1

Abstract:

Physics, being the study of relationships between entities, must attempt to provideclarification of such relationships. It was in this vein that M. Faraday introduced thenotion of the field to help describe the physical phenomenon uncovered during hisexperimental researches into electricity and magnetism. Since this graceful entrance,the use of the field to express phenomenon of the world has proven insightful andvaluable in other circumstances. There are however difficulties that arise with fieldtheory in both the ontological and epistemological regimes. Considering this, thequestion arises, are there other models capable of accounting for the necessaryphenomenon? For example, Newton’s law of Gravitation stood precariously underthe wobbly auspices of a model known as “action at a distance.” With its strikinglysimilar form, Coulomb’s law of electrostatic repulsion/attraction suggested alikewise account of interaction between charged particles. Contributions from thelater part of the 19th century found such a model unfit for application to E&M forseveral reasons. However, the 20th century engine of theoretical physics has found arelated model to have a potent relevance in the interpretation of observedphenomenon and mathematical accounting. Essentially, the classical field theory,despite its appeal, might not be the best way to account for the phenomena ofelectrodynamics; other possibilities will be explored.

Classical Electrodynamics: With Fields or Without?

An exposition of the necessity of fields in classical electrodynamics

James Augustin Hedberg

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Extracts:

“…and two times two is four is no longer life, gentlemen, but the beginning of death.”F. Dostoevsky, Notes from Underground

“And the LORD said unto Satan, Hast thou considered my servant Job, that there is none like himin the earth, a perfect and an upright man, one that feareth God, and escheweth evil? and still heholdeth fast his integrity, although thou movedst me against him, to destroy him without cause.”

Job, Chapter 2, verse 3

“O Sacred, Wise, and Wisdom-giving Plant,Mother of Science, Now I feel thy PowerWithin me cleere, not onely to discerne

Things in thir Causes, but to trace the wayesOf highest Agents, deemd however wise.”

Milton, Paradise Lost, Book 8, lns. 679-683

The magic and the most powerful effect of women is, in philosophical language, action at adistance, actio in distans; but this requires first of all and above all—distance.

Nietzsche, The Gay Science, Book II, # 60

“She got up and went to the table to measure herself by it, and found that, as nearly as she couldguess, she was now about two feet high, and was going on shrinking rapidly: she soon found out

that the cause of this was the fan she was holding, and she dropped ithastily, just in time to save herself from shrinking away altogether.”

Lewis Carrol, Alice’s Adventures in Wonderland

Mad let us grant him, then: and now remainsThat we find out the cause of this effect,Or rather say, the cause of this defect,

For this effect defective comes by cause:Lord Polonius, Hamlet, Act II, Scene II

PRIMUS DOCTOR.Si mihi licentiam dat dominus praeses,

Et tanti docti doctores,Et assistantes illustres,Très savanti bacheliero,Quem estimo et honoro,

Domandabo causam et rationom quareOpium facit dormire.

BACHELIERUS.Mihi a docto doctore

Domandatur causam et rationem quareOpium facit dormire.

A quoi respondeo,Quia est in eo

Vertus dormitiva,Cujus eat natura

Sensus assoupire.

CHORUS.Bene, bene, bene, bene respondere.

Dignus, dignus est intrareIn nostro docto corpore.Bene, bene respondere.

Moliere, The Imaginary Invalid, ThirdInterlude

I. Introduction: History and CausalityAs suggested by Bunge, scientific research can be dichotomized into two regimes: basic

research, that which is concerned with source theories and elementary empirical constructions,and based research, that which occupies the time and energy of the experimentalist. These tworegimes fuel each other by posing problems suitable for solution by the other. For example,analytical formulations suggested the existence of 21 cm radiation from the hydrogen atom (basicresearch), which was then detected by astronomers, thus confirming the abundance of thatelement in the universe (based research). The present study focuses its efforts on some problemsfound within the former of the two, that is, basic research. In what follows, a careful expositionof some of the theoretical foundations upon which the study of electromagnetism is based will beperformed. Specifically, the notion of the electromagnetic fields will be discussed and comparedto an alternative formulation of EM that has no direct reference to fields, namely the direct actiontheory.

That our most fundamental physical laws do nothing more than characterize relationsbetween entities will be taken as a starting point. It is then a matter of human decision whether ornot to offer an interpretation of the laws, and then if so, what the nature of the interpretation shallbe. Choices made in either direction, towards or away from interpretation can be found in theworks of many. Newton writes of his newly formulated law of universal gravitation in theGeneral Scholium of the Principia,

But hitherto I have not been able to discover the cause of those propertiesof gravity from phenomena, and I frame no hypotheses; for whatever isnot deduced from the phenomena is to be called hypothesis; andhypotheses, whether metaphysical or physical, whether of occultqualities of mechanical, have no place in experimental philosophy. [3](p. 547)

While an overly strict reading of this statement might inaccurately portray Newton’s efforts (itwould be simply untrue to say he posed not one hypothesis); nonetheless, it does offer insight intohis intentions. Consider, on the other hand, the concluding remarks made by Maxwell at the endof his Treatise,

Hence all these theories lead to the conception of a medium in which thepropagation [of light] takes place, and if we admit this medium as anhypothesis, I think it ought to occupy a prominent place in ourinvestigations, and that we ought to endeavor to construct a mentalrepresentation of all the details of its action, and this has been myconstant aim in this treatise. [2] (p. 493)

Maxwell fears not the hypothetical admission of the æther as a means to explain theelectromagnetic laws he has constructed. He even encourages the use of the imaginative facultyto aid in the interpretation or explanation.

In our midnight walks through this garden of explanation we become suspiciously awareof a constant companion; someone, like a shadow but more potent, follows quietly behind. If we

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stop to catch a glimpse of him, he slinks away, out of sight, faster than thought. Resuming ourcourse, again we feel his presence. His name? Causation. Yes, it is he who lurks behind everystatement of equality and model of explanation. It is he who instills fear in the hardest of hearts.Some, like our daring Scotsman, have attempted to drag him from the shadows in which he hides,thus exposing his true nature for all to see. “Causes and effects are discoverable, not by reason,but by experience,” [4] writes Hume in his enquiry. Others, in the context of modern physics,have chosen to exile him from garden completely. In any case, no matter how one addressescausality, it seems that there will never be a time when we can completely cease to concernourselves with the notion of cause, either with a welcoming handshake or by a spiteful disregard.

Let us now turn to the specifics of the physical problem at hand, that being theformulation of classical electrodynamics (CED) with and without the notion of a field. To begin,we shall briefly trace the roots of the problem from its inception.

The phenomenon associated with electric force has been observed since the days ofancient Greece. The curious properties of two minerals, amber (¥lektron) and the magnetic

iron ore (≤ l€yow µagn∞tiw) attracted the notice of several ancient authors. However, unlike

celestial motions, no quantitative studies were recorded for many centuries. It was not until thelate eighteenth century that the phenomena of electricity and magnetism would be well enoughanalyzed to allow the formulation of a codified law. At the able hands of Charles Coulomb, theattractive and repulsive powers belonging to the “electric fluid” were found to obey the inversesquare law. From the Mémoires de l’Académie Royale des Sciences, Coulomb writes,

It results then from these trials that the repulsive action which the twoballs exert on each other when they are electrified similarly is in theinverse ratio of the square of the distances. [16]

Thus the principle force law of electrostatics was found to be:

F ∝

q1⋅q

2

r 2(1.1)

It should be noted that this law was determined solely through experimental,that is, in the termsused above, based research. Since then, Coulomb’s Law has occupied a preeminent position inboth basic and based scientific studies. Unlike Newton, Coulomb did not hesitate to posit ahypothesis regarding the phenomena embodied by his inverse square law.

Whatever be the cause of electricity, we can explain all the phenomenaby supposing that there are two electric fluids, the parts of the same fluidrepelling each other according to the inverse square of the distance, andattracting the parts of the other fluid according to the same inverse squarelaw. [5] (p. 58)

More relevant to the present tract however, are the implications bound within this law for thequestion of causality. The immediate structural similarity between Coulomb’s Law andNewton’s Universal Law of gravitation suggests that they must share some commonalities. Inaddition to sharing a functional (inverse square) nature, both laws, as stated, assume

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instantaneous action a distance (AAAD) as the means by which the law operates. As of the latenineteenth century, the state of gravitationally relevant observations had yet to find any conflictwith a force communicated in that manner. However, experimental electrical researches soonproduced effects that indicated something more was needed. Gauss writes in a letter,

I would doubtless have published my researches long since were it notthat at the time I gave them up I had failed to find what I regarded as thekeystone, Nil actum reputans si quid superset agendum: namely, thederivation of additional forces – to be added to the interaction ofelectrical charges at rest, when the are both in motion – from an actionwhich is propagated not instantaneously but in time as is the case withlight. [6] (p. 4)

In these thoughts, Gauss expresses the shortcomings of the action at a distances principle.Inherent in the AAAD principle are certain implications for causality. An acceptance of AAADcarries with it the abandonment some of our most elementary and intimate causal instincts, mostobviously, that of contact interaction. The world seems a safer place when the only way to makesomething move is by displacing it with something else.

Fittingly, the next major advances in CED addressed this issue in pointed fashion. Theseadvances are the product of two dissimilar researchers. On the one hand, Michael Faraday,whose exhaustive experimental work prompted the seminal idea of the electromagnetic fieldwhich in turn lead to a dramatic increase in the status of CED research; and on the other, JamesClerk Maxwell, who constructed the mathematical apparatus necessary to deal with CED and thenew notions of a field. Again we see the fruitful interaction between based and basic research.

Thus Faraday, with his penetrating intellect, his devotion to science, andhis opportunities for experiments, was debarred from following thecourse of thought which had led to the achievements of the Frenchphilosophers, and was obliged to explain the phenomena to himself bymeans of a symbolism which he could understand, instead of adoptingwhat had hitherto been the only tongue of the learned. ¶ This newsymbolism consisted of those lines of force extending themselves inevery direction from electrified and magnetic bodies, which Faraday inhis mind’s eye saw as distinctly as the solid bodies from which theyemanated. [7]

These physical lines of force were the constituents of the newly postulated entity, theelectromagnetic field. Both Maxwell and Faraday utilized the concepts of the fields, and theyboth adopted similar, yet slightly different interpretations regarding its nature and ontologicalqualities. Faraday writes,

The term line of magnetic force is intended to express simply thedirection of the force in any given place, and not any physical idea ornotion of the manner in which the force may be there exerted; as byactions at a distance, or pulsations, or waves or a current, or what not.[8] (p. 202)

6

Like Newton, Faraday was clearly hesitant to offer hypothesis about that which was not presentedclearly by experimental observation, namely, the specific nature of these physical lines of force.Maxwell had in mind perhaps a field endowed with a more substantial existence. He speaksplainly of the energy and momentum associated with the electromagnetic field. As seen above inthe passage from the Treatise on Electricity and Magnetism, Maxwell clearly encouraged theconstruction of a mental image to model, in some fashion, the mechanism of electromagneticphenomena.

And these lines must not be regarded as mere mathematical abstractions.They are the directions in which the medium is exerting a tension likethat of rope, or rather, like that of our own muscles. [7]

In these conceptualizations of the electromagnetic field, there is, as always, hints to theimplications for causation. Prior to the field formulations, the nature of causality, at least in theregime of electricity and magnetism, was essentially limited to a direct interaction between twocharged particles. After introducing the field, a third entity was present in any representation of acausal chain.

(+) (–) has now become (+) { Field } (–)

The field represents, as Gauss characterized above, the missing “additional forces”needed to solve some issues that arose when charges were no longer stationary. Not only doesthe incorporation of fields satiate Gauss’ concerns, it also assuages certain fears regardingcausality in physics. While many of the nineteenth century ‘electricians’ were content, evenperhaps happy, with the AAAD formulation, others were not. [17] As seen above, Faradayintroduced his physical lines of force as a means to explain the mechanism he saw as an activeelement in his electrical researches.

II. Some Problems with Field TheoryLet us now look at some of the reasons why a field theory might be an inadequate

formulation for electrodynamics. As will be seen in section VI, a complete solution to theinhomogeneous wave equation involves two independent solutions, one of which will eventuallybe discarded. Such results are commonplace when dealing with differential equations. Forexample, consider the equations of a motion for a simple trajectory of a classical particle,

y = 1

2gt2 . The fact the time component is squared allows for the trajectory to occur in either

direction in time, explicitly: y = 1

2g ±t( )2

, in which a negative time value will have the same

result as a positive value. Thus this formula obeys a certain symmetry with respect to time; it isleft to be determined by the exact nature of the physical system which solution is the physicallyrelevant one and which can be dispensed with. In the case of a classic cannon ball trajectory,

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namely a parabola, the choice is clear, since one solution involves the cannon ball reaching itsdestination in the tower of the neighboring castle, while the other solution places the ball along apath into the ground underneath the feet of a startled gunner. Both solutions are equally valid; wesimply choose one over the other based on the system at hand.

The Maxwell Equations, as formulated with the incorporation of fields, likewise exhibittime symmetric properties. This is evidenced by the two solutions presented after solving theinhomogeneous wave equations. One portrays a light wave propagated radially from anaccelerated charge in the direction of increasing time, that is, forward in time (called the retardedsolution); the other solution, equally valid mathematically, leads to the construction of wavetraveling backwards in time away from the accelerated charge (called the advanced solution).The equal validity of the two solutions mathematically, presents the physicist with the dilemma ofchoosing one solution over the other. What criteria should he use in making his decision? In thecannon ball example above, the choice was clear. The present case however, presents moresubtle difficulties.

On the surface, the choice seems perfectly clear; we choose the more physically realisticsolutions, that being the one in which the light waves are propagated forward in time. Lettingeven the most elementary physical intuition serve as a guide, one would expect the radiation froman accelerating charge to reach a distant point at a later instant. This is in accord with allexperience and physical feeling. On the other hand, choosing the advanced solution woulddescribe waves that converge from infinity onto the source charge and crossed the distant(observation) point before they reached the source. This is has been called “manifestlyunrealistic,” [6] and for good reasons too. Considering the degree of imagination required to“see” the standard, forward propagating electromagnetic fields, the mental agility needed tocomprehend such entities traveling backwards in time surely limitless.

Figures 1 and 2 depict the world lines, (suppressing y and z) of two point charges, A andB. The dotted lines connecting them represent the radiation emitted through an acceleration atpoint A. Figure 1 portrays the standard representation. Note the slope of the wave is +1 thus

amounting to a null-vector, or light likeseparation between the two particles

( sAB2 = 0 ). Figure 2 however, portrays

a light line with slope of –1. As perthe nature of such Minkowskidiagrams, this amounts to the raytraveling backwards in time. Suchphenomena, being completely opposedto every causal instinct we, asparticipators in the nature world, have,are quickly tossed aside as physical

absurdities, that, although mathematically sound, have no part in a consistent description of

t

x

a b

A

B

Figure 1

t

x

a b

A

B

Figure 2

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nature. However, it is on these intuitions and assumptions alone that the advanced solutions areignored. The concept of a field plays no part in this decision. Also, it offers no explanation as towhy such causal principles should guide a description of nature. Field theory provides no answerto the question, ‘why is there an electrodynamic arrow of time? The selection of the retardedsolution can be seen as an ad hoc imposition, rather than then a conclusion arrived at through asound, deductive inference.

A second problem that arises in the context of a field theory based electrodynamics isknown as the problem of infinite self-action or energy. The problem is a indirect consequence ofCoulomb’s law. Remembering equation (1.1) we note following failure in the potential as the

value for r becomes increasingly small: limr→0

e2

r= ∞ . It can be shown that if the charge is caused

to move from rest by a pulse type force, say a nasty blow by blunted object, the ensuing equationof motion will have the following form,

x t( ) = F

me

3mc2

2e2t⎛

⎝⎜

⎞

⎠⎟

(2.1)

with F representing the force and all other constants taking their usual values.[6] Having such asolution indicates that the velocity of the now moving charge will increase without bounds as tincreases. Such a physically peculiar result is generally indicative of a problem, however subtle,somewhere in the original formulations and assumptions. Indeed, the singularity created by thestructure of Coulomb’s force law has grave consequences.

In addition to the two aforementioned conflicts, there also exists the problem of just howshould one conceive of a field. In a subchapter called ‘scientific imagination,’ Feynman pointsout the difficulties of such a project.

When I start describing the magnetic field moving through space, I speak of theE- and B- fields and wave my arms and you may imagine that I can see them.I’ll tell you what I see. I see some kind of vague shadowy, wiggling lines—hereand there is an E and B written on them somehow, and perhaps some of the lineshave arrows on them—an arrow here or there which disappears when I look toclosely at it...I cannot really make a picture that is even nearly like the truewaves. [11]

Considering also Maxwell’s earlier mechanistic interpretations of the nature of a field, it becomesclear that even the best scientific minds are still unclear about many basic ontological questionsregarding the nature of an electromagnetic field. That is, no clear mental picture can be drawn inwhich the field appears limpidly and uniquely. This being noted, what place does such aconstruction have in a physical system? It was originally introduced in the hopes of clarifying.Faraday’s lines of force were, as Maxwell puts it, “the key to the science of electricity.”(AAAD)However, when firmly scrutinized, the exact nature of a field shifts and squirms like Proteusunder Menelaus’ grasp.

Having thus pointed out some of the reasons why to and why not to involve the notion ofa field a field in the physical framework for electrodynamics, let us now proceed and perform

9

some relevant mathematical formulizations. The formulations that follow will rely heavily on theLeast Action Principle and for this reason a short description of the principle should be helpful.

III. The Principle of Least Action Suggested by Pierre Louis Maupertuis around the middle of the 18th century, the

Principle of Least Action (LAP) offers a pregnant insight into the character of physical law. Improved by the mathematical wizardry of Euler and Hamilton, the rather loose formulation ofMaupertuis has become a powerful apparatus for handling many physical systems, from optics tomechanics to, as we shall soon see, electrodynamics.  Also known as a variational principle, the

action principle can be stated with the following equation, δS = 0 , where S indicates a valueknown as the action of the system. For example, in the system of classical mechanics, theLangrangian, (KE-PE), serves as the quantity of action. Other systems are found to have differentformulations of the action quantity. From the chosen form of S, the standard equations of motionof the desired system can be obtained. Some of the benefits of a least action formulation are thefollowing, (a) it condenses the central statements from a theory into a single expression, (b) it canbe facile method for determining equations of motion, (c) it transmits all of its invariances into itsmathematical offspring, i.e. a Lorentz invariant action S will produce Lorentz invariant equationsof motion, (d) it implies conservation laws.[1]

Let us look at an elementary example that utilizes the LAP before diving into the moredifficult matters of CED.

Consider a particle dropped from rest in a uniform gravitational field. In this case, theaction S will be determined by the Langrangian, L, where L = (Kinetic energy) – (potentialenergy), or,

L =

m

2y2 − mgy (3.1)

Here y represents the height off the ground, and y is its speed. Thus,

δS=δ Ldt

t1

t2∫ = δm

2y2 − mgy

⎛⎝⎜

⎞⎠⎟t1

t2∫ dt = myδ y − mgδ y( )t1

t2∫ dt (3.2)

Noting that δy = ddtδy , the velocity term from (3.2) becomes, after integration by parts,

my

d

dtδy dt

t1

t2∫ = my]t1

t2 − myt1

t2∫ δy dt (3.3)

The first term on the right hand side of (3.3) will be zero, since the path varies not at theendpoints. Continuing on, the inclusion of the spatial term yields,

δS = δ −my − mg( )

t1

t2∫ δy dt = 0 (3.4)

Since δy represents an arbitrary variation of y, eq. (3.4) will only be zero for all cases if the

quantity in parenthesis is always zero, that is,

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−mg − my = 0 (3.5)

Thus we arrive at the expected result for this simple example, namely, Newton’s second

law for a body falling in a uniform gravitational field: y t( ) = − 1

2gt2 . It can then be shown that

any other solution besides the one offered in (3.5) would cause the integral in (3.2) to not be anextremum and thereby would be an invalid solution based on the requirements of the LAP.

The simplicity of the above example offers a clear insight into the workings of the LAP.While not an explicit statement of conservation, such notions are not far removed. Using resultsfrom Emmy Noether’s potent work of 1918, it can be seen that the time symmetries displayed inthe Lagrangian result in a conservation of energy rule for the system at hand. [10,1]

IV. Classical Electrodynamics from FieldsNow we may begin the process of formulating classical electrodynamics with and

without field. We will start with the field theory. Essentially, by finding the correct Lagrangianand applying the LAP, the standard equations of motion and the ME will be found.Starting from the least action principle,

δS = δ Ldt∫ = 0 (4.1)

a formulation of L must be found that will lead to the equation of motion for a charged particlemoving in electromagnetic fields. The sought after equation is none other than the Lorentz ForceLaw,

d

dtm

0γ v( ) − q E +

v × B

c

⎛⎝⎜

⎞⎠⎟= 0 (4.2)

The task now is to find an L such that upon insertion into the Lagrange equations,

d

dt

∂L

∂vi

⎛

⎝⎜⎞

⎠⎟=∂L

∂xi

(4.3)

will lead to the Lorentz Force law. Let us try the following for L.

L = −m0c2 1− β 2 −U (4.4)

Checking the first term,

d

dt

∂∂v

i

−m0c2 1− v2

c2( )⎛⎝

⎞⎠ =

d

dt−m

0c2 1

21− v2

c2( )− 12 −2v

c2( )⎛⎝⎜

⎞⎠⎟=

d

dtm

0γ v

i( ) , (4.5)

we see that this indeed returns the first term of (4.2). The second half (the U term) of (4.4) mustnow also satisfy (4.3) and return the electromagnetic portion of (4.2), namely, the Lorentz Force.

Starting with Lorentz’ law for the force on a charged particle in a field,

F = q E +

v × B

c

⎛⎝⎜

⎞⎠⎟

11

but,

E = −∇ϕ − 1

c∂A∂t

and B = ∇ × A (4.6)

Therefore,

F = −∇ϕ −1

c

∂A

∂t+

v × ∇ × A( )c

⎛

⎝⎜

⎞

⎠⎟ .

Using A × B × C( ) = B A ⋅C( ) − C A ⋅B( ) ,

F

i= −∇ϕ +∇

v × A

c

⎛⎝⎜

⎞⎠⎟−

1

c

∂A

∂t− A v ⋅∇( )

= −

∂∂x

i

ϕ − 1c

v ⋅A( ) − 1

c

d

dt

∂∂v

i

A ⋅ v( )⎛

⎝⎜⎞

⎠⎟.

This, however, is in the requisite form to yield the second term in (4.4), U.

U = qϕ − q

cA ⋅ v (4.7)

The Langrangian will then be represented by,

L = −m

0γ −1c2 − qϕ + q

cA ⋅ v( ) (4.8)

This quantity must now be converted to the standard 4-D covariant form. For this tohappen, we must use the relativistic intervals and velocity formulations:

ds = cd ′t =

cdt

γ(4.9)

and

v

idt =

dxi

dtdt = dxi . (4.10)

Thus, taking Ldt, (4.8) transforms to

Ldt = −mcds − q

ccdt,dxi( ) ⋅ ϕ, A

i( )

= −mcds − q

cAαdxα (4.11)

Now that the 4-D Lagrangian is known, it can be used in equation (4.1) in the formulationof the least action principle.

The Lorentz invariant 4-D action is then found to be,

S = −mcds − q

cAαdxα

a

b

∫ (4.12)

Applying the Least Action principle, (4.1), and using ds = dxαdxα as well as the

12

4-velocity, uα = dxα

ds= γ ,−γβ

1,−γβ

2,−γβ

3( ) , where β = v

c,

0 = δS = − mc

dxαdδxα

dsa

b

∫ +q

cAαdδxα +

q

cδ Aαdxα (4.13)

A quick integration by parts on the first two terms will be helpful.

− mcdxαdδxα

dsa

b

∫ = −mc δxα dxαds

|ab − duαδxα

a

b

∫⎛

⎝⎜⎞

⎠⎟(4.14)

and,

q

cAαdδxα

a

b

∫ =q

cAαδxα |

ab − δxαdAαa

b

∫⎛⎝

⎞⎠ (4.15)

Since δxα = 0 at a and b, as required by the LAP, we arrive the following for (4.13)

= mcduαδxα

a

b

∫ +q

cδxαdAα −

q

cδ Aαdxα (4.16)

= mcduα

dsδxα +

q

c

∂Aα

∂xβ

∂xβ

dsδxα −

q

c

∂Aα

∂xβ

dxα

dsδxβ⎛

⎝⎜⎞

⎠⎟a

b

∫ ds (4.17)

= mcduα

ds+

q

c∂β Aα − ∂α Aβ( )uβ⎛

⎝⎜⎞

⎠⎟a

b

∫ δxαds (4.18)

Thus, as in the previous LAP example above, see eqs. (3.4) & (3.5), we see that the only

way for (4.18) to be zero for any arbitrary δxα , is if the term in parenthesis,

mcduα

ds+

q

c∂β Aα − ∂α Aβ( )uβ⎛

⎝⎜⎞

⎠⎟(4.19)

is itself equal to zero.Thus, incorporating the Field Strength Tensor, F, which is anti-symmetric,

mc

duα

ds=

q

cFαβ uβ (4.20)

and remembering that ds = cdτ , we arrive finally at the following:

mcwα =

q

cFαβuβ . (4.21)

This is the 4-D Lorentz Force equation. ( wα is the 4-acceleration)

There now appears the field-strength tensor, Fαβ = ∂α Aβ − ∂α Aβ , or, more explicitly,

13

Fαβ = ∂α Aβ − ∂β Aα =

0 Ex

Ey

Ez

−Ex

0 −Bz

By

−Ey

Bz

0 −Bx

−Ez

−By

Bx

0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

(4.22)

or, with contravariant indices,

Fαβ = ∂α Aβ − ∂β Aα =

0 −Ex

−Ey

−Ez

Ex

0 −Bz

By

Ey

Bz

0 −Bx

Ez

−By

Bx

0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

(4.23)

These matrices follow from the relations between E,B and the vector potential, A,namely,

E = −∇ϕ − 1

c∂A∂t

,B = ∇ × A (4.6)

Now we are in the position to extract the Maxwell Equations from these formulations. Since,

Fαβ = ∂α Aβ − ∂β Aα (4.24)

therefore, by permutation of the indices,

∂γ Fαβ = ∂γ ∂α Aβ − ∂γ ∂β Aα

∂α Fβγ = ∂α∂β Aγ − ∂α∂γ Aβ

∂β Fγα = ∂β∂γ Aα − ∂β∂α Aγ

(4.25)

Summing the three equations of (4.25), and realizing that ∂α∂β ⇔ ∂β∂α , we arrive at,

∂γ Fαβ + ∂α Fβγ + ∂β Fγα = 0 , [ α ,β ,γ can take any value of 1,2,3] (4.26)

Now, using the relations from (4.6), and the corresponding matrices in (4.22), we will be able toextract the two source-less M.E. from the field-strength tensor. If,

α = 0

β = 1

γ = 2

→ (4.26) will return ∇ × E( )

3+ 1

c∂

tB( )

3

similarly,

α = 0

β = 2

γ = 3

→ (4.26) will return ∇ × E( )

1+ 1

c∂

tB( )

1

and finally,

14

α = 0

β = 1

γ = 3

→ (4.26) will return ∇ × E( )

2+ 1

c∂

tB( )

2.

All together, these three will yield the familiar Faraday’s Law,

∇ × E + 1

c∂

tB = 0 (4.27)

For

α = 1

β = 2

γ = 3

→ (4.26) yields ∇ ⋅B = 0 (4.28), the second source-less ME.

