that in the real world of observable consequences it is impossible to exert more action than the equivalent of what is sustainable as reaction; that is: Precept: Observable action is limited by sustainable reaction. This precept has been developed and supported here solely through theoretical arguments. Is there any empirical evidence for it? Yes, there is ... but let us return to that topic after first considering a theoretical counter-argument. It has been suggested [6] that for a very short time after the masses are released no measurable motion of either of them will occur, so during this interval there is no operational meaning to the "force reduction" claimed here. That is, there is a least-count displacement of the masses that is physically detectable, and for interaction times less than required to produce such a displacement it may be that the full formula force should be used. Of course one could with equal plausibility employ a parallel argument to assert the opposite conclusion, but it is more satisfactory, since this is a matter of physics, to appeal to empirical facts. Consider the Neal Graneau experiment already mentioned. The factual outcome was that the armature jumped up a distance concurrent with the impulse (time integral of force) predicted by the Ampere force law, supposed to act from the instant of initiation of current flow in the circuit, but omitting arc-gap currents and treating forces produced by currents within the external conductive part of the circuit at the full force-formula value. If the full formula force had acted within the arc gaps, there could have been no analytic distinction between solid conductors and gaps, so that the gaps might as well have been filled with metal. Had the gaps been filled with metallic conductor, they would have been of zero width at both top and bottom. This would have corresponded to the case of symmetrical gaps -- which by a corollary of the above-mentioned shape-independence theorem [6] would have resulted in force cancellation and no motion of the armature, in contradiction of the observed facts. So, the full formula force cannot have acted in the arc gaps. Instead, the force that acted physically may be presumed to be Fred. The force reduction factor in the arc gaps was [ml(M+m)], where the force exerter mass m was the mass of a stream of ions in the arc plasma (a very small fraction of a gram), and the force exertee (armature) mass M was several grams, so that m 0 are real, /3 is the normalized velocity of the solitary solution. Eqs. 29 are then transformed into a system of four real ordinary differential equations

(1 - /3)A+'

KA_ sint:4>

(1 + /3)A_'

KA+sin t4>

[(1- /3)¢+' - Ilk [(1+ /3)¢-' +Ilk+

a(A!

a(A:

+ 2A:)] A+ + 2A!)] A_

KA_ cos t4> -KA+ cos t4>

(33)

422

..z "J:======:::_-_-_-_'V\JVVVV\/VVVV\lntensit~f"--___

t, L'\:~_....;c::1>::a....---------------1

J__

Intensity

Ac$

,L>-r:;..¢::,_;;::,,,,,,,. _____

___,;:i ________

t2> t,

----1

..z .. z

Figure 3: Realization of an in-gap soliton in a finite periodic structure. At t = t1, a properly choosen input pulse is injected into the periodic structure. A part of the input pulse is reflected at t = t 2 , while the other part is transmitted as an in-gap soliton into the periodic structure. The intensities are given in arbitray unit. Notice that the linear counterpart is discussed and illustrated in Ref. (136].

where t4>= + - -The prime denotes the derivative with respect to x. By experimenting with different products of A+, A_, and cos t4>, two first integrals of Eqs. 33 can be found:

(i - P2 )A+A- cos t4>+ ':"(1- P)A!

+~

[(1 - P)(3- P)At

(34)

+ (1 + P)(3 + P)A~] = C2

where C 1 and C 2 are integration constants. These integrals can be used to reduce Eqs. 33 to a single differential equation for A!. All other variables can then be expressed as functions of A+. The following scaling is obeyed by Eqs. 29 and Eqs. 33: The intensities are proportional to lal-1 , whilst the space and time variables scale as K.- 2 . Thus, the solutions depend only on the normalized velocity P, the normalized frequency-detunning de/"-, and the normalized nonlinear refractive index a/ K.. This scaling law is applied in all illustrations of this section.

3.3

In-Gap Solitons The so called "In-gap" solitons can be derived by applying the boundary conditions lim A_(x) = 0

x-+oo

and

lim A_'(x) = 0

x-+oo

(35)

to Eqs. 33. This boundary condition can be approximately realized by injecting a finite periodic structure with an appropiate input pulse, as illustrated in Fig. 3.

423

Eqs. 35 imply C1

= 0 and

C2

= 0, and

Eqs. 33 are then reduced to

[(A!)']2 A_

+ + + - (36) In order to ensure A+ and A_ to be real, one has to assume l,BI < 1. In these equations Arc.sin, Arccos denote the multivalued inverse trigonometrical functions, whilst 0 denotes ( + + -)x=o· The general solutions of Eqs. 36 can be expressed by elliptic functions, which are generally periodic in x. Under suitable initial conditions, the elliptic functions are unimodal and can be given in terms of hyperbolic functions cosh and tanh, so that the following solitary solutions result. Case A: IMI< K (1 - ,82 )1/2 [

...!L2(l±P)~

lol

3-P

cos 2 8 ] sin 8+cosh(2()

2

sgn( o:) [(,8tan 6)( + (3 6

arcsin ~ 1ty1-p2

1tcos8 X

~i ± 1)arctan (i~:~!tanh ()] =f i+stfC 8

2

0

l 1r + t/Jo

(161 < 1!:. 2) 1tcos8

J1-p2

1/2

J1-p2

(z_ ,8.£..t) no

(37) It represents a form-stable pulse travelling with speed ,Bc/n 0 • The pulse form is also determined by an additional parameter 6, which depends on the frequency detuning M. An example of Eqs. 37 is represented in Fig. 2. Case B: t:Jc= -sgn(o:)K(l -,8 2 )1/2 [

...!L 4(l±P)~

lol

3-P 2

__i_] l/2 1+,:2

sgn(o:) [-~( + ( 3

~i ± 1)arctan(] 2

=f l+s~(o),r

+ t/Jo

(38)

no 1-p2

21t z-Pct

Case Bis the limiting case(}-+ -,r /2 of case A. Note that the pulse tails are no more exponential. The main properties of in-gap solitons are: 1. While the instantaneous circular frequency d

Winstant

= - dt arg[E(z,t)]

(39)

424 : -0 4 : ,'. ,,, -0~2 I

,,

,, ,

I

,

I I

,,

u, C

,,

I

,,

-

CD C

"'O L..

C

3

.::I.

u

C

~

forward

intensity

2

Figure 4: Dependence of the peak intensities of in-gap solitons on the detuning parameter tJc and the normalized velocity {J. The normalized forward and backward intensities are lo/,clmax(Ai) respectively. Solid lines are contour lines sgn( o)t:Jc/ ,c const, while dashed lines are lines with fJ const. The boundary of the region permitted for in-gap solitons is also given.

=

=

of in-gap solitons changes with the intensity and can be outside the frequency gap, their circular frequency Wtail of the low-intensity tails d±

Wtail

c sgn( a) sin(} ✓l _ /32

= W - dt = WB + K no

(40)

is generally within the frequency gap as shown in Sec. 2.3. This suggests the notation "in-gap". The reason for the transparency of the nonlinear periodic structure, as illustrated in Fig. 2, which would be 100% reflective for linear waves, was already explained in Sec. 3.1. 2. Apart from an arbitrary common phase 1/Jo and an arbitrary initial location, in-gap solitons Eq. 37 possess two degrees of freedom. The two independent parameters can be either t:,k,and /3,or(} and /3,(}defined in Eqs. 37. The relation between the two peak intensities max(Ai) and the parameters t:,k, and /3, as well as the boundary of the region permitted for in-gap solitons are illustrated in a universal plot Fig. 4, which can also serve as a design plot for realizing experimentally the necessary initial conditions for in-gap solitons. 3. In-gap solitons with relatively low intensity are known to be stable against small perturbations and collisions [106,126,137). In Fig. 5, a white noise n with 0.9 < n < 1.1 is multiplied with the initial wave shape E(z, 0) of the in-gap soliton. The pulse is capable to evolve to an ideal in-gap soliton. In Fig. 6( a-b ), the in-gap solitons restore their initial shapes and velocities after collision.

425

(a) >,

(b) >-.

0.4

C Q)

en C

0.2

Q) -+-'

-+-'

C

2

-+-'

-+-' (/)

1

C

o.o 60

0

so

Figure 5: Stability of in-gap solitons. 10 % white noise was added into the initial wave shape. The spatial coordinate is ,c(z - ct/no), the temporal coordinate ,ed. The intensity corresponds to

lol(Ai +A:).

(a) Low-intensity in-gap soliton ( a = K., M = 0.5K., /3= 0. 7) remains stable; (b) High-intensity in-gap soliton ( a ==-K., M ==0.5K., /3==0. 7) decays.

In-gap solitons with relatively high intensity tend to be unstable against small perturbations and collisions [126,137). Fig. 6(c) illustrates an inelastic collision between two intense in-gap solitons. After the collision, the in-gap "solitons" radiate continuously energy and decay. A similar decay can also be observed at noise-perturbed intense in-gap solitons. Thus, in-gap "solitons" are not solitons in the conventional sense, which was mentioned previously in Sec. 3.1. 4. The in-gap solitons exist for both signs of the coefficient n 2 of the nonlinear refractive index due to its formation mechanism. They are in contrast to those of the familiar nonlinear Schrodinger equation, which base on a balance of groupvelocity dispersion and self-phase modulation. 5. The velocity l,Bc/nol of the in-gap solitons ranges from O to the speed light in the medium.

c/n0

of

6. The energy of an in-gap soliton is apart from a constant factor given by

+Joo

2 IE(z, t)I dz~

-oo

+/oo[ 2 ) 2 ] A+(z, t + A_(z, t) dz= -oo

1 4(1 - ,02) ('Tr ) 41r ,82 2 - 0 < 3101•

~ 3_

(41) There exists therefore a maximum energy, which an in-gap soliton can carry (Fig. 4). The maximum will be asymptotically reached for the conditions ,B= 0, 0 --+ -'Tr /2. An implication of this energy limit is that a pulse amplified in a nonlinear periodic structure will become unstable, after it reaches a certain threshold.

426 (a)

>---Cl.s ........, Cl-4

(fl

C Cl.i Q) ........, C) C) C

(b) a

'--+-' (.[)

C Q)

0.4

-+-'

C

o.o

30

(c)

>,..

........, 4 (fl

3

C 2 (l) ........,1 C 0

.3

Figure 6: Collisions between in-gap so\itons with velocities

/Ji 'f /J2- Tbe

spatial coordinate is

K(z - ct/no), the temporal coordinate Kc!- The intensity corresponds to \o\(A".-+ A~)pararneters are t,k 0.51

