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'Interaction of Photons with Electrons in Dielectric

'INTERACTION OF WITH ELECTRONS IN

DIELECTRIC MEDIA

MiroSlav M. Novak

Thesis submitted for the Degree

of Ph.D.

Faculty of University of NSW

June, 1981 UNIVERSITY OF N.S.W, mhl 23.^cP.62

LIBRARY ABS1!RACT

A possilDle modification of some of the properties of the electromagnetic wave by a medium in which it propagates, are studied.

Assuming the validity of the yhxwell equations, a new way of deriving the formula for the "black body radiation in a medium is presented.

The mechanism for the mass generation is investigated. It is shown that a medium may be replaced, for some purposes, by a .

FurthermDre, if the non-linear interaction terms between the electro- magnetic wave and the curvature of space-time are. included, it is found that the resulting photons acquire mass, proportional to the curvature scalar. This is further supported by the conclusions following from the analysis of the gauge theories. Some of the consequences of this non-linear interaction are then examined. CONTENTS

I. Introduction 1

IIo Therinodynamics of Electromagnetic Waves in a

Medium o o.o. o... 9

III. Alternative Theories 23

IV, Generally Covariant Approach „.. 35

V. Gauge Theories and the Spontaneous Symmetry

Breaking .... 50

VI. Some Aspects of Interaction and Experimental

Evidence . <.. 66

VII. Conclusion o.».. 78

Biography 8l I. INTRODUCTION

Theory of light has been attracting attention of many for a long time. Here, it suffices to "begin with the work of Newton, who

"believed that light consists of a stream of massive corpusciiles travelling with very high velocities. This result was accepted in his time as it successfully explained the then available experiments„

There was also another proposal, put forward by Huygens, who claimed that the light has wave-like properties. However, in the absence of decisive experiments that theory did not come to the fore for a long time. Towards the end of the XVIII century, new experimental data became available, which could not, any longer, be explained within the framework of the Newton theory. Consequently, the wave theory of light was then accepted as being the only correct theory describing the propagation of light. This situation remained, more or less unchanged, till the end of the last century. Then, once again, the wealth of experimental evidence precipitated a contention regarding the nature of light.

In those confusing times, Planck was forced to introduce, in his work on the black body radiation, the discontinuous aspect of the radiation field, in order to resolve the discrepancy between theory and experiment. As a result, the particle nature of radiation was reintroduced. Later, the name "" was given to these particleso

When investigating the photoelectric effect, this concept was successfully used by Einstein. However, as distinct from the earlier times, the particle theory, now, did not become the dominant one, but was considered to be complementary to the wave theory,, Put otherwise. depending on the question posed, the radiation phenomenon could he accounted for hy either the particle or the wave theories.

At present, the complementarity of these different aspects of radiation is well established. For instance, the classical wave theory can explain neither the photoelectric phenomenon nor the

Compton effect, "because these "belong to the particle domain. On the other hand, the occurrence of interference, diffraction, and polari- zation manifest the wave nature of radiation and, therefore, cannot

"be explained by the photon theory. Thus, in order to really comprehend radiation phenomena, this duality must "be inevitably acceptedo

In spite of differences, there is a close connection between the wave and particle pictures of radiation. Classically, the notion of and are inherent in the former, whilst and momentum belong to the latter. However, it is now customary to associate with the electromagnetic wave of angular frequency w , a particle with energy E given by

E = tl 03 (l) where 2 Trti = h is the . Although this relation was proposed by Planck already at the beginning of this century and is now well accepted, it meaning as to the physical connection between energy and frequency remains rather obscure /l/. Nevertheless, (l) in conjunction with one of the most fundamental principles of the special theory of relativity, viz., the principle of the equivalence of mass m and any kind of energy, E = mc^ , suggest that an effective mass can be attached to the photon. Now, according to that theory, the energy of a particle is given by

E2 = p2c2 + m^ c^ , (2) where p stands for the momentum, m refers to the rest mass of a ' o particle, and c is the vacuum speed of lighto As the rest mass vanishes in the case of a photon, its momentum has the value iiw/c .

In addition, there is another, equivalent expression for the photon's momentum. Thus, on substituting the relation "between the frequency v and the corresponding wavelength X, which in vacuo takes the form

vX = c , (3) together with (l) and (2), one ohtaines

P =ti k, w where k = 2'nX ^ is the wave vector. Consequently, (l) and express the relation "between the particle and the wave aspects of radiation. Notice that the ratio "between the two is given in terms of the quantum of . This, in essence, reflects the wave- particle parallelism which is now usually adopted when dealing with radiation effects.

The relations (l) and (U) remain formally correct also for particles with non-vanishing rest mass, as was first recognized "by

de Broglie /2/. From the similarity between the phase of a plane wave and the Hamiltonian action of a particle, he deduced that the

Fermat principle, valid in the domain of geometrical , is

analogous to the Maupertuis theorem in the analytical mechanics.

Since the Fermat principle has no meaning in the realm of the wave

optics, he concluded that the traditional mechanics is also an

approximation with the same range of validity a Thus, de Broglie

applied the wave concept of radiation to matter and was able to show

that (l) and {h) are, indeed, relativistically invariant also for

massive particleso Despite these similarities, massless photons and massive particles are not the same. Thus, in order for a particle to be observed, it must interact. Consequently, its energy and momentum will be changed in accordance -with the conservation laws. Then, high energy material particles are like high energy photons, because they change energy by changing their effective mass. This is the domain of the applicability of the geometrical optics, which, by definition, is valid in the region of comparatively small . On the other hand, the differences between photons and massive particles are very pronounced at the other end of the energy spectrum. The energy of material particles approaches, in that case, the rest energy, whilst the approaches zero.

Therefore, in this limiting case, the theory of radiation is effectively replaced by the static field theory. As a result, our conception of matter is represented, classically, by the particle nature, whereas that of radiation assumes the wave character.

Bearing this in mind, the present work attempts to clarify some aspects of interaction of electromagnetic radiation with a general medium. As the essential difference between real particles and photons is the value of their mass, it may prove instructive to study this concept in some detail. In the absence of the generally accepted theory, different possibilities will be investigated.

This work is divided into two parts: one, in which the assumption of the linearity is invoked in the problem at hand, whilst in the other this assmption is dropped and some of the consequences are then examined. It should be emphasized, at this point, that the proper treatment of interaction phenomenon should take into account the mutualceffect of-the radiation and a medium. However, the scope of this work is more limited, as it examines, to a large extent, only the effect of a medium on the radiation field„

With this in mind, we begin, in Chapter II, by studying the effect of the medium on the moving within it.

The main new contribution of that Chapter, already to be found in literature /3/, deals with the generalization of the Brownian motion to include a medium with the dispersion relation of the form encountered in the theory of a collisionless plasma. The philosophy of the approach parallels that of Einstein's original article The results are, evidently, different. It may be said, in view of that analysis, that the black body radiation inside a medium differs from the one in vacuo. Although this result has been known /5/ for some time, the method of its derivation is new. In the process, it was confirmed that, in the region where hv » kT, i.eo, in the domain of the validity of the geometrical optics, the exchanged momentum between the radiation field and medium is proportional to the refrac- tive index of the latter. The form of this momentum was derived by

Minkowski /6/ who substituted the Maxwell equations to the conser- vation relations. As these field equations are linear, it is implied that, within the specified restrictions, the conclusion of Chapter II may be looked upon as following from the linearity assumption.

Generally, it is agreed that the classical electrodynamics is only an approximate theory, valid when the accelerations of electrons are small. In an attempt to make it more accurate, one encounters the main difficulty in the problem of the electron structure. Thus, assumption of a point charge leads to an infinite self-energy, whilst the spatial distribution of charge is not compatible with the theory of relativity. In the treatment of this problem, Dirac /T/ proposed a radically different theory for the motion of electrons. As is well known, the potentials A^ in the field theory are not unique. Normally, one restricts the arbitrariness in A^ by the condition

9 A = 0. (5)

Dirac argues that a more powerful electrodynamic theory can be built, whilst abandoning the above condition. In the process, the gauge

invariance is destroyed. However, as the existence of the gauge

transformations signifies that there are more variables in the theory

than necessary, from physical point of view, the lack of gauge

invariance is not considered to be a serious drawback. On the contrary,

these superfluous variables in the theory without charges, are then

used to introduce the charges, rather than bringing in new variables

to describe them, as is done in the classical theoryo

Realising that the classical electrodynamics is an approximate

theory, we shall concentrate on some of the ways put forward as a

possible generalization. These are presented in Chapter III. ¥e

shall discuss the modifications of Mie /8/ in his attempt to find the

field equations inside an electron. Further, an interesting theory

of Born /9/ and Born and Infeld /lO/ is reviewed. It is a non-linear

theory which, for reasons to be mentioned, did not become completely

successful. Another theory, which is relevant to the present

investigation, belongs to the class of scalar- theories and

has been developed by Brans and Dicke /ll/. We shall conclude that

Chapter III with the results of a very interesting paper by Rainich

/12/ where a possible way of geometrizing the electromagnetic field

is presented. That Chapter covers a wide range of different approaches

in investigation of topics closely related to the present study -

the concept of mass and a possible modification of the Maxwell Theory.

As the concept of mass is one of the issues examined here, it may be instructive to study it within the framework of the general

relativity. It will be seen, in Chapter IV, that the effect of a

medium on the electromagnetic wave closely resembles the effect of

the curved space on the same wave. This may suggest, then, that,

for some purposes, a medium may be replaced by a curved spaceo

Consequently, if the interaction between the electromagnetic field

and that (curved) space, however small, is taken into account, then

it seems plausible to say that photons have acquired a..non-vanishing

mass. Put otherwise, allowing an existence of a non-linear interaction

is seen as a possible way for mass generation.

A somewhat similar conclusion, regarding the mass of the photon,

was obtained by Kibble /13/. From the analogy between the refraction

of a beam of radiation by an electron cloud and of a beam of electrons

by the radiation field, his study reveals that the mass-squared shift

is proportional to the particle density divided by the particle

energy.

Let us look now at the same problems from a completely different

standpoint, viz., using the formalism of the gauge theories. It will

be shown that due to the spontaneous synmetry breaking, via the

Higgs mechanism, photons, under certain conditions, may be looked

upon as being massive. In addition, the question of mass generation

using the dynamical symmetry generation is also considered. These,

as well as the related issues are the subject of Chapter V.

In view of the results obtained, we shall reexamine the question

of the electromagnetic field energy-momentum tensor in a medium.

We emphasize here that the question of the total energy-momentum

tensor is not the object of the present analysis. Rather, the medium-

induced modifications to the radiation field are examined. It is

then concluded that the previous results /S, 1^/ remain, in essence, unchanged^ Hovever, in view of the extensive analysis, they may appear in a clearer perspective. The necessary details are to "be found in Chapter VI. It should he noticed that the experimental evidence is in complete agreement vith the theoretical conclusions of that Chapter - the fact vhich should not "be easily discarded.

The concluding Chapter summarizes the important points of this study and, "by bringing together different fields of , evaluates the new results. II THERMODYNAMICS OF ELECTROMAGNETIC WAVES IN A MEDIUM

As part of our investigation of the interaction "between the electromagnetic field and matter, concentrate now on the thermodynamics of the prohlem. First of all, it is necessary to study the consti- tutive laws, which characterize the particular medium in q_uestion.

Recalling the way they were introduced to physics, these constitutive relations are not universal laws, although most materials can l3e treated as "belonging to one of the mutually very different groups.

We restrict our attention to a system which, from the point of view of thermodynamiss5 behaves as a simple fluid. Such a system is readily found, e.g., one component plasma. The constitutive relations can be deduced from thermodynamics and describe the effect that the fluid has on the electromagnetic field. In addition, they reflect the influence of the electromagnetic field on the motion of the fluid, which obeys the conservation laws involving the energy-momentum tensor.

Conventionally, the constitutive relations are defined by the ratios of the electric E and magnetic H fields to the electric displace- ment D and magnetic induction B, respectively. In general, when directional effects become important, they form and permeability . As our study is restricted to simple media, these quantities are considered to be constants. Evidently, in reality they depend on many parameters, in particular, frequency, temperature, density, field strengths. Although they are defined as the ratios of quantities occurring in the macroscopic Maxwell equations

(which do not take into account the detailed structure of matter), their theoretical description is accomplished by the microscopic Maxwell equations (which, by restricting their region of applicability, consider the charges in vacuo, thus completely disregarding the ponderable medium). The microscopic theory is more fundamental and is supposed to hold everywhereo The macroscopic equations are, by their nature, statistic averages of the microscopic fields and charges. The connection between the two theories has been established by Lorentz

/15/.

In many cases, the wavelength A of the electromagnetic radiation is much greater than the interparticle distance, hence, it is possible to analyse the wave propagation in a medium from the point of view of the macroscopic theory. As a result, the discreteness of the constituents is' ignored and only the smeared-out, average fields are

considered. This brings us directly to the ambiguity involved in the definition of the permittivity. This question has attracted a lot of attention and there exists diversity of opinions /l6, IT, l8/.

The difficulty involves the local S^oc* ^^ assumed, on one hand,' that E"lO^ C is equal to the external field E-V, 'and , 'o n the other hand, that the contribution of the Lorentz polarization term,

^itP/3, must also be taken into account. However, due to the complexity of the problem, the resolution is not a trivial matter. A detailed analysis /IT/ reveals that either conclusion is viable and the choice depends on the type of a medium in question. Thus, inside matter, where electrons are neutralized by a positive continuum, the local field E, = E. Furthermore, a medium consisting of a set of bound -loc electrons experiences polarization, therefore, the Lorentz formula

2 = E + P/3 should be used in the derivation of dielectric »loc constant. The index of refraction n is connected with this constant via the relation p n^ = e y valid in the high frequency domain. In order to appreciate the restrictions of the following

calculations, it was considered useful to draw attention to a rather

unclear differentiation in the dielectric permittivity and, in turn,

the index of refraction. As a consequence, the following discussion

is valid, strictly speaking, for optical . In this case,

the dispersion relation for many different media approaches that of

a collisionless plasma, as the frequency of the electromagnetic wave

is, often, much larger than the characteristic frequency of a given

medium.

Next, let us turn to the important topic of the black body

radiation. It deals, as is well known, with the equilibrium spectral

distribution between the thermal radiation and matter in an evacuated

cavity. The presence of matter is necessary to satisfy the equilibrium

conditions and its amount is normally minute. When, however, the

cavity is filled with matter, it affects this spectral distribution

and it is quite plausible to suppose that the density of radiation in

a cavity now differs from the one in an evacuated cavity. Thus, if

e(a),T) is the mean energy of harmonic oscillators of frequency co at the

temperature T, then

(e-^"/^^ -1)-^ (1)

with K being the Boltzmann constant, so that the energy density

u(a)) of the radiation can be expressed by

u(w)daj = e(a3,T) g(a))dijo , (2) where g(a})dd) refers to the density of radiation states in the

frequency interval w to oj + dw . This density of states is usually defined /19/ in terms of the wave vector k, as

g(k)dk = k2 dk . (3) In order to arrive at the density of states, in terms of frequencies,

recall that the vave vector in a medium k is related to that in vacuo k hy

k = n k^ , ik) where n is the refractive index of a medium. When (3) is rewritten

in terms of the frequency, we have

g(a))da3 = TT~2 k2(a)) do) . (5) do)

In accordance with the above mentioned proposition to consider

radiation interacting with a simple fluid, e.g., a collisionless

electron plasma, the dispersion relation for the electromagnetic waves

in such a region is k2 = iii n2 , (6) c2 where

n2 = 1 - (T) with w^p = UiT e^N^/m denoting the collective oscillations of electrons.

It is now clear, from (6), that k is a function of frequency o) .

Therefore, in order to evaluate (5), calculate first the derivative as

dk(a3) _ H + iiL £ (5) do) c c ^ 2

using (6) and (T). Substitution of (8) into (5), utilizing (6), leads to "~_o2 ~ 3 . T iH ^ g(a))da) = TT c n^ n + —^ ~ do) or, in view of (7), 0)2 g(a))da) = n doo . (9) 2 3 The same expression can be foimd in literature /20/ and. is correct, providing one takes into account the dn/do) contribution.

As IS often done5 however, this dispersion term is neglected, thus resulting in a non-dispersive formula

g(a))d(ji) = n^ 0)^ dw/iT^c^ .

Now, if (9), together with (l), is inserted into (2), we obtain the following modified Planck law

uM = n u^(co) , (10) where corresponds to the vacuum value of the energy density.

This result was also obtained, by different route, by Bekefi Z^/ and by Dawson /2l/. Furthermore, Case and Chiu /22/ used the fluctuation-dissipation theorem to arrive at the same conclusion.

A modified form of the Planck law in anisotropic media was described by Cole /23/, which in the isotropic limit reduces to (lO).

As it is important to establish the correctness of the radiation energy density, we dwelt on that subject for a while. We are now in a position to present the generalization of the Brownian motion to include dielectric media. In the process, it will be confirmed that the exchanged momentum between the electromagnetic wave and a medium, in the ho) >> KT limit, is proportional to the refractive index.

The following investigation parallels the work of Einstein /h/ developed in his studies of emission and absorption processes. An attempt was made by Skobel'tsyn /2k/ to apply the method to optically dense media. His work, however, appears to be inconclusive. In the present treatment, the original (Einstein) method will be adapted to include simple media. The extension to other phases of matter is trivial, if the knowledge of the radiation energy density is assumed. It should be pointed out, in order to avoid any possible

misunderstanding, that the Einstein method is based on a general hypothesis regarding interaction of radiation with matter. Thus,

the objects interacting with radiation (the Einstein molecules)

have no special intrinsic properties and their role could be equiva-

lently taken over by the Planck oscillators. Here, these objects

•will be referred to as elementary systems.

During any absorption (or emission) process, a momentum is

transferred to an elementary system. The velocity distribution of

such systems are entirely due to their interactions with the radiation

field. Thus, this distribution must be the same as the one, these

systems attain through their mutual collisions alone. Hence, it is

necessary that the average knetic energy of such a system in the

radiation field be equal to J KT. Suppose now that we have an ensemble

of these elementary systems, each of which can be in one of the two

possible quantum states Z^ and Z^ with the corresponding

e and e . Next, let the transition from state Z to state Z be m n m n allowed. That transition will result if the system in state Z^ absorbs

a quantum of energy of magnitude ^^ ~ ^m ' Similarly, the transition

from Z to Z will result when the system emits that quantum of energy, n m

Wow, there are two different processes an elementary system can undergo,

in order to make a transition. First, there is a process independent

of the external radiation field, viz., the so called spontaneous

emission. The probability for such a process to occur in time dt is

d,P = A dt , (11) nm

where A is the Einstein "A" coefficient. Second, a transition nm which depends on the stimulation by the external field. Processes of this type are kno-wn as Induced radiation processes. Thus, the probability dP that a system under the influence of the radiation

density u(v) makes the transition from Z^ to Z^ by absorbing the

energy quantum is dP = B^ u(v) dt» Similarly, the probability

for the transition from Z to Z to occur is dP = B u(v)dt. n m nm Again, B and B are the Einstein "B" coefficients. Thus, in ° ' mn nm those two cases, the transition probability is the function of the

radiation energy density u(v) surrounding our elementary system.

In other words, as the system is in a plasma, u(v) refers to the

radiation density inside the latter. This density is given by (lO)

and its use is imperative. We shall prove below that this choice

does not disturb the thermodynamic equilibrium conditions.

Note that the approach used by Skobel'tsyn is quite

different. Originally, he assumed that the radiation energy density

in a dielectric is the same as the one in vacuo, but, in the same

article, he was forced to make an ad hoc assumption that density ought

to be u(v)/n for n > 1.

In view of the above discussion, therefore, the probabilities

for upward and downward transitions can now be written, respectively,

as

dP = n B U (V) dt; dP = n B u (v) dt. (12) mn o nm o

Hence, the probability for the radiation induced processes in a

collisionless plasma is decreased relative to vacuo. This is so,

because, as the plasma density increases, the refractive index

decreases and, consequently, the amount of radiation density is

diminished, in accordance with (10)„ We are now in a position to study the effect of the radiations field on the Brownian motion of test particles embedded in plasma.

In other words, if a particle emits or absorbs an energy quantum hv , then it experiences a recoil. If the wave is spherical, then there is no recoil. The question now arises, as to how is the magnitude of the exchanged momentum modified by the presence of plasma. To simplify the calculations and, at the same time, to make the physics of the problem more transparent, it will be assumed that the particle of mass M is moving with velocity v in the x-direction and that the conditions are such that the higher powers of v/c can be neglected.

In a radiation field, the linear momentum Mv of this particle changes, in time x, due to the effect of the radiation force f = Rv,

R being a coefficient to be evaluated below. In addition, during the same time, the fluctuations of the action of radiation on the particle, cause a A to the latter. The magnitude and the direction of this momentum transfer are constantly changing, as a result of the fluctuations of the radiation pressure. As the radiation-, field must not affect the overall velocity distribution in a thermal equilibrium, the average momentum of the particle, at both ends of the time interval x , must be < Mv > . Therefore, we must have

<(Mv - RVT +A) > = <(MV)2 >. (13)

Furthermore, in order not to disturb this equilibrium, we shall assume the validity of the equipartition law, Mv2/2 =KT/2. If, in addition, terms containing T^ are dropped as being small and vA set equal to zero, since v and A are uncorrelated, then the above equation reduces to a2 = 2RkT ; T

This is the fundamental equation required by the theory of heat and a comparison with /U/ affirms that the presence of a plasma does not affect it.

Nov, A^/t and R are calculated and then inserted in (l^), to see if that relation is satisfied® Ho-wever, in order to calculate these quantities, using the method devised by Einstein, it is necessary to know the radiation energy density and the exchanged momentum. The former is given by (lO) and, as far as the momentum is concerned, "we assume it to be proportional to the refractive index of the surrounding medium. Then, if the equality (1^+) holds, our momentum assumption will be justified.

In order to determine the propagation of light in moving media, it is only required to apply the to the corresponding laws for stationary body ~ a prescription given by e.g., Pauli /25/. As the radiation laws are defined only for the inertial frame moving with the medium, in calculating constant R, the above prescription is followed. To distinguish the quantities referring to the rest frame, they will be marked with a dash. Thus, radiation per unit volume, in the frequency range dv, within a solid angle dSl , transforms according to

u'(v',(l)') = u(v) (1 - 2nBcos (D), (15) where 3= v/c and depends on the direction defined by the angle (|)' it makes with the x'-axis, and the angle ip' between its projection in the y*z*-plane and the y'-axiso To facilitate the transformtion of the argument of u'(v',(t)M, the transformation formulae for the Doppler effect and the aberration are needed. These, however, are veil known in the theory of relativity and, in the approximation applicable here, take the forms

v' = v(l - n 3cos (j) ) (16) and

cos ({)' = cos (j) - n 3 + n 3 cos^ (f) , (IT) respectively. Further, no relevant information is lost if we put

il^' = ilJ . (18)

The inverse transformation to that in (16) is

V = v'(l + n 3cos ([)') (19) which, on substitution in u(v) and subsequent expansion, gives

u(v) = u(v') + V n 3cos (|)' ^ (20)

A solid angle dO, is (in the present approximation) connected with

by

d^^/d^^' =1-2 n3cos cf)' 3 (21)

Thus, the substitution of (l9), (20), and (2l) in (15) results in

u(v') + v' n3cos cj)' ^^— (1-3 n 3cos (1)0. (22)

Making lise of the last relation and the assumption about the elementary processes, discussed earlier, allows the calculation of the average momentum transferred, per unit time, to the test particle.

Let, then, the number of elementary transitions from level m to level n be mn where ^ ./KT S = w e -^J^T^ + V e -en- m n

and w^, w^ are the statistical weights corresponding, respectively,

to levels m and n. An analogous form holds for the transition from

level n to level m.

As assumed earlier, the momentum of the exchanged photon in

plasma is given by the Minkowski relation. Therefore, in each

elementary process, the momentum nhv« cos cf)'/c is transferred to a

particle. Combining the upward and downward transitions, the total

momentum transferred to the particle is then expressed, using (23),

as

nhv (Scr^w B (e-^^/^^ u'(v',(j)')cos (j)' df^'/^"^- (2U) m mn

Substituting (22) into (2U) and putting Q = w^ B^^ g'^m/i^T ^

the total average momentum transferred, per unit time, to a particle

through the induced processes can be written as

n^ hv 9u(v) -hv/KT X u(vf )\ ^ Q(l - e ). (25)

For the sake of clarity, the dashes over symbols have been dropped here.

As the spontaneous emission does not have any preferred direction, the average momentum transfer, due to processes of this type, vanishes.

Consequently, the coefficient R can be written, finally, as

n^ hv / \ V 3u(v) R = u -o —f— Q(1 - ) . (26) J 9v Prior to utilizing this result, let us inquire atout the effect of the presence of a plasm on the momentum fluctuation ^ . Denote

"by 6 the momentum transferred to the particle during a single process and assume that its sign and magnitude vary in time in such

a way that the average value of 6 vanishes. Now, if refers to

the transfer of momentum through the i-th independent process, then the resultant transfer is A Because the average value of

each vanishes, it is necessary to write (assuming =5 )

F"= n"^ , (27) where N is the number of all elementary transitions in time T .

If the momentum corresponding to each elementary process 6= nhv cos

is substituted into (27), it then yields

F'= I ( . (28)

As the number of induced processes from level m to level n, in time T , in equilihrium, is N/2, it is, then, possible to write

N = 2 Q U(V)T , which, on combining with (28), gives

'E = I ( nhv , (29) T 3 C

As previously emphasized, the expressions (26) and (29) are

determned in terms of the radiation energy density u(v) which,

evidently, differs from its vacuum value u^(v). To show the validity of the claim that the momentum of the exchanged photon

inside plasma is proportional to the refractive index of the latter, we must show that the thermodynamic equilibrium is not disturbed when the radiation momentum is transferred to a particle. This, in turn, will "be so if, after calculating R and ^ in terms of the proposed energy and momentum densities of radiation, their subsequent insertion in (ll+) yields an identity. Therefore, let us express R in terms of vacuum energy density ^Q(v) and substitute the modified Planck law

(lO) into (26). The step by step evaluation proceeds as follows:

3u(v) 9 ( ^ ( ^ f \ 9n(v) ^ , . ^""o^"^^ where f \ Stt h v^ f hv/KT ^ s-1 (31)

Then, substitute (3l) in (30) and calculate the individual terms

n(v) _ Stt h 1 - nMv) P" hv /k T , n(v) e -1 and hv/KT ^^o^^^ &T h ne n(v) 3n - ~ hv/KT hv/KT , kT -1 e -1 and hv/KT 1 -n^ , n e )L ^jd^Ozz ^ h v^ — + n - 3 9v hv/KT 3n kT -1

Finally, hv/KT 8it h V- l-i^ ^ ne n - "r n+ —• u(v ) - - c3 hv/KT , 3n kT 8v e -1

Substituting for the square bracket in (26), the above expression leads to hv/KT , nhv ^o^"^ hv e -hv/KT + n^-1 Q 1 - e . (32) "" KT ehv/KT_^

In order that the Brownian particle is not internally affected by the radiation field, hv = - e^ must be much larger than kT. Effectively, restricting our considerations to the region of validity of the Wien Law, the above reduces to

In addition, when (lO) is inserted in (29), it results in

— ) ^o(v) Q. (3U)

Finally, if (33) and (3^) are substituted into (lU), the identity is, indeed, obtained. Therefore, it can be concluded that the Minkowski form of the momentum density, used in the preceding analysis, appears to be satisfactory as it fulfills the condition for the thermal equilibrium, at least in the region where hv >> KT, when surrounded by a collisionless plasma. Ill. ALTERNATIVE THEORIES

Large part of the present investigation is concerned with the concept of mass and a possible modification of the classical electro- dynamics. Therefore, it seems appropriate to review some of the earlier studies, which hear some relation to the present work. In doing so, the conceptual background will be discussed, which should prove useful for our purposes.

As already pointed out, one of the aims of this work is the study of the feasibility of generalization of the Maxwell equations when the electromagnetic wave propagates in a non-empty space. As will be seen in what follows, this is closely connected with the mass generation problem.

The first theory to be mentioned here, was proposed by Mie /8/.

As is well accepted. The Maxwell and Lorentz theories do not hold

inside the electron. Therefore, Mie has attempted to generalise the electrodynamics in such a way that the repulsive Coulomb forces are balanced inside an electron, whilst the field in the region outside the electron is unaffected by the proposed modifications.

It turns out, according to Mie, that it is possible to generate the mass from the field. Thus, the first set of the Maxwell equations

8 F^ + F + 8 = 0 (1) a 3y 6 yc' Y with F describing the electromagnetic field tensor, together a3 with the continuity condition

=0 (2) with denoting the four-, is retained in that theory. It follows from (2) that there must exist an asymmetric tensor H^^ such that

= H^'^ . (3)

In the Lorentz theory it is assumed that H^^ = F^^ .

According to Mie, however, H^^ has a real significance and its components are universal functions of F^^ and ,

In addition, the four-current density is then given "by

j™ = S„ (F^^ , ). (5)

LX

In view of these assumptions, the field equations acquire a new physical content. Thus, relations (k) differ from those of conventional electrodynamics, as H^^ now depends explicitly on (f)^.

That is, not only the potential difference, but also the potential itself acquires a real existence. By virtue of the above assumption,

(U) and (5)9 Mie introduced in the theory ten universal functions.

However, by invoking the energy principle, these unknown can be reduced to a single one, resulting in a noticeable simplification.

This unknown is often called the world function ). If this invariant function exists, then it is true to say that (6) 3 F d cf) yv a and, consequently, we can write

d L = H^"" (SF^^ - 2/ fi (1)^ ' (T)

One of the difficulties encountered at this stage is the determination of L. Quite generally, the only invariants that can be formed from

F^^ and are: the square of the vector A — ({) d)^' ; the square a oi of the of the tensor "1g ^ cxB ' square of the vector p ^ J 3 ^ag"^ ~ ^ ^ ^ » ^^^ square of the tensor of the fourth order with components E F^^ . It is then true to assert that

•L must he expressible in terms of these four invariants. The problem

of matter in the theory of Mie can, thus, he replaced by the

determination of the appropriate expression for L, As is well known,

if L is set eq.ual to the second of the above invariants, the Mie

theory reduces to the electron theory in the space containing no

charges. Evidently, L must be different from this invariant inside

the electron.

Turn the attention now to the determination of the energy-

momentum tensor Sd p„ . Using the notation of the general theory of relativity, together with the variation of g0 1^ p , Weyl /26/ simplified the calculations of Mie, who was using, obviously, the special theory

of relativity. Weyl showed that 0(> and F0( i_ ( D remain ordinary tensors, whilst and should be replaced by tensor densities Y and otfi H , respectively. Consequently, we have to write

8 Y^" = 0 (8) a' and

y" = . (9)

P

The action function is now written as

dx =0 (10)

and it is used to determine the energy tensor S^^ . Since L

is now independent of the derivatives of the g^^ , we can write

a3 6 L = L ^ 6 g otp > and, in turn,

S^ = AA^ g.^^ - I L , (11) where the assumption of a constant electromagnetic field is implied.

Now, if the variation of the field quantities with respect to the

infinitesimal coordinate transformations is substituted into the

general expression for 6L , the latter must vanish, resulting in

725/

9 -Zj BY BY B .V S ' = H ^ae" J ^a • (12) 3 g

A subsequent substitution into (ll) leads to the expression for

the energy tensor, already obtained by Mie using a different method,

S^ = H^^F . jS -^L . (13) a aB ^a 2 a ^

R Now, knowing the energy tensor S*^ , it is possible to determine the cx rest mass of the particle from

dT , (lit) "o = -

where d T is the spatial volume. In addition, using the mixed

B

components of S^ , the can also be determined within

the framework of the Mie theory.

Let us now discuss some of the reasons why this theory has not

succeeded. As it has not been possible to establish the form of

the world function Z which corresponds to the reality, the theory

allows the existence of elementary particles with arbitrary value of

their total charge. It may be possible, in principle, to choose

other expression for L, which could well, correlate with experiment, however, this situation is unsatisfactory, as we have no method of

determining the world function. Another difficulty, already pointed

out by Mie, seems to follow from his very assumption. As in the

Mie electrodynamics the absolute value of the potential plays an

important role, it follows that if cf) is a solution (the electrostatic potential) of the field equations, <\>+ constant does not satisfy them. To put it in other words, a massive particle cannot exist in a static external potential field. This is, perhaps, the main reason leading to a downfall of the Mie theory.

