D E T E C T I O N OF T - M E T R I C R A D I O S I G N A L S

H. L E M K E

Moscow, Obrutcheva 4-1-89, U.S.S.R.

(Received 2 January, 1980) Abstract. We examine the properties of electromagnetic radiation embedded in the metric (+ - + +) in interaction with ordinary charged particles, where the x-direction is the direction of a superluminal source. The absorption of the corresponding photons and their conversion into ordinary light is considered in lowest-order perturbation theory. Particular emphasis is on the radiation component which spreads with velocities of values around infinity. It is shown that ordinary receivers of electromagnetic radiation do not respond to such light.

0. Introduction

Whether particles with spacelike four-momentum (tachyons) exist in nature, or, if not, if they could be produced artificially, is a question that is still unanswered; there have been experiments (Davies et al., 1969; Baltay et al., 1970; Danburg et al., 1971; Ramana Murthy, 1971, 1973; Danburg and Kalbfleisch, 1972; Hazen et al. 1975; Ljubicic, 1975; Prescott, 1976; Perepelitsa, 1977; Bartlett et al., 1978), but they were not successful. It might well be that present techniques are unsuitable for the direct detection of such particles and that indirect searches might stand a better' chance. After all, detecting the electron itself was much harder than observing the consequences of its existence. Particles with spacelike four-momentum can be expected to emit radiation when they interact with other particles. A theory of the emission of electromagnetic radiation, given by Lemke (1975a, b, 1976, 1977) claims that the properties of this radiation can be obtained from the ordinary Maxwellian radiation properties by applying what we call a superluminal Lorentz transformation. As the axis of proper time of a spacelike particle is spacelike, the transformation that interchanges time and one direction in space will belong to the superluminal Lorentz transformations. In this way it can be explained why the emitted radiation will be embedded in the metric ( + - + +), the T-metric, the preferred spatial direction x being the motion direction of the superluminal source (Lemke, 1975a, b, 1976, 1977). The T-metric light cone is a 90~ of the ordinary light cone in the x - t plane (Lemke, 1975a, b, 1976, 1977). It will be opened in the positive direction of motion of the spacel]ke particle. The spatial direction that points away from the source of the spacelike particle can be defined as positive. The photons emitted by a tachyon will thus exhibit properties that are totally different from the properties of ordinary light: if this were detected, that would be direct evidence of the existence of superluminal objects. Only T-metric Astrophysics and Space Science 74 (1981) 253-264. 0004-640X/81/0742-025351.80 Copyright 9 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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H. L E M K E

photons emitted in the x-direction move at the speed of one; in all of the other directions the speed is greater than one. They always carry positive momentum kx but can have positive as well as negative energy, i.e., they can move backward in time. Thus a causality problem arises, which is examined on the basis of Lemke (1979) under Section 1. In Section 2 we derive the S-matrix for the interaction of T-metric photons with ordinary charges. The requirement of gauge invariance will play an important r01e. In Section 3 the kinematics of the interaction of T-metric photons and ordinary particles will be presented. Here the component of the T-metric light that has speeds of values around infinity (that is photon energy ~ photon momentum) is of particular interest. The ordinary charge either destructs the T-metric photon, or converts it into ordinary photons, or else cannot interact with it at all. The destruction rate and cross-section for conversion will be given under Sections 4 and 5. In Section 6 the possible influence of long wavelength T-metric light on a material body is considered phenomenologically.

