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Quantization: Deformation and/or Functor?

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Abstract

After a short presentation of the difference in motivation between the Berezin and deformation quantization approaches, we start with a reminder of Berezin’s view of quantization as a functor followed by a brief overview of deformation quantization in contrast with the latter. We end by a short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective.

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References

  • Agarwal, G.S., Wolf, E.: Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics I, II, III. Phys. Rev. D2, 2161–2186, 2187–2205, 2206–2225 (1970)

  • Basart H., Flato M., Lichnerowicz A., Sternheimer D. (1984). Deformation theory applied to quantization and statistical mechanics. Lett. Math. Phys. 8, 483–494

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Bayen F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization I. Deformations of symplectic structures. Ann. Phys. 111, 61–110; and II. Physical applications. ibid 111–151 (1978)

    Google Scholar 

  • Bayen F., Fronsdal C. (1981). Quantization on the sphere. J. Math. Phys. 22, 1345–1349

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Berezin, F.A.: The method of second quantization. Nauka, Moscow (1965) (English translation, Academic Press 1966)

  • Berezin, F.A.: Quantization. Izv. Akad. Nauk SSSR Ser. Mat. 38(5), 1116–1175 (1974) (English translation: Math. USSR-Izv. 8(5), 1109–1165 (1975))

  • Berezin, F.A.: Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR Ser. Mat. 39(2), 363–402, 472 (1975) (English translation: Math. USSR-Izv. 9, 341–379 (1976)) Mat. 39, 363–402, 472 (1975) (English translation: Math. USSR-Izv. 38, 1109–1165 (1975) 39, 341–379 (1976))

  • Berezin F.A. (1975). General concept of quantization. Comm. Math. Phys. 40, 153–174

    Article  ADS  MathSciNet  Google Scholar 

  • Berezin, F.A.: Feynman path integrals in a phase space. Usp. Fiz. Nauk 132(3), 493–548 (1980) (English translation: Sov. Phys. Usp. 23(11), 763–788 (1980))

  • Berezin F.A., Shubin M.A. (1991). The Schrödinger equation. pp. XI+555. Kluwer, Dordrecht

    Google Scholar 

  • Bonneau P., Flato M., Gerstenhaber M., Pinczon G. (1994). The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations. Comm. Math. Phys. 161, 125–156

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Bonneau P., Gerstenhaber M., Giaquinto A., Sternheimer D. (2004). Quantum groups and deformation quantization: explicit approaches and implicit aspects. J. Math. Phys. 45, 3703–3741

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. 212, 591–611 (2000) (math.QA/9902090)

    Google Scholar 

  • Cattaneo, A.S., Felder, G.: On the globalization of Kontsevich’s star product and the perturbative Poisson sigma model. Prog. Theor. Phys. Suppl. 144, 38–53 (2001) (hep-th/0111028)

    Google Scholar 

  • Cattaneo, A.S.: On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds. Lett. Math. Phys. 67, 33–48 (2004) (math.SG/0308180)

    Google Scholar 

  • Connes A. (1994). Noncommutative geometry. Academic Press, San Diego

    MATH  Google Scholar 

  • Connes A. (1985). Noncommutative differential geometry. Inst. Hautes études Sci. Publ. Math. 62, 257–360

    Article  MathSciNet  Google Scholar 

  • Connes, A.: A short survey of noncommutative geometry. J. Math. Phys. 41, 3832–3866 (2000) (hep-th/0003006)

    Google Scholar 

  • Connes, A.: Cyclic cohomology, noncommutative geometry and quantum group symmetries. Noncommutative geometry, Lecture Notes in Mathematics, vol. 1831, pp. 1–71. Springer, Berlin Heidelberg New York (2004)

  • Connes, A., Dubois-Violette, M.: Moduli space and structure of noncommutative 3-spheres. Lett. Math. Phys. 66, 99–121 (2003) (See also math.QA/0511337)

    Google Scholar 

  • Connes A., Flato M., Sternheimer D. (1992). Closed star-products and cyclic cohomology. Lett. Math. Phys. 24, 1–12

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem; II: The β-function, diffemorphisms and the renormalization group. Comm. Math. Phys. 210, 249–273 (2000) (hep-th/9912092); 216, 215–241 (2001) (hep-th/0003188)

    Google Scholar 

  • Connes, A., Kreimer, D.: From local perturbation theory to Hopf and Lie algebras of Feynman graphs, EuroConférence Moshé Flato 2000, Part I (Dijon). Lett. Math. Phys. 56, 3–15 (2001)

    Google Scholar 

  • Connes, A., Kreimer, D.: Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs. Ann. Henri Poincaré 3, 411–433 (2002) (hep-th/0201157)

    Google Scholar 

  • De Wilde M., Lecomte P.B.A. (1983). Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dirac P.A.M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Sciences Monograph Series No. 2. Yeshiva University, New York

