Zusammenfassung:
Diese Arbeit ist ein Beitrag zur Geometrischen Quantisierung.
Sie ist in drei Teile gegliedert: Der erste ist der klassischen Theorie gewidmet und zeigt, im Rahmen des allgemeinen Gelfand-Gindikin-Programmes, dass die Metaplektische Darstellung als Erweiterung einer Darstellung einer Unterhalbgruppe der Komplexifizierung der reellen symplektischen Gruppe gesehen werden kann.
Im zweiten Teil wird auf Jordan-theoretischem Niveau eine Darstellung der reellen symplektischen Gruppe angegeben, und im dritten Teil, wird aufbauend auf den zweiten, im neuen Zustandsraum ein projektiv-flaches Hilbertraum-Bündel angegeben. Es wird eine konkrete Realisierung des Shilov-Randes gewisser komplexer Strukturen angegeben, welche, als Anwendung, zu einer konkreten Angabe der Fasern über Randpunkten des metaplektisch-korrigierten Bündels, welches auf diese Randpuznkte erweitert werden kann, führt.
Bibliographie / References
- J. Duoandikoetxea, Fourier analysis, American Mathematical Society, 2001. [ELRM] A. Echeverría Enríquez, M. Muñoz Lencada, N. Román Roy, and C. Victoria Monge, Mathematical foundations of geometric quantiza- tion, pre-print, arXiv: math-ph/9904008.
- K.-H. Neeb, Holomorphic extension of unitary representations, Seminar Sophus Lie 3 (1993), 2734. [Ner96]
- E. Witten, Quantum background independence in string theory, pre- print, arXiv:hep-th/9306122. [Woo81]
- N.M.J. Woodhouse, Geometric quantization and the bogoliubov trans- formation, Proc. Royal Soc. London A 378 (1981), 119-139.
- L. Charles, Semi-classical properties of geometric quantization with me- taplectic correction, Comm. Math. Phys. 270 (2) (2007), 445480. [Con90]
- J. Arazy and H. Upmeier, Boundary measures for symmetric domians and integral formulas for the discrete wallach points, Integral Equations and Operator Theory 47 (2003), 375434. [Bar]
- B. Hall and W. Lewkeeratiyutkul, Holomorphic sobolev spaces and the generalized segal-bargmann transform.
- J.B. Conway, A course in functional analysis, Springer, 1990. [Duo01]
- J. Faraut and A. Korányi, Analysis on symmetric cones, Oxford Science Publications, Clarendon Press, Oxford, 1994.
- J. Hilgert, A note on howe's oscillator semigroup, Annales de l'institut Fourier 39, n 3 (1989), 663688. [HK]
- O. Loos, Bounded symmetric domains and jordan pairs, Mathematical lectures, University of Irvine, California, 1977. LITERATURVERZEICHNIS 113 [McC78]
- W. Kirwin, Coherent states in geometric quantization, Journal of Geo- metry and Physics 57 (2007), 531548.
- [BK80] , Complex extension of the representation of the symplectic group assotiated with the canonical commutation relations, Reports on Mathematical Physics 17 (1980), 205215. [BP77]
- [KMS75] P. Kramer, M.Moshinsky, and T. H. Seligman, Complex extensions of the canonical transformations and quantum mechanics, in group theory and its applications iii, E. Loebe Ed. Acad. Press, New York, 1975. [KN63]
- P. Kramer, Composite particles and symplectic (semi-)groups, Group Theoretical Methods in Physics: Sixth International Colloquium Tü- bingen 1977. Editor: P. Kramer, A. Rieckers, Lecture Notes in Physics 79 (1977), 180200. [KU77]
- S. Helgason, Dierential geometry and symmetric spaces, Academic Press, 1962. [Hil89]
- S. Kobayashi, Dierential geometry of complex vector bundles, Prince- ton Univ. Press, 1987. [Kra77]
- S. Kobayashi and K. Nomizu, Foundatios of dierential geometry, vol i, Interscience Publishers, 1963.
