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Compton scattering in a strong laser radiation field from two observers' point of view
Volume 106A, number 1,2
PHYSICS LETTERS
19 November 1984
COMPTON SCATTERING IN A STRONG LASER RADIATION FIELD FROM TWO OBSERVERS' POINT OF VIEW
Igor V. BELOUSSOV Institute o f Applied Physics, Academy o f Sciences o f the Moldavian SSR, Grosul Str. 5, Kishinev 277028, U$SR
Received 29 May 1984 Revised manuscript received 17 September 1984
Two experiments on Compton scattering in an intensive laser radiation field, carried out in two different inertial reference systems are considered. It is shown that the results of both experiments transformed in a certain fLxedreference system differ from each other, as a consequence of the relativity of time and of the way of describing the laser field accepted in this paper.
One can classify all the quantum processes of particle scattering and transformation in the field of intensive laser radiation in two large groups: (1) processes induced by the laser field which do not occur when the latter is absent; (2) processes which can also occur when there is no laser field. From the point of view of the experimental testing of the theory the second group processes are seemingly of most interest, since their probabilities under certain conditions have a resonant character determined by the quasidiscrete structure of the energy spectrum of the system "electron + laser field photons" [1-3]. One of the simplest processes of the second group of this classification is Compton scattering of incoherent photons on electrons in the presence of an external field of intensive laser radiation. This process has been investigated for the first time in ref. [2] on the ground of the Wolkov non-stationary solutions [4~5] of the Dirac equation in an external laser field, classically described by the aid of the four-potential A (r, t) = a cos w ( n x ) , an=n 2 =0,
- a 2,n 0 > 0 .
(1)
The use of Wolkov solutions is valid when the field (1) switches on and off through the Lorenz-invariant variable nx [6]. One can easily see that such a way of describing the external laser field contradicts the caus-
ality principle. Indeed, in a real experimental situation the source, generating the external field 0aser), acts during a certain t'mite interval of time T 1 - T 0, limited by the moments of its switching on and off T o and T1, respectively. If one assumes that the switching on and off of the field is carried out through the variable n x = n o ( t - n . r/no), then when t < T O the latter turns out to differ from zero in a certain spatial region, although the source having generated it has not yet been switched on. The approach, based on the idea of external field switching on and offin time [ 7 - 9 ] and therefore, in compliance with the causality principle, seems to be more adequate to reality. However, nowadays there are various opinions as far as the way of describing the external laser field is concerned [6-10]. For this reason it is worth revealing some effects that would be quite within the reach of experimental investigatin, in which the difference in physical consequences of two above-mentioned approaches would come out most sharply. In particular, it has been shown [8] that adiabatically slow switching on and off in time of the external laser field (1) gives rise to essentially different frequency characteristics of the Compton scattering resonant process than in the case when the switching on and off is carried out through the variable nx. In the present communication we consider resonant
Volume 106A, number 1,2
PHYSICS LETTERS
Compton scattering in the external field (1) from the point of view of two observers in two different inertial reference systems K and K'. Each observer is assumed to carry out the scattering experiment in his own reference system. Both observers consider the external field to switch on and off in time adiabatically slow. In doing so the observer from the K-system makes use of the time t and the observer from the K'.system counts the time upon his own dock, employing the time t'. The results of two observations are transformed to a certain reference system (e.g. to the K-system) and compared after the experiments have been carried out. Thus, below is meant not just one experiment on scattering, organized in one reference system and analyzed by two different observers, but two experiments, in which a process of the same type is investigated. One of them is carried out in the K-system and independently the second in the K'-system. For the sake of definiteness we choose the inertial reference systems K and K' so that in the K-system the four-vectors n and a have the form n = (1,0, 0, 1), a = (O,a 1 ,a2, 0) and the system K' moves with respect to K with constant velocity o in the z-direction. Let the electron be in rest in the experiment performed in the K-system when t ~ _ o o (Pi = 0, Pi is the free electron momentum before scattering). Then the scattered photon frequency in this reference system is determined from the formula [8] ~ f = ( s ~ + will + 2(sto/m,)sin 2 (½tgi)]} X [1 + 2(s6o/rn,)sin2(~tgf)
+ 2 (wi/m,)sin2(~O)] - l ,
(2)
which is obtained from the conservation law:
p,f+kf=P,i+ki+s6on
(s = 0,-+1,-+2 .... ),
(3)
where k a = (6oa, ka) and P , a = (c,a, Pa) are the fourmomentum of the photon and the four-quasimomentum of the electron in the t~-state respectively (a = i, f; wa = (k2)l/2;c,a = (p2 + m2)1/2, m 2 = m 2 _ ~e2a2). There and below the index i ( 0 means, that the corresponding quantity refers to the state of the particles before (after) scattering. According to ref. [7] the space components of the electron four-quasimomentum in the external field (1), which switches on and off adiabatically slow in time t, do not depend on the intensity of the field and coincide with the respective
19 November 1984
components of the free electron four-momentum pa
= (ca,pa) (ca = (p2a + mZ)l/Z;pf is the free electron momentum when t ~ +~). We consider now the experiment on Compton resonant scattering, carried out in the reference system K'. The quasienergy-quasimomentum conservation law in this system has the form: t
t
~
t
p,f+kf=P,i+ki+swn'
( s - 0,-+1,-+2 .... ).
