Non-linear scattering of light in the limit of ultra-strong fields

PRAKASH BHARTIA' A N D SREERAM VALLURI Ut~iversily qfRegincr, Regitlo, Sosk., Crrtlcrcin S4S OA2

Received November 23, 19772

The non-linear interaction of two electromagnetic waves in a vacuum is treated by the introduction of the Lagrangian for a strong electromagnetic field. The quantum mechanical effect of the scattering phenomenon between plane electromagnetic waves can be simulated within the framework of classical theory under the assumption of an effective non-linear interaction be- tween the electromagnetic waves. The promise of continuous, intense, and monochromatic polarized radiation from electron synchrotrons and storage rings motivates the study. The scattered fields are obtained and a ratio of the power in the scattered fields to that of the incident fields is estimated for a simple case.

L'interaction non lineaire de deux ondes electromagnetiques dans le vide est tl-aitee au moyen de I'introduction du lagrangien pour un champ Clectromagnetique fort. L'effet quantique du phenomene de diffusion entre ondesClectromagnCtiques planes peut Ctre simule, dans le cadre de la theorie classique, en introduisant une interaction effective non lineaire entre les ondes. L'Ctude est motivee par la perspective d'obtenir, avec les synchrotrons 5 electron et les anneaux de stockage, des rayonnements polarises, monochromatiques intenses et continus. Les champs diffuses sont calcules, et le rapport entre la puissance dans le champ diffuse et la puissance dans les champs incidents est estime dans un cas simple.

[Traduit par le journal] Can. J. Phys., 56, 1122(1978)

1. Introduction The work of Euler and Heisenberg (1) and Weiss-

kopf (2) shows that classical electromagnetism becomes a non-linear theory due to the probability of pair creation. Schwinger's (3) rederivation of the result with the use of the proper time technique provides a suitable gauge invariant basis for treating problems and has inspired later work. The non- linearity introduces corrections to the classical Lagrangian Lo = J(E2 - B2) which result in the non-linear Maxwell equations. Thereby, electro- magnetic fields interact with each other and the scattered fields can be calculated.

In this work we use the Lagrangian in the ultra- strong field limit to discuss the scattering between two focussed plane waves. With highly powerful sources it might become possible to generate ex- tremely high electromagnetic fields and thus provide concrete experimental evidence for their interaction. Radiation from electron synchrotrons and storage rings show strong evidence for such a development. The energies from the visible to the X ray or soft X ray are lower than the electron rest energy mc2 and the quasi-static approximation can be used. In such an energy limit, the radiation cannot see the structure of the interaction and some of the earlier papers in quantum electrodynamics have shown (4) that the interaction Lagrangian density can be just added to the classical Lagrangian density. It is well known (5)

that the ratio of the non-linear term L, to the classical value Lo increases at high fields (much greater than the critical field) logarithmically. We therefore felt that a semi-classical approach like that of the work of McKenna and Platzman (6) might be reasonably appropriate to investigate the non-linear term effects in the ultra-strong field limit. We show that the non- linear Lagrangian gives rise to a fluctuating current and that the charge density, unlike the case in the weak field limit, vanishes.

The fields F(E, B) are assumed to be slowly varying and certainly satisfy the conditions

up to the soft X-ray range of the spectrum. Slowly varying fields, as is known, cannot in practice create real electron-positron pairs unless there is pair production by a large number of soft photons. This is not considered in the present problem.

The non-linear electrodynamic effects can be described with the aid of a field dependent dielectric permittivity and magnetic permeability of the vacuum which are obtained from the electric dis- placement D and the magnetic induction H.

The non-linear source terms are introduced into the extended Maxwell equations. The scattered fields are obtained by solution of the reduced wave equa- tions in terms of integrals over the derivatives of the u

address: Defense Research Department, Depart- current distributions. The variables are evaluated at ment of National Defense, Ottawa, Ont., Canada. the retarded time. The expression for the scattered

ZRevision received April 18, 1978. power is derived and the ratio of the power in the

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BHARTIA AND VALLURI 1123

scattered fields to that of the incident fields is waves, they must be non-linear in contrast to the obtained. We make use of the system of units in linear Maxwell equations. It follows that the which K = c = 1 unless otherwise indicated. Lagrangian density cannot be a quadratic function of

the fields and therefore differs from Lo = +(E2 - 2. The Non-linear Maxwell Equations in the ~ 2 ) .