- G.M. Tuynman, Generalized bergman kernels and geometric quantiza- tion, Journal Mathematical Physics 28 (3) (1987), 573583. [Upm83]
- J. niatycki, Geometric quantization and quantum mechanics, Springer Verlag, 1980. [Seg81]
- APW91] S. Axelrod, S. Della Pietra, and E. Witten, Geometric quantization of chern-simons gauge theory, J.Di. Geometry 33 (1991), 787902. [Ara90]
- W. Kirwin and S. Wu, Geometric quantization, parallel transport and the fourier transform, Comm. Math. Phys. 266 (2006), 577594. [Lev69]
- V. Bargmann, Group representations on hilbert spaces of analytic func- tions, Analytic Methods in Mathematical Physics, Gilbert and Newton, Eds. Gordon and Breach, New York. [Bar61] , On a hilbert space of analytic functions and an assotiated in- tegral transform, part 1, Comm. Pure and applied Mathematics 14 (1961), 187214. [BK79]
- G. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton University Press, 1989. [Hal94]
- [Hal00] , Harmonic analysis with respect to the heat kernel measure, Bulletin (new series) of the American Mathematical Society 38 (1) (2000), 4378. [Hal02] , Geometric quantization and the generalized segal-bargmann transform for lie groups of compact type, Comm. Math. Phys. 226 (2002), 233268. [Hel62]
- Y. Neretin, Integral operators with gaussian kernels and symmetries of canonical commutation relations, Contemporary Mathematical Phy- sics, Translations, American Mathematical Society 175 (1996), 97135. [Rit]
- K. McCrimmon, Jordan algebras and their applications, Bulletin of the American Mathematical Society 84 (4) (1978), 612627. [Nee93]
- H. Upmeier, Toeplitz operators on bounded symmetric domains, Tran- sactions of the American Mathematical Society 280 (1) (1983), 221 237. [Upm85a] , Jordan algebras in analysis, operator theory, and quantum me- chanics, Conference Board, American Mathematical Society, 1985. [Upm85b] , Symmetric banach manifolds and jordan c £ -algebras, North- Holland, 1985. [Upm85c] , Toeplitz operators on symmetric siegel domains, Math. Ann. 271 (1985), 401414. [Upm86] , Jordan algebras and harmonic analysis on symmetric spaces, American Journal of Mathematics 108 (1986), 355386. [Upm96] , Toeplitz operators and index theory in several complex varia- bles, Birkhäuser, 1996.
- N. Jacobson, Lie and jordan triple systems, American Journal of Ma- thematics 71 (1) (1949), 149170. [Kir07]
- V.I. Arnold, Mathematical methods of classical mechanics, Springer, 1989. [AU03]
- J. Arazy, Operator dierentiable functions, Integral Equations and Ope- rator Theory 13 (1990), 461487. [Arn89]
- [Upm08a] , Projectively at hilbert bundles on jordan varieties, Integral Equations and Operator Theory pre-print (2008).
- A. Koranyi and J. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Annals of Mathematics 81 (2) (1965), 265288. [KW06]
- M. Brunet and P. Kramer, Semigroups of lenght increasing transforma- tions, Reports on Mathematical Physics 15 (1979), 287304.
- D. Levine, Systems of singular integral operators on spheres, Transac- tions of the American Mathematical Society 144 (1969), 493522. [Loo77]
- C. Villegas-Blas, The bargmann transformand canonical transformati- ons, Journal of Mathematical Physics 43 (5) (2002), 22492283. [Wit]
- M. Brunet, The metaplectic semigroup and the implementation of com- plex linear canonical transformations in quantum mechanics, Group Theoretical Methods in Physics: Sixth International Colloquium Tü- bingen 1977. Editor: P. Kramer, A. Rieckers, Lecture Notes in Physics 79 (1977), 512514. [Bru85] , The metaplectic semigroup and related topics, Reports on Ma- thematical Physics 22 (1985), 149170. [Cha07]
- R. Howe, The oscillator semigroup in the mathematical heritage of her- mann weyl, Proc. Symp. Pure Math., R.O. Wells, Ed. AMS Providence 48 (1988), 61110. [Jac49]
- B. Hall, The segal-bargmannbargmann`coherent state' transform for compact lie groups, Journal of Functional Analysis 122 (1994), 103151.
- [Upm08b] , Toeplitz operator algebras and complex analysis, Operator Theory: Advances and Applications: Operator Algebras, Operator Theory and Applications, Birkhäuser Basel 181 (2008), 67118. [Upm09] , Hilbert bundles and at connections over hermitian symmetric domains, Integral Equations and Operator Theory pre-print (2009).
- G. Segal, Unitary representations of some innite dimensional groups, Comm. Math. Phys. 80 (1981), 301342. [Tuy87]
- W. Kaup and H. Upmeier, Jordan algebras and symmetric siegel do- mains in banach spaces, Mathematische Zeitschrift 157 (1977), 179 200. [KW65]