o,

',p

~3

~~~~

\

(au 0 the out-gap solutions exist for any far-field intensity. This is also illustrated in Fig. 10. The threshold condition is

(56) which can be deduced from inequality (49). 5. For identical far-field parameters (~, Wtail, u) and normalized velocity (/3), "bistable" or two-state out-gap solutions can exist under suitable conditions (Fig. 9). This is due to the fact that both signs in Eq. 50 can be used, if the resulting I+ is positive. In this case, the smaller solution whose minimal intensity is smaller than the background intensity ~' corresponds to a "dark soliton". The larger solution, i.e. the "bright soliton", whose maximum intensity is larger than the back- ground intensity~' always exists. 6. As in the case of in-gap solitons, out-gap solutions also exist for both signs of the Kerr coefficient n 2 •

432

a)

and

/3min

/3max

(ao>O)

5 ........ ----..--CID---r-~---r----,.-------.----------,

o

0

"'

..,

.,, -

...

-

-

b)

-

{ima•

- - -

t-' min

..

R

---

(ao 0 is a symmetric smooth bell-shape function with f(-x) = f(x), /(0) = 1, and /(±oo) = 0 (see Tab. 2). By inserting the trial functions (60) into Eq. 58, it is possible to deduce a reduced variational problem:

6

j

Cdr

=0

(61)

where

C(r)

J!"; C(z, r) dz 1 1>z 1 1>z 1>+ (t4>< 1 >)cos 20 m,Po [Ilk- ,p(O)' + ,t,< t4>< l!J/,(0)' - ,t,
(' >w)cos( 2t4>(

with the momenta mi=

1-: 00

/

1 (x)

dx,

0

2';;,2 P~

(j

(62)

( 1 + ½sin 2 20)

= 1,2)

(63)

and the normalized real Fourier transform

(64) Examples off,

mi and Fare

given in Tab. 2.

435 Table 2: Examples of pulse shape functions f and their characteristic data defined in Eqs. (63,64). The prime denotes differentiation with respect to k.

f(x) 2)

exp(-2x sech 2x

F(k)

F'(k)

m1

m2

If

-.i. exp(-½k 2) 2

2

!!.k 2

!

2

sinh( fk)

2)

.

!!.k

II"

sinh( f k)

3

-¼exp(-½k l -

t~(

f k)

The reduced variational principle (61) results in a set of coupled ordinary differential equations, which can be partially reduced into algebraic equations due to symmetry properties of C. Thus, the following variational equations are obtained: {JC, fJq,(O)

= Q =>

{JC,= 0

fJz {JC,= 0

fJw {JC, fJty/>(l)

{JC, fJq,(l)

=>

q, = const - t4>cos 20

=>

1>F 1 (2t4>w) cos(2t4>)sin 20 21et4>


= 0 => fJPo {JC,

fJt4>(0)= 0 60

1 - z' cos 20

= 21ewF'(2~}

def

1/)1 - t4>cos 20,

=>

= 0 =>

+ po( 1 + sin 2 20 ) m1Po w 2

=

F'(k)

(67) (68)

=0,

(70)

-1eF(2t4>< 1>w) sin 2t4>,

(71)

t4>(0)'- z' t4>(l) + q,(l) + ~ COS20

= om 2P 0/(2mi).

=

29 ), sin:

(69)

C

0'

+

(66)

1>w) cos(2t4>)sin 20,

+1eF(2t4>w) cos(2~< 0>)cot 20 with p 0

= -,.(1

= Q => z' = cos 20,

{JC,

{JC,

(65)

Po= const,

The prime denotes differentiation d~F(k). Eqs. (67-69) lead to

w ty/>(l)

with

1

sin20 1 21ecos 2~< 0>F'(p 8 ) ' cos 2~(0) IC . peF'(pe) Sln 20• def

2

Pe = Po(l + . 2 20 ) Sln •

=0

(72)

with respect to r, except

(73) (74)

436 a)

b)

I:: ;::-...__

e.

S-

S-

) = ?p1 cos 20 'H(0, t:4>
and generalized momentum 0. The Hamiltosystem with generalized coordinate t:4>< nian system (76) possesses two free parameters. One parameter is p 0 , the normalized power of the pulse, the other is 7Pi/ K, which is related to the phases of the forward and backward components E± of pulse. Once Eqs. 75 are solved, all other variables O) by Eqs. (66, 69, 73, 74) and can be given in terms of 0 and t:4>(

¢'= flk

+K

[

sin 2( 0

O)):F(pe)+ cos 2t:4>(0) + t:4>( sin 20 pe:F'(Pe)]

(77)

which can be deduced from Eq. 70. Examples of phase portraits of Eqs. 75 are represented in Fig. 13. For well-behaved

437

0

.S

space

TO

".s

Figure 14: Propagation of an nearly solitary wave. It is a solution of Eqs. 75 and corresponds to a close path near the center C in Fig. 13(b ). The spatial coordinate is ,cz,the temporal coordinate is 2 + IE-1 2 ). ( "Pl the normalized time T d/no, and the intensity is lo/,cl(IE+l 0.5K, Po 0.89,

=

0(0)

=

= 0.1671",~(O)(0) = 0.571".)

=

functions F, i.e. functions with the properties

F(p) > 0 ,

liIBp-±ooF(p)

9(;!,Po) def J;!-3

= 0,

liIBp-±ooF' (p) = 0 ,

[F(po)-2(po - Po)F'(po)] unimodal for ~>3 and Po= const

m~x [Q(U,po)]def 9max(Po)> 0

po=faxed

'Po

(78) which is the case for Gaussian and hyperbolic secant functions, there are two distinct types of phase portraits: 1. If

ll/J1I/ K < 9max(Po)/./2, there

exist one center and one saddle point, which can be found by setting the left-hand side of Eqs. 75 to zero. The center corresponds to in-gap solitons given in Eqs. 37.

2. Otherwise, no critical point exists and thus no in-gap solitary solution is possible. A further conclusion which may be drawn from the phase portraits of Eqs. 75 is that the width w( T ), the amplitude P0 /w( T ), and the normalized velocity cos 20( T) of a nearly solitary wave, which corresponds to a close path around the center C 0 >), oscillate temporally around those values of an in-gap in the phase space (0, t:4>( soliton, at least for a sufficiently short time. Fig. 14 illustrates an example of the nearly solitary wave. The distance between the center point C and the saddle point S is approximately given by

438

for 9max(Po)~ v'2ll/J1 I/,c. It is a measure for the range where an oscillating in-gap soliton can exist. A curve discussion of 9max(Po)implies that 9max(Po)is a decreasing function of the normalized energy p 0 . The smaller p 0 is, the larger the allowed range l/J1for oscillating solitons will be. Oscillating solitons are thus more likely formed by low-energy pulses.

4

Nonlinear

phenomena

in active periodic structures

4.1

Systems with Constant Gain The previous section has dealt with a passive infinite periodic structure without gain. In this section, the nonlinear coupled-mode equations discussed in the previous section will be extended to incorporate the effect of gain. An active nonlinear periodic structure is adequate to describe a distributed feedback amplifier or laser. Similar to a conventional laser with end mirrors [140-146], an active nonlinear periodic structure presents a variety of complex phenomena, such as multiple stable states, symmetry breaking, self-pulsing and chaos. One possibility to incorporate gain is adding a loss/gain term to the one-dimensional wave equation (26), i.e. i, 8E 1 aP 18E -- 2 - -----= ---2 2 2

8z

c

2

8t

(79)

where i, is the conductivity. The polarization P is defined by Eq. 27 as before. By the same coupled-mode ansatz

(28) and the same approximation, the nonlinear coupled-mode equations with gain [9, p. 7879) [125) can be deduced:

+ !!:a.aE+ = +('Y + it:Jc)E+ + i,cE_ + io: ( IE+ 12 + 2IE- 12 ) E+ at aE_ _ !!a. aE_ _ az at - -(7 + it:Jc)E_- i1eE+- io: (IE- 12 + 2IE+ 12 ) E_ aE+

{

az

C

(80)

C

where 7 = -1, /(2€.onoc) is the gain coefficient for the amplitude and -y< 0 corresponds to loss. The Eqs. 80 are based on the assumption that the response time of the gain medium is shorter than the time scale of electric field E and polarization P. Also, the effect of nonlinear refractive index is assumed to be the most dominant nonlinear process. Only a few exact analytical results concerning solutions of Eqs. 80 are known. One remarkable result are the energy and momentum conservation laws

439

4 (a)

2

1.75

0 8

(b)

1.90

4

0 8

C/)

··(].)

-+-> C/)

~

1.95

{d)

0 10

(].)

·-

{c)

4

-+-> ~

1.93

5

IWVVV\ll/\/

-•~M\Af

,,:VvVV{V\f\/\/\[\f\j\j\f\lV\J\f\l\.J\f\f\t ..f\JV\r~fv1I\.f\f\f \j'-./\J·1v·!

-+-> ~ ~

+,)

0 10

~

0

( e)

5 0 20

{f )

10 0 250 .3.00

( g)

125 0 0

100

200

300

400

500

time =

Figure 15: The normalized intensities laLEil(½) (solid line) and L laLE:.1(-½) (dotted line) at both ends of the active DFB structure a.s a function of time. The given parameters are the corresponding normalized amplification coefficients 1L (KL= 1).

440

In order to solve Eqs. 80, one has generally to resort to approximate or numerical methods. The "solitary" solutions of Eqs. 80 in an infinite medium have been investigated by de Sterke el al. with the equivalent particle method [125]. They conclude that an effective particle picture, based on the evolution of two conserved quantities in the lossless medium, gives a good approximation of the results of detailed numerical simulation, at least at short time scales and in the low-intensity limit. In contrary to the study by de Sterke et al. [125] and to the previous section, the discussion in this section is restricted to finite structures, because a continuously amplified pulse in a periodic medium would tend to break down into multiple pulses (seep. 15), while in a finite structure, among other interesting features, stable pulses can be generated. Furthermore, reflections from both ends of the periodic structure will be neglected for the sake of simplicity, i.e. the boundary conditions (12) will be assumed throughout this section. An overview of the phenomena which occur in such a periodic structure is given by Fig. 15. Shown are the intensities at the ends of an active DFB structure, which are obtained by solving Eqs. 80 numerically. There are two relevant free parameters of the system: the coupling strength KL which is kept constant in Fig. 15, and the normalized gain ,L. As the gain ,L increases, the system passes through a series of different regimes. Details will be discussed in the following sections.

4-2

Steady States

For a gain , L below the lowest threshold (11L ~ 1.755) defined by Eq. 15, a stationary non-zero field cannot build up. Any field distribution E±(z, 0) will decay exponentially to zero, as showed in Fig. 15(a). As the gain , L increases slightly beyond the lowest threshold, a steady state is attained after a transient (Fig. 15(b) ). The absence of a beat pattern in the output intensities indicates that the steady state represents a single-mode state. The existence of a steady state in absence of saturation can be qualitatively understood as follows. At the beginning where the intensity is low, all modes with threshold gain 1 q < 1 grow exponentially. The gains experienced by these modes are larger than the output loss. After the intensity has increased to a sufficient magnitude, the nonlinear refractive effect begins to modify the DFB grating and thus to change the electric field distribution, as well as to reduce the effective gain for the

441

~

0.5 0.4

,IJ

0

G) 0. 3

0i

ID ~

~ 0i

0.2

0.1 0.0

I

0

I

'

I

I

I

I

1

I

I

I

I

I

I

'

I

2

I

3

I

I

I

I

I

4

I

I

I

I

I

5

frequency Figure 16: Transient power spectra of the output intensities IEll for 5 ~ ct/noL ~ 50. The power spectra are given in an arbitrary unit, the unit of the frequency axis is c/2n 0 L. The difference between the power spectra of IE!I and IE:I is invisible. The beat frequency (0.93c/noL) is approximately equal to the frequency separation (0.85c/noL) between the linear q = ±1 modes.

electric field under suitable conditions. As the effective gain decreases to the value of the output loss, a steady state is attained. An interesting fact is the single-mode operation. According to the linear theory which is characterized by Eqs. 15, there are two modes (q = ±1) with the identical threshold gain , 1 L. In the absence of nonlinearity, these two modes would be decoupled and oscillate simultaneously. If the nonlinear gain is taken into account, the semiclassical laser theory [147] predicts that the presence of one mode can inhibit the oscillation of the competing mode in the case of strong coupling [147, p.122-5], which results in a single-mode operation. In fact, the transient part (ct/n 0 L 0.

=

effect of the pulsed gain is simply to lift the power of the pulse from the power P0 at time t = 0- to e2-Yac P0 while leaving the pulse width and its phases unchanged. The dynamics for t > 0 can still be described by the Hamiltonian (76) with

Po(0+)

= e2-r cpo(0-), 0

0(0+) = 0(0- ),

1P1(0+)= 1P1(0-),

~(O)(0+) = ~(O)(0-).

(93)

The solutions of the Hamiltonian system were discussed in Sec. 3.5 and are illustrated in Figs. 13-14. If is properly chosen, a low-intensity pulse can turn into an in-gap soliton, while it causes an in-gap soliton to oscillate. If the energy of the pulse exceeds a certain threshold, the pulse will decay. A numerical solution of Eqs. 86 is illustrated in Fig. 24 for the following situation. At time t = -3n 0 / Kc, one applies a pulsed gain 6.9 S(ct/no + 3/ K) which transforms a soliton-like low- intensity pulse into a perfect in-gap soliton which was described in Sec. 3.3. This soliton has the parameters /3 = 0.67 and M = 0.5K. At t = 3no/Kc, another gain pulse 0.26S(ct/n 0 - 3/K) is applied, which causes the in-gap soliton to oscillate and to decay eventually, as predicted in Sec. 3.5 and illustrated in Fig. 5.

,ac

For a finite structure, the effect of the gain pulse is similar. Two cases can be distinguished. 1. If the fields at t

= 0+ are sufficiently weak, the linear

solutions (24) are valid, at least during the period where the amplitudes are small. According to Eqs. 24,

450

a) £20

b)

40

(/)

C

10

CL>

-~

30

0 Vl

C

Q)

20

C

space

"\

Figure 26: The same as Fig. 25 except for noise. (a) The intensity distribution. (b) The intensity at the boundaries z = L/2. That at z = -L/2 is similar. The spatial coordinate is again z/ L, the temporal coordinate ct/n0 L, the intensity represents 2 + IE- 12 ). Parameters of the periodic structure are 11:L 1, -yL lo/11:l(IE+l 1.90, o > 0.

=

=

the fields decay exponentially to zero, if --y 0 is smaller than any of the linear threshold gains given in Eq. 15. Otherwise, the fields reach non-zero steady states, as depicted in Fig. 15 and discussed in the previous sections Secs. 4.14.5. 2. If the fields at t = O+ are sufficiently strong such that the linear solutions (24) are not applicable, no analytical solutions are known. Our numerical studies show that the system evolves, even for moderately large IE±(z, 0+ )I, into the same non-zero steady states as in case 1. Examples are presented in Figs. 25-26: Fig. 25 shows a single weak Gaussian pulse

E+(z, 0+)

= 0.001 IaK,11/2exp

[

- (z-05L) O.li

which experiences a pulsed gain 6.9~(ct/n pulse

E+(z, 0+)

=I

a 11/2exp [K

0)

2]

at t

,

E_(z,0+)

= 0 and

(z-O.liO5L) 2] ,

= 0.

(94)

turns into a Gaussian

E_(z,0+)

= 0.

(95)

The pulse is amplified and escapes the finite structure at the boundaries. The transparency of the boundaries is due to the boundary conditions (12). Some energy is scattered back during the propagation and near the boundaries. The remaining field after the transience is sufficiently weak to be approximated by Eqs. (24). The transition to non-zero steady states is identical to those considered in previous sections and illustrated in Fig. 15.

451 2 1

C 0 CJ)

0

-1+-o---.-....-.---.-......-.---,-......-.---,-......-.---,-....,......,,.....,........,......,......,

0

1

2

time

3

4

5

Figure 27: The normalized periodic gain function -y(t)/-Yac·The temporal coordinate is t/T.

After the discussion of the effect of a pulsed gain on an initial Gaussian pulse, we now consider the effect of such gain on initial noise. This effect is illustrated in Fig. 26. The noise fields E± ( z, 0+) have the power spectrum

P(k)

= 1.5l1eL2 /al exp [-

(kL/2048)

2 ],

(96)

where k is the spatial frequency. The initial noise is rather white, because the differences between its power spectra P( kq) for the ten linear modes q = ±1, ±2, ... ±5 with the lowest linear threshold gains are less than 10%, while the gain 7 = 1.90/ L applied to the periodic structure is between the lowest linear threshold gain "'fq=±I = 1.75 and the second lowest threshold gain "'fq=± 2 = 2.52. The amplification and the propagation of the fields are similar to those of a Gaussian pulse. Noise spikes are amplified along the characteristic lines t = ±n 0 z / c + const and escape the structure at the boundaries. Only little energy is reflected back due to the reflectionless boundary condition (12). The widths of the spikes which escape at the boundaries of the structure can be shorter than those of the initial noise at t = 0+. The remaining fields have low intensity at the time t > n 0 L/c. They build slowly up to the steady state, which is shown in Fig. 15(b ). 4.6.2

Periodic

Gain

The other interesting system is the nonlinear periodic structure with a periodic gain function. Numerical study is performed for the gain function 00

7(t)

= "'tacL n=O

f(t - nT),

t/Tw for O < t < Tw { f (t) = 2 - t /Tw for Tw < t < 2Tw 0

(97)

otherwise

which is illustrated in Fig. 27. The results are essentially the same as those with constant gain (Fig. 15), except that the fields in the steady state are now temporally modulated. With increasing gain "'tac,the system undergoes a similar scenario from a decaying state to chaos. An

452 (al)

---

-1

goin=7.0

>-.

·-

-2

(/)

-C

Cl)

C

..__

-3

C>

O"l-4

0

-5

(a2)

0

50

200

150

100

time

---

-1

goin=7.0

>-.

-2

(/)

-C

Q.)

-3

C

..__ C>

O"l -4 0

-5

0

50

100

150

200

time 3

(bl) ~

--+-'

goin=7.2

2

en C

Q) --+-'

1

C

0

-J....r""T""T..,.....~~~;r.,::.r-r-,r-T'""'9""T""T""T""T""T""T""T'"T"'T""T""...--r-r-r-ir-T'""'t""T""T

0

50

100

......... "T"T""T'"T"'T""T""...--r-r-r-,r-r,""T""T""T""T""T""T""T"-,

150

200

250

300

tirne (b2)

3

goin=7.2

2:;--2 en C

Q)

c:1 0

--J..-,""T""l'....,...,...,....~~~~r-T",""T""T""T""T""T""T""T"T'"'T""T""r"T"""r-T"-.""T""T..,...,.""T""T"T"'T'"'T""T""~r-r-,~-r-,

0

50

100

150

time

........

200

250

"T""T"-.-r....,

30 0

Figure 28: The normalized intensities laLE:.(-L/2, t)I [Fig.(al), . .. ,(el)l and laLEi( +L/2, t)I (Fig.(a2), ... ,(e2)] at the ends of the structure for different normalized gains -YacL. The temporal coordinate is the normalized time ct/no. ( to be continued)

453 8

{cl)

goin=7.5

6

en c::

4

Q.)

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

c::

2

0 - .......... -~~.......... ~~~ 0 50

.....,.....,......-. ..................... 150 100 ~"T"'"T""

.........

..-.--,.....,.....,

........

...,...............,........,...

r-r-.....-,......-,"""T'""'T""'I

300

250

time

8

(c2)

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

200

goin=7.5

6

en c::

4

Q.)

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

c::

2

0

................ ,r;,;:::........,....,....,..._.......-, ........................... ......,... ................. .....-,.....-,........-.-.- .................... .,....,...T""T"".,..,...,l""T""'l ........"""T""T-r-r"'T""T""T"">

~

0

50

100

150

250

200

300

time 20

{dl)

goin=7.8

15

>-.............

en c:: 1 0 Q..>

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

c::

5 0

...,__......,,11;::;........,........,.....-;......-;:..,~~-i!J'.y.-r--,Jc.,...,l}-.,....,....;,,.....u:.....,....,,...;;J-y.i.::,...,..~...,:u;... .........~.....,.....,,....'f--,;

0

50

100

150

200

.........

--.--,..--?r-.,.......;~

250

.......

300

time 20

{d2)

goin=7.8

15

>-............. en c10 Q.)

............. C

5 0

-+-r....-. ........................ ....,_,-111-,,..,:.,,-,au.."'T"'T.........,.....,....,;..T'""T"'"IILr-, ........-"i'-,,-'-o-r...,....uf.....,....,.......,.ll-,,';

0

50

1 00

150

time Figure 28: (continued)

200

........ ~.,.....~~..,....,IL'r-r-r-,

250

300

454 40

(el) >... __.

gain=B.3

30

~ 20

... __.

gain=B.3

30

~ 20

0. The maximum gain 7ac is varied. In Fig. 28(a), the average gain

(7)

d t e

1

IT

Tio

7(r)dr

(98)

is below the lowest threshold gain 7q=±h as defined by Eq. 15, of an active DFB structure. For t > 3n 0 L/c, the fields in the periodic structure are weak and decay exponentially like exp [( (7) - 7q=±t )t], as predicted by Eqs. 24 of the linear theory. In Fig. 28(b), the average gain is slightly above the lowest threshold gain 7q=±t of an active DFB structure. The system attains a periodic steady state, whose period is equal to the period of the gain 7( t). The steady state is symmetric in the sense IE+(z, t)I = IE-(-z, t)IAs the gain further increases, the symmetry is broken. In Fig. 28(c), the system still attains a periodic steady state with the period T. However, it is not longer symmetric: IE+(z, t)I -=/IE-(-z, t)I- Broken symmetry is a common phenomenon in nonlinear systems. It occurs in numerous complex dissipative systems, such as in fluid dynamics and chemical reactions [151,152]. A further increase of the gain brings the system into a state similar to the intermittent state encountered previously, as illustrated in Fig. 28( d). If the gain is

455

a) 50

intensity

b) 6

0 ~4 (/)

C Q) +J

.S 2 0

~~........... ,.,____..,_......... 1,..........,Lr5

0

space

time

10

15

Figure 29: A single sweeping pump pulse excites the noise in the DFB resonator with end reflections. The spatial coordinate is z/ L, the temporal coordinate d/noL. Parameters are KL = 1, o > 0, T±

=

0.9.

2 + IE-1 2 ). (a) The intensity distribution lo/Kl(IE+l 2 2 (b) The emitted intensity lo/Kl{IE+l- IE-1) at the boundaries z z = -L/2 is similar and temporally shifted.

= L/2.

That at the boundary

sufficiently large, the system ends up in chaos, as shown in Fig. 28(e).

4.6.3

Effects of End Reflections

Up to now, the reflection from the ends of a finite periodic structure has been neglected. With end mirrors, the resulting system possesses both the property of a DFB resonator and a Fabry-Perot resonator. The influence of the threshold gains and the oscillation frequencies by the end reflection have been discussed in Ref. [9, 15, 19, 22, 23, 34] in the frame of linear theory. In this section, two representative examples are given for the nonlinear case. With end reflection, the boundary condition (12) is replaced by (99) where T± are the effective coefficients of reflection at z = ±L/2. The effective coefficients are generally complex numbers and are related to the refractive indices of the periodic structure and the surrounding medium, the length of the periodic structure, and the coupling coefficient [34]. For the sake of simplicity, they are assumed to be constant and independent of the incident intensity. In the first example, a passive resonator with reflectivities r ± = 0.9 is filled with noise, which has the identical parameters as those in Fig. 26. The noise field is then

456

a) 5 intensity

b)

0

0.6 >-

~0.4 en

·

... C

(1)

. ~0.2

L J l l '0.0 .µ.,.,i._""F=i..L,.::::;:::...---r-'~~-,.L..=,...-r-'-r-.~......,

2

0

4

.

time

6

8

10

Figure 30: A Gaussian pulse in an active DFB resonator with end reflections. The spatial coordinate is z/ L, the temporal coordinate d/noL. Parameters are KL= 1, o > 0, -roL = 0.102, r± = 0.9. 2 + IE-12 ). (a) The intensity distribution lo/Kl(IE+l 2 (b) The emitted intensity lo/Kl(IE+l- IE-12 ) at the boundaries z = L/2. That at the boundary z = -L/2 is similar and temporally shifted.

excited by the sweeping pump pulse with the gain function

,(z, t)

= 3K exp

[- (z -

ct/no+ L 0 _3L

) 2].

(100)

As shown in Fig. 29, a burst of noisy pulses is emitted from the periodic structure at t ~ l.5n 0 L/c, where the center of the pump pulse leaves the boundary z = L/2. A substantial part of the pulses is reflected by the boundaries and remitted with the period 2n 0 L / c. The second burst of pulses is more intense than the first burst in Fig. 29 because the reflected field is also amplified by the tail of the rather broad sweeping pump pulse. During the multiple reflections, no single-hump pulse is formed and deformations of the pulse shape are observable. The second example, shown in Fig. 30, concerns the propagation of a Gaussian pulse with the initial form E+(z, 0)

= I0Kl/

1 2

exp [ -

(z-05L) , 0 _2i 2

]

E_(z,0)=0.

(101)

in an active nonlinear periodic structure with mirrors at the ends. During the propagation and multiple reflections, the pulse becomes narrower but tends to split. The output peak intensity fluctuates when the pulse splits. For large time, e.g. t > 100n 0 L/c, the pulse decays and the system attains a chaotic steady state, which is similar to that shown in Fig. 15(e). Although the two examples discussed above are rather specific, the phenomena they demonstrate are quite typical. In particular for the general gain function 1 (z, t)

= ,Ac(z,

t)

+ ,o,

(102)

457

where ,Ac(z, t) is dominant only during a short period, the transient gain 1 Ac(z, t) picks up the field in its path and amplifies it. However, the resulting single or multiple pulses are unstable against the reflection at the boundaries and the perturbations during the propagation. No stable recurring pulses are observed in numerical experiments. The pulses can be narrowed by the nonlinear periodic structure, but in general they tend to become unstable. Conclusions

Two opposite nonlinear phenomena in a periodic active or passive medium with Kerr nonlinearity are studied in this article. For the passive nonlinear periodic structure, which may become a useful element in laser applications, exact solitary solutions, both "in-gap" and "out-gap solitons" are discussed. Numerical studies indicate that, under suitable conditions, the in-gap solitary waves are robust against small perturbations in the initial conditions and against collisions with another in-gap soliton whilst the out- gap soliton is less stable. Furthermore, the propagation of a pulse with a nearly in-gap soliton shape in the medium is investigated with the variational approach. Conditions of formation of ingap solitons, as well as the oscillations in pulse height, pulse width and pulse velocity during the propagation are addressed. For the active nonlinear periodic structure, which represents a limiting case of an index-coupled distributed feedback laser, typical phenomena of a dissipative system are demonstrated. As the energy pumped into the system is increased, the system with constant gain undergoes a series of scenarios, including null-field state, symmetric cw single-mode state, asymmetric cw single-mode state, periodic state, quasi-periodic state, intermittent states and chaotic states, as illustrated in Fig. 15. The active system with time-dependent gain undergoes a similar series of scenarios, which are shown in Fig. 28. Effects of the time-dependent gain and the end-reflections are also considered. While in-gap solitary wave can be formed in infinite structure by properly preparing the gain function and the incident wave, pulses bounded in a finite structure with mirrors at the ends are shown to be unstable. Acknowledgements

The menting J. F. to and Dr.

authors thank Prof. Dr. J. W. Blatter (ETH Zurich) for reading and comthis article. They are also grateful to Prof. Dr. P. de Forcrand for introducing the world of computational physics, and to Dr. D. Scherrer (ETH Zurich) W. A. Yong (Heidelberg, FRG) for stimulating discussions.

This study was supported by the ETH Ziirich, the Swiss National Science Foundation and GRD/EMD, Bern.

458

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THE BALANCE EQUATIONS OF ENERGY AND MOMENTUM IN CLASSICAL ELECTRODYNAMICS

J .L. Jimenez

1

and I. Campos

2

Departamento de Fisica Universidad Aut6noma Metropolitana, Iztapalapa Av. Michoacan y Purisima S/N Apartado Postal 21-939 Mexico, D. F., 04000, MEXICO 1

2

Departamento de Fisica, Facultad de Ciencias Universidad Nacional Aut6noma de Mexico Apartado Postal 21-939 Mexico, D. F ., 04000, MEXICO

1. Introduction

A physical theory is more than a consistent mathematical structure: it requires an interpretation that enables us to test it and to apply it to a partial understanding of the physical world. In general terms, controversy arises more frequently about interpretation than about the mathematical structure and its formal consequences. Moreover, the interpretation of a physical theory develops slowly as the theory is applied to exemplary cases, usually through models that specialize the theory in some respects, and feedback from experience is obtained. In fact, the mathematical structure of a theory, its interpretation, and its corroboration advance in a very complicated way, as the history of any particular theory can show. Likewise, a theory does not grow up in isolation, and therefore is influenced by advances in other theories and in turn influences other theories; of course a theory is also tested with the aid of other theories, which makes its acceptation or rejection a non-trivial issue. The basic components of a phy~ical theory are: i) its primitive concepts, that are not explicitly defined within the theory, and therefore must be learnt in context; ii) its axioms, that relate the primitive concepts, usually through differential equations; iii) explicit definitions in terms of primitive concepts and theorems deduced

464

465 from the axioms, and iv) an interpretation that links the theory to experience. Of course a theory may use some of these elements from other theories, which makes an explicit listing of all the assumptions perhaps an impossible task. In the following we deal with some interpretative problems of Classical Electrodynamics (CED) that refer to the balance of energy and momentum in electrodynamical systems, as well as to the associated problem of the localization of energy in the electromagnetic field. Since the conception of localized energy was proposed by Poynting in 1884, controversy has divided physicists into those convinced of the soundness, or at least the utility, of the idea, and those with different degrees of skepticism. Formally Poynting's theorem (in SI units) states that

8u

..... ..... .......... H) = -j · E.

- + V ·(Ex 8t

(1)

= ½(E · ..... D +ii• B) is interpreted .....

as energy density in the electromagnetic field; S = E x H, the Poynting vector, is interpreted as a local flux of field energy, and ..... ..... j •Eis the work done in unit time by the field on the current. This power may be ..... ..... ..... ..... ..... positive or negative, depending on the relative directions of j and E. E, D, B and ii are the electromagnetics fields. Some reasons for having doubts about the interpretation of Poynting's theorem as a true continuity equation for electromagnetic energy are the following. One problem is that an arbitrary divergenceless vector field can be added to it and equation (1) will still be satisfied. Thus alternative definitions of the energy flux have been proposed for a long time [1], and the Poynting vector appears only as the simplest possibility. u

.....

Another problem with the concept of localized energy is that in some static situations, for example a point charge and a magnet at relative rest, the theorem implies a flux of energy in closed paths [2]. If this flux is taken as a transport of energy in space and time, the transport must be through the fields, but these are static fields. This unobservable circulation of energy in static conditions gives energy a quality of quasi-substance that goes counter to intuition. Another objection usually raised against the interpretation of S as a local flux of energy is that, to some, it sometimes seems to point in the wrong direction [3]. For example, in the case of a conduction current in a wire the flux is perpendicular to the wire and not

466

in the direction of the current. However, for a convection current ( a moving charge) the energy flux is in the direction of the current; for some authors this is "more reasonable" [4]. As a last difficulty we mention a seeming violation of the energy balance [25]. In the classical theory of the electron, developed by Lorentz and others, the electron is modelled as a tiny spherical shell of charge. When the electron is moving with constant velocity ii, the momentum of the field calculated with Poynting's vector results ¾ ~ ii, where U0 is the field energy of the charge at rest. Thus the moving charge seems to have gained an energy For these reasons we 0 "out of nothing". consider necessary a careful revision of the formal derivations of Poynting's Theorem and its possible interpretations consistent with the general interpretation of CED. It will be necessary to deal with controversial points, but controversy is the flavor of research.

½U

2. An outline

of Classical

Electrodynamics

Before we engage in the particular discussion of energy and momentum balance in electrodynamic systems, we sum up the main features of CED as a field theory. The formal basis of CED was established by Maxwell in the last century. His interpretation, however, did not survive [5]. The theory introduces as primitive ..... ..... ..... ..... concepts, on the one hand, four vector fields, E, D, B, and H, and on the other ..... hand charge and current densities, p and j, respectively. The fields are known as electric field intensity, electric displacement, magnetic induction, and magnetic field intensity, respectively, and are related to the charge and current densities through Maxwell's equations, that expressed in modern notation and SI units are .....

V ·D

= p;

.....

V ·B

(2)

= O;

These equations refer to the total fields and the total charge and current densities, total in the sense that these fields include the fields associated with the charged bodies and any external fields, while the charges and currents includes the "free" as well as the "bound" charges and currents. The first pair of inhomogeneous e..... ..... quations link into a natural pair the fields (D, H), and relate it to the "sources", p

467 .....

and j. The second pair of homogeneous equations link into another natural pair the ➔

.....

fields (E, B). This is a consequence of the non-existence of magnetic monopoles, otherwise in the right-hand sides of the second pair we would find the monopole density and monopole current density. Also, the first pair of equations implies the continuity

equation,

V ·

J+ OtP = 0, which

expresses the conservation

of charge.

Since Maxwell's theory was originally proposed to account for electromagnetic phenomena

in macroscopic

bodies and media, it is really what we now call Macro-

scopic Electrodynamics, to distinguish it from Microscopic Electrodynamics, that takes into account the atomic structure of matter. Both theories have their own difficulties, as we will see. Since the number of equations mine all the components the equations

of the four fields, constitutive

D = e(E) E and ii =

µ(~)

B, where

is not enough to deter-

relations are introduced

e(E) is the permitivity

by and

.....

µ(B) the permeability of the medium, which in general may be complicated tensor functions, or even functionals, of the fields E and B. For linear media, however, e and µ are independent of the fields E and B, but may depend on thermodynamic variables, as density, temperature, pressure and time. These constitutive relations ..... ..... express the response of a medium to the fields E and B. We have also the relation ..... ..... ..... j = u E that states that a current density j arises in ohmic conductors as response ..... to an external field E. Since in nature there are not instantaneous responses, the most general relation between response !out and exciting force fin, consistent with linearity ( weak fields) and time invariance is,

J

oo

fout =

-oo

I

I

I

J(t - t )fin(t )dt .

I(t - t') is the impulse response function [6). In order to comply with causality the ..... ..... ..... ..... ..... response function must be zero fort< t'. !out may be D,H or j, fin may be E,B ..... or E and I ( t - t ') may be e , µ or u respectively [7, 8). However, for weak fields that vary slowly with time we can use simpler relations. 8 , sometimes called the "ether relations", and ii = IJO

.....

In vacuum we have D where e0 and µ 0

constants, related to another universal constant, the speed of light in through c- 2 = µ 0 e0 • Here "ether" does not refer to the "luminiferous to a class of inertial frames. We will use "ether relations" constitutive relations of free space [9, 10). Of course in other systems of units the permitivity

as a shorter

=e

.....

E are universal free space c, ether", but term for the

and permeability

0

of free

space may have other values. For example in Gaussian units they are both defined

468 as 1, so

D = E and ff= B. However,

what really matters is that the invariance of

the ether relations define a class of reference frames, the Lorentz frames, since they are related through Lorentz transformations [9, 10). That is, Maxwell's equations are generally covariant, but the invariance of the ether relations restrict the spacetime transformations to Lorentz transformations [9]. Thus a Lorentz frame, or ether frame, is one with respect to which Maxwell's equations and the ether relations maintain their form, or equivalently, the wave equation maintains its form, or the velocity of light remains constant. Since our basic equations relate the electromagnetic fields to the charge and current densities, and the constitutive relations link the fields (D,H) to the fields (E,B), the problems that can be posed are: given the sources, p and and specific

J,

constitutive relations, to find out the electromagnetic fields, and given the fields to find out the charge and current densities. Usually the constitutive relations are postulated on the basis of experimental for simple materials. A particular

work or microscopic models are proposed

case of the first kind of problem is when in a region there are no

"sources"; that is, p and

Jare zero in the

whole region. It happens that there are non

trivial solutions of Maxwell's equations, and these solutions can be cast, in general, into the form of a superposition of plane waves. Hence in regions where there are no "sources" there may be electromagnetic fields, and therefore electromagnetic energy. The general solution of this kind of problem is the sum of a homogeneous

solution,

J

i.e., a solution of Maxwell's equations when p and are zero, and a particular solution of the inhomogeneous equation, when p and are given and known functions

J

....

of rand t. It is this kind of problem that lends a basis for calling p and j sources of the electromagnetic field. However, the second type of problem is more common in macroscopic where p and

J are

often the unknowns.

Indeed, Maxwell considered

CED,

every kind

of charge and current, even in vacuum, as resulting from field processes. That is, for Maxwell and his followers, like Poynting, charge was rather an epiphenomenon of a polarizable all-pervading medium: the electromagnetic ether [5). This view lent plausibility to the conception that energy is localized in the fields, as potential energy is localized in a stressed medium. Thus, contrary to the modern concept of charge, for Maxwell charge was not a substance: "In most theories on the subject,

Electricity

1s treated

as a substance,

but

469

inasmuch as there are two kinds of electrification which, being combined annul each other, and since we cannot conceive of two substances annulling each other, a distinction has been drawn between Free Electricity and Combined Electricity"

[11). The advent of Electron Theory, however, changed this view. Helmholtz, Lorentz and others regarded charge as carried by "atoms of electricity", thus introducing a fundamental dualism of basic entities: charged bodies constituted by "electrons", and fields. In this way the Lorentz-Maxwell Theory gives a different reading to the equations: "electrons" are the real sources of electromagnetic fields, and the natural problem to be solved is that given the charged particles and their trajectories to find out the corresponding fields. Macroscopic bodies are regarded as assemblies of "electrons", and any macroscopic property, such as conductivity, permitivity or permeability, is to be derived by statistical averaging over these assemblies of point charges and their trajectories. Hence the Lorentz-Maxwell Theory, or microscopic CED, was conceived as the fundamental theory, from which macroscopic CED had to be derived. Yet there are some problems with this view, since microscopic CED cannot account for atable systems of charged particles, and the averaging must be done with unknown statistical distributions. Most work has been directed at obtaining general averaging methods, but at present we do not have a sufficiently general method [12). Only Quantum Mechanics (QM) and Quantum Electrodynamics (QED) can explain to a certain extent these systems. Also Stochastic Electrodynamics, which is based on the assumptions of microscopic CED and the existence of a real, omnipresent, stochastic electromagnetic field of spectrum rv w 3 has obtained some features of QM and QED, so it may account for dynamically stable configurations of charged points and fields [13). According to another point of view, microscopic CED is subsumed by macroscopic CED [14). That is, from a logical point of view the equations of microscopic ... CED result from those of macroscopic CED when p and j take the particular forms

p(r, t) = q6(r - ro(t)) and J = q 6(r - ro(t))v. Here 6(r - ro(t)) is Dirac's distribution, or delta function, and p and J represent a point charge q at position ro(t) moving with velocity v. On the other hand, in order to go from microscopic CED to macroscopic CED, it is necessary to average over assemblies of point charges. This averaging requires special techniques to blur the delta functions. One of the best techniques is that proposed by Robinson [12), that consists in taking the Fourier transform of the delta functions, introducing an appropriate cut-off

470 frecuency. In this sense microscopic CED does not entail macroscopic CED, which does not contradict the program of founding macrophysics on microphysics. Indeed, a phenomenological approach to macroscopic CED (i.e. independent of the atomic constitution of matter) complements the microscopic approach. Hence macroscopic CED must be consistently correlated to other macroscopic theories, as continuum mechanics and thermodynamics. In this way a charged point particle is considered a kind of degenerate charged body without internal degrees of freedom [10]. An important consequence of this view is that, given the great diversity of media and constitutive relations modelling them, there is not a general expression for the electromagnetic force on a macroscopic body, and the force on a point charge emerges

-

-

-

as a motivated postulate, the Lorentz force F = q(E + ii x B), in which the fields are only the incident or external fields, that are independent of the presence of the point charge q. In other words, the field contributed by the charge q itself, its self-field, is to be excluded explicitly from the Lorentz force. We have therefore two very different theories to explore: microscopic CED and macroscopic CED, and following the present usage we begin our discussion with the first theory.

3. Balance

of energy

and momentum

in microscopic

We have in this theory two systems in interaction:

E,D,B and ii,

with the ether relations

the Lorentz force density / form p

= q6(i

-= p(E-+

CED

point charges and the fields

D = eoEand ii

-

=

B; we have also

/Jo

v x B), where p is a charge density of the

- i 0 (t)). In the interaction

of a point charge with electromagnetic

fields appears a problem pointed out by Einstein [15]: the point charge has a finite number of degrees of freedom, while the fields have an infinite number of degrees of freedom. This difference is exhibited by different transformation properties of energy and momentum

of particle and self-field under Lorentz transformations,

as

will be seen below. Other difficulties arise from the divergencies in energy and stress in the self-field of a point charge. Let us discuss this point and fix our concept of field energy.

The notion of energy associated

with an electromagnetic

field emerges as the

work necessary to create a given configuration of charges and currents. Thus the energy of the electrostatic field of point charges qi fixed at positions Ti is the work expended to bring these charges from infinite distance to their final positions:

471

(3)

This may be called the mutual energy of the configuration, since the work required to create any charge qi at position Ti has been excluded. At this stage the question of the localization of this energy is rather artificial. However, we can generalize the above expression to continuous distributions of charge:

U

=

1 . ! 41reo 2

J

dqdq'

R

(4)

'

where R is the distance between the elements of charge dq and dq'. Applied to a spherical shell of charge of radius T O the expression results in

u =

1 1 q2 41reo 2 T 0

(5)

With the above interpretation, this energy is the work done to shrink a shell of infinite radius, with a charge q uniformely distributed over it, to a shell of radius T 0 • Since after the contraction of the infinite shell an electrostatic field, the self-field, appears where there was none, this energy may be associated with the creation of the self-field. This same energy can also be obtained by integrating the expression u = ½eo E 2 over all space, and therefore u is interpreted as an energy density. Obviously if we take the limit r 0 -+ 0, the self-energy becomes infinite. If we have several point charges, the integral of u gives the mutual energy plus the sum of the self-energies of the point charges. Thus the energy associated with the electrostatic field through the energy density u is the work spent in creating the field, which includes the work necessary to create the charges themselves. In a similar way, an energy density u = -21 B/Jo2 is associated with the magnetic field of a distribution of stationary currents. Thus we have an expression for the energy required to create electrostatic and magnetostatic fields, in the form of an energy density u =

½( e0 E 2 + :: ) , at the expense of other forms of energy. In order to deal with time-dependent fields, let us return to our basic equations. The ether relations permit us the elimination of the fields D and H, and Maxwell's equations become

-

-

472 V-E=-

1 -V µo

p

eo

~ af x B - eo ~ vi

V x f In microscopic CED we can set

J=

~

=J

(6)

+ f}B = 0 8t

pv, with p = ~i qic5(r- ri(t)).

There are at least two ways of deriving Poynting's theorem, which is usually interpreted as expressing conservation of energy in electrodynamic systems. One derivation [7] begins with the power density developed by the Lorentz force. Our system is a region of space where there are electromagnetic fields and a point charge. Then

f. V = p ( fez + V X Bez) . V = pv. fez = J. fez'

(7)

where we emphasize that the fields are external fields. Since these fields are independent of the presence of the charged particle, they must be solutions of the homogeneous Maxwell equations in the given region. The derivation proceeds then to substitute J as given by the Ampere-Maxwell law:

~

~

Here we find a /auz pas [16], since E and B are the total fields, which include the homogeneous solution, and the inhomogeneous solution related to the presence of the charge in the region. Yet let us continue the derivation, using the vector ..... ..... ..... ..... ..... identity V · (Eez X B) = B · V X Eez - Eez · V X B. Then,

(9) .....

and using Faraday's law, V x Eez

=-

8

!t , we obtain -ez

In order to get the usual result that appears in textbooks we must take the risk of being inconsistent, since further reduction of the preceding equation requires neglecting the distinction between the external fields and the total fields. Then

473 either we discard the inhomogeneous part of the total fields, or we postulate that the fields in the Lorentz force are the total fields. Both alternatives are possible only in the limit that the magnitude of the charge tends to zero, since in that case the inhomogeneous fields also tend to zero. With these reservations we finally obtain -j

-

2 1 0E 2 = - 1 -0B + -eo+ -V1

•E

2µ,o 8t

2

8t

P,o

-

·(Ex

-

B),

(11)

which under the assumption that the energy density of the time-dependent the same as that of the static and stationary fields becomes -j

--

·E

= -OU+ 8t

1 -V

P,o

·(Ex

B)

fields is

= -OU+ V ·(Ex- H)

(12)

8t

with the interpretation given in the introduction referring only to external fields. In other words, the theorem with its interpretation, as derived above, holds only for teat chargea . The other derivation [17] is more general and amounts to a formal transformation of Maxwell's equations, that is a mathematical theorem. It begins with the scalar multiplication of the Ampere-Maxwell law by the field E,

-

1 - · V x B- - eoE- · -8E = j- · E. -E P.o 8t

(13)

With the vector identity used before we get

- = j- · E,-

(14)

- - - 8E µo [- B · 8t - V · ( E x B)] - eoE · 8t = j . E,

(15)

1 - · V x E- - V ·(Ex -[B P,o

and substituting

-

- - eoE- · -8E B)]

ot

V x E from Faraday's law results in 1

-

oB

which with the identifications made in the other derivation yields

-o -1 ( eoE 2 8t 2

+ -B1

P,o

2)

+ -V1 µo

·(Ex

- = -1-t · E- = -ou+ -V 1 B) 8t P,o

·(Ex

B).

(16)

Now the total fields appear in a consistent way in the final result of the theorem, since it is a consequence of two of Maxwell's equations. Accordingly, the energy

474 density and the energy flux are those corresponding to the total fields, but now the power E is also the power developed by the total field E, which includes the inhomogeneous field associated with Hence the term E represents a self-

J.

J.

J·

interaction besides the external work, unless we assume that the self-field cannot do work on the charge. This may be at best an approximation, since the radiation reaction is usually regarded as a self-interaction

[18].

We have thus two versions of Poynting's theorem, that can be made to coincide only if the inhomogeneous part of the total field, which contains the self-field, is discarded. To see if the theorem can be consistently interpreted as an energy balance, let us apply it to some simple cases. This requires integrating the balance equation over the region we are interested in. It is usually argued that since it is in its integral form that the balance equation receives its physical interpretation, it is not really a drawback that the flux of energy, S = E x ii, is not unique. However, this non-uniqueness of the energy flux is still a drawback for the concept of localized energy, or transport of energy in space and time. Let us consider now a simple case, a point test charge in a given external field. Then the Lorentz force applies and reflects adequately the charge-field interaction, since the self-field is left aside. The integrated form of the balance equation is

di + i - -

dt

udv

V ·(Ex

= -qv.

H)dv

-

E

(17)

The volume integral of the divergence of Scan be changed to a surface integral of S. Since the radiation of a test charge is negligible, the contribution of this surface integral can be made zero by taking the enclosing surface far away from the charge. Then we have

di

dt

udv

= -qv

- - -·E

= -v

•F

=-

dl mv 2 dt 2

(18)

Therefore we have that the sum of the field energy and the kinetic energy of the test charge remains constant. Thus the charge gets kinetic energy at the cost of field energy. However, this simple case becomes unwieldy if the magnitude

of the

charge is great enough, so that the radiation field resulting from the motion of the charge cannot be neglected. Then the Lorentz force is just part of the charge-field interaction, the other part being the radiation reaction force, of order q2 • In this case the fields in the balance equation must be the total fields.

475 We have noted before that only two types of problems can be solved in CED: given a medium, if the charge-current distribution is given then the fields can be calculated from the field equations, or if the fields are given then the motion of the charge can be calculated from the Lorentz force and Newton's second law. The simple example sketched above corresponds to the second type of problem. Let us see what happens with the energy balance in the first type of problem. Now the fields and the charge do not constitute a closed or isolated system, since an external force must intervene to produce the given current, J(e), also known as the external current, subject to conservation of charge, of course. In this case the term -J(e) · E is work per unit time done against the field, so it is positive. Thus in a region where there is a charge driven by an external force, energy is injected into the region by this force at the rate -J(e) • E, where E is the total field, and this energy may change the energy density of the electromagnetic field inside the region, may escape from the region as radiation at a rate given by the surface integral of S = E x H, and may change the kinetic energy of the charge. Indeed, one of the fundamental problems of CED lies in understanding the term - j • E, when E is the total field [14]. The interaction between charge and field is intricate, since the motion of the charge under the action of the field alters the field with which the charge interacts. At present we have only an approximate expression for this interaction, one part being the Lorentz force and other the radiation reaction force.

- - -

--

-

The field is indeed a formidable system, and in order to be the site of localized energy, Maxwell endowed it with a state of stress. In analogy with continuum mechanics, from this stress a force density can be derived through f = V • T, where Tis the stress tensor. The balance of momentum for the charge-field system can be obtained by substituting p and j in the Lorentz force density from the inhomogeneous Maxwell equations, though in this way we run into inconsistencies similar to those mentioned in the derivation of Poynting's theorem from the Lorentz force. These can be avoided by a consistent manipulation of Maxwell's equations. After vector manipulations one gets [17]

-

-

-

- H) + p(E- + v x a --(Ex -at c2 where TM is Maxwell's stress tensor given by being the energy density, u

i

= ;

=

eoEx B can

2 + : .82 ). = ½(eoE 0

-

B)

= V · TM '

Tt/ = eoEiEj

(19)

+:

0

BiBj

- DijU, u

We see that a momentum density

be associated with the field, so that Eq. (19) can be written,

476

(20) where pis the mechanical momentum

density, whose time derivative is equal to the

Lorentz force density, if the fields are external fields. However, in the expression similar to the Lorentz force we have now the total fields and then we cannot link directly the theory with Newtonian mentum is associated

mechanics, since in this theory change of mo-

only with external forces.

It is important

to note that we

have two Lorentz forces, one that contains only external fields and expresses the ponderomotive effect of the fields. The other expression, derived from Maxwell's equations, involves the total fields, and therefore implies self-interaction. The role of this last expression is at present uncertain. This completes the view of the electromagnetic field as a generalized continuous mechanical system, endowed with energy and momentum

densities, and stress.

3.1 Covariant Formulation Relativity

theory emerges from imposing the invariance group of Maxwell's e-

quation and the ether relations, that is, the Lorentz group, to the structure of space and time. Thus Newtonian mechanics, being invariant under Galilean transformations of space and time, requires a deeper modification than CED. However, an explicitly covariant formulation of CED leads to the unification of several concepts, .....

.....

for instance space and time in the first place. Also the vector fields E and B merge into a second rank four-tensor, F"", the field tensor, and some laws merge into more general laws. Thus the balance equations of energy and momentum can be linked in a covariant formulation of CED. In this case we have a four-tensor of energy, momentum and stress, from which a four-vector force density is obtained by applying the fourdivergence operator:

(21) where

/" = p

(E-v .....+ v .....) .....

-c-,E

x B

,

(22)

477

and xv

=

900 = 1

911

(ct,£). µ and v are 0, 1, 2 or 3. We use the metric tensor 9µ.v, with =

922

=

933

= -1,

and 9 µv

=0

if µ -=/v.

!

The stress-energy tensor is symmetric [19],so that = cg,or S = c 2g. Thus a flux of energy is equivalent to a momentum density, or in other words "the inertia of a system depends on its content of energy" [20]. The electromagnetic field can also be set in this language, as well as the charge and current densities, so that

_!E

0

pµ.v =

!E C

C

_!E

Z

C

_!E C

-Bz

0

Z

y

Z

By

(23)

!E y

Bz

0

-Bz

!E z

-By

Bz

0

C

C

and j µ. = (cp, 1); Maxwell's inhomogeneous

equation become

apµ.v ·µ. axv - µoJ

With this notation the energy-momentum

balance can be written as

8Tµv - _pµ.v. 8xv

-

(24)

- _pµ.

Jv -

(25)

,

The energy balance corresponds to

8T 0µ = OU + ! V . S = _ 10 = -p 8x"' cot c while the momentum

E · V' c

(26)

balance is

i

= x, y, z,

(27)

or

(28)

478 In this way we obtain a generatization

of Cauchy's first law of motion of con-

tinuum mechanics: pX = V . T + pb,where p is the mass density, T the stress tensor, and b the body force. Thus a very general expression of the balance of energy-momentum

is

(29) This equation defines a closed or isolated system [21], that is, one in which all possible interactions have been taken into account. The subsystems of this closed system can be characterized

by partial stress-energy

8(Tfv

tensors, so that

+ Tfv + ...) = O.

(30)

axv

lz:"

Hence our equation 8 = - f µ expresses the fact that the electromagnetic fields do not constitute a closed system in the presence of a charge-current distribution, as do the free fields. Of course we can write formally a stress-energy

tensor such that

!l.lµv

_u£ _ _ fµ _ pµv • axv Jv, and in this way we recover the general conservation

(31) law

8(Tµv + tµv)

------=0 axv However, only certain charge-current One particular

(32)

distributions

case is a swarm of charged particles,

Tµv = I: miufuf, which after the application the assumption that each particle obeys duµ m dr

that is, the generalization

=

can be cast into this form. with a stress-energy

of the four-divergence

qFµv Uv

tensor

operator

and

(33)

of Newton's second law with the Lorentz force as external

force, gives the negative of the Lorentz force. Thus one obtains that the fourdivergence of the stress-energy tensor of the electromagnetic field plus the stressenergy tensor of non-interacting charged particles is zero. Again, it must be noted that this result is an approximation, valid only for test charged particles, since the interaction with the self-field has been neglected.

479 9.! Self-interaction and balance of energy and momentum We have insisted on the fact that the interaction of a charge with the total fields includes the interaction with the self-field. This self-interaction has two main aspects: the electromagnetic inertia and the radiation reaction force. In both aspects we find problems with the balance of energy and momentum. Let us take first the problem of the electromagnetic mass. Larmor conceived the electromagnetic mass 1n analogy with the aparent increase of mass that undergoes an object moving inside a fluid: the object is analogous to the charged particle and the fluid is analogous to the self-field. Here we find the germ of a velocity-dependent mass, that developed with the theory of relativity. Now, relativity theory unifies several concepts and laws in a four-dimensional space-time structure, as we have seen. In particular, energy and momentum merge into a four-vector, pµ = "Y(mc,m v), for the case of a neutral particle. Note that p 0 c = "Ymc 2 is the relativistic kinetic energy of the particle, that includes the restmass energy mc 2 • Since the field ( and the continuous medium) is characterized by a stress-energy-momentum tensor, the question arises if the four-momentum

(34) where ds 11 represents an element of a three-dimensional hypersurface, behaves as a four-vector. The answer is that in general it does not. A necessary condition for this to happen is that 8[;" = 0 everywhere. This is a consequence of Gauss's theorem in four dimensions, that in this case implies that the integral that defines pµ is independent of the hypersurface over which T1J" is integrated. Thus one ( spacelike) hypersurface may be all space at t = constant, and other hypersurface may be all space at t' = constant in other reference frame related to the first by a Lorentz transformation. Since in the presence of charges the four-divergence of the stress-energy tensor of the electromagnetic field is not zero, but equals the Lorentz force, pµ as defined above is not independent of the hypersurface, and therefore pu does not behave as a four-vector. Two approaches have been proposed to overcome this problem. One, initially proposed by Fermi and rediscovered by Rohrlich, [22, 23] consists in restricting the definition of pµ to hypersurfaces orthogonal to the worldline of the particle. The other consists in considering a closed system such that

480 8 :.,

(T:,~

+ T:

0 ~)

= 0, where

T:C,~is a cohesion tensor that annuls the Coulomb re-

pulsion that otherwise would make the charged particle to explode. This approach was proposed by Poincare, and the cohesion forces are known as "Poincare stresses".

If we take as a model of a classical electron a spherical shell of radius r uniformly charged, the energy associated

;r.

with the electric field is, as we have seen before,

2

U = 4 ;E:o This happens in the rest frame of the charge, where the associated momentum is zero, and then the four-momentum is po = J T 00 d3 x = U, P 1 = P 2 = P 3 = 0. A Lorentz transformation of the stress-energy tensor to a frame moving in the x direction gives

T,oo = L~L~Tµv =,../TOO_ {J,·y2(T01_TIO)+ {J2,.y2T11, but since in the rest frame T 01

= T 10 = 0,

= ½ co(E 2

and T 11

-

(35)

2E;),

Now, because of the spherical simmetry in the rest frame,

(37) and therefore

Thus

p 10 = -y (

U+ ~f3U)= -y ( mc 2

2

+~

In a similar way,

Again T 01 and T 10 are zero in the rest frame, so

mv

2)

(39)

481 Hence we find that although the momentum of a convection current is in the direction of motion, there seems to be a violation of the conservation of the purely electromagnetic energy, since an extra energy appears equal to one third of the rest electromagnetic energy. As noted above, one solution to this problem is to change the definition of electromagnetic energy-momentum P" = f T 011 d3 z (independent of the hypersurface only for free fields). The proposed definition is [23].

pµ

1

= _!_ c2

T1J"v du,

(42)

11

(u)

v;

where duµ = du is an element of a hypersurface that is always orthogonal to the world-line of the particle. The integral leaves out the volume of the charge, a sphere in the rest frame and the Lorentz transformation of a sphere in any other frame; this is what the notation f(u) means. This definition yields an energy-momentum four-vector given by

U

=;

1

udu -

(u)

~

1S·

2 C

Sdu

(u)

2 C

vdu,

(43)

(u)

.... = , l .... + ,1

P

C

T · vdu,

(u)

(44)

!

With this definition the factor in equation [41] is reduced to 1, and so the conservation of the purely electromagnetic energy is re-established, but at the cost of making the definition of energy and momentum surface-dependent. The other point of view, Poincare's, takes into account the fact that the spherical shell of charge is not a closed system: we have not considered the constraint forces that keep the charge distribution stable. Thus besides the electromagnetic energy we have energy associated with the non electromagnetic forces that make the existence of the charge distribution possible. Poincare showed that this energy can be accounted for as an energy Ucoh = pV, where pis a uniform pressure p = ½e0 E 2 inside the spherical shell that balances the Coulomb repulsion. Then

482

1

Ucoh =

4

e2

2 eo (41re0 ) 2

r4

•

3

3 1rr =

1 41reo · 3 · 2r e2

1

1

= 3 U.

(45)

f

Thus the total energy, electromagnetic plus non electromagnetic energy, is U, corresponding to a mass Therefore there is no violation of conservation of energy. What happens is that this energy is seen as purely electromagnetic energy in

f ¾.

f;;.

the moving charge, and therefore the electromagnetic mass is In other words, although the constraint forces do no work in the rest frame, they may develop power in other frames of reference, and so produce a flux of energy that, in this case, appears as the electromagnetic energy of the moving charge. This power is necessary to mantain the mechanical equilibrium of the charge distribution, and thus the total energy and momentum of the shell of charge constitute a four-vector if and only if it is in mechanical equilibrium

in every frame of reference.

This is the content of von Laue's Theorem

[24]. This view, however, is not

well accepted because the nature of these cohesive forces is unknown in the case of the electron, and therefore little progress can be made in the development of a fundamental classical theory of charged elementary particles, as conceived by Lorentz and others.

It is worthwhile to note that, if we define the total energy of the charge as its electrostatic

rest energy, the energy transfered

from the cohesive stresses is sub-

stracted from the energy of the charge in motion and then we obtain a factor 1 in the electromagnetic mass instead of the factor 4/3. Thus for a closed system we can have either factor and the electromagnetic energy and momentum constitute a four-vector independently of our space-like hypersurface [25]. In the case of the radiation

reaction force the problems are even worse.

We

find that just tapping a charge causes it to flee, with ever increasing speed in the non-relativistic theory, or approaching the speed of light in the relativistic theory. In any case the charge gets an infinite amount of energy of unknown origin. These unphysical predictions arise from the fact that the radiation reaction force is proportional to the time derivative of the acceleration, thus eluding Newtonian mechanics or its relativistic generalization. We cannot discuss here all the intricacies of this problem [21]. We can just give a practical rule: never use the radiation reaction force as the only external force.

483 4. The balance

equations

in macroscopic

classical

electrodynamics

We have noted that Maxwell's equations refer to macroscopic bodies in interaction with electromagnetic fields. As a consequence of this interaction, charge and current distribution may be generated in these bodies or materials, which in turn alter the electromagnetic fields. The charge and bound current that arise from the initially neutral material are called bound charge and bound current, to distinguish them from charges and currents that may be studded through the material; these are called free or true charges and currents. The bound charge and current densities are related to new fields ft and M, which characterize the response of the material through the equation

Pb= -V-

P

and

.... .... .... 8P ib = V x M + &t ,

(46)

.... and of course Pb and ib satisfy the continuity equation. Since the total charge and current densities are the sum of the corresponding .... .... .... bound and free densities, p = Pb+ PJ and j = ib + ii, we can rewrite Maxwell's equations in terms only of the free densities, defining new fields .....

and

.....

D1 = D

.....

+P

.....

= e0 E

.....

+ P.

(47)

Then the inhomogeneous equations become

.... V-D1=P1i

(48)

and

while the homogeneous equations remain the same. We can drop the subindex f from the fields Df and ii f if we bear in mind that these fields are related to the free charge and current densities, while the fields D and ii that appear in the ether relations refer to the total charge and current densities. In quasi-static conditions the response of materials is expressed by the fields ..... ..... ..... ..... ..... P and M through the equations P = XeeoE and M = Xm .II., where Xe and Xm are IJO the electric and magnetic susceptibilities, respectively. The constitutive relations, again in the quasi-static approximation, can be expressed in the form D = e eoE= .....

.....

.....

.....

.....

1

.....

13

.....

--

eoE + D = (I+ Xe)eoE, and H = -B = -IJo - M = (I - Xm).!l..., so e = 1 + I\.ve µ IJO IJO and ; = 1 - Xm, where now e is the relative electric permitivity and µ is the relative magnetic permeability. The absolute permitivity and permeability, the

484

relative permitivity and permeability, or the susceptibilities are equivalent ways of characterizing the different materials and media found in nature. These properties may depend on many factors, such as the thermodynamical state, the mechanical state of stress and strain, the fields in the case of non-linear media, the frequency of the fields in dispersive media, and even history in the case of materials that exhibit hysteresis. Such a complexity makes the development of a general theory a very difficult enterprise that requires a consistent composition of electrodynamics, thermodynamics, and continuum mechanics. In the following we limit our treatment to the simplest constitutive relations that suffice for the quasi-static approximation.

4.1 Energy

Balance

In the case of macroscopic CED we cannot derive the energy balance from the Lorentz force density, even though there are now no difficulties associated with the self-field, since we are dealing only with non-singular charge and current densities. The difficulties arise rather from the fact that the Lorentz force is just one of the forces acting on an element of volume of material. We can, however, transform Maxwell's equations into an identity analogous to Poynting's theorem in microscopic CED. Indeed, this is Poynting's original derivation, since he was a committed Maxwellian. We do not repeat all the steps; we just point out two important differences with respect to the microscopic case. One is the expression for the energy density that must be obtained from E • + ii . ~B. The t usual expression u = ½(E · D + B · H) can be obtained only if the medium is linear, that is, if e andµ do not depend on E and B, respectively. Besides, e and µ must be independent of time or of any factor that depends on time. These restrictions leave out many interesting media and rapidly varying fields. The other point to note is that now we have a term E, which expresses the power developed by the total electric field and the free current density. The final form of Poynting's theorem in macroscopic CED is

~f

-

-

-J, ·

au - - 8t +V-(ExH)=-i1·E.

-

j f includes the conduction current density and the convection current density

(49)

pv.

485

The theorem was applied by Poynting to the understanding of the conduction current in Maxwellian terms, that is, not as motion of matter but as the result of field processes [5]. This example is used in textbooks to illustrate the balance of energy in a conductor carrying a current, some commenting [2], [26] on the strange direction of the energy flow given by the Poynting vector, since now energy flows in a direction perpendicular to the conduction current, and not in the direction of the current as is the case with a convention current. Thus what now seems strange to the authors was quite natural for Poynting, who considered that "A conduction current then may be said to consist of the inward flow of energy with its accompanying magnetic and electromotive forces, and the transformation of the energy into heat within the conductor" [5]. We see here again the view of Maxwell and his followers that charge and current are effects of the fields, contrary to the present view that the fields are effects of the charges and currents. This same example can be seen in a different way. Since we have a stationary condition, :; = O; also, we enclose our whole system within such a large closed surface that the flux of Sis zero. Then we have -J, •E = 0. This seems absurd, but let us remember that metallic conductors obey ohm's law, that in its generalized form is ]f = u(E + E(e)), where E(e) represents the external electromotive force, in this case provided by a battery, and u is the conductivity. This is another example of a constitutive relation, in the quasi-static approximation, which characterizes the response of a metallic conductor to an external electric field. Then the balance reduces to

-

..... =J..... 3

jf •E d

z

/

(u E.....2 + u E.....· E.....(e ) )d3 z = 0

(50)

Or, JuE 2 d 3 z = - f E(e)•] d 3 z; the left-hand integral is Joule's heat, while the right-hand is the power developed by the external emf. Thus the energy dissipated as heat in the conductor is provided by the battery. We have therefore two interpretations of the same phenomenon. In one, energy is conceived as flowing from the battery, going into space and then flowing into the conductor, perpendicularly to its surface, where it is dissipated as heat, while at the same time produces the conduction current. In the other, the battery establishes a difference of potential in the conductor, or equivalently, an electric field of constant magnitude in the conductor.

486

That is, the battery puts the conductor in a state out of electrostatic equilibrium. This requires a constant input of energy. The electric field sets the free charges of the conductor into motion, thus producing the current. The current gives rise to the magnetic field, and to the Joule heat.

ft

Momentum Balance

We have seen in microscopic CED that a momentum density must be associated with an electromagnetic field. The time derivative of this momentum density is equivalent to a force density, which is part of the momentum balance. In macroscopic CED it is more difficult to obtain a general expression for the force density. Let us see how far we can go guided by the general principles. The aim is to obtain the equivalent of the Lorentz force density from the macroscopic Maxwell equations. Thus

.... .... .... EV-D = PIE .... .... HV · B = 0 .... .... 8D .... .... .... (V x H) x B - 8t x B = i1 x B

.... .... 8B

(V x E) x D

+ fit

....

x D

= 0,

and adding the four equations, with a rearrangement

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

EV· D-

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

D x (V x E)

+ HV · B- Bx (V

(51)

of some terms, we get

..... 8 ..... .....

x H)-

8t(D x B)

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

= pE + j

x B. (52)

It can be shown that, for linear and quasi-static responses, [27] [28] .............................. EV· D - D x (V x E)

..... ..... ..... ..... HV · B - B x (V x H) = ..........

1 D) - - eV. 2 .......... 1 V · (BH) - - µ V. 2

= V · (E

.....

E 2I .....

(53)

H 2I

..........

where ED and BH are tensor (dyadic) products and I is the unit dyadic. The last terms on the right can be expressed in the form

= V • (eE 2 I) - E 2 Ve ff 2 1 = v. (µff 2 1) - ff 2 v µ.

eV · E 2 I µV.

(54)

487

With these results we obtain ..........1 .................... 1 .......... 1..... ..... a..... ..... ............... - IE-D+BH -- IH-B)+-(E 2 Ve+H 2 Vµ)- -(D x B) = P1E+i1 x B

V ·(DE-

2

2

-

at

2

(55)

.... .... If we define Smom =DX Band .................... 1 .................... Te1e ={DE+ BH - 2 I (E · D + B · H)}

(56)

we can write the momentum balance as T"7 V •

T

ele -

.... oSmom

(57)

at .....

.....

.....

.....

Note that we have introduced Smom = D x B, different from the vector Sen = .... E x H obtained in the energy balance. This difference is important, since now the combination of the energy balance and the momentum balance in the fourtensor formalism of special relativity will lack the symmetry of the microscopic case. Indeed, this asymmetry has given rise to the Abraham-Minkowski controversy, discussed in the literature since long ago.

-

This controversy arose after Abraham's proposal of symmetrizing the stressenergy tensor, considering the momentum density Sen= E x ii as the true expression of the field momentum density. Indeed, the momentum balance Eq. (57) can

gt(E;/i>, to the

be transformed identically, adding and substracting

....

I {JS

TJ· Tele -

c2

....

-:

at = p E + J

.... X

B

+

{J

form

eµ - I .... .... c2 EX H.

at

(58)

(We are considering homogeneous media for which Ve and Vµ are zero). The last term on the right is known as Abraham's force. It is pointless to discuss which balance is the correct one since both are mathematical theorems derived from Maxwell's equations. The question is rather about interpretation. Experimentally the interpretation is related to the radiation pressure on material media. However, experiment cannot by itself settle the controversy. It is necessary first to clarify the meaning of these expressions for the momentum balance before any coherent interpretation of the experiments can be made. This is a hard task, since the incident field excites the response of the medium by setting its charges

488

into motion. Thus part of the energy and momentum of the field is transferred to the medium. Indeed, the momentum density Smom / c2 proposed by Minkowski has been interpreted as a pseudo momentum, which is conserved in homogeneous and isotropic media [29] [30]. The presence of inhomogeneities or interfaces, however, requires the generalization of these results. Some generalizations are based on the method of virtual work, used by Helmholtz in 1881, [28, p. 146] to derive the force density in a dielectric under electrostatic conditions. The gist of the method [8, 12, 28, 31] consists in expressing the change of the energy in time, due to a small velocity v delivered to the medium, in the general form

J -

dU dt=-

ueis

In the case of a dielectric the total energy

U'

=

(59)

dVf•v.

J

dV

J.jji . dD.

(60)

If the medium is linear and isotropic, then (61)

- -

After algebraic transformations, that include a change from E to D as the dependent variable, one obtains, for the linear initially isotropic medium,

dUe dt

-=-

The partial time derivative of

j eij

dV

{pE•v+-eo__!1_E·E· .... 21 8e8t· •

can be substituted

} 1

1

•

(62)

from the convective derivative (63)

On the other hand, the elastic properties of the medium are given by (64) where Ui is a displacement of an element, therefore u,·can be identified with v.I, ,., .. u,IJ is the symmetric part of the strain tensor, and Wij is the antisymmetric part, which represent local rigid rotations. With these elements one can write

489

(65) where 8Eij = -8ukl

0.ijkl

(66)

If we have a solid medium initially isotropic, Wjk and

are zero. Therefore

'Wik

(67) Then in this case I 8 -eoE·E·-e·· 2 , J at

I

= -eo{ek·E·E· •1 2 J ,

J

-e··E·Ek}•1 1

8v·1 ark

8v·1 .

I

+ -eoa.··k1EkE12 •1 arj

1

(68)

--eov·E·Ek 2 1 J Again, for a solid initially isotropic 1 I 8vi 1 8e 2 8e 2 -eoE - = -eoa.i ·k1EkE1- - -eoE v,-. 2

8t

2

8Tj

J

2

8r1

(69)

Therefore the time derivative of the total energy is

-dUe =dt

J .... .... + dV {pE · v 1 2

--eoE

2

I 8vi -eoa.· ·k1EkE12 IJ 8Tj

8e vi-} 8r1

(70)

Integrating by parts the second and third terms results in

dUe

dt = -

J

dV vi{pEi -

8 I 2 8e eoO.ijkl Bri (EkE1) eoE ari} 2 2 I

(71)

Then we can identify the electric force density

(72) where

(73)

490 is the electrostriction tensor. Analogously, for the magnetic force density one obtains /i

m

-t

= (J

....

1

2

8µ

x B)i - -2µ,oH -8Ti

8

+ -aTj

rmstr

(74)

ij

where

Tijmstr is the magnetostriction

= - 2I/3 ijkl

H kHI

(75)

tensor, and

8µ,ij

/3ijkl

(76)

= -8ukl

Finally, adding the electric force density and the magnetic force density, one gets

/i

= pEi + (J x B)i 7

....

1 - -eoE 2

2

8e 1 - - -µoH 8ri 2

2

8µ 8 ( eo - - -aijkl 8ri 8rj 2

E E k I+

µo /3 H H ) ijkl k I

-

2

(77) As we can see, this expression differs from the one obtained from Maxwell's equations in two terms. The term gtCDx B) does not appear in Helmholtz's deduction, while the term

(78) does not appear in the force density derived from Maxwell's equations. The point is that the fields in matter and the free charge and current densities do not constitute in general a closed system. We have seen that even in the simplest case of a small shell of charge in uniform motion the constraint forces play a role in the energy and momentum balances. Therefore the first and most important condition to obtain a true balance of energy and momentum is to identify clearly a closed system, and the subsystems we are interested in. Hence we cannot give general arguments in favor or against Minkowski's and Abraham's views. Indeed, other expressions for the momentum density can be derived that yield other vector fields of the type Senor Smom• For example Peierls (30] found that for refraction the momentum density in the medium is a kind of average of Senand Smom• On the

491

other hand, Lai et. al. [33)showed that this expression holds only for certain characteristic times giving at the same time other expressions for different characteristic times. Also, other expressions for the force density can be derived from Maxwell's equations. Sometimes these expressions are discarded as mere indentities, [31)they are equivalent to Maxwell's equations. We refer the reader to the work of de Groot and Mazur [32)who propose another form of the force density derived from Maxwell's equations and the conservation of mass. These points deserve clarification, but we do not pursue these issues further here. The details of such discussion are interesting but subtle and lie beyond the scope of this work. The complexity of most materials takes us beyond the electromagnetic theory, and a complete energy balance requires considering these systems as mechanical, thermodynamic, and electromagnetic systems. Therefore we expect that transformations of mechanical, thermodynamic and electromagnetic energies may occur in several ways. For example, polarization, magnetization and conduction are usually accompanied by a heat exchange. We also have phenomena like piezoelectricity and pyroelectricity in which stress or heating produce electric effects. Therefore we have phenomena subject not only to the laws of electrodynamics, but also to the laws of continuum mechanics and thermodynamics, if we treat them at a phenomenological level. Furthermore, we have thermoelectric phenomena that must be dealt with the methods of irreversible thermodynamics, since these phenomena appear in systems out of thermodynamic equilibrium. In these cases the energy balance is insufficient and the entropy production must be taken into account.

5. Conclusions

We have tried to present classical electrodynamics as a living theory that still offers challenging problems. At the microscopic level the interaction between a point, finite charge and the total field presents the problem of self-interaction. In its most general form, the problem is that the field is a (tensor) function of the charge-current distribution, while at the same time the charge-current distribution is a function of the field. In this way we have that the exchange of energy between _. _. _. _. a point charge and the total field takes the form j(E) • E(j). At present we have

492 .....

only approximations to this general problem. One of these is to assume that E is independent of This is the usual assumption when we calculate the power of the Lorentz force,

J.

However, this same formula has been extended to the total field by Lorentz, who in this way obtained an approximation to the self-interaction of a point charge and its self-field in the form of an expression for the radiation reaction force. The mathematical expression of this self-interaction leads to paradoxical interpretations, since some of these imply violation of energy conservation, while others imply violation of the principle of antecedence, or causality, as it is more commonly known [18]. At the macroscopic level classical electrodynamics also presents interesting problems. Here the challenge is the complexity of materials. This complexity involves the consistent application of classical electrodynamics, continuum mechanics, and thermodynamics. This combination of theories poses important problems of interpretation, as in the Abraham-Minkowski controversy, as well as problems of identification of the different subsystems that constitute a closed system. This identification is necessary to follow the different transformations of energy that may occur. An additional problem that appears in macroscopic CED is its relativistic generalization, since we do not have generally accepted relativistic generalizations of continuum mechanics and thermodynamics. As a final remark we note that energy and momentum conservation hold only in closed systems and no electrodynamical system is a closed system; other ( nonelectromagnetic) interactions must operate for the system to be stable. 6. References

1 . J. Slepian, J. Appl. Phys. 13 (1942) 512; C.S. Lai Am. J. 841; D. H. Kobe, Am. J. Phys. 50 (1982) 1163; P.C. Peters, (1982) 1165;; R.H. Romer, Am. J. Phys. 50 (1982) 1166; U. Schafer, Am. J. Phys. 54 (1986) 279; C. Jefries, SIAM Rev.

Phys. 49 (1981) Am. J. Phys. 50 Backhaus and K. 34 (1992) 386.

2 . R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics Vol. II (Addison-Wesley, Massachussetts, 1963), p. 27-8.

493

3 . W. Gough, Eur. J. Phys. 3 (1982) 83. The literature on the electromagnetic momentum density in static fields is extense. See for example: E. M. Pugh and G. F. Pugh, Am. J. Phys. 35 (1967) 153. R.H. Romer, Am. J. Phys. 34 (1966) 772, 35 (1967) 945. E. M. Pugh, Am. J. Phys. 39 (1971) 837. E. Corinaldesi, Am. J. Phys. 48 (1983) 83. H. M. Lai, Am. J. Phys. 48 (1980) 658, 49 (1981) 366. G. G. Lombardi, Am. J. Phys. 51 (1983) 213. N. L. Sharma, Am. J. Phys. 56 (1988) 420. J. M. Aguirregabiria and A. Hernandez, Eur. J. Phys. 2 (1981) 168. F. S. Johnson, B. L. Cragin and R. R. Hodges, Am. J. Phys. 62 (1994) 33. 4 . J.C. Slater and N. H. Frank, Electromagnetism (Dover, New York, 1969), p. 101. 5 . J. D. Buchwald, From Mazwell to microphysics (University of Chicago Press, 1985), p. 44.

6 . A. B. Pippard, Response and Stability (Cambridge University Press, 1985) p. 42. 7 . J. D. Jackson, Clauical Electrodynamics lnd. Ed. (Wiley, New York, 1975). 8 . L. D. Landau, E. M. Lifshitz and L. P. Pitaevski Electrodynamics of contin-

uous Media (Pergamon, Oxford, 1984).

9 . D. G. B. Edelen Applied Ezterior Calculus, (Wiley, New York, 1985). 10 . A. Kovetz, The Principles of Electromagnetism (Cambridge, 1990). p. 72 11 . J.C. Maxwell A treatise on Electricity and Magnetism (Dover, New York, 1954). p. 39. 12 . F. N. H. Robinson, Macroscopic Electromagnetism

(Pergamon, Oxford, 1973).

13 . L. de la Pena in Stochastic Proceues Applied to Physics and Other Related Fields, Proceedings of the Escuela Latinoamericana de Fisica, 1982 ed. A. Rueda and B. Gomez (World Scientific, Singapore, 1983). I

14 . M. Bunge, Foundations of Physics (Springer, Berlin, 1967). p. 177.

494 15 . A. Einstein, Ann d. Physik, 17 (1905) 132. English translation Arons and M. B. Peppard, Am. J. Phys. 33 (1965) 367.

by A. B.

16 . I. Campos and J. L. Jimenez, Eur. J. Phys, 13 (1992) 117. 17 . R.H. Good and T. J. Nelson, Clauical theory of Electric and Magnetic Fielda (Academic, New York, 1971). 18 . J. L. Jimenez and I. Campos, Am. J. Phys. 55 (1987) 1017. 19 . A. 0. Barut, Electrodynamics and Clauical theory of Fields and Particles, (Macmillan, New York, 1964) 20 . A. Einstein Ann d. Physik 18 {1905) 639. 21 . C. Moller, The Theory of Relativity (Oxford, 1962). 22 . F. Rohr Iich "Clauical Charged Particlea" (Addison-Wesley, Reading, Mass. 1965) 23 . F. Rohrlich, Am. J. Phys. 28 {1960) 639; 38 {1970) 1310. 24 . M. Von Laue, Ann. Phys. (Leipzig) 35 {1911) 524. 25 . I. Campos and J. L. Jimenez, Phys. Rev. D33 {1986) 607. 26 . L. Eyges The Clauical Electromagnetic Field (Addison-Wesley, Reading, 1972). G. Bekefi and A. H. Barret Electromagnetic Vibrationa, Wavea and Radiation (MIT Press, Cambridge, 1977). 27 . W. K. H. Panofsky and M. Phillips Clauical Electricity and Magnetiam (Addison-Wesley, Massachussetts, 1962). 28 . J. A. Stratton Electromagnetic Theory (McGraw-Hill, New York, 1941 ). 29 . J. P. Gordon, Phys. Rev. A8 {1973) 14. 30 . R. Peierls, Proc. Roy. Soc. London A347 (1976) 475. 31 . F. N. H. Robinson, Phys. Rep. 16 (1975) 313.