1. Causality The T-metric light cone consists of rays in four-space that are directed forward or backward in time away from their superluminal source. The rays, travelling into the past, make it possible to formulate a causality objection of the following kind. Let such a ray be modulated and carry information to a receiver, the receiver answers with a ray that travels into the past more slowly than the ray received, so that the laboratory gets the answer before it emitted the information. It is, of course, only possible to get the answer if the laboratory actually does thereafter emit the information; since this caused the receiver to answer. The laboratory is, however, still free to decide whether to emit the information or not; they can even destroy the antenna after receiving the 'answer'. Here a temporal paradox seems to arise (in principle, a closed causality loop has been constructed). However, if they do it, the answer could not be an answer to an information emitted by this particular laboratory. That such an interpretation can be in accord with causality was explained by Lemke (1979). In practice, the receiver would not answer, so that its answer (which, admittedly, can begin with a repetition of the information) is obtained much earlier than the information is emitted, since this answer could, at that early time, be regarded as useless and forgotten. If, however, the answer is obtained only a short time earlier, the inducing information can indeed arise at that period of time and, therefore, indeed be emitted. If the laboratory does not emit the information (which can already be contained in the answering signal from the receiver), it would also be satisfied, since the answered information is an information which has just been compiled for possible emission. Then, the satisfaction of the laboratory would be accidential, the receiver emitted the answer together with the information independent of the particular laboratory,

D E T E C T I O N OF T H E T - M E T R I C RADIO S I G N A L S

255

and the reason for the receiver's message must be sought elsewhere. If there is no such reason, the message cannot in any way exist and so the laboratory will later emit the information. After analysing old records of answers from the receiver, the laboratory might find that they recorded this information together with the answer some days earlier. 2. The S-matrix The field of free T-metric photons is described by a four-potential AK(t,r), satisfying the free T-metric wave equation. To establish the S-matrix S}bs for the absorption of T-metric photons by conventional electrons, we can make use of the ordinary S-matrix of spinor electrodynamics (Bjorken and Drell, 1966). For the T-metric is identical with the Lorentz-metric in the two-dimensional t - x space, and, therefore, the properties of the T-metric photons in this space will be identical with the properties of ordinary light. Consequently S}bs takes the form S}bs = ~ -- iqef f d4yff(y)[Ao(y)yo - A~(y)yx ++-"9 "]air,(y).

(1)

We see that the function under the integral is a product of the conventional transition current @ykq~i, which is a Lorentz vector, with A k, which transforms under the invariance group of the T-metric (Lemke, 1975a, b, 1976, 1977). The product is thus not a scalar interaction density, and we must specify a reference frame relative to which it has the simple form (1). According to the theory by Lemke (1975a, b, 1976, 1977), one can specify a source of the bradyon, and the rest system of this source must be identified with that frame. Let us now establish the transverse components in Equation (1). They must be linear in yy and Yz because the potential couples to the transition current of an ordinary spin-l/2 particle. In other words, the bracket must be of the linear form Aly~ezk, where etk is a not yet completely determined matrix. We determine etk by demanding gauge invariance. The field-strength tensor of the T-metric radiation F tk = 3htAk - ~ h k A t , where d ht = (3t, -3x, +3y, +6z), exhibits invariance under the gauge transformation Ak(y) = A'k(y) + dhkA(y) with 6 hk the differential operator just defined. The S-matrix Equation (1) must exhibit the same invariance. This is the case only if 6h%~ = 3~ holds, since only then, the gauge term :is of the form (61A(y))f(y), where jl(y) is the transition current. The latter satisfies 61]~(y)= 0, so that the gauge term can be re-written as a four-divergence, which does not contribute. The matrix etk is thus completely determined: et~ = diag(1,-1, 1, 1), the T-metric tensor. It follows that [Ao(y)yo - Ax(y)yx +-'" "] = A~,(y)y k ,

(2)

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H. L E M K E

where the symbol h again means the signs of the transverse components are to be changed. This symbol describes the whole difference between normal and T-metric photons in interaction with ordinary matter, in particular, the different properties of polarisation. Of course, an electron cannot emit T-metric photons; it can only emit and re-absorb ordinary photons. The complete S-matrix for the absorption of Tmetric radiation must thus be written in the form

S~bs = 6~ + iqef f d4y~i(y)[A~(y) - Ak(y)]yk~,(y),

(3)

where A h pertains to the electromagnetic T-metric field and A to the electromagnetic Lorentz-metric field. Ordinary matter converts T-metric photons into normal radiation, T-metric electromagnetic waves falling on a metallic mirror would turn into ordinary light. We will however see that, kinematically, electrons of a given momentum cannot always interact with a T-metric photon. This is an important difference from ordinary photon-electron interactions, in which kinematics allows for interaction at all values of the initial photon and electron momenta. T-metric radiation of certain wave-vector values could thus pass through matter without being subjected to any absorption or conversion.

