Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 021, 30 pages      arXiv:0902.3628      http://dx.doi.org/10.3842/SIGMA.2009.021
Contribution to the Special Issue on Deformation Quantization

Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

Miroslav Englis a, b and Harald Upmeier c
a) Mathematics Institute, Silesian University at Opava, Na Rybnícku 1, 74601 Opava, Czech Republic
b) Mathematics Institute, Zitná 25, 11567 Prague 1, Czech Republic
c) Fachbereich Mathematik, Universität Marburg, D-35032 Marburg, Germany

Received October 01, 2008, in final form February 14, 2009; Published online February 20, 2009

Abstract
For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of ''star-restriction'' (a real analogue of the ''star-products'' for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion.

Key words: bounded symmetric domain; Toeplitz operator; star product; covariant quantizationn.

pdf (383 kb)   ps (247 kb)   tex (32 kb)

References

  1. Akhiezer D.N., Gindikin S.G., On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 1-12.
  2. Arazy J., Ørsted B., Asymptotic expansions of Berezin transforms, Indiana Univ. Math. J. 49 (2000), 7-30.
  3. Arazy J., Upmeier H., Covariant symbolic calculi on real symmetric domains, in Singular Integral Operators, Factorization and Applications, Oper. Theory Adv. Appl., Vol. 142, Birkhäuser, Basel, 2003, 1-27.
  4. Arazy J., Upmeier H., Weyl calculus for complex and real symmetric domains, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), 165-181.
  5. Arazy J., Upmeier H., A one-parameter calculus for symmetric domains, Math. Nachr. 280 (2007), 939-961.
  6. Arazy J., Upmeier H., Invariant symbolic calculi and eigenvalues of invariant operators on symmetric domains, in Function Spaces, Interpolation Theory, and Related Topics (Lund, 2000), Editors A. Kufner, M. Cwikel, M. Englis, L.-E. Persson and G. Sparr, Walter de Gruyter, Berlin, 2002, 151-211.
  7. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  8. Berezin F.A., Quantization, Math. USSR Izvestiya 8 (1974), 1109-1163.
  9. Berezin F.A., Quantization in complex symmetric spaces, Math. USSR Izvestiya 9 (1975), 341-379.
  10. Bieliavsky P., Strict quantization of solvable symmetric spaces, J. Symplectic Geom. 1 (2002), 269-320, math.QA/0010004.
  11. Bieliavsky P., Cahen M., Gutt S., Symmetric symplectic manifolds and deformation quantization, in Modern Group Theoretical Methods in Physics (Paris, 1995), Math. Phys. Stud., Vol. 18, Kluwer Acad. Publ., Dordrecht, 1995, 63-73.
  12. Bieliavsky P., Pevzner M., Symmetric spaces and star representations. II. Causal symmetric spaces, J. Geom. Phys. 41 (2002), 224-234, math.QA/0105060.
  13. Bieliavsky P., Detournay S., Spindel P., The deformation quantizations of the hyperbolic plane, arXiv:0806.4741.
  14. Borthwick D., Lesniewski A., Upmeier H., Non-perturbative deformation quantization on Cartan domains, J. Funct. Anal. 113 (1993), 153-176.
  15. Bordemann M., Meinrenken E., Schlichenmaier M., Toeplitz quantization of Kähler manifolds and gl(n), n → ∞ limits, Comm. Math. Phys. 165 (1994), 281-296, hep-th/9309134.
  16. Burns D., Halverscheid S., Hind R., The geometry of Grauert tubes and complexification of symmetric spaces, Duke Math. J. 118 (2003), 465-491, math.CV/0109186.
  17. van Dijk G., Pevzner M., Berezin kernels and tube domains, J. Funct. Anal. 181 (2001), 189-208.
  18. Englis M., Berezin-Toeplitz quantization on the Schwartz space of bounded symmetric domains, J. Lie Theory 15 (2005), 27-50.
  19. Englis M., Weighted Bergman kernels and quantization, Comm. Math. Phys. 227 (2002), 211-241.
  20. Englis M., Berezin-Toeplitz quantization and invariant symbolic calculi, Lett. Math. Phys. 65 (2003), 59-74.
  21. Englis M., Berezin transform on the harmonic Fock space, in preparation.
  22. Englis M., Upmeier H., Toeplitz quantization and asymptotic expansions for real bounded symmetric domains, Preprint, 2008, http://www.math.cas.cz/~englis/70.pdf.
  23. Erdélyi A. et al., Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953.
  24. Faraut J., Korányi A., Analysis on symmetric cones, The Clarendon Press, Oxford University Press, New York, 1994.
  25. Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  26. Helgason S., Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. 12, Academic Press, New York - London, 1962.
  27. Hua L.K., Harmonic analysis of functions of several complex variables in the classical domains, American Mathematical Society, Providence, R.I., 1963.
  28. Karabegov A.V., Schlichenmaier M., Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49-76, math.QA/0006063.
  29. Lassalle M., Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact, Ann. Sci. École Norm. Sup. (4) 11 (1978), 167-210.
  30. Loos O., Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
  31. Neretin Yu.A., Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space, J. Funct. Anal. 189 (2002), 336-408, math.RT/9911020.
  32. Reshetikhin N., Takhtajan L., Deformation quantization of Kähler manifolds, L.D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 257-276, math.QA/9907171.
  33. Schlichenmaier M., Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization, Conference Moshé Flato, Vol. II (Dijon, 1999), Math. Phys. Stud., Vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, 289-306, math.QA/9910137.
  34. Upmeier H., Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, Vol. 81, Birkhäuser Verlag, Basel, 1996.
  35. Weinstein A., Traces and triangles in symmetric symplectic spaces, in Symplectic Geometry and Quantization (Sanda and Yokohama, 1993), Contemp. Math. 179 (1994), 261-270.
  36. Zhang G., Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (2001), 3769-3787.
  37. Zhang G., Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 195 (2002), 306-349, math.RT/0110212.

Previous article   Next article   Contents of Volume 5 (2009)