(4)
It can be obtained from (3) by replacing all the quantities by the corresponding quantities with a prime. In the experiment we consider the external field switching on and off adiabatically slow in time t'. For this reason the space components of the four-quasimomenturn P*a also do not depend on the intensity of the field and coincide with the respective components of e 0 the free electron four-momentum Pa (Pi and pf are the free electron four-momenta when t' ~ _oo and t' + oo respectively). However, by transforming the four-vector p',a to the system K, because of the variation of the zero-component of the free electron fourmomentum under the influence of the external field t t (ca ~ e,a), the dependence we mentioned appears [9]. Thus, if the free electron four-momenta Pa and 0 Pa are linked by the Lorentz transformation e
Pa =Lpa
(t~=i, f),
(S)
the four-quasimomenta P , a and p',a of the electron in the field (1) corresponding to them turn out to be not linked by the Lorentz transformations L. This conclusion contradicts in no way the relativity theory principles. On the contrary, it is a consequence of this theory and of the problem as formulated here, in which the switching on and off of the external field is carried out upon an noninvariant parameter - the time. Applying the inverse Lorentz transformation to (4) one arrives at the conservation law
L - 1 p ,' f + ~ f = L - l p ' . +* 1L
' - 1 ki+s~oL-ln'
(6)
which determines the four-momentum/~f = (~f, ~f) of the photon, scattered by the electron in the system K from the point of view of the observer from K'. Taking into account the relation L-lk~ =ki, L - 1 n' = n and (5), one finds from (6) (Pi = 0): ~ f = [s~(1 - o)(l +po) + wiYi(s)]
X [Yf(s) + 2(wi/m)p(l - o2)sin2(½d)] - 1 ,
(7)
Volume 106A, number 1,2
PHYSICS LETTERS
19 November 1984
Let us compare the formulae (2) and (7) at nonrelativistic values of the parameters o f the problem, when
in his own reference system, transforming afterwards the results o f his observations to a certain fixed reference system and compares them with those o f the other, he will come to see that the results do not coincide. The relative character of the scattering processes in the external classical field has been first discovered by Oleinik [9]. It is not difficult to get in the same way the expressions for the resonant frequencies o f the falling and scattered photons in the system K from the point o f view o f the observer from K':
- e 2 a 2 / 4 m 2 , p2/m2, ~ / m , co~/m, V ~ 1,
¢~i = rco(1 -- v)(1 + pv) Y i - l ( - r ) ,
where Ya(s) = 1 - po 2 - (1 - p)v cos Oa + 2(sco/m)p(1
-
0 2) sin2(~Oa),
13 = [1 -- (e2a2/2m2)(1 - 02)] -1/2, a=i,f;
~t = i, f.
s =0,+I,+2
.....