3. Kinematics

S-matrix Equation (3), in which the V-field is a solution of the Dirac equation with the corresponding interaction term, describes the conversion of a T-metric photon into ordinary photons by an electron, including all the radiative corrections to the electron current. The conservation of the total four-momentum reads

k+p =p'+k',

(4)

where k is the four-momentum of the T-metric photon, k' the total fourmomentum of the ordinary photons emitted and p, p' are the initial and the final electron momenta (leaving aside the processes of the creation of electronpositron pairs). As in Equation (3), this law is valid in the rest system of the source that produced momentum p. Let us attach the summit of the T-metric light cone to momentum p in four-momentum space (the positive-p0 axis is directed vertically upwards). For every value of p it intersects the positive sheet of the hyperboloid p20-p2 = m 2. The common surface corresponds to elastic transitions of the electron in absorbing a T-metric photon, i.e., to transitions without the emission of photons (k'= 0 in Equation (4)). Such transitions always lead to an increase of the

DETECTION OF THE T-METRIC RADIO SIGNALS

257

p~-component and would be a first-order effect in e2/hc. The final electron momentum cannot lie above the c o m m o n surface because k6 is positive-definite, but it can lie below, in which case k' cannot be zero. A light ray embedded in T-metric covers the distance

ug = (2 cos 2 o~-

(5)

1) -1/2

per unit time, where ~ is the angle that the ray makes with the direction of the superluminal source, 0 ~< a ~< 7r/4. Speed Ug varies between 1 and ~. If the radiation passes a distance of one light year in one hour or faster, ug must be greater than 10 4, which corresponds to the very small angular region ~10-8~ > ~r/4- ~/> 0. We will call radiation in the angular region a ~< ~-/4 instantaneous. Instantaneous radiation is characterized by very small frequencies t~ol= kug I ~ k, where k is the wave vector magnitude. The propagation four-vector k i can thus be chosen to lie nearly in the x-y plane of four-momentum space. Let us attach four-vector k" to an electron four-momentum p~ to see that k ~ cannot lie inside the electron's momentum hyperboloid if ]Pt is too small, i.e., if

21pl < k.

(6)

In this event the instantaneous photon can neither be destroyed nor yet converted, for the total final momexltum p ' + k' always lies inside the electron's momentum hyperboloid. In other words, electrons which satisfy Equation (6) cannot interact with the instantaneous radiation. For an electron gas with a thermal energy of 10-I eV per particle, we have p - 300 eV. Then, Equation (6) holds if the wavelengths of the radiation are smaller than the far ultraviolet, where quantum effects dominate. Let us establish the kinematic p-region in which electrons can destroy instantaneous photons without emitting ordinary light. For the instantaneous photons we have k~.-k~r = k02- 0. So we will state that kx = ky and choose the y-z plane so that kz = 0 (kx is always positive-definite). In this approximation, wave vector k includes an angle of exactly 7r/4 with the x-axis. Let us decompose p into the component pit= (Px, Py) in the plane of the in-coming radiation and the component Pz perpendicular to this plane. Moreover, the angle between PIP and the x-axis will be named y (let 3' be positive-definite). Fourmomentum conservation then yields this condition for destruction in the form

pllsin(Y+ 4 ) = - 8 9

and

pz=p'z.