495

32 . S. R. de Groot and P. Mazur, "Non-Equilibrium New York, 1984) p. 376-378.

Thermodynamica"

(Dover,

33 . H. M. Lai, W. M. Suen and K. Young. Phys. Rev. A25 (1982) 1755.

NON-ABELIAN

STOKES THEOREM

BOGUSLAW BRODA• Department of Theoretical Physics, University of Lodi Pomorska 149/159, PL-90-296 Lodi, Poland

1. Introduction The (standard) Stokes theorem is one of central points of (multivariable) analysis on manifolds (see [1] for an excellent introduction). Low-dimensional versions of this theorem, known as the (proper) Stokes theorem, in dimensions 1/2, and the Gauss theorem, in dimensions 2/3, respectively, are well-known and very useful, e.g. in classical electrodynamics. In fact, it is difficult to imagine lectures on classical electrodynamics without heavy use of the Stokes theorem. The standard Stokes theorem has also being called the Abelian Stokes theorem, as it applies to ordinary (i.e. Abelian) differential forms. Classical electrodynamics is an Abelian gauge theory (gauge fields are Abelian forms), therefore its integral formulas are governed by the Abelian Stokes theorem. But a lot of interesting and important physical phenomena is described by non-Abelian gauge theories. Hence it would be very interesting and also fruitful to have at our disposal a non-A belian version of the Stokes theorem. Since non-Abelian differential forms need different treatment, one is forced to use a more sophisticated formalism to deal with this new situation. The aim of this chapter is to present a version of the non-Abelian Stokes theorem in the framework of the path-integral formalism [2).

2. From Stokes theorem to Stokes theorem The (Abelian) Stokes theorem says that we can convert an integral around a closed curve C bounding some surface S into an integral defined on this surface. Namely,

i

A- ds =

L

curl.A- ndu,

(2.1)

where the curve C is the boundary of the surface S, i.e. C = as, A is a vector field, e.g. the vector potential of electromagnetic field, and ii is a unit outward normal at the area element du. More generally, in any dimension,

I w = I dw, laM JM e-mail: bobroda)= H(r)lcl>),

= -x'(t)Af(x(t))fa,

(3.5)

where the creation and anihilation operators satisfy the standard commutation ( - ) or anticommutation ( +) relations (3.6)

500

It can be easily checked by direct computation that (3.7)

where ta = r:satas. For the irreducible representation R, the second-order Casimir operator C 2 is proportional to the identity operator 1, which in turn, is equal to the number operator N in our Fock representation, i.e. if Ta-+ ta, then 1 -+ N = Ors at as. Thus, by virtue of Eq. 3.7, we obtain a constant of motion N important for our further considerations (3.8) [JV,.H]_= o. Let us now derive the holomorphic path-integral representation

for the kernel,

of the evolution operator U. Literally repeating the standard textbook procedure [4] (it should be noted that our approach is in the spirit of the ''physical approach" to the index theorem, see e.g. [5]), we obtain U (ii, a; r", r')

=

f

+i

exp { ii(r")a( r")

1~"

L(ii( T), a( T))dr} DiiDa,

(3.9)

where L(a(r), a(r)) is the classical Lagrangian of the form given in Eq. 3.3, the imposed boundary conditions are: a.(r 11) = a, a( r') = a, and Da.Da is a functional "measure". Depending on the statistics of Eq. 3.6, there are the two ( =f) possibilities

equivalent as far as one-particle subspace of the Fock space is concerned, which takes place in our further considerations. Let us confine our attention to the one-particle subspace of the Fock space. As the number operator JVis conserved by virtue of Eq. 3.8, if we start from the one-particle subspace of the Fock space, we will remain in this subspace during all the evolution. The transition amplitude Urs(r 11 , r') between the one-particle states llr) = atlO) and lls) = a~IO) is given by the following scalar product in the holomorphic representation Urs(T II

I

T ')

=

I

u(-a, o; T II IT

')

e-iia

e -cior

arOs

d-d .J:::: . .1-. a auuuu.

(3.10)

One can easily check that Eq. 3.10 represents the object we are looking for. Namely, from the Schrodinger equation Eq. 3.5 it follows that for the general one-particle state oratIO) ( summation after repeating indices) we have (3.11)

501

Using the property of linear independence of Fack-space vectors in Eq. 3.11, and comparing Eq. 3.11 to Eq. 3.1, we can see that Eq. 3.10 really represents the matrix elements of the parallel-transport operator. According to Eq. 3.9, we can finally put Eq. 3.10 in the following path-integral form: Ur, (r", r')

=

j exp {-a( r')a( r') + i 1~" L(ii( T), a( T))dr}

ar( r")ii, (r')DiiDa,

(3.12)

where the free boundary conditions are imposed. [One can also rewrite Eq. 3.12 in the symmetrized form [4).) For closed paths, x(t') = x(t") = x, Eq. 3.12 gives the holonomy operator Urs(x). Since Urr is the famous Wilson loop, it seems that this formula could have some independent applications. Interestingly enough, the Wilson loop, which is supposed to describe a quark-antiquark interaction, is represented by a "true" quark and antiquark field, a and a, respectively. So, the mathematical trick can be interpreted physically. Obviously, the "full" trace of the kernel in Eq. 3.9 is obtained by imposing (anti-)periodic boundary conditions in the case of (anti-)commuting fields, and integration with respect to all the variables without the boundary term. Analogously, one can also derive the parallel-transport operator (a generalization of the one just considered) for symmetric n-tensors (bosonic n-particle states) and for n-forms (fermionic n-particle states).

4. The non-Abelian

Stokes theorem

Let us now define a (boson or fermion) Euclidean two-dimensional topological field theory of the fields ,,fJ,'I/) in an irreducible representation of g on the compact surface S, dims= 2, as =J0, Sc M, dimM = d, in the external gauge field A with the classical action Sc1 =

j (iDii[JD 1'1/)+ ~,ijFijVJ) dxi

or in the parametrization

xi(u 1 ,u 2 ), r' $ u 1

r

=r

-

I\ dx1,

i,j=l,

$ r", O $ u 2

::;

... ,d,

(4.la)

1, with the action

2

Sc1 = ls Cc1('1/), '1/))du

=

ls

€AB (iD Ai[JDB'I/)+ ~i[JFABVJ)d2 u,

A, B

= I, 2,

(4.lb)

where The described theory possesses the following "topological" gauge symmetry: o'lj)(x) = 8(x),

o,lj(x)

= ii(x),

(4.2)

where 8(x) and ii(x) are arbitrary except at the boundary as where they vanish. The origin of the symmetry Eq. 4.2 will become clear when we convert the action

502

(Eq. 4.1) into a line integral. Integrating by parts in Eq. 4.1 and using the Abelian Stokes theorem we obtain or in the parametrization Sc1= i

1

las

;fJD-r1/;dT.

To covariantly quantize the theory we will introduce the BRS operators. According to the form of the gauge symmetry (Eq. 4.2) the operator s is easily defined by s1/;= ,

s1/;= x,

s = 0,

sx = 0,

sx = {3,

s/3= 0,

s{3= 0,

where and x are ghost fields in the representation R, associated to 8 and iJ, respectively, ¢ and x are the corresponding antighost, and /3, {3 are Lagrange multipliers. All the fields possess a suitable Grassmann parity correlated with the parity of ;fJand 1/;. Obviously s2 = 0, and we can gauge fix the action in Eq. 4.1 in a BRS-invariant manner by simply adding the following s-exact term: S'

=

=s

(ls(

¢61/;

ls

(/361/; ± ¢6¢

± ;fJ6x) d 2 cr)

(4.3)

± x6x + ;fJ6f3)d2 cr.

The upper (lower) signs stand for the fields ;fJ, 1/; of boson (fermion) statistics. Integration after the ghost fields yields the quantum action

If necessary, one can insert yg into the second term, which is equivalent to the change of variables. Thus the partition function is given by Z

=

J

eis

D;fJD1j;D/3D{3,

(4.4a)

with the boundary conditions: /3ias = f31as= O. One can observe that the job the fields /3 and {3 are supposed to do consists in eliminating a redundant integration inside S. The gauge-fixing condition following from Eq. 4.3 imposes the following constraints 61/; = 0, 6;fJ = 0. Since the values of the fields 1/;and -0are fixed on the boundary as = c, we deal with the two well-defined 2-dimensional Dirichlet problems. Another possibility, more singular, is presented in [2].

503

Accordingly, we can rewrite Eq. 4.4a in the form (4.4b)

where the integration is confined to the boundary as. One can say that, in a sense, we have BRS-quantized the Abelian Stokes theorem passing from the theorem formulated for the classical action to the theorem formulated for the partition function (path integral). One can observe that by virtue of the Abelian Stokes theorem for a closed curve C, C = as, Eq. 4.5b is essentially equivalent to Eq. 3.12, modulo some boundary terms. It appears that this "quantized" Abelian Stokes theorem is a prototype of our main theorem. At present, we are prepared to formulate a holomorphic path-integral version of the non-Abelian Stokes theorem. Strictly speaking, this is a particular version of the theorem ( actually, the best-known one) that is applicable to the case of the Lie algebra valued 1(2)-form, i.e., a connection (curvature) on 1(2)-dimensional space. In the parametrized form the theorem reads

j exp [-ii( r')a( r') + i [" =

j exp

[

-a(r',O)a(r',O)

+i

L(ii( T), a( T))dr] a, (r")ii, (r')DiiDa

11, 1

11

T

l

Cc1(a,a)drda 2 ar(r",O)a 8 (r',O)DaDa,

(4.6)

where L(a(r),a(r)) and Cc1(a(r),a(r)) are defined by Eq. 3.3 and Eq. 4.lb respectively. The measure on both the sides of Eq. 4.6 is the same, i.e., it is concentrated on the boundary as as in Eq. 4.5b, and the imposed boundary conditions are free. Eq. 4.6 can be also put in parametrization-free form using Eq. 4.lb and Eq. 4.3a. By virtue of our earlier analysis, the proof of the Eq. 4.6 follows immediately from the Abelian Stokes theorem applied to Sc1, whereas it follows from the holomorphic path-integral representation for the parallel-transport operator that the LHS of Eq. 4.6 really represents the LHS of the "operator" version of the non-Abelian Stokes theorem (Eq. 2.8). It should be noted that the surface integral on the RHS of Eq. 4.6 depends on the curvature F as well as on the connection A entering the covariant derivatives, which is reminiscent of the path dependence. of the curvature :F.

5. Final remarks In this final section we will concentrate on the two issues: further generalizations of the presented version of the non-Abelian Stokes theorem and possible physical applications of this theorem. The proposed theorem is a very particular, though seemingly the most important, non-Abelian version of the Stokes theorem. It connects a differential I-form in dimension 1 and 2-form in dimension 2. The forms are of a very particular shape,

504

namely, the connection 1-form and the curvature 2-form. Thus, the possible generalizations should concern arbitrary differential forms in arbitrary dimensions. Since there might be a lot of variants of such a theorem depending on a particular application, we will only confine ourselves to presenting a general recipe. The general idea is very simple. First of all, we should construct a topological field theory on 8M, the boundary of M, in the external gauge field we are interested in. Next, we should quantize the theory, i.e. build the partition function in the form of a path-integral, where auxiliary topological fields are properly integrated out. Applying the Abelian Stokes theorem to the (effective) action in the exponent of the integrand we obtain the "RHS" of the non-Abelian Stokes theorem. If we wish also to extend the functional measure to the whole M we should additionaly quantize the theory to eliminate the redundant functional integration. The main source of applications of the non-Abelian Stokes theorem is coming from topological field theory of Chern-Simons type. The path-integral procedure gives the possibility of obtaining skein relations for knot and link invariants. In particular, it appears that only the path-integral version of the non-Abelian Stokes theorem permits us to nonperturbatively and covariantly generalize the procedure of obtaining topological invariants [5]. As a by-product of our approach we have computed the parallel-transport operator U in the holomorphic path-integral representation (see Eq. 3.12). In this way, we have solved the problem of the saturation of Lie-algebra indices in the generators T This issue appears, for example, in the context of equation of motion for Chern-Simons theory in the presence of Wilson lines (see Ref. 6, where an interesting connection with the Borel- Weil-Bott theorem and quantum groups has also been suggested). Our approach enables us to put those equations in terms of a and a purely classically. Incidentally, in the presence of the Chern-Simons interactions the auxiliary fields a and a acquire fractional statistics, which could be detected by braiding. To determine the braiding matrix one should, in turn, find so-called monodromy matrix, e.g., making use of non-Abelian Stokes theorem itself. 0 •

6. Acknowledgments

The author is grateful to Professors P. Kosinski, J. Rembielinski, A. Ushveridze and to Mr. H. Dq,browski for interesting discussions. The work has been partially supported by the KBN grants 2P30213906 and 2P30221706p01. References

l.M. Spivak, Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus (Benjamin, New York, 1965). 2.B. Broda, J. Math. Phys. 33 (1992) 1511. 3.Ya. Aref'eva, Theor. Math. Phys. 43 (1980) 353 (Teor. i Mat. Fiz. 43 (1980) 111); N. E. Brahe, Phys. Rev. D22 (1980) 3090;

505 M. B. Mensky, Lett. Math. Phys. 3 {1979) 513; Path Group: Measurements, Fields, Particles {Nauka, Moscow, 1983) {in Russian); P. M. Fishbane, S. Gasiorowicz, and P. Kaus, Phys. Rev. D24 {1981) 2324; L. Di6si, Phys. Rev. D27 {1983) 2552. 4.L. D. Faddeev, Introduction to functional methods, in Methods in Field Theory (Les Houches, Paris, 1975), edited by R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976); L. D. Faddeev and A. A. Slavnov, Gauge fields. Introduction to Quantum Theory (Benjamin, New York, 1980). 5.B. Broda, Mod. Phys. Lett. AS {1990) 2747; Phys. Lett. B254 {1991) 111; Phys. Lett. B262 {1991) 288; Phys. Lett. B280 {1992) 213; Mod. Phys. Lett. A9 {1994) 609. 6.E. Witten, Commun. Math. Phys. 121 {1989) 351.

EXTENSION OF OHM'S LAW TO ELECTRIC AND MAGNETIC DIPOLE CURRENTS HENNING

F. HARMUTH

Department of Electrical Engineering The Catholic University of America Washington, DC 20064, USA

Abstract Ohm's law gives the connection between the electric field strength and a. monopole current density due to electric charge carriers with negligible mass. It works well if the charge carriers are electrons but deviations were observed early on when the charge carriers were ions, such as the sodium and chlorine ions in seawater with masses equal to a.bout 42000 and 65000 electron masses. The extension of Ohm's law to such charge carriers with non-negligible mass may be found in many of the 18 editions of the first volume of the renowned book by Abraham and Becker. A further extension to dipole currents became necessary when the importance of magnetic dipole currents for signal solutions of Maxwell's equations was recognized. Electric dipole currents were always part of Maxwell's theory, but the name polarization currents obscured their physical significance. The ability of the dielectric of a capacitor to carry electric dipole currents makes it possible to drive currents through a. capacitor, even though the dielectric is a good insulator for monopole currents. The extension of Ohm's law to dipole current carriers with finite mass explains the old puzzle why the analysis of charge transfer from one capacitor to another requires a fictitious resistor whose value can be made zero at the end of the calculation. Both the finite mass and the fictitious resistor a.void a singularity due to infinite current peaks.

1. Introduction Consider the modified Maxwell equations for a medium at rest with scalar constants for permeabilityµ and permittivity e, electric and magnetic field strength E and H, electric charge and current density Pe and ~, as well as hypothetical magnetic charge and current density Pm and &n:

507

curl H

8E

= e at + ~ 8H

=µat + gm

-curlE

edivE = Pe µdivH = Pm eE = D, µH ~

= /e(E)

gm= /m(H)

=B

(1)

(2) (3) (4) (5) (6) (7)

The magnetic charge density Pm is needed only if the much demanded but not decisively proven magnetic charges or monopoles exist 1 . The magnetic current density ~ is needed even if there are no magnetic charges, since the existence of magnetic dipoles is not disputed and their existence implies the existence of dipole CUITents and dipole current densities. The existence of electric dipole currents has always been accepted since it is hard to explain how an electric current can flow through the dielectric of a capacitor without dipole currents, since the dielectric is an insulator for electric monopole currents. Electric dipole currents are usually called polarization currents, which obscures their physical significance, but there is at least one book that discusses them thoroughly [2]. Instead of Eq.(6) we usually see Ohm's law~= uE, where the electric conductivity u is a scalar constant. The current density ~ varies without any delay like the field strength E, which implies that the mass of the charge carriers is neglected. The generalization of Ohm's law to charge carriers with nonnegligible, constant mass may be found in the books by Becker [3, §58], [4, §58]. A charge carrier with constant mass mo, velocity v, and charge e is pulled by an electric field strength E with the force eE. Newton's mechanic yields the equation of motion dv

mo-=

dt

eE-!v

(8)

where !v is a term representing losses proportionate to the velocity. The constant ! is usually referred to as Stoke's friction constant, but we do not have to decide here what causes the losses represented by !v. ff there are No charge carriers per unit volume, each with charge e and velocity v, we obtain a current density ~:

(9) 1 For

a discussion of experiments that can currently be explained only by magnetic monopoles see Barrett [1]. Magnetic charges and currents were part of electromagnetic theory long before Maxwell's equations. Maxwell's theory eliminated them in principle, but there were always publications that used them long after the theory became one of the foundations of physics [7]-[12].

508 5(1)

--......

~

08

----------

,~\O},, ... (;}j:/ ,,,

,,,

06

/ I I

04

I

/

I

I

0 2 ,' I I

0

1

2 8-

3

4

5

FIG.I. The normalized electric field strength represented by the step function S(t) and the lagging current density ge(t)/uEo due to a finite mass of the charge carriers.

Substitution of Eq.(9) into Eq.(8) and multiplication extended to charge carriers with constant mass:

~

mo

Tmp

dge dt

+ Tmp

= -,

e

U

=

with N 0 e yields Ohm's law

= uE

Noe 2

e

Noe 2 rmp mo

(10)

ff the term Tmpdge/dt can be neglected we obtain the usual Ohm's law with the conductivity u. The effect of the term d~/dt will show up primarily if the charge carriers are not electrons but ions. For instance, the sodium and chlorine ions in seawater have masses equal to about 42000 or 65000 electron masses. Let the field strength E in Eq.(10) have the time variation of a step function E(t) = E 0 S(t). To avoid the point t = 0 for which S(t) is not differentiable we consider the infinitesimally larger time t = +0. H we require the current density ge( +0) to be zero we obtain from Eq.(10) the solution (11) which shows the current density lagging behind the electric field strength E 0 S(t). Figure 1 shows the time variation S(t) of the field strength and the lagging current density. For fast moving charge carriers, such as in accelerators, the mass mo in Eq.(8) will increase with the velocity v. We may introduce the relativistic correction

mo

m=------

(1 -

v2

(12)

/c2)1/2

for the mass and use the magnitudes of v and E since the two vectors have the same direction: 2 ( 1- 2v )-

c

112

dv l -+-v= dt

Tmp

-eE

mo

(13)

Using normalized notation this equation assumes the form

(19e

p2)-112

'%;+ /3= 11, /3(9= +o) = o

= Noev, 9c = Noec, /3= v/c = 9e/9c, 9 = t/Tmp, 11= TmpeE/moc

(14)

where /3is either a normalized velocity v/c or a normalized current density 9e/9c· The current density 9c is the largest possible one with N 0 charge carriers per unit volume and the charge e per charge carrier. The initial value /3(+0) = 0 implies that we are interested in current densities 9e or velocities v that start at zero for t = 0. The nonlinear differential equation Eq.(14) can be solved analytically by a series expansion or be direct numerical integration. We will briefly show the analytical method for comparison but soon see that numerical integration is simpler. Using the series expansion

(15) we eliminate the square root from Eq.(14):

d/3 1 2 d/3 -+/3+-/3 -+···=11 d9 2 d9 The variable second term:

/3 is

(16)

represented by a series expansion that we terminate with the

(17) (18) For

/31
0

(23)

(24)

of Eqs.(21) and (24) into Eq.(17) yields the first order relativistic

f-1(8)= 110( 1 - e -9)

JJ

3 + 21 1103 e -9 ( 2

1 e -29) 8 - 2e -9 + 2

(25)

Figure 2 shows a plot of the nonrelativistic current density 9e(8)/ c, Eo according to Eq.(11) and the relativistic current density /3(8) = 9e(8)/gc obtained by numerical integration of Eq.(14) for a step function 11= S(8). The interval 3 < 8 < 4 is shown enlarged in Fig.3. The relativistic current density /3(8) according to Eq.(25) is shown for T/O= 1 by the dotted line that is too close to the solid line to show in Fig.2.

511 0.49

04

12

14 8-

36

18

4

FIG.3. Enlargement of the interval 3 < (J < 4 of Fig.2. The dashed line shows the nonrelativistic current density /3(9) = 9e(9)/gc according to Eq.(11), the solid line the relativistic current density obtained by numerical integration of Eq.(14) for 1J= S(fJ), and the dotted line the relativistic current density obtained by a series expansion according to Eq.(25) with 110= 1.

2. Induced Electric Dipoles With Constant Mass We distinguish between induced and inherent electric dipoles. A hydrogen atom is electrically neutral. An electric field strength will pull the electron in one direction and the proton in the other to produce an electric dipole. A current flows while this pulling apart is in progress. Any individual atom can be polarized in this way. A gas, liquid or solid consisting of many atoms can be polarized by making dipoles out of its atoms, but this will work only for materials classified as insulators. What is conventionally called an insulator is an insulator for monopole currents but a conductor for dipole currents. Vice versa, a good conductor for monopole currents will not permit the creation of electric dipoles since the field strengths will be insufficient. No dipole current will flow and the material must be classified as an insulator for dipole currents. In many cases both monopole and dipole currents can flow simultaneously. A partly ionized gas has positive ions and negative electrons to carry monopole currents, but there are also polarizable neutral atoms that can carry dipole currents. The dielectric of a capacitor is usually a very good insulator for monopole currents, which makes the dipole current dominant. But even the best insulator will permit a little leakage of monopole current. Is there a material that is an insulator for both monopole and dipole currents? A material consisting of neutrons only would be such a super-insulator 2 , but it exists at this time in neutron stars only. Another "material" that comes to mind is a perfect vacuum. But a capacitor with a vacuum instead of some dielectric material between its plates permits a dipole current to pass. We explain this fact by means of the concept of vacuum polarization. Since only dipoles are needed, the conservation law of charge is not violated by the creation of dipoles. The conservation of momentum is also possible since for any positive particle moving in one direction there is a negative particle moving in the opposite direction. The problem is only that all the known charge carriers have a mass. Does vacuum 2 Quadrupole

and still higher order multipole currents would not be conducted either.

512

polarization prove the existence of mass-free charge carriers? This is a perfectly valid question. Quantum mechanics provides an alternative. Heisenberg's uncertainty relation in the form flEflt > n permits the existence of a mass m = flE / c2 for a time t < flt. Hence, quantum mechanics permits that electric dipoles with mass are created from nothing, exist for a time t < flt, and then disappear again while new dipoles are created in their stead. Not everybody will be satisfied by this explanation but it is currently preferred over mass-free charges. We ascribe features to the vacuum or empty space to explain observed physical phenomena about as freely as our ancestors used gods for explanation. Upon closer study these feature have a tendency to be as elusive as the gods of old [5]. We note that vacuum polarization explained by dipoles with mass made unobservable by Heisenberg's uncertainty relation still produces an average, observable mass. In addition to the induced dipoles we also have inherent electric dipoles. Most molecules are already dipoles before any electric field is applied (e.g., H20, HCl, NH 3 ). The molecules of water vapor will thus rotate to line up with an applied electric field strength, in addition to having the negative electrons and the positive nuclei pulled in opposite directions. This orientation polarization is of primary interest for magnetic dipoles and will be considered later on in connection with magnetic dipoles. The following investigation is carried out for atomic hydrogen, but its extension to molecular hydrogen and other atoms is not difficult. We assume that the proton and the electron can be represented by two spheres that are pulled apart by an electric field strength while an elastic force-represented by a spring-holds them together. In terms of Bohr's atomic model the circular orbit of the electron is stretched by the electric field to produce an electric dipole in the direction of the electric field strength as a time average. One could go one step further and consider the spherically symmetric probability density function of the location of the electron stretched into the shape of an American foot ball by the electric field strength. However, the simple model of two charged spheres held together by a spring is sufficient to permit the use of Maxwell's equations and their modification well into the visible light region in terms of sinusoidal waves [3, §58], (4, §58]. The electric field strength E applied to a medium does generally not imply the same field strength at the location of a polarized molecule or atom since the surrounding polarized molecules produce a field strength of their own. For instance, the effective field strength F

(26) applies to a liquid or a cubic crystal without any permanent dipole moment, where No is the number of atoms or molecules per unit volume and ae is the electric polarizability (3, §26], (4, §26]. For a gas one may use the simpler relation

513 +

mp=1836me

FIG.4. The large mass mp of a proton relative to the mass me of an electron implies that the polarization of a hydrogen atom by an electric field strength produces a current that is almost exclusively due to the electron.

(27) For a description of the movement of the electron in our two-spheres-with-aspring model of the hydrogen atom we use the equation [2, §58], [3, §58] mo (

J

Noev dt

2

Noe Tmp = --~E

(31)

mo

For N 0 ev we write the current density ge of Eq.(9):

(32) Here up is the electric polarization current conductivity or the electric dipole current conductivity. Equation (32) is Ohm's law for electric polarization or dipole currents with constant mass. A comparison with Eq.(10) shows that the integral is characteristic for dipole currents, while a term d~/ dt occurs in Oh.m's law for monopole as well as for dipole currents if a delay caused by the need to accelerate the charge carriers to give them a velocity is taken into account. Consider once more Fig.4 to see that one cannot always distinguish between a dipole current and a monopole current. If we observe a hydrogen atom and see it become polarized we will conclude that a dipole current is flowing. The pulling apart of the negative electron and the positive proton may reach a maximum, but if a sufficiently large field strength is applied long enough the electron will be separated completely from the proton and we get an ionized gas whose moving positive and negative charge carriers create a monopole current. Hence, one cannot always distinguish clearly between dipole and monopole currents. Indeed, the breakdown of an insulator due to an excessive applied electric field strength always means that a dipole current has changed to a monopole current. Assume that O"p in Eq.(32) is obtained by measurement. If we can devise a method to obtain Tmp and Tp we can derive numerical results from Eq.(32). To determine Tp and Tmp we differentiate Eq.(32) and reorder the terms:

515

d2~ 1 d~ -+--+-~=-dt2 Tmp dt

1 r;

up Tmp

dE dt

(33)

Let E be an electric step function E 0 S(t) applied at the time t = 0. The right side of Eq.(33) is then zero fort > 0. Substitution of~ = ~oe-t/r yields:

T 1- -

r2 p 2Tmp

T2

~

•2 -

(34)

p

2Tmp

Consider first the aperiodic limit case Tmp

--q-Tp

-

-

1 2

(35)

which yields the solution

(36) A current density ~(t)

that is zero at t

= 0 is obtained

by chsing~

2

= 0: (37)

For the determination of ~ 1 we return to Eq.(32) and observe that ~( +0) implies f ~ dt = 0. The first derivative must thus have the value:

d~(+0) -----dt

upE Tmp

d~( +o) _ d(t/rp)

upErp Tmp

=0 (38)

Equation (37) yields:

(39)

516

Equation (37) assumes thus the form: q

1

= -, 2

(40)

This function is plotted in Fig.5. For Tmp/Tp = q < 1/2 one obtains instead of Eq.(36) the solution (41) which assumes the form

(42) if we require~=

0 fort=

0. For the determination of geo we rewrite Eq.(34)

(43)

and Eq.( 42):

(44) Equation (38) yields:

517

d~( +0)

=

~

~o

~(8)

upEo q

= ~o

= -upEo = (l

upEo

(-_!_ + _!_) = -~o ~

~

(1 - 4q2)- 112

- 4 q2)-1/ 2 (e-D/9 2

_

=!( q

(1 - 4q2)1/2 q

1 )-1/2Sinh8(-l 2 4q - 1

eD/81)

4q 2

-1)1/2

(45)

Figure 5 shows plots for Tmp/rp = q = 1/3, 1/4, 1/5, 1/6. Equation (34) becomes conjugate complex for Tmp > rp/2

(46) and we get

t > 0, q = Since the current density ge must be zero at the time t function E 0 S(t) is applied, only the solution g.

= g..e-t/29rp

sin

is physically acceptable. The constant

mp Tp

(

47)

= 0, when the electric step

:P( ~2)1/2't > 1-

gei,

T,

0

(48)

follows from Eq.(38):

~s

- upEo ( -

___,;;;_.__

q

1 )-1/2 1- 4q2

(49)

Figure 6 shows plots of

~ (8) =- 1 ( 1-upEo

q

1 )-1/2 e4q2

0 / 2qsin8

( 1--

l )1/2 4q2

(50)

518 05

10 0 -01

FIG.6. Plots of Eq.(50) for q

= Tmp/Tp = 1, 2, 3, 4, 5.

for q = I, 2, 3, 4, 5. We need values of Tp and Tmp to use Eq.(32). Consider a capacitor with two metal plates and hydrogen gas between the plates. This should be atomic hydrogen H rather than molecular hydrogen H2 • The value of up is not needed directly if an electric voltage Vo with the time variation of a step function VoS(t) is applied to the capacitor plates since the derivative of S(t) is zero for t > 0. The voltage V0 S(t) drives a current with a time variation as those of the functions in Figs.5 and 6. Matching the observed time variation of the current with plots according to Figs.5 and 6 determines T P and T mp directly from observation. In principle one should be able to calculate T P and T mp from one of the models of the hydrogen atom, but this calls for a transient solution of a problem of quantum mechanics. The solution applying to the Stark effect is of no help since this is a steady state effect and not a transient effect. Indirectly the value of up in Eq.(32) enters our experimental determination of Tp and Tmp even though the right side of Eq.(33) is zero. The number N 0 of the hydrogen atoms per unit volume depends on the pressure and temperature of the hydrogen gas between the capacitor plates. By changing pressure and temperature one can measure T P and T mp for various values of N O •

3. Induced Electric Dipoles With Relativistic

Mass

For nonrelativistic velocities we had deduced from Fig.4 that the contribution of the proton to the dipole current of the hydrogen atom or any other atoms as well was negligible. This does not hold for velocities Ve of the electron or vp of the proton close to the velocity of light, since the greater velocity of the electron implies that its mass me is going to increase faster than the mass mp of the proton. Conservation of momentum demands (51) Denoting the rest masses of proton and electron by mop and moe, and observing that the direction of Ve and Vp is always opposite, we obtain from Eq.(51): (i -

v:I c2) /2 i

(52)

519

0.025

0.08

0.02

0.06

t 0.0,5

-

t0.04

u

.!:=

:,ii'-0.01

~

0.02

0DOS O0.9

0.92

094

0.96

0 0.99

098

0.992

Vele-

0.994

0996

0.998

Vele-

025 0.8

0.2

f 0.6

r 0.15

-

~

u

;?-

>...0.1

0.4 0.2

0.05 O0 999 09992

09994 0.9996 0.9998 v~/c-

°09999 0.99992 099994 099996 099998

1

Vele-

FIG.7. Ratio Vp/ c of the velocity of the proton to the ratio Ve/ c of the electron of a hydrogen atom being polarized by an electric field strength according to Eq.(53); mop/moe 1836.

=

The velocity Vp of the proton is derived as a function of the velocity electron: Vp = mop [1 - (1 - m~e/m~P) v; /c 2] 112

Ve

of the

(53)

For Ve -+ c we get Vp -+ Ve. In this case the proton contributes as much as the electron to the dipole current. A plot of Eq.(53) is shown in Fig.7. We may see from Fig. 7 that the velocity Ve of the electron must be very close to the velocity of light before the proton contributes significantly to the current density. For instance, a proton velocity vp/c = 0.01 requires an electron velocity ve/c = 0.99852, which implies a ratio of the current densities 9e/ 9p = ve/vp = 0.99852/0.01 = 99.85; hence, the proton contributes about 1% to the current density. For vp/c = 0.1 we get ve/c = 0.9999853 and 9e/9p = 0.9999853/0.1 = 9.999853 and the proton contributes about 9% to the current density. We may extend Eq.(31) to electrons with relativistic mass increase by replacing the constant mass m 0 with moe m =me=

where the subscript e of me, moe, and

(1 - v~/c2)1/2 Ve

(54)

refers to the electron:

(55)

520 Since N 0 eve is the electric current density ~e due to electrons we may make the following substitutions to replace Ve by ~e: Ve=

gee,

v~ = (

Noe

c

9ee )2 = (9ee)2, Noec 9ec

9ec =Noec

(56)

The newly introduced quantity 9ec = N 0 ec denotes the limit for the possible current density with N 0 charge carriers per unit volume, each having the charge e. Equation ( 56) assumes the form

(57) We could again obtain a solution by means of a series expansion as we did for the monopole current in the Introduction, but it is easier to use direct numerical integration. The vectors gee and E are replaced by their magnitudes 9ee(t) and E(t). Furthermore, we divide Eq.(57) by Yee and write g for 9ee/9ec:

(58) We choose the initial current density at the time t

g( +o) = 9ee(+0)

= +0 equal to zero

=O

(59)

9ec

and obtain from Eq.(58) the initial derivative dg/dt:

dg(+O) _ d[9ee(+0)/9ec] _ O"pE(+0) dt dt

(60)

In order to eliminate the integral in Eq.(58) we differentiate, multiply with r. 2/rmp, and make the listed substitutions for q, ec, and 8: P

d2g + ! dg + 9 = ec [(l _ 2) 112 d(E/ Eo) _ (l _ 2)_ 112~ dg] 9 9 9 d82 q d8 q d8 E 0 d8 q = Tmp/Tp, For a step function E

ec = Eoup/9ec,

= Eo S( 8) we obtain

8 = t/rp,

E = Eof(8)

for 8 > 0 the differential equation

(61)

521

o

z

4 0_

6

a

10

FIG.8. Plots of g(9) according to Eq.(67) for q = 0.4 and ec

( Jlg2 + ! dg + g) (1 d8

q d8

92)1/2

=-

g dg/d8

= 1, 2, 3, 4, 5, 6.

ec

(62)

q

with the initial conditions from Eqs.(59) and (60):

g(+O) = 0,

dg( +o) --------d8

Eo _21 4q 2

(67)

(68)

= upEo/ Yee

Plots of these functions are shown in Figs.8 and 9. Not surprisingly, the amplitudes of the functions are proportionate to the values of ec. Corresponding relativistic plots obtained from Eq.(62) by direct numerical integration are shown in Figs.IO and 11. As expected, the peak of y( 8) never exceeds but only approaches 1, which corresponds to a current density Yee = N 0 ec or a velocity Ve = Vp = c. In addition to the reduction of the peak current densities we also see a significant stretching of the pulses of Fig.8 in Fig.IO and an increase of the periods of Fig.9 in Fig.IL

4. Magnetic Dipoles With Constant Mass The prototype of the magnetic dipole is the compass needle or the ferromagnetic bar magnet. In the presence of a magnetic field strength they try to rotate so as to line up with the field strength. The atoms or molecules of a gas usually have

523

q=2

2

-0.2

0-

· ·~ ·,.,_' :--':~--;{~~:..---·"··-'10 ,._ . --....:.,_ .....4 ·---·--· B

FIG.11. Plots of g(9) according to Eq.(62) for q = 2 and ec = 1, 2, 3, 4, 5, 6. tttttttt

tttttttt

H

H

FIG.12. Two magnetic charges ±qm with equal mass mo are pulled with the force ±qmH the distance ±s from their rest position.

a magnetic dipole moment-that is the sum of a number of components-which makes them like small bar magnets. In solids the atoms or molecules interact to yield the effects of para-, dia-, or ferromagnetism but there is no such differentiation for a gas. In addition to these inherent, experimentally well documented dipoles we have the hypothetical magnetic charge dipole that is produced by pulling a positive and a negative charge ±qm apart by a magnetic field strength just like the electric dipole discussed previously was produced by pulling apart a positive and negative electric charge by an electric field strength. This hypothetical dipole is easy to analyze and we will use it as a standard of comparison for inherent dipoles which we will be able generally to investigate by numerical methods only. We start with the model of Fig.12 that shows two magnetic charges ±qm with equal mass m 0 pulled with force ±qmH the distance ±s from the rest position. The magnetic charge qm has the dimension Vs. We may use the left side of Eq.(28) and keep in mind that the total magnetic dipole current density will be twice as large as that calculated for charges with one polarity according to Fig.12 for any velocity, not just velocities close to c:

1/2 and no oscillations for q < 1/2. The aperiodic limit q = 1/2 yields the special solution:

s(B) = HoqmTpTmp! [l _ (l + B)e-6] 81 = 82

=

mo 1, 8 = t/rp, q

q

= 1/2,

s( +0) = ds( +0)/ d8 = 0

(72)

Plots of s(8)m 0 /qmr';H for q = 1/2, ... , 1/6 are shown in Fig.13. These functions show the distance s the magnetic charges have been pulled from their original rest position s = 0 at t = 0. In addition to the distance s(t/rp) we will be interested in the velocity v(t/rp) of the charge qm:

(73) Figure 14 shows a plot of v(t/rp) for q = 1/2, ... , 1/6. ff the number of magnetic dipoles per unit volume is N 0 , each with two charges ±qm having velocities ±v,we get the magnetic dipole current density gm = 2N 0 qmv, where the factor 2 is due to the equal mass of the charge carriers for +qm and -qm in Fig. 12:

525 1/6

-- --- ------------

6 5

o

2

4 0_

6

a

10

FIG.13. The distances= s(t/rp) from the rest position s = 0 of a magnetic charge qm according to Eqs.(71) and (72) for q = 1/2, 1/3, 1/4, 1/5, 1/6.

0

2

4

a--

6

8

10

FIG.14. The velocity v = v(t/rp) of a magnetic charge qm being pulled by a magnetic field strength H from its rest position s = 0 according to Eq.(73) for q = 1/2, 1/3, 1/4, 1/5, 1/6.

~

gm(8) spHo

= 2Noqmv; = 2(l

= Noq!

Tmp/mo

_ 4q2)-1/2 (e-8/82 _ e-8/81),

gm(B) = 48espHo

Sp

8,

q = 1/2

q

< 1/2 (74)

With v = ds/dt we may rewrite Eq.(69) as Ohm's law for magnetic dipole current densities due to a hypothetical, polarizable magnetic dipole:

(75) Here Sp is the magnetic polarization current conductivity or the magnetic dipole current conductivity.

526

Q=I

roe ...

...........

E

...... 06 '1: er ~c::,04

........ _- ..________ 2 ........... __....

E

"' 0 2

0

2

4

a-

6

8

10

F10.15. The distance s = s(t/rp) from the rest position s = 0 of a magnetic charge qm according to Eq.(76) for q = 1 (solid line) and q = 2 (dashed line).

-----·-.......... 0

8

2

-0.1

10

...... ____

FIG.16. The velocity v(t/rp) of a magnetic charge qm being pulled by a magnetic field strength H from the rest position according to Eq.(77) for q = 1 (solid line) and q = 2 (dashed line).

For q > 1/2 we obtain decaying oscillations. Equation (71) is replaced by an equation with complex values of 81 and 62

(76) while Eq.(73) is replaced by: - -----1 ds(8) _ qmTmpHol (l - - 1 - ----';;.....___ Tp d8 m0 q 4q 2

v (8) -

gm= 2Noqmv

= 2spHo-q1 ( 1 -

)-l/

2 _ 912 . e q sin

(

1- -1 ) 4q2

1/

1 )-1/2 _ 9 2 . ( l ) 1/2 - 2 e / q sin 1 - 8 4q 4q2

Figures 15 and 16 show s(6) and v( 8) for q = 1 and q = 2.

2

8 (77)

527 a ttttttttttttt

H.B

b

ttttttttttttt

H.B

-VyQm

ttttttttttttt

tttlttttttttt

FIG .17. Ferromagnetic ba.r magnet in a homogeneous field (a) and its rep la.cement by a. thin rod with magnetic charges ±qm at its ends (b ).

We turn to inherent magnetic dipoles and investigate a (ferromagnetic) bar magnet of length 2R in the homogeneous magnetic field of strength H and flux density Bas shown in Fig.17a. Let mmo with dimension 1 Am 2 denote the magnetic dipole moment and J the moment of inertia with dimension Nms 2 = kgm 2 of the bar magnet. The equation of motion equals

J~~ = -mmoBsind

(78)

and the velocity of the end points of the bar has the value

v(t) = -R-

diJ

dt

(79)

which suggests to introduce a friction or attenuation term {mv into Eq.(78):

(80) The term sin iJ shows that one can obtain generally numerical solutions only of this differential equation, but an analytical solution is possible for small values iJ ~ sin iJ and we will investigate it first. Substitution of iJ = Ae-t/r into the equation (81) yields: 1 If

=

we write fflmoB fflmo/JH, the term mmo/J has the dimension Vsm and the symmetry with the electric dipole moment in Eq.{28) with dimension Asm is obtained, if we multiply that equation with the distances and obtain on the right side (es)E [AsmxV /m].

528

(82) For

(83) we get non-oscillating solutions

{)= A1e-t/ra + A2e-t/r4

(84)

corresponding to the solutions of Fig.14, while

(85) yields oscillating solutions corresponding to the ones of Figs.15 and 16 with complex time constants r 3 and r 4 :

[i+ 4~~;;B _l) j (

T4

=

1 2 / ]

-2!-:-~-B [1- j ( 4~~;;B -1)1/2]

(86)

We will analyze the aperiodic limit case:

The velocity v(t) of the end points of the bar magnet equals:

(88)

529 0.14

ed. --;;, 0.04 ,/

... >

OD2i

,

-····--.......... ....

··-..... "--·----------

,

2

3

t/tp-

4

5

FIG.18. The functions v(t)rp/ R (solid line) and v11(t)rp/ R (dashed line) for -Do= 1r/8 in the interval O < t < 5rp according to Eqs.(92) and (93).

We choose c1 and c2 in Eq.(87) so that the following initial conditions are satisfied at the time t = 0 19(0)= -Do v(O) = 0

-+

(89)

-+

and obtain:

D(t) = Do

(1-:P)

e-t/T,,

Do~ sin Do

(90)

A series expansion yields for small values of t/rp: (91)

The velocities v(t) and v11(t) have the values:

v(t)

= -Rd-0 = R-Dot e-t/rp & T2

~t

(

p

v,(t) = v(t) sin D =

e-t/T, sin [0 0

(

1+

:J

e-t/T,]

~)

(93)

Plots of v(t) and v11(t) are shown for 190 = 1r/8 in Fig.18. For larger values of '190 we rewrite Eq.(80) with the help of Eq.(70)

(94)

530 b

a

05

3n/4

7n/8

3n/4

7n/8

0.4

04

f 03

l 03

~02 >

~02 .... >

-

01 2

3

4

5

1/tp-

FIG.19. The function v(t)rp/R = d-0/d(t/rp) (a) and v11(t)rp/R = sin-Od-0/d(t/rp) (b) for q = 1/2 and n-.10 = 1r/8, 1r/4, 1r/2, 31r/4, 71r/8 in the interval O < t < 5rp.

solve it numerically for the initial conditions t?(0) = nt?o = t?n and dt?(0)/d(t/rp) = 0 by computer, and plot v(t)rp/ Ras well as vy(t)rp/ Ras shown in Figs.19a and b for q = 1/2 and nt?0 = 1r/8, ... , 71r/8 in the interval 0 < t < 5rp. The functions for ,?0 = 1r/8 in Figs.19a and b are essentially equal to the two functions plotted in Fig.18. Let us connect the velocities v 11(t) with the current density gm(t) of a magnetic dipole current. First we replace the bar magnet in Fig.17a by a thin rod with fictitious magnetic charges +qm and -qm at its ends as shown in Fig. I 7b. The magnetic dipole moment mmo equals 2Rqm. The charge qm must be connected with the magnetic dipole moment by the relation ffimo

qm [Vs]= µ 2R

[ Vs Am.2

Am~

l

(95)

where µ is the magnetic permeability, to obtain the dimension Vs for qm. The magnetic dipole current produced by such a bar magnet is 2qmvy(t). H there are No bar magnets in a unit volume, all having the direction ,?0 and velocity v 11(t) = 0 at t = 0, we would get the dipole current density ~(t) = 2N 0 qmvy(t)y/y for the current flowing in the direction of the y-axis. For many randomly oriented bar magnets we must average all velocities vy(t) in the sector 0 < ,? < 1r as well as in the sector 1r 0, the Jacobian matrix formed by the partial derivatives of the components Di 1 Di 2 • • •Di;~ a, of the prolonged equations pr 1~ a = 0 with respect to the variables (xi, u[k+ij) has a constant rank in Jk+ 1(E). For example, a constant coefficient system of linear differential equations is always non-degenerate. The importance of non-degenerate systems for us lies in the fact that if a given differential function

vanishes on the solution manifold n1+1 of a non-degenerate F can be expressed as a combination

F(xi ' u[k+ij) =

""""Hi1a ..,i; (xi , u[k+ij)D· ~

11

system ~ a

... D·lj ~ a

= 0,

then

(2.16)

O~j~l

of products of the components of pr 1~ a = 0 and some functions H~1 ···i~(xi, u[k+ij) on Jk+ 1(E) (see ref. [20]). Note that a differential function F on some jet space Jk(E) can be naturally regarded as a differential function on any J 1(E), l > k. Now suppose that fa = fa(xi) is a solution of (2.13). Then, by the very definition of a solution for a system of differential equations, for each (xi) E Rn and for any l > 0, the point

is contained in the prolonged solution manifold 'R~. Thus if a function F = F(xi, ulk+ij) vanishes on the solution manifold 'R~ for some p > l, then, in particular, the composition Fopr

i i f(x)=F(x,fa(x),

k+l

.

k+l

.

i 8fa(x') 8 fa(x') , ... , 8 xJ1· . . . a· )=0 8 xJ1. x11c+1

vanishes identically. However, the vanishing of the compositions F o pr k+lf (xi) for all solutions of the system does not necessarily imply the vanishing of F on any of the solution manifolds 'R~, p > l ( cf. ref. [25]). Nevertheless, with a slight abuse of terminology, we will say that a differential function F = F( xi, u[k+ij) vanishes on solutions of a kth order system ~ a = 0 if, for some p > l, F vanishes on the solution manifold 'R.~.

566

A generalized vector field v on E is called a generalized symmetry Backlund symmetry) of the system .6.a = 0 if the expressions prv.6.

4

=0,

a=l,2,

... ,m,

( or a Lie-

(2.17)

vanish on solutions of the system. The order of the symmetry is the order of the generalized vector field v. A point symmetry (or a Lie symmetry) of the system .6.a = 0 is an ordinary vector field v on E satisfying the defining equations (2.17) for a symmetry. Equations (2.17) simply state that the action of the vector field v on functions as defined in (2.12) maps solutions of the system .6.a = 0 into other solutions. The defining equations of symmetry (2.17) typically lead to a large, overdetermined system of partial differential equations, so called determining equations of symmetry, for the components of the vector field v. While these equations are usually computationally complex, the method for obtaining them is completely algorithmic, and, as one might expect, several computer algebras presently exist (see, for example, refs. [8], [9], [41]) that set up the determining equations for point symmetries and also attempt to solve them, often succeeding. Interestingly, one can also obtain information on the dimension and the Lie algebra structure of the space of point symmetries of a given system of differential equations without having to integrate the determining equations, see ref. (40]. It is easy to see that a total vector field

always satisfies the defining equations (2.17) of symmetry. Hence, on account of the prolongation formula (2.10), we can, without loss of generality, always replace a generalized symmetry by its evolutionary form. This effectively simplifies computations entailed in solving the determining equations by reducing the number of unknown functions from m + n to m. Note that there is a one-to-one correspondence between point symmetries and first order evolutionary symmetries whose components are affine functions in the derivative variables Ua i· ' A symmetry is called trivial if it vanishes identically on solutions of the system. Thus the group action associated with a trivial symmetry leaves solutions of the system fixed. Two symmetries are called equivalent if they differ by a trivial symmetry. In a complete symmetry analysis of a system of differential equations one clearly only needs to classify all classes of equivalent symmetries. One can usually accomplish this by simply using the prolonged equations to solve a certain set of the variables ua,i 1 ... i, in terms of the remaining variables and subsequently, with the help of these relations, expressing a symmetry solely in terms of the remaining variables.

567

Next suppose that A L1

~ cab,iih···i, (xi)ub

a=

~

. .

,Jl)2·

.

.. J1

= 0'

a=l,2,

... ,m,

(2.18)

O~l~p

is a system linear differential equations, and let

be a generalized symmetry of (2.18) in evolutionary form. Then the prolongation formula (2. 7) together with the defining equations (2.17) of a symmetry shows that the characteristic Q a satisfies ~ cab,iih···i 1 (xi)D · D · · · · D · Qb - 0 ~

)1

)2

)I

-

'

a -- I ' 2 '

. " • '

m'

(2.19)

O~l~p

on solutions of (2.18). Thus the characteristic of an evolutionary symmetry for a linear system, upon substitution of a solution of the system, yields a new solution. This observation suggests that the composition v Ql 0q2 of any two evolutionary symmetries v q1 and v Q2 of a linear system obtained by replacing the derivatives Ub,11 12 ···Ip in the expression for v qi by the corresponding derivatives D, 1 D12 • • • D,PQ~ of the components of vq2 is again a symmetry.

Proposition 2.1. Let ~a = 0 be a non-degenerate system of linear partial differential equations as in (2.18) with the evolutionary symmetries VQ1 and VQ2. Then the composition vector field vq1 0Q2 is also a generalized symmetry of the system ~a= 0. Proof. Let the orders of the symmetries Q~ and Q! be k1 and k2, respectively. The characteristic Q! defines the mappings Ql: Jk 2 +'(E) --+ J 1(E) by

Now the characteristic

Qa of the composition vector field VQ10 Q2 is precisely

It is easy to check by a direct computation that (2.20)

568 Thus, by using induction on (2.20), we have that ( cf. (2.19)) (2.21)

'Ra

We need to show that the right hand side of (2.21) vanishes on for some s > 0. Since the system D.a = 0 is non-degenerate and since v q 2 is a symmetry of D.a = 0, we have, on account of (2.16) and (2.17), that

L

Pr vq2 D.a --

H ba 'i1

···i·' ( x i

,

u [p+q]) D 11· • • • D i;· uA b ,

(2.22)

O~j~q

for some q > 0. Now a repeated application of the total derivative operators Di on equation (2.22) implies that (2.23) for all l > 0. In arriving at equation (2.23) we have used the fact that the prolongation of an evolutionary vector field commutes with the total derivative operators Di, For a proof of this basic fact, see ref. [33], chapter 5. Now due to the linearity of the prolonged equations pr I D.a = 0, equation (2.23) implies that the point ull+q)), ... (x i 'aQ2(xi ' ull+q))'&1D· Q2(xi a '

,

D·&1···&1. Q2(xi ull+q))) E ,ol a ' /"\,a

is contained in 'R.~, whenever (xi, ull+q]) E n':,q.Consequently, using the fact that vq1 is a symmetry of the system (2.18) we see that, for larges, the right hand side of (2.21) vanishes on as required. I

'Ra,

Next let be a kth order Lagrangian for a variational problem on E. If smooth functions la = la(xi) determine a local minimum of the fundamental integral

J

i

L(x 'la

.

(

i X

),

2

.

k

.

8la(x') a la(x') a la(x') 1 2 n a x1.1 ' a xJ .1 a xJ .2 ' ... ' a xJ .1 a x1.2 · • • a xJ1r. . ) dx dx ... dx '

then these functions satisfy the Euler-Lagrange equations 2

.

2k

.

i i 8la(x ) 8 la(x') 8 la(x') E(L)(x,la(x),a. '8 ·a. ,···,a ·a. x12 . . . a.x121r. )=0, xJ1 x11 x12 x11 b

569

where the Euler-Lagrange

operator Eb is given by (2.24)

ff a system

~a

=

0 of differential equations is the Euler-Lagrange expression for some Lagrangian L we say that the system arises from a variational principle. A differential conservation law for the equations ~ a = 0 is an n-tuple

of differential functions whose divergence vanishes on solutions of the system, that is, for some p > 0, (2.25) DiV' = 0 on

n~.

Conservation laws play a prominent role in physical theories and in the study of partial differential equations since they impose a priori constraints on the solutions of the equations. For example, suppose that our independent variables consist of the time coordinate t and the space coordinates x 1 , x 2 , .•• , xn. Now equation (2.25) takes the form (2.26) where T is called the density and Xi the flux of the conservation law. Let la = la(t, xi) be a solution of the system ~a= 0. Upon integrating (2.26) over a smooth spatial domain n and subsequently applying Gauss's theorem we have that

where ni is the outward pointing unit normal vector to the boundary an of the domain n. Thus the rate of change of the total density is determined by the boundary values of the flux. In particular, if la is a solution such that the inner product n

L Xi(t, xi, la(t, xi), ...

)ni

= 0,

if xi E

an,

i=l

vanishes on the boundary an,then the total density is constant in time. A conservation law is trivial if it does not impose any constraints on the solutions of the equations under consideration. There are two easily identifiable cases in which this happens. A conservation law Vi is called a trivial conservation law of the first kind if the n-tuple Vi itself vanishes on solutions of the system, and it is called

570

a trivial conservation law of the second kind if the divergence Di Vi = 0 vanishes identically at all points of JP(E) for some p. Two conservation laws are equivalent if their difference is a sum trivial conservation laws of the first and second kind. Note, in particular, that two conservation laws Vi and Wi are equivalent if there are differential functions Sii = -Sii such that •

W,

= v· + D 1-s•1 , •

•

■

i·

= 1, ... , n,

(2.27)

on solutions of the system. Next suppose that the system ~ a = 0 is non-degenerate, and let Vi be a differential conservation law for ~ a = 0. It then follows from the definition (2.25) of a conservation law and from (2.16) that we can express the divergence Di Vi as a combination D·Vi = ~ Hiii2 .. -i1D )1· D )2· ... D )I· ~a 7 (2.28) l ~ a 0$l$q

where the coefficients Htii 2 .. ·i1 are functions on some Jr(E). grating (2.28) by parts we can rewrite this identity as ~·1 b D·V -- Yib~ ' J

By repeatedly inte-

(2.29)

where Vi is equivalent to Vi, in fact, the difference Vi - Vi is a trivial conservation law for ~ a = 0 of the first kind. The expression (2.29) is called the characteristic form of the conservation law Vi, and the evolutionary vector field Y

a Uii,... a )a= Yi(b X i , Ua, Ui, 8

(2.30) Ub

is called the characteristic vector field for the conservation law Vi. Note, in particular, that any conservation law of a non-degenerate system is equivalent to one in the characteristic form. Characteristics for conservation laws can be effectively detected with the help of the Euler-Lagrange operators (2.24). In fact, the m tuple (Y 1 , ••. , Ym) satisfies equation (2.29) for some differential functions V1 if and only if the expressions (2.31) vanish identically. A generalized vector field

.a

V

a

= C-a. +a-a x• Ua

571

on E is called a divergence symmetry (or a variational symmetry) of a Lagrangian L if . . prv L + LDie' = DiB 1 (2.32) for some n-tuple Bi of differential functions. A divergence symmetry of a Lagr~gian L is always a generalized symmetry of the associated Euler-Lagrange equations. The converse, however, fails to be true in general. For example, scaling symmetries of Euler-Lagrange equations typically fail to yield divergence symmetries of the Lagrangian. It is easy to see that the evolutionary form of a divergence symmetry for a Lagrangian Lis again a divergence symmetry of L. Theorem 2.2. A generalized vector field v is a divergence symmetry of a Lagrangian L if and only if the evolutionary form v ev of v is the characteristic vector fi.eld for a differential conservation law for the associated Euler-Lagrange equations Ea(L) = 0.

This is the classical Noether's Theorem. Thus, all conservation laws of a nondegenerate system of differential equations arising from a variational principle can be identified by first finding all generalized symmetries of the system, then determining which ones of these yield divergence symmetries of the Lagrangian, and finally, for each symmetry v found in the previous step, transforming the product Va,ev~ a into a divergence expression. Under general regularity conditions on the system of differential equations Noether's theorem actually provides a one-to-one correspondence between equivalence classes of conservation laws and equivalence classes of div:ergence symmetries ( see ref. [34], chapter 4). However, as was already observed by Noether, this one-to-one correspondence fails for underdetermined systems, i.e., for systems whose components satisfy a differential identity. A prototype of this is provided by Maxwell's equations in potential form, which admit an infinite dimensional group of non-trivial symmetries, the group of gauge transformations. However, the conservation laws corresponding to the gauge symmetries are trivial, cf. section 3. Noether 's theorem can also be formulated directly in terms of the symmetries of the system of differential equations instead of those of the Lagrangian ( see ref. [3]),an advantageous feature when the Lagrangian has a complicated expression or is not readily available. Finally, when a system of differential equations does not admit an underlying variational principle, as, for example, is the case with a single evolution equation of the form

and when the Noether's theorem does not apply, the problem of classifying conservation laws is often approached directly by solving equation (2.29) for characteristics Yb (cf. refs. [1], [44]).

572 §3. Symmetries and conservation laws of Maxwell's equations. We start with establishing some notation. Our underlying space of independent variables is the Minkowski space M 4 with the coordinates xi, where x 0 = t stands for the time coordinate and x 1 = x, x 2 = y and x 3 = z stand for the spatial coordinates. In this section Latin indices always take values form O to 3. We raise and lower tensor indices using the Minkowski metric TJwith the components (TJii) = diag(-1, 1, 1, 1).

Let A be any two form (i.e., the components define the dual *A of A by

*A-· IJ Here

tijkl

stands for the permutation fijkl

= -Aij

Aji

1 = -ti1"klA 2

kl

are skew-symmetric).

We

(3.1)

·

symbol defined by

= 1,-1,0

according to whether (i,j, k, l) is an even, odd or no permutation of (0, 1, 2, 3). Let E = (E 1 , E 2 , E 3 ) and B = (B 1 , B 2 , B 3 ) be the electric and magnetic vectors. The electromagnetic field tensor Fis a two form on M 4 with the components a=

Interpreting the components JP(F) with coordinates

Fij

1,2,3.

as our dependent variables we obtain the jet spaces

where j < k. The source free Maxwell's equations in physical form are F[ij,k]

= 0,

F ij

,}

. --

0.

(3.2), (3.3)

Here the square bracket stands for skew-symmetrization in the enclosed indices. In terms of the field vectors E and B equations (3.2), (3.3) become the familiar equations

E,t

= curl B,

divE

B,t

= -curl

divB

E,

= 0, = 0.

573

On account of (3.2), the electromagnetic field tensor can be expressed as Fii

= A(j,i),

(3.4)

where A is the electromagnetic potential. We are thus lead to consider the additional jet spaces JP(A) with the coordinates {(xi' Aj, Aj,li, ... 'Ai,l1l2, ..-,lp)}.

Equations (3.3), written in terms of the potential Ai,

=0

A(i,i). J

'

(3.5)

are the Euler-Lagrange equations for the Lagrangian function

L

= - 21 A [.•,J·1A[ i ,j) .

(3.6)

Specifically, Equations (3.5), as a constant coefficient system of linear differential equations, are non-degenerate. Consequently, any differential conservation law for Maxwell's equations (3.5) arises from a divergence symmetry of the Lagrangian (3.6) via the Noether correspondence. Let (3.7) where Qij = Qij(x 1, p(k) ), be a kth order generalized symmetry of Maxwell's equations in the physical form (3.2), (3.3). Then, by the defining equations (2.17) of a symmetry, the characteristics Qij in (3. 7) satisfies the equations D[iQjk)

= 0,

(3.8)

on solutions of (3.2), (3.3), that is, on the prolonged solution manifolds 'R} C Jk+ 1(F) of equations (3.2), (3.3). Similarly, let a ws = Si aA/ (3.9) where Si= Si(xi, A(k)), be a kth order generalized symmetry of Maxwell's equations in potential form (3.5). Now the characteristic Si of the symmetry satisfies the equations (3.10) on solutions of (3.5), that is, on the solution manifolds 'R~ C Jk+ 2(A) of equations (3.5).

574

Point symmetries of Maxwell's equations. The conformal group C(l, 3) of the Minkowski space consists of those transformations of M 4 that leave the Minkowski metric T/ invariant up to a scalar factor. By Liouville's theorem (see, for example, ref. [11)) C(l, 3) is a 15 parameter local Lie group generated by the Poincare group 0(1, 3), by scalings and the by inversions

where (pi) is some fixed point in M 4 . The Poincare group 0(1, 3), the group of isometries of T/, is, in turn, generated by translations, spatial rotations, Lorentz transformations, and the discrete time inversion and space reflection. Next let

v

.a

= v'-a.x'

E c(l,3)

be an infinitesimal conformal transformation of M 4 . Then, using the defining equations of an infinitesimal conformal transformation, Vi,j

where p

= ½vi ,i, it

+ Vj,i = PT/ij'

(3.11)

is easy to verify that the components of the Lie derivative (3.12)

satisfy equations (3.2), (3.3) if the Fij do. It follows that the expression (3.12) for the Lie derivative actually determines the characteristic (3.13) of a point symmetry of Maxwell's equations. These are the conformal symmetries found by Bateman and Cunningham in refs. [4], [10]. Interestingly, in his original computations Bateman did not make an explicit use of Lie's infinitesimal method, but instead, found all transformations of the underlying Minkowski space leaving invariant a pair of integral equations associated with the equations (3.2), (3.3). Later, following the revival of interest in symmetry methods in the former Soviet Union, Bateman's and Cunningham's results were confirmed by lbragimov in ref. [17], where, by now exploiting Lie's infinitesimal method, the maximal group of point symmetries of equations (3.2), (3.3) was found to consist of the conformal symmetries (3.13) together with the dilations and rotations of the field vectors. Explicit expressions for the point symmetries in terms of the field vectors are given in Table 1.

575 Table 1 Point Symmetries and Associated Conserved Densities

Symmetry

Conserved

a ax 0

Po= ½(EpEfJ

-aax

Pa

-

0

-xo-

a ax 0

+X

0

a (. p..,(xfla az"'f

a a a - (. p"'f(Bfl __ - Efl--) 0 ax 0 aE-, aB-,

+ Efl- a

aE'Y

1ea ( tJa x axle - 2 E aEtJ lea

=

Density

+ BpBfl)

f.ap-,EfJB"'f

-xoPa

+ XaPo

Comments

Time translation / Energy

Spatial translation / Momentum Lorentz transformation / Center of energy

+ Bfl-)a

f.afJ-,XfJ P"'f

Spatial rotations / Angular momentum

tJa aBtJ)

x1eP1e

Scalings

(x1ex1e)Pi- 2(xleP1e)xi

Conformal transformations

+B

aB"'f

lea

X1eX axi - 2XiX axle

'!__ -

EfJ '!__) aB"'f a 1 +(4xi6/J"'f + 4xlPfY1 )(Ep+ Bp-)a aE"'f aB"'f -2Xle(."lefJ"'f(Bfl 1

tJa E aEtJ

aE"'f

tJa aBtJ

Dilations of field vectors

a a BfJ a EfJ - EfJ a BfJ

Rotations of field vectors

+B

Here Greek indices take values 1, 2 and 3, and f.atJ-,is the three dimensional permutation symbol.

576

One can similarly construct conformal symmetries of Maxwell's equations in potential form (3.5) by computing the Lie derivative of the potential A with respect to the elements of c(l, 3). We thus obtain the point symmetries with the characteristics vk

E c(l, 3).

(3.14)

Equations (3.5) also admit, as is easily checked, the additional point symmetries of dilations of the potential A and the infinite dimensional group of gauge transformations with the characteristics

= ,i,

(3.15)

S,i

where= (xi) is an arbitrary function on M 4 . Higher order symmetries. The first order generalized symmetries of Maxwell's equations (3.2), (3.3) were computed in ref. [38] (see also ref. [37]). Specifically, the computations consist of solving the defining equations of symmetry (3.8), where now the components Qij are, a priori, arbitrary functions of the variables xi, Fi;, and Fij,k. The first order symmetries were found to be generated, in addition to the point symmetries, by the duals of the conformal symmetries with the characteristics 1

*Qv,ij

= 2.€ijklQv,

kl ,

(3.16)

where Qv,ij, v E c(l, 3) is as in (3.13). It follows that the first order generalized symmetries of Maxwell's equations, under the bracket operation (2.11) of generalized vector fields, form a closed Lie algebra isomorphic to s0(6, C) x h, where s0(6, C) is the algebra of 6 x 6 skewsymmetric matrices with complex entries, and h is the two dimensional abelian algebra of rotations and dilations of the field vectors. Due to their linearity, Maxwell's equations also admit the infinite dimensional algebra U of generalized symmetries obtained by forming arbitrary products of the point symmetries ( cf. Prop. 2.1 ). Results in ref. [38] assert that Maxwell's equations (3.2), (3.3) admit no other first order symmetries besides the obvious ones (each symmetry in (3.16) is a composition of a conformal symmetry (3.13) and the rotation of field vectors). However, this is fails to be true for higher order symmetries: Surprisingly, as discovered by Fushchich and Nikitin in ref. [13], Maxwell's equations admit second order symmetries in addition to those found in the enveloping algebra U. To exhibit the symmetries found by Fushchich and Nikitin, and also to extend their results, we first describe a method for constructing symmetries of Maxwell's equations analogous to Hertz's method for constructing solutions of Maxwell's equations from those of the wave equation.

577

Proposition 3.1. Suppose that the differential functions Aij = Aij(x 1, p[k1), Aji = -Ai;, depending on the derivatives of the components of the electromagnetic field tensor up to order k satisfy the wave equation

on solutions of equations (3.2), (3.3). Then the Qij duals *Qij = *Qi/x 1, p[k+ 21) defined by and

(3.17), (3.18)

form the characteristics of generalized symmetries of Maxwell's equations of order k+2.

Proof. The proof consists of a straightforward computation verifying that the Qij and *Qij in (3.17) and (3.18) satisfy the defining equations of symmetry (3.8), and will be omitted. I Next we define the functions Gi, Gi, Hij, Hi;, 0 < i,j < 3, by

-

-

Proposition 3.2. The functions Fij, *Fii' Gi, Gi, Hij, Hi;, 0 < i,j < 3, satisfy the wave equation on solutions of Maxwell's equations (3.2), (3.3).

Proof. The result is immediate for Fi;, *Fii' Gi and Gi. The proofs of the result for Hij and Hij are almost identical. Thus we will only show that D1D1Hij = 0 on 'R,~.

We first note that Hij can be written as

(3.20) Next recall that

D 1D 1(KL)

= (D1D1K)L + 2(D1K)(D 1L) + K(D1D 1L),

578

where K, L are any differential functions. Thus, using the fact that the Fij and Gi satisfy the wave equation on 'R}, we have that and D,D 1(xkxkFij)

= (D1D 1xkxk)Fij + 2(D 1xkxk)Fij,l = 8Fij + 4xk Fij,k

(3.21) (3.22)

on 'R}. Hence, on account of (3.20), (3.21), and (3.22), we have that

vanishes on 'R~, as required.

I

Thus, with pq _ A (xi, A[k]) is any differential function depending on the derivatives of Ai up to some order k. The analogue of the symmetries in Proposition 3.1 for the potential A is given in the following Proposition. Proposition 3.3. Suppose that the differential functions Aij = Aij(x 1, A[kl), Aji = -Aij, depending on the derivatives of the electromagnetic potential up to order k, satisfy the wave equation (3.29)

on solutions of (3.5). Then the functions and

580

defined by and

(3.30), (3.31)

form the characteristic of a generalized symmetry of Maxwell's equations (3.5) of order k+l. One can construct two forms Aij satisfying (3.29) simply by letting, as in (3.23), Aij = -¢>8[it5j]' where¢> is now obtained _by replac~ng Fpq by A[q,p)in the expression for any of the functions Fk1, *F kl' Gk, Gk, Hk1, Hkl· Conservation Laws. We first derive conservation laws arising from the conformal symmetries of Maxwell's equations under the Noether correspondence. We start with a preliminaxy result that we will use repeatedly in the sequel. Proposition 3.4. Let Gij and Hij be any two forms, and let Kij be a two form with K[ij,ij = 0. Suppose that v = via/axi E c(l, 3) infinitesimal conformal transformation of M 4 • Then

(3.32) (3.33) k

(VJ,· +v

k ,]

.. .. · )Gi1H·k -- pGiJH·i],· i

(3.34)

w h ere p -- 2I v I ,I.

Proof. By the definition (3.1) of the dual we have that

*G ik* Hjk

-

-

1 .

4€zklm

Glm

€

jkpqH

pq•

(3.35)

Recall that where 8ff:/i is a generalized Kronecker delta. For the definition and basic properties of the generalized Kronecker deltas we refer to ref. [28). Thus (3.35) becomes

*G-ik*Hjk

1 = - !t5jpq 4 ilm G m Hpq

= _ !(81 t5Pq+ t5Jt51!q + t5lI t5Pq_)Qlm Hpq 4 i Im m ii mi 1 .Gm ' Him+ = -2(t5f

. Hmi) G'.1 Hil + G'm

1 . ' = -2(t5f Gm Him -

2G1 Hi1),

.,

581

which immediately yields (3.32). Next note that

+ K1i,k + Kk,,i) = 2Gkl Kik,l + Gkl Kkl,i,

0 = 3Gkl K[ik,ij = Gk 1(Kik,l

which gives (3.33). Finally, note that (3.34) is an immediate consequence of the defining equation (3.11) of an infinitesimal conformal transformation. I

Theorem 3.5. Let v = via/axi E c(l, 3) be an infinitesimal conformal transformation of the Minkowski space M 4 . Then vi determines a point symmetry of Maxwell's equations (3.5) with the characteristic (3.36)

The symmetries (3.36) are divergence symmetries of the Lagrangian (3.6), and yield, under the Noether correspondence, conservation laws with the components pi V

= vi p.kpik 1

- !vi p.kpik 4 1

(3.37)

•

Proof. We only need to derive expressions (3.37). By Noether's theorem it suffices to show that the product Sv,i Ei(L) can be written as a divergence .

= Div:,

Sv,i E'(L)

.

where the Vj are equivalent, as a conservation law, to the P~ in (3.37). We first integrate by parts to get Sv,iEi(L)

= (vkAi,k

+vk,iAk)A[i,j]j

= 2vk A [1,k] · A[i,j]. 1 + D·(vk

A k A[i,j] 1-) •

I

(3.38)

Note that, by (3.34), 2vk ,1· A[·., k]A[i,j]

= (vk ,1· +v 1'· k )A[·., k]A[i,i] = !vk 2

' k A[·1,1·1A[i,j] '

(3.39)

and, by (3.33),

=

2A[i,k]jA[i,j]

A[i,1lkA[i,j]_

(3.40)

Consequently, upon another integration by parts we have that 2vk A[i,k]A[i,i\

= D j(2vk = D·(2vk 1 = Dj(2v

A[i,k]A[i,j])

- 2vk,j A[i,k] A[i,j] - 2vk A[i,k)j A[i,j]

A[·1 k]A[i,ll)

- !vk 2 ,k A[·1 ]j:iA[i,j] - vk A[·1 1·1kA[i,j]

1

k

A[i,k]A

[.

1

·1

111

) -

Dk(

1

2v

k

A[i,i]A

1

[. :i

••JJ),

(3.41)

582

where we used (3.39) and (3.40) in establishing the equality between the expressions on the first and second lines of (3.41 ). Thus (3.38) becomes

The last term in the above equation is a trivial conservation law of the first kind (the characteristic vanishes on solutions of (3.5)). Thus, after the substitution Fij = A[j,i), equation (3.42) ( apart from a factor 2) yields (3.37), as required. The conservation laws in (3.37) are precisely the conservation laws found by Bessel-Hagen in ref. [5] (see also ref. [36]). Note that the conserved quantities obtained from (3.37) with vi = 6} combine into the stress-energy tensor of the electromagnetic field pkl Tji _- F·Jk pik - 1 vcip j kl .

4

Explicit expressions for the densities in (3.37) can be found in Table 1. Dilations of the potential

a

D

= Ai 8Ai'

as is readily checked with the help of the Euler-Lagrange operators ( cf. (2.30)), are not divergence symmetries of the Lagrangian (3.6), and consequently do not give rise to a conservation law via the Noether correspondence. The generalized gauge symmetries S4, (3.28), in turn, yield, via the Noether correspondence, the trivial conservation laws of the first kind

Thus non-trivial symmetries of Maxwell's equations may be associated with trivial conservation laws. This is a consequence, in accordance with Noether's second theorem (see ref. [32]), of the differential identity

satisfied by the Lagrangian (3.6).

The Zilch tensor.

In 1964, Lipkin in ref. [26] described a set of new conservation laws, combined into a single valence three tensor, the Zilch tensor, for the free electromagnetic field involving quadratic expression in the field variables and their first order derivatives. The discovery of the new conservation laws resulted in a series of articles. Kibble in ref. [23] was the first to extend the Zilch tensor to

583 an infinite sequence of quadratic conservation laws involving higher and higher order derivatives of the field variables. Fairlie (see ref. [12]) showed that Kibble's sequence arises from the translational invariance of the electromagnetic Lagrangian L, and he was able to further generalize Kibble's hierarchy of conserved quantities by considering the invariance of L under the full Lorentz group. The physical relevance of the Zilch tensor was investigated by Candlin in ref. [7]. At the same time Morgan (see ref. [30]), motivated by the discovery of the Zilch tensor, exhibited two infinite sequences of conserved quantities for the free electromagnetic field. Later, after deriving Maxwell's equations in physical form from a tensor Lagrangian, he and Joseph in ref. [31] concluded that the new conserved quantities appropriately arise from the invariance of the new Lagrangian. In refs. [42], [43] Steudel shows that a certain composition of any three divergence symmetries of a quadratic Lagrangian yields new divergence symmetry, thus shedding light on the occurrence of infinite sequences of conservation laws for the electromagnetic field. A refinement of Steudel's result is given in ref. [], chapter 5. We will attempt yet another interpretation of the sequences of conserved quantities associated with the Zilch tensor. For definiteness, we consider the two hierarchies of Morgan explicitly given by and,

(3.43) (3.44)

where A= (a 1 , a 2 , ... , ak) and B = (b1 , b2, ... , b1),k, l > 0, are multi-indices of the integers 0, 1, 2, 3 of length IAI= k, IBI= l. Lipkin's original Zilch tensor can be seen to be equivalent to \1ij,a • The fact that the divergencies

D 1T/

,A,B

=0

and

1 AB=0 D·V.· J i ' '

vanish on solutions of (3.2), (3.3) is a straightforward calculation based on the identities (3.32), (3.33) in Proposition 3.4. To analyze the hierarchy of conserved quantities in (3.43) and (3.44) we first need to discuss a method for constructing new conservation laws from old ones. Suppose that Vi is a differential conservation law of a system of differential equations ~ i = 0 with the characteristic Qi, and suppose that v is a generalized symmetry of the system ~ i = 0 in the evolutionary form. Then we claim that the n tuple (3.45) of differential functions again provides a conserved quantity for the equation~ In fact, by assumption,

i

= 0.

584

Thus Di(Vj)

= Di(prv(Vi)) = prv(Di Vi) = prv(Qi~i) = (prvQi)~i + Qi(prv~i),

(3.46)

where we have used the fact that the prolongation of an evolutionary vector field commutes with the total derivative operators. Now keeping in mind that v is a symmetry of the system ~i = 0 we see that the right had side of (3.46) vanishes on solutions of ~ i = 0, which proves the claim. Next let Va be the first order generalized symmetry of Maxwell's equations given by

Then, by (3.45), the quantities

Va=

1 a -*F---. 4 IJ,a 8Fij

Uij,A,

defined by

(3.4 7)

(3.48) are conserved on solutions of Maxwell's equations. Given two multi-indices A= (a1, ... , ak), B = (b1, ... , b1),we let the multi-index AB stand for the union of A and B,

Theorem 3.6. Let Tij,A,B, Vij,A,B and Uij,A be the conservation laws (3.43), (3.44) and (3.48) of Maxwell's equations (3.2), (3.3), respectively. Then the Tij,A,B are, up to a sign, equivalent to Uij,AB, if the sum IAI+ IBIis even, and the Tij,A,B are trivial, if the sum 1-41 + IBIis odd. The Vij,A,B are, up to a sign, equivalent to Uij,AB, if the sum IAI+ IBIis odd, and the Vij,A,B are trivial, if the sum IAI+ IBI 1s even. Proof. We start by showing that, for c

= 0, 1, 2, 3,

the conservation law

Tij,A,Bc

is

equivalent to -Tij,Ac,B• First note that by (3.32), T,.i 1 ,A,B

= F.·im,A pmj

,B

+ F.·im,B

pmj

,A

+ !2 F,mn,A pmn

,B bji •

(3.49)

Thus an integration by parts shows that T/ ,A,Bc

= -Tij

,Ac,B

+ Dc(Fim,Apmj

+ Dc(Fim,Bpmi,A)

,B)

+ Dc(~Fmn,Apmn,Bb)).

(3.50)

585

But

= 2Dk(Fim,Apm[i,Bc5!l) + Dk(Fim,Apmk,Bc5t) = 2Dk(Fim,Apm[i ,BcS!l)+ Fim,kApmk ,Bc5t+ Fim,Apmk ,kBcSt,

Dc(Fim,Apmj,B)

(3.51) and

+ ~Di(Fmn,Apmn,Bc5t).

= Dk(Fmn,Apmn,Bc5!ic5!l)

Dc(~Fmn,Apmn,Bc5})

(3.52)

The first and the third terms in the last expression of (3.51) and the first term in (3.52) are trivial conservation laws ( cf. (2.27)). Thus it follows from (3.50) that Tii A Be is equivalent to ' '

-

.

T,,J AB I 1

1

.

C -

-T,·JI A C, B I

+ F.·

lffl

k 1

kApm

.

1

Bc51C

+ Fim,kBpmk,Ac5t + ~Di(Fmn,Apmn,Bc5t).

(3.53)

But proceeding as in the proof of formula (3.33) one can show that F.im,kA

F mk ,B

3 z:;, =2 L'[im,k]A

pmk

,B -

1 F,

2

mk,iA

pmk

,B,

(3.54)

where the first term is again a trivial conservation law. Now it follows form (3.53) and (3.54) that Ti1,A,Bc and, consequently, Tii,A,Bc is equivalent to -Ti1,Ac,B• One can similarly show that the conservation law ¼i ,A,Bc is equivalent to - ¼i ,Ac,B. Note, in particular, that a repeated application of the above claim shows that the conservation laws Tii,A,B, ¼i,A,B are equivalent to (-t)IBI times Ti~AB

V/,AB

= Fim,ABpmj + *Fim,AB*Fmj, = Fim,AB*Fmj - *Fim,ABpmj,

and

(3.56)

respectively. Next let v a be as in ( 3.4 7). One can easily check that and Hence

.

1

.

(3.55)

.

-(V/ i , A)= Prv a (T.·' 2 1 , A,a - ½'1 , Aa ).

586 Thus pr v a(Ti 1 ,A) is equivalent to -½ 1,Aa· Similarly, one can show that pr Va(½1,A) is equivalent to Ti 1,Aa· Now a repeated application of this result in conjunction with the above claim shows that, for IAI+ IBIeven, the conservation law U/ ,AB is equivalent to ( -l)(IAI-IBl)/2r,.1

and, for

IAI·+ IBIodd,

z ,A,B,

it is equivalent to ( -l)(IAI-IBl+I)/2i,,:j

z ,A,B

.

We still need to show that Ti 1,A is trivial when IAIis odd, and that ½1,A 1s trivial when IAIis even. Given a multi-index A = (a 1, a2, ... , ak ), we let A1,1 and A2,1 be the multi-indices and where 1 < l < k. Now suppose that by parts yields the identity

IAI= k is odd.

A repeated integration

a,(F·zm,A

,A 2 , 1

of (3.49)

k

,.,,_j

-

.Lz,A -

1 ~( ~

2

-

l)l-ID

1 ,,

pmj

l=I

+ F im,A2,1 pmj

,A1,1

+ 21 F mn,A2,1

pmn

d)

A1,1Ui ·

(3.57)

Note that each the summand in (3.57) equals the sum of the second, third, and fourth terms on the right hand side of (3.50) with A = A 1,1 and B = A 2,1. Thus proceeding as in (3.51) - (3.54) we see tha~ each summand in (3.57) is a trivial conservation law, and, consequently, that T/A is a trivial conservation law, if IAIis ' odd. The proof that ½1A, for IAIeven, is trivial is almost identical to the above proof .

'

for Ti~A' and will be omitted. Good, in ref. [15], finds conservation laws of the free electromagnetic field by casting Maxwell's equations in a form similar to the Dirac equations, which then allows the construction of conserved densities following standard quantum mechanical methods. In the problem at hand this amounts to the following observation: The quantity

1 3 T---~F-·Q·· 2 ~ i,j=O

ZJ

ZJl

587 where Qij is the characteristic of an evolutionary symmetry of equations (3.2), (3.3), provides a conserved density for the free electromagnetic field. In fact, one can check that Tis the density of the conservation law (3.58) The expressions in ( 3. 58) can again be seen to arise as a special case of the the construction (3.45). In fact, if we let P~ be the Bessel-Hagen conservation law associated with vi E c(l, 3),

and if Qij is the characteristic law

of a symmetry w, then (3.45) yields the conservation (3.59)

The expressions in (3.58) are a special case of (3.59) obtained by letting the time translation.

vi

= b~ be

588 REFERENCES

1. L. Abellanas, A. Galindo, A generalized variational algebra and conserved densities for linear evolution equations, Lett. Math. Phys. 2 (1978), 399-404. 2. Ian M. Anderson, Introduction to the variational bicomplex, Mathematical Aspects of Classical Field Theory, Contemporary Mathematics 132, (M. Gotay, J. Marsden and V. Moncrief, eds.), Amer. Math. Soc., Providence, 1992, pp. 51-73. 3. I. M. Anderson,

The Variational Bicomplex, Academic Press, Boston (to appear).

4. H. Bateman, The conformal transformations of a space of four dimensions tions to geometrical optics, Proc. London Math. Soc. 8 (1909), 223-264. 5. E. Bessel-Hagen,

Uber die Erhaltungssatze

6. G. W. Bluman, S. Kumei, Symmetries 1989.

and their applica-

der Eletrodynamik, Math. Ann. 84 (1921), 258-276.

and Differential Equations, Springer-Verlag,

New York,

7. D. J. Candlin, Analysis of the new conservation law in electromagnetic theory, Nuovo Cimento 37 (1965), 1390-1395. 8. J. Carminati, J. S. Devitt, G. J. Fee, Isogroups of differential equations using algebraic computing, J. Symbolic Comput. 14 (1992), 103-120. 9. B. Champagne, P. Winternitz, A MACSYMA program for calculating the symmetry group of a system of differential equations, Preprint CRM 1278, Universite de Montreal, 1985. 10. E. Cunningham, The principle of relativity in electrodynamics London Math. Soc. 8 (1909), 77-98.

and an extension thereof, Proc.

11. B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry - Methods and Applications, Vol. 1, 2nd ed., Springer, New York, 1992. 12. D. B. Fairlie, Conservation laws and invariance principles, Nuovo Cimento 37 (1965), 897-904. 13. W. I. Fushchich, A. G. Nikitin, New and old symmetries Sov. J. Part. Nucl. 14 (1983), 1-21.

of the Maxwell and Dirac equations,

14. W. I. Fushchich, A.G. Nikitin, On the new invariance algebras and superalgebras of relativistic wave equations, J. Phys. A 20 (1987), 537-549. 15. W. I. Fushchich, A. G. Nikitin, Symmetries

of Maxwell's Equations, D. Reidel, Boston, 1987.

16. R. H. Good, Jr., Particle aspects of the electromagnetic field equations, Phys. Rev. 105 (1957), 1914-1919. 17. N. Kh. lbragimov, Group properties of wave equations for particles of zero mass, Sov. Phys. Dokl. 13 (1968), 18. 18. N. Kh. lbragimov, ton, 1985.

Transformation

Groups Applied to Mathematical

Physics, D. Reidel, Bos-

19. N. Kh. lbragimov, Group analysis of ordinary differential equations and the invariance principle in mathematical physics, Uspekhi Mat. Nauk 47 (1992), 83-144. 20. D. W. Kahn, Introduction

to Global Analysis, Academic Press, New York, 1980.

21. E.G. Kalnis, R.G. McLenaghan, G.C. Williams, Symmetry operators for Maxwell's equations on curved space-time, Proc. R. Soc. Lond. A 439 (1992), 103-113. 22. E.G. Kalnis, W. Miller Jr., G. C. Williams, Matrix operator symmetries and separation of variables, J. Math. Phys. 27 (1986), 1893-1900. 23. T. W. B. Kibble, Conservation

of the Dirac equation

laws for free fields, J. Math. Phys. 6, 1022-1026.

589 24. G. A. Kotelnikov, On the symmetry equations of the free electromagnetic field, Nuovo Cirnento 72B (1982), 68-77. 25. H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. Math. 66 (1957), 155-158. 26. D. M. Lipkin, Existence of a new conservation law in electromagnetic 5 (1964), 696-700.

theory, J. Math. Phys.

27. D. Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles, Dover, New York, 1989. 28. A.V. Mikhailov, A.B. Shabat, V. V. Sokolov, The symmetry approach to classification of integrable equations, in What Is Integrability (V. E. Zakharov ed.), Springer-Verlag, Berlin, 1991, pp. 115-184. 29. W. Miller Jr., Symmetry

and Separation of Variables, Addison-Wesley, Reading, Mass., 1977.

30. T. A. Morgan, Two classes of new conservation laws for the electromagnetic field and other massless fields, J. Math. Phys. 5 (1964), 1659-1660. 31. T. A. Morgan, D. W. Joseph, Tensor Lagrangians and Generalized conservation fields, Nuovo Cirnento 39 (1965), 494-503. 32. E. Noether, Invariante Variationsprobleme, Phys. Kl. (1918), 235-257. 33. P. J. Olver, Applications 34. L. V. Ovsiannikov, 1982.

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Nachr. Konig. Gesell. Wissen. Gottingen,Math.-

of Lie Groups to Differential Equations, Springer, New York, 1986.

Group Analysis

of Differential

35. R. Penrose, W. Rindler, Spinors and Space-time,

Equations,

Academic Press, New York,

Vol. 1, Cambridge, New York, 1984.

36. B. F. Plybon, Obsen,ations on the Bessel-Hagen fields, Arner. J. Phys. 42 (1974), 998-1001.

conservation

37. J. Pohjanpelto, First order generalized symmetries (1988), 148-150.

of Maxwell's equations, Phys. Lett. A 129

38. J. Pohjanpelto, Symmetries Minneapolis, MN, 1989. 39. J. Pohjanpelto,

laws for the electromagnetic

of Maxwell's equations, Ph.D. Thesis, University of Minnesota,

New second order generalized symmetries

of Maxwell's equations, in prep.

40. G. J. Reid, Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, European J. Appl. Math. 2 {1991), 319-340. 41. F. Schwarz, Automatically ing 34 {1985), 91-106. 42. H. Steudel, Die Struktur 1826-1828. 43. H. Steudel, Invariance 39 {1965), 395-397. 44. T. Tsujishita, 445-450.

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of partial differential equations, Comput-

der Invarianzgruppe fur lineare Feldtheorien, Zeit. Naturforsch. 21a, groups and conservation laws in linear field theories, Nuovo Cirnento

Conservation

laws of free Klein Gordon fields, Lett. Math. Phys. 3 {1979),

Applications

SIX EXPERlMENTS WITH MAGNETIC

CHARGE

V.F. Mikhailov 480082 Almaty, Sadovaja 8, Kazakhstan

More than sixty years have passed since Dirac [1]predicted the possibility of free magnetic charge. The history of physics, it seems, does not know another precedent where such a great amount of human effort has been given to the confirmation of firm theoretical predictions, and yet the problem remains unsolved. Under such circumstances, the title of this chapter could evoke considerable skepticism. But the results of all six experiments described here permit a categorical wording because these experiments uniquely indicate the existence of magnetic charge (monopoles) and also deny alternative explanations of the observed effects. The experiments of the Austrian physicist Felix Ehrenhaft (1879-1952) preceded our investigations .1 In 1930 Ehrenhaft reported that he observed an effect, the essence of which is that separate aerosol particles, obtained by means of dispersion of electrode arc material, and in the presence of a light beam in a homogeneous magnetic field, move like objects carrying a magnetic charge [3]. Ehrenhaft's experiments, in which he invariably reproduced the effect in confirmation of his first results, were published over a period of more than twenty years, but recognition of magnetic charge discovery was not forthcoming. Numerous measurements of magnetic charge by both Ehrenhaft and other investigators give the value in the range 10- 9-10- 14 gauss.cm, while Dirac obtained the value theoretically of gv = 3.29 x 10-s gauss.cm. Probably, this disagreement was a major reason for the decline in interest in Ehrenhaft's work soon after his death. His work was largely forgotten by the 1980s when it was taken up again by ourselves [4]. Ehrenhaft's legacy has been highly complicated . Vagueness of the experimental design, contradictory interpretations concerning the reported effect, and, moreover, complete negation by some investigators, induced us, in accordance with general interest in the monopole problem, to reproduce and continue investigations in the initial direction. Our first aim, naturally, was to convince ourselves of the existence of the effect itself. 1A full bibliography of his publications is contained in (2).

593

SIX EXPERlMENTS WITH MAGNETIC

CHARGE

V.F. Mikhailov 480082 Almaty, Sadovaja 8, Kazakhstan

More than sixty years have passed since Dirac [1]predicted the possibility of free magnetic charge. The history of physics, it seems, does not know another precedent where such a great amount of human effort has been given to the confirmation of firm theoretical predictions, and yet the problem remains unsolved. Under such circumstances, the title of this chapter could evoke considerable skepticism. But the results of all six experiments described here permit a categorical wording because these experiments uniquely indicate the existence of magnetic charge (monopoles) and also deny alternative explanations of the observed effects. The experiments of the Austrian physicist Felix Ehrenhaft (1879-1952) preceded our investigations .1 In 1930 Ehrenhaft reported that he observed an effect, the essence of which is that separate aerosol particles, obtained by means of dispersion of electrode arc material, and in the presence of a light beam in a homogeneous magnetic field, move like objects carrying a magnetic charge [3]. Ehrenhaft's experiments, in which he invariably reproduced the effect in confirmation of his first results, were published over a period of more than twenty years, but recognition of magnetic charge discovery was not forthcoming. Numerous measurements of magnetic charge by both Ehrenhaft and other investigators give the value in the range 10- 9-10- 14 gauss.cm, while Dirac obtained the value theoretically of gv = 3.29 x 10-s gauss.cm. Probably, this disagreement was a major reason for the decline in interest in Ehrenhaft's work soon after his death. His work was largely forgotten by the 1980s when it was taken up again by ourselves [4]. Ehrenhaft's legacy has been highly complicated . Vagueness of the experimental design, contradictory interpretations concerning the reported effect, and, moreover, complete negation by some investigators, induced us, in accordance with general interest in the monopole problem, to reproduce and continue investigations in the initial direction. Our first aim, naturally, was to convince ourselves of the existence of the effect itself. 1A full bibliography of his publications is contained in (2).

593

SIX EXPERIMENTS WITH MAGNETIC

CHARGE

V.F. Mikhailov 480082 Almaty, Sadovaja 8, Kazakhstan

More than sixty years have passed since Dirac [1] predicted the possibility of free magnetic charge. The history of physics, it seems, does not know another precedent where such a great amount of human effort has been given to the confirmation of firm theoretical predictions, and yet the problem remains unsolved. Under such circumstances, the title of this chapter could evoke considerable skepticism. But the results of all six experiments described here permit a categorical wording because these experiments uniquely indicate the existence of magnetic charge (monopoles) and also deny alternative explanations of the observed effects. The experiments of the Austrian physicist Felix Ehrenhaft ( 1879-1952) preceded our investigations. 1 In 1930 Ehrenhaft reported that he observed an effect, the essence of which is that separate aerosol particles, obtained by means of dispersion of electrode arc material, and in the presence of a light beam in a homogeneous magnetic field, move like objects carrying a magnetic charge [3]. Ehrenhaft's experiments, in which he invariably reproduced the effect in confirmation of his first results, were published over a period of more than twenty years, but recognition of magnetic charge discovery was not forthcoming. Numerous measurements of magnetic charge by both Ehrenhaft and other investigators give the value in the range 10- 9 -10- 14 gauss.cm, while Dirac obtained the value theoretically of gv = 3.29 x 10- 8 gauss.cm. Probably, this disagreement was a major reason for the decline in interest in Ehrenhaft's work soon after his death. His work was largely forgotten by the 1980s when it was taken up again by ourselves [4]. Ehrenhaft's legacy has been highly complicated. Vagueness of the experimental design, contradictory interpretations concerning the reported effect, and, moreover, complete negation by some investigators, induced us, in accordance with general interest in the monopole problem, to reproduce and continue investigations in the initial direction. Our first aim, naturally, was to convince ourselves of the existence of the effect itself. 1 A full bibliography of his publications is contained in [2).

594 Experiments

This series of experiments involve direct observations of microparticle (aerosol) motion in a light beam in the presence of various magnetic and electric fields. The experimental apparatus (the prototype of which was used by Millikan in measuring electric charge) has been described in detail in [5-7]. In these experiments ferromagnetic aerosols were used prepared by electrospark sputtering of current interrupter iron contacts in argon atmosphere. The pressure of the argon was one atmosphere. The obtained iron aerosol particles have a spherical form with sizes 10- 5 -10- 6 cm [2, 8]. Homogeneous magnetic fields were obtained with Helmholtz coils. 1st Experiment:

Experiment

with homogeneous

magnetic

field

These first observations of the ferromagnetic aerosols (iron, nickel and cobalt) performed in an argon atmosphere confirmed Ehrenhaft's earlier reports. Against a background of particles in Brownian motion and which are not reacting to the magnetic field, other separate particles move in the magnetic field along its lines of force. The reversal of the field, H, causes the reversal of the particles' motion. The motion ceases when the field is switched off. ➔

-H -

b

-

t

'

Fig. 1. Illustration of magnetic-field-sensitive particle trajectories (a) in a changing magnetic field; (c-c) are the light beam boundaries; (b) magnetic field reversals of direction. Particles move in a downward direction due to the slow gas flow (during the trajectory scanning time).

If the field is alternating, separate particles simultaneously vacillate along the field lines, each with an individual amplitude of vacillation (Fig. 1). An increase or decrease of field strength or luminous flux intensity causes the particle velocity to increase or decrease respectively.

595

.... The number of particles moving in the direction of H, up to measurement error, equals the number of particles moving in the other direction. These observations confirmed the Ehrenhaft interpretation of magnetic charge existence. Measurement of the magnitude of that charge was addressed in the following experiment. 2nd Experiment:

Quantitative

measurements

of magnetic

charge

Ehrenhaft obtained quantitative measurements of magnetic charge by comparison of two forces - the magnetic F9 and the gravitational F-r. Therefore he was compelled to use large (lo- 4 -10- 5 cm) sedimenting particles. By stopping the falling particles with a homogeneous magnetic field he was able, by the balance of forces, to measure the magnetic charge. But this method implies that all parameters of the particles (size, density, etc.) are known and therefore demands considerable measurement effort. The difficulties in implementing the method were the source of numerous errors. We used another method. Among the particles which demonstrate the magnetic charge effect there are also a large number of particles carrying an electric charge [5]. The presence of these other particles permitted a comparison of the relative .... .... magnitude of the forces of the magnetic, H, and the electric, E, fields influencing the particles. By this means, the electric q and the magnetic G charges of the particles could be determined. The design of this experiment in shown in Fig. 2. Here, the electric field is parallel to the gravitational force, and the magnetic field is parallel to the light .... .... beam, 4>,and E .l H. The frequencies of the switching fields are in the ratio of 2: 1 and a precise fixed phasing is maintained between the fields. This method permits the complete exclusion of error in track identification because such tracks are only observed in the mode of the fields. It also permits the particles to be distinguished moving along and against the vector direction of the magnetic field intensity, i.e., to determine the sign of the magnetic charge. If a particle has both electric charge q and magnetic charge G, the forces influencing the particle can be expressed as: FE= KvE

= qE;

(1)

from which it follows that: G

q-

EvH

HvE ·

(2)

This ratio indicates that the relation of the magnitude of the magnetic charge to that of the electric charge, ~ , for a specific particle does not depend on the Stokes'

596

H 3Hz

t

£

a

t b

l.5Hz.

--

-

£

q,

C

--

--H

d

Fig . 2. Field intensities over time for H (a) and E {b). (c) shows particle trajectories for a single electric field period; shown are two particle trajectories with magnetic charge of opposite sign. {d) is a photograph of a track of a binary charged particle.

coefficient, K, which is defined by the particle size and which has been the stumbling block of most of Ehrenhaft 's opponents (2). In this experiment more than 6,000 particle tracks were photographed and analyzed for various regimes corresponding to various relations of ~ . It became apparent that the numerical values of the particle ~ ratio could be expressed with respect to the values of the particle velocities VH and ve, which were found experimentally using the fine structure constant, o = 1~7 , in the empirical formula:

G q

no

6 '

where n = I, 2, ... (5,7]. The smallest value of magnetic charge is then for n and g = e, the electron charge, and using Eq. (3) the charge "quantum" is: g=-

oe 6 .

(3)

=

I

(4)

597 Thus, (5)

so a "quantum" of charge defined by us is rv o 2 times smaller than the magnetic charge of Dirac's monopole [l]. However, its numerical value, obtained with Eq. (4) as g = 5.84 x 10- 13 gauss.cm 2 , is in good agreement with evaluations made earlier by Ehrenhaft, Ferber [9] and Shedling[l0]. In this series of experiments we studied in detail some regularities of the dual-charged aerosol particle motion in electromagnetic fields. We attempted to demonstrate contradictory evidence against the magnetic charge hypothesis [8]. For example, the behavior of particles of various molecular composition in a variety of gases were investigated; and electric and magnetic fields of various configurations, including a magnetic field from a linear conductor's current (Fig. 3), were applied. We even observed the magnetic charge phenomenon when the temperature of the gas was T = 77K (the boiling temperature of liquid nitrogen) [11]. Dependence of particle velocity as a function of magnetic and electric field strength and of luminous flux density has been demonstrated for three points of the optical spectrum corresponding to the gas laser lines: A1 = 4400 A, A2 = 4480 A and A3 = 6328 A. In summary, the effect has been observed as stable and its main characteristics have been studied. The main experimental results are shown in Table I. There is a complete parallelism between particle behavior in an electric and a magnetic field. Without already knowing, it is impossible to observe a particle under these conditions and to determine which field is being used. 3rd Experiment: current

Magnetic

charge in a magnetic field of a line conductor's

We turn now to consider the experiment using a magnetic field of a line conductor's current (Fig. 3). In the presence of this field ferromagnetic particles move along a circular arc (along the force line) and at the same time displace to the conductor axis [6]. The change of current direction induces a reversal of particle velocity along a circular arc, but not along the radius. At the same time one can observe particles which move both clockwise and counterclockwise. The absence of a conductor carrying magnetic current does not permit the experiment with electrically charged particles which would be symmetrical to those described. However, under very high photographic magnification some particles exhibit wavy lines with an amplitude 10-100 r, where r is the particle radius. It is plausible to conceive that this is the plane projection of the helical (spiral) path of the particle, the axis of the spiral being along the magnetic force line. Supposing now that a

598 Table I

...

Uniform electric field, E

Uniform magnetic

field,

H

E the particle moves along the line of force with constant velocity.

At constant H the particle moves along the line of force with constant velocity.

Reversal of the field E direction causes the reversal of the particle velocity, and field shut-down causes the particle to stop. On increasing E, the particle velocity, at constant ~, rises nonlinearly ( the velocity "saturation effect" occurs) [8].

Reversal of the field H direction causes the reversal of the particle velocity, and field shut-down causes the particle to stop. On increasing H, the particle velocity, at constant ~, rises nonlinearly ( the velocity "saturation effect" occurs) [8].

The number of particles moving along the direction of the vector E and in the opposite direction is equal. The effect is observed at any angle between the light beam axis and the direction of the intensity of the E vector.

The number of particles moving along the direction of the vector H and in the opposite direction is equal. The effect is observed at any angle between the light beam axis and the direction of the intensity of the H vector.

1. At constant

2. At constant E the particle velocity rises linearly with increase of the luminous flux intensity ~ [8].

At constant H the particle velocity rises linearly with increase of the luminous flux intensity ~ [5,7 1 8].

3. The velocity of some particles, at constant E and constant ~, changes unevenly. Cases of velocity reversal (recharging) have been observed.

The velocity of some particles, at constant H and constant ~, changes unevenly. Cases of velocity reversal ( recharging) have been observed [7].

4. For the case where E is vertical, the particle in the gravity field can be brought to equilibrium by changing the value of E. In this case E is single-valued at any point in space. Cases are observed in which spontaneous disturbance of the equilibrium ifi restored at another value of the field strength E (recharging, classical experiment of Millikan).

For the case where E is vertical, the particle in the gravity field can be brought to equilibrium by changing the value of H. In this case H is single-valued at any point in space. Cases are observed in which spontaneous disturbance of the equilibrium is restored at another value of the field strength H (recharging, Ehrenhaft [2]).

5. At constant intensity of the light beam and constant E, the particle velocity rises with a decrease of the light wavelength ( according to measurements taken with: Al 4400 A, A2 4480 A, A3 6328 A).

At constant intensity of the and constant H, the particle with a decrease of the light (according to measurements Al 4400 A, A2 4480 A,

=

=

=

...

=

=

light beam velocity rises wavelength taken with: A3 6328 A).

=

599

I mm

cp 2

3

1

Fig . 3. Photograph of a particle track over five path reversals for magnetic charge in the field of a line conductor with temporal variation of the current (/ = 6 A, frequency , v = 2 Hz) . The current conductor, (1), is perpendicular to the field of view; (2) is the crossing path of the light beam 4>; (3,4) is the terminal screen: (3) is copper, (4) is glass. H 2 )(f x R).

= (ci

particle carries an electric charge and, at the same time, is a source of magnetic current the direction of which coincides with the force lines, then these observations, at least qualitatively, agree with a model of electric charge interaction with magnetic current. These qualitative observations are not included in Table I, but should be when they become quantitative.

600

4th Experiment: chamber

Observation

of the magnetic charge effect in a diffusion

The previous experiments confirmed the existence of a magnetic charge effect with no contradictory observations. The results cannot be described as the behavior of dual-charged particles in electric and magnetic fields. Nevertheless, although this interpretation does not contradict the experimental results, the results do not exclude the possibility of the existence of other unknown but more conventional mechanisms which could imitate a magnetic charge effect (MCE). One such might be associated with structural inhomogeneity of a particle's surface which might correlate with the particle magnetic moment [10]. Such conditions could easily occur in the case of a solid particle. However, a liquid surface cannot have any stable nonuniformities, and it is also hard to imagine a gas-kinetic mechanism imitating the MCE in a liquid drop. Therefore we decided to surround a ferromagnetic particle with a liquid shell and, as a consequence, to eliminate contact of the particle's surface with the surrounding gas. Absence of the MCE under observation of such objects would show unambiguously that a magnetic charge is not participating in the effect seen. Thus the primary aim of this experiment was to search for additional arguments in favor of other interpretations of the effect observed. An easy way to surround a solid particle with a liquid is to condense saturated water vapor on the solid's surface in a diffusion chamber [12]. A water-receptive particle surface (we used Fe30 4 [7]) is enclosed completely by a condensed drop and the coated particle falls in a gravitational field with a speed convenient for observation. If such a coated particle still exhibits the properties responsible for the MCE, the magnetic field ii perpendicular to the force of gravity should reflect its trajectory from the line of free fall. To avoid....ambiguities when identifying tracks, it is necessary to periodically reverse the field H (scanning). In this way one can reveal and account for any systematic effects (e.g., photophoresis). Under such conditions a particle exhibiting the MCE must show an oscillation along the magnetic field strength line. The first observations made in the diffusion chamber show that the M CE is still present. Ferromagnetic particles (increased in mass six order of magnitude due to the saturated water vapor condensation) fall into the microscope field of view in accordance with the expectation if the MCE is exhibited. A photograph of such a trajectory is shown in Fig. 4. Thus, the test decided in favor of a magnetic charge effect. This experiment also permits measurement of the magnetic charge. A simple derivation produces the following formula:

601

0

H

y I

0

-2

2· 10 cm

Fig . 4 . Photograph of a particle trajectory in the diffusion chamber. 4> is the light beam; OY is the force by gravity direction; OH is the force line of the magnetic field. The scanning frequency of the magnetic field is v = 5 Hz, and its amplitude is H = 2. 7 gauss .

g=

l81ra,3

-

H

-VH./vs'

21p

(6)

where H is the magnetic field strength, T/ is the viscosity, 1' is the free fall acceleration, p is density, vs is the fall velocity and VH is the velocity of oscillation of the particle along the magnetic field strength line. During this experiment the tracks of 428 particles were photographed and processed. A charge distribution histogram was obtained and approximated by a log-normal distribution using the minimization program FUMILI (12, 13]. The most probable value of magnetic charge obtained is: (7)

It is evident that the charge of Dirac's monopole, 9D = 3.29 x 10- 8 gauss.cm 2 , is within the range of Eq. (7). The log-normal distribution suggests that among the observed events are rare cases where a measured value is considerably greater than the mean. This suggests that among our measurements there may be charges which are a multiple of the elementary (Dirac) charge .

602

5th Experiment: Detection of a discriminating to light of various polarizations

magnetic

charge response

This experiment continues a series of experiments which established the existence of an apparent magnetic charge effect (MCE) in ferromagnetic aerosols. The idea for this experiment was prompted by Barrett's theoretical investigations of electromagnetic phenomena due to fields the underlying algebraic logic of which obeys symmetries higher than the conventional or U(l) symmetry, i.e., higher symmetry fields than those of "Maxwell's theory" or U(l) theory [15, 18-22]. Barrett's theory applied to MCE provides an explanation for the apparent appearance of a magnetic charge in ferromagnetic aerosol particles. Furthermore, an analysis of the phenomenon from this group theory point of view results in some experimentally testable consequences [20-22]. According to this group theory treatment of radiation-matter interaction by Barrett: (i) Magnetic monopoles can exist in fields of SU(2) symmetry, but not in fields of U ( 1) symmetry. There can be positive and negative magnetic charge in fields of SU(2) symmetry, but not in fields of U(l) symmetry. (ii) Ferromagnetic aerosol particles behave as if their field description is of SU (2) symmetry form because of (a) the spherical boundary (cavity) conditions of the particle, and (b) the large spin exchange coupling offered by ferromagnetic compounds in solution. The aerosol particle's spherical boundary conditions (SBC) and the spin exchange coupling (SEC) provide the conditions for fields of SU(2) symmetry form due to the resulting compactification of degrees of freedom (i.e., higher order transformation symmetry) offering a lower energy state. This compactification of degrees of freedom of U ( 1) fields into fields of SU(2), or higher, group symmetry results in the particle behavior of a "pseudo" monopole or instanton. (iii) The directionality of the interaction between such a particle, constrained by SBC and with the availability of SEC, and light is dependent on the polarization handedness (chirality) of the light. The existence of SBC and SEC conditions in ferromagnetic aerosol particles also provides the conditions for field degrees of freedom compactification, resulting in the substitution of SU(2) for U(l) symmetry group rules describing the particles' behavior with respect to external field influences. Therefore, Barrett offered the following predictions. (For ease of referral, a ferromagnetic aerosol particle with SBC and SEC and exhibiting MCE behavior will be referred to as an SU(2) aerosol particle.):

603 A. The irradiation of SU(2) particles with linearly polarized light (LP, U(l) symmetry field) will result in no increase in particle movement over that detected in the previous series of studies [2, 5-7, 11-12]. The resulting condition can be referred to as the pseudo-magnetic monopole's ground state. B. The irradiation of SU(2) particles with circularly polarized light (CP, U(l) symmetry field) will result in increased particle movement over that detected in the previous series of studies. The resulting condition can be referred to as the pseudo-magnetic monopole's excited state. In other words, in case B the aerosol particles will be attracted or repelled by the external magnetic field more strongly than in case A. The prediction was that the effective magnetic charge in case B would be larger than in case A. These assertions were tested in the following experiment. In this experiment and, as before, the LP light source was a single-mode laser ( A = 4400 A, 25 mW) and the aerosols were generated by an electric spark source with iron contacts. The motion of the submicron size (lo- 5 -10- 6 cm) ferromagnetic particles generated was observed in a homogeneous magnetic field (Helmholtz coil). The light polarization was changed by conventional methods using a double refraction quarter-wave plate, e.g., for = is ±2 x

off erromagnetic particles by laser light beam in a magnetic field

It is clear that it is not possible to interpret this effect as due to ordinary optical levitation in a laser beam [23], because we observed the effect only in the case of ferromagnetic particles and only in the presence of magnetic fields. Moreover, the fact that the effect is observed when 0 = 0 dismisses possible explanations, such as

610

photophoresis and light pressure, because the the contrary light beams balance (the ray returns upon itself [24]). Therefore we offer again an interpretation of the effect within the context of a magnetic charge formulation. This interpretation is based both on our previous investigations and those of others: a magnetic charge is created on a ferromagnetic particle in an intensive light beam and its value is proportional to the light beam intensity, ( as described above). Thus, a ferromagnetic particle in a motionless gas medium and subjected to a gravitational force, F'Y, and in the presence of a laser light beam and vertical magnetic field, Hy, acquires a magnetic charge +G or -G. The force acting on the magnetic charge, G, is: (14) Fa= GHy. The occurrence of a balance:

Fa= F'Y

(15)

is realized only for the two conditions: The magnetic charge is +G and the magnetic field, Hy, is directed upwards. ii. The magnetic charge -G is and the magnetic field, Hy, is directed downwards. 1.

In both cases the particles will entrain and be concentrated by the laser light beam near an upper boundary due to the diametrical distribution of the light intensity (Gaussian). The entering of a particle from the boundary progressing towards the axis of the light beam causes an increase in the force, Fa (because of an increase in the magnetic charge, G), and a particle propels itself towards the upper light beam boundary. As soon as the horizontal magnetic field, Hx, is switched on, the ensemble of particles must commence motion along (against) the magnetic field intensity iix. This effect was observed in the experiment reported above. We thus observed for the first time particle separation dependent on the sign of the magnetic charge, ±G, and there is no alternative interpretation of this effect. b. The accumulation of ferromagnetic particles near the screen

If both the particles' motion towards the screen and a short-range repulsive force, Fr, from the screen and acting on the particles take place, an accumulation of particles near the screen will be observed (in conformity with Fig. 8). We still do not understand the nature of this repulsive force but we determined that it is a nonlinear function of the screen-particle distance and its magnitude sharply increases for distances of the order of< 10- 2 cm (Fig. 7). The experimental

611 +

G-

- G-

F;;y

~!,'

1)

;:;

-

Fi -- q,

cp

Fi H9

Hx

Hx Hy

Fig. 8. Two different versions of the observed effect. S is the screen; D is the diametrical of the light beam density, 'P.

distribution

data deny that this force is an electric force, so perhaps it is due to the specifics of magnetic charge interaction with matter [25). None the less, we see that for certain distances the magnetic force, Fax, is equal to the repulsive force, Fr, because the particles stop motion and accumulate at this point. Therefore, let the first force, the magnetic force, be Fax=

(16)

ngHx

and the second force, the repulsive force (because of a formal similarity of this effect with Meissner's and according to Fig. 7) be Fax=

A . 2 r

(17)

Fax=

Fr,

(18)

Then for stationarity

or ngHx

A

= 2r

,

(19)

where n is an integer, g is an elementary magnetic charge, A is some constant and r is a particle-screen distance. Thus, the particle-screen distance is a function of the total magnetic charge, ng, of a particle (when H = constant and 4>= constant).

612

For two neighboring equilibrium points we have

(20)

Therefore, (21) and the ratio ~ is the ratio of integer numbers (by definition). J These formulas can be applied to the incident ray shown in Fig. 6 (see Table II). In the middle column of Table II, the ratio ~ is given rather well by integers (mean J errors are smaller than 10%) and the magnetic charge of a particle is indeed a multiple of the elementary magnetic charge, i.e., G = ng if Eq. (17) applies. Table II

-ni

r, in relative units

n

n·]

7.5

108 2.15

= 2 + 7.5%

11.0

54 2.25

= 2 + 12.5%

16.5

27 1.47

= 3/2

- 2%

20.0

18 1.44

= 3/2

- 4%

24.0

12 1.83

=2-

8.5%

32.5

6 1.95

=2 -

2.5%

45.5

3

1.58

= 3/2 + 4%

57.0

2 1.97

80.0

=2 -

1.5% 1

The value of the magnetic charge can also be obtained. If a particle has both electric, q, and magnetic, G, charge, then on the basis of the experimental evidence

613

described above we can provide the following system of equations: A 2 r-i A GHx +Eq = 2 r-1 GHx=

whence G

E

= q-H-. 'X

--

(22)

I r2

:f-1 rj

(23)

'

where Eis an electric field intensity, EIIHx, and q is the electric charge of a particle. In the experiment reported here, E = 650 v /cm = 2.17 CGSE and H = 9.44 gauss, ri = 60 and ri = 4 (relative units). Thus, for q = e = 4.8 x 10- 10 CGSE, we obtain G = 5 x 10- 13 gauss.cm- 2 . This result coincides with the results of early measurements of the magnetic charge of ferromagnetic aerosols [2-10, 12, 24], and this agreement suggests that the nature of the observed phenomena in all these experiments should be the same. Note that the result obtained in its magnitude coincides with the condition of vertical force balance. In our case, Fay = GHy = 5 x 10- 13 x 10 = 5 x 10- 12 dyn. Therefore, F-, = 5 x 10- 12 dyn and this value is specifically for a particle with radius r

=

(

3F. ) 1/3

~

a= 981 cm.s-

2•

= 5.38 x 10- 6 cm, where Hy = IO gauss, p = 7.8 g.cm-

3

and

In our experiment the particle radius is 10- 5 - 10- 6 cm [2, 7].

Commentary And so the results of all six experiments are uniquely explained within the framework of the magnetic monopole concept. The values of particle magnetic charge obtained in these experiments divide into two groups. The values may be expressed as (24) This empirical formula provides a wider spectrum of values than Dirac's formula [1], with which it may be compared. However, it is in fine agreement with a series of experiments [3, 5, 9, 10]. There are a number of interpretations of magnetic monopoles [26], and the concept requires further effort and attention. A similar problem arose concerning electrically charged particles (see [27] and the Millikan-Ehrenhaft polemics) which still has not been satisfactorily solved. Perhaps a solution requires a new conception of what charge means.

614

A discriminating effect induced by circularly versus linearly polarized light, and predicted by Barrett's group theory approach [18-22] to higher symmetry forms of electromagnetism, has been confirmed by experiment. An effect induced by the chirality, or handedness, of polarization-modulated light (the dependence of the magnetic charge sign on the chirality of light) has also been discovered. This is the first step in the comprehension of a magnetic monopole creation process and the first successful collaboration of theory and experiment. The effect of ferromagnetic aerosol particle accumulation in a light beam near a screen in a magnetic field (screen effect) was discovered for the first time. This effect results in particle separation depending on the sign of particle magnetic charge, ±G. The physical nature of the screen effect (i.e., the nature of the repulsive force) is presently unknown and is an object of further investigations. Nonetheless, this effect permits experiments with large numbers of similarly charged particles, i.e., opens up the possibility of studies of collective phenomena. And a final remark. Magnetic charges (monopoles) are experimentally observed only in the presence of two components: light and ferromagnetic particles. It seems, therefore, that magnetic charges are created as a consequence of an interaction between photons and ferromagnetic particles, and, moreover, such charges cannot exist without these physical conditions: without light a particle loses magnetic charge almost instantaneously. According to Barrett, a ferromagnetic microparticle provides the conditions for SU(2) symmetry forms which are necessary for magnetic charge creation. Perhaps such conditions existed by themselves in an early universe but are absent now and must be specially prepared. Thus from this perspective a search for relic monopoles makes no sense. Appendix: Interpretation of the repulsive force in the "screen effect" in the 6th Experiment: Magnetic charge and optical levitation of ferromagnetic particles in a magnetic field

The "screen effect" is interpreted here as due to screen heating in the region of the lighted area, with this area becoming a heat radiation source. Let R be the radius of the heated screen "spot" and 0 be the radiation density (W.cm- 1 ) of the incident lasers. In Fig. Al we have: dSo = RdRdcp , dS

= dSocos[90 -

where cpis the azimuth. But

a]

= dS0 sin[a]

,

(Al)

615

particle

""' "' X

2

(x

' ' "

" ' ' " ' " ' ' ' " " ' ' 1n +R ) ' 2

' ' '

"

R

"

a

dSo

' (dR)

Screen

'\

Fig. Al

sin[a] = (x2 +~2)1/2 Therefore, X

dS

= RdRdcp(x2 + R2)1/2

(A2)

Thus, the element of heat density flux from dS0 towards a particle is: dS

=

~ibodS

x2 + R2

=

~iboxdR2 dcp

2(x2

+ R2)3/2

The projection of the OX axis is: . ~ibodcpx2 dR 2 dibx = dibsma = (x 2 + R 2) 2 2

(A3)

The total heat flow along the axis OX is:

(A4) where

~

is the transformation ratio of light into heat.

616

However, dT 4>x= -K dx '

where T is temperature (A5) we obtain

(A5)

and K is the gas thermal conductivity. R2 7r~4>o 2 R2 X +

From (A4) and

dT

= -K-d

X

'

or

dT

= _ 1r~4>oR2

dx x2

K

+ R2

'

and the integral is:

T When x

= 0 then

T

= To.

=

1r~4>oR x K arctg R

+C .

(A6)

The temperature of the screen is C

T

= To -

1r~4>oR ---arctg K

= To.

x

-R.

Thus (A6a)

The formula (A6a) is the low end of the temperature change of the gas along the direction OX. The temperature jump on a particle (letting the heat conductivity of a particle be negligible) is: tl.T

[ = - 1r~4>oR K arctg

xR+ r -

x]

(A7)

arctg R

The Taylor series expansions of arctg are: x

arctg

+r R x

arctg R As x

>>R, we can

,:

0-t------------------..( SIIOT START

0.0

SHOT OUT

TIME L:J ..J N N

=>

:e

0

+------------------SIIOTSTART

0.0

SIIOT O~T 0.4

0.0

TI\IE (ms) 1.2

~lu1.1.lc\'olt:igc for Rl'II' 12

Fig. 11. Muzzle voltage measurements.

The implication of Kirchhoff's law is that the muzzle voltage should increase as the armature accelerates along the rails because v increases and this is contained in e AB. Measurements show no velocity-dependent increase in the muzzle voltage. Typical traces of Vm are reproduced in Fig. 11, which were obtained by Stainsby and Bedford [28]. In these measurements the armature was a traveling electric arc in air between the rails. This explains the large voltage drop of 200 V across the armature. The arc pushed against a plastic projectile and accelerated it for the first 0.9 ms of Fig. 11. Then a sudden large increase in muzzle voltage occurred due to the arc leaving the railgun and dissipating the remaining energy stored in the inductance by what is known as the muzzle flash.

662 The velocity-dependent part, eAB of Eq. (25), should have become prominent in the second half of the 0.9 ms record of Fig. 11, but there is no indication of it. Most investigators now appear to be resigned to the absence of eAB in the muzzle voltage measurement. For example, in the description of one of the most powerful railguns built, Holland et al. [29] simply state: "A voltage probe located at the railgun muzzle measures the arc-voltage drop as function of time." Equation (25) assumes that there are no e.m.f.s induced in the rail portions AE and BF and in the voltmeter branch EF. These conductors do not carry current. Consider an element in the rail portion AE. From Eq. (24) it is obvious that field theory obtains the motionally induced e.m.f. with the help of a motionally induced electric field strength v x B. Special relativity requires that v be the relative velocity between the moving charge and "the observer" . In the case considered here the observer is the muzzle voltmeter, which is stationary with respect to the rails. In AE the magnetic flux density is perpendicular to the railgun plane and the only charge velocity in the rail would be the drift velocity of the conduction electrons. But when no current flows, field theory assumes that the drift velocity is zero, i.e. v = 0 in Eq. (24). Hence according to field theory no e.m.f.s can be induced in the current-free branches AE, BF, and EF. In the active rail branches there is current flowing and the conduction electrons move with a finite velocity v with respect to a stationary observer, which can be the breech voltmeter. But now it is found that, with a vertical flux density and axial charge motion, the vector product of Eq. (23) gives an induced electric field which is transverse to the conductor axis. This cannot be detected by the Vb voltmeter. Field theory, therefore, suggests that no e.m.f.s can be induced in the conductor branches AD, BC, and DC. Only in the armature does the special theory of relativity permit the induction of a back-e.m.f. In that case v must be taken as the charge velocity in the direction of the armature velocity with respect to the stationary voltmeters Vb and Vm. This represents convection of the conduction electrons with the armature metal, and it is not the electron drift velocity due to current flow. In summary, field theory underestimates the induced back-e.m.f. in the railgun and it is unable to explain why the muzzle voltage is not a function of armature velocity. Both these problems have been resolved with Neumann's law of dynamic induction, Eq. (9). It is important to remember that Neumann's law does not stand for reciprocal interaction, but for a one-way action of cause and effect. The cause is a current element and the effect is an induced e.m.f. in a conductor element which may, or may not, carry current. Apply Eq. (9) to the three elements dm, dn, dl of the railgun circuit of Fig. 12. Only the first two elements carry the current i. Without current, matter element dl

663

-2

r '"

.:

2

,-·

_!___

.·

Fig. 12. Inductive interactions

-- 0, zo > 0), save for a multiplicative constant, is the FWM expressed in the coordinates, u and v, i.e.

(5) Note that the FWM has a single independent parameter that affects its shape, i.e. we can rescale the spatial coordinates to obtain

(6) where (P1,u 1 ,vi)

=

(f3p,{3u,{3v).

670

2

0.5

1.5

0.4 0.3

1 0.2 0.5 0.1 0

0 -0.1 6

-0.5 6

10

10

0

-10

u

0

-10

u

Figure 1: The real part of the FWM Gaussian envelope for parameter values: /3zo= 0.5 (left), and f3zo= 2.0 (right). Axes labels are dimensionless coordinates, f3p, and /Ju. The FWM, "PF, consists of a plane wave, eif3v, propagating in the negative z direction and a Gaussian envelope with complex variance, (zo+iu)/2/3. Graphs of the real part of this complex Gaussian envelope are shown in Fig. 1 for two different values of {3z0 . The wave propagates indefinitely in the positive z direction without broadening or decaying. At the wavefront (u = 0), the pulse envelope has a transverse profile which is a real Gaussian with a constant variance of zo/2/3. Along the z-axis, and, as well, along the hyperbolic surface defined by p2 = + u 2 , the pulse envelope falls off in magnitude eventually as 1/lul.

z5

3. Finite

Energy

Superpositions

While possessing most of the characteristics sought by Brittingham and outlined in the introduction, FWMs have infinite energy and are not therefore physically realizable. They are, however, useful as a basis to construct finite energy solutions of the scalar wave equation. In the following, we follow Ziolkowski's [36] development of a finite energy superposition leading to a solution which he has termed the modified power spectrum (MPS) pulse. Since the FWMs are solutions of the homogeneous scalar wave equation, it follows

671

that if a function is formed as

VJFE = fo00 F(/3)7PF d/3

(7)

where F is an arbitrary function, then VJFE is also a solution of the same equation. Expanding this we have 1

- ----VJFE - 47r1·cZQ + 1,U .)

looF (/3)

e-pfJ

0

d

/3

(8)

where p = p 2 /(zo + iu) - iv. This expression resembles that of a Laplace transform in the new variable p, which makes it convenient to choose suitable weighting functions, F, in designing finite energy superpositions. Furthermore, it can be shown [36] that if F(/3)/./P is square integrable, i.e.,

{oo F2(/3) d/3 < oo

lo

(9)

/3

then 1PFE possesses finite energy. An example of a weighting function that yields a finite energy superposition with rather interesting properties is

F(/3) =

{

0

where a and b are arbitrary real constants. pulse is ~PS

/3 > b otherwise

41rie-a((J-b)

1 ------e

1

(10)

The resulting expression for the MPS

-bp

(zo + i u) (p + a) 1

p2

+ (a - iv)(zo + iu)

eibv e-bp

2 /(zo+iu)

(11)

which is illustrated in Fig. 2. The form of the MPS pulse differs from the pure FWM only by the fractional term 1/(p + a). Once again, we rescale the coordinates, (p1, u 1, v1) = (bp, bu, bv) to obtain (12)

in which it becomes apparent that the MPS pulse contains two independent parameters affecting its shape: ab and bzo. As was the case for the FWM, a plane wave travels in the negative z direction and is modulated by a Gaussian envelope in p which has the same complex variance but with parameter b replacing /3. The pulse is therefore concentrated on the zaxis and does not broaden as the wave propagates. The additional fractional term

672

1

0.3

0.8 0.2 0.6 0.4

0.1

0.2 0 0 -0.1 6

-0.2 6 10

0

-10

10

u

0

-10

u

Figure 2: The real part of the MPS pulse Gaussian envelope for parameter values: bzo = 0.5 (left), and bzo = 2.0 (right). Axes labels are dimensionless coordinates, bp, and bu. present in the expression for the MPS pulse however, causes the wave solution to decay eventually. This is seen by observing VJMps at the pulse center which is 1

1

"PMPs I(p=O'u.=O) = -ZQ (a - 2'lZ. )

(13)

