Geodesic
Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field
Teoretičeskaâ i matematičeskaâ fizika, Tome 26 (1976) no. 1, pp. 16-34 
N. B. Narozhnyi; A. I. Nikishov. Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field. Teoretičeskaâ i matematičeskaâ fizika, Tome 26 (1976) no. 1, pp. 16-34. http://geodesic.mathdoc.fr/item/TMF_1976_26_1_a1/
@article{TMF_1976_26_1_a1,
     author = {N. B. Narozhnyi and A. I. Nikishov},
     title = {Solutions of the {Klein-Gordon} and {Dirac} equations for a~particle in a~constant electric field and a~plane electromagnetic wave propagating along the field},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {16--34},
     year = {1976},
     volume = {26},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1976_26_1_a1/}
}
TY  - JOUR
AU  - N. B. Narozhnyi
AU  - A. I. Nikishov
TI  - Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1976
SP  - 16
EP  - 34
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1976_26_1_a1/
LA  - ru
ID  - TMF_1976_26_1_a1
ER  - 
%0 Journal Article
%A N. B. Narozhnyi
%A A. I. Nikishov
%T Solutions of the Klein-Gordon and Dirac equations for a particle in a constant electric field and a plane electromagnetic wave propagating along the field
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1976
%P 16-34
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1976_26_1_a1/
%G ru
%F TMF_1976_26_1_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

A complete set of solutions is found to the Klein–Gordon and Dirac equations for the case of a constant (in space and time) electric field along which a plane electromagnetic wave propagates. The solutions are labeled by the numbers $p_1$, $p_2$, $p_3$, which become the conserved three-momentum when the field of the wave is switched off. These solutions are related by an integral transformation to previously obtained solutions labeled by conserved components of the momentum: $p_1$, $p_2$, $p_-=p_0-p_3$. In contrast to these last solutions, the $\psi_{p_3}$-solutions arc everywhere finite and can be explicitly classified with respect to the sign of the “frequency” as $x_0\to\pm\infty$. It is also shown that the solutions $\psi_{p_-}$ too can be classified with respect to the sign of the “frequency”. This means that they can be used in the usual manner to describe matrix elements. Propagators are obtained in the Fock–Schwinger and Feynman representations. It is shown that not only the total but also the differential probabilities of pair creation by the field are independent of the field of the wave if they are expressed in terms of Lorentz and gauge-invariant quantities.

[1] P. J. Redmond, J. Math. Phys., 6 (1965), 1163 | DOI | MR | Zbl

[2] V. P. Oleinik, UFZh, 13 (1968), 1205; 14 (1969), 2076

[3] V. P. Oleinik, ZhETF, 61 (1971), 27 | MR

[4] V. G. Bagrov, Yu. I. Klimenko, V. N. Shapovalov, V. M. Shakhmatov, G. G. Ekle, Izv. VUZov, fizika, 1973, no. 7, 95 | MR

[5] A. A. Borgard, D. Ya. Karpenko, UFZh, 19 (1974), 227

[6] A. I. Nikishov, ZhETF, 57 (1969), 1210

[7] J. Schwinger, Phys. Rev., 82 (1951), 664 | DOI | MR | Zbl

[8] V. A. Fock, Sow. Phys., 12 (1937), 404 | Zbl

[9] I. A. Batalin, E. S. Fradkin, TMF, 5 (1970), 190 ; Международный семинар. Функциональные методы в квантовой теории поля и статистике, препринт ФИАН No 140, 1971, стр. 15 | MR

[10] H. R. Lewis, Jr., W. B. Riesenfeld, J. Math. Phys., 10 (1969), 1458 | DOI | MR | Zbl

[11] I. A. Malkin, V. I. Manko, ZhETF, 59 (1970), 1746; V. V. Dodonov, I. A. Malkin, V. I. Man'ko, Physica, 59 (1972), 241 | DOI

[12] A. I. Nikishov, Problemy teoreticheskoi fiziki, «Nauka», 1972, str. 299 | MR

[13] A. I. Nikishov, Avtoreferat dissertatsii, Moskva, 1971

[14] G. Beitmen, A. Erdeii, Tablitsy integralnykh preobrazovanii, t. 1, «Nauka», 1969 | MR

[15] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, t. 2, «Nauka», 1966 | MR

[16] G. Beitmen, A. Erdeii, Tablitsy integralnykh preobrazovanii, tom 2, «Nauka», 1970

[17] R. P. Feynman, Phys. Rev., 76 (1949), 749 | DOI | MR | Zbl

[18] N. B. Narozhnyi, A. I. Nikishov, YaF, 11 (1970), 1072

[19] A. I. Nikishov, TMF, 20 (1974), 48 | MR

[20] A. I. Akhiezer, V. B. Berestetskii, Kvantovaya elektrodinamika, Fizmatgiz, 1959 | MR

[21] A. I. Nikishov, Nucl. Phys., B21 (1970), 346 | DOI