Another interesting proposal for the nonlinear theory of the electromagnetic field was put forward hy Born /9/ and Born and Infeld

/lO/. Recall that the field equations are determined "by the principle of the least action. The Lagrangian of the Maxwell theory

( L = - E^)) leads to infinities which violate the principle of finiteness. In order to restore it, the assumption of the upper limit of the field strength is invoked, resulting in the Born modification of the action function

1 + ^ (h2 - E2) - 1 (15) where 3 has the of a field strength and may be called the absalute field. In order to take the guess out of this expression,

they /9jlO/ start from the principle of general covariance, i.e., requiring that all laws of nature are covariant under all space-time transformations. Thus, it is necessary to find a transformation group which is larger than the Lorentz group of the special theory of relativity. They proceed to show that the Lagrangian (15) belongs, indeed, to the group of . Consequently, the relation between the Born theory and that of Maxwell is analogous to that between the mechanics of Einstein and Newton. Thus, having found the Lagrangian density, the field equations as well as the conservation laws can be determined. From the action principle, the field equations are

3 /^F^l'^ =0 ^ (16) 3 H*^^ = 0 , V where F*^^ is the dual tensor and the relation between H^^ and

F^^ is similar to that of the dielectric displacement and magnetic induction to the field strengths in the Maxwell theory. In order to gain some insight into this theory, consider the electrostatic field. It is then found /lO/ that the solution for the potential is

= f- f(^) , (17) o o where dy . _ /e f(x) = v^TT y- X

Thus, (17) represents a modification to the Coulomb law, which, of course, regains its established form if r >> r^. The advantage of the potential (IT) is its finiteness as x ^ 0. It is found that the potential has the maximum at this point which is given by

(l)(o) = 1.85 e/r^, (19)

Using the conservation laws, the field mass is determined. Its value is found to be 2 (20) in = 1.21+ e ' which is finite. In order to complete the theory, the value of the critical field 3 is calculated. In view of (20), this value is

3 = 9.18 x 10 e.s.u. (21)

This large magnitude of the critical field justifies the use of the

Maxwell equations in most cases. However, for the wavelengths of the order of r^, the modifications are mandatory.

The philosophical superiority of the theory which assumes only one physical entity, viz., the electromagnetic field, resulting in a mass being a secondary (derived) concept, is unquestioned. Yet, the reason why this theory has not been generally accepted can be traced to its unsuitaloility to generalization encompassing the quantum domain.

According to the Mach principle, as is well known, the only meaningful motion is that relative to the rest of the matter in the Universe. There are two types of fields which transmit the long-range forces, viz., the gravitational and electromagnetic fields. In line with the Mach principle. Brans and Dicke /ll/ built a theory assuming the existence of still another kind of long-range forces, produced by scalar fields. Their theory is a generalization of the general theory of relativity. The resulting gravitational effects are in part geometrical and in part due to the postulated scalar interaction. The inertial masses of the elementary particles are not the fundamental constants, but arise from their interaction with the surrounding field. Consequently, the particle's mass can be determined by measuring the gravitational acceleration Gm/r^ .

Put otherwise, the gravitational constant G is related to the scalar field c|), which is, in turn, coupled to the mass density in the

Universe.

167T G L + R d^x = 0 , (22) where L is the Lagrangian density and R is the scalar of curvature.

The proposed generalization of the above takes the form

1677 L + (1)R - 03(8 (j) 9°' (f)/(j)) d'^x = 0, (23) UC where to is a dimensionless constant and it is assumed that G ^ is proportional to (j) so that the Lagrangian density of this scalar field is included in (23). Now, in order to obtain the wave equations for (p, we vary cj) and (j) in (23)» This equation can be expressed as 3 + 2w ^^ where • is the generally covariant d'Alembertian. Thus, the source term for the generation of (j) waves is expressed by the of the energy-momentum tensor T of matter alone, what agrees with the content of the Mach principle.

Further, in order to find the field equations for the metric field, the and the corresponding first derivatives are varied this time, leading to

^v - ^ Syv « = - "F "uv - F ^'a^^e^ - ^ Sv > '

In this equation the terms on the RHS have the following meaning: the first term denotes the source, as in the relativity theory; the second term refers to the energy-momentum tensor of the scalar field; the last term arises from the existence of the second derivatives of the metric tensor. To see the point of contact with the Einstein theory, consider the limit w °° . The relations

(2^+) and (25) then reduce to the well known results

C] (1) = 0 (26) R -is R = Stt G T,^, . yv ^yv yv

The theory of Brans and Dicke is not widely accepted.

Although it can explain the deflection of light, the gravitational redshift, and the perihelion rotation of the orbit of Mercury, the computed results differ from those of the general relativity. Yet, as there is not a sufficient experimental accuracy to determine the 31o correct theory, it is too early to choose -between the two. It is also very difficult to establish the numerical value of the coupling parameter w,

It seems pertinent, at this point, to mention an interesting conclusion of the Brans-Dicke theory: It predicts the time- dependence of the gravitational constant. Although a rather unconventional result, it seems to he based on sound physical reasoning.

It follows from the argument that in the Mach picture there is no direct way in which the mass of two particles at different space- time points can be compared. However, there is no ambiguity if the mass ratios are involved. Thus, using the characteristic mass of , ("fec/G)^, the mass ratio of a particle /27/

m(G/|Eic)^ (27) should be uniq.uely defined and, further, provides the means to compare the masses at different space-time points. Now, if fe and c are assumed to be constants and, in addition, if the number

(27) varies, then it may be asserted that either m or G, or both, are space-time dependent. In principle, there is no difference between the theories involving constant mass of a particle or the constant value of G, although they may lead, formally, to a different structure of the theory. The metric tensors, corresponding to either alternative, are then also different /ll/ and their relation is expressed by a confomal transformation.

Finally, we shall recapitulate here a very interesting proposition, put forward by Rainich /12/ and, later, further developed by Misner and Wheeler /28/ in their study of the space theory matter.

Thus, there exist two basic tensors commonly used in physics: the electromagnetic field tensor and the Riemann tensor. The former comlDines, in the domain of the special theory of relativity, the electric and vectors, whilst the latter determines the curvature of the space which, in turn, accounts for the gravita- tional phenomena. Clearly, these tensors have quite different roles. Thus, from the curvature of space, described by the Riemann tensor, one can deduce the gravitational properties - we say that the gravitation can be geometrized. Qn the other hand, the electro- magnetic tensor is commonly understood to be superimposed on this space and is treated independently of the space. There were several attempts to include the electromagnetic tensor in the geometry of

space, of which the most successful appears to be the model proposed by Weyl /26/. In essence, the Riemann tensor was replaced by a more general curvature tensor, describing a different space geometry.

It was then conjectured that one part of this tensor belongs to the gravitational domain, whilst the other one describes the realm of .

Not wanting to change the curvature tensor, Rainich /12/ turned his attention to the energy-momentum tensor which connects the electromagnetic tensor with the Riemann tensor. As a consequence of that investigation, it became clear that under certain (Rainich) assumptions the electromagnetic field is completely determined by the space curvature. In a nutshell, Rainich starts by comparing the electromagnetic and Riemann tensors at a given point, without taking into account the appropriate tensor fields. Put otherwise, the algebraic properties of the tensors of the second rank are analyzed first. As is well known from the special theory of relativity, we have three types of vectors: time-like, null, and space-like, depending on the value of their scalar product. Further- more, there are three kinds of planes: those containing two-, one-,

and none- of the null vectors. It is then argued that only the first

and the last case have any connection with reality, as the two

corresponding invariants of the tensor are not simultaneously zero.

Now, when the four, mutually perpendicular unit vectors i, j, k, 1,

with the square of each of the length +1., are introduced, the

electromagnetic tensor can "be written in the form

f(x) = Xri(jx) - j(ixn + yfkCjx) - ;^(kx)'l , (28) — -L — where f(x) denotes F x^ , and y and X are real and imaginary ot numbers, respectively.

Next, according to the general theory of relativity, the

energy-momentum tensor at a given point can be obtained from the

Ricci tehsor This tensor is given by - g g^'^ R, where

R is the scalar of curvature. If there is no matter in the region

in question, then the just mentioned tensor must be equal to the

electromagnetic energy-momentum tensor, leading to the equation

= FV (29)

This is a clear indication that the electromagnetic energy tensor

can be expressed in terms of the geometry of the surrounding space.

This relationship is further studied by Rainich. He then finds

that if (28) is to satisfy (29), the constants X and y satisfy

X sin (p y = cov/2 cos cj) , (30)

where

20)2= It now emerges that the electromagnetic tensor is not determined hy the Riemann tensor uniquely, as the scalar (f> remains arbitrary.

Put otherwise, in view of (30), there is an infinite number of electromagnetic tensors satisfying the given curvature tensor.

Therefore,

in order to complete the proposed theory, it is necessary to eliminate this arbitrariness. Recall that so far only the algebraic properties of tensors were considered. Now, by studying the differential properties of the energy-momentum tensor, it is possible /12/ to reduce this arbitrariness to one constant of integration. In order to remove even this ambiguity, the analogy and difference between the theory of electromagnetic field and the theory of analytic functions of complex variables are analyzed. It is then found that, assuming the Rainich conditions are satisfied, the electromagnetic field is completely deterinined by the curvature of the space-time. The Maxwell equations thus reduce to geometric expressions connecting the Ricci tensor and its rate of change.

Developing the idea of Rainich further, Misner and Wheeler

/28/ allowed for multiple connected topologies and showed their compatability with the Riemann space. The charge is then interpreted in terms of source free Maxwell field in vacuo which is, however, trapped in the "worm holes". In addition, the gravitational mass originates entirely in the energy of the electromagnetic field.

Therefore, mass, as well as charge, are certain representations of the concept of curved space. This theory is not without difficulties, as it can explain only irasses larger than kg - and these magnitudes are, simply, unknown. Elementary particles and real masses are excluded, on the assumption as belonging to the realm of micro- physics . IV GENERALLY CO VARIANT APPROACH

In essence, "we si:ud.y tlie interaction of elecbromagnetic waves with, matter. Commonly, this phenomenon is dealt with in a linear approximation and in many cases that is sufficient to explain the results of experiments. However, from the theoretical point of view, the linear theory is only an approximation which, strictly speaking,

should be replaced "by a more general, nonlinear theory. This idea has already been touched upon by the present author /29/ and is

further developed here. It stemmed from an inquiry into the feasibility of the assumptions /3/ used to describe the relative merit of the

Abraham and Minkowski tensors. The premise of that work, viz., that an electromagnetic wave propagating through a medium is

influenced by the latter and that this effect is manifested via the

generation of massive photons, is further analyzed.

This brings us to the most familiar, yet, the least understood

concept in physics. I am referring to the notion of mass. Our

comprehension of this issue is minute, although, recently, some

information has been obtained from elementary particle theory.

An excellent monograph on the subject of mass, from a general

standpoint, has been written by Jammer /30/, who not only covers the historical development of this concept but also explains the

difficulties encountered, at the time of writing. Therefore, we

shall not review the related literature, as it would not be possible to add anything of substance to Jammer's book, but concentrate,

instead, on the more recent developments, with the aim of bringing a new insight to the old issue. The idea to be elucidated bears some resemblance to the ^ch principle. It will be recalled that the latter suggests that, roughly speaking, the mass of a body is due to the influence of all stars in the Universe on that body. Expressed in mathematical terms, this dependence takes the form

m(x) =5 Z(x) , (1) where m(x) is the mass of the particle, C - a coupling constant and

Z(x) is the mass generating field. The central idea is, therefore, not to consider the mass of a particle as an intrinsic, fixed quantity but, rather, to think of it as a variable quantity depending on the field in which it moves.

The Mach principle is not universally accepted and opinions about its validity differ /3l/. There are many who dismiss this idea altogether on the that it is not testable. On the other hand, as the general theory of relativity is built on this concept, and some of the predictions of that theory were experimentally confirmed, it is difficult to see that the connection between the two could be coincidental. In any case, we shall not go into this debate here, but simply assume the validity (in the absence of any proof to the contrary) of the relation (l).

To take the effect of all stars in the Universe, Einstein introduced the concept of curved space into physics. Let us start, then, with modifications of electrodynamics in such a space.

Using the generally covariant expressions, we can write the Maxwell equations in an arbitrary system of coordinates as /32/

3F +9 =0 Va3 a 3y 3 ya (2) 1 / -^013 -01 = g F ^ = J , / where F^'^ is the electromagnetic field tensor and ^^ is the four-current density. Clearly, in every local inertial frame the above reduce to the standard Maxwell eq.uations in vacuo. On account of simplifications, introduce a system of coordinates (also known as time-orthogonal system) in which

(3)

In addition, introduce the antisymmetric tensors and B by

and the vectors if' and E^' by

pa^ = ^ = . (5)

The corresponding covariant expressions are

F = g g F^^ = g g., = B = (Ua) ^kk

= Say = = Pa = ^a ' ^^a)

This allows us to write the Maxwell equations (2) in the form

-i (/=iD") = (6)

-is (^D") = P .

In order to rewrite these in a more familiar form, introduce the

dual vectors B and H corresponding to the tensors B^^ and h"® by the standard prescription, resulting in

H^ = h23 H^ = H^i H3 =

Finally, on substituting (T) into (6), ve obtain the Maxwell equations in the form

V X E = - 3 (/IJ B). V X H - -i. 8 (V^D) = PU /-g /_g (8) V«B = 0 V • D = p o

The connection between the electromagnetic waves in a medium and the same waves in vacuo in a curved space now emerges. Thus, if g in the above equations does not depend on time, and if we put

e = y = , (9) then the field, responsible for the space curvature, behaves like a medium with the dielectric permittivity and magnetic susceptibility given by (9). In the following, it will be assumed that inverse of this statement is also valid. In other words, in studying the electro- magnetic wave propagating in a dielectric medium, one may, equivalently, replace that medium by vacuo, providing a.non-vanishing space-time curvature is simultaneously introducedo

In support of the above idea, investigate the effect of the intense electromagnetic field on the geometry of space-time. Thus, if the Minkowski metric is denoted by n^^ , the metric produced by the intense electromagnetic field can then be written as

g (r,t) = n + h , (r,t) . (10) The perturbation h^^ can be obtained from the linearized

Einstein equations /32/

P = - - 5 ' ^^^^ where G denotes the gravitational constant and T ^ is the energy- momentum tensor defined by

T = i- (F ^ F - J n J^^F J. (12) Viv ^^ y vy yv a6

Now, to calculate the perturbation h (r,t) consider a simple yv ~ example. Thus, an intense radiation beam normally incident on a mirror is reflected and travels in the same line with its direction reversed. For the argument's sake, this process may be imagined as a head-on collision of a particle with vanishing mass m (representing an electromagnetic wave) and a very massive particle of mass M

(describing the mirror). Fin'thermore, if the initial four^omentum p^^ of a particle with mass m has the components

^mi ~ ^ ^^^ ^ ' (I3a) then the final momentum is given by

pV^ = (E , 0, - p sin 0, -p cos 0 ). (l3b) mi m

Similarly, the initial and final four-momenta corresponding to the other particle are, respectively,

Pjk " ' ^ 0, - p cos 0 ) and

^Mf = ^^M' P P cos 0 )

In addition, if the involved particles are assumed to be point-like, then the appropriate energy-momentum tensor is given by /33/ i^o.

= ^mi ^mi {r - Y . t)0(-t) + ^mf^ sHr - v ^ t)e(t) + E . E mi mf

m mi pL P." (15) E . 63(r-y t)e(-t) + ^^^ 63( t)e(t), mi Mf ~ ""

"Where, e.g., E^^ describes the final energy of a particle with mass

M, V denotes the appropriate velocity, and 6(t) is the step function.

Next, as is well known /33/, the solution of (ll) is given by

f S (r',t - r - r yv ~' - ), h (r,t) = kG d^r' (16) y V ~ ' r - r' where

S (r,t) = T (r,t) - ij, T^ . (IT)

At distances much greater than the soiirce dimensions, the above

can be approximated by

h (r,t) S d^ + C.C., (18) yv ~' r yv ~ where S (k,(n) is the Fourier transform of S (r,t) given by yv ~ yv 1 S (r,t) = S (k,,,) e ^^^ d3k. (19) dco yv ~

Utilizing (15), (iT), and (l8) and performing the required operations,

we find that the metric perturbation is given by

/ . \ U / • \ §11 ^ p (ml) ^v(mi} yv e(r-t) + E . (1 - V . cos e) mi -ml

P P. - ^^ •P.(mf) ^v(mf) (t-r) + E „ (1 - V . cos 0 ) mf mf

M2 Py(Mf) ^^v(Mf)

V - iMi ^^^ ® ^ Ul.

Far away from the source, all components of h^^Cjst) except h^^

and h^^ (= - ^^^ ^^ transformed away /33/ and (20) reduces to

2G h^^ (r,t) = 0(r-t) E^ - p cos e E^^ + p cos

(21) M' e(t-r) E^ + p COS 6 - p cos 6

h,. (r,t = 0) '12

We can, therefore, conclude that, although using a rather crude example, an intense radiation field introduces a non-zero curvature to the space-time metric. Certainly, a more realistic example could "be chosen, "but the additional complexity of the calculation did not "Warrant it, as we were only interested in finding out if the metric pertur"bation is finite. Consequently, the following working hypothesis is proposed: as the intense electromagnetic wave

generates a curvature of the space-time, it must have an active

gravitational energy. Let us investigate if such an assertion can

"be incorporated in the standard theoretical framework.

In order to arrive at the Maxwell equations in a curved space,

the Principle of general covariance is invoked and applied to the

corresponding equations in the flat space. However, terms

containing the curvature tensor are normally neglected. Let us,

therefore, study what will happen if these terms are included in the

theory. Thus, consider a space-time , described hy the

metric tensor g . In addition, allow an electromagnetic wave, ]-l V characterised "by the four-potential A^ , to travel through and, also. h2. interact with that . Furthermore, for the sake of simplicity, assume that the electromgnetic wave is weak and does not perturb the metric. The Lagrangian for this system can then he written as

where the first term refers to the free electromagnetic field, described by the electromagnetic tensor F^^ , the second term is the

Lagrangian of the curved space with R^^ being the Ricci tensor, and the last term specifies the coupling, with X being the proportionality constant.

We now enquire as to the form of the field equations which are determined by the above Lagrangian density L . For this purpose, a variational calculus is employed. In order to use the Euler-

Lagrange equations of motion

gV _ = 0 , (23)

the variation of (22) is needed. Ihus, if g^^ are subject to the variation

a: + 6 g ,

^yv ^yv then the relevant transformation formulae are required. Ihese, however, can be calculated /25, 33/» We shall show here explicitly the calculation of SF^""; other transformation formulae will only be stated. Thus,

apUV ^ gVB) ^ ^^^ g^^ (2lt) From the equality ) = o, it follows that

. vg VY e|

Substitution of (25) into (2l|) leads to

^^ - S - F g^ . (26)

The folio-wing are also required

= I g^^ (27)

and

6R = 0. (28)

Consequently, from the knowledge of the results involved in the

variational calculus, we obtain, after some algebra, the equations of

motion in the form

G +XA2G +yRAA+YA2 +YA2 =-KT . (2Q) yv ^ yv ^ y V ^ ;y;v Sv ^ ;y;v yv ^^^

The meaning of the symbols in the above equation is as follows:

G - the , semicolon denotes the , yv ' ' and = A A^. Taking the trace of that equation result in

R = -3 xQA^. (30)

On the other hand, if the four-potential A^ are varied in the Lagrangian density, then the modified Maxwell equations are obtained

F^*^ + ^ R A'' = 0. (31) ;v K

Substituting (30) into (3l) leads to

pyv _ ^2) A^ = 0, (32) ;v K and the resulting nonlinearity, due to the coupling of the electromagnetic radiation with the curvature of space-time, is clearly exhibited. Let us now reflect on the intermediate results. First of all, it has been shown that an electromagnetic wave in a curved space can also "be treated as if it were in a medium. As a consequence, it was assumed that the inverse statement is also valid, viz., that a medium through which the electromagnetic wave travels may be replaced by a curved space. In addition, it was shown that if a medium is replaced by a high intensity radiation, the Minkowski metric is perturbed, thus, again, leading to a deviation from a flat space. There, it appears that, for our purposes, the exact nature of the medium was not required as long as it was assumed that the relevant region was curved.

Subsequently, some questions involving the radiation-medium interaction may be replaced by the appropriate studies including the interaction with the curvature of the space-time. Obviously, this kind of problem is rather complex as it introduces nonlinear terms in the field equations.

A possible way of deriving the field equations is through the calculus of variations, as already indicated. Starting with the

Lagrangian density of the form

L = i R, (33) one is led to the

R -Jg R = -8ttGT. (3^) yv ^ ^yv yv

These are nonlinear partial differential equations. As the gravitational fields carry energy and momentum, the effect of gravitation on itself is represented by the nonlinear terms. This may be contrasted with the Maxwell equations, which are linear.

Equations (S^l) represent ten apparently independent relations.

However, in view of the Bianchi identities (R^^ - I g^-R) ^^^ = 0, (35) the situation is different. There are, in fact only six independent equations, so that four out of ten unknowns are undetermined. This is the result of a general coordinate transformation involving four arbitrary functions .'). There is an analogous ambiguity regarding the four-potential in the Maxwell theory.

To elucidate the meaning of the Einstein equations, write the metric tensor as

S yv = n...pv. + h^.yv. . (36) The Ricci tensor, linear in h^^ , is then written as /33/

h^ X _ iIaS .ifjLUL 4. hyK (37) yK 9 x^ 9x^ 9x^ 9x A

Then, if y K is defined as

t = 1 (R - 1 g R^ _ + In ) , (38) yK 8ttG ^ yK ^ ^yK X yK yK X the exact Einstein equations can be written as

R^l^- Jn R^^n = - SttG (T + t ). (39) yK W ^ yK yK The interesting feature of the theory now emerges: as t^^^ may be interpreted as the energy-momentum pseudo-tensor of the gravitational field, the RHS of (39) - the source term - depends on the values of the components of the metrical tensor g^^ .

Turn the attention now to the electrodynamics of massive spin one particles. Again, starting with the Lagrangian formalism, one arrives at the equations of motion. Thus, if the Lagrangian density is of the form

L = - P l67r a3 Btt a

also known as the Proca Lagrangian, then the equations of motion have h6.

the form

8 .F^^ + A^ = 0 , where the is associated with the rest mass of a

particle hy y = m^c/t . As is well known, the Proca equations

describe a wave aspect of massive spin one field. The formulation

of such a field, in terms of its particle properties, is available

and was first worked out by Duffin /3h/ and Keraner /35/. Guided by

the similarity with the Dirac equation of electron, they arrived at

d^ ^^ ilj + ip = 0 (k2)

together with the commutation rules for B's

533+333= 3 6 +3

These are, however, nothing else but the 10 x 10 matrices. In analogy

to the Dirac theory, we therefore define a ten component in

terms of the field strengths E and H and, also, include the term

proportional to the four-potential. Hence, if the spinor has the form

ip = (E^, H^, - y -yV)^ k = 1,2,3

then we can write (k2) in the form

0 0 0 0 curl 0 \ + y I ip = 0, - curl 0 0 0 0 0 -uV A

where I is 10 x 10 unit and the following substitutions were

used 0 82 0 0 -^3 0 0 ; curl = 3 0 3t 3 ^t = 0 0 0 ^t UT. When y is set equal to zero, then (1+2) reduces to the free JVlaxwell equations. There is, however, a fundamental difference between the massless and jmssive electrodynaioics, as the latter is not gauge invariant. As is well known, there are two types of gauge invariance in electrodynamics: global gauge invariance, expressed as

ifi(x) —^ •i/;(x) e^®^ with X being a constant, and local gauge invariance

^ieX(x) ^ (U6) where e denotes the charge of a particle and A is now a function of space-time. As we shall return to the gauge theories later, it suffices to say now that the physical meaning underlying (1+5) and (k6) is rather different. Thus, the global invariance is closely related to the and, consequently, its violation undermines the very basics of the theory. On the other hand, a consistent theory is available even if the local gauge invariance is broken.

Let us now investigate the similarities between the Maxwell equations in a curved space (allowing for their interaction with the curvature) and the Proca equations in the Minkowski (flat) space.

Clearly, in both cases we are dealing with ten algebraic equations.

However, whilst the Proca equations obey the Lorentz invariance, the gravitational field equations do not; they are invariant under a broader group of all point transformations. In addition, due to the occurrence of mass term in the Proca equations, these are not gauge- invariant, whilst the gauge invariance holds for the Maxwell equations, provided the coupling with the curvature tensor is excluded. The reason for this exclusion should already be apparent. As seen above, the Maxwell equations, when coupled with the s'urrounding curved space-time have the form F + ^ R A^ = 0 . (31)

On the other hand, the Proca eq.uations, which are defined in the flat space, have the form

F^^ = 0 . (1,1)

The similarity "between these two equations is rather striking.

Therefore, it is plausible to identify the mass of the neutral vector meson y as "being proportional to the scalar of curvature R. In other words, equation (31) "was o"btained by considering the coupling of the electromagnetic wave with the curved space, assuming the validity of the Maxwell theory. On the contrary, the Proca equations are derived in the , by adding the mass term explicitly to the

Maxwell Lagrangian. Hence, the following identification seems possible

y2=X.R. (1,7) K

Assuming the validity of (Ut), it follows that a mass of a neutral meson vanishes if R = 0, i.e., if the space surrounding that meson is flat. Now, as is well known /36/, the Proca equations are the unique generalizations of the Maxwell equations. As a consequence, we may be able to assert that in a non-empty space, where the curvature is not vanishing, a photon acquires a mass, as a result of its interaction with the surroundings.

Before discussing the and the symmetry breaking in the massive field theories, let us consider some analogies between the

Maxwell and Einstein theories. A theory of general relativity is based on a powerful principle, which refers to general covariance. Concisely, it demands that equations preserve their form under the point coordinate transformations. In addition, when the metric tensor g^^ equals to the Minkowski tensor n^^ , then these equations adopt the form which is accepted in the special theory of relativity. However, it was pointed out /3T/ that this principle, by itself, does not have any physical significance. The meaning of the principle reflects only the validity of an equation in a gravitational field, if such an equation is valid in the absence of those fields. The presence of the gravitational field is then described by the additional terms ct involving g^^ and the . On the contrary, the

Lorentz invariance demands that no additional terms (in particular the velocity of the reference frame) enter into the transformed equation. Therefore, one may say that general covariance does not imply the Lorentz invariance. In other words, the principle of general covariance is not an invariance principle, but, as it is restricted to the gravitational field, it is also known as the dynamic symmetry /38/. This brings us to the point of contact with the dynamic symmetry in electrodynamics, viz., the local gauge invariance.

According to that symmetry, the fields i|;(x) and the electromagnetic potentials A transform as / \ , / \ ieX(x) ^{x) ^ iliix) e (I^Q)

\ ^^^^ *

Now, as the 9QL , in general, does not obey (U8), an equation of the form

(a a^ + m2) H^) = 0 (^9) a is not gauge-invariant. In order for (^9) to be gauge-invariant, it must be derived only from i|;(x) and the respective gauge-covariant derivatives which transform like the fields,

D^Mx)—^D^Mx) e^^^^^^ . (50)

The above equation is then written in a gauge-invariant form as

(D^ + m^) ip(x) = 0. (51)

In analogy, the Lorentz invariant equation woi£Ld be invariant under the point coordinate transformations, provided only tensors and their covariant derivatives, which include the affine connection and transform as tensors, are used. Evidently, there is a close correspondence between the general covariance of the theory of gravitation and the local gauge invariance in the Maxwell theory.

This section aimed to provide some insist into this relation.

So far the possibility of replacing the medium by a curved space was studied. It appears that, if the nonlinear interaction is allowed in the theory, the photons may be described by the Proca equations. Essentially, the argument is dependent on the choice of the metric: if the Riemann metric is adopted, the IMaxwell equations are modified by the inclusion of some additional nonlinear terms, whilst if the Minkowski metric is preferred, the photons appear to behave as if they possessed a non-vanishing mass. A similar conclusion, for particles with spin 0 and was recently obtained /39/ using the supersymmetry technique. They found that a masslessness of a particle in the de Sitter space or a massiveness of the same particle in the

Minkowski space are, in fact, just two different gauge choices of one and the same theory. V. GAUGE THEORIES AND THE SPONTANEOUS SYMMETOY BREAKING

Earlier, we have studied a possibility of attaching a non-

Tanishing im,ss to the photon, as a result of the nonlinear interaction between the electromagnetic wave and the curvature of space-time. In

the process, we have discussed some of the similarities and differences

"between the notions of the Lorentz invariance and the General Covariance,

As it was shown, the medium, in which that wave propagates, may "be

replaced "by a curved space.

However, in order to confirm the correctness of this supposition,

an analysis dealing with the gauge theory is required. The reason

"being, as already mentioned earlier, is that, classically, the

lykxwell equations in a medium are invariant under the gauge trans-

formations, whilst the Proca equations, due to the appearance of

the mass term, violate that invariance principle. Before invoking

the concept of the spontaneous symmetry "breaking, as a mechanism for

the mass generation, let us reflect upon the gauge theories.

It is a common practice to adopt the Lagrangian formalism in

field theories. Of prime importance, in such a description, is

the action integral A, defined as /hO/

A = dT , (1) where dx is the four-dimensional volume element and the Lagrangian

density is the function of the fields and their gradients. The field

equations then follow from the Euler-Lagrange equations of motion

gV _ iJl = 0 . (2)

Ma"" 3 ^^ Next, it is well estalDlished that the symmetry principles in physics are closely linked with the conservation laws. Every symmetry principle is built on an assumption that a particirLar quantity is not measurable. For instance, the lack of the absolute position in

space, implies translational invariance. In fact, one can construct

transformations on the fields, which leave L invariant, for every

conserved quantity., So it follows that the energy, momentum, and

are conserved if L is the Lorentz invariant.

Now we are interested in transformations which do not affect the

space-time point - the so called internal symmetries. The simplest

transformations of this kind are defined by

U(r Uv

Where X is a phase, independent of the position x. A transformation

of that kind is called a global gauge transformation. Physically, the

invariance means that the phase of the field cannot be determined and

is purely arbitrary. Hence, this phase cannot be fixed within the

region of the experiment and any change in A must be the same at all

times. The group of transformation (3) is also referred to as the

Abelian U(l) group - the group of unitary transformations in one

.

There is another kind of transformations, with larger syimnetry,

known as the local gauge transformations, which are defined by

where the phase X(x) is now dependent on x. However, complications

arise, as the gradient of the field now does not transform as the field

itself. This implies that e.g., the Lagrangian density of a free electron field is not invariant under As already indicated, the invariance of the Lagrangian under (U) can be restored if a new, the so called covariant, derivative is introduced. It has the property that it transforms as the field itself, and is defined vy

D E 3 + ie A , (5) a a a ' where A is the gauge field which transforms according to ot

A^(x) -i X(x). (6)

This is well known from electrodynamics and it represents the non-

uniq.ueness of the potentials, as the gradient of an arbitrary scalar

can be added to the vector potential. In addition, the definition (5)

is often referred to as the minimal coupling.

The interpretation of gauge invariance was extended beyond that

of the phase transformation by Yang and Mills /Ul/, to describe an

internal symmetry. Thus, a phase transformation was generalized to a

rotation in the internal space. By studying the similarities between

the gauge theories and the general theory of relativity, it was shown

/1+2/ that the gauge invariance may be expressed using elements of

geometry. It seems pertinent to ponder over this issue for a while.

Thus, in the general relativity we study the motion of particles

in a curved space-time . To facilitate this investigation,

a concept of a free-falling is often used. Such

a frame is well defined only in a limited region of space-time.

Because the gravitational field is inhomogeneous, it is impossible

to extend a free-falling, frame of reference far from a given world

point. Hence, in the vicinity of such a point the particles appear

unaccelerated, despite the presence of gravitation field. As a

consequence, a particle can then be described at each point in space- time, with respect to the free-falling frame, as if the gravitational

field was absent. The effect of the gravity appears when the motion

of a particle is described from different non-inertial frames and

one inquires about the relation between the two. On the other hand,

the gauge theory deals with the internal properties of a particle,

like charge (global invariance) or isotopic spin (invariance under

SU(2) group).

Thus, the abstract internal symmetry space is quite

different from the ordinary physical space-time. Consequently,

in view of /U2/, one may identify the internal symmetry space with

the local free-falling frame of reference, which is used to describe

the motion of a particle through the external field. An observer,

located inside such an internal space, would notice a rotation of

the phase of the wave function, describing a particle. The angle

of rotation would then depend on the strengths of the interaction

with the external gauge field as well as on the distance the particle

travelled through that field, ^^erefore, in analogy with the gravi-

tational field, this external gauge field determines the modifications

of the coordinates in the internal space, as the particle moves from

point to point. Hence, the external gauge field specifies the

behaviour of the internal coordinates at different points in the

space-time.

There is a close connection between the gauge invariance and the

Lorentz invariance. The mutual interdependence is clearly seen in

the following example. Consider two observers moving in different

inertial frames. For a given (the same) problem in electrodynamics

they both choose the Coulomb gauge. As is well known /Us/, such a

gauge is not Lorentz invariant. Nevertheless, the electromagnetic fields, as measured in those inertial frames, are connected by the

Lorentz transformation. This implies, therefore, that the Lorentz transformation not only changes the space-time coordinates in the inertial frames, "but also affects the rotation in the internal space.

