It is shown, using techniques inspired by the method of orbits, that each non-zero mass, positive energy representation of the Poincaré group can be obtained via contraction from the discrete series of representations of .
Nous montrons, en utilisant des idées provenant de la méthode des orbites, que toute représentation massive et d’énergie positive du groupe de Poincaré peut être obtenue par contraction de la série discrète de .
@article{AIF_1993__43_2_551_0, author = {Cishahayo, C. and Bi\`evre, S. De}, title = {On the contraction of the discrete series of $SU(1,1)$}, journal = {Annales de l'Institut Fourier}, pages = {551--567}, publisher = {Imprimerie Louis-Jean}, address = {Gap}, volume = {43}, number = {2}, year = {1993}, doi = {10.5802/aif.1346}, zbl = {0793.22005}, mrnumber = {94e:22023}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1346/} }
TY - JOUR TI - On the contraction of the discrete series of $SU(1,1)$ JO - Annales de l'Institut Fourier PY - 1993 DA - 1993/// SP - 551 EP - 567 VL - 43 IS - 2 PB - Imprimerie Louis-Jean PP - Gap UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1346/ UR - https://zbmath.org/?q=an%3A0793.22005 UR - https://www.ams.org/mathscinet-getitem?mr=94e:22023 UR - https://doi.org/10.5802/aif.1346 DO - 10.5802/aif.1346 LA - en ID - AIF_1993__43_2_551_0 ER -
Cishahayo, C.; Bièvre, S. De. On the contraction of the discrete series of $SU(1,1)$. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 551-567. doi : 10.5802/aif.1346. https://aif.centre-mersenne.org/articles/10.5802/aif.1346/
[AAG] De Sitter to Poincaré contraction and relativistic coherent states, Ann. Inst. H. Poincaré, 52 (1990), 83-111. | Numdam | MR: 91e:22027 | Zbl: 0706.22018
, , and ,[DBE] Quantum mechanics and coherent states on the Anti-de Sitter spacetime and their Poincaré contraction, Ann. Inst. H. Poincaré, 57 (1992), 403-428. | Numdam | MR: 93m:81053 | Zbl: 0770.53048
and ,[D] Contractions of Lie groups and applications to analysis, in : Topics in modern harmonic analysis, Vol.I, 483-515 (Instituto Nazionale di Alta Matematica Francesco SEVERI, Roma 1983). | MR: 86e:22015 | Zbl: 0551.22006
,[DR1] Contractions of rotation groups and their representations, Math. Proc. Camb. Phil. Soc., 94 (1983), 509-517. | MR: 87a:22022 | Zbl: 0532.22014
and ,[DR2] On contractions of semisimple Lie groups, Trans. Amer. Math. Soc., 289 (1985), 185-202. | MR: 86g:22019 | Zbl: 0546.22017
and ,[E] Théories classique et quantique sur l'espace-temps Anti-de Sitter et leurs limites à courbure nulle, Thèse de Doctorat, Université Paris 7, décembre 1991.
,[EDB] Phase space quantum mechanics on the Anti-de Sitter spacetime and its Poincaré contraction, preprint 1992. | Zbl: 0810.53070
and ,[Fr] Elementary particles in a curved space, Rev. Mod. Phys., 37 (1965), 221-224. | MR: 31 #3193 | Zbl: 0125.45305
,[GH] Poincaré Contraction of SU (1, 1) Fock-Bargmann Structure, J. of Physics A, Math. Gen., 25 (1992), 1549-1573. | MR: 93k:81128 | Zbl: 0824.22018
and ,[Gi] Lie groups, Lie algebras and some of their Applications, Wiley, New York, 1974. | Zbl: 0279.22001
,[IW] On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U. S., 39 (1953), 510-524. | MR: 14,1061c | Zbl: 0050.02601
and ,[Ki] Eléments de la Théorie des Représentations, Editions Mir, Moscou, 1974.
,[Ko] Quantization and unitary representations, Lecture Notes in Math., 170, Springer Verlag, Berlin, 1970. | MR: 45 #3638 | Zbl: 0223.53028
,[LM] Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, 1987. | MR: 88c:58016 | Zbl: 0643.53002
and ,[Ma] On the analogy between semisimple Lie groups and certain related semi-direct product groups, in : Lie Groups and Their Representations, I. M. Gelfand (Ed.), 339-364 (Akadémiai Kiadó, Budapest 1975). | MR: 53 #13478 | Zbl: 0324.22006
,[MN] Contractions of representations of the Sitter groups, Commun. Math. Phys., 27 (1972), 167-180. | MR: 46 #8611 | Zbl: 0236.22021
and ,[Pe] Generalized Coherent States and their Applications, Springer Verlag, Berlin, 1986. | MR: 87m:22035 | Zbl: 0605.22013
,[R] Variétés Symplectiques et Quantification, Thèse Orsay, 1969.
,[Ra] Representations of a semi-direct product by quantization, Math. Proc. Camb. Phil. Soc., 78 (1975), 345-350. | MR: 52 #8341 | Zbl: 0313.22014
,[Sa] Contraction of Lie groups, J. Math. Phys., 2 (1961), 1-21. | MR: 23 #B259 | Zbl: 0098.25804
,[Wo] Geometric Quantization, Clarendon Press, Oxford, 1980. | MR: 84j:58058 | Zbl: 0458.58003
,Cited by Sources: