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Ali, S. Twareque; Antoine, J.-P.

Coherent states of the \(1+1\) dimensional Poincaré group: Square integrability and a relativistic Weyl transform. (English) Zbl 0702.22024

Ann. Inst. Henri Poincaré, Phys. Théor. 51, No. 1, 23-44 (1989).
The definition and usual properties of square integrable representations of a locally compact group are generalized and extended to transitive homogeneous spaces of the group, as well as some associated structures such as Wigner map, coherent state decomposition. This is applied to the construction of relativistic coherent states in the case of \(1+1\) space- time dimensions. In this case, the Wigner representation, while being non square integrable in the usual sense, is square integrable when the quotient by time translations is performed. Finally, a relativistic Weyl transform is derived in this case.
Reviewer: G.Loupias

Cited in 1 Review
Cited in 5 Documents

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
22D12 Other representations of locally compact groups
81R30 Coherent states

Keywords:

square integrable representations; locally compact group; homogeneous spaces; coherent states; Wigner representation
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References:

[1] J.R. Klauder and B.S. Skagerstam , Coherent States - Applications in Physics and Mathematical Physics , World Scientific , Singapore , 1985 . MR 826247 | Zbl 1050.81558 · Zbl 1050.81558
[2] A. Perelomov , Generalized Coherent States and Their Applications , Springer , Berlin , 1986 . MR 899736 | Zbl 0605.22013 · Zbl 0605.22013
[3] J. Dixmier , Les C*-algèbres et leurs représentations , 2 e éd., Gauthier-Villars , Paris , 1969 . MR 246136 | Zbl 0174.18601 · Zbl 0174.18601
[4] S.A. Gaal , Linear Analysis and Representation Theory , Springer , Berlin , 1973 . MR 447465 | Zbl 0275.43008 · Zbl 0275.43008
[5] M. Duflo and C.C. Moore , J. Funct. Anal. , Vol. 21 , 1976 , p. 209 . Zbl 0317.43013 · Zbl 0317.43013 · doi:10.1016/0022-1236(76)90079-3
[6] A. Carey , Bull. Austral. Math. Soc. , 15 , 1976 , p. 1 ; Rep. Math. Phys. , Vol. 14 , 1978 , p. 247 . MR 430146 | Zbl 0418.22004 · Zbl 0418.22004 · doi:10.1016/0034-4877(78)90047-2
[7] A. Grossmann , J. Morlet and T. Paul , J. Math. Phys. , Vol. 26 , 1985 , p. 2473 ; Ann. Inst. H. Poincaré , Vol. 45 , 1986 , p. 293 . Zbl 0571.22021 · Zbl 0571.22021 · doi:10.1063/1.526761
[8] M. Flensted-Jensen , Ann. of Math. , Vol. 111 , 1980 , p. 253 . MR 569073 | Zbl 0462.22006 · Zbl 0462.22006 · doi:10.2307/1971201
[9] H. Schlichtkrull , Invent. Math. , Vol. 68 , 1982 , p. 497 . MR 669427 | Zbl 0501.22019 · Zbl 0501.22019 · doi:10.1007/BF01389414
[10] S.T. Ali , Rivista del Nuovo Cim. , Vol. 8 , # 11, 1985 , p. 1 . MR 821194
[11] E. Prugovecki , Stochastic Quantum Mechanics and Quantum Spacetime , D. Reidel , Dordrecht , 1984 . MR 741681 | Zbl 0536.60100 · Zbl 0536.60100
[12] S.T. Ali and H.D. Doebner , Quantization, topology and ordering , in Lect. Notes in Physics , Vol. 278 , Springer , Berlin , 1987 . MR 915802
[13] A. Grossmann , G. Loupias and E.M. Stein , Ann. Inst. Fourier , Grenoble , Vol. 18 , 1968 , p. 343 . Numdam | MR 267425 | Zbl 0176.45102 · Zbl 0176.45102 · doi:10.5802/aif.305
[14] I. Daubechies and A. Grossmann , J. Math. Phys. , Vol. 21 , 1980 , p. 2080 . MR 579204
[15] I. Daubechies , A. Grossmann and J. Reignier , J. Math. Phys. , Vol. 24 , 1983 , p. 239 . MR 692298
[16] J.C.T. Pool , J. Math. Phys. , Vol. 7 , 1966 , p. 66 . Zbl 0139.45903 · Zbl 0139.45903 · doi:10.1063/1.1704817
[17] S.T. Ali and E. Prugovecki , Acta Appl. Math. , Vol. 6 , 1986 , p. 1 . MR 837718 | Zbl 0603.22010 · Zbl 0603.22010 · doi:10.1007/BF00046932
[18] S.T. Ali and E. Prugovecki , Acta Appl. Math. , Vol. 6 , 1986 , p. 19 . Zbl 0603.22011 · Zbl 0603.22011 · doi:10.1007/BF00046933
[19] S.T. Ali and E. Prugovecki , Acta Appl. Math. , Vol. 6 , 1986 , p. 47 . Zbl 0603.22012 · Zbl 0603.22012 · doi:10.1007/BF00046934
[20] R. Kunze , Pacific J. Math. , Vol. 53 , 1974 , p. 465 . Article | MR 374330 | Zbl 0321.22012 · Zbl 0321.22012 · doi:10.2140/pjm.1974.53.465
[21] A. Borel , Représentations des Groupes Localement Compacts ; Lect. Notes in Mathematics , Vol. 276 , Springer , Berlin , 1972 . Zbl 0242.22007 · Zbl 0242.22007
[22] H. Moscovici , Commun. Math. Phys. , Vol. 54 , 1977 , p. 63 . Article | MR 442152 | Zbl 0407.22011 · Zbl 0407.22011 · doi:10.1007/BF01609836
[23] A.A. Kirillov , Elements of the Theory of Representations , Springer , Berlin , 1976 . MR 412321 | Zbl 0342.22001 · Zbl 0342.22001
[24] H. Moscovici and A. Verona , Ann. Inst. H. Poincaré , Vol. 29 , 1978 , p. 139 . Numdam | MR 513686 | Zbl 0392.22008 · Zbl 0392.22008
[25] H. Scutaru , Lett. Math. Phys. , 2 , 1977 , p. 101 . MR 507248 | Zbl 0395.22009 · Zbl 0395.22009 · doi:10.1007/BF00398574
[26] A.M. Perelomov , Commun. Math. Phys. , Vol. 26 , 1972 , p. 222 (reprinted in Ref. 1); ibid. , Vol. 44 , 1975 , p. 197 . MR 363209 | Zbl 0243.22016 · Zbl 0243.22016 · doi:10.1007/BF01645091
[27] I.M. Gel’fand and N. Ya. Vilenkin , Generalized Functions , Vol. 4 , Academic Press , New York , 1964 .
[28] J.-P. Antoine , J. Math. Phys. , Vol. 10 , 1969 , pp. 53 , 2276 . Zbl 0172.56602 · Zbl 0172.56602 · doi:10.1063/1.1664761
[29] E. Nelson , Annals of Math. , Vol. 70 , 1959 , p. 570 . MR 107176
[30] P. Bona , Czech. J. Phys. , Vol. B 33 , 1983 , p. 837 .
[31] S.T. Ali and S. De Bièvre , in XYIth International Colloquium on Group-Theoretical Methods in Physics (Proceedings, Varna , 1987 ); Lecture Notes in Physics , Vol. 313 , p. 201 ; H.-D. DOEBNER, J.-D. HENNIG, T. D. PALEV Eds., Springer , Berlin , 1988 . Zbl 0675.22010 · Zbl 0675.22010
[32] S. De Bièvre , J. Math. Phys. , vol. 30 , 1989 , p. 1401 . Toronto , 1988 (preprint). MR 1002242 | Zbl 0686.46049 · Zbl 0686.46049 · doi:10.1063/1.528321
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