To finish this formulation, we must now modify the action S as given in (4.12). The two termsthat exist in that equation only account for the action of the free particle and the action pertainingto the interaction between the field and the particle. There must be added a term that accounts forthe action of the field itself, that is, the field without reference to other charged particles. Theaction will then take the following form:

S = S

m+ S

fm+ S

f (4.29)

where,

Sm= action of particle = −mc ds∫

Sfm= action of field and particle = −

q

cAαdxα = −

1

c2Jα AαdΩ∫∫

Sf= action of field

(4.30)

Having already exposed the first of these two terms above, we will now determine the third term,that is, the action of the field. It has been found that this term will take the following form:

S

f= −

1

16πcFαβ FαβdΩ∫ (4.31)

with dΩ = c dt dx dy dz . Such a form is essentially a Lagrangian Density term and is needed to

allow for certain properties of fields in general, such as the satisfaction of the principle ofsuperposition.Combining the two action terms that mention the fields in the following manner,

S = S

f+ S

fm= −

1

16πcFαβFαβdΩ∫ −

1

c2Aα J αdΩ∫ (4.32)

and applying the LAP,

0 = δS = −

1

c

1

8πFαβδFαβ +

1

cJαδ Aα

⎡

⎣⎢

⎤

⎦⎥dΩ∫ (4.33)

15

after some integration by parts and index manipulations, we can construct the following,

∂β Fαβ = − 4π

cJα (4.34)

In a manner similar to that above, (See eq. (4.26) and following), the permutation of the indicesof (4.34) will eventually present the second pair of ME:

∇ × B −

1

c

∂E

∂t=

4πc

J (4.35)

and

∇ ⋅E = 4πδ , (4.36)These are Ampere’s Law and Gauss’ Law, respectively.

These equations as formulated, demonstrate conservation of charge, satisfy the waveequation, and are gauge independent.

Thus we have seen that the least action principle, when used in conjunction with theconcepts of fields, can result in the equations of motion and the M.E.

With a similar mathematical apparatus the same results can be derived without directmention of electromagnetic fields.

V. Classical Electrodynamics sans FieldsIn the 1930’s A. D. Fokker and others proposed an alternative to the action formulation

as presented above. Their version, known as the Fokker action, requires no mention of theconcepts of fields. Instead, S will be seen to contain two terms, one that generates the non-electromagnetic motions, i.e. the mechanics, while the other is responsible for the electromagneticinteractions. The form proposed by Fokker is as follows.

S = − m

adx

aαdx

aα∫

a∑ + e

ae

bδ∫∫

a<b∑ x

a− x

b( )2⎡⎣⎢

⎤⎦⎥

dxaα

dxbα ≡ S

0+ S

1(5.1)

where, a,b,… serve as the labels for the charged particles,

x ≡ x

0, x

1, x

2, x

3( ) is the 4-spacetime point, with c = 1, x0 = t,

α runs from 0, 1, 2, 3,e are charges, m are masses,and δ(f(x)) is the normal Dirac delta function.

Most immediately, we can notice the absence of any field terms in eq. (5.1). Instead, the twoterms, hereafter referred to as S0 and S1, develop what has since been called the direct actionformulation. S0 is recognized as the standard, relativistically invariant form of free particle

motion, as seen also in first term of the action S used in the preceding section, eq. (4.12). The

second term, S1, has taken the place of two terms found in (4.29). Let us examine the S1 term ingreater detail. Firstly, the appearance of the Dirac delta function must be noted. The argument ofthe delta function contains the separation between two world points of two charged particles,squared. More explicitly,

16

x

a− x

b= c2 t

2− t

1( )2− x

2− x

1( )2− y

2− y

1( )2− z

2− z

1( )2. (5.2)

The delta function will provide a non-zero value, iff xa - xb = 0. This implies that the twoparticles must be connected by a null-vector, i.e. the interaction, if it exists, between a and b mustpropagate at the speed of light. Let us continue by deriving the Lorentz force law and the M.E.from the direct action formulation.

If we define

A

bαx( ) = e

bδ x − x

b( )2⎡⎣⎢

⎤⎦⎥∫ dx

bα(5.3)

then

S

1= e

aA

bαx

a( )dxaα∫

b≠a∑

a∑ (5.4)

also, if we let Aα x( ) ≡ A

bαx( )

a≠b∑ , then

S

1= e

aAα x

a( )dxaα∫

a∑ (5.5)

To use the least action principle for this quantity, the variation δS must be incorporated into (5.5).

The variation x

aα→ x

aα+ δx

aαwill result in the following,

δS = e

aδAα x

a( )dxaα + Aα x

a( )δdxaα⎡⎣ ⎤⎦∫

a∑ .

Considering also,

δAα =

∂Aα

∂xβ δxβ ,

therefore,

δS = e

a∂β Aα( )δx

aβdx

aα + Aαδdx

aα⎡

⎣⎤⎦∫

a∑

Since δxaα returns a zero when evaluated at the endpoints, integration by parts on the second term

above yields,

= e

a∂β Aα( )δx

aβdx

aα − dAαδx

aα⎡

⎣⎤⎦∫

a∑ .

Using,

dAα =

∂Aα

∂xβ dxβ ,

therefore,

δS = e

a∂β Aα( )δx

aβdx

aα − ∂β Aα( )δx

aαdx

aβ⎡

⎣⎤⎦∫

a∑

17

= e

a∂α Aβ − ∂β Aα( )δx

aαdx

aβ⎡

⎣⎤⎦∫

a∑ .

Remembering

Fαβ ≡ ∂α Aβ − ∂β Aα , (5.6)

where F and A were solely defined in terms of the direct action term, (5.3),we can see that

δS = e

aFαβ x

a( )∫ δxaαdx

aβ

a∑ (5.7)

Thus, applying the least action principle,

0 = δS = δS0+ δS

1= m

a

d 2xaα

dτa

2+ e

aFαβ x

a( ) dxaβ

dτa

⎛

⎝⎜⎜

⎞

⎠⎟⎟δx

aαdτ

a∫a∑ (5.8)

will return the following,

m

a

d 2xaα

dτa

2= −e

aFαβ x

a( ) dxaβ

dτa

, (5.9)

which is indeed the Lorentz Force Law.With methods analogous to those in the previous section, the two source-less ME will be

arrived at by consideration of the anti-symmetric nature of Fαβ . To obtain the other two ME, a

different approach will be used.

To begin, use the D’lambertian operator ( ≡ ∇2 −

d 2

dt2) on the

A

bαas defined above in

(5.3).

A

bαx( ) = e

b δ x − x

b( )2( )∫ dxbα

(5.10)

Noting the helpful Dirac identity,

δ x2( ) = 4πδ 4 x( ) (5.11)

equation (5.10) can also be written as

A

bαx( ) = 4π e

bδ 4 x − x

b( )( )∫ dxbα

(5.12)

with

δ 4 x − x

b( ) = δ x0− x

b0( ) ⋅δ x

1− x

b1( ) ⋅δ x

2− x

b2( ) ⋅δ x

3− x

b3( ) (5.13)

Letting

18

J

bαx( ) = e

bδ 4 x − x

b( )dxbα∫ (5.14)

equation (5.12) can be formulated quite succinctly as,

A

bα= 4π J

bα(5.15)

The exact meaning of the J

bαterm has to be determined. Taking the time component of J, that is,

J0, one obtains,

J

b0x( ) = e

bδ 3 x − x

b( ) δ x0− x

b( )dxb0= e

bδ 3 x − x

b( )∫ (5.16)

which is nothing more than the charge density, ρ . The other components yield:

Jbi

x( ) = ebδ 4 x − x

b( ) dxbi

dxb0

dxb0= e

bv

bαx

b0( )δ 3 x − x

b( )∫ (5.17)

which is the 3-D current density. Here v is the velocity of a charge at the given point.

Thus, equation (5.15) will generate the other two ME, namely Gauss’ Law and Ampere’s Law.

What remains now is to provide a more meaningful interpretation regarding the physical

significance of the Aα used in the preceding derivations.

Starting with the Aα as determined by the DAE, a slight mathematical manipulation will prove

insightful.Let us start by noting a valuable property of the Dirac delta function.

δ f x( )( ) = 1

′f xj( )j

∑ δ x − xj( ) (5.18)

where f x

j( ) = 0 and ′f x

j( ) ≠ 0 . If

f x( ) = x2 − a2 = x + a( ) x − a( ) (5.19)

and thereby

′f x( ) = 2x (5.20)

the following identity is obtained:

δ x2 − a2( ) = 1

2 aδ x + a( ) + δ x − a( )⎡⎣ ⎤⎦ (5.21)

This allows us to reformulate eq. (5.3) as

Abα

x( ) = 1

2d 4 y∫ Jα y( ) δ x

0− y

0− x − y( )

x − y+δ x

0− y

0+ x − y( )

x − y

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪(5.22)

or, with a different notation:VI. Solving the Wave Equation

To see what this means, let us now solve the inhomogeneous wave equation.

19

Now we must solve the inhomogeneous wave equation,

Aα = 4πJα (6.1)

Define G to be

G x, ′x ;t, ′t( ) = 4πδ x − ′x( )δ t − ′t( ) (6.2)

(The notation has switched)Thus,

Aα = d 4 yG x − y( ) Jα y( )∫ (6.3)

Let ′x = 0 and ′t = 0 for the moment without any loss of generality.

G x,t( ) = −4πδ x( )δ t( ) (6.4)

With a 4-D Fourier transform,

G x,t( ) = 1

2π

⎛⎝⎜

⎞⎠⎟

4

d 3k∫ dω∫ g k,ω( )ei k ⋅x−ωt( ) (6.5)

and

δ r( )δ t( ) = 1

2π⎛⎝⎜

⎞⎠⎟

4

d3k dω∫ ei k ⋅x−ωt( )∫ (6.6)

since,

δ x( ) = 1

2πe± ikxdk

−∞

∞

∫ (6.7)

Using (6.2) in combination with (6.5) and (6.6),

ik( ) ⋅ ik( ) − 1

c−iω( ) −iω( )⎡⎣ ⎤⎦ g k,ω( ) = −

1

π(6.8)

Thus,

g k,t( ) = −

1

πc2

ω 2 − k 2c2(6.9)

and finally,

G x,t( ) = 1

2π

⎛⎝⎜

⎞⎠⎟

4

−1

πc2⎛

⎝⎜⎞⎠⎟

d 3k∫ dω∫ei k ⋅x−ω t( )

ω 2 − k 2c2(6.10)

After some complex integration and other teeth pulling-like operations, it can be seen that (6.10)results in two possible solutions.

20

G +( ) x, ′x ;t, ′t( ) =δ ′t − t −

x − ′x

c

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

x − ′x(6.11)

hereafter called the retarded solution, and, due in part the fact that δ x( ) = δ −x( )

G −( ) x, ′x ;t, ′t( ) =δ ′t − t +

x − ′x

c

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

x − ′x(6.12)

hereafter called the advanced solution. Notice that the only difference lies in the second nestedterm in the delta function. The retarded solution bears a minus while the advanced solution tauntsus with the addition symbol.In our relativistic notation, these two solutions will be written as:

G+ x( ) = δ t − r( )

r(6.13)

and

G− x( ) = δ t + r( )

r(6.14)

Upon insertion separately back into (6.3), that is, one time for each solution, we have thefollowing:

A

α

ret x( ) = d 4 y∫δ t − r( )

rJα y( ) the retarded potential (6.15)

A

α

adv x( ) = d 4 y∫δ t + r( )

rJα y( ) the advanced potential (6.16)

Together, it is clear that these two quantities will equate with twice the value for Aα as found in

(5.22), which was arrived at through the direct action formulation of CED.In the hopes of preserving our intuition of causation and the obvious arrow of time in CED, theadvanced solution is generally set aside as meaningless. As discussed above, this choicehowever, can be considered ad hoc and lacking clear, deductive reasons.