(8)
(9)
In this case the largest contribution in the section o f the process we treat is yielded by the terms, corresponding to the value s = 0 [2,8]. One has: (¢~f - cof)ls= 0 ~- ucoi(e2a2/4m2)(cos 0 i - cos Of). (10) The result we have obtained has the following meaning. Let us assume that the observer in the reference system K carries out an experiment concerning Compton scattering in an external laser field on a resting electron when t -+ _oo. The detector o f photons, which is in the same reference system, will register an electromagnetic radiation with frequency ~f. Let now the observer from K' carry out in his reference system a similar experiment on an electron, which has the energy m(1 - 02) -1/2 and momentum (0, 0, - m o o - u2)-1/2) when t ' --, _oo; this electron is in rest in the reference system K. Then the detector from K will register photons with frequency t~f. The difference between the quantities t~f and ~of is due to the fact that the scattering process (i.e. the process in which the interaction between the electron and the external field switches on and then offin time) studied by the observer in reference system K is different from the point o f view o f the observer from K', because of the relativity o f time. For this reason, if each of the observers watches the scattering process
&f = rco(1 - v)(1 + o v ) y f l ( r ) , r = 1, 2, 3 .....
(11)
These quantities differ from the corresponding expressions for the resonant frequencies ~ i and cof (given in ref. [8]) received by the observer from K who watches the Compton scattering process in his own reference system. When the conditions (9) hold, the following approximative equality hold: ~aa - 6o~ = vr6o(e2a2/2m2)sin2(~Oa)
(a = i, f). (12)
The author wishes to thank Dr. Yu.G. Shondin for helpful discussions concerning this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
V.P. Oleinik, Zh. Eksp. Teor. FIX. 52 (1967) 1049. V.P. Oleinik, Zh. Eksp. Teor. Fix. 53 (1967) 1997. M.V. Fedorov, Zh. Eksp. Teor. Fix. 68 (1975) 1209. D.M. Wolkov, Zh. Eksp. Teor. FIX. 7 (1937) 1286. V.B. Berestetskii, E.M. Lifshits and L.P. Pitaevskii, Relyativistiskaya kvantovaya teoriya, part 1 (Nauka, Moscow, 1968). H. Mitter, Acta Phys. Austr. Suppl. 14 (1975) 397. V.P. Oleinik and V.A. Sinyak, Opt. Commun. 14 (1975) 179. I.V. Belousov, Opt. Commun. 20 (1977) 205. V.P. Oleinik, in: Kvantovaya electronika, eds. M.P. L~tsa et al. 15 (Naukova ctumka, Kaev, 1978) p. 88. V.I. Rims, Tr. FIX. Inst. Akad. Nauk SSSR 111 (1979) 5.
9
PHYSICS LETTERS
19 November 1984
COMPTON SCATTERING IN A STRONG LASER RADIATION FIELD FROM TWO OBSERVERS' POINT OF VIEW
Igor V. BELOUSSOV Institute o f Applied Physics, Academy o f Sciences o f the Moldavian SSR, Grosul Str. 5, Kishinev 277028, U$SR
Received 29 May 1984 Revised manuscript received 17 September 1984
Two experiments on Compton scattering in an intensive laser radiation field, carried out in two different inertial reference systems are considered. It is shown that the results of both experiments transformed in a certain fLxedreference system differ from each other, as a consequence of the relativity of time and of the way of describing the laser field accepted in this paper.
One can classify all the quantum processes of particle scattering and transformation in the field of intensive laser radiation in two large groups: (1) processes induced by the laser field which do not occur when the latter is absent; (2) processes which can also occur when there is no laser field. From the point of view of the experimental testing of the theory the second group processes are seemingly of most interest, since their probabilities under certain conditions have a resonant character determined by the quasidiscrete structure of the energy spectrum of the system "electron + laser field photons" [1-3]. One of the simplest processes of the second group of this classification is Compton scattering of incoherent photons on electrons in the presence of an external field of intensive laser radiation. This process has been investigated for the first time in ref. [2] on the ground of the Wolkov non-stationary solutions [4~5] of the Dirac equation in an external laser field, classically described by the aid of the four-potential A (r, t) = a cos w ( n x ) , an=n 2 =0,
- a 2,n 0 > 0 .
(1)
The use of Wolkov solutions is valid when the field (1) switches on and off through the Lorenz-invariant variable nx [6]. One can easily see that such a way of describing the external laser field contradicts the caus-
ality principle. Indeed, in a real experimental situation the source, generating the external field 0aser), acts during a certain t'mite interval of time T 1 - T 0, limited by the moments of its switching on and off T o and T1, respectively. If one assumes that the switching on and off of the field is carried out through the variable n x = n o ( t - n . r/no), then when t < T O the latter turns out to differ from zero in a certain spatial region, although the source having generated it has not yet been switched on. The approach, based on the idea of external field switching on and offin time [ 7 - 9 ] and therefore, in compliance with the causality principle, seems to be more adequate to reality. However, nowadays there are various opinions as far as the way of describing the external laser field is concerned [6-10]. For this reason it is worth revealing some effects that would be quite within the reach of experimental investigatin, in which the difference in physical consequences of two above-mentioned approaches would come out most sharply. In particular, it has been shown [8] that adiabatically slow switching on and off in time of the external laser field (1) gives rise to essentially different frequency characteristics of the Compton scattering resonant process than in the case when the switching on and off is carried out through the variable nx. In the present communication we consider resonant
Volume 106A, number 1,2
PHYSICS LETTERS
Compton scattering in the external field (1) from the point of view of two observers in two different inertial reference systems K and K'. Each observer is assumed to carry out the scattering experiment in his own reference system. Both observers consider the external field to switch on and off in time adiabatically slow. In doing so the observer from the K-system makes use of the time t and the observer from the K'.system counts the time upon his own dock, employing the time t'. The results of two observations are transformed to a certain reference system (e.g. to the K-system) and compared after the experiments have been carried out. Thus, below is meant not just one experiment on scattering, organized in one reference system and analyzed by two different observers, but two experiments, in which a process of the same type is investigated. One of them is carried out in the K-system and independently the second in the K'-system. For the sake of definiteness we choose the inertial reference systems K and K' so that in the K-system the four-vectors n and a have the form n = (1,0, 0, 1), a = (O,a 1 ,a2, 0) and the system K' moves with respect to K with constant velocity o in the z-direction. Let the electron be in rest in the experiment performed in the K-system when t ~ _ o o (Pi = 0, Pi is the free electron momentum before scattering). Then the scattered photon frequency in this reference system is determined from the formula [8] ~ f = ( s ~ + will + 2(sto/m,)sin 2 (½tgi)]} X [1 + 2(s6o/rn,)sin2(~tgf)
+ 2 (wi/m,)sin2(~O)] - l ,
(2)
which is obtained from the conservation law:
p,f+kf=P,i+ki+s6on
(s = 0,-+1,-+2 .... ),
(3)
where k a = (6oa, ka) and P , a = (c,a, Pa) are the fourmomentum of the photon and the four-quasimomentum of the electron in the t~-state respectively (a = i, f; wa = (k2)l/2;c,a = (p2 + m2)1/2, m 2 = m 2 _ ~e2a2). There and below the index i ( 0 means, that the corresponding quantity refers to the state of the particles before (after) scattering. According to ref. [7] the space components of the electron four-quasimomentum in the external field (1), which switches on and off adiabatically slow in time t, do not depend on the intensity of the field and coincide with the respective
19 November 1984
components of the free electron four-momentum pa
= (ca,pa) (ca = (p2a + mZ)l/Z;pf is the free electron momentum when t ~ +~). We consider now the experiment on Compton resonant scattering, carried out in the reference system K'. The quasienergy-quasimomentum conservation law in this system has the form: t
t
~
t
p,f+kf=P,i+ki+swn'
( s - 0,-+1,-+2 .... ).
(4)
It can be obtained from (3) by replacing all the quantities by the corresponding quantities with a prime. In the experiment we consider the external field switching on and off adiabatically slow in time t'. For this reason the space components of the four-quasimomenturn P*a also do not depend on the intensity of the field and coincide with the respective components of e 0 the free electron four-momentum Pa (Pi and pf are the free electron four-momenta when t' ~ _oo and t' + oo respectively). However, by transforming the four-vector p',a to the system K, because of the variation of the zero-component of the free electron fourmomentum under the influence of the external field t t (ca ~ e,a), the dependence we mentioned appears [9]. Thus, if the free electron four-momenta Pa and 0 Pa are linked by the Lorentz transformation e
Pa =Lpa
(t~=i, f),
(S)
the four-quasimomenta P , a and p',a of the electron in the field (1) corresponding to them turn out to be not linked by the Lorentz transformations L. This conclusion contradicts in no way the relativity theory principles. On the contrary, it is a consequence of this theory and of the problem as formulated here, in which the switching on and off of the external field is carried out upon an noninvariant parameter - the time. Applying the inverse Lorentz transformation to (4) one arrives at the conservation law
L - 1 p ,' f + ~ f = L - l p ' . +* 1L
' - 1 ki+s~oL-ln'
(6)
which determines the four-momentum/~f = (~f, ~f) of the photon, scattered by the electron in the system K from the point of view of the observer from K'. Taking into account the relation L-lk~ =ki, L - 1 n' = n and (5), one finds from (6) (Pi = 0): ~ f = [s~(1 - o)(l +po) + wiYi(s)]
X [Yf(s) + 2(wi/m)p(l - o2)sin2(½d)] - 1 ,
(7)
Volume 106A, number 1,2
PHYSICS LETTERS
19 November 1984
Let us compare the formulae (2) and (7) at nonrelativistic values of the parameters o f the problem, when
in his own reference system, transforming afterwards the results o f his observations to a certain fixed reference system and compares them with those o f the other, he will come to see that the results do not coincide. The relative character of the scattering processes in the external classical field has been first discovered by Oleinik [9]. It is not difficult to get in the same way the expressions for the resonant frequencies o f the falling and scattered photons in the system K from the point o f view o f the observer from K':
- e 2 a 2 / 4 m 2 , p2/m2, ~ / m , co~/m, V ~ 1,
¢~i = rco(1 -- v)(1 + pv) Y i - l ( - r ) ,
where Ya(s) = 1 - po 2 - (1 - p)v cos Oa + 2(sco/m)p(1
-
0 2) sin2(~Oa),
13 = [1 -- (e2a2/2m2)(1 - 02)] -1/2, a=i,f;
~t = i, f.
s =0,+I,+2
.....
(8)
(9)
In this case the largest contribution in the section o f the process we treat is yielded by the terms, corresponding to the value s = 0 [2,8]. One has: (¢~f - cof)ls= 0 ~- ucoi(e2a2/4m2)(cos 0 i - cos Of). (10) The result we have obtained has the following meaning. Let us assume that the observer in the reference system K carries out an experiment concerning Compton scattering in an external laser field on a resting electron when t -+ _oo. The detector o f photons, which is in the same reference system, will register an electromagnetic radiation with frequency ~f. Let now the observer from K' carry out in his reference system a similar experiment on an electron, which has the energy m(1 - 02) -1/2 and momentum (0, 0, - m o o - u2)-1/2) when t ' --, _oo; this electron is in rest in the reference system K. Then the detector from K will register photons with frequency t~f. The difference between the quantities t~f and ~of is due to the fact that the scattering process (i.e. the process in which the interaction between the electron and the external field switches on and then offin time) studied by the observer in reference system K is different from the point o f view o f the observer from K', because of the relativity o f time. For this reason, if each of the observers watches the scattering process
&f = rco(1 - v)(1 + o v ) y f l ( r ) , r = 1, 2, 3 .....
(11)
These quantities differ from the corresponding expressions for the resonant frequencies ~ i and cof (given in ref. [8]) received by the observer from K who watches the Compton scattering process in his own reference system. When the conditions (9) hold, the following approximative equality hold: ~aa - 6o~ = vr6o(e2a2/2m2)sin2(~Oa)
(a = i, f). (12)
The author wishes to thank Dr. Yu.G. Shondin for helpful discussions concerning this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
V.P. Oleinik, Zh. Eksp. Teor. FIX. 52 (1967) 1049. V.P. Oleinik, Zh. Eksp. Teor. Fix. 53 (1967) 1997. M.V. Fedorov, Zh. Eksp. Teor. Fix. 68 (1975) 1209. D.M. Wolkov, Zh. Eksp. Teor. FIX. 7 (1937) 1286. V.B. Berestetskii, E.M. Lifshits and L.P. Pitaevskii, Relyativistiskaya kvantovaya teoriya, part 1 (Nauka, Moscow, 1968). H. Mitter, Acta Phys. Austr. Suppl. 14 (1975) 397. V.P. Oleinik and V.A. Sinyak, Opt. Commun. 14 (1975) 179. I.V. Belousov, Opt. Commun. 20 (1977) 205. V.P. Oleinik, in: Kvantovaya electronika, eds. M.P. L~tsa et al. 15 (Naukova ctumka, Kaev, 1978) p. 88. V.I. Rims, Tr. FIX. Inst. Akad. Nauk SSSR 111 (1979) 5.
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