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


SIGMA 5 (2009), 036, 21 pages      arXiv:0803.4201      http://dx.doi.org/10.3842/SIGMA.2009.036
Contribution to the Special Issue on Deformation Quantization

Three Natural Generalizations of Fedosov Quantization

Klaus Bering
Institute for Theoretical Physics & Astrophysics, Masaryk University, Kotlárská 2, CZ-611 37 Brno, Czech Republic

Received May 19, 2008, in final form February 14, 2009; Published online March 25, 2009

Abstract
Fedosov's simple geometrical construction for deformation quantization of symplectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does not have to be of Weyl/symmetric or Wick/normal type. (3) The initial geometric structures are allowed to depend on Planck's constant.

Key words: deformation quantization; Fedosov quantization; star product; supermanifolds; symplectic geometry.

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References

  1. Batalin I.A., Fradkin E.S., A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B 122 (1983), 157-164.
  2. Batalin I.A., Fradkin E.S., Operator quantization of relativistic dynamical systems subject to first class constraints, Phys. Lett. B 128 (1983), 303-308.
    Batalin I.A., Fradkin E.S., Operator quantization and abelization of dynamical systems subject to first-class constraints, Riv. Nuovo Cimento 9 (1986), 1-48.
  3. Batalin I.A., Fradkin E.S., Operatorial quantization of dynamical systems subject to second class constraints, Nuclear Phys. B 279 (1987), 514-528.
    Batalin I.A., Fradkin E.S., Operator quantization of dynamical systems with irreducible first and second class constraints, Phys. Lett. B 180 (1986), 157-162, Erratum, Phys. Lett. B 236 (1990), 528.
  4. Batalin I.A., Fradkin E.S., Fradkina T.E., Another version for operatorial quantization of dynamical systems with irreducible constraints, Nuclear Phys. B 314 (1989), 158-174, Erratum, Nuclear Phys. B 323 (1989), 734-735.
    Batalin I.A., Fradkin E.S., Fradkina T.E., Generalized canonical quantization of dynamical systems with constraints and curved phase space, Nuclear Phys. B 332 (1990), 723-736.
  5. Batalin I.A., Grigoriev M.A., Lyakhovich S.L., Star product for second class constraint systems from a BRST theory, Theoret. and Math. Phys. 128 (2001), 1109-1139, hep-th/0101089.
  6. Batalin I.A., Tyutin I.V., An Sp(2) covariant formalism of a generalized canonical quantization with second class constraints, Internat. J. Modern Phys. A 6 (1991), 3599-3612.
  7. Batalin I.A., Vilkovisky G.A., Relativistic S matrix of dynamical systems with boson and fermion constraints, Phys. Lett. B 69 (1977), 309-312.
  8. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61-110.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  9. Berezin F.A., Quantization, Math. USSR-Izv. 38 (1974), 1109-1165.
  10. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  11. Berezin F.A., Quantization in complex symmetric spaces, Math. USSR-Izv. 39 (1975), 341-379.
  12. Bering K., Almost parity structure, connections and vielbeins in BV geometry, physics/9711010.
  13. Bordemann M., The deformation quantization of certain super-Poisson brackets and BRST cohomology, in Conférence Moshé Flato 1999, Quantization, Deformation, and Symmetries, Vol. II, Editors G. Dito and D. Sternheimer, Math. Phys. Stud., Vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, 45-68, math.QA/0003218.
  14. Bordemann M., Waldmann S., A Fedosov star product of the Wick type for Kähler manifolds, Lett. Math. Phys. 41 (1997), 243-253, q-alg/9605012.
  15. Cattaneo A., Felder G., Tomassini L., From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002) 329-352, math.QA/0012228.
  16. Cattaneo A., Felder G., Tomassini L., Fedosov connections on jet bundles and deformation quantization, in Deformation Quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys., Vol. 1, de Gruyter, Berlin, 2002 191-202, math.QA/0111290.
  17. Dirac P.A.M., The fundamental equations of quantum mechanics, Proc. Roy. Soc. London A 109 (1925), 642-653.