Strong Field Limit The Lagrangian for any arbitrarily strong electro- In order that the field equations should be able to magnetic field which satisfies the conditions shown

describe the process of scattering of electromagnetic by [1.1] is given by Akhiezer and Berestetsky (5).

This Lagrangian was originally derived by Euler L , is obtained from the original Euler-Heisenberg and Heisenberg (1) and is a function of the two Lagrangian [2.1] for the case when E 2 - B2 is large independent invariants E2 - B2 and EB. We differ and E B is small, -0. In the case of either zero from the notation of Akhiezer and Beresteksy (5) in electric or magnetic fields, L , again reduces to the that we use B instead of H. following equations (5).

In the case of weak fields, L , reduces to a B2 B2

2 (e2)2 1 [2.5a] L,(E = 0 ) - - - In -2

C2.21 L1 =xw ;;p C(E2 - B2)' 6n 2 Fcrit a E2 E2 + ~ ( E B ) ~ ] + ... [2.5b] L , (B = 0 ) = ---1n - 6n 2 F,,~:

e2/4n = a is the fine structure constant, m the mass of the electron, and E and B are the electric and magnetic fields. This Lagrangian was used by quite a few authors to study the non-linear interaction between electromagnetic fields in vacuum. The Maxwell equations that result from such a choice of the Lagrangian are non-linear and describe well the interaction of low energy photons. McKenna and Platzman (6) have used these equations to derive the counting rate for the scattering of light by light in

For the case of the initial fields being the sum of the fields due to two plane waves, L , in [2.4] corres- ponds to the case when the polarizations of the plane waves E , , E~ are parallel to each other.

In the case of perpendicular polarizations the situation is exactly reversed; E2 - B2 = 0 and EB is large when the intensities are large. The Lagrangian in this case is obtained again from [2.1] and is represented as:

the presence of classical static fields and a semi- a EB classical approach to derive the cross section for the C 2 s 6 1 L --(EB) - 2n ln- scattering of light by light in the absence of external

Fcri:

fields, which was already derived quantum mechanic- Equations [2.5] and [2.6] contrast strongly with ally by Karplus and Neumann (7). [2.2] for the weak field case.

In the case of strong electric and magnetic fields The electric displacement D and the magnetic

with polarization such that E*B* = 0, the Lagran- gian L , is given as:

-aE2 - ~2 ~2 - ~2 C2.41 L, = -

6n 2 In

Fcri? + terms in m, e

The terms independent of E, and B are neglected. Fcri , is the critical field strength and is given by Fcrir = m2/e.

induction H are determined from. the equations:

[2.7a] D = aL/aE; H = -aLjaB

where

[2.7b] L = Lo +L1

for the case of parallel polarization and

[2.7c] L = Lo + L2

when the polarizations are perpendicular to each other. Straightforward differentiation gives for:

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1124 CAN. J. PHYS. VOL. 56 , 1978

[2.9b]

Equations [2.8] and [2.9] differ strongly from each other as well as from the expressions for D and H in the weak field case, which for arbitrary polarization states are:

D and H are related to E and B by the following equations:

[2.1 l a ] Di = E ~ ~ E ~

[2.11b] Hi = pik-' Bk Our notation in [2.11b] differs again from that of Akhiezer and Berestetsky (5) and conforms with that of

Adler (8), who has derived expressions for D and H instead of D and B as done by Akhiezer and Berestetsky. A comparison of the formulas of [2.8] with those of [2.11] shows that:

and a comparison of the formulae of [2.11] with those of [2.9] shows that:

a E .B . EB l---(1+ln7)] 271 B~~ Fcr i t

in the limit of strong field intensities. For two plane The fields have to be almost infinite for [2.14b] to waves with polarization vectors parallel to each hold, which is also true in the case of perpendicular other, E2 - B2 # 0. However, for polarizations and is almost improbable in physical

situations. In the weak field limit

We have, 1 a 2 [2.14b]

[2.15b] pik-' = ijik +- - cik = pik-' = 0 45n in4

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BHARTIA AND VALLURI 1125

An extended set of Maxwell's equations occurs The non-linear terms are therefore included in two due to the inclusion of the non-linear terms; these source terms p and J. Since the non-linear terms in are of the form : the Lagrangian L , and L , are <<Lo, it is convenient

[2.16] and appropriate to use a power series expansion in v x E = - a ~ / a t -a/6n and a/2n to discuss the effects of scattering. [2.17] v x H = a ~ / a t Therefore, the series expansion in the case of

[2.18] parallel polarizations is :

V . D = O

[2.19] V . B = O [2.24a] E = E, --E, + ... a 6n

McKenna and Platzman (6) have shown that no scattering takes place between photons travelling in [2.24b] B = Bo - -Bf a + ... the same direction. The extended set of Maxwell's 6n

equations can be rewritten for c1 I I E , as: and for perpendicular polarizations is: [2.20a] v x E = - a ~ / a t

[2.25a] a

E = Eo + - Ef + ... [2.21a] V x B = aE/at + 4 n J ( - a / 6 ~ ) 2~

and for E, I E~ as:

[2.20b] v x E = - a ~ / a t

[2.21b] v x B = a ~ l a t + 4nJa/27c

[2.22b] V E = 47cpa/27c

[2.23b] V . B = 0

where :

Eo and B, are solutions of [2.23a] and [2.23b] with the source terms p = 0 and J = 0 and we specify them in the next section as the initial fields.

The fields Ef and Bf are the solutions of [2.20] to [2.23] when p = p , = p(Eo7 B,) and J = J, = J(E,, B,) and satisfy the wave equations:

[2.23d] I They can be solved by the use of Green's functions and are given by Tralli (9).

where R = Ir - r'l, t , , , = t - R is the retarded time, and d3r' is the source volume element.

In the far field approximation, Irl >> 1.

[2.30] R = lr - r'l ct r

The total power radiated by an electron in a circular orbit is known. Hence, the intensities can be estimated.

For 2Eele,,,,, N 6 to 8 GeV and R,le,t,o, - 12.7 m (10, 11) and for observed luminosity - lo3' cm-2 s-' at E = 3 GeV, the intensity can be very high and -1018-1019 W/cm2. It is possible that the above intensities could be improved by a higher order of

magnitude in case the proposed storage rings PEP (Stanford) and PETRA (Hamburg) are operated.

The initial fields are chosen carefully to simplify the calculations and their strengths, polarizations, spatial and time dispersion are completely specified in the following section.

3. The Initial Fields The initial collimated beams are represented by

plane waves. It is also assumed that the initial fields which generate the source terms are a sum of two fields with propagation vectors in opposite directions to each other. In essence we consider the scattering

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1126 CAN. J. PHYS. VOL. 56, 1978

of two electromagnetic fields in the very strong field The initial fields polarize the vacuum and generate limit. The two impinging waves produce a standing a fluctuating current which again produces the wave pattern which in turn generates the non-linear scattered fields Ef and B,. The initial fields are sources. represented as :

2

[3.2] B, = F,v , x E , cos (w , t - k , v ) + F2v2 x E , cos (w2t - k 2 r ) = C Fivi X E~ cos $ i i = 1

We suppose that F, and F2 are real and in the center of mass system of the two incident beams we have

The real expressions for the initial fields are due to the fact that they appear non-linearly in the formulae for the scattered fields. E , and E , are the unit polarization vectors such that

due to the transversality conditions. For beams of equal intensity F , = F2, as is assumed in the present problem. The information about the initial fields is therefore completely specified. It is also reasonable to assume that the initial fields are plane waves (6, 12) though at the focal point it is not exactly true.

The source terms are now obtained from [2.23c] and [2.23d] since the initial fields are known from [3.1] and [3.2]. We find that the source term p, = 0 and J, is proportional to the amplitude of the initial field, unlike the case of the weak fields as will be shown in the following section.