Pat otherwise, the Lorentz transformation also implies a gauge transformation. To illustrate this important point further, consider the following example. In /hh/ the transformation law for the four-potential is given "by

^a ^a^. ^T)

R Where L^ is the Lorentz matrix. Evidently, the four-potential is not a real four-vector as, under the Lorentz transformation, an extra term is required. That term is interpreted as being due to the rotation in the internal ^pace. The effect of this term on the

Lorentz invariance is now obvious. Thus, using the Lorentz gauge implies 3 A°'= 3' A'cx . a a

However, (7) will only be satisfied if

3 3°^ X = 0 , (8) a which is precisely the constraint on X in the Lorentz gauge. The meaning of (8) is clear now: in order for 3^ A^ to be treated as the Lorentz invariant, the rotation in the internal space must be eliminated. The reason for the non-invariance of the Coulomb gauge is now self-evident. Unlike conservation, there are other symmetries in Nature that are not exact. It is well established by now that if the physical vacuum is not invariant under the symmetry group, an exactly symmetric Lagrangian will yield a spontaneously "broken symmetry. To illustrate this point, consider a Lagrangian density describing the dynamics of a scalar field Hx) interacting with an electromagnetic field

yv y

This Lagrangian, in view of its construction, is invariant under the local gauge transformation U(l). The first term describes the energy contained in the electromagnetic field, the second term is a minimal coupling term, and the last two terms are interpreted as referring to the potential energy. Now, the behaviour of the potential energy

V((})) depends on quantities y and X. Thus, V((p) has a minimum only if

X > 0, In order to determine the value of this minimum, we need to know the sign of y^. Thus, if y^ is non-negative, the minimum of V((j)) is at (j) = 0, the well known classical solution. However, if y^ is negative, the minimum of V((j)) is at c{) ^ 0 and there are many solutions which yield the required minimum. For a moment, assume that y? -is negative and choose one of the possible solutions in the form

^(x) (a ^ . (lO)

Substituting (lO) into (9) yields

L = - J B2 + s!^ B2 + \ fa, B(x)l -Xa2-e2(x) - 1 yv 2y -y ^

- + i b2 e2(2a3(x) +32(x)). (H)

In the process we have used

A (x) = B (x) + -i- 3 e:(x) . a a a e a Evidently, the Lagrangian density (ll) is not invariant under the gauge transforinations. It contains only massive particles, thus, displaying the particle aspect of the theory. To put it succinctly, the spontaneous synmietry breaking is responsible for the mass of the gauge particles. The above described mechanism is known as the Higgs mechanism 7^5/•

A common occurrence, in the many-body systems, iS the existence of collective modes of excitation. For instance, photons in a superfluid helium, the spin waves in the ferromagnet. The description of these collective phenomena is facilitated by the introduction of a

degenerate ground state. Thus, in the BCS model of superconductivity

7^6/, the ground state is characterized by a nonvanishing isospin density which, in turn, determines the energy gap 7^77• Hence, the

ground state is degenerate and the symmetry is broken. Put otherwise,

in this theory, the photon acquires a mass, manifested by e.g., the

Meissner effect, through coupling to the Cooper pairs. An analogous

situation arises in the theory of a ferromagnet. At temperatures

T > T^, the ground state of the ferromagnet is invariant under the

rotation, however, this invariance breaks down for T > » as the

spins point in the preferred direction. Again, as the ground state

does not possess the full symmetry of the interaction describing the

system, the spontaneous symmetry breaking occurs.

In the light of the above discussion, consider an interaction

of electromagnetic wave with a medium. As most of the characteristics

of a medium are described by considering only valence electrons, we

restrict our analysis by replacing a medium by an electron distribution.

Alternatively, consider a medium irradiated by an intense radiation and investigate the interaction iDetween such a system and a test radiation beam. If the medium in question is not in thermal equilibrium, the corresponding electron density may he written in the form

n(r,t) = n + a(r,t) (13)

•w O

Let us now introduce the notion of the order parameter by considering the following example /hQ/, In the mean field theory of the ferro- , the direct interaction between the spins is viewed as a two- step process: 1, different spins generate the ; 2. this magnetization then determines the behaviour of the individual spins. The magnetization, which is a macroscopic quantity, can be referred to as an order parameter, as, apart from determining the behaviour of each micro-system, it also gives the degree of order - in analogy to that concept in the phase transition theory.

Let us now construct the Lagrangian density for laser-medium system, using the just mentioned order parameters, in analogy to the

Ginzburg-Landau theory of superconductivity /h9/o A possible form in the absence of the electromagnetic field is

L = + ilJ id^ - ^ -iVi^ - V (15) Next, introduce the interaction with the medium through the minimal coupling. Ttie Lagrangian for the complete system then becomes

1 L = (E^ - E^hl Stt

(iV - eA)i|;* (-iV - eA)i|; 2m - V ip^ ip. (16) where ij; is the order parameter. To see the effect of the symmetry

"breaking, transform the order parameter, in analogy to the Higgs picture, as

ie6(r,t) ^ = OIQ + (IT)

Substitution of (IT) into (l6) leads to

L = ^ (e2 - H^) - 1 Sir 2m

-(a^ + jp^)^ V - (a^ + $, (18) where the following relations have been used

B = A + V 6 $ = (!) - 9, 6. (19) ^ ~ A. "C

Although the Lagrangians (l8) and (l6) are formally different, it must be emphasized that they both describe the same physical system.

One is derived from the other by a transformation, which in no way

can change the physical content. Although the latter form is not

gauge invariant, it readily lends itself to the important inter- pretation: it describes the interaction of the massive vector field

By with the massive scalar field ^^ (r,t)~ . In other words, the

electromagnetic field, as a result of its interaction with the medium,

far from the thermal equilibrium, has acquired a real mass. Despite

the formalism of the elementary particle physics being involved, the phi'iosophy of this result is quite novel. Thus, in paxticle physics the symmetry is "broken even in the absence of the electromagnetic field, and is due purely to the auxiliary Higgs field. In the present case, however, the symmetry is broken as a direct consequence of the interaction between the electromagnetic fields. So far, we have seen how a photon can acquire a nonvanishing mass, using the fomalism of the Higgs model.

Let us, now, diverge for a while, whilst considering the interaction of radiation with a medium. As is well known, the electrodynamics is based on the Maxwell equations and the Lorentz equations, describing the motions of electrons. Although not widely appreciated, this is not an exact theory. The application of the theory to various inter- action processes often makes use of the perturbation approximation.

Commonly, only the first non-vanishing order in the electric charge

(coupling parameter) is taken into account, as it is, generally, assumed that the magnitude of the higher orders in the expansion is progressively smaller. Occasionally, however, one includes these higher orders in the calculation and these are then termed the radiative corrections. These are quantum effects which are closely linked with problems of self-charge and self-mass. Although these corrections are small in comparison with the first order, they are of fundamental importance from the theoretical point of view. It is still a major unsolved difficulty of the theory that the infinities, resulting from self-interaction, cannot be satisfactorily accounted for. In order to avoid this difficulty, the following argument is used: as the observable quantities, the total mass and the total charge, are finite, the renormalization procedure is employed. In essence, this method involves separation and subsequent subtraction of these divergent quantities, leading to the finite measured values, A further question arises when the additional terms in the expansion are included.

Although it is generally assumed that the series converges, it is rather improhahle as the number of the Feynman diagrams corresponding

n to each higher order term increases rapidly with the power of e .

The reasons for these unsatisfactory features of the theory are not known. It may be possible that the expansion in powers of e^ is not allowed. Or, there may be another possibility, involving a radical modification on the more basic level. However, whatever the changes in the improved theory may be, they must, in the zeroth order approximation, include its present form, which in many cases leads to highly accurate results.

¥e are now in a position to investigate another mechanism for mass generation, which is often called dynamic symmetry breaking.

This phenomenon was already. Investigated /50/ and, as it may improve our understanding of the concept of mass, the pertinent details of that model will now be recalled.

The essential difference between the Higgs model and the one being analyzed now is the driving mechanism of the instability.

Thus, in the former case its role is assumed by the negative term in the Lagrangian density, arising from the postulated scalar field; in the latter case, the radiative corrections are responsible for the symmetry breaking.

Using the method of Jona-Lasinio /5I/, the effective potential is introduced. Then, the minima of this potential determine the exact vacuum states of the theory. Thus, the knowledge of the effective potential will yield the necessary information about the symmetry breaking. With Jona-Lasinio, consider the modifications to the dynamics of the scaxal field ^(x), l^y inclusion of its interaction with an external source j(x)

(j)) (|) ) + (f)(x) J(x) (20)

The effective action S(cj) ) is then defined "by

S(6 ) = W(J) - J(x) (p (x) d'+x , (21)

where (j, , the classical field, is given "by

< 0 (22) = <5J(x) < d 0 J

and V(j), the generating functional, is expressed in terms of the

transition amplitude as

,iW(J) _ = <0 0 > (23) J

It follows from (21) that we can write

& S = - J(x) (24)

Now, the effective action can "be expanded in powers of momentum as

S = d X (25)

where is the effective potential. According to the perturbation

theory, one can express the mass and coupling constant in terms of

functions in (25) as

d^V ,2 - (26) d (j);

X = dV (27) Now, assmaing that the Lagrangian density has an internal symmetry and the source j(x) is absent, the appearance of the spontaneous symmetry

"breaking signifies that the quant-um field ^ has a non-zero vacuum

expectation value. It follows, from (22) and (24), that

= 0 0. (28) 0 ^c

Further, if we require that the vacuum expectation value "be trans-

lationally invariant, then, in view of (25), the above simplifies to

7dct)7 c = ° • (29)

If the new quantum field, having a vanishing vacuum expectation value,

is defined by

(})' = (j) - < (j) > , (30)

then the classical field obeys

= (f)^ - <(f) > , (31)

where < cj) > is the value of (j) at which the potential is minimum. C ^ - - ... - , . ...

Therefore, in order to use (26) and (2?), in calculations of mass and

coupling constant, they must be evaluated at < cf) >

Next, consider a massless charged meson field interacting with

the electromagnetic field. The corresponding Lagrangian density is

£ = - i- - e K^^^f. - ^ ^2)2 , (33)

where cf)^ and cl)^ are real fields. In order to eliminate the

contribution of certain classes of the Peyranan graphs to the effective

Lagrangian, the Landau gauge is used. After some calculation /50/, the

effective potential is found to be in the form •n p 6 (33) where M is an ar"bitrary renormalization mass, which we chose to be

the location of the minimuin <(()>• As there is no a priori reason

to the contrary, we restrict oin? analysis to the case where the

magnitude of x is of the order of e ^ , This restriction is shovm

to "be valid /50/. Consequently, (33) is simplified to yield

4 X 3e 23 In (34) ^ = 4: ^ c + ^ <(b>' ¥e are now in a position to determine A in terms of e. Since < c}) >

is the minim-ujn of V, i.e. v(< c}) > ) = 0, it is straightforward to

obtain

X = e\ (35)

The effective potential can now be written, in view of (35) > as 1+ 3e^ V = 1 (36) )4 TT' ^>2 2

This, in essence, is the dynamical model for the mass generation.

From now on, there is an overlap with the Higgs model. Therefore,

following the standard procedure, we find that the mass of the scalar

meson is given by

m 2 = v.. ( < ^ > ) = < cj) >2 (37) S 8 7T2

and that of the photon, which now becomes massive, by

m^ = e^ < ^ > ^ . V

It is essential to note that the mass of scalar and vector mesons

is expressed in terms of the higher order corrections in e. Therefore,

within the restrictions imposed throughout this calculation, it is valid to say that the radiative corrections are responsilDle for the degeneracy of the vacuum state and, consequently, are instrumental in mass generation. Put otherwise, if only the first order corrections in the interaction process are dealt with, it is correct to assume that photons are massless. However, in the circumstances where it "becomes necessary to include the higher order radiative effects, it seems plausible to attach a non-vanishing mass to the photons. VI. SOME ASPECTS OF INTERACTION AKD EXPERIMEIWAL EVIDMJCE

Reader, who has caxef-ully followed the preceding pages, may feel

that the somewhat different issues, which were studied, present a rather disjoint picture. Two different approaches were adopted to

verify whether the results were consistent. It must be clear that

the concept of mass is rather complex and its complete understanding

is still quite far, For instance, the ambiguity involved in determining

the form of the Lagrangian density is not satisfactory, if the proposed

theory is to "be successful. Nevertheless, within the imposed

restrictions, the general relativity as well as the gauge theory

approaches lead us, independently, to the conclusion that, under

certain conditions, the originally massless photons behave as the

neutral vector mesons. Let us now apply this suggestion to some

specific problems and, then, compare the theoretical predictions with

the experimental results.

In the meantime, as was argued above, the Maxwell equations inside

a medium may be replaced by the corresponding equations in a curved

space. Or, on the other hand, if the Minkowski (flat) space-time is

preferred, than these equations should be generalized and substituted

by the Proca equations /52/. Assume, for a moment, that this supposition

is valid and investigate what would be the conclusions derived from this

premise. Thus, as the mass of the photon is now finite, there would

be three states of polarization. Thence, apart from two transverse

polarizations, there also exists a longitudinal one. Furthermore, the

faster the longitudinal photon moves, the weaker is the associated

electric field /36/, thus, implying only a rather weak interaction with

a medium. In addition, the electric field diminishes exponentially with distance and the corresponding flux lines fade away, even in vacuo. Similarly, magnetic field is als.o affected and the associated

lines are compressed around equator. In addition, the velocity of

energy propagation is now frequency dependent, as can be seen from

the following argument. A free electromagnetic wave, corresponding to

a massive photon, may "be described by

with (u>/o)2 - k2 = (2)

where y is the rest mass of the photon. Then, from its definition,

the group speed is v = d(jo/dk, and, on using (2), we obtain S

V - ^ (0.2- , (3) ^ CO

in accord with the experimental results /53> 36/.

Consider now a static field, i.e., a case when w= 0. It then

follows from (2) that k = ±i y . Subsequent substitution into the

wave equation (I) implies that the spherical static field decays

exponentially, i.e., A = A^ e"^^ (A^ r""^). On selecting k = iy ,

we have a behaviour similar to that displayed by the Yukawa model of

nuclear binding. Thus, one of the consequences of adopting a non-

vanishing mass for photons is a deviation from the Coulomb law.

As the photon is now assumed to have a non-vanishing mass, a

mass term has to be added to the Lagrangian density describing the

Maxwell field. The thus obtained Lagrangian density, in the absence

of charges, becomes £ 1 p /B + ^ A A" (4)

^P IbTT a3 8 a

and is termed the Proca Lagrangian density. The reciprocal length

y in (4) is associated with the rest mass of the photon by y = m^c/fe The Proca equations of motion axe then written as

F^^ + A^ = 0 . (5)

Thus, (5) together with the definition of the electromagnetic field tensor, P^^ = ^^ A^ - A^ , represent the covariant Proca equations for massive vector field. It should also be noted that these are the only possible linear generalizations of the Maxwell equations. In addition, as a result of the inclusion of the coupling constant y , the potentials now acquire real physical characteristics.

That is, in contradistinction to the potentials in the Maxwell theory, they now become observable.

As already indicated, the massive photon has three degrees of polarization. It appears that the result of Hora /54/> dealing with an exact Maxwellian beam in vacuo or in homogeneous medium, may be related to the Proca description. Studying the intense irradiation, that work concentrates on the non-linear effects, whilst relying on the quivering motion of electrons in laser field. It is then found that an unexpected longitudinal component appears, as a result of the exact description of high intensity beam. It seems necessary to include this component in the description of the non-linear force emission of electrons in radial direction from the laser beam. In addition, it is also required to explain the internal diffraction of the beam.

Next, it seems appropriate to mention here the Schwartz-Hora effect /55/, as the resemblance with the present work is quite intriguing.

The effect arises when an electron beam traverses a solid slab which is simultaneously irradiated by a crossing laser beam. An analogous sit-uation with the electron he am'being replaced hy the electromagnetic wave was considered above. The electron diffraction pattern, then, is mod-alated by the colour of laser photons. As the existence of plasmons in solids is well established /43/, Hora /56/ used this result to extend it to the electron wave modulation by plasmons. In that work, the differences between the classical bunching and the quantum mechanical modulation were studied. It was shown that a superposition of two electromagnetic waves differing by Aw , leads to a beating wavelength 0) X = - A Aoj 0 Further, in investigating the similarity between the light waves and

the matter waves, it was found that in the latter case one must include

an additional beating wavelength

no) where A ^ is the de Eroglie wavelength. It was pointed out that this

difference is related to the second order term in the momentum expansion.

The result was then used to describe the side shift at the total reflection -

the Goos-Haenchen effect. It was then concluded that matter waves allowed

the rigorous mathematical description of the wave bundle /57/» whilst

the use of light waves represented only an approximation.

One of the topics which for a long time has occupied the mind of many and is still not completely understood, as the number of papers

appearing recently can prove, is the ambiguity regarding the correct form of the energy-momentum tensor in a medium. Although this problem has been studied by the present author /5, 14/, the background of this work is different and it seems appropriate to include it here. An

extensive literature survey, which was covered in that work, will not be repeated now. Here, we shall concentrate on the effect of the medium on the electromagnetic wave propagating in it, drawing on our knowledge from the previous sections. In other words, to emphasize the point which may "be easily misunderstood, we are interested, exclusively, in the properties of the electromagnetic field, modified "by the presence of the non-empty surroundings. Commonly, one is interested in the total energy-momentum tensor, which includes the field as well as the mechani- cal contrihutions. It can then he shown that, from mathematical stand- point, i.e., "by formally altering the form of the matter tensor, one arrives at the conclusion that all these descriptions are equivalent

/58» 59/» This may "be so, "but only formally. In other words, the physical content of electrodynamics of Abraham, Minkowski, and others differ, as argued hy Brevik /6o/ and Novak /l4/« One of the reasons

in support of that latter claim is an excellent correlation with the

experimental findings, as will be detailed shortly.

Consider then the energy-momentum tensor of the free massive

field. As is well known, it can be formed from the Lagrange density.

Thus, if the Hamiltonian density is defined by

9 ^ P f E = ^^ 9 A - Lp, (6) t a then the covariant formulation of (6) leads, in view of the Lorentz

transformation properties of ^ , to the canonical energy-momentum

density tensor

9(9^ A^)

Substituting (4) into (?) and making use of the field equations (5),

the components of the above tensor become = 47r

Stt ^(E X H A^Y + I • (A^ E)] (S)

; (E X H ) +y2 A V + (V + VH). - 8,(VE.) " . 477

In order to eliminate the unwanted divergencies in (s), three

dimensional space integration results in the following expressions for the energy and momentum of the Proca field

d^ X = St1 t L(E^ + H^ + ^2(v2 + A2)] d^x 1 ^(E X H)^ +y2 A^ Vjd3 X . (9) d^ X 4tt .

The shortcomings of the canonical tensor are well known /19, 3/.

In an attempt to meet them, Belinfante /61 / proposed a symmetrical form for the . Recently,Hehl /62/ gave the more general

symmetrization procedure, of which the Belinfante method is a special case. The primary concern in defining the syiranetrical energy- momentum tensor is to modify the canonical tensor T^^ in such a way that the total distribution of energy and momentum densities in the field be given by UV ^ ^PV ^ ^pv (10) where the divergenceless tensor T^^ must be chosen in such a way that the conservation laws involving T^'^ still hold. It is then found that

yv ^ ^ T X P VIV ^V s 4-n (11)

Since Xo o Xo 3 e d X = T d X, (12)

T ^^ gives no contribution to the total energy and momentum, s Consequently, Tj^^ may be interpreted as the spin energy momentmi tensor.

Next, the total angular momentum density tensor J ^^^ is generally written as

= M^^" . (13)

i.e., as a sum of an orhital and spin N angular momentum density tensors. However, this last subdivision may be shown /63/ to be unique only when the rest frame of the particle can be defined.

And a meson, having a non-vanishing rest mass, satisfies this condition,

The total angular momentL"Lm density tensor, which is defined as

J^l^^ = x^ - (14)

is, in view of being a , conserved, as its

divergence, J^^^ , vanishes. It is, therefore, evident that the

contribution by T ^^ ensures the conservation of J^^^ .

Taking into the account the expressions for T^^ , (s), and that

for Tyv , (11), the tensor 6 can be expressed in the component s form as

e°° = 1 I? (V^ O IT (15) = = iC(ExH). v].

The influence of the mass term on the energy-momentum tensor is now

quite explicit. As y is assumed to be, in most cases, small, the

deviation of the above results from those corresponding to the Maxwell

field will be small. Nevertheless, these results are of utmost

importance from the theoretical point of view. Next, using the Proca equations, the linear momentum derived from the canonical tensor becomes

Y (E.A) d^x. (16) 4^1 J The corresponding expression, obtained from the symmetric tensor QPV

, takes the form S = { A(v.E) + (E X H) } d^x . (17) 471 T

The integrated total angular momentum can be written, therefore, as

J = X X A(V.E) + (E X H) d^ X. (18) 4tt

Using standard vector relations, the above integrand yields

X X V (E.A) - (E. V)A + A(V. E) (19)

Now, in view of (l6), the above is interpreted as the moment of the canonical momentum and the spin momentum densities. Consequently,

substitution of (19) into (I8), followed by an integration, results in 1 J = .(x X v) d3 X (E X A) d^ X. (20) 4 E A + 4tt J

Thus, the modification of J due to the symmetric tensor 6 is now clear.

In accord with our supposition that the Maxwell equations in a medium may be replaced by the free Proca equations, let us apply now the reasoning required in the massive electrodynamics to the actual, proposed forms of the energy-momentum tensors. There are many different forms of this tensor, which were put forward by various workers. Their advantages as well as disadvantages were already discussed /3, 60/. Here, to emphasize, we are interested in the mutual relation between the two of them, viz., the tensors of

Minkowski and Abraham. This choice is, to certain extent, governed iDy the availability of the experimental data.

Thus, Minkowski, applying the macroscopic Maxwell equations to the energy and momentum conservation laws, arrived /6/ at the ener,gy- momentum tensor of the form

q yv 1 F H'^ + i n^^ P H^^ ^ "" 4tt ' aX a3 (21) yv yv where H is obtained from F "by substituting E D and H B.

In an attempt to overcome the deficiencies /3,25/ of the Minkowski tensor, all arising from the asymmetric character of that tensor,

Abraham /64/ proposed a different form

H^^ + i n^^F H"^ + (ey - Y^ J , (22) A " 4tt where e and y are, respectively, a dielectric permittivity and magnetic permeability and v ^ denotes the four-velocity of the medium,

Furtherinore 5 the last term in (22) c an i.) 0 v/r itten as

(ey-DPg V, (vl^ H^^ + v^ E S^^^ (23) with F = F ^ v^ . Evidently, the Minkowski tensor differs from a ap . , the Abraham one by the expression (23), which, in the rest frame of

the medium, can be written in the component form as

S°° =0 = ) sp sp '

s" = E >< H . (24) sp 4 ttC ~

Clearly, the only non-vanishing component of S^^ is the space- . - - - . - . ^ - - time one. Let us now inquire about the role of this tensor in the energy-momentinn tensor pro"blein. It is immediately olDvioiis that for frequencies at which ep may be replaced "by unity, the space-time components of also vanish. The magnitude of frequencies at which sp . . this statement applies, depends on the medium in question. For instance, in a collisionless plasma, the frequency domain is given "by w >>

(Wp - plasma frequency); in a solid, frequencies larger than those of the electronic vibrations, i.e., in the optical frequency domain, are required. This is, however, the region where WKB approximation applies, i.e., a domain of the geometrical optics. In this case the electromagnetic wave may be replaced by particles with no internal structure.

These considerations, together with the Belinfante symmetrization yv procedure, suggest that the tensor S^^ , defined as

= ^r - ^^ (25) may be identified as the . In the (optical) frequency limit, S^^ may be neglected and, as the effect of the medium on the electromagnetic wave is minute, the energy-momentum tensor of the latter is given by the Minkowski expression. On the other hand, at the opposite end of the frequency spectrum, the electromagnetic wave is considerably modified by travelling in a non-empty space.

This influence is reflected in the Abraham form for the energy- momentum tensor. Furthermore, it is accepted that the energy and spin localizations are related by the angular momentum conservation.

In addition, they (mass and spin) are the unitary representations of the Poincare group. From the arguments detailed in the previous section, it follows that mass is not an intrinsic concept, but arises hs a result of the interactions. In virtue of the just presented discussion, it seems plausible to extend this conclusion also to the concept of spin. Thus, high frequency electromagnetic wave does not

"see" the mediiom - a conclusion represented ty ey approaching unity.

In this case, then, the electromagnetic wave is described by spinless photons, as is well accepted. In this limit, the spin tensor S^^ - • . , . sp vanishes. If, on the other hand, the frequency of the electromagnetic wave is of the order of the characteristic frequency of a medium in question, then the interaction term cannot be neglected. The spin tensor Jr is, in this case, non-vanishing, what amounts to saying that photons display spin. This is connected with polarization currents.

It is to be noted that they give no contribution to the total energy and momentum fluxes, since

Sl^ d^x = d^x. (26)

In the limit of low frequency, the corresponding force is also finite and, in the rest frame, can be written as

f = —I-L a (E X H) . (27) ^ sp 4 -IT C t ~ - ^ ^ ^

In virtue of the above arguments, this force will almost vanish at high frequencies, whilst it will increase towards the low frequency end of the spectrum.

Let us now corroborate this result by turning our attention to the experimental evidence. Although the available data are not in abundance, nevertheless, they well correlate with the above predictions.

It should also be pointed out that there is a general agreement as to the correctness of the experimental findings. Briefly, the experiments of Jones and Richards /65/, Jones and Leslie /66/, and Ashkin and

Dziedzic /67/, to mention just a few, were all performed using the optical frequencies. The obtain results are in accord with the Minkowski form for the energy-moment\im tensor. On the other hand, the experiments of Hakim and Higham /68/ and Walker et al. /69, 70/ were all conducted at the low frequencies. The conclusion they reached agrees with the Abraham result.

It is pertinent to mention one other point. The Minkowski momentum is often referred to as a pseudo-momentum /71, 72/ of the medium. This does not disagree with our result. As was already emphasized, in this work we study the effect of a general medirni on the electromagnetic wave propagating in it and assume that the conditions are such that the effect of the wave on the medium can

"be neglected. Then, as was shown in the investigation of the black

"body radiation /5/, in the domain of the geometrical optics, the exchanged momentum between the electromagnetic wave and the medium is given by the Minkowski expression. In this frequency range, this is the total momentum of the photon, which is carried away by the medium, as a pseudo-momentum.

A possible way to support the predictions about the momentum of the electromagnetic wave in a medium would be to devise an experiment covering a large frequency range, preferably with a high intensity source. At the frequency extremes we would then expect the results as detailed above, whilst for intermediate frequencies one would obtain the average value of those at both ends of the range. In other words, the momentum g of the electromagaetic wave of intermediate frequencies propagating in a medium would""be given by

g = (g^ + %)/2 . (28)

This result can, indeed, be recovered from the work of Peierls

/72/ if the correction term of the second order is dropped. Klima and Petrzilka

/75/found the same expression for the wave in a collisionless plasma and

Hora /74/ further substantiated it by showing that the Fresnel formulae can be derived by using that momentum. VII CONCLUSION

It remains now only to synthesize the intennediate results of the

individual sections, in order to obtain a clearer picture of some

aspects of interaction phenomena.

Accepting, implicitly, the linearity of the field equations, the

new derivation of the "black body radiation inside a medium was presented,

At the same time, it was found that the magnitude of the momentum

exchanged between the radiation field and a medium is, in the region

of validity of the Wien law, proportional to the refractive index of

the latter.

There is a sufficient evidence that the classical electrodynamics

holds only as long as the wavelengths are long compared to the "radius"

of the electron. When the field contains shorter wavelengths, this

theory breaks down. It is, therefore, desirable to modify the theory

in such a way, as to widen its region of applicability. Some

interesting, and very different proposals of others were discussed.

The concept of mass, which was interpreted as arising from the

higher order interaction, played a dominant role in this work. It

was seen that, first of all, a medium may be replaced by a curved space.

If the non-linear interaction between the electromagnetic wave and the

curvature of that space is taken into account, one is naturally led

to the non-linear photons, whose mass is proportional to the scalar

of curvature.

In support of this result, the framework of elementary particle

physics was invoked. The detailed analysis of a simple experiment

then confirmed that a photon, in a medium, may, under certain

conditions, acquire a non-vanishing rest mass. The underlying

philosophy is similar to that occurring in the theory of super- conductivity, where photon is considered to be massive. This mass generation, in either case, is a result of the symmetry "breaking and a subsequent loss of the gauge invariance.

In the light of this result, it is suggested that, for most media, at optical frequencies, the energy-momentum tensor of the electromagnetic field inside that medium is given "by Minkowski,

However, when conditions are such that the photon may be considered massive, then the Abraham result seems more appropriate.

In conclusion, when conditions are such that the Maxwell theory may be considered to be exact, then the corresponding photons are massless. However, if this is not the case, e.g., in the presence

of the strong fields, which substantially modify the character of

the electromagnetic wave, then this theory must be altered to include

the non-linear interaction terms, thus, giving rise to the non- vanishing mass of the photons. ACKNOWLEDGMENT

The author would like to express his gratitude to Professor

H. Hora, for his constant interest and enthusiasm displayed while this work was in progress. BIBLIOGRAPHY

1. E. Schxodinger, Proc. Roy. Soc. A232, 435 (1955)

2. L.de Broglie, Phil. Mag. 47, 446 (1924)

3. M. M. Novak, Fort. d. Physik 285 (1980)

4. A. Einstein, Phys. Z. 18, 121 (19^7)

5. G. Bekefi, Radiation Processes in Plasma, Wiley, 1966

6. H. Minkowski, Nachr. Ges. Wiss. Gottingen, p.53 (19O8)

7. P.A.M. Dirac, Proc. Roy. Soc. A209, 291 (1951)

8. G. Mie, Ann. d. Phys. 37, 5II (I912); 1 (I9I2); 40, 1 (1913)

9. M. Born, Proc. Roy. Soc. 243, 410 (1934)

10. M. Born, L.Infeld, Proc. Roy. Soc. 144, 425 (1934)

11. C. Brans, R.H. Bicke, Phys. Rev. 124, 925 (I96I)

12. G.Y. Rainich, Trans. Am. Math. Soc. IO6 (1925)

13. T.W.B. Kibble, Phys. Rev. 1^, IO6O (I966)

14. M.M. Novak, Can. J. Phys. 1317 (1980)

15. H.A. Lorentz, Encyklopedia der mathematischen Wissenschaften,

Band V2 Heft 1

16. D.R. , Proc. Cambridge Phil. Soc. 25, 47 (1929);

143 (1932)

17. C.G. Darwin, Proc. Roy. Soc. AI46, 17 (1934)

18. V.L. Ginzbiirg, The Propagation of Electromagnetic Waves in

Plasma, Pergamon, 1964.

19. J.D. Jackson, Classical Electrodynamics, Wiley, 1975*

20. J.Dawson, C.Oberman, Phys. Fluids 5, 517 (1962)

21. J. Dawson, Adv. Plasma Phys. 1, 1 (1968)

22. K.M. Case, S.C. Chiu, Phys. Rev. II70 (1970)

23. K.D. Cole, Austral. J. Phys. 67I (1977) 24. D.V. SkolDel'tsyn, Voprosy Matematicheskoi Fiziky, Nauka,

Leningrad, 1976,

25. W. Pauli, Theory of Relativity, Pergamon, I967.

26. H. Weyl, Space-Time-Matter, Methuen, 1922.

27. P.A.M. Lirac, Proc. Roy. Soc. AI63, I99 (I938)

28. C.¥. Misner, J.A. Wheeler, Ann. Phys. 2, 525 (1957)

29. M.M. Novak, Optics Comm. (in print)

30. M. Jammer, Concepts of Mass, Harvard University Press, I96I.

31. H. Bondi, Assumption and Myth in Physical Theory, Cgjnbridge, I967.

32. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, San

Francisco, 1972.

33• S. Weinberg, Gravitation and Cosmology, Wiley, 1972.

34. R. Duff in, Phys. Rev. III4 (1938)

35. N. Kemmer, Prov. Roy. Soc. A173« 91 (1938)

36. A.S. Goldhaber, M.M. Meto, Rev. Mod. Phys. 277 (1971)

37. E. Kretschmann, Ann.d.Phys. 575 (I917)

38. E.P. Wigner, Symmetries and Reflections, Indiana UP, 1967-

39. R.Marnelius, B. Nilsson, Phys. R^v. D20, 839 (1979)

40. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields,

Pergamon, 1971-

41. C.N. Yang, R.L. Mills, Phys. Rev. 96, 191 (1954)

42. C.N. Yang, Phys. Rev. Lett. 33, 445 (1974)

43. W. Heitler, The Quantum Theory of Radiation, Oxford, 1954.

44. J.L. Bjorken, S. Drell, Relativistic Quantum Fields, McGraw-

Hill, 1965.

45. P.W. Higgs, Phys. Lett. 12, 132 (19^4); Phys. Rev. Lett.

1_3, 508 (1964); Phys. Rev. 1^, II56 (I966).

46. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys.Rev.108, 1175 (1957) 47. Y. Nambu, Phys. Rev. II7, 648 (I960)

48. H. Haken, Rev. Mod. Phys. 67 (I975)

49. V.L. Ginzburg, L.L. Landau, Zh.Exp.Theor.Piz. 20, IO64 (1950)

50. S.Coleman, E. Weinberg, Phys. Rev. 1888 (1973)

51. G. Jona-Lasinio, Nuovo Cim. 34, 179O (I964)

52. A. Proca, Compt. Rend. 193> 832 (1931 ); I366 (l935;

202, 1490 (1936); 203, 709 (1936)

53. G. Feinberg, Science I66, 879 (I969)

54. H. Hora, Physics of Laser Driven Plasmas, Wiley, I98I.

55. H. Schwarz, H. Hora, Appl. Phys. Lett. 349 (I969)

56. H. Hora, Proc. Sec. Int. Conf. on Light Scattering in Solids,

ed.M. Balkanski, p.128, I97I.

57. H. Hora, Optik I7, 409 (I960)

58. M. Kranys, Can. J. Phys. 1022 (1979)

59. G.A. Ma-ugin, Can. J. Phys. _58, II63 (I98O)

60. I. Brevik, Phys. Rep. 133 (1979)

61. P.J. Belinfante, Physica 6, 887 (1939)

62. P.W. Hehl, Rep. Math. Phys. 9, 56 (1976)

63. J.M. Jauch, F. Rohrlich, The Theory of Photons and Electrons,

Addison-Wesley, 1955.