(7)

We see that Pll is bounded by k/2 from below (Equation (6)) and can be arbitrarily large. There are two values of y for which destruction can happen. We have y~ =3'2 = 57r/4 for Pll = k/2, that is, Pll is antiparallel to k, and "/1 =

258

H. L E M K E

7~-/4- e and y2 = 3n-/4 + e in the limit pll~oo, that is, p becomes perpendicular to k. In absorbing the photon, the electron's direction of motion in the x - y plane changes by an angle of 2 ( 7 v / 4 - ~ 1 ) or 2(yz-3ir/4). At the lower limit of Pfl, at which Pit must be antiparallel to k, the electron's direction in the x - y plane makes a transition into the opposite direction and becomes parallel to k. The higher PlI, the smaller the change of the electron's direction. One can use a rectilinear electron current of a certain value of p (produced by, e.g., a cathode-ray tube) to detect instantaneous radiation of a certain value of k. To satisfy Equation (7), the value of 3' must be appropriately chosen. This angle changes uniformly because the Earth rotates. The unequal satisfaction of Equation (7) is a necessary condition for the occurrence of the conversion process, in which the electron emits photons. This less probable process is kinematically possible only if four-vector p + k lies inside the electron's momentum hyperboloid. For the light-cone opening downwards with its summit at the end of four-vector, p + k must be able to intersect this momentum hyperboloid. The line of intersection bounds the region of the possible p'-momenta. As the whole light-cone is cut off, t h e f i n a l photons can be emitted in every spatial direction. Let us again examine the case of instantaneous radiation. Because then k0 = 0, the total energy of the photons emitted will be bounded by

k'o < P o - m = p Z / ( p o + m ) ,

(8)

which is a rather small energy for non-relativistic electrons. If only one photon is emitted, the photon energy has not only an upper bound but also a non-zero lower bound. We shall now examine this more probable case. To find the exact upper bound, we denote the distance of vector p + k from the electron's momentum hyperboloid in the x - y plane by K and the maximal photon energy than can be emitted by k ' . The fact that four-vector p ' = (P0 - k L, P - K - k~, 0) must have the length m yields

k;

=

pK -

89

po--p+K

For great K - p we find k itn - ~lp 2/ Po which is about the same as Equation (8). For small K < p , km = Kp/(pO--p +K), which equates with k ' = •P/Po in the non-relativistic limit p ~ P o . This value of k" is significantly smaller than the momentum deviation K. In the relativistic limit we have to distinguish the two cases K ~ Po - P ~ mZ/2P and K ~>P 0 - P. In the former case, k " is proportional to K but greater than K by a factor of 2p2]m 2 (although k ' ~ p ) . In the latter case, k" is independent of K and just about equal to p. For K < p and, in the relativistic case, the angular dependence of k' shows a strong maximum in the direction of the total initial three-momentum p + k.

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DETECTION OF THE T-METRIC RADIO SIGNALS

4. Destruction Rate

Absorbing a T-metric photon, the electron either remains on-shell or becomes on-shell after emitting ordinary photons. The former process is a destruction of radiation, the latter a conversion into ordinary photons. The two processes have the diagrams shown in Figure 1 in lowest-order perturbation theory. Let us examine diagram (a) and calculate the rate of destruction. The T-metric electromagnetic wave can be described by the four-potential

r~ 4~r A t ( y ) = e ~/-2kx-ff A t [exp (iky) + exp ( - iky)] ,

(9)

where F At is a unit volume in the t-y-z hyperspace. A t is normalized to one photon per unit volume. We substitute Equation (9) into Equations (1)-(2) and calculate S}bs with free plane-wave electron spinors. This gives