  18. Dolgushev V.A., Lyakhovich S.L., Sharapov A.A., Wick type deformation quantization of Fedosov manifolds, Nuclear Phys. B 606 (2001), 647-672, hep-th/0101032.
  19. Donin J., Classification of polarized deformation quantizations, J. Geom. Phys. 48 (2003), 546-579, math.QA/0205211.
  20. Eckel R.K., Quantisierung von Supermannigfaltigkeiten à la Fedosov, Ph.D. Thesis, University of Freiburg, 2000.
  21. Fedosov B.V., Formal quantization, in Some Problems in Modern Mathematics and Their Applications to Problems in Mathematical Physics, Editor L.D. Kudryavtsev, Moscow Phys.-Tech. Inst., 1985, 129-136.
  22. Fedosov B.V., Quantization and the index, Sov. Phys. Dokl. 31 (1986), 877-878.
  23. Fedosov B.V., A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213-238.
  24. Fedosov B.V., Deformation quantization and index theory, Mathematical Topics, Vol. 9, Akademie Verlag, Berlin, 1996.
  25. Fisch J., Henneaux M., Stasheff J., Teitelboim C., Existence, uniqueness and cohomology of the classical BRST charge with ghosts of ghosts, Comm. Math. Phys. 120 (1989), 379-407.
  26. Farkas D.R., A ring-theorist's description of Fedosov quantization, Lett. Math. Phys. 51 (2000), 161-177, math.SG/0004071.
  27. Fradkin E.S., Fradkina T.E., Quantization of relativistic systems with boson and fermion first and second class constraint, Phys. Lett. B 72 (1978), 343-348.
  28. Fradkin E.S., Linetsky V.Ya., BFV approach to geometric quantization, Nuclear Phys. B 431 (1994) 569-621.
    Fradkin E.S., Linetsky V.Ya., BFV quantization on Hermitian symmetric spaces, Nuclear Phys. B 444 (1995), 577-601.
  29. Fradkin E.S., Vilkovisky G.A., Quantization of relativistic systems with constraints, Phys. Lett. B 55 (1975), 224-226.
    Fradkin E.S., Vilkovisky G.A., Quantization of relativistic systems with constraints, CERN Preprint TH 2332-CERN, 1977.
  30. Gadella M., del Olmo M.A., Tosiek J., Geometrical origin of the *-product in the Fedosov formalism, J. Geom. Phys. 55 (2005), 316-352, hep-th/0405157.
  31. Gelfand I., Retakh V., Shubin M., Fedosov manifolds, Adv. Math. 136 (1998), 104-140, dg-ga/9707024.
  32. Geyer B., Lavrov P.M., Fedosov supermanifolds: Basic properties and the difference in even and odd cases, Internat. J. Modern Phys. A 19 (2004), 3195-3208, hep-th/0306218.
  33. Grigoriev M.A., Lyakhovich S.L., Fedosov deformation quantization as a BRST theory, Comm. Math. Phys. 218 (2001), 437-457, hep-th/0003114.
  34. Groenewold H.J., On the principles of elementary quantum mechanics, Physica 12 (1946), 405-460.
  35. Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992.
  36. Karabegov A.V., Deformation quantizations with separation of variables on a Kähler manifold, Comm. Math. Phys. 180 (1996), 745-756, hep-th/9508013.
  37. Karabegov A.V., On the dequantization of Fedosov's deformation quantization, Lett. Math. Phys. 65 (2003), 133-146, math.QA/0307381.
  38. Karabegov A.V., Schlichenmaier M., Almost-Kähler deformation quantization, Lett. Math. Phys. 57 (2001), 135-148, math.QA/0102169.
  39. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  40. Kravchenko O., How to calculate the Fedosov star-product (exercices de style), math.SG/0008157.
  41. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99-124.
  42. Neumaier N., Universality of Fedosov's construction for star products of Wick type on pseudo-Kähler manifolds, Rep. Math. Phys. 52 (2003), 43-80.
  43. Xu P., Fedosov *-products and quantum momentum maps, Comm. Math. Phys. 197 (1998), 167-197, q-alg/9608006.
  44. Rothstein M., The structure of supersymplectic supermanifolds, in Differential Geometric Methods in Theoretical Physics (Rapallo, 1990), Lecture Notes in Phys., Vol. 375, Springer, Berlin, 1991, 331-343.
  45. Vaisman I., Fedosov quantization on symplectic ringed spaces, J. Math. Phys. 43 (2002), 283-298, math.SG/0106070.
  46. De Wilde M., Lecomte P.B.A., Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), 487-496.

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