and so the amplitude of the propagating wave eventually falls off as the inverse of the distance. For small values of z however, the MPS pulse envelope should behave much as the FWM, i.e. the amplitude of the envelope at the pulse center is approximately constant. The distance at which the square magnitude of the wave at the pulse center falls to half its initial value is z = a/2 and so the transition between these two regions can be controlled by the choice of parameter a.

4. Physical

Realization

The previous sections have dealt with the design and analysis of solutions to the homogeneous scalar wave equation. The FWMs and MPS pulses can in theory propagate in a source free region of space, however it is not readily apparent how they or suitable approximations might be launched in a practical experiment. In this section, we develop an approximate Huygen's reconstruction of the desired localized wave.

673 To begin, consider the Green's function solution to Eq. 1, i.e. the solution for an impulsive source term, s = c5(r )c5(t), g(r, t)

1

= -c5(t 41rr

- r/c)U(t)

(14)

where r = lrl and U is the unit step function, U(t) = 1 for t > 0 and zero otherwise. Now a wave solution for an arbitrary source distribution can be expressed as a convolution in all space-time with the Green's function:

'1/J(r,t)

fo s(r', t')g(r

=j

00

- r', t - t') dt' dV'

(15)

where the volume integral is over all space. Inserting the expression for g and performing the t-integration, we have 1

'1/J(r,t) = / -s(r, 41rR

,

t - R/c) dV

,

(16)

where R = Ir - r'I- Eq. 16 expresses a wave as a volume integral involving the source distribution, s, and the propagator, 1/ 41rR. Now leaving this aside for the moment, consider Kirchoff's solution to the wave equation for a source-free region of space, V, bounded by the surface S (see for example [20, pages 40-42]), 1

1{

'1/J(r,t) = 41r ls

1 [ a¢]

R 8n'

a

1

['1/J] 8n' R

1

+ cR

[a¢] a 8t'

R}

8n'

dS

,

(17)

where n' is an inward-pointing unit vector normal to the surface S and the notation [~] means to evaluate cJ!, ¢( r', t') and then replace t' with the retarded time t - R/ c. If we take the volume, V, to be all space to the right of the plane z = z', then we can simplify Eq. 17 as

_1_ { [8¢] (z - z') ['1/J] - (z - z') [8¢]} ls 41rR 8z' R cR 8t'

'1/J(r,t) = /

dx' dy'

2

(18

in which S is the plane z = z'. Furthermore, for points near the propagation axis and far from the plane z = z', we can make the approximation that (z - z') ,....,Rand 1/ R tends to zero. In this case, Eq. 17 reduces to

'1/J(r, t)""

l

{ [ -8¢

- 1

s 41rR

8 z'

l

- -1 [ -8¢ c 8t'

l} , , dx dy

(19)

which has the form of Eq. 16 with a source distribution given by

s(r, t ) = c5( z - z)I

(

1 8'!p) = c5(z- z I )-8'1p -8'1p- -8z C 8t 8u

(20)

674 In other words, the wave solution, 'lj;, can be reconstructed approximately in a hemispherical region to the right of a plane transverse to the direction of propagation by a source distribution concentrated on the plane. Returning now to the MPS pulse illustrated in the previous section, we have 81PM_ps .

au =

1,

r

l(

bp2

zo + i U)2

-

l

a - iv p2 + (a - i V)(zo + i U) '1/)Mps

and so we can reconstruct approximately the MPS pulse in the region source distribution on the plane z = 0 as sa(P, t)

=

z

(21)

> 0 from the

1,

p2

[

+ (a -

ict)(zo - ict)

bp2

(zo - ict)

a - ict ] 2 p + (a - ict)(zo - ict)

2

eibct e-bp

2 /(zo-ict)

(22)

for which it should be clear that sa, at any particular value of p, is a broad-band temporal signal. This is illustrated in the graphs of Fig. 3 for a series of normalized radii and two different values of the parameter ba. The source distribution, just as the MPS pulse itself, is not separable in space and time-the only separable term that doesn't depend on p is the sinusoidal term eibct. In order to launch an MPS pulse using the method just described, it is necessary to make a further approximation by discretizing the source distribution. The resulting array of point sources must be individually addressable, i.e. a series of unique excitation signals is required to be made available to the source elements. Ziolkowski [37, 38] has used this method to reconstruct acoustic MPS pulses from a 25-element planar array. The resulting beams, although degraded somewhat over theoretical predictions, exhibit the expected near-field extension and reduced beam broadening over the CW-driven array.

5. Green's

Functions

for Cylindrical

Geometry

As an alternative to the approximate Huygen's reconstruction explored in the previous section, it may be possible to launch localized waves from sources that have cylindrical symmetry [28, 29, 30]. To do this, it is useful to find a Green's function in the cylindrical characteristic coordinates, (p, u, v ). We wish to find 9u(P, u, v) that is the solution to Eq. 2 with the impulsive source term, s

6(p) 21rp

= -6(u)6(v)

where the term 6(p )/21rp is a two dimensional impulse on the z-axis.

(23)

675

p=O 8=10

p=O 8=2 0.5.....---------,.--------, Qi----

-0.5 -5

0

5

-0.1 -5

0

t

t

p:28=2

p:28=10

0.5

0.1

-0.5 -5

-0.1 -5

0

5

0

t

p:58=10 0.1

0.5

ov-....__ _____

,

0

-0.5'------....__ __

Figure 3:

5

t

p:58=2

-5

5

0 t

5

-0.1----------5

0

5

t

Excitation signal, sa, required to launch an approximate MPS pulse is shown for radii bp = 0, 2, and 5 and for dimensionless parameters bzo = 1 and ba = 2 (left) and ba = IO (right).

676 We begin by performing Fourier transforms in the u and v coordinates and a zero order Hankel transform [3] in the p coordinate, on Eq. 2,

(24) in which Gu is the Hankel-Fourier transform of 9u, K is the Hankel transform variable or spatial frequency corresponding to p, and Wu and wv are the Fourier transform variables corresponding to u and v respectively. Now 9u is obtained by inverting the Fourier- Hankel transform: (25) where Jo is the zero order Bessel function of the first kind and where the integrals with respect to Wu and wv are interpreted in a principal value sense. Employing Eq. 3.352-7 from [8], (26) where sgn(u) is one for u > 0 and minus one for u < 0. Now, using Eq. 6.631-6 from [8], we obtain 00 __ l_sgn(u) -iwv(v+p2/u) d (27) 9u - 167r 2 e Wv U -00

1

in which the integral is recognized as a shifted delta function. Furthermore, we observe that if we had reversed the order in which the two inverse Fourier transforms were performed then the resulting expression would have u and v interchanged. The Green's function is therefore expressed in either of two equivalent forms: (28)

Since 9u contains both retarded and advanced components they must be constrained to ensure that they are causal by setting it to zero for time less than zero, i.e.

9c(p,u,v)

= U(t)gu(p,u,v) = U(v

- u)gu(p,u,v)

(29)

and now 9c is the causal Green's function, i.e. for the retarded wave alone.

6. Localized

Wave Solutions

from Line-Sources

Localized waves are solutions of the homogeneous wave equation and cannot be launched directly from simple source configurations. The approach taken by Ziolkowski et al. [37, 38) and outlined previously involves an approximate Huygen's

sn reconstruction based on a discrete array of point sources arranged on a plane perpendicular to the direction of propagation. We now pursue a different approach to the problem of generating localized waves by making use of the Green's functions developed in the previous section. Consider a source of the form

8(p) = -e

s(p, u, v)

i/3

v

21rp

h(u)

(30)

where h is a function of the distance from the wavefront, u. For a wave propagating in the positive z direction, eif3v represents a plane wave propagating in the negative z direction, and the term 8(p)/21rp is a unit impulse concentrated on the z axis. The propagated wave is VJL(P,u, v)

= s(p, u, v) ®p,u,v9c(P, u, v)

(31)

where ®p,u,v refers to convolution over all space-time. The convolution in p is trivial, and so we have VJl=

_!..11"' 00

81r

h(u - u')

-oo

1

u') O(u' + p2/v') du' dv'

U( \

ei/J(v-,I)

-oo

V

(32)

Performing the u' integration yields eif3v

1Pl= -

81r

looU(v' +

e-if3v'

p 2 /v')

-oo

and recognizing that the argument, v' otherwise, we obtain: 1

h(u

+ p 2 /v',

+ p 2 /v') IV 'I dv'

is positive for all v' > 0 and negative

1Pl= -eif3v h(u + p2/v') 81r lo

=

Now consider the case where h(u) positive. We now have

1/J= l

-i{3v1

oo

r

(33)

e

v'

dv'

1/(u - izo), for z 0 and

(34)

/3 both real and

foo 1 e-if3v' dv' 81ri(zo+ iu) lo v' - vP eif3v

(35)

where vp = - p 2 / ( u - iz 0 ) is a simple pole in the complex v'-plane. We perform the integral in Eq. 35 by integrating along the pe-riphery of the entire fourth quadrant in the complex v'-plane. The pole, vp, is enclosed only when u < 0, and so we have ,.,, o/l

=

eif3v

87r7,.( ZO + 1,U . )

[-

loo 0-ioo

e-if3v' dv'

1 I

V -

Vp

- 2rri U(-u)

e-i/3vp]

(36)

678

0.25 0.2 0.15 0.1

5

p

0 -10

u

Figure 4:

The real part of the complex envelope corresponding to a field propagated by a line-source for parameter value f3zo= 1. Axes labels are dimensionless coordinates, f3p, and f3u.

and with a change of variables and the aid of Eq. 3.352-4 in (8] we obtain

1Pl=

. eif3v

.

e-/3p2/(zo+iu)

41ri(zo+ iu-)

[~E1 (-f3p2/(zo + iu)) - 'TriU(-u)] 2

(37)

where E 1 is the exponential integral function [l]. The complete solution, for this particular line-source, is then a product of a FWM and a term involving the exponential integral:

(38) We note the similarity in form between this result and that reported by Vengsarkar et al. [32] for optical waveguide localized waves. In the latter, the FWM solution is modified by a term due to the waveguide boundary which is analogous to the line source in the current analysis. For large transverse radius, E1(e) rv and so the most slowly decaying 2 component of the field falls off as 1/ p . This solution is illustrated in Fig. 4 where the real part of of the envelope is plotted as a function of the dimensionless coordinates introduced earlier. The wave is suppress in the region u > 0 when compared to the

e-E.;e,

679 FWMs illustrated in Fig. 1. This is likely a result of demanding causality in the Green's function formulation of the previous section.

7. Discussion

This chapter has examined certain packet-like solutions of the scalar wave equation and their underlying focus wave modes. It was shown that the FWM is a natural, exact, non space-time separable solution of the homogeneous scalar wave equation in moving characteristic coordinates. Localized waves are derived from superpositions of the basic FWMs. Ziolkowski's MPS pulse was illustrated as an example of a LW possessing finite total energy and a scheme based on an approximate Huygen's reconstruction was used to launch these pulses. An alternative approach was developed for launching localized waves employing a causal, cylindrically-symmetric, Green's function and a source aligned on the propagation axis. The fields of application for these novel wave phenomena are numerous. The success of many imaging and communications systems is critically dependent on the ability to maintain a collimated beam or pulse train along the direction of wave propagation. Localized waves, by spreading energy over spatial and temporal frequencies, have the potential to develop narrow beams of focused energy beyond that which is currently achieved using monochromatic or narrow-band beams. Medical ultrasound imaging, therapeutic procedures, range finding, secure communications, radar and sonar are just a few of the applications that stand to benefit from the development of LW-based systems. The employment of LWs in practical applications is impeded however by the technical challenges associated with the engineering of suitable antenna or transducer arrays. In conventional arrays, a monochromatic signal is distributed to individual elements by simple, and often passive, circuit elements that perform phase shifting and attenuation. In sharp contrast to this, arrays designed to launch LWs must be constructed with elements that are individually addressable. Multiple independent waveform generators must be employed to deliver a large number of unique temporal waveforms to the array. The costs associated with such a construction may be prohibitive in many applications. However, as the interest in LW phenomena continues to grow, further developments that lead to simplified array design are likely to follow. We have presented one such possibility here with the introduction of a source aligned on the propagation axis.

Acknowledgments

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. The author wishes to thank Dr. Rod Donnelly of the Faculty

680

of Engineering and Applied Science, Memorial University of Newfoundland for many helpful discussions and observations.

References l.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. (Dover, New York, 1970). I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, J Math Phys 30 (1989) 1254-1269. R. N. Bracewell, The Fourier Transform and Its Applications 2nd edition, (McGraw-Hill, New York, 1986). J. N. Brittingham, J Appl Phys 54 (1983) 1179-1189. G. A. DesChamps, Electronics Lett 7 (1971) 684-685. J. Durnin, J Opt Soc Am, A4 (1987) 651-654. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys Rev Lett 58 (1987) 1499-1501. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. (Academic Press, New York, 1980). E. Heyman, B. z. Steinberg, .and B. Z. Felsen, J Opt Soc Am A4 (1987) 20812091. E. Heyman and B. Z. Steinberg, J Opt Soc Am A4 (1987) 473-480. E. Heyman, IEEE Trans Antennas Propag 37 (1989) 1604-1608. E. Heyman, Wave Motion 11 (1989) 337-349. E. Heyman and L. B. Felsen, J Opt Soc Am A6 (1989) 806-817. P. Hillion, J Math Phys 19 ( 1978) 264-269. P. Hillion, J Appl Phys 60 (1986) 2981-2982. P. Hillion. J Math Phys 28 (1987) 1743-1748. P. Hillion, Wave Motion 10 (1988) 143-147. P. Hillion, J Math Phys 29 (1988) 2219-2222. P. Hillion, Phys Rev A 40 (1989) 1194-1197. D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964). J. Lu and J. F. Greenleaf, IEEE Trans Ultrason, Ferroelec, Freq Contr 39 (1992) 19-31. J. Lu, T. K. Song, R.R. Kinnick, and J. F. Greenleaf, IEEE Trans Med Imaging 12 (1993) 819-829. H. E. Moses, J Math Phys 25 (1984) 1905-1922. H. E. Moses and R. T. Prosser, IEEE Trans Antennas Propag 34 (1986) 188-196. H. E. Moses and R. T. Prosser, Proc Roy Soc Lond A422 (1989) 343-349. H. E. Moses and R. T. Prosser, Proc Roy Soc Lond A422 (1989) 351-365. P. L. Overfelt, Phys Rev A44 (1991) 3941-3947. M. R. Palmer and R. Donnelly, J Math Phys 34 (1993) 4007-4013. M. R. Palmer, Proc IEEE Canad Conj Elec Comput Eng (1993) 43-46.

681

30. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, J Appl Phys 65 (1989) 805-813. 31. H. Shen and T. T. Wu, J Appl Phys 66 (1989) 4025-4034. 32. A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, J Opt Soc Am A9 (1992) 937-949. 33. T. T. Wu, J Appl Phys 57 (1985) 2370-2373. 34. T. T. Wu, R. W. P. King, and H. Shen, J Appl Phys 62 (1987) 4036-4040. 35. R. W. Ziolkowski, J Math Phys 26 (1985) 861-863. 36. R. W. Ziolkowski, Phys Rev A39 (1989) 2005-2033. 37. R. W. Ziolkowski, D. K. Lewis, and B. D. Cook, Phys Rev Lett62 (1989) 147-150. 38. R. W. Ziolkowski and D. K. Lewis, J Appl Phys 68 (1990) 6083-6086. 39. R. W. Ziolkowski, IEEE Trans Antennas Propag 40 (1992) 888-905.

ANALYTICAL AND NUMERICAL METHODS FOR EVALUATING ELECTROMAGNETIC FIELD INTEGRALS ASSOCIATED WITH CURRENT-CARRYING WIRE ANTENNAS

Douglas H. Werner Applied Research Laboratory 1he Pennsylvania State University P.O. Box 30 State College, PA 16804, USA

Wires with currents that oscillate Give rise to fields which radiate. At the speed of light they propagate, As James Clerk Maxwell did postulate And Heinrich Hertz would demonstrate. D. H. Werner

1. Introduction Considerable advancements in methods for both the analytical and numerical treatment of wire antennas have been made since the pioneering work of Maxwell [l] and Hertz [2]. In spite of this progress there remained, until recently, several wire antenna problems for which no closed form or exact solutions had been found for the vector potential or electromagnetic field integrals. Surprisingly, this has been the case even for such elementary wire geometries as cylindrical dipoles and circular loops. This chapter is primarily intended to provide a comprehensive summary of recent advances in the area of analytical as well as numerical methods for evaluating various fundamental electromagnetic field integrals associated with current-carrying wire antennas. Particular attention will be given to the derivation of exact expressions for several of the more commonly occurring and basic integrals found in antenna theory. Included among these integrals are the thin-wire generalized exponential integral, the cylindrical wire kernel integral, and the uniform current cylindrical dipole and circular loop vector potential integrals. In the past, these integrals were either evaluated by timeconsuming numerical techniques or by making certain convenient but restrictive assumptions. The solutions to the integrals presented in this chapter are mathematically exact in the sense that no assumptions or approximations were made in their derivation. These exact expressions may be used for evaluating antenna near fields to any desired degree of accuracy, while computational efficiency is achieved through the use of several recurrence relations.

682

683 In addition to these analytical expressions for the vector potentials and fields, several formulations are presented and discussed which are amenable to efficient numerical evaluation. Many electromagnetic field integrals contain sharply peaked or singular integrands, especially for situations in which the field point is in the vicinity of the source point. Consequently, extreme care must be exercised when numerical methods are employed to evaluate these integrals. Direct numerical integration may require a large number of terms in order to adequately sample the integrand in the interval where it varies most rapidly. Under these circumstances, a procedure may be followed for effectively isolating and removing the troublesome "singularity." This procedure is based on finding a decomposition of the integrand which has a slowly varying part that may be efficiently evaluated numerically and a "singular" part which, in many instances, may be integrated analytically. This singularity extraction technique has been applied in order to obtain numerically convenient forms for many of the electromagnetic field integrals discussed in this chapter.

2.

The Cylindrical Antenna Vector Potential

The solution of any antenna problem consists of finding the electric field E and the magnetic field H that are created by an impressed current distribution J . An auxiliary vector ~tential is often introduced in order to simplify the solution for E and H with a given J [3,4]. This vector potential is a consequence of the fact that the divergence of the magnetic field H is zero, that is

V ·H

=

(1)

O

Since H is a solenoidal field, i.e. it only has circulation, then it may be expressed in terms of the curl of some other vector function as 1 H=-VxA µ

-

(2)

where A denotes the vector potential. It can be shown that the associated electric field may be expressed in terms of the vector potential by

E = -i

w

A

+

_I_

V (V ·

A)

(3)

lWµE

where £ andµ represent the permittivity and permeability of the medium, respectively. Eqs. (2) and (3) suggest that only a knowledge of A is required in order to determine the electromagnetic fields which radiate from a particular antenna. The cylindrical wire antenna is one of the most fundamental and, as a consequence, commonly used of all antennas. Many antennas are constructed from cylindrical wires arranged in various configurations. The geometry for the cylindrical antenna under

684 consideration is illustrated in Fig. 1. The length of the antenna is 2h and its diameter is 2a. A circumferentially uniform current I(z') is assumed to be present on the surface of the cylindrical antenna. The cylindrical coordinates of the source point and field point are (a,(p'Jp 2-v 2) def,'dv

-ex>

O

(13)

686 1 integral has a closed-form Eq. (13) may be further simplified by recognizing that the (pJ132-v2)dv

(14)

-CX)

where use has been made of the fact that (15)

The zeroeth order Bessel function present in Eq. (14) may be expressed in series form as (16)

Furthermore, the series representation (17)

may be used in conjunction with Eq. (16) to arrive at ex,

n

1o(aJf32-v2)=LL cnk(/3a/2)2n13-2k(iv)2k n=Ok=O

(18)

where (-l)n (n!)2

( nk) = _n! k! (n-k)!

(19)

(20)

687 Substituting Eq. (18) into Eq. (14) and integrating term by term yields K(z-z')

=

L Ln 00

n=O k=O

-k l2 /3

( )2n Cn1c f32a

k(z-z')

(21)

where

ii f 00

½t(z-z')

=

(iv)2k eivlz-z'I HJ2>(p ✓f32-v2)dv

(22)

-00

By recognizing

a2k eivlz-z'I

=

(iv)2k eivlz-z'I

(23)

az2k

it follows that the integral in Eq. (22) may be written as (24)

A reapplication of the identity in Eq. (10) with p'

=

p leads to (2S)

Hence, using this form of I2k(z- z'), it may be readily shown that then = 0 term of Eq. (21) corresponds to the standard thin-wire kernel previously defined in Eq. (8). Retaining then = 0 and n = 1 terms results in an approximation of Eq. (21) given by K 1 (z-z')

=

K0 (z-z')

a2 (1 {1 - 2 2r 0

+

i/3r0)

(26)

688 Representations of this type are sometimes referred to as "extended" thin-wire kernel approximations (7,9]. For values of n > 1, approximations of the cylindrical wire kernel resulting from Eq. (21) rapidly increase in complexity due to the evaluation of the higher-order derivatives required in Eq. (25). An alternative exact expression for the kernel of the cylindrical antenna was found more recently by Wang [10]. The derivation of this exact representation depends on the fact that the cylindrical wire kernel Eq. (6) may be expressed in the following way: Tr

fe ,,, O R

= _!_

K(()

-iJIR'

1

def,' = --

1

(

d

-i/3C-rr) d C

Tr

Je

-iJIR'

def,'

(27)

0

where

R' = JR2 -2pacoscf,

1

(28)

R =

Jc2+

d2

(29)

d

Jp2

a2

(30)

1

(31)

=

+

C = z-z

The next step in the derivation concerns the determination of a suitable expansion for the integrand e-i/JR'contained in Eq. (27). This may be accomplished through the use of Lommel's expansions for (Jx+y)±vJv (Jx+y) [11], namely (Jx+yrv 1y(✓x+y}

00

I:

=

(-y~7)m (rxrcv•m> 1v•m (Ji)

m=O

(✓x+yr Jy(✓x+y}

= -½

·

00

=

I:

(y/~)m (Jx)cv-m>Jv-m(v'x)

m=O Setting "

(32)

in Eq. (32) and "

=

m.

½ in Eq. (33) leads to

(33)

689 (34)

(35)

The relations involving half-integer order Bessel functions (36)

(37)

(38)

may be used to further reduce Eqs. (34) and (35) to

,:;--;-;-.+};

cos (yX Ty

=

LJ

m=O

(-y/2)m

jm-1{/i}

m!

(/ir-t

(39)

(40)

If we let x

= (/jR) 2 and y =

-2/j 2pa cos1 (/32 p a)mcosm ..'TD ,a VJ m! (/3R)m-l m=l

L

(43)

where ~ 2!1 is the spherical Hankel function of the second kind of order m-1. Integrating both sides of Eq. (43) with respect to 'from O to -rand dividing by -r yields

.!.

je

'TT

O

-i,9R,d fl2R) 2n h2n
0 and k > 0

(56)

For the case in which k =0, Eq. (54) reduces to

A00

=

n (2n - 1) D (n, 0) (½)

(57)

But since 1 ' ( ~)

=

n =0

(2n -3) (2n)

(58)

(/,)

'

n > 0

and 1, D(n,0) =

it follows that

n =0

D (n -1,0) n > 0 (2n -1) (2n) '

(59)

693 -1

, n

0

=

(60)

-

l

(2n)2

A

n-1 o

, n>O

which provides us with a useful recurrence relation for calculating the coefficient AuoThese coefficients are important because they provide the initial values for the recursion given in Eq. (56). The recurs10ns developed for the coefficients Ank may be used to derive a set of recurrence relations for the cylindrical wire kernel expansion in Eq. (53). This is accomplished by first dividing both sides of Eq. (53) by /j to obtain a unitless quantity and then introducing the notation

!

2n

oo

K(()

=

~

L L

(61)

Qnk(()

n=O k=O

where (62)

By using Eq. (62) together with Eq. (56), a recurrence relation in terms of k for may be easily obtained. The form of this recurrence relation turns out to be

Qnt(()

=

1 Qnk-1(() (/3R)

(2n+l-k)

(2n+k) i (2k)

'

n > 0 and k > 0

A similar procedure is followed to find the associated recursion for

0 correspond to higher-order associated wire surface at 0, corresponds to higher-order associated integrals when m =2n +k. By introducing the change of variables r=z- z', the integral in Eq. (73) may be transformed into

,2 Im=

J

e -RiJ3R

,.

(74)

where ( 1 =

z-h

(75)

(76)

If we define e-i/:IR

1

R

@R)m

u=-----

dv

=

d(

(77)

(78)

then integration by parts may be used to obtain a recurrence relation from Eq. (74). This recurrence relation is given by

1 Im=---(m-l)@d)2

(79)

where

697 (80)

(81)

which requires a knowledge of the initial three integrals I0 , I 1 and I2 . One approach for evaluating the integrals I0 , I 1 and I2 involves using a Maclaurin series expansion of the complex exponential function. This approach has been used by Harrington [14] to approximate I0 by retaining the first few terms in the series expansion. Here, we present the complete series expansion for I0 as well as for I 1 and I2 • Before expanding the integrands it is advantageous to express I0 , I 1 and I2 in the form

C2e -il3CR-Ro> Io = e -i/3Ro --dC R Ci

f

(82)

1

C2e -il3CR-Ro>

/3

C1

I1 = e-i/3Ro -

f --f

-il3(R-Ro)

_e__ R3

,.

132

(83)

R2

C2 I2 _- e -il3Ro- 1

d(

d(

(84)

where R0 -- ✓z 2

+

(85)

d2

such that IR - RoI < h . The Maclaurin series expansion for the exponential function is represented by 00

e -il3CR-Ro> =

also we have

L

(i/3)" n=O n!

(Ro-R)"

(86)

698 (87)

The series in Eqs. (86) and (87) may be combined to produce

(88)

Finally, substituting this expansion into the expressions for 10 , 11 and 12 given in Eqs. (82), (83) and (84), respectively, yields the following exact representations [6]: 00

T = "'O

e -i~Ro ~ ~

n ~ ~

B

nk

F

k-l

(89)

n=Ok=O

(90)

00 n -i~Ro 1 ~ ~ B F I2 - e 2 ~ ~ nk k-3 f3 n=Ok=O

(91)

where 8 nk

(-i/3l (i/3Ro)n-k = ------

k! (n-k)!

(92)

(93)

Computationally useful recurrence relations exist for the coefficients Bn1cdefined in Eq. (92). These recursions are given by

699 B

nk

= -

_l ( n + l - k) B Ro k o k-1

'

n > 0

and k > 0

(94)

and 1, n

=

0 (95)

i/3Ro

--

B0-1

n

o , n > 0

Closed form solutions can be obtained for the integrals F_3 , F_2 , F_1 and F 0 while a recurrence relation can be used to determine the higher-order integrals Fm for m > 1. That is

(96)

(97)

ln

F_l

l

[(,2•R, , '1 0 t + Rt

>

(98)

=

Ri - C2 , '1

en[R1

F0

Fm

=

-

( 1]

=

2h

< 0

(99)

(100)

where the options for F_3 and F_1 are intended to avoid loss of precision. Numerical integration schemes may also be employed to evaluate 10 , 11 and 12 . However, these integrals are difficult to evaluate directly using numerical techniques

700 because their integrands are sharply peaked. This problem can be avoided by extracting the "singularity" from these integrals. Following this procedure results in

h

fo =

f

F_1 +

(101)

R

-h

h

I 2

==

_!_F - J_ F - _! F 132

-3

/3

-2

2

+ -1

f-h e

_!_ 132

-i/3R +

(/3R)2/ 2 + i/3R - l dz' R3

(103)

The integrals contained in Eqs. (101), (102) and (103) may be more efficiently and accurately evaluated numerically since their integrands are much smoother. The remaining terms appearing in Eqs. (101)-(103) correspond to integrals which have sharply peaked integrands but may be evaluated in closed form. The exact expression for the kernel and the vector potential may be used to find corresponding exact expressions for the electric fields produced by a uniform current cylindrical wire antenna. As previously stated in Sec. 2, the electric field may be expressed in terms of the vector potential by E = _l_

V (V·A) - iwA

(104)

lWµE

The vector potential associated with the cylindrical antenna shown in Fig. 1 is z-directed, which implies that

A

=

A/p,z)z

(105)

where h

Az(p,z)

=

4~

f

I(z ') K (z-z ') dz 1

(106)

-h

This property of the vector potential may be used in conjunction with Eq. (104) to show that (107)

701 where (108)

(109)

We next make use of some properties of the cylindrical wire kernel, namely [15]

- a K(z-z')

a K(z-z ') az'

= -

az

- & K(z-z')

az2

(110)

-

& K(z-z') az12

(111)

= -

in order to show that

!

Az(p,z}

= -

:.,.

j

[I(z'} K(z-z'}J:::~h

-Jl (z '} K(z - z '} dz') 1

(112)

-h

-

cJ2 2 Az{p,z}

az

= -

~

j[I'(z')K(z-z')

+

l(z')- d K(z-z') dz

4 7T

-f -h

l 11(z '} K (z - z '} dz

'l

]z'-h z, = -h

(113)

Eqs. (112) and (113) are valid for any arbitrary current distribution on the surface of the antenna. In particular, for an assumed uniform current distribution of l(z') = Iu, Eqs. (112) and (113) may be employed in order to reduce Eqs. (108) and (109) to the convenient forms

702

E (p,z) p

Ez(p,z)

=

=

E

uo

- J_ ~ K(z+h)l 132 ap

ei'"/2 [- 1 ~K(z-h)

132

ap

Euo ei'" 12 [ J_ ~K(z-h) 132 az

- J_ _E__K(z+h)] -iwA/p,z) 132 az

(114)

(115)

where (116)

(117)

For the uniform current cylindrical wire antenna of arbitrary radius, it can be shown using Eq. (53) that

_!_ ~ K(z±h) 132

Bp

-i/3~ =

oo

2n

L L

1 e 3 Ank (/3P) (/3R±) n=O k=O

. {(/3 p )2 [(2n

+ k+

1) +

iJ3R±]

(/3R±)2n•k

(/32

p a)2n

- 2n (/3R±)2 }

(118)

(119)

where (120)

Furthermore, the expression for the z-component of the electric field Eq. (115) contains the vector potential A 2 which may be evaluated using the exact representation Eq. (70). Recurrence relations may be derived for Eqs. (118) and (119) by following an analogous procedure to that introduced in the case of the cylindrical wire kernel. Let

703

(121)

(122)

where -il3~

Q

2

DpQnk(z±h)= - 1- e An1c(~l n 1 {(J3p}2[(2n+k+l)+ iPR:1:] 3 (/3p) (/3R:l:) R:I: (/3R:1:l - 2n (/3R:1:)2}

(123)

(a 1 DzQnk(z±h) =/3(z±h) e -il3~ An1c ~ )2 n --[(2n+k+l) (/3R:l:)3 R:I: (/3R:l:)k

+ i/3R:1:]

(124)

As before, Eq. (56) may be used along with Eqs. (123) and (124) to obtain the following set of recurrence relations: D Q P

n1t

=

(2n+l-k)(2n+k) i(2k)

D zQnk

1 r(Pp) 2[(2n+k+l)+j/1R:t]-2n(PR:t) 2 10 Q (PR:t) (Pp )2[(2n+k) + i/lR:t)- 2n(P~) 2 P n k- t

l

= (2n + 1 - k)(2n +k) 1 [ (2n +k + 1) + i/3R:1: D i (2k) (/3R:1:) (2n +k) + i/3R:1: zQnk-I

(125)

(l 2 6)

which are valid for n > 0 and k > 0. In addition to Eqs. (125) and (126), we also have n

=

0 (127)

2-] D Q _1_ (_/3p_al2[_(/3_p_}2 ___ [(2_n_+ 1_) _+i/3_R_:1: ___ ]-_2n_(/3_R_:1:)_ , n >0 2 P n-l0 (2n)2 ~ (/3p )2[(2n-1) +if3R:1: ]- 2(n-1) (/3R:1:)

704

,

n

=

0 (128)

1 - (2n)2

(/3p a)2[(2n+ 1) + if3R:tl R, (2n-1) + if3R:tDzQn-10

, n > 0

An exact expression for the magnetic field produced by a uniform current cylindrical antenna may be obtained by making use of the fact that (129)

Substituting Eq. (105) into Eq. (129) results in

H

=

Hl/>(p,z}

4,

(130)

where (131)

Using Eq. (106) in combination with Eq. (131) and assuming a uniformly distributed current lu leads to (132)

where (133)

In order to find an exact representation for the magnetic field component H 41, an exact expression must be found for the integral contained within the brackets of Eq. (132). This may be accomplished by first substituting the exact series expansion for the cylindrical kernel Eq. (53) into the bracketed term in Eq. (132), yielding the result

h

f /3 ap -h

_!___E_ K(z-z ')

oo

dz 1 = -

2n

L L

n=0 k=0

An1c! __E_ {(/32 p a)2n 12n•k ( p ,z,h)}

/3 ap

(134)

705 where the terms I2n+k have been defined in Eq. (73). simplified by making use of the identity

Eq. (134) may be further

(135)

in order to show that (136)

where t1Jln•k = ;;

)

li.,,k - i(P p) l2n•k•I - (Pp) (2n + k+ 1) li.,,k•l

(137)

Finally, Eqs. (132), (134) and (136) may be used to arrive at an exact representation given by oo

Hl/>(p,z) =

8uo L

2n

L

An1cep2p a)'hl

"12n•k (p,z,h)

(138)

n=Ok=O

The integrals I2n+1c,I2n+k+l and I20 +1c+2 appearing in Eq. (137) may be efficiently computed using the recurrence relation presented in Eq. (79).

4.2. Numerical Formulations for the Uniform Current Cylindrical Antenna Electromagnetic Fields Exact expressions for the uniform current cylindrical antenna vector potential and associated electric fields were derived in the previous section. These expressions are in a computationally efficient form and may be used to achieve any desired degree of accuracy. In this section, several formulations of the uniform current vector potential and electric field integrals are developed and discussed which are suitable for numerical evaluation. Included in this discussion will be a procedure for evaluating certain important derivatives of the cylindrical wire kernel required for electric field calculations. These formulations are also useful for the efficient as well as accurate computational modeling of cylindrical wire antennas. The expression for the exact cylindrical wire kernel possesses a singularity which must be properly treated in order to evaluate the cylindrical antenna integral equation. Common procedure is to extract the singular part of the kernel which results in a slowly varying function that is amenable to efficient numerical integration. Various techniques for evaluating the integral of the extracted singularity have been reported. Schelkunoff (16] transformed the singular part of the kernel into an elliptic integral of the first kind and showed that the asymptotic behavior in the immediate vicinity of the singularity is

706

an integrable logarithmic term. Pearson [17] pointed out that by expanding the elliptic integral in a series form, the logarithmic term which gives rise to this asymptotic behavior is extracted. However, the resultant infinite series is complicated by the presence of additional integrals which must be evaluated numerically. Wilton and Butler [18] also avoided the singular nature of the kernel by directly replacing the singularity with the integrable asymptotic logarithm term and then extracting it from the kernel. The disadvantage of this approach is that the remaining integrand, although very smooth, cannot be evaluated at the singularity because the logarithm function as well as the elliptic integral are unbounded at this point. Also, evaluation at all other points requires double numerical integration. In another reference, Butler [13] presents a different form of the extracted integrable singularity and expands it in a highly convergent power series which is valid in the vicinity of the singularity for thin wires with piecewise constant current. For thin wires in which the radius is much less than the wire length and wavelength, the reduced kernel approximation is widely used [5]. This form is independent of azimuthal variation and hence requires evaluation of only a single integral. When the current is uniform, the integrable singularity can be evaluated analytically. Some computationally efficient and accurate alternatives for computing a common integral of the extracted singularity associated with the vector potential of a uniform current cylindrical wire antenna have been presented in [ 19] and will be summarized here. An intermediate approximation, which is valid for thin wires with piecewise constant current, is shown to be more accurate than the classical thin wire approximation while maintaining computational simplicity. For thicker wires, it is shown that the term containing the extracted singularity can be cast into a single integral form which is amenable to numerical integration. Also, a highly convergent exact series solution to the integral of the extracted singularity is presented which is valid for all cases except in the vicinity of the singularity. The results given here can be directly implemented in a moment method solution in which the unknown current is represented by pulses or by more complicated basis functions which contain a constant term, such as trigonometric or polynomial [20,21]. The vector potential and kernel associated with the cylindrical wire (segment) illustrated in Fig. 1 have been defined in Eqs. (5) and (6) respectively. The integrand in Eq. (6) contains a singularity which may be extracted by expressing the cylindrical wire kernel in the form [ 16] K(z-z')

=

217' 1 d,1,.' '+' __ 217' 0 R'(z-z',')

f __

e -i~R'(z-z',(/)')

+

R

1(z-z

1

_

1

,')

d'

(139)

where R 1(z-z

1 , 4

(166)

A useful approximation may be obtained from Eq. (166) by retaining the logarithmic term and the first two terms in the series expansion. This leads to

/(a,O,h)

= 2 en(!) + 4

[:f- [:f

(167)

18

An exact expression for the integral I may be found which is valid for p ¢ a or when p = a and Iz I > h. The first step in the derivation of this exact solution is to make use of the fact that the singular part of the cylindrical wire kernel may be expressed in the form [ 19] 7r

1 1r

£

d ct,1 R 1(z-z

1,,f,')

- _!__l_ k F (~ , k) , 0 7r [pi 2




2

is a complete elliptic integral of the first kind and k

2fpi ........... __

= ___

_

✓cz-z')2 + (p+a)2

An infinite series representation for F (; , integrand in a binomial series CX)

(1 - k 2 sin2 q>f½ =

L

k)may be obtained

(-½)(-l)"k2nsin2n q>

n=O n

and then using termwise integration together with the fact that

(170)

by first expanding the

(171)

712 Tr

(172)

The resulting series expansion is [24]

F (;, k)

I )2 -½ k2n 2 n=O n

L 00

= 1T

(173)

(

which converges provided that O < k < 1. Substituting Eq. (173) into Eq. (168) and integrating term by term with respect to results in an expression for I which is given by

r

00

I ( p, z, h)

=

I

2 C2

L (-½)f

1

2/pi

k 2n...1 d C

, p

~

(174)

a

C1

n=O n

where k

2/pi

Jc2 + (p +a)2 =

We next introduce the change of variables x series representation

/(p,z,h)

=

en[x 2 +u 2 ] X1 + U1

+

(175)

= - ,.....,,....----'---'_ -_-_-_--=-

tl(p+a)

in order to arrive at the exact

E(-½) 1Pa)2n 2

n=l

n

(

2

p

F.{p,z,h),

+

p,.

a

(176)

a

where

(177)

U =

✓1

+ X2

(178)

713 (179)

(180)

A recurrence relation exists which provides a computationally efficient method for calculating the integrals F n defined in Eq. (177). The form of this recurrence relation was found to be

where (182)

(183)

112= ✓1

Xi

+

(184)

An approximation of I may be obtained by retaining the logarithmic term and the first two terms in the series expansion Eq. (176). The resulting expression is

(185)

When p

= a,

Eq. (176) reduces to (186)

where

k= ---Jc2

2a +

(2a)2

(187)

(188)

714 (189)

The corresponding expression for Eq. (185) when p

/(a,z,h)

= ln

[ X2 x1

+ U2] + +

u1

.!!. [ X2 32 u2

-

= a is given

~i u1

+

_1_[ x~ 64 u2

by

x~l u

(190)

1

which is primarily useful when ~/a > 4 and z/ ~ > 1. At the point z=0, four methods were used to evaluate the uniform current vector potential integral of the isolated singularity associated with the cylindrical wire kernel. Eq. (156) was computed using a three point Gaussian quadrature numerical integration technique, while Eqs. (164), (165) and (167) give the intermediate approximation, the thin wire approximation and the three term approximation of the power series expansion derived by Butler [13], respectively. Plots of the relative percent error for the various methods versus the segment length-to-radius ratio, ~/a, are shown in Fig. 2. As a basis of comparison, Eq. (156) was numerically integrated to a sufficiently high degree of accuracy. Clearly, the intermediate approximation and the three term Butler series have lower percent errors than the thin wire approximation across the entire range of !:../a. This becomes significant as ~/a approaches 4 (thicker wires) where the error associated with the thin wire approximation exceeds 1%. The three term Butler series proves to be extremely accurate and has the lowest error of the four methods for very thin wires (~/a .z:, 30). However, the three point Gaussian quadrature and the intermediate approximation also give acceptable errors in this range. For thicker wires, the three point Gaussian quadrature is superior and, because no assumptions were made in modifying the integral to the form given in Eq. (156), ~/a can be extended below the ratio of 4 and still achieve very accurate results. When z is not in the immediate vicinity of the singularity, the Butler series expansion is no longer valid, but can be replaced by the three term approximation of the exact series representation defined in Eq. (190). Also, a three point Gaussian quadrature numerical integration of Eq. (155) is valid as well as the intermediate and thin wire approximations of Eqs. (162) and (163), respectively. Contour plots of the relative percent error as a function of ~/a and z/ ~ for the various methods of computing the integral are shown in Figs. 3 through 6. For all cases, the percent error decreases as z/ ~ or ~/a increases. The contour plot for the thin wire approximation (Fig. 3) depicts the highest errors across the entire range and exceeds 1% when z/ ~ is less than 2 and ~/a approaches 4. As shown in Fig. 4, the intermediate approximation is much more accurate and the error remains below 1% for most of the smaller ~/a and z/ ~ values. The contour plot of the three term approximation of the exact series representation is shown in Fig. 5. The percent error decreases rapidly as z/~ or ~/a increases and the error never exceeds 1% when ~/a > 4. Also, note that the error associated with the three term approximation decreases more rapidly for thinner wires and larger z/ ~ than the errors associated with the other approximations. For the three point Gaussian

715

L

0

L L

w -+-'

C

(1.)

uL (1.)

Q_

1 0.0 ---.,..---:-.,.......,..--,--,-..,...,..,.----,,---------. -.-.-.-. -,_ 1 .0 •'• 1 .0e-1 1.0e-2 1 .0e-3 1 .0e-4 1 .0e-5 1 .0e-6 1 .0e-7 1 .0e-8 -JPointGQ 1.0e-9 ~ Thin Wire 1.0e-10 --+Intermediate -aBuller (J terms) 1 .0e-11 . . . . . . . .. . . . ..... . 1.0e-12 1 .0e -1 3 ,___..._.........., .............................. ____ ___,______.._ ................... ...___...__.........__.._-......-.... 1 1000 10 100 •

•

•

•

•

...

•

•

•••

-

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

■

••••

·•·

.........

.

~/a Fig. 2. Relative percent error versus /l/a associated with various techniques for evaluating the integral in Eq. (149) when z=O [19).

10 . .. ·.·. ...... . . ·.·

9 8 7

o (224)

=

in [

R1- ( 1] R3 - (3

,

( 1

s

o

F0 -- h

(225)

(226)

J:

!

An alternative representation for the initial three integrals J ~, and J may be found in which the sharply peaked nature of the integrands is avoided by extracting the "singularity" in each case. This yields the results 1

JI 0

=

F (/3h)f +

-I

2

e -iJ3s -1 (/3s)

1dt

(227)

728

f1

1 1 ( /3h) e -i/3s + i/3s - 1 JI = /3 F_2 - iF_1 + 2 -1 (/3s)2 dt

5_+_(/3 2/_2_+_i/3 _e_-i_l3 ____ s)_ ____ s_-_1dt

(228)

(229)

(/3S)3 where (230)

The integrals contained in Eqs. (227), (228) and (229) may be efficiently accurately evaluated using numerical integration schemes, such as Gaussian since their integrands are slowly varying. We next introduce a computationally efficient procedure for evaluating the form found in Eq. (217). The variable change w = /3R may be used convert Eq. (217) into a more convenient form given by

f

as well as quadrature, integrals of in order to

W3

J

2 m

= --

1

(/3h)

e -iw

--

Wm

dw

(231)

WI

where (232)

(233)

If we define u = e-iw

dv

= -

1

Wm

dw

(234)

(235)

then integration by parts may be used to obtain a recurrence relation from Eq. (231).

729 This recurrence relation was found to be (236)

The values of the integrals J; and J determined from

f

required to initiate this recursion may be

[ -iPR:J

J02 _---e I (Ph)

-iPR1)

-e

(237)

(238)

where 00

E 1 (ix) =

f ~t dt = -Ci(x) -it

+

X

iSi(x) - i TT

2

(239)

is the exponential integral and X

Si(x)

=

J sin t dt 0

(240)

t

X

Ci(x) = y +

tnx

+

£(costl) dt

(241)

represent the sine and cosine integrals respectively. The Euler-Mascheroni constant 'Y which appears in Eq. (241) is defined by [24]