S~bs= q ~

m

V

~/

47r ~(p')ehyku(p)(21r) 4 64(p ' - p -- k). 2k~-At

The transition rate per unit volume and particle is given by V f d3p'(21r) -3 IS~OSI2/VAt. The unit volume F At will give us the density of the photon flux ]ph(X)=ug/V; because of Ax/At=ugCOSO~ one finds 1~FAt= jph(x) cos ~. The rate of destruction dN/dt follows from multiplying with the incident intensity of photons of given momentum. Let us denote the spectral intensity by W(kx, cos (b, q~) dk0 dZktr/(2~r)3, using spherical co-ordinates in (k0, kt0-sfface: k0 = kx cos 4~, ktr = kx sin ~b, and r the azimuth in the plane perpendicular to x. We find

2m2f

dN dt = O~-D~ff d3xjph(X)Pel(X)8[(p + k) 2 - m 21 x X[u'ehykulZW(kx, COS~b,q~)

kZ~dkxdcos~b,

(10)

where p' = p + k expresses the four-momentum conservation (note that Equation (9) is a component of the wave group, see Lemke, 1975a, b, 1976, 1977).

+

+

+...

Fig. ]. (a) Absorption of a T-metric photon by an electron and (b) conversion of a T-metric photon into ordinary radiation in lowest-order perturbation theory.

260

H. L E M K E

Let us summate with respect to the final spin states and average with respect to the initial spin states. This can be done by calculating a well-known trace (Bjorken and Drell, 1966), and it gives

1 ~ Ju'e~yku 12= (eh)i(eh*) k 2 ~ [PiPk, + P ~ i -, - gik(PP , -- m2)].

(11)

ss'

As Equations (1) and (2) are gauge invariant, the result in Equation (1 t) is also gauge invariant. In accord with this, a density matrix 0 ~k must be substituted for (eh)i(eh*) k if the photon state is partially polarized. We know from ordinary electrodynamics that p~k equates with half the metric tensor for unpolarized radiation. That is, the average with respect to the polarization states in the case of unpolarized radiation is achieved by substituting ~diag(1,-1, 1, 1) for (eh)i(eh*) t~, which gives us

1~1

1

2 vo, 2 ~ss' [u'ehykur = ~ P

,

ph = ~

1

(12)

(pph + p k h)

Let us calculate this expression for instantaneous radiation, kx = ky and k0 = 0 (that is, & - 7r]2 in Equation (10)). It is easy to see that condition (7) is equivalent to

py + kx = - Px.

(7')

From this, we find that

_lyt

1

(13)

We can now specify Equation (10) to the case of instantaneous radiation. To this end we integrate the g-function with respect to cos q~ and find that

dN c~ f d3xjph(x)p~T(x)(l+P~]w(kx, O,r dt - 2~./2 j \

dkx

(14)

'

where we made use of the fact that cos ~b - 0 and kx/k - 1/~/2. Non-relativistic electrons correspond to p 0 - m and P z ~ m , so that an increase in the zcomponent of the electron momentum only slightly increases the absorption rate. (For given Pz, the energy P0 depends on k and 3' (see Equation (7)), a dependence which can, however, be neglected in the non-relativistic case.)

DETECTION OF THE T-METRIC RADIO SIGNALS

261

5. Cross-Section o[ Conversion

The contribution of diagrams Figure l(b) to the S-matrix Equation (3) becomes 4~r

m Sfi = ty V /-p op" r ~

•

i e'

vZ, ~ gx/.~[l (2"B')4 ~4(p _~_ k - p' - k') x

15'+fc'+rn ~h+~ ( p , + k,)2 _ m2

p^- k ~t + m ~,] -(p _ k,)2 _ m~ e j u.