4. The Non-linear Source Terms and the Scattered Fields We assume that the propagation vectors of the two incident beams are along the z axis in opposite direction

to each other. We compute the non-linear source terms and derive expressions for the scattered fields in this section. From

[2.24] and [2.25] we have in the following cases:

(4 E l I I ~2

p vanishes due to [3.4], unlike the problem for the weak fields.

P.21 J = - 2(F1 w1/47t) [ E , (cos 4 , tan 4 , + cos 4 , tan + , ) I

where 4 , and 4 , in the center of mass system are:

[4.3a] 4 , = w,t - k l r = w,( t - z ) [4.3b] 4 , = w2t - k,r = w,( t + Z )

J is therefore in the opposite direction to that of the incident fields.

(b) E l 1 E 2

again due to the transversality conditions.

[4.5] J = (- 1 1 4 ~ ) F, w , [ v , x ( E , tan 4 , cos 4 , - E 2 tan 41 cos + , ) I

J is in a direction different from that of the

electric and magnetic fields in the case of perpen- dicular polarizations. Equations [4.2] and [4.5] show also that the current is fluctuating in space-time and is only proportional to the intensity of the field, unlike the problem for weak fields.

From the expression for J, aJ/at (r', t - R) and V x J ( r l , t - R) can be derived by straightforward calculation. The scattered fields can be obtained from [2.28] and [2.29] by integration over the source volume. The cross sectional area A of the beams is assumed to be small and ~ 7 t h ~ . We assume that the interaction of the two beams is over a length L -- 21 or 3h. We make the approximation that the source can therefore be considered to be almost linear and along the direction of the incident beams as shown in Fig. 1.

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BHARTIA AND VALLURI

f Z

FIG. 1. The shaded region indicates the interaction of the two beams over a length L 2. 2 1 or 3h.The cross sec- tional area A of the beams is assumed to be small and ~ n ? , ~ .

The source therefore can be considered to be almost linear around the direction of the incident beams. The point P is at an angle 0' with respect to the source. The polarization vectors E l and E2 are in the xy plane.

In the calculation of the scattered electric field the following term appears.

[g (rl, t - R ) ] / R = [g (rl, t - R ) I. I / C4.61 - Sv, [g (r', t - R)] / R d3rt = 47tr

a2 [ E t - R) dz' dxt dy' - V x 6B(rr, t - R ) dx' dyl dz' I

The retarded time ( t - R) can be approximated as:

C4.71 f r e t N t - Ir - 1.11 = t - ( r 2 + 1.1'- 2,.,.~)1/2

where 0' is the angle between the field vector u and u' - z since the source is almost along the z' direction. The integrand is now independent of variables x' and y', the integrations over which give the cross sectional area A of the cylindrical source. Hence:

L

c4.81 - s [g (rt, t - R) a

A [SoL $ 6 ~ d l ' - &so v1 x 7 6Br(zr, f r e t ) dz' v' 4 ~ 1 . az

a = - L [ S o L $ 6 E dz' --v1 x 6B1(z', t,.,)

I 4 ~ t I. at

We calculate the integral Jk(aZ/at2) 6E dz' first for the two different polarizations.

(4 E l 1 1 E 2

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1128 CAN. 1. PHYS. VOL. 36, 1978

where: Eo = Eo (z', fret) and B = Bo (z', t,,,). For 8' # 0,

[4.10a] ~ L $ ~ o d z ' = - ~ l o 1 2 ( r , c o s ~ , + ~ ~ c o s ~ ~ ) d z ' S,'

&,{sin [w,t - o , r + o,(cos 8' - l)L] - sin (w,t - w,i*)) COS 8' - 1

c2{sin [wit - wlr + O,(COS 8' + l)L] - sin (w,t - w r)) + -- cos 8' + 1 ' - 1 w,t - w,r + w,(cos 8' - 1)L sin [w,(cos 8' - l)L/2]