    Google Scholar 

  • Dirac P.A.M. (1950). Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148

    MathSciNet  MATH  Google Scholar 

  • Dito J. (1990). Star-product approach to quantum field theory: the free scalar field. Lett. Math. Phys. 20, 125–134

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dito J. (1992). Star-products and nonstandard quantization for K-G equation. J. Math. Phys. 33, 791–801

    Article  ADS  MathSciNet  Google Scholar 

  • Dito J. (1993). An example of cancellation of infinities in star-quantization of fields. Lett. Math. Phys. 27, 73–80

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dito J. (1993). Star-produits en dimension infinie : le cas de la théorie quantique des champs. Thesis, Université de Bourgogne, Dijon

  • Dito, G. (=J.): Deformation quantization of covariant fields. Deformation quantization (Strasbourg 2001), IRMA Lect. Math. Theor. Phys. 1, 55–66 (2002) Walter de Gruyter, Berlin (math.QA/0202271)

  • Dito, G., Sternheimer, D.: Deformation quantization: genesis, developments and metamorphoses, Deformation quantization (Strasbourg 2001). IRMA Lect. Math. Theor. Phys. 1, 9–54 (2002) Walter de Gruyter, Berlin (math.QA/0201168)

  • Douglas, M.R., Liu, H., Moore, G., Zwiebach, B.: Open string star as a continuous Moyal product. J. High Energy Phys. 0204(22), 27 (2002) (hep-th/0202087)

    Google Scholar 

  • Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2001) (hep-th/0106048)

    Google Scholar 

  • Drinfeld V.: Quantum groups. In: Proceedings of ICM86, Berkeley, vol. 1, pp. 101–110. American Mathematical Society, Providence (1987)

  • Fedosov B.V. (1994). A simple geometrical construction of deformation quantization. J. Diff. Geom. 40, 213–238

    MathSciNet  MATH  Google Scholar 

  • Fedosov B.V. (1996). Deformation quantization and index theory. Mathematical Topics 9. Akademie Verlag, Berlin

    Google Scholar 

  • Fedosov B.V. (2000). The Atiyah-Bott-Patodi method in deformation quantization. Comm. Math. Phys. 209, 691–728

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Flato M., Fronsdal C. (1988). Composite electrodynamics. J. Geom. Phys. 5, 37–61

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Flato M., Frønsdal C., Sternheimer D. (1999). Singletons, physics in AdS universe and oscillations of composite neutrinos. Lett. Math. Phys. 48, 109–119

    Article  MathSciNet  MATH  Google Scholar 

  • Flato M., Hadjiivanov L.K., Todorov I.T. (1993). Quantum deformations of singletons and of free zero-mass fields. Found. Phys. 23, 571–586

    Article  ADS  MathSciNet  Google Scholar 

  • Flato M., Hillion P., Sternheimer D. (1967). Un groupe de type Poincaré associé à l’équation de Dirac-Weyl. C.R. Acad. Sci. Paris Sér. A 264, 82–85

    MATH  Google Scholar 

  • Flato M., Hillion P. (1970). Poincaré-like group associated with neutrino physics, and some applications. Phys. Rev. D1, 1667–1673

    Article  ADS  MathSciNet  Google Scholar 

  • Flato M., Sternheimer D. (1994). Topological quantum groups, star products and their relations. St. Petersburg Math. J. (Algebra Anal.) 6, 242–251

    MathSciNet  Google Scholar 

  • Fronsdal C. (1978). Some ideas about quantization. Rep. Math. Phys. 15, 111–145

    Article  MathSciNet  ADS  Google Scholar 

  • Frønsdal, C. (with an appendix by Kontsevich, M.): Quantization on curves, math-ph/0507021 Lett. Math. Phys. (to appear)

  • Gelfand I., Retakh V., Shubin M. (1998). Fedosov manifolds. Adv. Math. 136, 104–140

    Article  MathSciNet  MATH  Google Scholar 

  • Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79, 59–103 (1964); (II) ibid 84, 1–19 (1966), (IV) ibid. 99, 257–276 (1974)

  • Gerstenhaber, M.: On the deformation of sheaves of rings. Global Analysis (Papers in Honor of K. Kodaira) pp. 149–157, University of Tokyo Press, Tokyo (1969)

  • Gerstenhaber, M., Schack, S.D.: Algebraic cohomology and deformation theory. In: Hazewinkel, M., Gerstenhaber, M. (eds.) Deformation theory of algebras and structures and applications, NATO ASI Ser. C 247, pp. 11–264, Kluwer, Dordrecht (1988)

  • Groenewold A. (1946). On the principles of elementary quantum mechanics. Physica 12, 405–460

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ínönü E., Wigner E.P. (1953). On the contraction of groups and their representations. Proc. Nat. Acad. Sci. USA 39, 510–524

    Article  ADS  MATH  Google Scholar 

  • Jimbo M. (1985). A q-difference algebra of \({\cal U}(\frak{g})\) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kodaira K., Spencer D.C. (1958). On deformations of complex analytic structures. Ann. Math. 67, 328–466

    Article  MathSciNet  Google Scholar 

  • Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003) (q-alg/9709040)

    Google Scholar 

  • Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999) (math.QA/9904055)

    Google Scholar 

  • Kontsevich, M.: Deformation quantization of algebraic varieties. Lett. Math. Phys. 56, 271–294 (2001) (math.AG/0106006)

    Google Scholar 

  • Kostant B. (1970). Quantization and unitary representations. Lect. Notes Math. 170, 87–208

    Article  MathSciNet  Google Scholar 

  • Kostant, B.: On the definition of quantization. In: Géométrie symplectique et physique mathématique (Colloq. Int. CNRS, N° 237), (éds) pp. 187–210 CNRS, Paris (1975)

  • Kostant B. (1977). Graded manifolds, graded Lie theory, and prequantization. Differential geometrical methods in mathematical physics. Lect. Notes Math. 570, 177–306

    Article  MathSciNet  Google Scholar 

  • Mathai, V., Melrose, R.B., Singer, I.M.: Fractional analytic index. math.DG/0402329

  • Minlos, R.A.: Felix Alexandrovich Berezin (A Brief Scientific Biography). Lett. Math. Phys. 74(1), 5–19 (2005) (Original publication in: Contemporary Mathematical Physics, American Mathematical Society Translations, Ser. 2, 175, Amer. Math. Soc., Providence, RI, pp. 1–13 (1996))

  • Moyal J.E. (1949). Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc. 45, 99–124

    Article  MathSciNet  MATH  Google Scholar 

  • Nadaud, F.: Generalized deformations, Koszul resolutions, Moyal Products. Rev. Math. Phys. 10(5), 685–704 (1998) Thèse, Dijon (janvier 2000)

    Google Scholar 

  • Nest R., Tsygan B. (1995). Algebraic index theorem. Comm. Math. Phys. 172, 223–262

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Nest R., Tsygan B. (1995). Algebraic index theorem for families. Adv. Math. 113, 151–205

    Article  MathSciNet  MATH  Google Scholar 

  • Nest R., Tsygan B. (1996). Formal deformations of symplectic manifolds with boundary. J. Reine Angew. Math. 481, 27–54

    MathSciNet  MATH  Google Scholar 

  • Omori H., Maeda Y., Yoshioka A. (1991). Weyl manifolds and deformation quantization. Adv. Math. 85, 225–255

    Article  MathSciNet  Google Scholar 

  • Omori H., Maeda Y., Yoshioka A. (1992). Existence of a closed star product. Lett. Math. Phys. 26, 285–294

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pankaj Sharan. (1979). Star-product representation of path integrals. Phys. Rev. D 20, 414–418

    Article  MathSciNet  Google Scholar 

  • Segal I.E. (1951). A class of operator algebras which are determined by groups. Duke Math. J. 18, 221–265

    Article  MathSciNet  MATH  Google Scholar 

  • Sternheimer, D.: Deformation quantization: twenty years after. In: Rembieliński J., (ed.) Particles, Fields, and Gravitation (Łodz 1998) pp. 107–145. AIP Press, New York (1998) (math.QA/9809056)

  • Sternheimer, D.: In retrospect: a personal view on Moshé Flato’s scientific legacy. In: Conférence Moshé Flato 1999, Mathematical Physics Studies, vol. 21, pp. 31–41. Kluwer, Dordrecht (2000)

  • Sternheimer, D.: Quantization is deformation. In: Fuchs, J. et al. (eds.) Noncommutative geometry and representation theory in mathematical physics, Contemporary mathematics, vol. 391, pp. 331–352, American Mathematical Society (2005)

  • Weyl, H.: The theory of groups and quantum mechanics. Dover, New York (1931) edited translation of Gruppentheorie und Quantenmechanik, Hirzel Verlag, Leipzig (1928)

  • Weyl H. (1927). Quantenmechanik und Gruppentheorie. Z. Physik 46, 1–46

    Article  ADS  Google Scholar 

  • Wigner E.P. (1932). Quantum corrections for thermodynamic equilibrium. Phys. Rev. 40, 749–759

    Article  ADS  MATH  Google Scholar 

  • Woodhouse N.M.J. (1992). Geometric quantization. 2nd edn. Oxford Science Publications, Oxford Mathematical Monographs. Oxford University Press, New York

    MATH  Google Scholar 

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  1. Institut de Mathématiques de Bourgogne, Université de Bourgogne, BP 47870, F-21078, Dijon Cedex, France

    Daniel Sternheimer

  2. Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Daniel Sternheimer

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Sternheimer, D. Quantization: Deformation and/or Functor?. Lett Math Phys 74, 293–309 (2005). https://doi.org/10.1007/s11005-005-0028-4

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