64. M. Abraham, Ren. Circ. Mat. Palermo 28, 1 (1909)

65. R.V. Jones, J.C.S. Richards, Proc. Roy. Soc. A221, 480 (1954)

66. R.V. Jones, B. Leslie, Proc. Roy. Soc. A36O, 347 (1978)

67. A. Ashkin, J.M. Dziedzic, Phys. Rev. Lett. 30, 139 (1973)

68. S.S. Hakim, J.B. Higham, Proc. Phys. Soc. _80, I90 (I962)

69. G.B. Walker, B.G. Lahoz, G. Walker, Can.J.Phys. 2577 (1975)

70. G.B. Walker, G. Walker, Can.J.Phys. 2121 (1977)

71. J.P. Gordon, Phys. Rev. m, 14 (1973)

72. R. Peierls, Proc. Roy. Soc. A347> 475 (1976)

73„ R. Klima, V.A. Petrzilka, Czech.J.Phys.B22, 896 (1972) 74. H. Hora, Phys. Fluids I7, 1042 (1974) X'o o h I- i t- lit*. (lucio)

Interaction of Photons .with Electrons in Dielectric Media

MmosLAV M. NOVAK*)

Department of Theoretical Physics, University of New South Wales, Kensington, N.S.W 20?iZ, Australia,

Abstract

The question of the field energy-inoinentum tensor in a medium is not new and many Avorkeis, in the past, attempted to find the answer to it. Nevertheless, there was no general agreement about the form of .sueh a ten.5or, tiius resulting in a confusion involving the very fundaments of plij'sirs. Although the present work uses well established theories, the underlying philosophy is completely novel. Investigations are mainly centered around the eollisionless plasma as, in that state, develop- ment of the argument is most transparent. On the b.isis of a simple theoretical reasoning, it is, first of all, postidated that the momentum of the photon in a plasma is given by the Minkowski expression. This hypothesis becomes more viable as it directly leads to the accepted form for the field energ}' density. Furthermore, atilizine the concepts of stimulated and spontaneous emission and absorption, it. is confirmed that, in the equilibrium (betv/een radiation and plasma), the transferred momentum has the value given by the Minkowski theory. However, as will be seen, this result is valid onlj' in the region of high fre- quencies. Put otlierwise, when conditions are such that the laws of geometrical optics apply, tlie momentum of the photon is, to a good approximation, given by the ^Minkowski theory. In order to arrive at a more general result, valid in the domain of wave optics, a similarity between the dispersion relation (describing the wave propagation througli a- medium) and the relat ivistic-i energy-momentum of a particle moving through vacuum, is explored. Accepting the equivalence of these two relations implies that some of the properties of radiation in a meclium, can also be described by a massive particle travelling through empty space. In other words, the task of ana- lysing the behaviour of electromagnetic waves in a medium, caii be replaced by the analysis of the free neutral vector meson field. Adoption of this equivalence results in the field energy-nioinentum tensor being symmetrical. A thorough study of the field theory then provides the basis for interpreting the Abraham tensor as the sum of the "orbital" and "spin" tensors. The former is directly connected M'ith the energv transport, whereas the latter one is not. Realising that the magnitude of the s^in term increases towards the lower end of the frequency spectrum provides the resolution of the Abraham-Minkows- ki dilemma: the energy-momentum tensor, corresponding to the electromagnetic wa\'e in a me- dium, is that given by Abraham, while the one of Minkowski is only an approximation valid at hioh frequencies. , ' Although this conclusion stems from studies involving collisionless isotropic plasma, it is in ex- cellent agreement with the experimental data,

*) Present address:, Department of Natural Philosophy, Universitv of Glasgow, Glaso-ow r:>SOO United Kingdom ... - o , connection between the energy and the frequency remains rather obscure [1]. Never- Contents theless, (1.1) in conjunction with one of the most fundamental principles of the special theory of relativity, viz. the principle of the equivalence of mass m and any kind of I. lutrodnction • ' _ 2Sl> energy [2], E — mc^, suggest that an effective mass /ico/c- can be attached to the photon. II. Covariant Electromagnetism 291 Now, according to the special theory of relativity, the energy E oi a particle is given by III. The Energj'-Momentum Tensor in a Meditim. . . 298 (1.2) 1. Origin of the confusion 298 2. Theoretical considerations .-..•..... 302 where p stands for the momentum and «?o refers to the rest mass of a jiarticle. Of course, 3. Experimental results ^ . . 319 the rest mass vanishes in the case of a photon, hence its momentum has the value hejlc. IV. Electromagnetic Waves in a Plasma 322 In addition, there is another, equivalent expression for the photon's momentum. Thus, the substitution of the relation between the frequency v and the corj'esponding wa\'e- V. Dual Nature of Radiation 332 length which in vacuo takes the form VI. The Proca. Equations 340 Vri. Conclusion 349 . vX^c, (1.3)

Bibliography i ,351 together with (1.1) into (1.2) results in p = JiJa, or

(1.4)

I. Introduction where k — is the wave vector. Hence, (1.1) and (1.4) emphasize the relation between the corpuscular and the wave aspects of radiation. It is noticed that the ratio between The idea that light has purely imdiilatory character was generally accepted in the last the two is given in terms of the quantum of action. This, in essence, reflects the wave- •entury. The corresponding theory was able to explain empirical facts quite well and, corpuscular parallelism, which Einstein apjslied to particles with zero rest mass — hus, superseded the older, corpijscular theory of Xewton. As the new expenmental data photons. beeanie available, the theory was altered, but this change Avas only gradual. However, The relations (1.1) and (1.4) remain, however, formally correct also for corpuscules with owards the end of that century, the understanding of the ])hysical phenomena was, non-vanishing rest mass, as was first recognized by de Bkoglie [-3]. He noted the simi- jnce again, thrown into coniusion. There was an accumulation of the experimental larity between the phase of a plane wave, t) = AQ exp {i{Et — p • r)/7<), and tlie evidence which could not be accounted for by the existing theory. Hamiltonian action of the particle. Thus, he deduced that the Fermat principle, valid Thus, Planck, in his work on the black body i-adiation, was forced to introduce the dis- in the domain of geometrical optics, is analogous to the i\laupertuis theorem in the ana- •ontinuous as]-iect of radiation, in order to I'esolve this discrepancy between theoiy and lytical mechanics. Since the Permat theorem has no meaning in the i-ealm of the v.ave •xperiment. This meant that the corpuscular nature of radiation was j'cintroduced. optics, he concluded that the traditional mechanics is also an approxin)ation with the Einstein iised ajid further expanded this concejjt, when investigating the photoelectric same range of validity. Therefore, de Broglie applied the wave concept of radiation to •ffect. The name "photon" was intioduced to denote these corpuscules. jNTevertheless, matter and showed that (1.1) and (1.4) are, indeed, relativisticalh' invai-iant also for Ins did not mean that the corpuscular (photon) theory became (as was the case in the massive particles. inie of Newton) the dominant one. As distinct from the exclusiveness of the previous Sul-sequentty, utilizing the definitions for the phase c.^ and the group Vg speeds, given by heoi'ies, the I'adiation was now to be understood as having two distinct, complementary the wave theoi y as 'spects: undiilatoiy and corpuscular. Jn other words, depending on the question ])osed, t'p = cojk Vy = dco'idk, (1.5) lie radiation-induced effects could be accounted for by either the wave or the])article lieory. The complementarity of these different characters of radiation is now we!i esta- he arrived at the well known group speed theorem In view of its derivation, it is valid )iished. Foi' instance, the classical wave theory can explain neithei' the photoelectric in the geometrical optics approxi7nation and expresses the relation between the s^jced jhenojnenon nor the Compton effect, because these apparently belong to the paiiicle. of a particle and the phase speed Vp of the associated wave — the so called matter wave iomain. On the other hand, the occurrence of interference, diffraction, and polarization nanifest the wave nature of radiation and, tlierefore, cannot be explained by the photon c^jv. (l.G) iieory. Thus, in order to really comprehend the radiation phenomenon, this duality nust be inevitably accepted. It is then easily shown [4] that the group speed of the \p (matter) waves connected with the free particle is equal to the speed of the latter. Hence, a photon travelling in vacuum [n spite of diffei-ences, there is a close connection between the undulatory and corpus- appears as a particle whose speed (and also group speed) is equal to the phase^peed of the 'ular pictures. Classically, the notion of frequency and wavelength are inhei'ent in the corresponding wave, as expected. "ormer, wliile energy and momentum belong to the latter. However, it is now customary Despite these similarities, masslcss photons and massive particles are not the same. Tims, o associate with the electromagnetic wave of angular frequency to, a particle M'ith energy in oi-der for a particle to be observed, it must interact. Consequently, its ei^icriry ;;n.d E given bv momentum will be changed in accoi'dance with the conservation laws. Thtm, hi(;;i eiKM^n- E:=h0J, (1.1) material particles are like high energy photons, because they change ejiergv

2rr7( -- h is the Planck constant. Although (1.1) was pi'oposed by Planck already their effective mass. This is the domriiji of ilu^ uppJiraluIity of (lu^ gcouirfric:!] oi) 1 >'-?,'inning of tSus cculury and is riow well aece])ted, its meaning as to a 5>hysical which, })y definition fJJ, is valid in iJic icnion wa vc'lcsi^t hs. On other h^piej. the differences between photons and particles are very pronounced at the other end of the • , To complicate matters even further, other forms for' the electromagnetic energy-mo- energy spobtnnn, i.e., in the low energy limit. The energy of material particles approaches, mentum tensor have been put forward as well. Thus, EINSTEIN and LATJB [21] examined in this case, the rest energy, whereas the photon energy (and frequency) approaches zero. the problem and arrived at a completely different result from those of MINKOWSIA Therefore, in this limiting case the theory of radiation is effectively replaced by the static and ABRAHAJI. Another proposal is due to BECK [22], who arrived at a conclusion sii\ii- field theory. Thus, classically, our conception of matter is represented by the particle lar to that of MARX and GYORGYI [23]. nature, while th&,t of radiation assumes the wave character. Erom the in depth analysis of the properties of a field in a medium, DE GROOT and It is important to remeinber this differentiation related to the energy sxsectrum, as it SuTTOBP [24] obtained two different expressions for the energy-momentum tensor. The will, in the following, provide one of the crucial points of the analysis. first one was derived by the statistical averaging of the relevant microscopic quantities, The main investigation of the present work is directed towards the electromagnetic while the second one was found by subtracting the total energy-momentum in the ab- energy-momentum tensor in a medium. This tensor is very ixnportant as it jDrovides the sence of the field from the one when the field Avas present. basic information related to a system in question. It is directly involved in the various , It would appear that tliis confused situation should be easy to resolve experimentally. conservation laws. When written out explicitly, its components can be identified as This, however, is not so. Results obtained from the few ex]3eriments actually performed follows: the time-time component is equal to the field energy density, the space-time do not correlate. Thus, the data collected by JONES and RICHARDS [26], and in the more components describe the and the momentum density, and its space- accurate, recent version of the same experiment by JONES and LESLIE [2()], lead to the space components comprise the electromagnetic stress tensor. conclusion in agreement with the Minkowski form of the momentum. This finding is also The form of the energy-momentum tensor corresponding to a field in vacuo is well suiDported by the experiment of ASHKIN andDziEDZic [27]. Nevei'theless, the empirical established. However, when such a field is embedded in a medium, the form of the ten- data from WALKER et al, [25] and WALKER and WALKER [20] suggest that the photon sor is, generally, not agi^eed upon. In view of its great significance, this may come as a momentum in ,a medium is given by the Abraham value. In addition, )-esults obtained siu^jrise to many, especially when it is realized that the principal work on the subject in the static limit by HAKIM and HIGHAM [30] confirm the Abraham theory. was done more than sixty years ago. The distinction between the various proposed ten- On the basis of the brief desci'iption of the theoretical and experimental results, it is sors is rather subtle and the resolution of the existing confusion is by no means trivial. seen that the confusion is rather profound. The obvious difficultj^ is pai'tly due to the As there is scarcely any coverage of this dilemma in the books, the large majority is not coniplexity of the question of the nature of the photon moinentum in a medium. Eui- even aware of the existence of the problem. thernioi-e, there is a rather large variation in the conditions under which experiments Amongst those who studied the topic, oijinions differ. Their research is exclusively di- were performed. For instance, in the Jones and Richards experiment, visible light rected towards the dielectric medium with the refractive index n larger -than unity. (v Hz) was used and liquid imder investigation had the refractive index of Complexity of the problem is reflected in many attempts by various workers to find the the order of unity. On the other hand, WaUter et al. employed very low frequency energy-momentum tensor of a field in a medium. Thus, the first proposition put forward {v < 100 Hz) and a ceramic sample had very high dielectric constant (?^'3200). It is, was due to Minkowski. Making use of the macroscopic Maxwell equations in the energy therefo7'e, not sui'pi'ising that such a wide range of the involved variables produces con- and momentum conservation laws, he arrived at a field energy-momentum tensor which tradictory results. was not synunetrical. This lack of synunetry was rather disturbing as it led to some In order to comprehend this, rather peculiar, beliaviour, it is necessary, fii-st of all, to peculiar consequences. Therefore, Abraham (and Hertz) suggested a synmietrized ver- consider the simplest case. As is well known, a niedium can be in any of the four states: sion. As, a consequence, the value of the momentum density differed now by the factor • solid, liquid, gas, and plasma. Each of these states is characterized by the diffei-ent set of of n-. Both derivations seems to be based on sound physical principles, yet they lead to variables. Therefore, in order to simiDlify the structure and to ejnphasize the physical different conclusions. It is important to find the correct expression as it is closely con- nature of this fundamentalproblem, we shall concentrate our attention on the case of the nected with the concept of the ponderomotive force. Although both yield the same mag- l^lasma state. It will be assumed that there is no external magnetic field present and that nitude for this force, the directions predicted by them are mutually opposing. Thus, a > collisions among electrons are rare and, hence, can be neglected. In other words, our main numbei- of workers set out to resolve this controversy. In his original article on rela- effort will be directed toward an isotropic, collisionless plasma with the dispersion re- tivity, PAULI [2] appeared to favour the Abi'aham tensor but, later, aj^parently being lation of the form convinced by the argument due to VON LAUB [5], reverted to the Minkowski proposition. E((O) = 1 - (C<)//£O2) , (1.7) That argument required that the velocities of the energy propagation (in a medium) must y satisfy the Einstein addition of velocity theorem, and only Minkowski tensor was able to where e{o}) is the frequency dependent dielectric permeability, co^- = 4:7tNZq^/m is the fulfil this condition. Hence, the von Laue criterion had a profound influence on further plasma frequency, co denotes the frequency of the incident wave, and NZ, q, m represent developments regarding the field energy-momentum tensor. Tiais question was also the total number density, charge, and mass of the electrons, respectively. The reason analyzed by MOLLER [7] and he opted too (apioarently) for the non-symmetric (Min- for this selection is due to the weak nature of the interaction among jjlasma particles. kowski) tensor. On the other hand, LANDAIT and LIFSHITZ [(§] adopt the version due to This, consequently, considerably simplifies the procedure. Once this relatively simple Abraham. In fact, most authors now tend to accept for the electromagnetic eneigy- case has been well understood, the extension to other states is then more likely to lead to momentum tensor in a dielectric the form given by Abraham. Amongst these are BALAZS the correct conclusion. Eurthermore, most of the results obtained from the investigation [P], SKOBEL'TSYN [lO], BURT and PEIERLS [22], GORDON [12], PEIERLS [23], JONES [14], of a tenuous plasma will also apply for the case of and metals when the in- and SnocKLEY [25]. Some others, like e.g., HAUS [25] and ROBINSON [27], advocate the cident frequency is approximately in (or above) the ultraviolet range, since (1.7) may be necessity of additive correction terms to the Abraham tensor, furthermore, BEEVIK [25], shown [5] to be valid at high frequencies for matter in any state. GINZBURG [29], and MIKURA [20] advance arguments intended to show the equivalence As may be deduced by now, the central task of the present work involves the analysis of of both proposed tensors. the interaction of an electromagnetic field with a dielectric inedium. Hence, to facilitate the inquiry, the following scheme is proposed: The field theory, which provides the n. Covariant Eleclroinagnetism ,' necessary background information for this type of ]3roblem, is briefly outlined in the nest chapter. The four-dimensional formalism of the special theory of relativity is used The notion of the energy-momentum tensor is closely linked with the field equations. throughout that chapter. Starting from the covariant form of the Maxwell equations, Thus, it is appropriate to recall these equations at this stage and to show how such a the energy momentum tensor of the total system is derived employing the concept of the tensor is derived. The special theory of relativity is used throughout, as its formalism is Lagrangian density. It is then found that part of this tensor refers to the ele~6tromagnetic most convenient. By utilizing only four-scalars, four-vectors, and tensors, the theory field, while the other part describes the backgj-ound niedium. This "splitting-up" pro- clearly manifests the Lorentz invariance. In other words, it is form independent under cedure of the total tensor is, however, not unique. As a consequence, this is responsible the Lorentz ti-ansformations. for the above ambiguity in selecting the form of the field energy-momentum tensor. A comprehensive review and an analysis of the various proposed foriais of that tensor The summation iconvention, i.e., summation over the repeated indices will be employed. are presented in Chapter III. As already indicated, they lead to different conclusions Furthermore, Greek indices run from 0 to 3, while-Rojnan ones from 1 to 3. In accord which are then also examined. In addition, the correlation between the theory and ex- with this convention, the time coordinate of the position four-vector becomes = ct, isting empirical facts is discussed. while .space components are denoted by {xi, a;,, .^3). A four-vector position can, thus, be expressed concisely as = {ct, x). The metric tensor is diagonal and its components are As mentioned above, the present analysis is centered on the fourth state of matter, i.e.,. gOo _ gii „ _ gzz — The quantities denoted by a prime v/ill refer to the iner- plasma. There are various kinds of electromagnetic waves that such a fetate can support. tial frame which moves with a uniform velocity with respect to the iinprimed frame. It is, therefore, instructive to include some details about the wave propagation and-also Hence, the four-vectors, as measured in both frames, are connected by a 4 x 4 matrix, to recall the essential features of the supporting iiiedium. The problem of defining dielec- as tric permittivity of such a vsystein is a rather complex one and it has attracted much attention [SI, 32, 33]. These questions are treated in Chapter IV\ A"'^ • (2.1) Elementary considerations, which often help in understanding the dual natui'e of radia- i.e., the elements of a four-vector transform like the coordinates. Similarly, a tensor of tion are examined in Chapter V. It will be found that, within the WECB approximation, the rank two' transforms as this simple theory is in .agroouient with the accepted I'csults. The de Broglie group velo- = , . (2.2) city theorem is utilized in order to arrive at some conclusions of that chapter. Purther, it is of interest to inquire into the momentum transfer from the radiation field to a We are now in a position to consider the electromagnetic field. As is well known [50], the particle moving in a plasma. The method used by Einstetk [34\ in analysis of the black" scalar potential 9? and the vector potential ^ are the compjonents of a four-potential A"". body radiation is, first of all, adapted to include the media with the refractive index Although they are not directly measurable, and are introduced in order to simplify the differing from unity, along the lines attcmj)ted by.SKOBET.'TSYX [55]. It is then applied calculations, their direct relation to the observed field strengths is well established in derivation of the momentum transfer, thus directly leading to the accepted value [^36, 37, 35] of the radiation energy' density inside plasma. Let us emphasize again that E - —— - V(p H^ Fx A. these results are ap)piicablc to the case of an isotropic, collisionless plasma. A possible c (2.3) extension to the collisional plasma is also suggested, in analogy to the treatment of a nonlinear force by Hoba [39]. When written in the component form, they become It was xDointed out by Colje [40, 41], Lee [42], and later by Kltma and Petrzilka [43] that the dispersion equation for the transverse electromagnetic waves in plasma is for- - = - d^A^) H^ = -(52^3 mally identical with the energy-momentum relation of a massive particle. In other Avords, E, = -(aoi.2 _ = -(a^i - d^A^) ' ' this suggests that the photon, with vanishing rest mass in vacuo, when ^propagating in a plasma, acquires a non-zero effective rest mass. Put othez'wise, a photon in a plasma • behaves as a neutral vector meson in vacuo. Hence, in order to resolve the energy-mo- where = mentum controversy, it is instructive to resort to the Proca field. The electrodynamics It is convenient to combine these components in a matrix form. Thus, defining the anti- for such a field is developed by modifying the results of classical electrodynamics. It symmetric field tensor Fi^" as leads to a'tensor which is syjiimetrical [44, 45]. A careful analysis of the work by Belin- FANTE [4G, 47] shows that the symmetrizing term, which is normally added to the ca- Fi" = di'A' — d'Af, * (2.4) nonical energy-momentum tensor, may be interpreted as the spin angular momentum. As the time-time and space-time components of the canonical and symmetric energy- we obtain the matrix expressed in terms of the field strengths momentum tensors ai^e the same [4S], it is deduced that the spin angular momentum is responsible for the additional linear momentum which, however, is not connected 0 with the energy transport — in accordance with a similar proposal put forward bv • = (2.5) Bopp [49]. 0 -E, The concluding chapter of the present work briefly reviews the main findings of this 0 , investigation. Tt is argued there that the comyjlete field energy-momentum tensor is, In terms of the field tensor F'"', the homogeneous Maxwell equations take the form in fact, tliiil proposed T)y Abvaham with the Minkowski version Ijcing only a])pi-oxiuiately -j- ef'F'-^ -i- ^'F^f = 0. (2.0) valid fit Inuli fre<\ui>u(''vos. Gcjuatiori^ . t/i. oiig/i t/i« cJuiil fieia tensor whicli is clcfi- neci as and jr rofer.s to tUe l^jif^iangia-n aeilBity. Xhis iviiietioii, B1>1>\"\<'<1 to our Hy»Vo»i\, «.on«»isV.rt of three contributions A J -i- A,„ + A (•2.14) J^l^' = JL yfivxa p (2.7) J"- The first term, Af, refers to the action of the field alone, i.e., to system in the ab>ei,(.c where the , is equal to , • of any charges. If one knows this term, the field equations can be easily [5] obtained. The 1 for any even permutation of indices, Lagrangian density corresponding to this term nnist Ije cpiadratic in the field, in ordt: that the field equations remain linear. Purthermore, as this function nuist be Lorentz — 1 for any odd permutation, invariant, it cannot involve coordinates ex])licith'. lir addition, it cannot be expressed in terms of the four-potentials, as these are not uniquely determined. In view of the.se 0 if two indices are equal. resti-ictions, the only Lorentz invariant quadratic forms are and The hojnogeneous field equations in terms of the dual field tensor assume the form However, the latter scalar changes sign under the inversion. Therefore, only the former scalar, Fi^'F^,,, constitutes the free field Lagrangian. Thence, the action A^, cor- = (2.8) responding to the free field, takes the form Mathematically sjjcaking, (2.0) and (2.8) are equivalent. However, as in an ordinary space the electric field vector E is polar and that due to the magnetic field 11 is axial, it seems more correct [2] to represent the electromagnetic field by i"'"'. The inhoniogeneous pair of the Maxwell equations can also be written in the relativisti- which, in the three dimensional representation, becomes cally invariant form. Thus, in terms of F"" and the four-current f, these take on the form 1 _ 7/2) civ dt. (2.15a) 4T: (2.9) In order to find the Lagrangian density for the free field, it is only necessary to compare The space components of the four-current, j = nv, form a vector in three dimensior.s, the (2.12) with (2.15a). One then obtains so called cui-rent density vector. The time component is the product of the charge den- sity o and the vacuum c. The relation between these components of the four-current is expi'essed through the (2.16)

V = (2.10) The second term in (2.14), A,^, represents that i)art of the action which depends exclusi- vely on the properties of the particles, i.e., the system in the absence of the field. If In a continuous medimn we think of the charges as smeared over space-time nianifold. there are several particles present, then the general form for the action is [5] Pour-current then takes this effect into account. Of course, if there are no charges present, the right hand side of (2.9) vanishes. In the macroscopic formulation it is, however, ne- = —y nic f ds. (2.17) cessary, because of polarization and magnetization effects, to introduce the additional field tensor 77''". This tensor is obtained from Fi"" by the transformation E B and In other words, J,,, is equal to the sum of the actions I'eferring se2)arately to each particle. H -> B. The covariant form of the macroscopic Maxwell equations is then The last term in (2.14), Af,,,, describes the interaction between the components of our system, viz. between the field and niatter. Thei'cfore, it nmst contain information re-1 471 garding both subsystems. Hence, it is found convenient to represent matter bv the elec- (2.11) tronic charge, q, while the field is determined by the four-potential, Ai^. The action term then takes the form The definitions of the four-current the four-potentialand the field tensors F^" and II''\ in conjunction with the Maxwell equations conclusively establish the covariance (•2AS\ of the field equations {oO\. Turn attention now to the covariant fornmlation of the conservation laws.Tt will be If discrete charges are replaced by the continuous (p, then (2.18) assumes assumed here that the system under investigation consists of an electromagnetic field the form and that there also are some charged particles moving in it. Hence, we shall examine the interaction between these two subsystems, resorting to the Lagrangian formalism. Of (1^.19) prime importance, in the description, is the action integral A, defined as [5] Combining (2.15), (2.17), and (2.19), the action for the whole system (with a continuous, Jldt, (2.12) charge distribution) then becomes where the four-dimensional volume element is

dx = cdt dx dy dz (2.1-3) ^ / - i: / ^'-if (-20) With this in mind, the dectromagnetic Lagrangian density becomes In general, this tensor is not symmetric. Furthermore, the component corresponding to the energy density is not always positive [45]. These facts have caused some difficulties. ^ — L. ^ L o A'' (2.21) To make this tensor more transparent, consider first the free electromagnetic field. The mjt "" c ' second term on the right hand side of (2.21) then vanishes (as there are no charges pre- sent). The free field Lagrangian, I = — l/lG.-r is then substituted in (2.28) to give Having found the Lagrangian density, the field equations can be readily obtained. Thus, utilizing the Euler-Lagrange equations of motion Tt" = [2.29) 471 (2.22) This expression becomes clearer when it is written in the thi'ee-dimensional form. Thus, substituting (2.5), the matrix form for the field tensor F^", together with the free field together with (2.21), one arrives at the .field equations. To elucidate the meaning of Lagrangian (2.16) into (2.24) gives us the following components for the electromagnetic (2,22), the required ox)erations are first performed. After some calculation, we obtain energv-momentum tensor

S'F 4- — i =0 (2.23)

As is easilv i-ecognized, these are nothing else but the inhomogeneous Maxwell equations (2.9). To obtain the conservation of the current density, we proceed to take the diver- Toi = -L{ExH)i+ ^ {AiE) (2.30) 4:71 gence of (2.23), thus yielding

T'O = — (E X n)i + —

However, it is recalled that the product of a symmetric operator and the anti- It is seen that all expressions differ from the commonly accepted ones for the energy and symmetric tensor F^,. vanishes, resulting in , > momentum densities by the addition of the last term. This difficulty, however, is usu- ally obviated. It is recalled that the components of the energy-momentum tensor refer (2.24) to the densities. However, in order to obtain energy and nromentum of the field, they must be integrated over all space. As it is assumed that the fields are localized, the contri- the earlier obtained continuity equation. butions from the added terms in (2.30) vanish when the integration is performed, thus We are now in a position to present the derivation of an expression for the energy and leading to the standard expression for the field energy and momentum. momentum of the electromagnetic field. In doing so, it is found convenient to utilize the Knowledge of the energy-momentmn tensor enables one to wi'ite down the conservation Hamiltonian formalism. Thus, in the particle mechanics the canonical momentum j^i laws. Thus, the Poynting theorem, which expresses the conservation of energy and mo- is defined by mentum, can be written [2] in the covariant form as = (2.25) crqi d/ff' = 0. (2.31)

and the Hamiltonian density then becomes In analogy, the vanishing of the four-divergence of the angular momentum density implies that the orbital angular momentum of the field is conserved. That is, if the an- (2.26) gular momentum tensor is cq-t Mr^'" Ti^'x" — . (2.32) then w-here qi refers to the time derivative of the generalized coordinates qi. Similarly, the o^Mr'" = 0 (2.33) field canonical momentum is obtained from (2.25) by the appropriate replacement for (ji- Thus, it is then defined by expresses the approj^riate conservation law. However, in order for (2.33) to hold, the direct calculation requires that T'^'' must be symmetric. Ae was already noticed, and is == ' (2.27) also clearly exhibited bj?" comparing T®'- and in (2.30), the tensor 7''"' does not satisfy this symmetry requirement. Hence, the angular momentum, with J/t-""" as defined what means that the field momentum is canonically conjugate to A,. In analogy, the by (2.32), is not conserved. In addition, according to the definition of T"'\ (2.29), it <'X{>rcssion for the field Hamiltonian density can be obtained. The result is then known depends explicitly on the potentials. However, it is known that tlie potentials are not uniquely determined. Consequently, the tensor 7'''" is not gatige invariant [o?'.']. I^Iore- as She canonical energy-momentum tensor [48] ovcr, direct calculation sho^vs that its trace, T"', does not vanish. As the just considered field is free (in view of (2.10)), it is difficult to accept the tensor T'" o^A' (2.28) as physically siL'nifkant. TheTcfore, it ^\r,s neccssnry to find another cncri^v-ntonicntuni tensor, which woidd remove the above mentioned deficiencies. Here lies the oricrin of the symmetnc energy-momentum tensor The pi^escription for such a tensor was first total energy and momentum integrals over space-like volumes {4S\. Pauli has, further- 11 given by Belinfante 47]. Thus] to symmetrize the term d^A' in (2 29) is re- more, emphasized that for the unique localization in space-time of the energy and mo- placed, utilizing (2.4), by - FK Then, the tensor T"^ assumes the form mentum, the gravitational theory is necessary. The introduction of the gravitational field gives, then, an unambiguous meaning to the energy-momentum tensor. The total angular momentum density of the field is now defined, using in terms of Tf" — (2.34) the tensor ilf''"", as 4:71 ^ O^'^'x" ~ Qi'^'x'. . . (2.40) The last term in above can now be rewritten as In view of the symmetry of the total angular momentum is conserved, which is mathematically expressed by ^ , = 0. (2.41) 2.35) 4yi 47r I It is assumed that the field^is free. That is, as there are no charges present, the Maxwell The total angular momentum is again obtained by the integration of the corresponding , equations have the form (2.9), d.F'f = 0. Thus, a term A^dxF^'* can be added to (2 35) density over space-like volumes. Consequently, we can write ~ ' , , .. - So far, the considerations were mainly directed towards a free, i.e., non-interacting electromagnetic field. If the charges are included, interaction occurs. Consequently, the Tr = Ati (2.36) Lagrangian for the Maxwell field is also changed and is now given by (2.21). This, in turn, implies that the energy-momentum tensor is no longer divergenceless. Then, The symmetric energy-momentum tensor is now defined as instead of (2.39), we now write - , (2.42) Qt^" ~ J^nv 'V liv c S ' or, written in terms of the field tensors it becomes The right hand side of (2.42) describes'the interaction between the electromagnetic field, here represented by and the four-current jx, describing the source. This coupling term is known as the Lorentz force density. The integral of this force density over the = (2.37) space volume then gives the time rate of change of the total four-momentum of all particles. On the other hand, the space integral of describes the time rate of change It is clear that the potentials are now not explicitly involved in the definition, therefore of the field four-momentum. Therefore, the conservation of the momentum of the whole S 07W1?' n invariant [5]. If the field energy tensor (2.5) is substituted in' ' system (field and particles) is written as ' (^.37), then can be written in the component form as + = (2.43)

H^) OTt where the Lorentz force p = 1/c F'^j^. Thus, the sum of the total momentum and energy of the system is conserved. The Poynting theorem can also be interpreted as follows: the time-time component of (2.38) the Lorentz force density multiplied by c is j • E, The space volume integi-ation then describes the conversion of tlie electromagnetic energy into the mechanical or heat energy. Hence, it must be balanced by the corresponding decrease of the field energ3^ 0ij = — E^E^ + - ± + H^) 4jr Physically, this is equal to work done by the field per unit time. Thus, it represents the rate of increase of the mechanical energy. In analogy, the considerations of a similar It isii^iediately obvious that the expression for the energy and momentum densities • nature apply to the linear momentum. given by the tmie-time and space-time components of 6)"% respectively, are in accor- Landau and Lifshitz [5] proceed along slightly different lines. First of all, they deter- dance with the accepted formulae. Furthermore, the space-space components re- mine the field energy-momentum tensor. Then, the corresponding tensor for the par- present the taken with the negative si

= . (2.39) The total tensor must be symmetric. From the fact that the tensor representing the particles, 6/", is symmetric, they conclude that also the field tensor 0/" must be sym- Of course, (2.39) is nothing else but the covariant form of the conservation of energy and metric. . • . momentam. In other words, it is the four-dimensional equivalent of the Poyntin- theo- ' The above discussion has centered about the system with the Lagrangian density having rem. It is to be noted that the comparison of (2.29) with (2.37) yields the equalit/of the one, (2.16), or two, (2.21), terms. A more accurate treatment is presented by Rohrlich where the three-term Lagrangian is used, which makes it possible to investigate such while the time-space, component represents the energy flux effects as radiation reaction, self-energy, etc. The derivation of the field energy-momentum tensor inside a macroscopic medium pre- sents a con^plex problem. We saw above that it was possible to divide the tenspr corre- = (3.4) sponding to the total system, consisting of field and particles, into two parts, each re- presenting the separate component of the system. However, when the macroscopic me- dium is involved, this separation cannot be established uniquely. The main reason for the where S — 6/471 {E X H) is the well known Poynting. vector. implied ambiguity lies in tne impossibility to differentiate as to what is electromagnetic Now, on calculating the space-time component of the (3.1) tensor, we have \ and what is not. The following chapter examines the different proposals put forward to avoid this dilemma. = X = ^ X = V •

In view of the relationship

III. The Energy-Momeiituia Tensor in a Medium we obtain, therefore, the expression 1. Origin of the confusion ' (3.5 a) The considerations of the preceding chapter apply, essentially, to the electromagnetic' 9.xf c- field in vacuo interacting with the charged particles. Results arising from such analysis are well established. Tliese do not hold, however, \\'hen the field energy-momentum for the momentum density gr.v/- This, however, would appear to contradict the Planck tensor inside a macroscopic body is discussed. Judging on the basis of a number of pro- principle of inertia posed forms for such a tensor, it is clear that neither is the question trivial nor is the in- volved subtlety sufficiently appreciated. It juay appear surprising that a problem as fundamental as this has still not been satisfactorily resolved, although the main work on,, } the subject was completed decades ago. Numerous articles appearing in the literature associating a momentum density g with each energy flux S. testify to the complexity of the pioblem. The question of the electromagnetic energy- Similarly, a contradiction arisesialso in conncction with the principle of equivalence of. momentum tensor v>-as almost completely solved by Helmholtz, (cf. [57]), decades be- energy and mass, E = mc~. This is not surprising, as the latter principle is just an inte- fore the controversy originated. Using the principle of virtual work in conjunction with grated form of the Plaiick principle of inertia [i?]. The asymmetry of tensor S^i'' al.^^o the macroscopic field equations, Helmholtz found the force which acts on a medium in must give rise to uncompensated mechanical torcjues wln'ch, cleai'ly, are entirely un- the static field. physical [2]. : The above deficiencies in the Minkowski tensor, all arising from the asymmetric charac- Amongst the early workers investigating this subject was Minkowski. By a straight- ter of that tensor, are rather disturbing and, in an attempt to overcome them, a diffe- forward application of the macroscopic Maxwell equations to the energy and momen- rent form for the energj^-momentum tensor (it will bo denoted here by iS.i"'') Avas x'>i"o- tum conservation laws, he arrived [2] at the energy-momentum tensor of the form posed by ABRAHAM [62]

(3.1) g'-'^F^JP- -f - g^'^F^pU^P -f (^^i - 1) gi^^Q^^.f (3.6) 4.T 4:71

Substitution of the components of the field strength tensors H*'" into (3.1) results which, evidently, differs from Sm'"' by the addition of the last term, in which in the space-space part of this tensor being expressed in terms of the field variables, thus becoming

ik S M (3.2) with = F^pvi^, and where is the four-velocity of the medium. The components of 4:71 this new tensor are

Recall that = g^yF^^g^, and that is obtained from F^" by substituting ED and II —> B. The influence of the medium is taken into account by introducing con- ^ + + - +II B)] cepts of the tensors and of dielectric permittivity and magnetic permeability, respectively. Hence, we write .Dj = fjVc^/c and Bi — [XijcHk' It is generally accepted [2] that the tirnc-time, component of the tensor (3.2) refers (3.0 a) to tlie field energj' density 1 D ^ 11' li), (3.3) {!<: • n -f- //./;). Srr Hrr. The space-time components form a quantity which, again, should be identifiable with It turned out, however, that, although the tensor Sb'*^ was symmetric, some conclusions the momentum density of the field, while satisfying drawn from it by that author were incorrect [10]. The question was further examined by MABX and^GYOEGYI [23]. On considering the _ S ' 9a (3.7) field and the dielectric as forming a closed system, they also found (similarly to Beck) that the field energy-momentum tensor, , as given by Abraham, had to be supplemented in accordance with the Planck principle. • by a matter tensor. The covariant form of the Marx and Gyorgjd tensor was Using the prescription (2.42)', for the formation of the force density in the field, we then have, on comparing, the resulting expressions for the Minkowski and Abraham tensors, 1 f 1 AS^/" = — ^ + J g'^F^pH^^ -h {sf^ - 1) Ur (3.12) the relation • , • - V £// 147C N ' f - f I ^^ - 1 ' •• .oos where {Eil - 1) FJI^ -f - F^^H'^^ while z // = Im' (3.9) In the rest frame of the medium, the components of the radiation tensor (3.12) are is the corresponding time componemt. It is clear, from inspection of (3.8), that the Abra- ham force density differs from the Minkowski one by the very small additive term. The macroscopic four-force density is to be regarded as a statistical average of its Bi^ = - B,^^ = ^/o = (3.13) 1 microscopic equivalents. Therefore, the relation f^^' = 8a.''" implies that the macrosco- X3ic energy-momentum tensor, in analogy with the four-force, is equal to the averaged Although the tensors of Beck (3.11) and of MARX and GYORGYI (3.12) appeared to be microscopic version, because the symmetry of the tensor is preserved under the required quite different, it was shown by SCHOPE [53] that, at least in the case of a pure radiation operation. It appears that it was this reasoning that initially inclined PAULI [2] towards field, tliey were, in fact, identical. Nevertheless, while BECK [22] advanced arguments in favouring the synmietric tensor. support of the Minkowski tensor being talvcn together with additive terms due to matter, EINSTEIN and LATJB \21] arr-ived at an expression for the energy-momentum tensor which MARX and GYORGYI [23] accopted the AbrahaTn vei'sion when further supplemented by was completely different from the two previously discussed. Starting from simple a cont.ribution again arising from presence of matter. examples, they first calculated the force exerted by electric, or magnetic, fields on a In an extensive analysis of the question of tlie energy-momentum tensor, DE GROOT and material volume. They then obtained the force due to the electric dipoles. In other words, SuTTORP: [24] adopted an approach which was purely microscopic, i.e., one which was they assumed that the interaction between matter and an electromagnetic field was related ito the behaviour of point particles. According to their thesis, this was the only mediated through some hypothetical elementary electric and magnetic particles. The rigorous way to arrive at the solution of the problem. Thus, they began with a microsco- observed force density then had to consist of two contributions, viz., a part depending pic force law for a constituent particle. The total electromagnetic field acting on such a of the velocity of those particles, and a i^art which did not. Essentially, this separation X^article is, of course, equal to the external field, which the system is subjected, modi- was to be understood as a splitting of the total observed force into a surface force fied by contributions of the neighboiiring particles. Put otherwise, the field may be seen (1 — (i Xi2'ext)/c and a volume force (j X Hxnt)!^. Consequently, the corresponding as made up of two parts. First, there is the external field together with the addition of energy-momentum tensor, was written as ' the field due to the surrounding particles — this field varies only veiy little over the atomic distances. Second, there is the intra-atomic field, due to the constituent particles EiD, + HiB, - - d^'^iE^ + m of the atom. Obviously, that field varies quite drastically over the atomic dimensions. 4:71 (3.10) Utilizing this concept, de Groot and Suttorj;) defined an atomic field energj^-momentum tensor tj"^. Then, from the behaviour of a particle in that field, the atomic material (EXH),. energy-momentum tensor was obtained. They thesi assumed that the atomic con- 471 servation laws had the form / It is to be noted that this tensor was not completely determined, as its time-time compo- dpt''^ = 0, = tf^ + (3.14) nent was not specified. Another'proposal regarding the energy-momentum tensor was put forward by BECK [22]. The tensor thus obtained was, naturally, symmetric, hence implying-the law of In order to reconcile the validity of the velocity addition theorem (which agreed with conservation of angular momentum. If then is the (density) angular momentum the conclusion of Minkowski), on one hand, with the Planck principle of inertia (satis- tensor, we have fied by the Abraham tensor), on the other, he introduced what he called the radiation = m^^y = xH^y — xH'^y. ' ' (3.15) tensor Sp^"' of the form In order to change from densities to observable quantities, a space integration at con- stant time is applied, thus, in turn, leading to a conserved quantity. where represents the matter tensor. This radiation teiisor may be put in a covariant On performing a statistical averaging on such microscopic systems, de GROOT and SUT- form . - TORP [2-i] obtained a macroscopic conservation law ^f the form , • 1 VgV^ S.^iSI^SM"' (3.11) v^vp SM"^ (3.16) The total macroscopic energy-momentum tensor Sq"^ is symmetric and, in analogy with argument, it is, nevertheless, based on sound physical principles of classical physics its microscopic counterpart consists of the sum of field and material tensors. Although and it leads to correct results. the tensor xS'g"^ is an average of t"^, this does not apply to the tensors making up the Sg"^. Consider, then, two identical and parallel channels with no external forces acting on In other words, the macroscopic field (matter) tensor is not the mean of the microscopic them. Let each of these channels contain an identical medium, A beam of radiation can field (n\atter) equivalents. Thus, according to the above workers, the averaging proce- be then chosen in such a Avay that it passes through the given medium in the first dure applies only to the total energy-momentum tensor. channel, but by-passes it in the second. (The selected paths are assumed, for simplicity, The symmetry of fulfilled in isotropic media, breaks down when the system is ani- to be parallel.) Now, it is obvious that the speed of energy propagation in the medium is sotropic. This is clearly seen when the tensor is written in the three-dimensional for- smaller than that in the empty sjmce. Hence, the time a photon takes to travel a certain malism distance is diffei-ent in the two channels. Put otherwise, a photon (and so the mass- /I 1 associated with its energy) in the first channel requires a longer time to cover that o ik EiD, + - 4:71 lengtli tlian does a photon in the second channel. Suppose, the medium does not move (3.17) while a wave is travelling through it. Then, the displacement of the centre of mass of each channel should be different, in spite of the fact that, no external forces are present. This, however, contradicts the velocity invariance. Therefore, the medium, during the 8:r passage of an electromagnetic wave through it, must have been displaced by the same amount. It means, in turn, that the medium has a certain momentum and, due to the where 31 — B — H is the magnetization vector. - conservation of the momentum of the total system, value of that momentum in the In polar substances, where 31 — 0, the tensor So''" coincides with the Einstein and Laub mediinn is different from that in vacuo. Put otherwise, as the displacement veJocity of the tensor as seen on comparing (3.17) with (3.10). centre of mass of the sj'stem is the same in both cases, its position of the first channel is De Groot and Suttorp {24\ proposed also an alternative energy-momentum tensor. shifted further from the initial one. This, thei'efore, implies that the medium itself nuist That tensor {So'"'') was defined as the difference between the total energy-momentum also jnove in the same direction. Hence, it may be said, the momentum is transferred to tensor in the presence of the electromagnetic field and the corresponding tensor, when the medium in the direction of motion. This, however, would not be 2:)ossible had the such a field was absent. The last of these wks, of course, identical with the material electromagnetic field momentum in a medium been that given by the Minkowski ex- tensor. Assuming linear constitutive relations, the new tensor had the following compo- ])ression (3.5). (It has been tacitly assumed here that tJie refractive index of the medium nents in the rest frame ; \ is greater than unity.) 1 The above may be expressed quantitatively, as follows. If Xc is the :r-cooj-dinate of tlie 'Sz,^^ = - centre of inass of tlie system, W — the mass of the medium, and tv — the relativistic ^ mass equivalent of the energj^ carried by the wave, then the velocity of the centre of mass of the wave-medium system is (3.18) 4 = {W + d,x,)l{W + I..), (3.10) E.D + H-B + E-^T^ + b^T^ where z^ and X2 are the coordinates of the centre of mass of the material medium and tlie Stt Avave-field, respectively. If, further, {8tXc)Q is the value of the velocity of centre of mass before the interaction, then where V stands for the specific volume, T for the temperature, and the electric ji: and magnetic suscepti])ilities x are defined sls P ^ xE and 31 = x^- ^^ is assumed that the - {8tXc)o - wc/{W + w)., (3.20) derivatives of the material constants occurring in (3.18) vanish and the medium is iso- tropic, then it is immediately seen, on comparing (3.18) with (3.6a), that Sd"" is identical Prom the momentum balance, it then follows that with S/'-". All of the above described energy-momentum tensors occur frequently in literature. In W diXi {hv/c) — jj, (3.21) the section below, the just outlined forms of these tensors will be examined and the argu- where p denotes the momentum associated with the electromagnetic field inside the ments put forward to support them will be discussed. medium. Let {dtXc)i be the velocity of the centre of mass, while the wave-medium inter- action is taking place, i.e., while the wave travels throtigh.tlie medium. Then, com}>inincr 2. Theoretical considerations (3.19) with (3.21) leads to

We shall now examine the applicability of the various tensors to the description of (O-^-c)i = ({hvfc) —20+ {ivcl7i))l{W + w), (3.22) different physical phenomena. Also, as conclusions reached in numerous papers, pub- lished in "many diffei-ent journals, are quite contradictory, it will be instructive to in- whcj-e fifX, = c/n is the group speed of the wave motion. That speed, on the assumptioji vestigate, at this stage, the different approaches employed in an attempt to resolve the of a simple medium, is identical with tlie energy (and, hence, mass) t rans].)ort. pr<'S(>iit. dik-iuiua. Now the j)rinciple of invariance of the velocity of the ceritj-e of uiass requires that PropotuMils of a syniin(>trical field euergy-monientuiii tensor often refer to a simple ihuU'.'ht v P.A».A7,s {^V Although slnunnng froiu a ratiier naive iCL "txii t.jio r&icvanC cxuressiori 7TitO (3^33) ind on applying the principle of mass- volume integration of that force results in f^'^'^^gy equivalence (iv = hv]c^), we obtain f dV = f (Ex II) dV/c\ p = hvjcn, (3.24) From this, it directly follows that the electromagnetic momentum density in a non- dispersive linear medium is {B XII){c^. which is the Abraham expression for the field momentum inside the medium. The momentum exchanged between the field and the medium, is of course, equal to The above considerations can be summed up in the following way. The total momentum difference between the field momentum before and after the interaction has taken place. of a system (radiation plus matter) can be uniquely split into two parts [9], with one part Haus then argued that the momentum density possessed by the electromagnetic wave referring to the change in the field momentum and the other desciibing the increase in packet travelling in a medium need not be necessarily given by once the said ' the mechanical momentum. It can be then concluded that only the symmetric field exchange is taken into account. energy-momentum tensor simultaneously satisfies the momentum conservation and the Furthermoi-e, it is shown that in the case of normal incidence at a sharp boundary be- centre of mass theorems. tween the vacuum and the medium, there are no singularities in the force density Essentially the same thought experiment was also used by other workers. Thus, e.g., Consequently, there aj'e no surface forces. This implies that all of the momentum is then BURT and PJETERLS [11] employed this argument for illustrative purposes. They pointed imparted to the bulk of the medium. Hence, he concludes that all of the momentum is, out, however, that the controversy Avas not adequately resolved by the above reasoning, contained in the incident, reflected, and refracted wave, as none is left on the stu'face. as the assumption of a rigid medium was involved -i- the concept incompatible with the These considerations apply to a medium with negligible electi-o- and magneto-striction. special theory of relativity. ' ' Thus, finallj^ it appears avS if neither cja, given by (3.7), nor from (3.5), could rep- A similar analysis was also put forward by J ONES [14]. Utilizing rather simple concci)ts, resejit the wave momentum. According to Haus, the correct value for the momentum he attempted to find out how the momentum of radiation inside a medium was distribu- is given by (3.7) supplemented by terms describing a mechanical momentum, attribu- ted between its electromagnetic and mechanical parts which then led to the conclusion table to a medium. In addition, the energy flux density must also be increased by a that the total momentum associated with the radiation field should be'that given by contribution from tlie medium, ' . the iMinkowski expression, i.e., y = nkvjc. If, then, refers to the group refractive in- The conclusions reached by Haus, essentially, agree with results of SHOCKLEY [15]. That dex, the fraction p/n^ {p — momentum in vacuo) may be (according to Jones) regarded author, using an ingenious model, succeeds in interpreting tlte connection between the as residing in the wave and, hence, constitutes the electromagnetic part of the total macroscopic laws and the structure of the medium. The electromagnetic theory is, first momentum, while the part n2-)[l — {Ijnvg)) {n is the conventional phase refractive in- of all, expanded to includc the magnetic equivalents of elcctric charges thus, naturally, dex) is, according to him, to be associated with the mechanical momentum present in the leading to the magnetic analogue of the Lorentz force [54]. Consequently, the hidden '••body. It should be noticed that, in order to arrive at this conclusion, Jones introduces linear momentum is introduced. Its application to the rolativistic energy-momentum . an extra (group) refractive index, n,„ thus, complicating the issue. Besides, in the ex- tensor is discussed [io] and, in conjunction with the centre of mass theorem, points to the periment of JOKES and LESLTK [^6'], one of the conclusions was that the refractive index conclusion that, in stationary matter, the electromagnetic momentum density is correctly should be given by the ratio of the pliase, rather than group, speeds. The above analysis given by the Abraham expression. This is supported by the experiment of JAJMES [55] by Jones seems, therefore, rather unsatisfactory. whose results definitely reject the Minkowski form. A more profound approach is that of HAUS [I(5J. In order to arrive at the expression for The physical realitj^ of tlie magnetic dual of the Lorentz force was previously discussed the electromagjietic momentum density in a linearly polarizable medium, he seioarat^s by COSTA DE BEAUKEOAKD [56] in his attempt at analyzing the GOILLOT [57] experiment. the total system into a part describing the motion of a mechanical continuum and Although the theoretically deduced effect was qualitatively similar to that observed in another part corresponding to electromagnetic and thermodynamic properties. Then, the the exj)eriment, its magnitude .was much smaller. In Costa de Beauregard's thought requirement for the mechanical momentum conservation of the continuum takes the form experiinent, a ferromagnetic tubular test body and an axial wire with a constant density of charge are considered. Calculations of the force acting on the test body [5

y if^y -P) (1 - M (3.27) (3.33) — ifty y n'\i{7i - ft) {I - ftn) -{n According to (3.Ga), the Abraiiam tensor in the rest frame takes the form [76'] Let us next consider, together with Skobel'tsyk [10], the second limiting case, viz., u nijn^ that of a "dust-like" medium. In that case, all stresses vanish and the corresponding (3.28) tensor is then a product of the momentum density and the velocity of the medium. This i^ijn —u assumes that the mass of each of the constituent particles is approaching infinity, while where u is the energy density. Similarly, in view of (3.3), (3.4), and (3.5), the Minkowski their corresponding number density is rather small. Consequently, in this approximation, tensor in the rest frame is written as the refractive index may be taken to be independent of the intensity of radiation (as the particle mass is very large). In addition, the radiation is not reflected at the incidence. u inu\ When an electromagnetic beam enters the medium, the surface of the latter moves to- Sm^^ = wards the source with a mechanical momentum [j?(9] \iuln —ti j' (3.29) = (w — 1) hvjc. (3.34) Transforming and according to (3.26) and using (3.27) leads to It should be reinarked that the same expression is obtained from both the Minkowski 'n{l -f _ 2ft 1(1 _ 2l3n -f ft^) and the Abraham tensors. \i{l _ + -[n{l + ft-^) - 2ft] (3.28a) Applying now the conservation equations (3.19), (3.21), and (3.23), the momentum p^ of the material component of the system becomes and

^ J{n- ft){l -nft) i{n-ftf i^i — (w — 1) hv/cn. (3.35) (3.29a) n'^ \i{i-nftf - ft) - nft) This momentum, however, is equal to the difference betVeen the medium and surface momenta. Thus, addition of (3.34) and (3.35) leads to the expression These two matrices then describe the corresponding two forms for the energy-momentum tensor in a moving inertial frame. These expressions will now be compared in model = {n- — 1) hi'lcn (3.3G) situation describing the limiting behaviour of matter, viz., the cases of a rigid body and of a rarefied medium. for the momentum of the medium. In the first of th(;se (the case of rigid body) the total mechanical tensor is represented Furthermore, if the particle number density in the absence of the field is then, at hy the stress tensor, whereas in the latter case it is rewrescnted by the product of the som<; later time t, it becomes !i\(.ni(Mituin density and tlio velocity of the medium. Tlius, in the fornier case, the £>(0 f k>- AQ {N^ — L)UO{T)LD^. ^ (3.38) Conscqueritlv, = thus invplyiuo; the symmetry of 1 ron. the requuthat li the energy-momentum tensor of the total system be syunuetno, it follows tha. also .ne The mechanical energy-momentum tensor may then be obtained in the form [10] mechanical tensor must be symmetric. Hence, Skobel'tsyn concludes that the coirect expression for the field is that given by Abraham. • It should be noted that, initially, von Laue also [67] accepted the reasoning of Abraiiam. However, in a later aiticle [G], he developed an argument which (so he believed 1 re- (3.39) n- — 1 jected this tensor and replaced it by the Minkowski one. Calculations of Scheye [68] — — {n'^ — 1) even further supported this idea. The following considerations led to this change. n It is well known that, in general, when plane electromagnetic waves ])i-opagate tln ougli a On subtracting the value of TJ''^ in the absence of the field from (3.39), we'obtain an medium, their phase si)eed is different from their group speed. Further, if the absorj^tion expression for that part of the tensor wliich is related to the field in the medium is low, tlie latter speed may be identified with the spr>ed of energy pro- pagation. \'oN Laue n" [C] required the gi'oup velocity to satisfy the velocity superjDOsition Voit) theorem due to Einstein, and proceeded to show tliat only the Minkowski tensor ful- (3.40) filled this condition. Moller [7] developed a more general version of the same argument. Thus, if is the electromagnetic energy-momentum tensor, then the velocity of the -{n^- - 1) Uo{t) energy propagation is defined by = icS^'jS'K Moller finds the condition for this velocity to transform like a particle velocity to be where (^QC — {ii^ — 1) ujQ(fin. In the limit of infinitesimally small number density, the = 0, (3.45) above tensor reduces to 7^2—1 /O i Wo(0 (3.41) where /t'"' Si" + \Ic- and represents the four-velocity. He then sho\^-s tiiat \i —n only the Minkowski tensor satisfies (3.45) and, thus, concludes that this tensor must gi\-e an adequate description of electi'oniagnetic phenomena in matter. A subsequent transformation to the moving inertial frame yields Skobel'tsyn [70], nevertheless, doubts the validity of this criterion. Next, using tlie thermodynamics of irreversible processes, Tang and IMeixner [60] ana- (3.42) 13-zed the influence of electromagnetic waves on a simple fiuid. The total energy-mo- mentum tensor obtained by them consisted of three contributions: a constant fieki term, The energ3^-momentum tensor corresponding to the whole system is then constructed, an oscillating field term, and a constant matter term. Noting t!iat, in the absence of the in analogy to above, resulting in ^ field, the velocity of the. enei-gy transport does not vsatisfy tlie addition of velocities theorem, they s])]it the total energy flow in the separate parts and then examine the von (n/? - 1)^ i{7i — /5) (1 — nft)\ Laue undMoller transformation criterion for these parts. They cora-luded tliat ^Molier's (3.43)- treatment was incomplete and suggested that the Abraham and Minkowski tensors lead to the same result, provided the matter contribution is j)ro])erl3' accounted for. Together with Skobel'tsyn {10, we can now compare the two expressions. (3.33) De Groot and Sitttorp [24], however, claimed that what Tang and Meixner [60] did. and (3.43), for the total energy-momentum tensor applicable in each limiting case. It is in fact, disj)roved the very validity of the von Laue criterion. Recently, this last state- interesting to note that those forms differ from each other only by the factor Ijn-. ment was strongly criticized by Brevik [/(Sj. We can now see the reason for considering two limiting cases of media, in which the Let us concentrate now on static fields inside a medium, hoping to obtain at least son>e interaction with field is to be investigated. For, clearly, the splitting of the total tensor correct results. We adopt a variational approach of Brevik and assume that the field of the system into contributions corresponding to the field and the matter, is not simple energy is known, A similar method as also used by I^andau and Lifsciiitz [5"]. [17, 18, GO, 66]. Skobel'tsjm, therefore, in an attempt to overcome this difficulty, con- The free energy of an electrostatic field in a medium is defined, in the usual M tly [5], b\- siders two extreme model situations, viz., these of rigid and "dust-like" 7iiedia. This makes it possil>le to divide up the total energy-momentum tensor uniquely into field and matter E • DDV. (3.4G) parts, provided that the field energy-momentum tensor satisfies S.-r

(3.44) Now, the rate of change of the field free energy may be interpreted as equal to mechanical work J /• V dV {v — velocity, / — force density). One thus obtains the expression for (in accord with the discussion in Chapter 11).' , - the force density Skobel'tsyn, further, claims [10] that the momentum density S^'^ is obtained from the 1 ^ „ „ 1 invariance of the velocity of centre of mass together with the momentum conservation IA = QE - ^ EIE.VSI, + — 8,{ED, - E,B). (3.41 law, while to determine the component one uses (3.44), (in the form = 0). 8.-r I a' = so (3.47) on applying the above variational procedure, we are led to the Helmholtz force

de fn = qE - -^ EWs + (3.52) ^71 dp,. o;T (3.48) Comparing it with (-3.6ca) shows n t^ K fi. " where the last term refers to the dependence of permittivity on the mass density tenso,. h. the statio\i„.i/. Chi It., fo, ot Sr? ^ the Abraham Using the Clausius-Mossotti relation, (e — l)/(£ + 2) = const, q^, we then calculate the excess pressure inside a dielectric as [-30]

(3.53) even while the external forS^Tr^f Th^'^ r"" bimonot electric dipok.si„ a „,edian/:d«bkZ "hen there is a distri- On the other hand, the excess pressure connected with (3.51) is il>eii, given the torque, there is an I ^ Pohrizat.on vector P = D-E tl>e torc,„e density r fl x / r^;;'™ 'X'"' 1 (3.54) work. Putting the rate of chan'e ort e W ^ eontr,l,„t,on to the total meehanica tnne. lead, to the Minkow.Ki The experiment to test the above n\entioned expression for the excess pressure in the presence of the field has, in fact, been performed by Hakim and Higham [30], and will be discussed in the last section of this chapter. It will be sufficient to say here, that it (3,49) has firmly established the validit^^ of (3.53). In view of that last finding, it can be asserted, that the Einstein-Laub (and also de Avhile the stress tensor takes on the form Groot-Suttorp) tensors yield results which are (at least for static fields) incompatible with experiment. ik Consider now a plane wave travelling in an isoti-opic and homogeneous medium and E,D, - . D 471 (3.50) follow the arguments of Laxdau and Lifsiiitz [,§] about the construction of the energy- momentum tensor from the inacroscopic considerations. Those autlibrs assumed that the total energy density is the sura of the electrostatic and magnetostatic energy densities and that the same also applies to the stress tensor. If the frequency of the incfdent M'ave is Juuch lower than the characteristic frequency of electronic vibrations, then, for not too strong fields, the relations between thefield intensities and corresponding inductions Yet another distinction ben'^, th!' ! quite natural, remain linear. Hence, the above described construction of the energy-momentum tensor equation is recast in ten ^^ he ^^ balance is yalid. satisfied only by the Minkowski t n "^^^^^ t^ll^tj '' ^^^ ^^ It is found from experiments that the electromagnetic wave travelling through an insula- after integration, has been added to it the ea nli? / . vanishes tor does not produce, in the first approximation, any heat, as it undeigoes an elastic scat- well. This means that the Abraham In^tv^ f ' ^^e Abrahan. tensor as tering. This, therefore, means that the time component of the four-force vanishes. Conse- in the Minkowski case it is r^sen^^^^^^ mc udes the effect of torques, while, quently, the continuity equation is d(V + P • S == 0 and, from the expression for the tially, one of the n.ain CfS] ^^^^^^ ^^^^^ ^' energy density, zi = (E • IJ -h II • B)/8:z, one obtains the energvflux as S' = c(E x //)/4.-r. Extending the validity of the Planck law of inertia, S = c^f/, the expression for the ^.s in the time-inde- momentum density becon^es g {E X II)lc. Hence, the enelgy density, energy flux, and uiomentum density constitute the Abraham tensor, as is seen fronr(3.6a). ;tKs quite sufficient to consider only one pro,^^^^^^ Further, if the mechanical part of the energy-moTuentum tensor is then A\ e can write Jation of the force density from the LualK?^]' ^' . ^^e caleu- for the so called Kelvin forceX.sit3 equations then leads to the expression: (3.55) Of course, the electrostrictive and magnetostrictive effects are not considered here. A {P . V) E. Combining (3.55) with the corresponding relation for the field part of the total tensor! (3.51) d^S/"' = —allows to interprete -f w'") as the total tensor describing a closed system. This tensor is synmietric and so are its constituents. b^^^^ (3.49) .smts, but is For Brevik [25], however, tlie argument does not end there. He believes that, when ixo- perly interpreted, the Abraham and Minkowski toisors will be equivalent in the naajo- nty of situations. He advocates tlie .synunetry of tlie total energy-momentum tensor of Ta tbat case, ifis ^^ the wiiolo system but its separation, merely motivated by convenience, into two asyns- metrical parts is not well suj);)orled and remains ratiu«r dtihions. to oqu.tt« fho ^AbiaJia^TT JVIh.kowski results. Kloiuentarv considemtions in svipport of iJioposal cl^v'^t^on^^ Observed to be acting on the Abraham xvevo pvit iovsvavO. hvliUKT aAdPEiERLS [ /i]. It is accepted that, as the wave propagates through a medium, i 1 r^sfon fh^n^r J " Reaction to this force results in a mechani- the motion of atoms of that medium is affected, so that these may acquire additio- c sties.. On the other hand, the meaning of the Minicowski force is rather unclear some nal momentum. In their discussion of the total momentum, bodi authors, however, A other, paper deahng with the radiation forces in dielectric media, written by GORDON did not consider this mechanical contribution. In other words, their remarks aj^ply to a demonstrates that the true nxomentum density of the e]ecti;magnetic field a medium which, effectively, does not disturb propagation of light in it. on-dispersive medium is, indeed given by the Abraham expression. Gordon adopt^^ a In a more recent paper, TEIERLS [13] rectifies this shortcoming. He begins with a dis- oJ H^f " ^^^^^^^^^^ polarizahle medium, such as a gas cussion of the forces acting on the atoms in a medium. In agreement with the result of hea^ atoms. He further, assumes that the corresponding permittivity only sli^rhlly found previously by Gordon [12], the electromagnetic momentum density, he asserted, 'Sdv d!^:^^'1 h propertie^s of the medium are co n"^ is equal to the Abraham value. He arrives at this conclusion by realizing that most re- for e i^;"^^ polanzabihty .. xXow,, in the rarified gas, the ponderomotive fractive media are non-magnetic, so that it is permissible to write /.'oi/micro = lorce is smipiy the Lorentz force, which can be written as [i^] scribing the macroscopic magnetic induction as an average of the microscopic intensity i/micro- Assuniingthemicroscopic validity of the Poynting vector and E^i^ro = E, leads then directly to the conclusion Qf = E X ll/'c-. (E.fyjE-i-l^Xlf (3.56) c dt In order to determine the total momentum, the value of its other, i.e., mechanical com- ponent part must be found. This is obtained by considering the force with which the Y^'atoms^rr'^ ^^^^ density due to field acts on an atom. Thus, if d is the atomic , then this force is

(3.60) n — 1 .^^mech = {Ex 21). (3.57) where E^n and are the effective field strengths at the site of the atom. Peierls then argues that, in a non-magnetic dielectric, iieif may be replaced by the macroscopic field intensity B. In addition, the difference between the effective and external electric field Prom the laws of conservation of energy and momentum, Gordon then obtai ns is proportional to the polarization. Making use of the Maxwell equations and calculating = 0 + gmcci,, the force on all atoms in a given volume, he finally obtains for the density of mechanical momentum the expression thus mterpretmg g ^ E X Jl/in.c as the electromagnetic momentum density. Hence the possible separation of tlie total moruentuui into field and matter parts leads in this — i fi'mech — E X //. (3.61) inodel, to a complete solution. Although he proceeds with generalization of the gaseous 2c- state to include lluids and solids, it docs not appear that his model will lend itself to an easy generalization, as it basically applies only to media with low polarizations. Cleai-ly, j/mccn represents the momentum density in a medium while an electromagnetic A (.niierent resolution of the present dilemma is advocated by GINZBURG [7.91 Altlioucdi wave is travelling within it. Peierls proceeds then to derive expression for the total he consiaei-s that no objection can be made against the Abraham tensor, which is mo"e momentum density iundamental, and correct in the general case, he, nevertheless, attempts to explain why {71' - 1)^" he quantum mechanical analysis of the problem leads to the accepted conservation (3.62) aws. ihus, first of all, he derives the laws, using the Hamiltonian method, and proceeds n 07l then to comparc them with the results of macroscopic electrodynamics. From the point of view of the latter, the conservation laws whei'e he assumes that the light pulse is wide and short. He obtains (3.62) l)y adding to- gether the mechanical and the field momentum densities. The factor 1/5 in the last term of (3.62) results from assuming that all dipoles arc randomly oriented. That value is ik (3.58) found to be valid in the region of optical fre(j[nencies, when the motion of atoms during and the cycle can be neglected, i.e., when the atoms may be treated as stationary. A value of 1/3 (instead of 1/5) was proposed by LAXDAU and LIESIUTZ [, (3.63) moving matter. Consider a nearly monochromatic pulse impinging on a dielectric slab characterized by wL'li'cdlSion^^ this result is very different from the Min- constants e and /.i. It is assumed that the wave energy is conserved, i.e., the kinetic energy associated with the transferred momentunr is much smaller than the rest energy of the medium. Wave incident from vacuum is partially reflected and partially transmitted at f made use of the the boundaiy. There is also another term in the momentum balance equation, viz., one describing the momentum imparted to the surface. By writting down the momentum lace, ^Miile assuiuu.g tliat a dielectric is lossless, they obtain for the surface force density^ conservation law at incidence, it is found [17] that the sum of all surface momenta gives the total momentum transferred to the slab. It is, in fact, equal to the difference be- (3.G4) tween the initial and the final momenta of the waves, i.e., those just before an entry and just after an exit. jhere E refers to the incident wave. It is to be noted that, as this force is due to dioole Assuming that the total momentum of a wave pulse in a medium is (xlivjcn, and that the

^RIB'SRAR^^^^^ ^ ^^ ^^^ - ^^ • transferred momentum is ^hvjcn, where the coefficients a and are related to e and it tne ADiaham ar d the Minkowski expressions give only the first order dependence is found that [77], in view of the center of mass theorem, a — ^ = I. Thus, there remains a part (a — /S) hvjcn — hvjcn of the total projiagating momentum which is not associated Tne aeri^tion [72] of the surface force (3.64) does not take into account' tL'iLnce -^^-nal field and which puJE" with matter and may, therefore, be regarded as corresponding to the field. However, the Tretn ^ ^^ere are two different momentum fluxes expression for oc is not known and this makes it impossible to attach a definite value to the vdodtV W t la of th"'" ' e ectromagnetic wave, propagating with the group surface momentum. If it is assumed that this value vanishes at normal incidence, then Lnn ^II „ momentum of the pressure wave, travelling with the speed of oc = 1/2(£ + fji) and the total momentum travelling through a medium is sound. Above authors [72] explicitly consider the former case only ^ "" ^^ " generally assumed that the hv , piopcitits of the latter do not change during the interaction. Naturally, this is not the (3.65) ^ " ^^^^^^ - --dium with a variablc^^olanz Uh yt Such a anatioa is, in genera , associated with changes in the parameters (e.g., shape, vo- In his earlier treatment [ 7-3], PEIERLS considered normal incidence and only those media X ^^^ --'^ing change in the dipole eneigy inav grckt^y in which the difference bet\veen jx and unity could be neglected, to that the refractive ?o fn I hence, the corresponding radiation moment.nn. ft Ts then index was given by n = ]/£. Consequently, for media where n — 1 is small, the results Con" n f ' in polarizability is proportional to (3.62) of Peierls and (3.65) of Robinson are identical. Consequently depending on the sign of the deformation forces n>ay either enhance As ali'eady indicated, the Helmholtz method of virtual v,'ork was generalized by PE>'- or dinumsh effects due to the ordinary radiation pressure EIELD and HAUS [cf. 77]. They included moving, dispersive media, by treating the mecha- ^I'f^oaoh is e.nployed by ROBTXSOX [77]. Although he realizes the nical aspects of the system from a relativistic standpoint. They claimed their so obtained comp.exity of problein at hand, he claims that the solution to it already exists and that solution to be complete. The essence of their argument is as follows. t has been dismissed by PENFIELD and HAUS (cf. [77]). This is done by extending the The radiation field together with the surrounding medium form a system which may be pnnc.ple of virtual work, used originally by Helmholtz. Robinson, for reasons stat?d in divided into two parts, according to the way in which the energy and momentum are his articlc, describes that approach in a "digested" form. The macroscopic properties of matter, m his view, are described by dielectric permittivity and magnetic permea}>ilitv. stored. In one part, this is related to the motion of the medium, while in the other, it is He emphasizes hoM^ver, the importance of the electro- and magneto-strictive term's connected to mechanical or electromagnetic stresses. Clearly, these subsystems are not arguing that the effect of the electronic structure of matter on the electroma.aietic closed and, hence, there must be some mutual interaction. The conservation laws have, phenomenon cannot be completely described by the constitutive relations therefore, the forms Manipulating the Maxwell equation, Robinson arrives at an identity when he replaces o + d^g^ ^ f, + = 4>, (3.66) and J, in oL ^ j x B, by F - I) and Fx II- d,D, respectively. In Regions where polari: zation P and magnetization J/ vector densities vanish, the obtained equation is then where is the momentum flux tensor, (/ the momentum density, and cj} refers to the interpreted as a momentum balance equation. power transfer density. The tensor T^' includes, however, not only the electromagnetic but also the mechanical (e.g., electro- and magneto-strictive) variables. This, in"turn, Different considerations yield different identities. However, a unique solution cannot should be understood as pointing to the feasibility of splitting up the total momentum oe revealed by merely manipulating the field equations. Penfield and Haus consider the into its field and material contributions. This follows from the fact that the rate of possibility of extracting some relevant information from the general conservation laws. change of the mechanical'momentum density must now include not only the spatial Ihe solution for particles moving in vacuo is rather well known and this helps the ana- lysis. ^ derivatives of fZ'^"' but also dig^'. The most natural interpretation for the latter lerni is. then, to identify it with the rate of change of the field momentum density. Tlic thro,'v of relativity leads to the velodty invariance of the ccntre of mass of the Regarding the Abraham-Minkowski controversy, Penficid and flans coucludcd that f/\ syshMM. d Ihrcnrvgy of a \vuv mass altrihntcd lo this wave hvjcr. although not referring fo the motion of matter, is important in <-aici//afing Forces rxcrtc ! by tho Held on inaterinl pnvticlcs. Robinson agrees that, in some case, the Minkowski wlicre nioiuentuni density may enable one to find the total momentum tranpferred to a me- dium, however, he advocates [17] the Abraham formula as the rigorciisly correct one. A variational approach using the Hamiltonian principle was recently presented by Mikura [20]. Although it is well knoAvn that the Lagrangian density cannot be defined is interpreted as the field energy-momentum tensor. uniquel^^ its form is suggested intuitively and, then, justified a posteriori. Thus, the To fi'nd the conservation of the total energy and momentum, IMikl'Ra [20] defines the material part of the Lagrangian density is tensor for the total system, although not uniquely, as the sum of tlie tensors correspond- ing to the subsystems and, then, combining (3.70) and (3,71), obtains c.S^' = 0. (3.67) It is to be noted that the definitions of S,/" and aS/" are not properly justified on the physical basis. As already indicated, the total tensor may be split into two parts in Here, is a four-vector of mass flow density, i.e., W^ {qv, ioc), Q{Xf,) is the mass several different ways, if the formal modifications are ipade in the conservation laws density, — the velocity of particle. Further, is the" non-electromagnetic for the sufjsystems. Mikura, then, attempts to find a physical criterion which would internal energy and s refers to the entropy. make this tensor splitting into material and field parts unambiguous. He j^ostulates, Another component part of the Lagrangian density is due to the field. It is usually w ritten therefore, that: 1. the part of the tensor which has a form a of pure material tensor as (2.16) represents the material component is the total tensor; 2. the jJart identical with the field tensor in vacuo is associated with tiie field contril)ution to S'^"', 3. tensors witli non- (3.68) 16.T ^^ • vanishing space-space couif)oneiits, while the medium is stationary, must belong to the field tensor. He then finds that both the Abraham and JMinkowski tensors satisfy these The interaction Lagrangian density is proposed on a basis of rather intuitive arguments. requirements. Thus, the values of tlie electromagnetic momentum flax density are the If and are polarization and magnetization tensors, respectively, then this den- same in both cases (if the medium is stationary), although the momentum densities are sity takes the form different in the two cases. liecause the radiation pressure is determined bj' the flux density, rather than })y the mouientum density, the result is the same, regardless of the ffi = -H — - - ^ - A^o'JI"^*. (3.69) choice of the (Abi-aham or Minkowski) tensor. -Mikui'a, nevertheless, favours the Minkowski expression which, in view of results of ex- |)ei'iin(>)!t.s [2-5, 27] should i-epresent the true radiation momentuui density. The radiation The first term of (3.69) describes the usual interaction between free charges and the pressure is tlien given by the piwluct of this momentum density and the velocity of field, while the second and third terms refer to the electric dipole-dipole and the field- light in matter, i.e., (/> X /J)/47r??., in disagreement with the generall}' accepted result dipole interactions. The last two terms are the magnetic analogues of the preceding two {E X 11)14:7171. Moreover, Mikm-a claims that there is no need for the mechanical pressure, ' teriiis. Further, lU"^* denotes tensor dual to M'"'. The ])arameters « and /?, wliich depend {D X B — E X II)/4rjin, as, in tiie steady state experiments, the time average effect of on OQ and s, characterize the assumed linear properties of a medium, viz., £=14- 4.-t« the electi'o- and magneto-strictive forces is balanced by the elastic hj'-drostatic force. and ,u = (1 - Tliat pressure could be, however, im])ortant in ti'ansient phenomena. Tlie Hamiltonian principle is next ax)plied to derive the ec^uations of motion. Mikura Mikura's discussion of the acoustic radiation pressure, caused by the density waves, then shows that the sum of the above postulated component parts of the Lagrangian shows it to be smaller than the electromagnetic radiation pressure by a factor EQ'IQVS- density leads to the well established formulae for the equations of motion. These are the (r, — speed of sound in a medium). As this factor is rather small, the mechanical pressure hydrodynamic equations for the material, the constitutive relations, and the field equa- arising from the density waves is negligible in comparison with the true electromagnetic tions. Clearly, they are mutually independent and, thus, the energy-momentum tensors pressure. derived for different subsystems are also mutualty independent. In other words, the. Thus, according to Mikura [20], although there is no unique way for splitring the total energy-momentum conservation law for the material part is derivable from the hydro- tensoi- when the mouientum flux density, instead of the momentum density, is dynamic equations, w hile the corresponding law for the field is obtained from the field taken into account in calculations, the radiation pressure in matter can be uniquely equations. Thus, the conservation law for the complete system is a direct consequence determined. of the conservation law for individual subsystems. Noting the fact that the group speed of light waves is reduced in matter and asserting An explicit calculation leads to the following form for the material part of the energy- the equivalence of the mass and energy, together with the conservation of the totc5 momentum conservation law energy and momentum, Thorxber [74] was led to postulate the existence of transverse forces on matter. Siuiilarly to many workers before him, he was supporting the Abraham (3.70) hypotliesis. While analyzing the radiation pressure on a multilayer medium, Akxaud [7o] generali- Here, the right hand side describes the effect of the field on the material medium, i.e., it zed results obtained previously bj^ Haus [76]. ^ represents, in fact, a force density. Arguments based on the adiabatic invariance and the energy quantization are used bv Similarly, the field part of the energy-momentum conservation law is derived as Joyce [7C] in the analysis of the radiation force. From the operational stand-point be distinguishes among the impulsive-, the system-, and the energv-earrvincr momentmn. Furtlier, he differentiates between bare and clothed particles and shows that the theorv (3.71) applies not only to photons but also to phonons [77]. In fact, the theorv eneompis^Js both zero and non-zero rest mass particles. Thus, he argues that the Abraham momen- It may be possible, in order to throw some light on the energy-momentum tensor con- tum refers to a bare particle, whereas that of Minkowski describes the clothed particle. troversy, to" regard the asymmetric canonical tensor and the Minkowski tensor as. As the essential point of clothing and localization is often confused when interpreting describing the same quantity. Indeed, NOVOBATZKY (cf. [23]) has shown that, in the free observations, it is, clearly, necessary to distinguish carefully between these concepts. It field, both tensors differ only by the divergence term, so that we may write would appear [76\ 77] that, as clothed particles are the only ones observed, the corre- sponding theory must lead to the Minkowski result. The classical relativistic field theory served BEWAR [75] as a vehicle to analyze the (3.74) splitting of the total energy-momentum tensor into its components and also to find the force density acting on a medium in any state. In analogy with the canonical and physi- cal momentum for a particle, Dewar distinguishes between the canonical and physical However, this does not hold in the case of an open system. Therefore, it is not possible energy-momentum tensors. [23], in general, to identify the canonical energy-momentum tensor with a physical He, first of all, obtains tlie conservation equations for the whole system using the No- tensor supposedly symmetrical. This proposition hinges upon the fact that the equation ctlier theorem. This, as is known [7.9], relates the invariance of equations of motion to the of conservation of angular momentum applied to a spinless (classical) system leads to sj-mmetry transfoi-mations on the system. It employs two symmetry postulates. The the requirement that the energy-momentum tensor be symmetric [84]. first of these re(iuires the Lagrangian density X to ])e invariant under space-time trans- An interesting way to approach the taks of finding the cnergy-inomentum tensor was lations. The second postulate requires X to be invai'iant under rotation. It is then shown suggested by BOPP [49], His investigation is based on the following principles: that the angular momentu-m is conserved and the energy-momentum tensor is synnue- 1. the field energy flux is proportional to the corresponding momentum; 2. the spin of the radiation field contributes to the total momentum, but not to the energy flux. These trized by the method due to BELINPAIS^TE [47]. requirements were questioned by HEHL [6'-5] as they, apparently, did not obey the rela- The thus obtained synmietrical energy-momentum tensor O'"' is then split up in two tivistic invariance. Nevertheless, BOPP [5^] succesfully defended his original arguments. alternative ways. The first of these, DEWAR [78] calls the pliysical split-up. It has the Thus, the total energy-momentum tensor consists of the linear and the spin parts. The advantage that the bactkground energy-momentum tensor is not modified by the presence latter then corresponds, in the macroscopic pliysics, to the phenomenon of magnetiza- of fields, as all interactions arc embodied in the term describing the force density on the tion. In other words, the spin-angular momentum, as well as the magnetizing current in medium. Hence, the following equations apply to the subsystems macrophysics, are not linked with the energy trans23ort.

= fir, /a-)' (3.73)

Here, f,;' is the force density acting on the backgrotmd medium, also known as the pon- 3. Experimental results deromotive force density, wlnleis the physical force density acting on the A:-th sub- system. In tlie p)-eceding pagfvs the arguments put forward by vaiious workers A\-ere discussed The second \^'ay of splitting up the tensor involves identifjdng the canonical energy- purely from the theoretical poijit of view. It appears that the present situation is un- clear, although more woi'kers tend to favour the Abialiam tensor. However, the final momentum tensor \\'ith tlie background mediTun. DEWAR [7f cuLitions are, \p,)fo\ 1 unaU-\y, not gi-ncral. api)lying only to veiy dilute gasc^s in which the the mirr-or, the absoi-bed light was I^an^sJLOI•iued iruo /R-at, thus generating e(m\-ech'()n fli'ci ls of Uk' ))ic V!\nishiu!.'!y stnnll. ruiMv-nts. 'rh<' diffficntiation b('tw(>cn fhc fon-cs coricsponding to this e/fccf nad iho ratlintian farcas waw, iiovvc-vcr, possibly duo to the unequal time of response in both pei-iencecl forces in the divection away fronx tVie tHxis sxippoilViig Mi»iUoNvtsW\ cases. Now, in order to improve the accuracy, a highly reflecting mirror was used. As a prediction. consequence, the minor heating was reduced by the factor of 400. Furthermore, the The theoretical explanation of this, as well as of the Jones-Richards experiment, was tung?;ten lamp of the Jones-Ricliards experiment was replaced, as the source of ilhi- given by Gordon [12] and later, including the hydrodynamic time response of the liciuid mination, by a He-Ne laser. This served to minimize the uncertainty due to the width surface, by Lai and Young [94]. Also, Kats and Kontorovich [05] calculated the in- of the spectral distribution of the source. The data thus obtained supported the earlier fluence of laser beam on a liquid surface and, whilst including the non-linear effects, conehision (as to the validity of the Minkowski expression) with an accuracy of appro- obtained results in agreement with the experiment. ximately 0.05%. All the above mentioned ex[)eriments, measuring the radiation force, employed high In addition, it was found that n was the conventional, i.e., the phase- rather than the (optical) frecjuencies. They led to the conclusion [25, 26, 27] that correct vahies were ob- group-refractive index. Moreover, the Jones-Leshe experiment showed no dependence tained when accepting the Minkowski tensor. of the radiation pressure on the plane of polarization for angles of incidence up to 20°. Let us now turn our attention toward the other extreme — the region of very low fre- This was contrary to the Peierls prediction (3.63). quencies and, in the limit, that of a static field. The electrostatic field interacts witli a liquid dielectric and, in turn, causes a compression At the beginning of this century Poynting [n the poles of a magnet as a torsional pendulum. The varying electric can ex])]ain the driving forces and stabilizing torques acting on dust particles. field with fixMjucncy of about 0.3 Hz was then ap])lied to the sam]>le. Whenever that field was exactly in phase with the natural oscillation of the ])e'ululum, the mechanical A similar experiment v,'as j)erfornied by A.shkix [03], Avho investigated the behaviour of niici'on-size latex spheres in a continuous laser beam. In order to avoid the photo- forces arising from the interaction of the })olarization current v.'ilh the magnetic phoresis, i.e., the temperature gradients caused by light, particles were suspended in a field would be ex])ectcd to enhance the oscillations. In the case of ])hases being transjiarent liquid. Furthermore, the jrawer densities were 10^ times higher than those opposite, oscillations were dam{)ed. If the })olanzation current was iji phase with the varying electric field, the conduction ctjrrent force cancelled itself over the comjilete reported by Rawson and IMay [91] and .May et al. [92]. cycle. It was found, when tlie spheres were submerged in a liquid with refractive index smaller The force density ])iedieted from the Abraham tensor is {(••a — 1) di{E X I!);Cr, which than tliat of the spheres, that, in addition to being accelerated along the direction of the is extresnely small, except at low frequencies. In tlie case of a constant magnetic field beam, thev were a Iso drawn toward the center of the laser beam. and a non-magnetic samj)le, it can be expressed as X/'o^^-This force, which does Whenever the index of refraction of the spheres was lower than that of sunwmding not arise in the Minkowski theory, was, in fact, confirmed by experiment [25]. Also, mediuni (such as the case of air bubbles in the mixture of glycerol and water), the spheres when this experiment was modified [2.9] so as to include the case of time varying were again accelerated along the direction of the beam but were forced away from the magnetic field, the results obtained were again in agreement with those of the Abraliam beam axis. theory. A similar behaviour was postulated to apply also -to atojns and molecules in a laser In this chapter we have reviewed the main arguments leading to differcjit electromagne- field. tic energy-momentum tensors. Although several distinct forms were proposed, thenutin In another experiment by Ashkin and Dziedzic [27], the effect of a 4 kW pulsed laser discussion centered around the Abraham and Minkowski tensor's. Theoretical anui- beam on a free surface of a medium was studied. It was found that, after about 390 ns, ments, as well as the experimental data, wei-e examined and it was found that they did a positive surface developed, thus indicating an outward force around the area of not always correlate. Most authors, nowadays, tend so favour the Abraham tensor,'as it the beam entry into a' medium. This also confirmed that, during that time interval, the is considered to be the more fimdamental of the two. Furthermore, in view of the sug- force acting on a surface must have exceeded the surface tension. Following an addi- gestion by Bofp [49] and on realizing that the niagnetization (or spin) effects are. essen- tional 700 ns, the lens disappeared, presumably under the delayed influence of surface tially, low frequency phenomena, one can say that experiments are, in general, in agree- tension. ment with the Abraham theory. Further, when the surface was irradiated from within the liquid, it experienced out- In most of the following, the energy-momentum tensor in a plasma medium will l>e ward forces. It was concluded, therefore, that both the entrance and exit surfaces ex-' duced electric field. Thus, electrons oscillate in the direction of the electric field vector which, m turn, is parallel to the direction of propagation. This kind of j^lasma electrorl oscillations is known as the longitudinal oscillation. Parameters associated with th^r^ IV. Electromagnetic Waves in a Plasma behaviour are called the electron {co^) and the ion {Dp) plasma frequencies and these are defmed by

than urn y. It seen, however, that [le conH,', larger The plasma state is capable of supporting a number of thermally and externally induced «.r,ed. Therefore, in an atten,j,t to Xd"om' ^"thors oscillations. For instance, low frequency electromagnetic waves in the presence of an njents w,ll revolve around the plas.na st^ A T a'gu- external magnetic field, also known as the hydroiuagnetic Avaves, can proparrate in a always well understood and J ft o, id f'urtI, " is ™t plasma with speeds often many times smaller than that of light in vacuo. Further not below^"'' longitudinal waves occur in plasmas, as Mill be seen

In most cases, the wavelength / of the electromagnetic radiation is much greater than the mterparticle distance in plasma. Therefore, it is possible to analvze the wave pro- pagation m a plasma from the point of view of the phenomenological electrodvnamic« Ju^tSf -f ^-harged a.,d ne,,.™, particles In so iar as the concepts of charge density and current density / are involved, 'the plas- Often, at least one of a be rt ded^fh^-'-rges ma IS considered as an equivalent to a continuum. In other words, the discctenels of to reproduce typieal plasma ch.aracteSti^s iff li'ghly mobile. However, in order tlie p.asma constituents is ignored and, hence, the Maxwell equations are used to descril,e ween the eharged plisn.a part? eft "f I'® - smeared-out, average fields and not the microscopic, interparticle fields. If these macro- cl.mens.ons of tho plas.na ,'egion. Th s for in\tan '''tual pl.ysical scopic, electric and magnetic field vectors are E and If/with /> and B denotincr the eharges fixed and the co,H:cnt,atio, of't "'etallic solid, witii its positive corresponding inductions, then the Maxwell field equations yield order of ,0- e„,-3, ean oft^, .Veha'^cT^'t-'^^'i™" ^he of a hquid plasma. J„ the pres'ent' io k Ltove } XE=~1 VxH + l ej), man,ly towards tennons, gal-ous plaint' ' "'ill be directed (4.4)

A ote that, as the mapetic permeability of a ten uous plasma is practically equal to unity, ti e magne ic induction B was rep aced, in (4.4), by the n^agnetic field //. Xow, to make ^^ , fJt t' ? f' /^dependence of the electric induction I) and current density J. In absence of any external magnetic field and for an isotropic plasma, we have D = eE gE, (4.5) vIZV Tl^ "" '''''' the dielectric permittivity and the conductivity of the

Pi = Th-kT, A^^t^^nZ' " ^^^ - that

fhe case of til" where f,., and ff„ are now tensor quantities corresponding to « and

Jensor'"""""* "" "" « ^'"gle, eon,plex pe.-.nittivity

= £ki — 4:T.i(T/,ila), (4.6) ^mmmmimEwhich, in general, depends on all coordinates npproxiination, the plasinii is able to support four different types of oscillations. A very which, after substitution in thv; above, yields clear exposition pertinent to this topic was given by DENISSE and Delcroix [.96^1 and we shall follow it rather closely here. VVy = —ikTiyViyVyH\,Jiloi. Let us first eliminate the )iiagnetic field strength froin (4.4). Assuming that all variables With the assumption that the magnetic field can be split into two components, one, Bj, are simple harmonic functions of time (i.e., are proportional to e''"') we get normal and the other, BL, parallel to the direction of propagation, the momentum con- 4m servation equations can be written out more explicitly. Thus, the substitution of (4.13) F xPxE = ~ o>j + /->. (4.7) f and (4.12) into (4.11) leads to conservation equations in the component form

But, the current density j can be wiitten as i(j)T)yinyVy^ = V/l y{E ^ 4" ^ " ^^ ? ^^^ T) ^ ^, -^y H y{V, " V j, — (4.14) j + TiiqiVi. (4.8) iOiVylHyVyy = Tlyq^{Ey " V y ^ B l) ^ V^iTlymyiVey — Viy) — VynTlyWyVyy Hence, on substituting the plane monochromatic wave in (4.7) and utilizing (4.5) and iOjVynlyVy, == T> yQ y{B, Vyjlr) ^ " ^ i " V y J} y f H yV y , 4" i k^Ji yW y VV y j OJ . (4.8), we obtain When the above equations describe electron (ion) momentum conservation, then Uie negative (positive) sign before the electron-ion collision term is used. In conjugation ({Jc^c^/co^) - 1) Ey == (4.9) with (4.9), they form a set of nine homogeneous linear equations with nine unknown. These unknowns are components of vectoi s E, Vi, and v,. Hence, in order that such a set has a solution, the formed from the coefficients of the individual tenus nmst wiiere i = ]/'—!. Thus, it is clear from (4.9) that E^ is always lagging with respect to be equal to zero. If this is the case, thei-e results an equation of the fourth order in k. /-, whilst and Ey can lag or advance relative to the respective current components, It, therefore, means that the system has four solutions which descril^e four, geiicially depending on whether {k-c^/co-) — 1 is negative or positive. different, modes of proi)agation: the-oi'dinary, extraordinary, electron, and ion modes Next, the conservation equations ai-e determined. It will be assumed that the pertur- ^Moreover, there are two conjugate solutions for each of k-, which means that the bation of the plasma variables caused by the passage of the wave, is small and can be (identical) waves propagate in the o{)posite directions. When setting uj) this determinant, neglected. the collision frequency is, for simplicity, put equal to zero. Then, the first colunui refers If the symbols n^ and denote the fluctuations of the number density of electrons and to the variable second column — to third column — to Vi,., and so on, foi' the io)is, respectively, then the corresponding conservation equations are written'as other two components. In addition the fii'st equation of (4.9) and (4.14), referring to Stne + r • = 0; + P • TiiVi - 0. (4.10) electrons and ions, occu})y tlie first, second, and third rows, respectively. The y- and components follow in the same order — up to I'ow 9. This determinant is written out in The momentum conservation equations are then, on neglecting the effect of the magnetic [96], Using the rules applicable to , a more manageable form describing the field of the wave on particles, general dispersion relation is obtained

^'ef'-'ei^tVe) = ^'egeiE + X i?o) " + Pen + Pei k-c- Q- 1 — 0 0 T'in^ii^trd = n,q,{E + X B^) - Vpi + + P,,. ^^ COx We assumed here that the phase speed of the wave is nuich greater than the thermal 0 speed; for otherwise, these momentum equations would be incomplete. The terms CO CO with subscripts e, n, i, and their permutations, refer to momentum transfers in collisions' -^0 0 III involving electrons, neutral particles, and ions. It is assumed [96] that thej- can be O) expressed as -0/ 0 0 Pen = Pin = — i ' Pei = —— Vj) ' (4.12) P • = —P 0 0 0 0. (4.15) ei — ic • CO The symbol v, with appropriate subscripts, refers to the collision frequency between the 0 0 corresponding plasma constituents. Although this frequency is assumed here to be a a) constant, in general it also has a tensor character. Further, following {9G\ the pressm-e O 2 gradient is written in terms of the average thermal speeds Vy [971, where Vy^ = ^y.T^jviy^ 0- as COx FF2 VPy = DlyVyVny, 1 1 — CO with subscript y referring to either electrons or ions. Asserting [96] that the particle conservation may be written as k^Vr riy = khyVy.Jco CO In the dispersion relation (4.15), symbols and refer to the electron (ion) cyclotron frequency resulting when the magnetic field is split into its transverse and magnetic wave, upon entering the plasma, encounters different particle behaviour, thus longitiidmal components. The relation (4.15), although appearing to be rather compli- leading to an analogy with a doubly refracting crystal. cated, readily lends itself to simplification. To generalize this investigation still further, the external magnetic field will be next Consider first a plasma in absence of any external magnetic field, H^ = 0. In this case assumed to be in an arbitrary direction. The resulting dispersion relation becomes, both electron and ion cyclotron frequencies vanish. The dispersion relation then reduces accordingly, very involved. In this case, the transverse waves are coupled to the longi- tudinal ones, leading to a rather complicated behaviour [96\ This interaction, in turn, I'esults in four distinct modes of propagation and one can no longer talk either about a At-AJ, = 0, (4.16) p\n-c transverse or a pure longitudinal wave. Thus, onty mixed modes exist and their where Aj,, Af^ are 3x3 subdeterminants of (4.15), i.e., discussion is not immediately necessary for our present task. Tt follows from the preceding discussion that, in the case when the transverse magnetic Ar 0 0 field is zero, there are two transverse and two longitudinal modes. Utilizing tliat part of the determinant which corresponds to the transverse waves implies that these modes 0 Ar 0 = 0. of propagation do not depend on the thermal motion of particles. 0 Consider now a plane monochromatic wave normally incident on a plasma. A simple In view of its derivation, (4.16) expresses the relation between w and /,', i.e., the dispersion manipulation of the Maxwell equations then leads to relation, in the absence of the external field. Thus, to find a non-trivial solution of (4.16), we have either A^^ = 0 or A^ = 0. Consider the former case first. It consists of a pro- AE -f (o>2 |(fo) 0. (4.17) duct of two klentical determinants of order three. Due to the construction of (4,16). the first determinant, corresi>onds to the .r-component, whilst the second one, As the electromagnetic field is transverse, the above equation applies to both E^ and Ey describes the ^/-component. To emphasize again, the direction of propagation is along components. The third component can be found from the z-axis. If the first determinant is now set e(iual to zero, all components other than i'f.-. vanish. This, tlierefore, means that the thus obtained result informs us about f(fo) = 0. (4.18) the ;r-componcnt of the field and of the i)artiele velocity. Put otherwise, if this deter- minant vanishes, we say that the Avave is polarized in tlie cr-direction. If i{c)) 0, then = 0 satisfies (4.18) and the obtained answer is associated with the Similarly, wlien the sccond determinant, Ar^^K is set equal to zero, the obtained solution pure transverse waves. On the other hand, if ?(w) = 0 and E. =j= 0, one refers to a longi- lUl f fy, Vjy immediately implies that the wave is ])olarized in the ^-dii-ection. tudinal wave in an isotropic, homogejieous plasma. The root of e{v)) = 0 leads, gencralh', It is clear, tliei'cfore, that the equation A^' = 0 determines the transverse cliaracteristics to a complex frequency, meaning that the wave is evanescent. The associated damping of the electromagnetic wave. is in most cases only slight, since the imaginaiy pai't of 6(0) is small relative to the real Concentrate now on the second equation implied in (4.16), i.e., A^ = 0. In that case, the j)art. I'his aj)proximation is valid for a sim})le i)lasi]\a, so that 5(0) — 0 may then be only non-zero roots are E,, ?,V:, and r,-,. This then means that the wave is polarized in the replaced })v direction of propagation and, lience, it is a longitudinal wave. Because the equation (4.19) corresponding to the developed determinant is quadratic in h'^, there are two roots, thus implying tAvo types of longitudinal wave. The solution of (4.19) has a real root, the so called plasma frequency, mentioned above. The above concjusions will be mainly used in the following chapters of the present work. If the plasma oscillations are simple harmonic, with the phase factor exp (i{coi — k • r)). However, to help us form on overall picture of the wave propagation in ])lasma and to the frequency cj and the wave vector h are unrelated. This, clearly, is so, as (4.19) deter- deepen our insight into the restrictive assumptions used, other types of waves are in- mines only the frequency. Therefore, the grouyj speed of these waves is zero, thus im- troduced in the next few paragraphs. plying no energy transport. This presumed behaviour (which does not agree with the The presence of a magnetic field parallel to the direction of tlie wave proi)agation, does experimental data) results from our neglect of the spatial dispersion. The ratio of the not affect the longitudinal waves, as a glance at (4.15) will reveal. Both the longitudinal Debye length and the wavelength in the medium determines this effect and its small electron and ion cyclotron frequencies have now non-zero values, ensuring a coupling magnitude ])ermits, often, to omit it. between the above mentioned waves which then leads to two distinct modes of ])ro- In the approximation used here, the longitudinal waves are unconnected with the trans- pagation. The j^lasma behaves like an anisotropic, doubly refracting medimn and the two verse waves Hence, two types of Avaves can be regarded as independent, except in waves travel with different velocities. This lends itself to a simple physical interpre- a region where the wave frequency has a value close to that of plasma frequency. The tation: the plane electromagnetic wave can be considered to consist of two (right- and separation of the field into transverse and longitudinal modes cannot, however, be left-) circularly polarized waves. The properties of those separate waves are cletermined from their interactions with plasma particles. The electric field of an electromagnetic effected in an anisotropic plasma. wave alters the orbiting trajectories of electrons and ions (induced bj' the static magnetic As implied above, the permittivity is a rather important concept, as it establishes the field) in a way dependent on the relation between the wave and the cyclotron frequencies. link between the electrics induction and the electric field. It has been, generally, assi^mcd Thu^, the left circularly polarized wave rotates in the same direction as would the ions ['12], that either the effective field E^n is equal to the external field C, or it is modified under the iufluoncc of the static external magnetic field. The other ])olarization agrees by the presence of the Lorentz j^olarization term, 4.-/7'^- differentiation, however, \\in» tbi- din-ction , the electro- between those two possibilities proves to be not a trivial matter. The question has attracted a lf){ (>f attmil ion and (Iier(< exists (livri-sil v of oiu'nioiis [•>/. J7. .W. /]. aVie u\ai;in aluivgo of j)}iisntii vfinishos. Jt niL-iiiis that tho charge of tlio free eJectron.s As there is no polarization in the absence of tlie field, tlicn 22 ^ ~ = ^ Jiiust soinvhoxv bv iicutralized. Two possible ways in which this may happen have been the permittivity can be expressed as puttovward: by a positive continuum and by the discrete positiA^e ions. Difficulties arise in attempting to determine the conditions under which these two approaches may be equivalent. ^ " co-^ \7n M A detailed analysis reveals that J^eff = B inside a medium in which electrons are neutralized by a positive continuum. On the other hand, a medium consisting of a set where n^ and iii refer to electron and ion densities, respectively. On account of then- of bound atoms obeys the Lorentz formula mass, ions usually have only small influence of the permittivity and are, therefore, often neglected. Thus, the approximate expression for the permittivity becomes 4rn Krr = E P. IT £ = 1 — {(Op-jco-), (4.25)