2 cos 8' - 1

+ EZ . 2 cos {wlt - w,r + [w,(cos 8' + l)L/2]) . sin w,(cos 8' +XI

cos 8' + 1 2

[4.10al SoL$ E~ in E2 - B~

= - F I W I E 1 X - { 1

F c r i t (cos 8' - 1) [sin 4 , 0

+ sin 4, In F - 1 , - cos 8' - 1 1:

- (cos 8' - 1)' a2 cos 2klz1 dz' + 2

cos 8' + 1 xz S, cos 4, 1 4 ~ , ' cos 4, cos 4, 1

+ cos 8' + 1 Fcrit2 o

- (cos 8' + 1)' a2 cos 2klz1 (cos O f - 1) S, cos ($2

dz' + 2 cos 4, -

Therefore the sum of [4.10a] and [4. lob] give I,. Also for E, I I E,,

L

+ tan 4, cos 4, I - sin 4, (2 + ln 4F12 cos 4, cos 4, - tan+, cos+,) :] = 0 F c r i t 2

:t2 soL V1 X E l 2 ~ , ' cos 4, cos 4, [4.12a] - 6 E d z t = - F , w ,

cos 8' - I F c r i l Z

- cos 8' + 1 ~ ( C O S 8' + 1) sin 4, cos 4, cos 4, tan 4,

cos 8' - 1

- E2 [sin 4, (4 + In 2 ~ , ' cos 4, cos 4, + sec2 4,) F c r i tZ

~ ( C O S 8' - 1) L C O S ~ ~ - I -

cos 8' + 1 sec2 4, + 2 tan 4, cos 4, - l o

cos 4, tan 4, 0 cos 8' + 1

2(cos 8' - 1) sin 4, cOs 81 - 1 1 a2 joL cos 42 1) + C O S ~ ' + 1 C O S ~ ~ ~ ~ ~ ~ ~ ~ ----A cos 8' + 1 a at2 cos2 4,

Also

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BHARTIA AND VALLURI

L E B + tan $2 cos +,I I + v l x ~ ~ [ 2 sin $2 (1 + i in 7) + tan b1 ~ 0 ~ 4 ~ 1 I :} o F c r i t

Ef can be obtained from [4.8] to [4.12] and for parallel polarizations and 0' # 0 is:

+ sec2 +, - dz '

4~~~ cos 4, cos 4, + cos 0' + 1

+ sec2 4, - F c r i t

The integrals are more conveniently evaluated in the calculations for the emission power due to the scattered fields which involve an integration over time.

For 0' = 0, i.e., along the axis of propagation,

A 8F12 cos ( o l t - wlr) C4.141 Ef(r, t ) = - F , o 1 L o l ~ , cos ( o l t - o l r ) - sec2 (wlt - o l r )

4nr F c r i t Z I Equation [4.14] shows that due to scattering, the incident wave F l ~ l cos ( o l t - o l r ) is modulated by a

kernel function K(r, t ) given by

C4.151 8F12 cos ( o l t - o l r ) - sec2 ( o l t - wlr)

Fcrit2 I The forward scattering can be interpreted as causing a change in the dielectric constant of the vacuum. In

the case of perpendicular polarizations,

A C4.161 Ef = F 1 ~ o l 2 [ ~ {[sin (o,l - o l r + 26,) - sin ( o l r - wlr)]

x [5 + In (2Fl2/FCri?) - 2/z0 + 66,] + sin ( o l t - o l r + 26,) In [cos2 ( o l t - o l r + 26,)]

5 - sin ( o l t - wlr) In [cos2 (colt - olr) ] +

cos2 ( o l t - o l r )

- {[sin ( o l t - o l r + 26,) - sin (wit - o,r)] [5 + ln (2F1 '/FCrif) - 2/60 + 66,] 26, + o l L

3 - + sin (colt - wlr. + 2%) In [cos2 (wlt - w ~ r + 26011 + e o s 2 (wit - + 260)

2F12 cos2 ( o l t - wlr + 26,) + 26, In - sin ( o l t - w,r)ln [cos2 ( o l t - olr) ] + 3

