Quantization in complex symmetric spaces
Izvestiya. Mathematics , Tome 9 (1975) no. 2, pp. 341-379.

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

By means of the method described in the author's paper “Quantization” (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116–1174), we construct the quantization of a classical mechanics whose phase space is a classical complex symmetric space. We establish the important qualitative differences between the quantization of such mechanics and the quantization of ordinary mechanics with plane phase spaces: for all the spaces considered, except for the sphere, Planck's constant is bounded above. Moreover, in the compact case Planck's constant takes on only discrete values. Bibliography: 17 items.
@article{IM2_1975_9_2_a6,
     author = {F. A. Berezin},
     title = {Quantization in complex symmetric spaces},
     journal = {Izvestiya. Mathematics },
     pages = {341--379},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {1975},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a6/}
}
TY  - JOUR
AU  - F. A. Berezin
TI  - Quantization in complex symmetric spaces
JO  - Izvestiya. Mathematics 
PY  - 1975
SP  - 341
EP  - 379
VL  - 9
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a6/
LA  - en
ID  - IM2_1975_9_2_a6
ER  - 
%0 Journal Article
%A F. A. Berezin
%T Quantization in complex symmetric spaces
%J Izvestiya. Mathematics 
%D 1975
%P 341-379
%V 9
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a6/
%G en
%F IM2_1975_9_2_a6
F. A. Berezin. Quantization in complex symmetric spaces. Izvestiya. Mathematics , Tome 9 (1975) no. 2, pp. 341-379. http://geodesic.mathdoc.fr/item/IM2_1975_9_2_a6/

[1] Berezin F. A., “Kvantovanie”, Izvestiya AN SSSR. Ser. matem., 38 (1974), 1116–1174 | MR

[2] Berezin F. A., General Conception of Quantisation, Preprint, ITP-74-20E, Kiev, 1974

[3] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[4] Khua Lo Ken, Garmonicheskii analiz funktsii mnogikh kompleksnykh peremennykh v klassicheskikh oblastyakh, IL, M., 1959

[5] Veil G., Klassicheskie gruppy, ikh invarianty i predstavleniya, IL, M., 1947

[6] Berezin F. A., Gelfand I. M., “Neskolko zamechanii k teorii sfericheskikh funktsii na simmetricheskikh rimanovykh prostranstvakh”, Tr. Mosk. matem. ob-va, 5, 1956, 311–352 | MR

[7] Kobayashi S., Nomizu K., “On Automorphysm of the Kählerian Structure”, Nagoya Math. Journ., 11 (1957), 115–124 | MR | Zbl

[8] Berezin F. A., Karpelevich F. I., “Zonalnye sfericheskie funktsii i operatory Laplasa na nekotorykh simmetricheskikh prostranstvakh”, Dokl. AN SSSR, 118 (1958), 9–13 | MR

[9] Berezin F. A., Metod vtorichnogo kvantovaniya, Nauka, M., 1965 | MR

[10] Bellman R., Vvedenie v teoriyu matrits, Nauka, M., 1969 | MR | Zbl

[11] Murnagan F., Teoriya predstavlenii grupp, IL, M., 1950

[12] Berezin F. A., “Neskolko zamechanii o predstavleniyakh sootnoshenii kommutatsii”, Uspekhi matem. nauk, XXIV:4 (1969), 68–88 | MR

[13] Monastyrskii M. I., Perelomov A. M., “Kogerentnye sostoyaniya i ogranichennye odnorodnye oblasti”, Dokl. AN SSSR, 207:6 (1972), 1303–1305 | MR | Zbl

[14] Lieb E. H., “The classical limit of Quantum Spin Systems”, Comm. Math. Phys., 31 (1973), 1–14 | DOI | MR

[15] Berezin F. A., “Kovariantnye i kontravariantnye simvoly operatorov”, Izv. AN SSSR. Ser. matem., 36 (1972), 1134–1167 | MR | Zbl

[16] Berezin F. A., “Kvantovanie v ogranichennykh kompleksnykh oblastyakh”, Dokl. AN SSSR, 211:6 (1973), 1263–1266 | MR | Zbl

[17] Gindikin S. G., “Invariantnye obobschennye funktsii v odnorodnykh oblastyakh”, Funktsionalnyi analiz i ego primeneniya, 9:1 (1975), 59–61 | MR