It, therefore, follows that, when considering a fully ionized plasma, the correction term where cop^ = Anqhi^jm defines the plasma frequency. 4rrP/3 ]nay be neglected [32, 33]. As already indicated the conductivity is zero under the above assumptions. This is so, The main contribution, in determining the value of dielectric i:)ermittivity, arises from the because here we refer to a collisionless plasma. Hence, electrons do not transfer energy motion of charges. The effect of neutral particles is important only when the ionization but merely oscillate in the field. These conditions change when collisions are included. is very small. In the case under consideration, it may be, therefore, neglected. In order to Then, a term proportional to the frictional force, af, is added to the right hand side of obtam accui-ate expression for e and a, one must start with the Boltzmann equation. (4.21). This force is caused by the momentum transfer occurring daring collisions. On However, as most of the present work applies to the case of cold plasma, an approximate repeating, essentially, the same arguments as above, we obtain the result derivation of the permittivity will be sufficient. The high conductivity of plasma is well known. It signifies that, although macroscopi- e = 1 (4.2G) cally neutral, a plasma consists of moving free charged particles. The total current den- vj{(0 — ?Veff) ' sity j;, due to this motion, is then written as where rprr is the effective (-ollision frcqiRUUiy. The complex permittivity implies the existence of a non-vanishing conductivity and, jt = qlJr^ - k > (4.20) k=l hence, the absorption of energy. It must be sti'cssed, however, that (4.20) is oiily an approximation, and the exact result, obtained from the Boltzmann tlieory, cannot be where q is the electronic charge, and are the time derivatives of the electron and reduced [33] to it. Nevertheless, the relation (4.20) is quite adequate for our purposes. ion position vectors, respectively. The question now arises as to the validity of applying the classical theoi-y of mechanics Now, the equation of motion of the electron in a collisionless, isotropic plasma is to the motion of electrons and ions. It is found [33], that the classical theory is valid if hco where 27ih is the Planck constant. Another condition for the applicability 777.i\. = qEf^e^"'^ = qE, (4.21) of the classical theory is that the ])lasma be regarded only in a non-degenerate case. When analyzing the propagation of the electromagnetic Avaves in j_)lasma (or-, generally, where E denotes the mean macroscopic field, in agreement with the above discussion. in a medium) the concepts of the refractive and absorption indices become more promi- The solution of (4.21) is nent. Let us elaborate this point and find out the influence of these indices on some -iqE/mar) + r,'{t), ' (4.22) variables pertinent to propagation. Thus, the electromagnetic wave equation in a medium with a constant permittivity tensor can be written [

(i) — _ n (qE/Moy^) + (4.2.S) If a plane electromagnetic wave, E = Eq exp (i{cot — k • r)) is substituted into (4.27), one obtains the dispersion relation and M stands for the ion mass. Xow, by definition, the total current density is A:2 = {co''lc~) £{(0). (4.2S) .e - 1 \ jt =j-r io^P = + i E. (4.24) To obtain the relation between E and II, the plane wave expression is inserted into the 4:71 two Maxwell equations (4.4) which involve the curl operator. This yields

Substitution of (4.22) and (4.23) into (4.24) shows then that the conduction vanishes and a)e{oj) E = —ck X II )H^ck X E. (4.29) the polarization vector can be written as In order to find the condition for transverse waves, (4.29) is (scalarly) nuiltiplied by k, q^E n . \ P = giving k ' E ^ 0 and k • II 0. It is usually assumed that e{co) 4= 0, otherwise,the m Mj longitudinal waves may appear, as indicated above. ^\Tien tho wave vector is com])lex, honogeneo.., those two ^^^^^ ^^ The wave vector h whose direction indicates the direction of propagation, is related to the wavelength by k = 2:;r/A. Therefore, taking into accoimt (4.32), the wave vector in a^ medium is related to that in vacuo by

h = nk, (4.3G)

There is a close connection between (4.36) and the concept of the phase and group ^"elo- citics. Thus, the phase velocity is defined as describing the propagation of phase. In other words, it refers to the phase differences between oscillations observed at two different TI, ^T ' " (4-31) points. Hence, when the frequency cj and the wave vector h are given, the phase speed -Llius, the simple harmonic wave i.s modulatrH ^ . is expressed as connected with the absorption process As74 ' , which is medium changes by a factor ofTov; 'tt fe LcT ; - ' ' r ^^^^ - (4.37) ^ow, utilizing (4.31), the relation bet™ i . vacuum wavelength), '-P k vacuo becomes me wavelength 2 in a medium to that in This speed is primarily used in the treatment of standing waves and in the inter- •;. = ii. ference and diffraction phenomena, as phase differences are, then, of particular impor- re (4.32) tance. The signal, or modulation of the wave, on the other hand, travels (when absorption is small) with the group velocity. The group speed, Vg, is, in a dispersive medium, different (4.30) ^(1:6) tis to "" ^^ Thus, compar-; from Vp and is (Icfincd by doj or == - • "'-Ik- •

•ina e = n^ ~ a^ 2710C = Hence, in vacuum, which is, of course, non-disjiersive to electromagnetic waves, OJ (4.33) ^p = Vg c and the signal pi'opagates without distortion. To empliasize again, the phase Whence, velocity has nothing to do with pro]iagation [JOl], but gives us, essentially, the distri- bution of phase in sjiace. On tlic other hand, tlie group velocity refers, in the present (4.-34) a])proxiniation, to the energy transport. Howevei-, when absorption is high, the signal ind is strongly frctjuency dependent, bcco))ies distorted, and the concept of the gi'oup velo- cit^r then loses its meaning [7<77]. Let us now apply these concepts to an isotropic, homogeneous ])lasma. Consider an electromagnetic wave of frequency w > cOp. Then, assuming absence of absorption, it follows from (4.30) that the wave vectoi- is real. Its subseqtient substitution in (4.37) and ^^ ^ ^oilisionless plasma with and a = 0 yields parameters e = 1. _ (4.:J8) then deteiunines the pliase and group speeds of waves, i-espectivelv. Thus

__ c t\ = cn. (4.39) a 0 ^ ~ n if CO > (Or which means, as n < in plasma, that Vp > c and v, < c. However, as the phase velo- 0), 1/2 r 1 n - 0, a = - 1 city is not associated with the energy transport, it does not contradict the theory of OJ' if w < cop (4.35) relativity. At frequencies much larger than the plasma frequency, both speeds apnroach / the vacuu)n value of light, as is obvious from (4.35). n 0, = 0 if OJ = 0)^. As already indicated, when oj < cop, the value of the wave vector is ii7iaginary, the wave exponentially decays, penetrates to a certain thickness, known as the skin depth, and then totally reflected. Hence, when a> > the wave travels with^t Stcn atlL whiS ^ It ^^'equency. The above described behaviour can be explained in relatively simple terms. At very lo^^• freciuencies, electrons in a plasma have enough time to move in such a wav as to scjxen the plasma interioi- from the oscillating external electric field. In otlier WOKIS. it Iben, according to the Mawxell ec,nations I ' v O o } ^ ' clJcctcd when ro g from the Maxwell equations that the real current density, caused by the liiotion of a consequence,! tnu.svcrso rlX™ ^' magnetic field vanishes. As cliarged particles, is greater than the displacement currentdensitv. At verv hio-h'f,v' '•ondni.L. longitud--ncierthese quencics, on the other hand, electrons cannot move fast cnorigti to follow t ip.' field aiKl. tlins. roc unahh- to srre-n tfir pinsnvi inter/or. In otlwr wonh. (lie rr.i! c ur/-- sniiill coinpaiccl with the density, due to tlie inertia of The wave packet, on tlTe other hand, is ciefined as toevr^a tUe resvxlt, oi a, ol the ^i^ntvni.s. Therefore, the behaviour of an electromagnetic wave of high frequency nearly monochromatic waves and is seen as occupying some finite regioi^ vrv space in pJasnia approaches the one in vacuo. o ^ j- It is described mathematically by It of interest to inquire how the propagation is affected when there is a small absorp- tioii. liius, m the limit |£| > d^ccr/co, relations (4.34) reduce to A{x, t) = Aijc) (5.1) ]/2\7Zj J 1/2 — oo 1 — where it must be remembered that w is a function of k. The harmonic term contains in- (4.40) formation about the velocity of propagation of phase and it, therefore, belongs to the undulatory domain. C0„- 1/2 • Next, the momentum density of an elementary radiation field is given by .Abraham as 1 [CO- + I'lfj; hv (5.2) The corresponding phase and group speeds are then, respectively, Qa = cn and by Minkowski as nhv (5.3) 1/2 g.M = 1 — 4- i'l j (4.41) In a plasma, where 7i < 1, it holds that c[l - oj/i(CQ-^ + virr)]'^-' (5.4) 1 — + ' {7.1/ < go < f/.i»

Compai ing (4.41) with (4.39) confirms that, in the presence of absorption, the phase where r/o = ''"V'c- is, thei-cfore, led to suggest that, in a piasuui, g^; refers to a "bare" s})eed is decreased, whereas the group speed increases. a]id 17 , a "dressed" particle, since a "bare" particle influenced by (i.e., interacting with) Considerations in this cliapter should serve to provide sufficient information to form a a n\edium is icferr(>d to as a "dressed" one. As a particle has, in general, a medium- background for analysis in following chapters. Some of the features presented here will dep(>iuient eff(!ctive mass, only that latter kind is of experimental significance. Hence, 'be further developed there. it is proposed that the "bare" particle be associated with the concept of a photon, where- as the "dressed" one be linked with that of a wave packet. Putting it in another way, the radiation, wi)en interfering witii matter, assumes some of tlie characteristics of the V. Dual Nature of Radiation latter. Thus, the initially pure, e3ectron\agnetic w^ave !)ecomes now modulated by the surrounding medium. However, when a photon is absorbed by a constituent of matter, The complementary character of tlie corpuscular and imdulatory aspects of electro- the wave packet description combines the properties of the material constituent with magnetic radiation is, nowadays, readily accepted. Only the two aspects taken together those of the aljsorlxjd photon. can reveal the complete nature of that radiation. The above discussion distinguishes l)etween the complementary meanings of the photon Very often, in treating radiation phenomena, concepts of the photon and the wave and the wave packet for radiation in a medium. It is, hence, conjectured that the mo- packet (meant to represent the two complementary kinds of radiation) are used inter- mentum density g.u applies to a ])lioton and to a wave packet, whenever an electro- changeably. This is, of course, quite permissible in the empty space. It is believed, how- magnetic field inside a medium with the I'cfractive index less than unity, is considei'ed. ever, that such a (random) alternation between these two notions cannot be validly The above proposition is in agreement with the work of HORA [39, 103] and KLTMA and applied v.hen the radiation propagates in a material medium. Thus, SKOBEL'TSYN [3-5] PETKZILKA who found the total radiation momentum in a plasma medium to be refers to a "quantum photon" and a "classical photon", in order to discriminate between given by the inherent differences of the t\vo aspects. It is a common mistake to confuse the corpus- cular and undulatory characters of radiation, when it is inside a medium. It should be, 9 = — {(l.i + g.u)- (3.5) therefore, emphasized that these two notions represent two distinct (but complemen- tary) aspects of the interaction phenomenon. Tliis represents, of course, the average of the two momenta: that of the ])hoton and that Let us now clarify this dissimilarity between the photon and the wave packet from yet of the wave packet. In addition, if the last term, in the Peierls expression (3.G2), which another point of view. The former concept arises in the quantization of the field. Thus, arises from the presence of dipoles, is neglected, then the total momentum is again given if and di^ are the operators which, respectively, create and destroy a quantum of by (5.5). Although this interpretation is somewhat arbitrary, it, nevertheless, leads to energ3'' ficok and wave vector A', then these quanta are known as the photons [102]. The satisfactory conclusions. number of photons with the v/ave vector k excited in the cavity is determined by the Consider now radiation ^propagating in a medium and assume the conditions to be such eigcmvalue Sb of the appropriate number operator Sk = d^'^^dk. It is to be noted that the that laws of geometrical optics apply. This means that the wavelerigtli is much smaller notion of the energy exchange is of prime importance in this definition. In addition, the than the typical dimensions involved in the problem. In this case, therefore, \vaves are value of S{i will vary depending on the surrounding medium. treated as corpusCules travelling along the ray trajectories, eompleteh' lacking in any 21. Zi.'itschriff. „!''ortscl>!'il (o tier Ch.N sili", ird't fi undulatoiy characteristics. The role of phase and group speeds in this asymptotic- The conservation of momentum and center of mass theorems were already discussed in picture becomes clearer from the following considerations, due to PAEK [104]. The scalar^ d Alcmbert wave equation Chapter III. They were applied there to the analysis of the photon momentum inside an optically dense {n > 1) medium. Although that approach is rather simplistic, it will be used here to cover also the domain of optically thin (n < 1) media. The qualitative ana- (5.6) lysis is, essentially, the same as before and, hence, it is not presented here. Nevertheless, V it is to be noted that only the Minkowski result sinuiltaneously satisfies both the con- degenerates in the limit of X 0. Let us assume a solution of (5.6) to have the form servation of momentum and the velocity invariance theorems. There is some connection between the above approach and that discussed by JFOCK {107\ who di-aws analogy, from the purely formal stand^^oint, between classical particles and ip — ae^^, (5.7) optical waves. Furthermore, HORA [100] and CARTER and HORA [110] applied this simi- larity to investigate the Goos-Hanchen effect for matter weaves. In addition, arguments where S the eikonal. In such a case, (5.6) will be satisfied by presented {)y JOYCE [76, 77] attempt to reformulate classical mechanics so as to encom- pass particles of vanishing as well as non-vanishing rest masses. An application of matter waves to investigate the propagation of light in fibres w^as presented by ARXAUD {VSf = ^nK (5.8) C" To illustrate the similarity between those two types of waves, he referred to the quantuju mechanical AHARONOV-BOHM effect [111, ii2], which has been experimentallj'- This, in turn, means that the surfaces S = const, describe surfaces of constant phase verified [113]. The classical explanation of this effect was given by BOYER [114]. Basi- and, consequently, determine the phase velocity. cally, the AIIARONOV-BOHM effect predicts the physical significance of the electro- On the other hand, the Hamilton-Jacobi eikonal magnetic potentials, and also suggests that their role, in quantuiu mechanics, is similar to that of the i-efractive index in classical physics. To verify this proposal, a siinple experi- {VSy^ = 27n(E - F(x-)) (5.9) mental set up is used to observe diffractive effects between matter and light waves. Thus, consider a row of metallic channels wliich are at gradually increasing potentials. A determines the group velocity which is necessary in order to find the time a particle plane matter wave (s\u>h as that corres])onding to electrons) entering those channels at spends in an optical region. This is connected with the work of LEWIS [105], who divided one end, will suffer a deflection ujion exit, despite the absence of any classical forces. the transport equations into tw'o groups: those describing intensity and those referring Similarly, a light wave passing through an oj)tical pliase ari'ay (a row of channels with to the pliase change. increasing optical (]cnsiti(?s) will be also defJecled. Moreover, if the dispersion of the light A different approach is adopted here. By extending the Presnel theorem of stationary waves inside this 0])tica] system is the same as that for tlie ujatter wa,vcs in vacuo, then phase, DE BKOGLTE [106] arrived at the relation between the speed v of a massive par- the r(\su]taut deflection of the Avave will be equal. ticle and the phase velocity Vp of the corresponding wave This correspondence ])etween inattei" and optical waves will be further elaborated in the next chapter. Now, let us returji to the question of the electromagnetic momentum in a vi\ = C-. (5.10) plasma and investigate the important topic of the })lack body i-adiation. That last phe- nomenon descril)es, as is well known, the equilibrium s})ectral distribution between the Purthermore, he found tliat the group speed of the wave, associated with a particle, is thermal radiation and iiiatter in an evacuated cavity. The presence of matter is necessarv equal to the particle speed. Relation (5.10), known as the Broglie group velocity theo- to satisfy the equilibrium conditions and its amount is U)iimte. When, however, the cav- rem, may be interpreted as pointing to some form of connection in the particle and ity is filled with a 2:>lasma, it affects this spectral distribution and it is quite natural to phase velocities. Of course, this theorem also applies to particles with vanishisig rest suppose that the density of radiation inside plasma differs from the one in vacuo. Thus, mass. Then, it expresses the constanc}^ of the product of the corpuscular and undulatory if t(v, T) is the mean energy of oscillators of frequency v at the temperature T, then speeds. However, as the energy transport decreases in a medium, the seiJaration into those two .aspects is more obvious. - l)-i (5.12) In view of (5.10), it should be remarked that and the energy density u{p) of the radiation can be expressed by (5.11) QM • gA = Go' u{v)dv ^ e{v,T)g{v)dv, ' (.5.13)

whicli is a relation connecting the Minkowski, Abraham, and vacuum values of the where (/(r) dv refers to the density of radiation states in the frequency interval j- to momentum densities. Therefore, (5.11) can be looked upon as describing the link be- V + dv. This density of states is usually defined, in terms of the wave vk^tor as tween the corpuscular and undulatory aspects of the momentum. Although a direct phy- sical significajice cannot be attached to the undulatory momentum, this need be no g{k) dk = (5.14) cause for concern, as this analysis ap])lies only in the WKB (i.e., effectively geometrical optics) a])proximation. It will be seen below that tlie Minkowski value of tlie niomen- In order to arrive at the density of states, in terms of frequencies, (5.14) is transfo! lucd. tum (representing photons) does, indeed, satisfy tlie condition of equilibrium between utilizing (4,36), into radiatloT\ niKl a )lasma (or, generally, a medium with n < whereas, as expected, the $71 j 1), g{v) dv —- —— nv~dr. AV-rab;nn VAIIH- »1 1. Aow, if (0.15), together with (5.12), is inserted into (5.13), we obtain the following modi- fied Planck law for an isotropic, collisionless plasma In view of the above discussion, therefore, the probabilities for upward anddownwaifl transitions can now be written, respectively, as (5.1S) u(v) nvoiv), (5.16) dP = dt; dP - nB„„Vo{v) dt. where corresponds to the vacuum value of the energy density. The result (5.16) was Hence, the pi'obability for the radiation induced processes in a collisionless ])lasma is obtained in the literature by a completely different method by BEKEFI and DAWSON decreased relative to vacuo. This is so, because, as the ])lasnia density increases, tiie [37]. CASE and CKIU [117] used the fkictuation-dissipation theorem to arrive at the same refractive index decreases and, thus, the ainount of radiation energy density prcsejit is conclusion. Although OSTER [276'] also obtained (5.16), his radiative transfer theory was diminished, in accordance with the expression (5.16). ci-iticized by CHOXYN [118] as violating the Second Law of thermodynamics. In addition, Investigate now the motion of a test particle under the influence of the radiation field. the trans])ort ec^uation [116] differs from that due to MARTYN [119]. A modified form To simplify the calculations and lo make, at the same time, the j)hysics of the problem of the Planck law in anisotropic media was described by COLE [28], which, in the iso- more it will be assumed that the particle of mass M is moviiig with velocity tropic limit, reduces to (5.16). V in the .r-direction and tliat the conditions are such that tiie higher ]>owers of r.-c can Neglecting the longitudinal waves, INXMAN [120] developed the thermodynamics for be neglected. In a radiation field, the linear momentum Mv of this jjarticle clianges, in electroniagnetic waves in plasma. He found that, as the plasma frequency inci-eases, the tune T, due to the effect of the radiation force / = —Rv, Ji being constant to be eva- number-, free energy-, entropy- and energy-densities are all decreasing from their vacu- luated below. Also, during the same time, the fluctuations of the action of radiation on um values. The correctness of the formula for the radiation pressure given by Inman the particle cause a momentima transfer A to the latter. The magnitude and the direction was, however, disputed by MORE [121]. of this momentum transfer are constantly changing. As the radiation field nuast not We are now in a position to develop further the method used by EIXSTEIX [34] in his affect the overall velocity distribution,.the avei'age momentum of the particle, at botli investigation of euiission and absorption processes. An attempt was made by SKOBEL'- ends of the time interval T, inust be Mv. Therefore, wo. must have TSYX [.3-5] to apply this method to optically dense media. His work, however, appears to be inconclusive. In tlie present treatment, the original (Einstein) method will be'adapted {Mv — JlvT + A)- = {Mv)K (5.19) to the x^lasma domain. During any al)Sorption (or emission) process, a momentum is transferred to an elemen- We shall work in the aj)proximation in w hich all terms containing rJ can be neglected. tary system in the i)lasma. The velocity distributions of such s3'stems are merely due to Further, in order not to disturb the thermodynanu'c equili})riuui, we shall assume the their interactions with the radiation field. Thus, this distribution must be tlie same as the relation (l/2)i1iy = (l/2)x7' to be valid. The above e(juation then reduces to one these systems attain through their mutual collisions alone. Hence, it is necessary that the average kinetic energy of a system in the radiation field be equal to {l/2)y/T. Suppose now that there are two possible quantum states z„, and with the corresponding — = 2Ry/J\ (5.20) energies t:,,^ and £,,. Next, let the transition from state z^ to state z„ be allowed for the T system. That transition will then result if the system in state absorbs a quantum of Tims, this is tlu? fundamental ecjuation re') = u{i>) — cos , (5.21) plasma, w(r) refers to the radiation density inside the latter. This density is given by (5.16) and its use is imperative. (That this choice does not disturb the'thermodynamic where vjc and u'{v', ') depends on the direction defined by the angle 4>' it makes equilibrium conditions will be shown below.) Never the less, SKOBEL'TSYN [JO] originally with the a;'-axis, and the angle r/ between its projection in the ^'z'-plane and the t/'-axis. ro lacilitate the transformation of the argument of u'{v', 4>'), the formulae for the Dopp- As the spontaneous emissions do not have any prefen-ed directions, average momentum ler effect and the aberration are needed. These, however, are well known in the theory tx-ansfers, due to these processes, vanish. of relativity and, in the approximation applicable here, take the forms Thus, finally, the coefficient H can be written as

v' = 5/(1 — 70 cos (jy) (5.22) n~hv V du{v) B = u{v) — (5.32) ana y cv cos = cos (}) — n^ + n^ cos- (5.23) Prior to utilizing this result, let us inquire about the effect of the presence of a plasma respectively. Further, no relevant information is lost if we put on the momentum A. Denote by d tlie momentum transferred to the i^article during souie process and assume that its sign and magnitude vary in time in such a way that •y/ — ip. the average value of d vanishes. JN^ow, if d^ I'efers to the transfer of momentum through (5.24) The inverse transformation to that in (5.22) is the t-th independent process, then the resultant transfer is zl — ^ (5,-. Because the average value of each <5j vanishes, it is necessary to write (assuming di" = d-) r = )''(! -f-??/S'cos (5.25) == m', (5.33) which, on substitution in W(J') and subsequent expansion, gives where N is the number of all elementary transitions in time T. If the momentum corre- sponding to each elementary process, <5 = ??/«)-cos ^/c, is substituted into (5.33), it n{v) — ?/()•') -f v'7ip cos (j)' (5.26) 8v' then yields /7ikvV = (5.34) The solid angle dQ is (in our approximation) connected with dQ' by 3

dQ/dQ' = l — cos c^'. (5.27) As the number of induced processes from level m to level n, in time r, in equihbrium, is ZV72, it is, then, possible to write Thus, the substitution of (5.25), (5.2G), and (5.27) in (5.21) results in A" = 2Qn{v) T, ,, du{v') u{v') -j- v'niS cos (f)' (1 — cos4>'). (5.28) Avliich, on combining with (5.34), gives cv' zP nln Making use of the last relation and the assumption about the elementary processes, dis- Qui (5.35) cussed earlier, allows calculation of the average momentum transferred, per xmit time, to the particle. Let, then, the niuubcr of elementary ti-ansitions from level rn. to level n be i\s already previously emphasized, the expressions (5.32) and (5.35) are determined in terms of the radiation energy density zi{v) which, of course, differs from its vacuum value ^ dQ'IAn, (5.29) An attempt will now be made to show the validity of the claim that the photon momentum inside plasma is correctly described by the Minkowski relation. For that where purpose, we must show that the thermodynamic equilibrium is not disturbed when the S = Wrne-""'!"^' + tv„e-'nlxT radiation momentum is transferred to a particle. This, in turn, will be so if, after cal- and iv„, are the statistical weights corresponding, respectively, to levels ???. and n. culating B and Zl- in terms of the proposed energy and momentum densities of radiation, An analogous form holds for the transition from level n to level m. their consequent insertion in (5.20) yields an identity. Thus, substitution of (5. IG) in (5.32) results in As suggested earlier, the photon momentum in plasma is given by the Minkowski re- lation. Therefore, in each elementary process, the momentum nhv cos (p'jc is transferred nhv u^{v) , hv e""/'''^ to a particle. Combining the upward and downward transitions, the total momentum B = + — 1 Q{1 — (5.36) transferred to the particle is then expressed, using (5.29), as c- e""/"'^' - 1

-ejy.r e-'"!"'^') f u'{v',') cos<^W/4rr. (5.30) Now, if conditions are such that livjy.T ^ 1, then (5.36) reduces to

Substituting (5.28) into (5.30) and putting Q — {l/'S) w,„B,„^e the total averaged mo)nentum transferred, per unit time, to a particle thi'ough the induced processes is 3 y.T (5.36a)

n'^hv • V c!u{v) Furthermore, when (5.16) is inserted in (5.35), the result is - ^ (5.31) 3 ?)• zl- Voiv) Q. The clashes f>vcr syuiVols imve Ix'cn dropped )i'>r<'. for the sake of cuuity. ^ / J-^ii^t/ly^if (.5.:}-) mo «.if,>st/tu(eci iiUo (5.20), an idcjitiT^ relation is, indeed, wave ])ropagates Ihiough a collisionles« plasma coWisionul jAawiwa \122\, ov a d 'ntitr^^^^^^^ thez-efore concluded that the Minkowski form of the moiuentuni dispersive fluid it interacts with its surroundings and, thus, takes on sonie of tVie oUth 'f n P-^'^'i^ng analysis, appears to be satisfactory, at least in the region properties of the medium. Thus, for instance, the effect of a collisioiiless plasma on tiie o Ligh frequencies and withni a colJisionless plasma. On the other hand, the use of the field is contained in the ojp term in (6.1), since, in that case, y 1. Heiue, a p]-cr()n Abraaam nionientv.ni density does not satisfy (5.20) and, therefore, nuist be considered inappropriate. which, before entering a plasma region, had a zero rest mass, acquires, v/hilst travelling- inside it, an effective rest mass of Aojp/c-. This is readily obtained from (6.1) and (0.2). Vvlien interparticle collisions are present in a plasma, the above analysis should still Although it is normally assumed that photons are massless particles, this requirement is apply provided that wo include the effect of those collisions in our calculations One not necessary from the point of view of the special theory of relativity. What is really possible way of domg this is to introduce a eom])lex refractive index (4.26) which then important, is that r,, the speed of light in vacuo, remains constant in moving ineitial takes mto accorait the absor])tion due to collisions. It is, then, sufficient, for < frames. This, however, would point to a particle witli a vanishing rest mass. On the to replace the value of the lefractive index in the above relations by the corresponding other hand, the sp(>ed of a massive photon must be less than c and ought to de])end on magnitude of its complex counterpart. This is, essentially, the approach used bv Hoka itsenergy, hv, and, hence, its frequency. This would imply that electromagnetic waves of biJJ whilst treating the interaction of laser radiation with an inhomogeneous plasnui different frecpiencies would propagate with different speeds. In fact, it has been sug- in .his chapter, results, valid in what is, essentially, a WKB approximation, were pve- gested that liigh frequency waves should propagate a( higher speeds than the lo^v fiv- . sensed. In the following pages, a more general approach will be adopted to facilitate our quency ones. .As the waves of ])) 0gressively lower fivquencies are considered, the limit inquiry into the form of the electromagnetic energy-momentum tensor in a plasma. is eventually I'eached when the propagation S])eed drops to zero. This happens A\lien h(o — he., in view of the above discussion, when the fi'equency of the incident wave is equal to the plasma frequency. Ex])ei-iuiental results [123] obtained from pu-sars VI. The Proca Equations agree with the just ])r(,\sented int{>)"pretation. Therefore, it seems quite i)lausible to re])lace an electromagnetic wave in a medium by Results obtained in the previous chapter should be understood as the asymptotic an ecjuivalent massive {)hoton in vacuo. It will Ix^ recalled that the principal difficulty solutions to the wave optics viewed from the classical standpoint. It appears, in this in establishing the correct form of the energy-momentum tensor in a macroscojyic ]>ody lignt, that the only expression for the momentum exchange satisfying (5.20) the mo- lay in the impossibility of splitting that tensor uniquely into the field and the matter mentum exchange between field and a ])article, is the Minkowski one. It is to 'be noted, parts. That separation beconses possible here, if we are willing to accept the aho^•e however, that no generally applica})le'conclusion about the energv-niomentum tensor interpretation. can be reached through utilizing the method of the preceding chaj^ter, as that formu- lation applies only in the case of geometrical optics. Furthermore, it is not relativisticallv The consequences of replacing propagation of radiation in a mediiim b}' motion of a CO variant. photon with a nonvanishing rest mass are rather intiiguing. Thus, it follows from the Maxwell theory that the photon can be polarized in two directions. Howe\'er, as the In order to arrive at a reliable result, some other, more rigorous, method must be used. photon has.now an effective mass, the generalization of the Maxwell equations leads to Thus, the procedure to be presented here, resorts to the formalism of the covariant field tlieory, which encompasses the realm of wave phenomena, as well as the corpuscular the theory of Proca equations. Consequently^ there should occur now all three states of lavo approach, just presented. polarization which are predicted bj^ the latter theory. Thence, apart from trans- verse ]iolai-izations, there also exists a longitudinal one. Furthermore, the faster the Quite generally, a medium, in which the phase speed is frequency dependent, is known as longitudinal photon moves, the weaker is the associated electric field [124], thus, im- a dispersive one. In such a case, the relation between the angular frequency w of the plying only a rather weak interaction with a medium. In addition, the electric field di- wave and the corresponding wave vector h assumes the form {oO'] minishes ex])onentially with distance and the corresponding flux lines fade away, even in vacuo. Similarly, magnetic field is also affected and the associated lines are com- CO- = X--C- + lOphf (6.1) where ])ressed around efjuator. Using methods of classical electrodynamics [50], we proceed now'to analyze this problem more rigorously. Although there will be some overlap with Chapter II, as far as the CO- — (Oq- presentation is concerned, the conclusions we reach here will be, in general, quite diffe- and (Op is the plasma frequency, whilst coq describes the characteristic excitation fre- rent. i quency of the medium. In a plasma, a>o vanishes. The expression (6.1) is only approxi- It is necessary, at this stage, to discuss the problem of the photon spin. I'hus, aceordinir mate, as it is derived under the assumption that the dispersion is determined solely to , the total angular momentum of the I'adiation field is e(|ua] [72-5] by electrons. to the sum of the spin and the orbital angular momenta. The spin of the plioton is. ge- nerall}^ accepted to be unity and to lie in the direction of j^i'opagation. This was veri- In the special theory of relativity, the relation between the energy E and the momentum p of a particle, having rest mass m^, is given by fied by observation [^T^]. We shall, therefore, assume here tliat tlu^ spin of the massive photon is also unity. Hence, in view of the above discussion, the Maxwell field in a me- (6.2) dium is, essentially, replaced by a neutral vector meson field in vacuo. Then, it is ajiplied to the question of the energy-momentum tensor. A quick comparison of (6.1) with (6.2) reveals a striking similarity between both relations. Thus, a free electromagnetic wave corresponding to a massive photon may be deseri};ed Thus, it was suggested [40, 41] that the propagation of a wave in a medium may be by treated analogously to the motion of a massive particle in vacuo. Put otherwise, as the witJi where F''^ the electromagnetic field tensor, is given by (2.4). Thus, (6.10), together with - F = (6.4) (2.4), gives us the covariant Proca equations for a massive vector field, and is should be noted that it is the only possible linear generalization of the Maxwell equations [126].* It is to be noted that, as a result of the inclusion of the coupling constant//, the potentials the phase speed is now acquire real physical characteristics. In other words, in contradistinction to the potentials in the JNlaxwell theory, they now become observable [127]. (6.5) If we, further, assume the validity of the Loi'entz gauge, d^A'' = 0, then (6.10) may be rewritten as 4rr whilst, for the group speed, = dco/dl-, we have the expression OA'' + = — f. (6.11)

^V = - - (6.6) where the four-dimensional Laplacian operator (also known as the d'Alembertian) is defined bv • = When time independence oiAf" is assumed, so that the field may be A glance at (G.O) shows tlmt also for such a massive field, the dependence of the energv regai-ded as static, the solution of (6.11) leads to the Yukawa expression for the potential, propagation on frequency is in accordance with experimental findinc^s [m 124] as alreadj' discussed. Let us consider, for a moment, the case of a static field, i.e., co 0. It then follows from In three dimensional notation, the modified Maxwell equations are (0 4) tlia. k = ±7//,. Consequent substitution into the wave equation (6.-3) implies that the spherical static field decays exponentially, i.e., = 1/r) as alreadv indi- A oc IXE^ --dJI, + cated Un selecting k ^ ifi, M^e have a behaviour similar to that displayed bv the Yukawa C C c mo(ie. ol nuclear binding. Thus, as one of the consequences of the photon being massive F E = 4.-ro — P -11 = 0 (6.12) we m.ust have a deviation froui the Coulomb law. Hence, in view of our supposition, the' lieJd 111 a colhsionless, isotropic plasma should decay exponentially with distance. The same conclusion M-as also o])tained by COLE [40], in his study of wave proimgation in an i. a,V + y • - 0, rxA^ vv 4- ^ ^t^i = —ju-E. inhomogeneous medium. He replaced fo by and k by in the dispersion relation C [G and arrived at the Klein-Cordon equation These conqnise the set of fifteen Proca equations and may be viewed as forming two pliysically distinct grou])S [727]. Thus, four top ecjuations represent, apart from the coup- ~ di^ = (G.7) ling constant /<, the Maxwell field through the six vector {E, Jf): the bottom three equa- In the case of a spherically symmetric static field, the solution of this is clearly tions, on the other hand, describe a four-gradient of a scalar potential. A closer inspection reveals that those groups may be seen as corresponding to the transverse and longitudi- 4> = -J . (6.8) nal waves, respectively. It is generally accepted that, in the Maxwell model, the vectors E, If, and /» are all This Yukawa type behaviour applies to the case of an isotropic plasnia, in view of the mutually orthogonal; Furthermore, the scalar potential may be made to vanish there, form of the dispersion relation (6.1). On introducing, in addition, a magnetic field, the so that the vector potential A is ])arallel to E. Assuming that the same should apply also solution of the Klein-Cordon equation, in the static limit, takes the form of the Coulomb in the Proca theory, for fields corresponding to transverse polarizations, we obtain from potential, thus, effectively, removing the shielding effect of c/coj,. the Lorentz transformation relations [2] that [J27] There is another interesting point reported by COLE [40]. He noticed that the parameter Jl\ _ ck c-o)p also occui's in the solid state literature, where it is referred to as the London pene- (6.13) tration de])th. However, as we saw above, it also distinguishes a plasma without a \E\ ~ (o magnetic field from one with the field. In other words, it distinguishes between super- Similarly, in the case of longitudinal waves, the vectors E, yl, and k are mutually parallel conducting and conducting states. and, further, II — 0. Therefore, the following relations hold As the photon now is assumed to have a nonvanishing effective rest mass, a mass term has to be added to the Lagrangian density f. The thus obtained Lagrangian becomes (6.14) CO 1 _ . 1 . Now, utilizing the dispersion relation, or — k-C" = pre-, it becomes clear tliat the four- !- F pF^P ^—lA^^ — A A' (6.9) vector {(0, kc) must be time-like. Hence, it is possible to transform it to the rest frair.e of the massive photon. (In the case of a massless photon this transformation, of course, and is termed the Proca Lagrangian. The reciprocal length in (6.9) is associated with tlie rest mass of the photon by // m^clti. The Proca equations of motion are then written fails, as the four-vector is then light-like.) Thence, in the heavy photon's rest frame k = 0. Ccmsequently, (6.13) yields U = 0 and (6.14) gives F = 0, Because, in ^^oth cases, as (ci. 2.9) the vectors £Jand are parallel, the resulting /i'-field is just the expression of mutuai 4.-T 4-= — f, (G.IO) traMsformability between transverse and longitudinal waves. Tliis, in turn, a direot cotiseqijciicc of the fact that the state of polnrizution is not the I.orentz inynri;U!t [/:'/!. J/o/-f'o\W, iho lon^itiuiiiuil wiiva projyn^ntvti in tJio dilection of IheTomnitjii uave vector JL is ^euciallv acceptedThat, ))V f'^r the largest part ot the \Jnweise in phvHuv.v ^taVe. \%ith tba velocity equal to that of the transverse wave [i^/"]. Longitudina] and transverse Therefore, it'seems ciuiLe natural to turn our attention towards cosmic phenonvena waves must he, therefore, treated as being equally important. However, this appears to From the definition of the refractive index, n^ ^ 1 - (r^p^Vo^-), it, clearly, follows that lead to the conclusion that an extia degree of freedom must be taken into account in values of n at radio frequencies are much smaller than those at oj.tical frec(uenc!es inu>, discussing the black body radiation. In fact, as a consequence, a factor of 3/2 should now the variation of propagation of an electromagnetic wave through piasu..a, is, ex-icier.tiy appear in the Planck foi'mula. But, as explained in [127], the transition between the Max- -. of energy is, at the same time, carried away by the transmitted longitudinal wave. The Although he was looking into the problem of the photon mass in vacuo, a task entirely effect of a longitudinal wave on the matter in its path is veiy small [12H], so that vir- different from ours, his results are directly applicable here. tually no transfer of momentum to the obstacle occurs. Because the electric field is so Whilst exj)laining the FREUXDLICH [130] red shift, TER HAAR [131] concluded that the small (according to (6.14), it is proportional to ucjo)) the interaction occurring between effect of electrons on the passing electromagnetic wave is stronger tlian the corresponding this type of wave and mattei' can be, for most pui-poses, neglected. In other M'ords. tiie effect caused })V photons. approacli of the longitudinal photon to equilibrium is slow. Thus, for example, as- very There is still another effect observed on the astronomical scale, viz., that of quasar red suming the rest mass of a photon = g and an angular freciuency in the 0])tical m^ co shifts which arc uot caused by the Doppler velocities order to explain this region in a physical cavity, it caji be estimated [727, that the ti)ue necessaiy foi- the [132]. hi 128] phenomenon. PECKER ct al. introduced the concept of an inelastk photon-photon transverse photons to transform into longitudinal ones is of the order of years. This [133] interaction. They claim that a redshift results from the slowing down of emitted photons transformation effect is, indeed, minute. by the radiation" field sturounding the source [134]. Putting it differently, they suggest Furthermore, in order to avoid the conti-adiction with the black Ijody radiation law, that the interaction of a tiansverse background photon with a (transverse) emitted GOLDHATJEK and NIETO [72-^] suggested that the longitudinal photons may have a lest photon may he mediated hy a longitudinal ])hoton — thus assuming that photons sliould mass of several GeV so that the interaction length is so small that the effect cannot be, have non-vanishing rest luasses. This explanation was queried by "WOODWARD and essentially, observed. Their reasoning follows from the rule known in pai'ticle physics YornoRAU [//>Jj as not fully meeting the requirements of the "tired light" hypothesis. which states that the range of a propagated interaction is inversely proportional to the Another anomalous rc^d shift effect was observed by COLDSTEIX W hen the satellite mass of the particle responsible for its transmission. [13()], Pioneer-G reached its perihelion behind theStm. MERAT etal. [137] have analysed this GERTSENSHTETN and SOLOVEI [120], taking into account that even the best conductors shift, which was synuuetrical with respect to the center of the Sun, and concluded that are not ca.pable of reflecting these waves, introduced the concept of the skin depth the effect was due to the interaction of the photons emitted by Pioneer-G with the radia- corresponding to the longitudinal waves. This is related to the skin depth d^ for the tion emanating from the Sun. These anomalous shifts dv, and the associated energy los- tninsverse waves by ses, could be interpreted, so it was suggested, as being caused by the non-elastic photon- ^ (6.15) ])hoton scattering. ^L {iojficf St, In the just discussed explanations of the non-velocity red shifts, it was assumed that the mass of a photon, although minute, was still finite. However, the effect of the inter- whej-e the equality holds for low conductivity. Due to the factor {coj/uc)-, the transparency stellar medium was not taicen into account. Thus, a riiodified explanation can be piit of the stellar medium (which is in ])lasma state) to the longitudinal photons is nearly forward. As the collisionless plasma is a good approximation [138] in analyzing the wave perfect, th\is, explaining the lack of their observation. It was, furthei', pointed out by phenomena in the interstellar s[)ace, it can be asserted that a photon (originally mass- WIGNER (cf. [124]) that the longitudinal waves cannot be responsible for cooling of the less) ac(piires a non-vanishing mass when propagating through such a medium. And only stars, as the rate of their prodUvCtion must be minute. He then concluded that the ther- then it interacts in the way proposed above. This last reasoning is in agieement with the modynamics of a system in question would be barely affected by the presence of .the supposition of this work. longitudinal photons. Turn the attention now to the energy-momentum tensor for the free massive field. Re- It is im])ortant to emphasize, at this point, that although the above mentioned workers calling that the free field is one with no charges or currents present, we see that (0.11) assumed the ]jhoton )nass to })e non-vanishing in vacuo, the underlying philosophj'^ of the becomes simplv pi'esent work is diametrically opposite. We assume the standard Maxwell field in vacuo, im})lying a zero rest mass of a photon in empty space. Only the study of fields inside • += 0, (C.IG) matter requires an inti'oduction of massive photon electrodynamics. In that case then, or, written in terms of the electromagnetic field tensor, we have tlie coupled (Maxwell) field is replaced by the free (Proca) field. Put otherwise, the ana- S^F^^ + fi-^A^ == 0. (G.17) Ivsis of the behaviour of a photon in a medium, where the dispersion relation, co- — k-c? -f- o)p~, holds, is replaced by investigation of the properties of a neutral vector meson in As a consequence of the mutual dependence of and F^'" which stems from (6.17) and vacuo. (2.4), the // =r 0 theory cannot be gauge invariant [46?]. This needs an explanation, sincc Next, let us apply this interpretation to some of the available experimental observations. gauge invariance is an impoi'tant concept. The vector potential, introduced as an auxi- liaiy Juatheniatical field to help to simplify the analysis of physical field, is not directly a sum of an orbital Mand spin angular momentum density tensors. However, measurable. However, in order that correct values of field strengths are obtained, one may always add gradient of a scalar A without affecting those, directly measurable, field this last subdivision may be shown to be unique only when the rest frame of the particle variables. The vector potential itself, however, is not uniquely defined. The transfor- can be defined. Thus, a meson, having a non-vanishing rest mass, satisfies this condition. mation Therefore, it lends itself for the analysis of the energy-momentum tensor of the electro- magnetic field in a plasma. Ai"-h d/'A-' (6.18) Our primarv concern in defining the symmetrical energy-momentum tensor is to - is th en termed the gaug? transformation. Hence, in order that a field cjuantitv has a modify T"" in such a way that the totafdistribution of energy and momentuui densities phj'sical meaning, it nmst be invariant imder gauge transformations (6.18). In this con- in the"field, described by be given by nection, we may recall the idea of Aiiaeoj^ov and Bohm [111, 112] to consider the vector 0"" == T^^ -f T.r (6.23) potentials jrather than the field tensor J^f") as the true physical field quantities. It can be shown [22-5] that the theoretical requirement of the gauge invariance implies the van- A divergenceless tensor T/" may be then interpreted as the spin energy-momentum ishing photon mass. Nevertheless, Coester [23.9] was able to construct a theory of a tensor, and must be chosen in such a way that conservation laws involving T'"' still hold. neutral vector meson which, whilst not being gauge invariant, remained in a'ccoid It is then found that with observations. T.f^v ^ JL (6.24) Xow, as is well known, the energy-momentum tensor can be formed from the Lagrange 471 density. Thus, if the Hamiltonian density is defined by It is to be noted that T/" gives no contribution to the total energy and jnomentum since Pf (6.19) J QHD-X = J T-'D^X. (6.25) the covariant formulation of (6.19) leads (in view of theLorentz transformation propei- ties of .J^^} to tlie canonical energy-momentum density tensor (2.28). Substitution of the The total angular momentum density tensor Proca Lagrangian density (6.9) into (2.28) and utilizing the corresponding field equations = - a-"©'-' (0.26) (6.12), yields the following expressions for components of this tensor is now conserved, as its divergence, d^J^^J, vanishes. Thiis, the contribution of Tf ensTU-es tlie conservation of TOO _ + _ 1/ . (VE) To elucidate the meaning of (6.23), we express it in a component form, making use of the 8.-T 47t Proca equations (0.12), as

T'' - ~ [(^Xi/), + iX'A.V + V . {A,E)] (6.20) (6.27)

rfiO _ [{E X Jf)i + -f (P + VII)i - d,{VEi)]. = eoi ^ X U). -f 471 471

Generally speaking, the energy density corresponding to (6.9) is not positive definite Similar expressions for the massive vector field were obtained by ScHRODI^'GEK [ 727]. At

[48]. least, for a relatively loose coupling between the tensor and the potential fields, there Components T^^ and T^'' in (6.20), corresponding to the energy and momentum densities, will be no mixed terms'in (6.27). As 12 is assmned to be small, the above expressions A-ery respectively, are clearly seen to differ from the accepted ones. Nevertheless (cf. Ciiap- closely resemble the ordinary ]\Iaxwell field. ter II), by a three dimensional space intergation, the followingex^jressions for the energy There is one other point worth mentioning. Using the field equations (6.12), the linear and momentinn, associated with the Proca field, are obtained momentum derived from the canonical tensor T''" becomes

1 <7 = l ''(E • A) d^x. (6.28) 4rr. (6.21) The corresponding expression, obtained from the synimetric tensor O-"', takes the form Jt^^cPx = X U)i + d^x. = — J • E) + {E X //)} rPx. (0.29) The shortcomings of the canonical tensor were discussed in Chapter II. In an attempt to meet them, Belinfaxte [4G] ])roposcd a synunetrical form for the tensor density. Thus, The integi-ated total angular momentujn can be written, therefore, as the total ai\

'/ = ^ y X (A(I'. E) -h {E X If)} d^v. {n.-M)) (6.22) Weto7 rolnt icjii.-^ [.joi, t/io intu^i nuciof (6.':\0) assuTiies the form ' frame, the corresponding exjoression. takes the fornn X (/ . A) - (E . F) A + A(r . E), (G.31)' c,{ExU). (0.3.>) /sp 4.10 Nmv, in viev/ of (6^28), tlie above is interpreted as the moment of the canonical momen- / • ?Ar/if ^ T! momentum density. Consequently, substitution of (6.31) into Now, I'ecall that, in the low frequency domain, the value of the dielectric pennittivity (b.30) lollowed by an mtegration, results in is considerably different from that of the vacuum. On the other hand, it i;s ^^•el accepted that tlie value of the permittivity, at high frequencies, asymptotically approaches unity J = ~ J E. {XX V) AcPz + ~ f {Ex A) d^x, as a'.dance at (4.25) will reveal. Therefore, this behaviour also affects the magnitude of (6.32) Thus, it may be deduced that the magnitude of f,p will almost vanish at mgh ire- quencies, whilst"^it will increases for lower frequencies. in accord with the relevant exx)ression of Bopp [^.9]. Althoucrli this work concentrated mostly on the case of a collisionless, isotropic plasma Tliis point essentially, concludes the derivation of'the field energy-momentum tensor nevertheless, the above interpretation is in excellent agreement with the experimental lor the neutral vector meson in vacuum. Thus, it is found that the total tensor is svm- results obtained by using liquid and solid dielectrics. Thus, the conclusion reached here metnc and consists of an "orbital" part, which determmes the energy and momentum, predicts a vanishing spin energy-mouientum tensor (6.33) in the region of liigh frequencies. ami a spin part, which does not contribute to either, but is important in calculations Therefore, in view of the above discussion, the field energy-momentum tensor is quite oi tile total angular momentum. correctly given by the Minkowski theorv, when the frequency is higli. This conclusion A remark about the energy-momentum tensor (6.27) is in order here. The time compo- is well supported by experiments [25, 26, 27]. At the other end of the frequency spectrum, nent oi tlie vector field (I', ^1) vanishes in the ease of a transverse field, as was noted in (6.33) does not vanish and, thus, the field energy- momentum tensor is then a sum oi oonmients following (6.12). Therefore, its contribution to in (6.27) is nil Moreover and S':;, directly leading to the form put forward by ABK.UfAM. The experimental the space-time component] of is now and the wave is, therefore, nearly' [E X evidence su'pplied by [2S, 29, 30] well supports tliis conclusion. pure transverse wave. ' 'J' This concludes our discussion of tlie Proca theory of neutral vector mesons and of its Hence, in accordance with the thesis of this work, the same conclusion must also appK- applications to the problem of the electromagnetic energy-momentum tensor in a me- to an electromagnetic field in an isotropic, collisionless plasma. Put otherwise, in order dium. In fact, we have, essentially, completed the task attempted in tliis work. It only tliat a tensor have a physical meaning, it must be synnnetric renuiins for us to summarize and synthesize our findings, placing them in proper per- Xext, it is well accepted [IS, 23, 78] that the AlinkoVski tensor readilv adjusts it- spective. This will be done in the final chapter, where also some extensions for future work seii to the canonical formalism. Recall, further, that that tensor (similarly to the cano- will be suggested. mca^tensor) is not synnnetric. Tims, in analogy to the above discussion, the tensor — ' ^- "" — ought to be interpreted as the spin energy-momentum tensor. Ex- S l VIL Conclusion pressed covariantly, it becomes In the preceding cliapters, we dealt with a few as[)eets of the interaction jilicnomena = {i)u - 1) g^-o^v (6.33) between the ele^c-tromagnetic wave and a medium. Our main effort M'as devoted to and, in the three dimensional form in the rest fi-ame deducing the correct expression for the field energy-momentum tensor in a medium. As discussed in Chapter 111, many different suggestions had been pieviously put forward and we siiowed that the emerging picture reflect CM! a rather confused situation with no, sp - } consensus of opinions as to which form of that tensor was correct. This was shown to (6.34) be partly due to some ver}' diffcn'ent interpretations being given to certain physical OiO e.u — 1 S7> E X H. quantities. 4.TC In vi(;w of the complexity of the problem, it was considered necessary to make a few It follows, from (6.34) that tlie only non-vanishing component of is the space-time ajiproximations, in order to elucidate the physics of the question at hand. I'hus, the one. Let us inquire, then, about its effect on the total tensor which, in view of the defi- main effort was directed to the case of an lsotro]>ic, collisionless plasma, i.e., a collision- nition of S^;;, is identical with the Abraham form Recalling the classical meson less plasma in the absence of an external magnetic fiekL Although this resulted in a • theorj^ presented above, we see that all high frequency phenomena can be approxima- ratlier artificial problem, it helped to clarify the distinction betv/een the suggested tely described by the Maxwell theory. Put otherwise, the characteristic differences be- tensors. Then, as a further step towards generalization, collisions could be included, as tween two theories become pronounced onty in the low frequency limit. the above method is readily adaptable. xV somewhat similar conclusion can be reached when investigating the propagation of the According to the present writer's knowledge, all relevant woiks, hitherto, deal with the electromagnetic radiation in a medium. lu the process, this medium becomes, generally interaction of the electromagnetic field with a medium whose refractive index is greater speaking, magnetized. On a microscopic level, this effect corresponds to the spin of the than unity. It was, therefore, considered essential to concentrate instead on inter- field quanta. Furthermore, it is known [5'] that the magnetic effects are nuich smaller actions with media whose refractive index is smaller than unity. Although the most tlian dielectric effects and that they belong, essentially, to a low frequency domain.'Let obvious example of such a medium is a plasma, we know that similar considerations also us apply now this consideration to the force corresponding to (6.34). Thus, in the rest apply to a gas, liquid, or a solid, provided that the frequency of the incident radiation is very much larger than the characteristic frequency of a medium.

. 7.\irt--clirl!.lr. il. r PK It proved illustrative to consider the radiation from both points of view - those of corpuscules and waves. The differentiation between these two aspects turned out to be Acknowledgements more profound when radiation was inside a medium. Within the boundaries of geometri- cal optics approximations, it was found (in Chapter V) that the momentum of a photon in The author would like to express his gratitude to — a plasma js given by the Minkowski expression. In the process, the method of Einstein, Professor H. Hora, who introduced him to the subject and whose constant enthusiasm made this to investigate the interaction of radiation with matter, was expanded and used, in tui-n, investigation very rewarding; to suivport the IMinkowski result. It should be emphasized, however, that tliis conclusion J. R. Shepanski, w ho worked through the manuscript and whose comments proved to be of great IS valid only in the region of high frequencies. There is some experimental evidence, viz., value; that of JONES and RICHARDS [25], JONES and LESLIE and ASIIKIN and DZIEDZIO which tends to support this finding. D. T. Holmes, \vho conscientiously read the following pages and contributed towards an early To investigate the behaviour of the low frequency electromagnetic field in a medium, the completion of this work. simikrity of the relativistic form of tlie energy-momentum relation for a massive particle to the dispersion ecjuation for the wave motion was invoked. Although the preliminary Bibliography steps were inti-oduccd earlier [40,43], the resulting neutral vector meson theory had not been previously applied to the pro!)lem of the form of tho energy-momentum tensor. It E. ScHRoniNGER, Proc. Royal Soc. A 232, 435 (1955). should be em]ihasized that, although the theory itself is not new, the underlying philo- W. PACLI, Theory of Relativity, Pergamon Press, 1958. sophy is certainly entirely novel. FurtJicrmore, it should be stressed that, throughout this L. DB BKOGLIE, Phil. Mag. 47, 446 (1924). work, the mass of the photon in vacuo is assumed to be zero. However, when the radia- 0. COSTA JXE BEAUREGARD, Precis of , Academic Press, New York, 1966. tion field propagates in a medium, it may be represented by photons having non-zero D. LANDATT, E. ]\L LIFSIRITZ, The Classical Theory of Fields, Pergamon Press, 1971. rest mass. The coupling between the radiation field and a medium is responsible for the M. VON LAUE. Z. Phys. 12S, 387 (1950). existence of the massive photons-like behaviour. C. JLOLLER, The Theory of Relativity, Oxford, 1952. Therefore, the meson theory, together with the symmetrization procedure of BELIN- L. 1). l.ANUAR^, E. M. LIFSHITZ, Electrodynamics of Continuous Media, Pergamon, 1960. FANTE was utilized to determine the form of the energy-momentimi tensor. The X. BALAZ.S, PJiys. Rev. 91, 408 (1953)"! work of Bopp [^.9] was instrumental in drawing an analogy between the tensors T"" and D. V. SKOBEI/TSYN, Uspekhi Fiz. Nauk 110, 253 (1973). of classical elcctrodjmamics, on one hand, and those suggested by Minkowski and M. G. BURT, R. PEIICBLS, Proc. Royal Soc. A 333, 149 (1973). J. P. GORUOX, Phys. Rev. A 8, 14 (1973). Abraham, on the other. It was then found that the Abraham form of the tensor is, gene- R. -PEIERLS. Proc. Roval Soc. A 347, 475 (1976). rally, the correct one. The Minkowski tensor was definitely ruled out in the low fre- R. V. JoxES, Proc. Royal Soc. A SCO. 365 (1978). ' quency limit. This agrees witli results of tlie work of WALKER et al. and WALKER W. SHOCKLEY, Proc. N. A. S. tiO, 807 (1968). . and WALKER. [29]. Further, the results obtained by HAKIM and HIGIIAAR [30], in the H. A. HAUS, Physica 43, 77 (1969). static limit, definitely supported the Abraham conclusion. (It is instructive to notice that F, N. H. ROBIXSON, Phys. Rep. 10, 313 (1975). in experiments [28, 29], the used frequencies were in the range 5—100 Hz and, thus, 1. BREvnc, Mat. Fys. :\Icdd. Dan. Vid. Selsk. 37, No. 11 and 13 (1970). obviously, did not satisfy the WKB-condition). Although SHOCKLEY and JAMES [54] did V. L. GIXZBXJRG, Soviet Physics-Uspekhi Hi, 434 (1973). not actually conclude that the value of the momentum was that given by the Abraham Z. MIKURA, Phys. Rev. A 13, 2265 (1976). A. EIXSTEIX, J. LAUR, Ann. Phys. (Leipzig) 26, 532, 541 (1908); 27, 232 (190S). expression, they, nevertheless, definitely excluded the Minkowski result, while using the F. BECK, Z. Phys. 134, 136 (1953). frequency of 5 kHz. G. MARX, G. GYOROYI, Ann. Phys. (Leipzig) 3G, 241 (1955). The formal equivalence of the jueson theory in vacuo with the Maxwell theory in a me- S. R. KE GROOT, L. G. SUTTORP, Physica 37, 284, 297, (1967); 30, 28, 41, 61, 77, 84 (1968). dium was derived while utilizing the concept of a collisionless plasma. The first step R. V. JoxES, J. C. S. RICHARDS, Proc. Royal Soc. A 221, 480 (1954). towards generalization would be to apply the above method to the case of a collisional R. V. JONES, B. LESLIE, Proc. Royal Soc. A 3G0, 347 (1978). plasma. No real difficulty is envisaged and, it appears that, the extension would be A. ASHKIX, J. I\L DZIEDZIC, Phys. Rev. Letters 30, 139 (1973). straightforward. Further, although, there is qualitative agreejne.nt with the experiment, G. B. WALKER, D. G. LAHOZ, G. WALKER, Canad. J. Phys. 53, 2577 (1975). it would be informative to extend the above method to apply to any state of matter. G. B. WALKER, G. WALKER, Canad. J. Phys. 2121 (1977). S. S. HAKIM, J. B. HTGHAM, Proc. Phys. Soc. 80, 190 (1962). Moreover, the interaction of a high density field, e.g., that of laser, with a medium would C. G. DARWIN, Proc. Royal Soc. A 14R>, 17 (1934). certainly warrant further investigation. B. B. KADO^ITSEV. Soviet Physics JETP G, 117 (1958). To summarize, the correct form for the field-energy momentum tensor has been given by V. L. GINZBUKG, The Propagation of Electromagnetic Waves fti Plasma, Pergamon, 1964. Abraham. The Minkowski form for that tensor, on the other hand, a])pears to represent A. EINSTEIN, Phys. Z. 18, 121 (1917). a fair approximation in the region of high frequencies, where the spin contributions may D. V. SKOBEL'TSYN, Voprosy INhxtematicheskoi Fiziki, Nauka, Leningrad, 1976. be neglected. G. BEKEFI, Radiation Processes in Plasma, Wiley, 1966. J. M. DAWSON, Adv. Plasmas Phys. 1, 1 (1968). K. D. COLE, Austral. J. PJiys. 30, 671 (1977). H. HORA, Phys. Fluids 12. 182 (1969). K. D. COLE, Austral. .J. Phys. 24, 871 (1971); 26, 359 (1973). K. D. COLE, Phys. Letters 39 .V, 278 (1972): 41 A, 325 (1972). A. P[ivs. Letters A, 2S3 {1971); 41 A, 32:> (1972). C-^-^J Ij- T- A. i'jrriM^ii-jrA. C2:toI,. J. li 23, 1190 (1973). I; I'-.ITI K. ('. WKSTFOLO, Phil. Mag. 41, 509 (1950). , CO..BEJV, J. Sai£\viis^aisii, Phys. Rev. ;>8, 953 (1940). '\F'>! \ L. BUILLOUIN, Wave Propagation and Group Velocity, Academic Press, New York, 1960. V Classical Charged Particles, Acldison-Wesley, 19G5. . I IN'J] \V. HEITLER, The Quantum Theory of Radiation, Oxford, 1954. IJOJ i'. J. BELKVFANTJS, Piiysica (>, 887 (1939). H. IIORA, Pliys. Fluids 17, 1042 (1974). F. J. BELINFA-N-TE, Physica 7, 449 (1940). ^ , [;/-'/] 1). PARK, Ann. Phys. (Leipzig) 27, 319 (1971). W. FAULT, ROV. Mod. PJiys. 13, 203 (1941). \10->] R, .^r. LEWIS, Arch. Ration. Mech. Analysis 20, 191 (1905). - [49-] F. Boiri', Ann. Phys. (Leipzig) 11, 35 (1903). [106] L. DE BROGLIE, Non-linear Wave Mechanics, Elsevier, Amsterdam, 1900. [56*] J. D. JACKSO^^ Classical Electrodynamics, Wiley, 1975. [107] \'.A. FOCK, Electromagnetic Diffraction and Propagation Problems, Pergamon Press, 1965. [51] F. X. H. PvOBiNSON, Macroscopic Electromagnetism, Pergamon 1973 [lOS] J. A. ARNAUD, j. Opt. Soc. Amer. 65, 174 (1975), \o2] Pv. GRAMMEL, Ann. Phys. (Loip/ig) 41, 570 (1913). [100] H. IIORA, Optik 17, 409 (1960). [53] H. G. ScnoPF, Ann. Phys. (Leipzig) 9. 301 (19G2). [110] J. L. CARTER, H. HORA, J. Opt. Soc. Amer. 61, 1640 (1971), [-5-/] \V. SHOCKLEY, R. P. JAMES, Phys. Lev. Letters IS, 870 (19G7). [111] Y. AHARONOV, D. Boiar, Phys. Rev. 115, 485 (1959). [oo] R. P. JAMES, Proc. N.A.S. (51, 114!) (1908). [112] Y. AHARONOV, D. BOHM, Phys. Rev. 123, 1511 (1961), [5u] 0. COSTA DE BEAOREGART), Phys. Lettex's 21, 410 (19GG). [///] R. G. CHAMBERS, Phys. Rev. Letters 5, 3 (1960). [J7] CH. (JOILLOT, Piiys. Letters 21,"408 (196G). [114] T. H. BOYER, Phys. Rev. D S, 1667, 1679, (1973). O. COSTA UE BEAUREGARD, NUOVO Cimento 4S B, 293 (1967), [115] J, DAWSON, C. OBERMAN, Phys. Fluids 5, 517 (1962). [50] O. COSTA DE BEAUREGARD, Ph3^s. Letters 24 A, 177 (1967). [116] L. OSTER, Astrophys. J. 13S, 761 (1963); 144, 836 (1966). [60] O. COSTA DE BEAUREGARD, Fonnd. Physics 2, 111 (1972). [117] K. M. CASE, S. C. Cmu, Phys. Rev. A 1, 1170 (1970). [6T\ A. PAPAPETROU, PHI'L Mag. 40, 937 (1949). [IIS] W. M, CRONYN, Astrophys. J. 144, 834 (1966). [62] G. GYORGYI, Amer. J. Phys. 28, 85 (19G0). [119] D. F. JMARTYN, Proc. R. Soc. A 193, 44 (1948). [63] J. AGUDIX, Phys. Letters 24 A, 761 (19G7). [120] C. L. TNMAN. Astrophys. J. 142. 201 (1965). [64] D. V. SKOBEL'TSYN, Uspekhi Fiz. Nauk 122, 295 (1977). [121] R. M. MORE, J. Pliys. A 9, 1979 (197(5). [65] R. L. DEWAR, Phys. Fluids 13, 2710 (1970). [122] V. A. PETRZILKA, (3zech. J. Phys. B 26, 151 (1876). [66^] \V. L. JONES, Rev. Geophys. Space Phys. 9, 917 (1971). [72-3] G. FEIXRERG, Science 166, 879 (1969). [6'/J vox LAUE, Das Relativitatsprinzip, Braunschweig, 1911. [121] A. S. GOLDHABER, ^L NIETO, Rev. .Afod. Phys. 43, 377 (1971). A. ScHEVE, Ann. Phys. (Leipzig) 30, 805 (1909). [125] J. M. JAUCH, F. Ro/IRLTOH, Tlie Tlieory of Photons and Electrons, Addison-Weslev, 1955. [60] C. L. TA^-G, J. IVlEixx'ER, Phvs. Fluids 4, 148 (1961). [12(7] :\L A. GINTSBUHG, Soviet Astron. AJ 7, 536 (1964). [70] G. H. LIVEN'S, Phil. ^Lig. .12, 162 (1916). [i27] L. BASS, E. SOIIKODINGER, Proc. Royal Soc. A 232, 1 (1955). [77] R. PEIERLS, Proc. Royal Soc. A 35R>, 141 (1977). [72.S'| I. Yu. KORZAREV, L. B. OKUX, Soviet Physics-Uspekhi 11, 338 (1968). [72] H. K. AVONG, K. YOUNG, Atner. J. Phy.s. 45, 195 (1977). [120] M. E. GERTSEXSHTEIN, L. G. SOLOVEI. JETP Letters 9, 79 (19A). [73] G. A. ASKAR'YAN, JETP Letters 9. 241 (19C9). [130] E. FixLAV-FREirxDLicH, Phil. Mag. 45, 303 (1954). ['/i] K. K. TnoRNBER, Phys. Letters 43 A, 501 (1973). [131] 1). TER ILAAR, Phil. Mag. 45, 320 (1954), [7,5] J. A. ARNAUD, Optics Comm. 7, 313 (1973). [132] li. ARP, Science 174, 1189 (1971). [76] B. JOYCE, Amer. J. PJiys. 43. 245 (1975). [7.3.)] J. C. PECKER, A. P. ROBERTS, J. P. VIGIER, Nature 237, 227 (1972). [77] W. B. JOYCE, Phys. Rev. i> 9, 3234 (1974). [131\ J. C. PECKER, W. TAIT, J. P. VIGIER, Nature 241, 338 (1973). [76] R. L. DEWAR, Austral. J. Piiys. 30, 533 (1977). [/JiiJ J. F. AVOODWARD, W. YOUR(;RAL'. Nature 241. 338 (1973). [7.9] E. L. HIEL, Rev. :\lod. Phys."^23, 253 (1951). [136] R. :^L GOLDSTEIN, Science 160, 59S (1969). [SO] E. J. POST, Phys. Rev. 118, 1113 (1960). [137] P, MERAT, J, C. PECKER, J. P. VIGIER, Astron. Astrophys. 30, 167 (1974). [6'Z] C. ECKART, Phys. Rev. R>4, 920 (1938). [735] I. LERCHE, Adv, Plasma PJiys. 2, 47 (1969). [.^'I'J R. KLIJIA, V. A. PETRZILKA, CV.ecli. J. Phys. B 22, 896 (1972). [139] F. COESTER, Phys, Rev. S3, 798 (1951). [6'^^] R. KLT:JRA, V. A. PETRZILKA, Ann. Phys. (N.Y.) 92, 395 (1975). [<5i] P. A. .STURKOCK, Ann. Phys. (N.Y.) 4^ 300 (1958). [] F. HEIIL, Ann: Phys. (l.eipzig) 2,',, 215 (1970). [

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