F c r i t 2 C O S ~ (colt - o l r )

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1130 CAN. J. PHYS. VOL. 56, 1978

with

For 8' = 0

2F12 cos ( o l t - o l r ) 2 I - cos2 ( o l t - o l r )

-- 2F12 cos ( o l t - o l r ) " sin ( o l t - o l r ) 0 1 Fcrit - + 211

In their work on non-linear interactions, McKenna and Platzman (6) have shown that in the weak field limit, for the initial field as the sum of two plane waves, no scattering other than forward scattering would have been described, unlike the case for strong fields. The scattered fields contain terms proportional to the interaction length. From [2.12] and [2.13] we have for the phase velocity u p : up = T/E 2 1 . Also from [2.12], for Fl 1: 1O2FCril, 6.5 - Zjp-' - in accord with the result of Erber and Tsai (13). B, can now be calculated from [2.29] for the two different polarizations:

(a) for parallel polarizations and 8' # 0 .

[4.17b] 8 A o l

B,(r, t ) = - - Flvl x cos (colt - o l r + 6,) sin 6, 4x1.

For 8' = 0

[4.17c] B, ( r , t ) = 0

showing that along the axis of propagation the magnetic field vanishes. (b) for perpendicular polarizations and 8' # 0

C4.18~1 B,(r, t ) = - - 4A01 Fl(c2 - z l ) cos (colt - o l r + 6,) sin 6, 4x1.

and for 8' = 0

[4.18b] B, (r , t ) = 0

showing that [4.17c] is also true for perpendicular polarizations. The time averaged power d P radiated into the solid angle dQ by the scattered fields is estimated. We give

the expression for the radial component of the time averaged Poynting vector and multiply by r 2 dQ to get the average energy per unit time radiated into dQ. For parallel polarizations,

( E , x B,) dt r2 dQ / Xdt

The average incident power in beams 1 and 2 is equal to F12A/47t, hence the ratio of the emission power to the incident energy flux can be estimated.

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BHARTIA AND VALLURI 1131

The expression in the brace brackets after integration is just a constant since wL is fixed, is obtained in terms of the functions Ci and Si, and

+ $ + Ci (2wL) - Ci (0) - 2 Ci (3wL) + 2 Ci (0) + 2 Ci (wL) - 2 Ci (0)

where (14) sin t

s i ( ~ ) = -;+ SO ?dl

is the sine integral,

Ci(x) = C + l n x + S o X cos :- l dt

is the cosine integral, and C is Euler's constant E 0.577. For h = 3000 A, L = 3h, r = 1 m, and A = lo-' cm2 (the source dealt with is not an ideal source for which A = xh2 1: x x lo-' cm2),

for perpendicular polarizations :

[4.21a] d P = A2 2

5 + 66, - + In F c r i t

+ sin SO(l - cos 26,) 6 + ins)] ( Fcr i t

The quantity in the square brackets when integrated over 0' or equivalently over So gives:

where C and the Si (x) are as defined previously. ratios are certainly non-negligible and far greater Again the ratio of the scattered to the incident power than that for the weak field case. is estimated : Quantum electrodynamics is one of the founda-

[4.22] tions of e- - e+ storage ring physics. The non-linear

RI = pLscatt/pLinc = effects of vacuum polarization can be tested through For X rays with h = 1 A and A = (10-10-10-12 electron synchrotrons and storage rings. It is inter-

cm2) the ratio is larger by a factor of 10' to lo4. esting to note that the phenomenon of the charged vacuum in strong electrostatic fields has been

5. Conclusions We have studied the scattering of electromagnetic

light waves in the visible to soft X-ray region in the ultra-strong limit of the Euler-Heisenberg Lagran- gian and examined closely the differences with the problem for the weak fields. The radiated fields are derived and shown to be phase shifted for both electric and magnetic fields. The consideration of the Euler-Heisenberg Lagrangian in its complete form without any approximation might lead to the generation of high harmonics and is an interesting problem worth further pursuit.

The emission power is estimated for both polariza- tion cases and compared with the incident flux. The

recently well understood as pointed out by Reinhardt and Greiner (15).

