Geodesic
Projection-iteration method of solution of the equations of quantum field theory and its connection with the theory of renormalization. The equations of quantum field theory and improperly posed problems of mathematical physics
Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 2, pp. 167-183 
D. Ya. Petrina; A. L. Rebenko. Projection-iteration method of solution of the equations of quantum field theory and its connection with the theory of renormalization. The equations of quantum field theory and improperly posed problems of mathematical physics. Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 2, pp. 167-183. http://geodesic.mathdoc.fr/item/TMF_1980_42_2_a1/
@article{TMF_1980_42_2_a1,
     author = {D. Ya. Petrina and A. L. Rebenko},
     title = {Projection-iteration method of~solution of~the equations of~quantum field theory and its connection with the theory of~renormalization. {The} equations of~quantum field theory and improperly posed problems of~mathematical physics},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {167--183},
     year = {1980},
     volume = {42},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1980_42_2_a1/}
}
TY  - JOUR
AU  - D. Ya. Petrina
AU  - A. L. Rebenko
TI  - Projection-iteration method of solution of the equations of quantum field theory and its connection with the theory of renormalization. The equations of quantum field theory and improperly posed problems of mathematical physics
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1980
SP  - 167
EP  - 183
VL  - 42
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1980_42_2_a1/
LA  - ru
ID  - TMF_1980_42_2_a1
ER  - 
%0 Journal Article
%A D. Ya. Petrina
%A A. L. Rebenko
%T Projection-iteration method of solution of the equations of quantum field theory and its connection with the theory of renormalization. The equations of quantum field theory and improperly posed problems of mathematical physics
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1980
%P 167-183
%V 42
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1980_42_2_a1/
%G ru
%F TMF_1980_42_2_a1

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

The equations for coefficient functions of the scattering matrix are considered in the Euclidean space-time of the dimension $d=4$. It is shown that they can be defined correctly in the pair of functional spaces of the translationally-inyariant functions. The problem of finding the solutions for such equations is stated improperly, therefore the iteration method of solution results in the arising of ultraviolet divergencies. A projection-iteration method of solution is constructed which leads to the renormalized perturbation theory series. Renormalized perturbation theory series is proved to be a quasisolution of the input equations.

[1] R. Iost, Obschaya teoriya kvantovannykh polei, «Mir», 1967 | MR

[2] Konstruktivnaya teoriya polya, Sb. statei (per. s angl.), «Mir», 1977 | MR

[3] S. S. Ivanov, D. Ya. Petrina, A. L. Rebenko, EChAYa, 7:3 (1976), 647 | MR

[4] G. V. Efimov, Nelokalnye vzaimodeistviya kvantovannykh polei, «Nauka», 1977 | MR

[5] N. N. Bogolyubov, O. S. Parasyuk, Izv. AN SSSR, ser. matem., 20 (1956), 585 | MR | Zbl

[6] N. N. Bogoliubov, O. S. Parasiuk, Acta Math., 97 (1957), 227 | DOI | MR

[7] O. S. Parasyuk, Izv. AN SSSR, ser. matem., 20 (1956), 843 | MR

[8] K. Hepp, Commun. Math. Phys., 6 (1967), 161 | DOI

[9] S. A. Anikin, O. I. Zavyalov, M. K. Polivanov, TMF, 17 (1973), 189

[10] S. A. Anikin, M. K. Polivanov, TMF, 21 (1974), 175 | MR

[11] V. A. Scherbina, Dokt. diss., MI AN SSSR, 1975

[12] W. Zimmermann, Commun. Math. Phys., 6 (1967), 161 | DOI | MR

[13] O. I. Zavyalov, TMF, 28 (1976), 27 | MR

[14] V. A. Scherbina, TMF, 14 (1973), 342

[15] R. P. Feynman, Phys. Rev., 74 (1948), 939 | DOI | MR | Zbl

[16] F. T. Dyson, Phys. Rev., 1949, 1436

[17] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, «Nauka», 1974 | MR

[18] V. K. Ivanov, DAN SSSR, 145 (1962), 270 | Zbl

[19] S. S. Ivanov, D. Ya. Petrina, A. L. Rebenko, TMF, 19 (1974), 37

[20] D. Ya. Petrina, A. L. Rebenko, Preprint ITP-78-165E, Kiev, 1979

[21] Bogolyubov N. N., Shirkov D. V., Vvedenie v teoriyu kvantovannykh polei, «Nauka», 1977 | MR | Zbl

[22] D. Ya. Petrina, TMF, 4 (1970), 389 | MR

[23] S. S. Ivanov, D. Ya. Petrina, A. L. Rebenko, TMF, 23 (1975), 160

[24] A. L. Rebenko, Preprint ITP-74-41E, Kiev, 1974

[25] T. Hadamard, Le probleme Cauchy, Paris, 1932

[26] A. N. Tikhonov, DAN SSSR, 39 (1943), 195 | MR

[27] N. S. Kurpel, Proektsionno-iterativnye metody resheniya operatornykh uravnenii, «Naukova dumka», Kiev, 1968 | MR

[28] V. K. Ivanov, V. V. Vasin, V. P. Tanana, Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, «Nauka», 1978 | MR