(15)

Let us call the second dynamical factor M~. The probability for transitions into the phase space unit of the final state per unit time and volume then becomes m2

(47r)2

p,

d w = a2 Pop6--V 2k'2kx A t F (27r)4 64(p + k -

~ 2 d3k' d3p ' - k') 1v1~

~

. (16)

The cross-section do- for conversion can be defined by dividing dw by the density of target particles 1/V, the density of incoming particles A t [ A t F A x and the relative velocity lug- v] in the electrons' source rest-system

d6=dwVAtF

Ug COS O~

[Ug-V l

(t7)

using A x / A t = u~ cos a. The phase space integration in Equation (16) can be done to give dw = a 2

m2 k' po Vkx A t e IM~ 12da~, p0 + k 0 - Iv + kl cos t~'

(18)

where fl is the angle between k' and the total incoming three-momentum p + k; the dependence of k' on/3 is of the form k' = {[(p + k) 2 - mZ]/(po + k0 - IP + k] cos fl),

(19)

which shows that the last numerator in Equation (18) disappears if the condition of destruction (e.g., Equations (7) or (7')) is satisfied. ~f the condition for destruction is nearly satisfied and p + k lies within the hyperbolic momentum sheet of p, then only infrared photons can be emitted and the /~'-terms in the amplitude M~ can be ignored. Ms decays into the product of the amplitude for process Figure l(a) and the amplitude for the emission of infrared photons by a baryon M~ ~ ff'~hu [ p e ' [pk'

p'e'] p'k'] "

(20)

262

H. LEMKE

Let us form IM~I2 and average over the initial polarization states (see section 4) and sum over the final polarization states to get dw -

o~2

2

( p , p h ) dflk, ~ - r r2t K

2po Vkx A t F

x

2pK

x (P0-IP[ cos v)(K0-IKI cos fi) m2 ] (K0-[K[ cos fi)2 ,

m 2

~P0-IP[ cos u) 2 (21)

where short-hand p + k = K has been used and ~, is the angle between k' and p. In the relativistic case the angular distribution of the radiation emitted peaks in the direction of p and p + k - p'. In Equation (21) one can carry out the angular integration (approximating the length of K in the bracket by m). This gives us a2 dW-2poVkxAtF(PP

, h 16 )~(ta~20

1),

(22)

where 4 sinh 2 0 = - (p - K ) 2 / m 2 ~ 2k~r/m 2. We see that it is possible to approximate K by p' so long as ktr r 0. Because of the factor (K 2 - m2) -1 in Equations (21) and (22), the conversion cross-section tends towards infinity if the momentum of the final photon tends to zero. This divergence is nothing other than the well-known infrared divergence. When the second-order vertex correction to diagram Figure l(a) is taken into account, it vanishes.

6. Collective Phenomena

The ordinary Compton condition shows that k ' - k for k ~ m in the frame relative to which the initial electron is at rest. That is, the recoil of the electron can then be ignored and the scattered wave will have the same frequency as the incident wave. At wavelengths which are large compared to the electron spacing in the medium, the radiation amplitudes of all the electrons will add up coherently. One can see from Equation (19) that this is also true of T-metric waves with ktr = 0. Such waves possess all the properties of ordinary Maxwellian waves (Lemke, 1975a, b, 1976, 1977). Hence, the case of instantaneous radiation (neglectable frequency, k 0 - 0 ) will be of particular interest; Equation (19) gives k' = pxm-+py kx P0

(it is P0 - m)

(23)

DETECTION OF THE T-METRIC RADIO SIGNALS

263

for kx = ky in the limit kx ~ IPx +Pyl (we see that k' no longer depends on/3), k' is essentially smaller than k and the recoil of the electrons cannot be ignored; k' depends on the electron momenta. Since we can ignore the energy of the incoming radiation, the energy of the induced radiation is provided by the medium. According to Equation (6), p must be large enough to be able to interact with instantaneous radiation of wavelength 1/k. If p is much larger than k, then the change in p (and the more the momentum of the ordinary radiation emitted) is relatively small. This would be the limit in which a classical field description should apply (if the intensity of the instantaneous radiation is not too small). Electromagnetic T-metric waves have unusual polarization properties. The electric and magnetic field vectors satisfy the relations

H

=

[llg, E],

E = [H, u~],

(24)

which only turn into the Maxwellian relations at utr = 0. They show that the magnetic field vector H is perpendicular to E and lies in the plane perpendicular to u~, as normally. But E lies in the plane perpendicular to Ug, h which plane differs from that plane if utr does not disappear. In particular, this is the case for instantaneous waves, where [kH h] = [k, E] = 0.

(25)

This means: (1) E is parallel to k (or Ug), the direction of propagation of the wave, (2) H is perpendicular to k and lies in the plane spanned by k and the x-axis. By means of a superluminal transformation one can show that the ratio IEI/[HI can take on every value (though IE] and IHt are of course finite). A model for the emitter of a monochromatic T-metric wave would be a superluminal current: which moves in one direction of space (the x-direction) and periodically changes its speed. The instantaneous wave cannot oscillate in time, because its frequency is negligibly small or zero. The T-metric light-cone shows that the duration of the wave depends on the spread in a - 7r/4 and the distance from the emitter. In the planes perpendicular to the plane spanned by k and the x-axis, H and E are constant. The field-strengths vary with wavelength 3, = 27r/k in a direction perpendicular to these planes (that is, in direction H). In a system of parallel metallic rods oriented so that they are parallel with k and lie in the plane spanned by k and H, vector E induces a voltage E 9 1 (1 = length of a rod) that varies from rod to rod with wavelength A. The conductance of the metal is however smaller than usual, because only about half the free electrons can be influenced. In connection with Equation (23) we saw that the induced radiation emitted by the medium possesses a spectrum even if the instantaneous radiation is monochromatic, for the electron momenta have a spectrum. The electrons

264

H. LEMKE

o b v i o u s l y emit the p h o t o n s incoherently. T h e angular distribution of the induced intensity I will be of dipole f o r m , I ~ E ~ sin 2/3, w h e r e / 3 is the angle b e t w e e n the direction of o b s e r v a t i o n and the direction of the i n c o m i n g w a v e (which direction is parallel to the inducing electric field v e c t o r E). C o r r e s p o n d i n g l y , the induced radiation will be polarized in the plane s p a n n e d by k and k'.

Acknowledgement I would like to thank m y friend D. Miller for reading the manuscript.

References Baltay, C., Feinberg, G., Yen, N. and Linsker, R.: 1970, Phys. Rev. D1,759. Bartlett, D. F., Soo, D. and White, M. G.: 1978, Phys. Rev. D18, 2253. Bjorken, J. D. and Drell, S. D.: 1966, Relativistische Quantenmechanik, B-I, Mannheim. Danburg, J. S. and Kalbfleisch, G. R.: 1972, Phys. Rev. DS, 1575. Danburg, J. S., Kalbfleisch, G. R., Borenstein, S. R., Strand, R. C., Vanderburg, V., Chapman, J. W. and Lys, J.: 1971, Phys. Rev. D4, 53. Davies, M. B., Kreisler, M. N. and Alv~iger, T.: 1969, Phys. Rev. 183, 1132. Hazen, W. E., Green, B. R., Hodson~ A. L. and Kass, J .RA 1975, Nucl. Phys. B96, 401. Lemke, H.: 1975a, Nuovo Cimento, 27A, 141. Lemke H.: 1975b, Lett. Nuovo Cimento 12, 342. Lemke, H.: 1976, Nuovo Cimento 32A, 169. Lemke, H.: 1977, Phys. Lett. 60A, 271. Lemke, H.: 1979, Phys. Lett. 72A, 409. Ljubicic, A., Pavlovic, Z., Pisk, K. and Logan, B. A.: 1975, Phys. Rev. DII, 696. Perepelitsa, V. P.: 1977, Phys. Lett. 67B, 471. Prescott, J. R.: 1976, J. Phys. Geophys. Nucl. Phys. 2, 261. Ramana Murthy, P. V.: 1971, Lett. Nuovo Cimento 1,908. Ramana Murthy, P. V.: 1973, Phys. Rev. D8, 3990.