Dittrich (16) has considered the radiative correc- tions to the one loop effective potential and has indicated the formal analogy between the Lagrangian for ultra-strong fields and the radiative corrections to the coulomb potential. The physics of very strong electrostatic fields permits the study of phenomena not encountered in ordinary atomic physics.

Becker and Mitter3 (17) have considered vacuum

3The authors came across a more recent reference to the work of Mitter in the Preprint Registry of the International Centre of Theoretical Physics, Trieste, Italy, but cannot obtain the Preprint at the moment. The paper is entitled, "Intense Fields and Quantum Electrodynamics" (Univ. Graz, Austria).

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polarization in laser fields and pointed out its significance from the point of astrophysics or near surface of heavy nuclei. They pointed out that the increase in available laser intensities, though rapid, is not enough for the magnitude of the effects to be observed. Tsai and Ng (18) have considered the pair creation by photon-photon scattering in a strong magnetic field and also hinted a t its astrophysical importance, especially for pulsars. The method makes use of the explicit 'momentum' representation of the electron propagator in a homogenous magnetic field.

The incorporation of the electromagnetic non- linearities of matter into Maxwell's equations has therefore shown promise in the solution of boundary value problems. The major experimental tools for confirmation could possibly be the more than 20 machines for the high energy sources of synchrotron radiation in the world. Two more such facilities (19) are proposed to be scheduled around 1980 by the National Science Foundation and the Energy Research and Development Administration for production of an intense broad band of electro- magnetic radiation.

grateful to Mrs. Brigitte B. Stange for her patient typing of the work. This research was supported by a National Research Council of Canada Grant, A-8863.

1. H. E U L E R : ~ ~ W. HEISENBERG. Z. P h y s 98,714(1936). 2. V. S. WEISSKOPF. K. Dan. Vidensk. Selsk. Mat. Fys. Medd.

16, No. 6 (1936). 3. J . SCHWINGER. Phys. Rev. 82.664 (1951). 4. A. ACHIESER. Phys. Z. Sowjetunion, 11,263 (1937). 5. A. I. AKHIEZ and V. B. BERESTETSKY. Quantum elec-

trodynamics. Interscience Publishers, New York, NY. 196i. pp. 780-795.

6. J . MCKENNA and P. M. PL,ATZMAN. Phys. Rev. 129. 2354 (1963).

7. R. KARPLUS and M. N E U M A N N . Phys. Rev. 80,380(1950). 8. S. L. ADLER. Ann. Phys. 67,599(1971). 9. N. TRALLI . Clilssicill electromagnetic theory. McGraw Hill

Book Company, Inc., 1963. p. 296. 10. Y. PEREZ, J . P. JORBA, and F . M. RENARD. P h y s Rep. 30'2,

l(1977). 11. E . E. KOCH, C. K U N Z , and B. SONNTAG. Phys. Rep. 29C,

153. (1977). 12. L. I. SCHIFF. Quantum mechanics. McGraw Hill Com-

pany, Inc., New York, NY, 2nd ed. 1955. pp. 101-102. 13. T . ERBER, and W. Y. TSAI. Phys. Rev. D, 12, 1132 (1975). 14. I . S. GRADHSTEYN and I. M. RYZHIK. Table of inegrals,

series, and products. Academic Press, New York, NY. 1965. pp. 928.946.

15. J . REINHARDT and W. GREINER. Rep. Prog. Phys. 40, 219 (1977).

Acknowledgments 16. W. J . DITTRICH. ~ h y s . A , 10,833 (1977).

~h~ authors would like to thank D ~ , james ~ l ~ i ~ 17. W. BECKER and H. J . MITTER. Phys. A , 8.1638 (1975). 18. W. Y. TSAI and Y. J . NG. Phys. Rev. D, 16,286(1977). for reading of the manuscript and several 19, G, pREsENT, physics N~~~ i n 1977. Public Relations Divi-

helpful suggestions. They are also grateful to the sion. American Institute of Physics, New York, NY. Pub. referee for-several helpful comments. They are R-281. 1977.

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