Geodesic
Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 90 (1992) no. 3, pp. 388-406 
I. Yu. Krivsky; V. M. Simulik. Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 90 (1992) no. 3, pp. 388-406. http://geodesic.mathdoc.fr/item/TMF_1992_90_3_a4/
@article{TMF_1992_90_3_a4,
     author = {I. Yu. Krivsky and V. M. Simulik},
     title = {Dirac equation and spin~1 representations, a~connection with symmetries of the {Maxwell} equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {388--406},
     year = {1992},
     volume = {90},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1992_90_3_a4/}
}
TY  - JOUR
AU  - I. Yu. Krivsky
AU  - V. M. Simulik
TI  - Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1992
SP  - 388
EP  - 406
VL  - 90
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1992_90_3_a4/
LA  - ru
ID  - TMF_1992_90_3_a4
ER  - 
%0 Journal Article
%A I. Yu. Krivsky
%A V. M. Simulik
%T Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1992
%P 388-406
%V 90
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1992_90_3_a4/
%G ru
%F TMF_1992_90_3_a4

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

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli–Gürsey symmetry operators (without the $\gamma_5$ operator) of the Dirac equation with $m=0$ form the same representation $D(1/2,0)\oplus D(0,1/2)$ of the $O(1,3)$ algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations $D(1/2,1/2)$ and $D(1,0)\oplus D(0,0)$ of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group $O(1,3)$ are investigated. Analogous $O(1,2)$ and $P(1,2)$ invariance algebras are established for the Dirac equation with $m>0$.

[1] Sallhojer H., Z. Naturforsch. A, 41 (1986), 463–470 | MR

[2] Ljolje K., Fortschr. Phys., 36:1 (1988), 9–32 | DOI | MR

[3] Simulik V. M., TMF, 87:1 (1991), 76–85 | MR | Zbl

[4] Simulik V. M., “O novom metode nakhozhdeniya sokhranyayuschikhsya velichin dlya elektromagnitnogo polya”, Kvantovaya teoriya polya i fizika vysokikh energii, MGU, M., 1989, 185–190

[5] Simulik V. M., Vopr. atomn. nauki i tekhn. Yaderno-fizich. issled. (teoriya i eksper.), 1(9), 1990, 79–82 | MR

[6] Gursey F., Nuovo Cim., 7:2 (1958), 411–415 | DOI | Zbl

[7] Mignani R., Recami E., Badlo M., Lett. Nuovo Cim., 11:12 (1974), 568–572 | DOI

[8] Lomont J. S., Moses H. E., J. Math. Phys., 3:3 (1962), 405–408 | DOI | MR | Zbl

[9] Borgardt A. A., ZhETF, 34:2 (1958), 334–341 | MR

[10] Moses H. E., Nuovo Cim. Suppl., 7:1 (1958), 1–18 | DOI | MR | Zbl

[11] Lomont J. S., Phys. Rev., 111:6 (1958), 1710–1716 | DOI | MR | Zbl

[12] Fuschich V. I., Shtelen V. M., Spichak S. V., DAN USSR. A, 1990, no. 3, 36–40 | Zbl

[13] Mack G., Salam A., Ann. Phys., 53:1 (1969), 174–202 | DOI | MR

[14] Fuschich V. I., Nikitin A. G., Simmetriya uravnenii kvantovoi mekhaniki, Nauka, M., 1990 | MR

[15] Krivskii I. Yu., DAN USSR. A, 1988, no. 7, 9–13 | MR | Zbl

[16] Krivskii I. Yu., Simulik V. M., O lagranzhevom podkhode dlya elektromagnitnogo polya v terminakh napryazhennostei i zakonakh sokhraneniya, Preprint KIYaI-85-13, IYaI AN USSR, Kiev, 1985 | MR

[17] Krivskii I. Yu., Simulik V. M., UFZh, 30:10 (1985), 1457–1459 | MR

[18] Krivskii I. Yu., Simulik V. M., TMF, 80:2 (1989), 274–287 | MR

[19] Krivskii I. Yu., Simulik V. M., TMF, 80:3 (1989), 340–352 | MR

[20] Krivskii I. Yu., “Teorema Neter o zakonakh sokhraneniya dlya nelievskikh preobrazovanii invariantnosti”, Teoretiko-gruppovye issledovaniya uravnenii matematicheskoi fiziki, IM AN USSR, K., 1985, 134–139 | MR