y =

fim N-.

00

(£

.!. - tnN)

= 0.57721. ..

(242)

n=l n

Finally, the recurrence relation given in Eq. (236) together with Eqs. (237) and (238) may be used to show that

730

The exact expression for the triangular current vector potential, which is based on the expansion for L 1 given in Eq. (213) may be used to find corresponding exact expressions for the electric and magnetic fields. For an assumed triangular current distribution we have (244)

I' (z ')

I 11(z 1)

= -

I sgn (z ') , h

Iz I I
0 { -l,z 1 (p,z)

=

f\

0

[- 1

f3

-

a

L 1 (p,z, -h) - -1 - a L 1 (p,z,h) ap f3 ap

]

(260)

where (261)

An exact expression for the integral h

_!__E._L 1 (p,z,h)

f3 ap

=

f

_!__E._ (1-z 1/h) K(z- z 1) dz 1 f3 ap 0

(262)

733

may be derived by differentiating the expansion for L 1(p,z,h) given in Eq. (213) term by term with respect top. This leads to the following exact representation of Eq. (262) 1 a a L 1 (p,z,h) fJ

ap

~

= -

2n

L L

n=O k=O

Ank(/32P a)m X2n+k(p,z,h)

(263)

where X2n+k(p,z,h) = (1-z/h) w1i+k(p,z,h) + Win+k(p,z,h)

and use has been made of the identities (266)

(267)

As seen in the case of W~n+k,a similar situation exists for W~n+k in which the integrals J2~+k, Jin+k+1 and Jin+k+ 2 may be efficiently computed using the recurrence relation of Eq. (236). A "thin-wire" approximation of the vector potential Eq. (208) may be found by retaining only the zero order term in Eq. (213). This leads to Az(p,z)

=

µ~ {O-z/h) Jd(p,z,h) - (l+z/h) Jd(p,z,-h) 47T (268)

The corresponding "thin-wire" expressions for the electric and magnetic field components are EP(p,z)

=

Eto (/3p) [ {J/(p,z,h) + J/(p,z,-h)}

-i {Jj(p,z,h) + J21 (p,z,-h)}]

(269)

734 EzJ4 dipole such that the length-toradius ratios are 100, 10 and 4 in Figs. 16a, 16b and 16c, respectively. The near field behavior in the three cases is similar, each displaying the peaks in the proximity of the gap and the ends while exhibiting small values in the central region of the wire and at larger values of z/X.. Highest intensities are observed for the thin wire case, ll./a = 100. A vector plot of the total electric field of a quarter-wave dipole is shown in Fig. 17 at an instant of time, t = 0. With a length-to-radius ratio of ll./a = 4, the dipole can be classified as moderately thick. Selected arguments of z/X.and p/X. in the ranges 0 < z < A and 2a < p < A are used. The plot clearly shows the absence of EP along the z = 0 axis and the stronger intensities which exist in the very near region of the dipole. The shape of the electric field vectors indicate a transition from the local field, in which

740

0 ...,

w

---..,__ N

w

1 o- 2

L......L.._.__............,L...... ...............................

0.0

0.1

_.__

0.2

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

0.3

..J

0.4

z/A (a) 10 2 1 01

0 ...,

w

---..,__

10O 10- 1

Q.

w

10-

2

1o- 3 1 0-

4

L........................... ....1.... .................... ..._,._......_

0.0

0.1

0.2

..................... '----'- .................... _.

0.3

0.4

z/A (b)

o......

:r: ---..,__

10°

~

:r:

1 0-

1

1 0-

2

L........................... ....1.... ................_..._,._......._

0.0

0.1

0.2

...=""'

.................... L......L.. ..........

0.3

0.4

z/A (c) Fig. 14.

Magnitude of the near field components versus z/),. for fixed values of p using the exact series solutions: (a) IE1 / E1o I, (b) IEp/E1o I and (c) IH41/Hio 1- The dipole has a length ~=>JS and radius a= l.25xlo· 2>,,.such that ~/a= 10.

741

.... ; ...............

0 ....,

................ ;.. . - .. - .. ~

w

"w

N

1 0-

......._. ........... ......__._.......,__ ........., __ ......___ ................... 2 ..........

0.0

0.1

0.2

0.3

0.4

0.3

0.4

Pl" (a)

10 1 1 OO 0 ....,

w

"w

10-

1

10-

2

10-

3

Q.

0.0

........

0.1

0.2

Pl" (b) 10 1

0

-1 0-

......._............ ......__. __ 1 ..........

0.0

........__

0.1

.........,__

................._.............. _._

0.3

0.4

(c) Fig. IS.

Magnitude of the near field componentsversus pl).. for fixed values of z using the exact series solutions: (a) IEzfE1o I, (b) IEp/E1o I and (c) IH./H.oI- The dipole has a length /l=)./8 and radius a= l.25xl0·2). such that/l/a= 10.

742

0 ....,

w

"w

N

1 0-

1

L--

.................... ___.____._~

0.0

......... __.___.____.__

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

0.2

0.1

0.3

z/A (a)

10 1

0 ....,

1 o0

w

"w

N

1 0-

1

1 0- 2

0.0

0.1

0.2

0.3

0.4

0.5

z/A (b) 1 o0

10-2 ................................................................................................................. ~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

z/A (c) Fig. 16.

1Ezf E10 I versus zl}.. for a quarter-wavelength dipole at fixed values of p. radius ratio is (a) 4/a= 100, {b) !:a/a= 10 and (c) !:a/a =4.

The length-to-

743 the lines begin and end on the antenna, to the radiation field, where the lines have become detached from the antenna and close upon themselves. Although not shown, the vector plots obtained for thinner dipoles have similar features. The general shape of the electric field lines shown in Fig. 17 are in agreement with results found in the literature for thin oscillating dipoles [15,35,36].

1 .1

1.0

0.9 0.8 0.7 0.6 0.5

,
h

(305)

and for

Iz I
h

(308)

The exponential integral representation of Eq. (308) is then (309)

Expressions for the electric field components due to a thin uniform current dipole may be easily obtained from Eqs. (114), (115), (8) and (290). The resulting field quantities are

751

EP(p,z}

=

Euo

-(kp}

E/p,z}

= Euo

-k(z-h)

ei-rr/2 {

e -ikr •

(kp} [l + ikr.J -(krJ3

[1 + ikr_]

e i-rr/2 {

e-ikr_} (kr_) 3

(310)

e -ikr • k(z+ h) [l + ikr.J -(krJ3

[1 + ikr_]

e-ikr_

(kr_)l

-

E 0 (p,z)} o

(311)

where (312)

and

Eu is defined in Eq. ( 116).

6.

Exact Expr~ions for the Vector Potential and Electromagnetic Fields of a Uniform Current Circular Loop Antenna

0

Exact expressions for the vector potential and corresponding electromagnetic field integrals have not been available for the uniform current circular loop antenna. In the past, various assumptions have been made in order to obtain closed form approximations to these integrals [27]. In this section, the derivation of exact expressions will be presented for the uniform current circular loop vector potential as well as electromagnetic field integrals. It will be demonstrated that the well-known results for small loops and for the far-field may be obtained as special limiting cases of the more general exact series expansions. The geometry for the circular loop antenna of radius a is defined in Fig. 18. The source point and field point are designated by the spherical coordinates (r' = a, 8' = 1) and (r, (J, ) 90°, l

,

n-10

and -1

'

n

=

1

(351) -

1 D2 lo ' n > 1 (n-1)2 n-

The familiar small-loop approximations for the fields [27] may be obtained directly from Eqs. (344)-(346) by retaining only the first term (n = 1) in each series and recognizing that R ~ r as a~ 0. Under these conditions, Eqs. (344)-(346) reduce to the well-known results Hr(r,9)

H9(r,8)

=-

E (r,9) "'

=

i/3 a 2 Iucos9 2r 2

[l l_le -i/Jr + -.

1/3r

l l.

(/3a)2 lusin9 [ 1 1 4r 1 + if3r - (/3r)2 e-ipr

=

(/3a)2 T/ Iu sin 9 [ 1 4r

(352)

+ -. 1 -

e -,pr

(353)

(354)

1/3r

A procedure for evaluating the vector potential integral associated with a uniform current circular loop antenna has been presented in this section. However, one important point which should be emphasized is the fact that the basic mathematical analysis techniques developed in this section may be readily applied to finding exact expressions for vector potential integrals resulting from other, more complex, current distributions as well. For instance, it is possible to obtain exact expressions for the vector potential integrals which result from a more general Fourier series representation of the loop current given by

758 00

I_,,=

L

(355)

10 cos (ncf,)

n=O Since the current on the loop is ¢-dependent in this case, it follows that the vector potential must necessarily have the form

(356)

where

(357)

27T A9(r,O,)= µacosO 41r

fO I_,,sin(cf,-cf,)

-i/:JR1

e

R

d cf,1

(358)

1

(359)

It can be shown, using the integration techniques presented in this section, that exact expressions may be obtained for Eqs. (357)-(359) with an assumed current distribution of the type defined in Eq. (355). The resulting series representations are

ArCr,0,cf,) =

oo m 2 ar sin0/2)m-l ~ n(/3 µ/3asinO ~ L.J L.J l sin(ncf,) _ _.:,__ __ ..:............ __ 0 4i m= 1 n= 1 [(m+n)/2]! [(m-n)/2]! m-n = 2k k = 0, 1, ...

(360)

759 =

00 m 2ar sin9/2)m-l µ/3acos9 l 0 sin(ncp) _ n(/3 ____;_ ____ _ [(m+n)/2]! [(m-n)/2]! 4i m=l n=l m-n = 2k

LL

(361)

k = 0, 1, ...

00

m

m(/32ar sin9/2)m-l A.,,(r,9,cp) = JJ13_a l 0 cos(ncl>) [(m+n)/2]! [(m-n)/2]! 41 m=l n=0 m-n = 2k k = 0, 1, ...

L L

7.

(362)

Summary

The main objective of this chapter has been to provide a summary of analytical as well as numerical methods for evaluating various fundamental electromagnetic field integrals associated with current-carrying wire antennas. An effort has been made to make this chapter as self-contained and comprehensive as possible with an emphasis on recently developed techniques, both analytical and numerical. The formulations included in this chapter have been selected because of their ability to be used for the accurate modeling of wire antennas while maintaining computational efficiency. The derivations of three different expansions for the cylindrical wire kernel integral have been outlined in Sec. 3. The properties of each of these expansions are compared and discussed. A set of recurrence relations for efficient evaluation of the cylindrical wire kernel have also been derived in Sec. 3. An exact solution of the vector potential and corresponding electromagnetic field integrals associated with a uniform current cylindrical dipole antenna has been presented in Sec. 4.1. It has been demonstrated that the uniform current vector potential may be expressed in terms of a series involving a generalized exponential integral and higherorder associated integrals. A three term numerically stable forward recurrence relation has been found which may be used for the efficient computation of the higher-order integrals. Various techniques have been discussed for the evaluation of the initial three terms (integrals) required for the recurrence relation. These techniques include the use of exact representations, such as Maclaurin series expansions, and numerical integration procedures, such as Gaussian quadrature. In addition to the exact solutions presented in Sec. 4.1, Sec. 4.2 discusses several formulations of the uniform current vector potential and electromagnetic field integrals which may be efficiently evaluated numerically. These formulations are based on the application of singularity extraction techniques. Exact expressions for the vector potential and electromagnetic field integrals· for a dipole with an assumed triangular current distribution have been derived in Sec. 4.3. These exact formulations were used to investigate near field behavior of traditional thin wire as well as moderately thick wire dipoles. This theory is extended in Sec. 4.4 to

760

include mathematically exact solutions for the electric fields of a cylindrical wire dipole which has a sinusoidal current distribution and arbitrary radius. Existing methods for evaluating the well-known generalized exponential integral associated with the uniform current thin dipole vector potential are summarized in Sec. 5. Particular attention is devoted to a recently found exact solution to this integral. This exact series representation of the generalized exponential integral converges rapidly in the near field region of the antenna and, therefore, provides an alternative to numerical integration. An asymptotic small-argument expansion of the generalized exponential integral was presented which is convenient for numerically modeling thin straight-wire antennas. These exact expressions of the generalized exponential integral have been shown to be useful in the computation of impedance matrix elements in several method of moments formulations. Finally, in Sec. 6, analytical techniques originally developed for evaluating the cylindrical wire kernel integral have been successfully applied to solving the vector potential integral associated with a uniform current circular loop antenna. Two equivalent forms of exact series expansions are derived for the vector potential integral. It is shown that the familiar small-loop approximations as well as the classical far-field expression may be obtained as limiting cases of the more general exact series solutions of the vector potential integral. The analysis for the uniform current loop was then generalized to loops having an arbitrary current distribution represented by a Fourier . . cosme senes.

Acknowledgments I would like to express my deepest appreciation to my wife, Dr. Pingjuan L. Werner, for her technical assistance, encouragement and the many sacrifices she has made in support of my research. The continued guidance and inspiration provided by my mentor Professor Anthony J. Ferraro of The Pennsylvania State University is gratefully acknowledged. It is a pleasure to acknowledge the technical contributions, programming assistance and graphical illustrations provided by my graduate students Julie A. Huffman and Scott E. Metker. I am also grateful to Ors. James K. Breakall and Ray J. Lunnen of The Pennsylvania State University, and to Ors. Richard W. Adler and Gus K. Lott of the Naval Postgraduate School for their support of this work. I would like to thank George H. Hagn and Professor James R. Wait for their helpful discussions relating to this work. Finally, thanks are especially due to Ms. Shelby L. Cosio for her word processing skills.

References [I] [2] [3]

J. C. Maxwell, Proc. Royal Soc. London 13 (1864) 531. H. Hertz, Wied. Ann. 31 (1887) 421. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design (John Wiley & Sons, New York, 1981).

761

[4] [5] [6] [7]

[8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27]

R. F. Harrington, Time-HarmonicElectromagneticFields (McGraw-Hill, New York, 1961). R. W. P. King, Proc. IEEE 55 (Jan. 1967) 2. D. H. Werner, IEEE Trans. Antennas Propagat. 41 (Aug. 1993) 1009. A. J. Poggio and R. W. Adams, Approximations/or terms related to the kernel in thin-wireintegralequa.tions(Lawrence Livermore Laboratory, Livermore, CA, Dec. 1975), UCRL-51985. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1965). G. J. Burke and A. J. Poggio, NumericalElectromagneticsCode (NEC)- Method of Moments (Naval Ocean Systems Center, San Diego, CA, Jan. 1981), NOSC Technical Document 116. W. Wang, IEEE Trans. Antennas Propagat. 39 (Apr. 1991) 434. G. N. Watson, A Treatiseon the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, U .K., 1962), p. 140. M. Abramowitz and I. E. Stegun, Handbook of MathematicalFunctions (Nat. Bur. Stand., Washington, DC, 1972), p. 439, Eq. 10.1.17. C. M. Butler, IEEE Trans. Antennas Propagat. AP-23 (Mar. 1975) 293. R. F. Harrington, Proc. IEEE SS (Feb. 1967) 136. S. A. Schelkunoff and H. T. Friis, Antennas Theoryand Practice (John Wiley & Sons, New York, 1952), p. 369. S. A. Schelkunoff, AdvancedAntenna Theory(Wiley, New York, 1952), pp. 140142. L. W. Pearson, IEEE Trans. Antennas Propagat. AP-23 (Mar. 1975) 256. D.R. Wilton and C. M. Butler, in MomentMethods in Antennas and Scattering, ed. R. C. Hansen (Artech, Boston, 1990), pp. 58-77. D. H. Werner, J. A. Huffman and P. L. Werner, Accepted for publication in IEEE Trans. Antennas Propagat. (Nov. 1994). E. K. Miller and F. J. Deadrick, in Numerical and Asymptotic Techniques in Electromagnetics,ed. R. Mittra (Springer-Verlag, New York, 1975), Ch. 4. Y. S. Yeh and K. K. Mei, IEEE Trans.Antennas Propagat. AP-15 (Sept. 1967) 634. R. L. Burden and J. D. Faires, NumericalAnalysis (Prindle, Weber & Schmidt, Boston, 1985). A. H. Stroud and D. Secrist, Gaussian Quadrature Fonnulas (Prentice-Hall, Englewood Cliffs, New Jersey, 1966). L. C. Andrews, Special Functionsfor Engineers and Applied Mathematicians (MacMillan, New York, 1985), p. 110. W. H. Press, S. A. Teukolsky, W. T. Vettering and B. P. Flannery, Numerical Recipes in Fonran, TheAn of ScientificComputing(Cambridge University Press, Second Edition, New York, 1992). E. C. Jordan and K. G. Balmain, ElectromagneticWavesand RadiatingSystems (Englewood Cliffs, New Jersey, 1968). C. A. Balanis, Antenna Theory (Harper & Row, New York, 1982).

762 [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

A. T. Adams, B. J. Strait, D. E. Warren, D-C Kuo and T. E. Baldwin, Jr., IEEE Trans. Antennas Propagat. 22 (Sept. 1973) 602. A. T. Adams, T. E. Baldwin, Jr. and D. E. Warren, IEEE Trans. Electromag. Compat. 20 (Feb. 1978) 259. R. W. P. King and T. T. Wu, Radio Science 1 (Mar. 1966) 353. L. D. Scott, D. V. Giri and S. A. Long, IEEE Trans. Antennas Propagat. 21 (Mar. 1973) 213. C. W. Harrison, Jr., C. D. Taylor, E. A. Aronson and M. L. Houston, IEEE Trans. Electromag. Compal. 12 (Nov. 1970) 164. D. H. Werner, Proceedings of the IEEE Antennas and Propagation Society International Symposium 3 (19-24 June 1994) 1590. S. Singh and R. Singh, IEEE Trans. Microwave Theory Tech. 40 (Jan. 1992) 168. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison Wesley Publishing Co., London, 1962). F. M. Landstorfer and R. R. Sacher, Optimisation of Wire Antennas (Research Studies Press LTD., Letchworth, England, 1985). R. W. P. King, The Theory of Linear Antennas (Harvard Univ. Press, Cambridge, MA, 1956). R. W. P. King, R. B. Mack, and S. S. Sandler, Arrays of Cylindrical Dipoles (Cambridge at the University Press, London, 1968). R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968). R. Mittra, ed., Numerical and Asymptotic Techniques in Electromagnetics (Springer-Verlag, New York, 1975). R. Mittra, ed., Computer Techniques for Electromagnetics (Hemisphere Publishing Corporation, New York, 1987). R. C. Hanson, ed., Moment Methods in Antennas and Scattering (Artech House, Boston, 1990). E. K. Miller, L. Medgyesi-Mitschang and E. H. Newman, Computational Electromagnetics (IEEE Press, New York, 1992). S. Weinbaum, J. Appl. Phys. 15 (Dec. 1944) 840. J. R. Wait, Canad. J. Phys. 30 (1952) 512. Staff of the Computation Laboratory, Tables of Generalized Sine and Cosine Integral Functions (Harrard Univ. Press, Cambridge, MA, 1949). R. W. P. King and C. H. Harrison, Jr., Antennas and Waves: A Modem Approach (The MIT Press, Cambridge, MA, 1969), Appendix 1. D. H. Preis, IEEE Trans. Antennas Propagat. AP-24 (Mar. 1976) 223. P. L. Overfelt, IEEE Trans. Aruennas Propagat. AP-35 (Apr. 1987) 442. D. H. Werner, IEEE Trans. Antennas Propagat. 42 (Aug. 1994) 1201. D. H. Werner, P. L. Werner, J. A. Huffman, A. J. Ferraro and J. K. Breakall ' IEEE Trans. Antennas Propagat. 41 (Dec. 1993) 1716.

TRANSMISSION AND RECEPTION OF POWER BY ANTENNAS DALE M. GRIMES

Departmentof ElectricalEngineering PennsylvaniaState University University Park PA 16802-2705, USA CRAIG A. GRIMES

D'!f}artmentof ElectricalEngineering University of Kentucky Lexington KY 40506-0046, USA ABSTRACT This review paper provides both a qualitative and a quantitative description of antenna operation, including complete analytical and partial numerical field solutions for the transmission and reception of power. Emphasis is on the mathematical physics of antennas and how the physics determines antenna design. Only biconical antennas are both useful for technological applications and support an exact analytical electromagnetic solution, in the sense that no electromagnetic approximations are made, although approximations are necessary for numerical evaluations. A basic theorem is that all electromagnetic fields may be described by sums over multi poles. Using this theorem, analytical solutions of balanced, biconical antennas of arbitrary arm lengths and cone angles are presented in a way designed to emphasize both similarities and differences between transmitting and receiving antennas. Solutions appear as weighted sums over special functions: associated Legendre functions of the zenith angle, spherical Bessel functions of the normalized radius, and trigonometric functions of the azimuth angle. The antenna terminals are confined within a vanishingly small sphere that is centered at the apexes of the cones. The radiation problem is formulated and the solution is obtained; the solution permits calculation of fields at all radii, all current modes on the arms and caps, and the energy and momentum radiated by a transmitting antenna and absorbed by a receiving antenna from an incoming plane wave. Among other things, it is shown that the receiving current modes with m > 0 are necessary for electromagnetic momentum to be conserved during reception. Detailed calculations of comparative solutions show that the terminal receiving and transmitting admittances are identical if identical voltages occur at the terminals, as predicted by the reciprocity theorem. The transmitting antenna analysis starts with a terminal input voltage and determines the input impedance and the fields. The receiving antenna analysis starts with an incoming plane wave, traveling in the positive y direction and polarized in the z direction. It is incident upon a biconical antenna whose antenna axis is the z axis. Solutions are expressed as functions of arm length a and cone angle 1/f for both transmission and reception.

TRANSMISSION AND RECEPTION OF POWER BY ANTENNAS DALE M. GRIMES

Departmentof ElectricalEngineering PennsylvaniaState University University Park PA 16802-2705, USA CRAIG A. GRIMES

D'!f}artmentof ElectricalEngineering University of Kentucky Lexington KY 40506-0046, USA ABSTRACT This review paper provides both a qualitative and a quantitative description of antenna operation, including complete analytical and partial numerical field solutions for the transmission and reception of power. Emphasis is on the mathematical physics of antennas and how the physics determines antenna design. Only biconical antennas are both useful for technological applications and support an exact analytical electromagnetic solution, in the sense that no electromagnetic approximations are made, although approximations are necessary for numerical evaluations. A basic theorem is that all electromagnetic fields may be described by sums over multi poles. Using this theorem, analytical solutions of balanced, biconical antennas of arbitrary arm lengths and cone angles are presented in a way designed to emphasize both similarities and differences between transmitting and receiving antennas. Solutions appear as weighted sums over special functions: associated Legendre functions of the zenith angle, spherical Bessel functions of the normalized radius, and trigonometric functions of the azimuth angle. The antenna terminals are confined within a vanishingly small sphere that is centered at the apexes of the cones. The radiation problem is formulated and the solution is obtained; the solution permits calculation of fields at all radii, all current modes on the arms and caps, and the energy and momentum radiated by a transmitting antenna and absorbed by a receiving antenna from an incoming plane wave. Among other things, it is shown that the receiving current modes with m > 0 are necessary for electromagnetic momentum to be conserved during reception. Detailed calculations of comparative solutions show that the terminal receiving and transmitting admittances are identical if identical voltages occur at the terminals, as predicted by the reciprocity theorem. The transmitting antenna analysis starts with a terminal input voltage and determines the input impedance and the fields. The receiving antenna analysis starts with an incoming plane wave, traveling in the positive y direction and polarized in the z direction. It is incident upon a biconical antenna whose antenna axis is the z axis. Solutions are expressed as functions of arm length a and cone angle 1/f for both transmission and reception.

764

1. Introduction 1.1 /ntroductory Comments Modal theory provides analytical antenna solutions that are complete, exact, and unique. Solutions are expressed as an infinite series of terms in a modal expansion. All analyses are for an arbitrary but single frequency field with radian frequency m and wavenumber k. The solution forms contain full information about the fields, and hence surface currents, real and reactive powers, exchanged energy, and the kinematic property of linear momentum [1,2,3,4]. Solution forms permit assigning physical interpretations to the coefficients of modal terms that, in turn, may provide physical insights to the understanding, operation, and continued development of antennas. The solutions are left in the form of sums over an infinite series of weighted, spherical special functions, although selected samples have been numerically analyzed and are described here. Analytical solutions of these equations are complete. However, since numerical solutions require series truncation, numerical results include both truncation and round off errors. Numerical evaluation of the series was first done for very small and very large cone angle transmitting antennas [l], and then for certain selected angles [2]. Summarized information is available in review form [3]. Computer evaluation has been extended to transmitting antennas with a wide range of cone angles, from large to small, and made available [5,6,7]. In this work we use eico,time dependence. The physics and electrical engineering conventions are to use, respectively, e-iox and ei 0 modes.

I 3 Solution regions defined We divide space into three regions, see Fig. 1. The source or sink region is a uniform, spherical volume centered at the origin and of radius b, which, in turn, is a small fraction of a wavelength; kb « I. It is centered at the origin of coordinates and satisfies:

Source or Sink Region

r < b ; 0 ~ 8 ~ 1t ; 0 ~

~

21t

(1)

We define interior and exterior regions by constructing a virtual, spherical shell of radius a centered at the origin. Within the ranges 'If< 8 < 1t - 1/f the shell is conceptual; within the ranges O ~ 8 < 1/f and

7t -

v, < 8 ~ 1t it

coincides with the cone caps. The interior

766 region is the concentric volume between radii b and a that excludes the antenna arms; the exterior region is the rest of space:

InteriorRegion ExteriorRegion

b < r< a;

a < r;

lfl< 8< 1C-1/f; 0~ 8

~ 1C ;

0~

0~

q,~ 2x (2)

q,~ 2x

(3)

The major difference between transmission and reception is the energy source. For transmission, the product kb is so small that circuit techniques may be used to describe power transfer from the source to the interior region [1]. Since neither the source nor the z oriented antenna has azimuth angle dependence, the fields are independent of q,. We seek to find the transmitted power, the surface currents, and the radiated fields. For reception, the antenna remains along the z axis, but a propagation direction is needed for an incident, z polarized plane wave. We chose a unit magnitude electric field intensity wave traveling in the positive y direction. We seek to find the absorbed power, the current densities on the antenna surfaces, the scattered fields, and the momentum transferred from the field to the antenna. For both transmission and reception, solutions are expressed as functions of the configurative parameters a and 1/f and, for receiving antennas, the terminal impedance.

zaxis

Figure 1. Sketch of the modeled biconical antenna. Source region has radius b, much less than either a or A.. Cones extend from radius b to radius a, with half angle y,.

767

2. Special functions [ 12] A basic modal expansion theorem, applicable to spherical coordinates, is that any field function may be written as sums of weighted products of special functions, where each special function is a function of only one coordinate. For example the radial component of all possible electric fields may be expressed in the form:

where Rt(kr) are combinations of spherical Bessel and Neumann functions, 0'f(cos8) are combinations of associated Legendre functions of the first and second kind, and cllm(q,)are trigonometric functions. The task in any radiation problem is to obtain the correct weighting of functions so the sum gives the unique solution to the problem of interest.

2 .1. Azimuth functions The azimuth fields satisfy the one dimensional wave equation in rectangular coordinates. Solutions include the unknown degree m where:

cbm(q,)= Amcosmq,+ Bmsinmq,

(4)

and Am and Bm are unknown constants. The boundary condition that both internal and external regions occupy the full 21t radian sweep of q,is met if mis a positive integer.

2 .2. Radial functions The radial functions satisfy the spherical Bessel equation. Bessel functions of order v and have the form,

Solutions are spherical

(5) where er= kr, Av and Bv are constants, and jy(cr) and j_v(cr) are spherical

Bessel

functions of order +v. If v = l, an integer, the two solutions of Eq. 5 are dependent. Since a second order differential equation requires two independent solutions, for integer orders we retain jt(cr) as one solution and introduce spherical Neumann functions, Yt(cr), as the second solution: Rt(cr)

= Atit(cr) + BtYt(cr)

(6)

Useful function combinations for integer orders are Hankel functions of the second kind:

(7) Related functions occur in a special form so often that it is convenient to give them special symbols:

768 (8)

jj

Equations 7 and 8 define

(cr), y; (cr), and h; (cr).

2.3 Zenith angle functions The zenith angle functions satisfy the associated Legendre equation. Solutions are the associated Legendre functions of order v and degree m: P~(cos9)

(9)

and ~(-cos9)

where m and v of the Legendre functions, respectively, are the degree and order of the trigonometric and Bessel functions. Since the solutions require either even or odd functions of cos8, and since functions P~(+cos8) are neither, we construct functions of definite parity: 1

~ (cos8) = 2 {P~ (cos8) + ~ (-cos9)}

1 M~ (cos8) = 2 {P~ (cos8) - ~ (-cos8) },

(11)

and write the zenith angular dependent functions as E>~(cos8) = AvmMm(cos8) + BvmL~(cos8)

(12)

With non-integer values of v, Pv(+l) = 1, and Pv(-1) are logarithmically singular. For integer values, Pt(+l) = 1 and Pt(-1) = (-1) 1, and the two Legendre functions are dependent. The required second solutions are Legendre functions of the second kind, Q1(cos9). The functions Q1(cos8) are singular on both ±z axes; the only order of Qt of interest here is l = 0, where 1 { 1+cos8} Qo(cos8) = 2 ln -= In cot(8/2) 1-cos8 dQo(cos8) d8

1 sin8

(13)

(14)

3. Transmitting antennas [ 1,2,3] Two important parameters for antenna operation are the voltage difference between equal radii points on the two arms, and the ann currents as a function of radius. Since the source has rotational symmetry, the fields depend only on the zenith angle and the radius

769

and there is no azimuth angle dependence. Voltage difference, V(r), may be expressed as the integral of the electric field intensity between the two positions: 7t-Y,

(15) fEa d8 "' The current at each radius r is related to the integral of the circumscribing magnetic field as: V(r) = ~

21t

l(r)

= cr sinv,J k H, dct,IB=y,

(16)

3 .1 Source region To obtain the proper solution within the source or sink region we note that circuit analysis is useful whenever element sizes and the distances between them are much less than a wavelength. Since it has already been stipulated that kb « 1, it follows that the source may be treated as a lumped element The source voltage and current follow, and are given by the equations [ 1]: V(0) = Lim V(b)

(17)

1(0) = Lim l(b)

(18)

b~

b~O

H, H,

Since V(0) and 1(0) are neither zero nor infinite in the limit as b becomes vanishingly small, it follows from Eqs. 15 to 18 that both Ea and near the source are proportional to 1/b. All terms in the expansions for Ea and that vary as r to a power less than minus one are required to have zero coefficients, otherwise the resulting voltage and current would be singular at the source. Since the functional part of terms that vary as r to a power more than -1 vanish, they contribute nothing to the source voltage or current. Therefore, near the source, only functions that are proportional to 1/rare significant Spherical Bessel functions of order v satisfies the asymptotic condition that (19)

and both functions are present in the field expressions. Therefore only the value v = 0 meets the power conditions of Eqs. 15 to 18 and there are no negative order Bessel functions in the interior region; all negative order expansion terms have zero coefficients and play no role in the field expression. The asymptotic limit of the integer order functions for small values of radius are: Lim·

-

l (J

O'=>O Jt( cr) - (2i+ 1)! !

The zero order radial and related functions are

~Yt(cr)

(U-1)!! 0 t+1

(20)

770 . • sincr .• coscr Jo(cr) = Yo(cr) =and yo(cr) = - Jo(cr) = -cr cr

(21)

32 Internal region A multimodal expansion for all possible fields is given elsewhere [13, Eqs. A5-A10]. From a center driven source and with the antenna of Fig. 1, the electric field satisfies the symmetry condition that E2 (x,y,z)

= E2 (x,y,-z)

(22)

Equation 22 is satisfied by odd parity Legendre functions, and not by even ones. Therefore only functions Mv(cos0) are present; functions Lv(cos0) have multiplying coefficients of zero. Exclusion of fields from the z axis by the antenna arms results in non-integer and zero orders of spherical Bessel and Legendre functions. Using suppressed time dependence and selecting the l = 0 mode for separate handling, the full solution talces the form [13, App.A]: 00

Er= "L.lv(v+l)rv jv(cr) --Mv(cos0) v>O cr

(23)

00

·• (cr) d.My + ---[CoJo(cr) dQo(cos0) Ea= "L.lfvJv v>O d0 d0

·•

• + Do Yo(cr)]

(24)

00

. (cr) dMv dQo(cos0) [COJO . (cr) + D o Yo(cr)] T\H cp= -1. "r L.I vJv - z. --v>O d0 d0

(25)

where T\is the impedance of space and r v are the coefficients of a set of TM fields that are present in the interior region and vanish near the source. Co and Do are the coefficients of TEM fields; since the radial component is multiplied by the order number, in this case it is zero. The angular field components are proportional to the 0 derivative and the derivative of Po(cos0) is zero. The derivative of Qo(cos0) is non-zero, see Eq. 14. The characteristic admittance G of a biconical transmission line for the TEM mode is defined to be [1] 7t

G=----T\ln cot(llf/2)

(26)

It follows from Eqs. 15, 24 and 26 that the TEM electric field intensity of Eq. 24 may be written

771 Tl G V(r) E9=---

(27)

21tr sin8

It follows from Eqs. 16 and 25 that the TEM magnetic field intensity of Eq. 25 may be written

H =

4>

l(r) 21tr sin8

(28)

It is convenient for what lies ahead to express V(r) and l(r) in terms of their values at the outer antenna termination, r = a. In those terms, after some algebra, the expressions may be put in the form of transmission line equations [1,4] V(r) =

vgz){G cos[k(a-r)]+

i Y(a) sin[k(a-r)]}

l(r) = V(a){Y(a) cos[k(a-r)]+ i O sin[k(a-r)]}

(29) (30)

where Y(a) is the TEM admittance at the termination, r = a. The load admittance YL presented between the conical apexes of the antenna is known or can be measured. The ratios are, by definition, l(a) Y(a) = V(a)

and

1(0) YL = V(0)

(31)

In terms of known parameters, the TEM admittance versus position is Y(r)

G

Y L - iG tancr = G - iYLtancr

(32)

Combining the above shows that the field quantities of Eqs. 24 and 25 may be written as 00

~r .•( ) dMv +

Ea-_ LJ vJv v>O

(J

d8

Tl G V (r) 21tr sin8

(33)

00

t1H4> = -l

.~

. dMv LJrvJv(cr) +

v>O

d8

Tl l(r)

(34)

21tr sin8

3 .3 External region Initially it is most convenient to impose boundary conditions on both the radial and angular functions. In the limit as r => 00 the fields become proportional to e-i0 /cr. Since the asymptotic limit of Hankel functions at large radii is

772 ;l+le-io

ht(cr)=--

(35)

cr

it follows that the radial dependence is proportional to spherical Hankel functions for all radii. Since fields are regular on all exterior axes, integer orders of Legendre polynomials are used; all other Legendre functions are irregular. Including these results and imposing the antenna symmetry condition between upper and lower space, the resulting functions

are: 00

~

ht(cr)

Er= ~/.(/.+

l)Dto~l --

Pl (cos8)

(36)

cr

l=l 00

Ee= L,Dto~l l=l

hjdPl

(37)

d(J

(38)

where ~l are unknown field constants and Dto has been introduced for later convenience 2(21.+1) l.!6(l,2n+ 1) Dto = 2ll.(l+ 1) [(l.-1)/2] !2

(39)

n is any integer, and 6 represents the Kronecker delta function, defined by

f 1 for l 6(l,2n+ 1)

=

odd,

~

(40)

lofor I. even, Problem solution, which appears later in this chapter, requires evaluation of YL and each of the infinite sets of tenns ~l and r v-

4. Receiving antennas [4] 4.1 Incident plane wave The incident, z polarized plane wave travels in the +y direction with phase exp[i( wtky )] . Expanding the spatial phase factor in spherical Bessel functions of integer order and associated Legendre polynomials of order I. and degree m, it is shown elsewhere [4] that

e-iky = exp(-ia sin8 sin(/))= [

r I,

cosmf/)- i

le:2 me:0

rf

sinmf/J] [~

lo:1 mo:1

it(a)P"/(cos9)]

(41)

m

where the symbols e and o in the summation indicate only even and odd values, respectively, where C

_ 2m(U+ l)(i-m)!U(m)O(i+m,2n) tm - 2 1i(l+l)[(l+m)/2]![(l-m)/2]!

(42)

U (m) is the step function

roform
0, We next define the related parameter Dtm, see[4] D

_ 4(2i+ l)(i-m)!U(m)O(i+m,2n+ 1) lm - 2l.t(i+l)[(i-l+m)/2]![(i-1-m)/2]!

(44)

We note that Dto of Eq. 39 is the same as them = 0 portion of Eq. 44. The symmetry requirements and Eqs. 41 through 44 lead to the radial field components [4] Er= [

r l: r f 1

cosmf/)- i

lo:1 me.O

t1Hr = [

lo:1 mo:1

rr r l:

le:2 mo:l

cosmf/)- i

sinmf/)] [ i(i+ 1) Dtm ji(cr) cr

1

sinmf/)] [ i(i+ 1) Ctm jt(cr) cr

P'7(cos8)] P'7(cos9)]

(45)

(46)

le:2 me:O

Equation 22 still applies since the driving field is z directed and nothing in the antenna rotates the polarization. Examination of the equation forms shows that this occurs if (i+m) is odd for TM modes, Eq. 45, and even for TE modes, Eq. 46. The angular fields follow from the radial ones [ 14, 15].

4 2 Scattered field Since the antenna conductors lie along lines of constant coordinates and antenna symmetry matches that of the incoming wave, the formal description of the scattered wave has but two differences from that of the plane wave: (a) to be of the form of Eqs. 45 and 46 in the limit of large values of cr, the spherical Bessel functions j1(cr) are replaced by spherical Hankel functions of the second kind ht( cr), and (b) since different modes will be

scattered differently, infinite sets of unknown scattered field constants, CJ.tmand ~tm, are needed, respectively, for the TE and TM modes. In these terms, the radial components of the scattered fields are described by Eqs. 47 and 48: l-l

00

Er= [ I.

l-l

00

I. cosmq, - i I.

lo:l

me:O

11Hr= [ I.

I. cosmq,- i I.

lo:I mo:1

l

I. sinmq, ][i(l+l)

P't'(cos0)]

h (cr)

l-l

00

h (cr) cr

le:2 mo:l

l

00

I. sinmq,] [ l(l+ 1) ~tmDtm

CJ.tmCtm l

cr

le:2 me:O

P1(cos8)]

(47)

(48)

4 .3 Total field, external region

The sum of Eqs. 45 and 47 and of Eqs. 46 and 48 are the radial components of the total external fields. The angular field components follow directly, and are described by the following equations [4]: l-l

00

Ee= [

I, lo:I

l-l dPm cosm 0, we conclude that they are necessary to conserve momentum. They are, therefore, essential for power reception to occur and the m = 0 terms cannot by and of themselves be a complete solution to the receiver problem.

789

8. Conclusions This paper reviews the modal solutions of biconical transmitting and receiving antennas. The solutions are formulated as similarly as possible. Solutions are in the form of infinite sums over weighted products of spherical special functions. Solutions permit the calculation of directivity and input impedance, momentum and power flow to the antenna, all current modes on the arms and caps, and all electromagnetic fields including near and scattered ones. The transmitter is driven by a small, spherical source. The receiver is driven by a y directed, z polarized plane wave. It is affirmed that although the m > 0 modes contribute no useful load power, they must necessarily be present for momentum to be conserved during power absorption. A spiral reactance-resistance curve is shown for a 5° cone, as is radiated power versus arm length with a unit input voltage. Receiving surface currents associated with the three lowest interior modes and two lowest exterior modes are sketched. The lowest degree modes are m = 0; the next higher ones have m = 1 and describe a generated magnetic moment that acts, in accordance with Lenz's law, to decrease the magnetic flux that passes through the antenna anns.

9. References 1. S. Schelkunoff, Advanced Antenna Theory (John Wiley, 1952) pp.32-49. 2. C-T Tai, J. Appl. Phys. 20 (1949) 1076-1084. 3. J. R. Wait, in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, 1969) Chp.12. 4. D. M. Grimes, J. Math. Phys. 23 (1982) 897-914. 5. V. Bad.ii, Numerical Analysis of the Biconical Transmitting Antenna, Doctoral theses, Electrical and Computer Engineering Dept, Pennsylvania State University, August 1988. Available from University Microfilms, Ann Arbor MI, USA. 6. V. Badii, K. Tomiyama, D. M. Grimes, Appl. Comp. Electromagnetic Soc. J., S (1990) 62-93. 7. D. M. Grimes, V. Badii, K. Tomiyama, Radio Science, 26 (1991)101-109. 8. R. E. Collin, in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, 1969) Chp. 4. 9. R. W. P. King, in Antenna Theory, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, 1969) Chp. 9. 10. R. W. P. King, C. W. Harrison, Proc. IRE 32 (l 944) 18-34. 11. T. Morita, Proc. IRE 38 (1950) 898-904. 12. S. Schelkunoff, Applied Mathematics for Engineers and Scientists, 2nd ed., (Van Nostrand, 1965) 13. D. M. Grimes, C. A. Grimes, in Advanced electromagnetism: Foundations, Theory and Applications, T. W. Barrett, D. M. Grimes, eds. (World Scientific Press, 1995) 14. S. Ramo, J. R. Whinnery, T. VanDuzer, Fields and Waves in Communication Electronics, 2nd ed., (John Wiley, 1984) Chp.7. 15. W. K. H. Panofsky, M. Philips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962) Chp.13. 16. D. M. Grimes, in Essays on the Formal Aspects of Electromagnetic Theory, A. Lakhtakia~ ed., (World Scientific Press, 1993) p.310-356. 17. A. Einstein, in The World of the Atom, H. A. Boorse, L. Motz, eds. (Basic Books, 1966) p.901.

790

Appendix: A table of integrals [4] Where n is any integer, Tt-Y, B) . BdB- 2sinv,L~(cosljl)[ciP'2(cosljl)/dljl] Itv _ - JPm( t cosv/J\Lm( sm 1 v cos y, l(l+l)-v(v+l)

=

K tv

2 L ~(cosljl) l(l+l) - v(v+l)

[ (l-m+ l)P

t'~1(cosljl) - (l+ l)cos 1/fJ>1(cosljl)]

t:(l

u

+m,2 n )

6(l+m,2n)

7t-Y, m m m( D\Mm( D\ . BdB 2sin1f/P t(COSl/f)[dMv(cosljl)/dljl] t:(l l) = t cosv 1 v cosv 1sm = u +m,2 n+ .,, v(v+ 1) - l(l+ 1)

Jp

2P7(cos1/I) - -_____,;:~~v(v+l) - l(l+l)

(v-m+ l)Lv':1 (cos 1/f)6(l+m,2n+ 1)

Where the Kronecker delta function is extended to include non-integer values: D\Lm( D\ . dB- 2sin v,L~(cos9) d2r.;1(cos9) t:('l ) Ivv _- TtJ-L.,,m( >i, COSv1 v COSv1Slil 8 u /\,,V .,, 2v+ 1 dvd 11' 7t-Y, 2 . m D\ m D\ 1 dMy(cosv, t:('l ) JMm( D\Mm( D\. BdBS1Ill/fdMy(cosv K vv _ v COSv1 >i, COSv1Slil - u /\,,V .,, 2v+ 1 dv d 11'

1t-y, Jsin8d8{~ y, Tt-Y, Jsin8d8 y,

dPm dr.;1

m2PmLm + Y}=v(v+l)ltv d8 d8 sin 8

.1

dL m dL m m2L mLm {-v _>i. + . ; A.}= v(v+l)lvv d8 d8 sm 8

7t-Y, dPm dMm fsin8d8{-l_v+_ y, d8 d8

m2PmMm l v}=i(l+l)Ktv sin28