VII. How to make sense of it all: Wheeler-Feynman Absorber TheoryHaving taken our escapade into the land of mathematical adventure, we can again resume a verbaltack and hopefully offer an account of some of the results arrived at above. Stated briefly, it was

21

shown that in a formulation of CED without a direct use of fields, both the retarded and theadvanced solutions to the ME were required to complete the formulation; however, incorporatingthe notion of a field into the picture allows for, one could even say demands, the completeremoval of the causally troublesome advanced solution. There are then two choices available:

1. Use Fields in the construction of CED, and accept only the components that areconsistent with our causal intuitions, or,

2. Abandon the use of Fields in the original formulations and accept what appear to beacausal implications.

An attempt to rectify the situation was offered by John Wheeler and Richard Feynman in the1940s. It has been since called the Wheeler-Feynman Absorber Theory of Radiation (WFAR).Essentially, the WFAR adopts the second choice and makes it acceptable by incorporating theinteraction between a typical charge and all other charges in the universe. In doing so, the theoryhas a net result that is found to be consistent, but not based on, the common notions of temporalcausality.

As stated, their mission was,

…to go back to the great problem of classical physics—the motion of asystem of charged particles under the influence of electromagneticfields—and to inquire what description of the interactions and motion ispossible which is at the same time (1) well defined (2) economical inpostulates and (3) in agreement with experience. [14]

Their physical foundations were,

(1) There is no such concept as “the” field, an independent entity withdegrees of freedom of its own

(2) There is no action of an elementary charge upon itself andconsequently no problem of an infinity in the energy of theelectromagnetic field

(3) The symmetry between past and future in the prescription for thefields is not a mere logical possibility, as in the usual theory, but apostulational requirement. [14]

It must be made clear that the intentions of the WFAR were not to completely erase the notion ofthe field from physics but were instead to find a complimentary theory that did not rely on themin the usual sense. Seen below is a Minkowski diagram of the emission and absorption ofelectromagnetic radiation in the context of the WFAR. Without getting entrenched in the details,the theory involves the radiation being emitted by the emitter in the form of a half-amplituderetarded wave as well as a half-amplitude advanced wave. The interaction that ensues between inthe emitter and absorber generates more waves which, when considered in conjunction with theoriginal waves will produce the needed radiation to be consistent with experience.

22

The crucial difference between this and the prior field theories lies in the recognition that theadvanced waves are a necessary part of the formulation. Such a stipulation carries heavyconsequences when considering the causal implications; as with most time reversing phenomena,paradoxes are present†.

Considering the WFAR as a model of explanation, it seems slightly more intimidatingthan the usual field models. The less than lucid Fokker action (eq. (5.1)) offers insight only tothose well versed in such notations while field theory can be portrayed quite well to even themost untrained eyes. To fully comprehend the WFAR model, one must be comfortable adoptingbackwards causality and non-instantaneous action at a distance without dissent. Initially, suchnotions are difficult to accept. However, if we cleanse ourselves of some of our more humanisticinclinations, the difficulties become more tractable. Can it be deduced, for example, in a fully apriori manner, that backwards causation is indeed impossible? Or, as the above mathematicalformulations suggest, should the possibility of such a thing be formally acknowledged.

By way of a conclusion, let us point out some ontological concerns in regards to the fieldand non-field systems. By accepting fields into a formulation, to what degree must they be givenan independent existence? Or, should they be accepted only as a model useful in the constructionof a physically meaningful system? In so deciding, one must carefully weigh the consequences ineither direction. In adopting fields, the scientist must be ready to deal with the troublingconsequences seen above in order to benefit from the friendlier model. And conversely, bycasting the fields aside, there are other equally worrisome conflicts. This suggests a compromisemuch like that implied by the wave-particle duality in quantum physics. Simply accept bothinterpretations as valid and use one over the other depending on the situation.

† See reference [14] for an example and its supposed resolution.

Figure 3: Adiagram ofthe WFARmodel.

23

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[2] Maxwell, James Clerk, A Treatise on Electricity and Magnetism, Vol. II, Dover, 1954, NewYork

[3]* Newton, Isaac, Philosophiae Naturalis Principia Mathematica, U. of California, 1962

[4] Hume, David, An Enquiry Concerning Human Understanding, Hackett publishing company,1977

[5] Whittaker, Sir Edmond, History of the Theories of Aether and Electricity, 1987 AmericanInstitute of Physics.

[6] Hoyle, Fred; Narlikar, Jayant V; Lectures on Cosmology and Action at a DistanceElectrodynamics, World Scientific Publishing Co, Singapore, 1996

[7]* Maxwell, J. C. On Action at a Distance, From the Proceedings of the Royal Institution ofGreat Britain, Vol. VII

[8] Hesse, Mary B.; Forces and Fields; Thomas Nelson and Sons, 1961

[9]* Cramer, John G.; Generalized absorber theory and the Einstein-Podolsky-Rosen paradox;Physical Review D, Vol. 22, No. 2, 1980, p. 362

[10] Byers, Nina; E. Noether's Discovery of the Deep Connection Between Symmetries andConservation Laws, ISREAL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 12,1999

[11] Feynman, Richard; The Feynman Lectures on Physics; Vol. II; 1989, California Institute ofTechnology.

[12] Landau, L.D. and Lifshitz, E.M.; The Classical Theory of Fields, 1975

[13] Jackson, J.D.; Classical Electrodynamics; 3rd Edition, 1999

[14]* Wheeler, J.A. and Feynman, R.P.; Classical Electrodynamics in Terms of DirectInterparticle Action; Reviews of Modern Physics; Vol. 21, No. 3, 1949

[15]* Wheeler, J.A. and Feynman, R.P.; Interaction with the Absorber as the Mechanism ofRadiation; Reviews of Modern Physics; Vol. 17, No. 2 & 3, 1945

[16] Magie, W.F.; A Source Book in Physics; Harvard University Press, 1963

[17] Leung, Peter; unpublished notes, 1967

[18] Leung, Peter, Lecture Notes from PH 632, PSU