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Formality Theory pp 21–59Cite as

Deformation Quantization and Formality Theory

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Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 2))

Abstract

This chapter will be devoted to the theory of formal deformation quantization. We will recall the description of a quantum physical system in terms of a non-commutative algebra of operators (quantum observables) on a Hilbert space.

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References

  1. T. Ali, M. Englis, Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17, 391–490 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Quantum mechanics as a deformation of classical mechanics. Lett. Math. Phys. 1, 521–530 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  3. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization I-II. Ann. Phys. 111(61–110), 111–151 (1978)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. F.A. Berezin, M.A. Šubin, Symbols of operators and quantization, in Proceedings of International Conference, Tihany, 1970, Hilbert space operators and operator algebras, p. 1972

    Google Scholar 

  6. M. Bertelson, P. Bieliavsky, S. Gutt, Parametrizing equivalence classes of invariant star products. Lett. Math. Phys. 46(4), 339–345 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Bertelson, M. Cahen, S. Gutt, Equivalence of star products. Class. Quantum Grav. 14, A93–A107 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. P. Bongaarts. Quantum theory: a mathematical approach. Springer, to appear

    Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. M. Bordemann, Deformation quantization: a survey. ed. by J-C. Wallet. in International Conference on Noncommutative Geometry and Physics. vol. 103 J. Phys.: Conference Series, 2008

    Google Scholar 

  11. M. Bordemann, M. Brischle, C. Emmrich, S. Waldmann, Subalgebras with converging star products in deformation quantization: an algebraic construction for \(\mathbb{C}P^n\). J. Math. Phys. 37(12), 6311–6323 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. N. Bourbaki, Lie Groups and Lie Algebras (Springer, Berlin, 1989)

    MATH  Google Scholar 

  13. M. Cahen, S. Gutt, M. de Wilde, Local cohomology of the algebra of \(C^\infty \) functions on a connected manifold. Lett. Math. Phys. 4, 157–167 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. A.S. Cattaneo, D. Indelicato, Poisson geometry, deformation quantization and group representation, in Formality and Star Product, London Mathematical Society Lecture Note series, ed. by S. Gutt, J. Rawnsley, D. Sternheimer (Cambridge University Press, Cambridge, 2004), pp. 81–144

    Google Scholar 

  15. L. Chloup, Star products on the algebra of polynomials on the dual of a semi-simple Lie algebra. Acad. Roy. Belg. Bull. Cl. Sci. 8, 263–269 (1997)

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. P. Deligne, Déformations de l’Algèbre des Fonctions d’une variété symplectique: comparaison entre Fedosov et DeWilde, Lecomte. Selecta Math. N. S. 1, 667–697 (1995)

    Google Scholar 

  18. P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964)

    Google Scholar 

  19. V. A. Dolgushev, A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold. Ph.D. thesis, MIT, 2005

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. B.V. Fedosov, Deformation Quantization and Index Theory (Wiley-VCH, Weinheim, 1996)

    MATH  Google Scholar 

  22. M. Flato, A. Lichnerowicz, D. Sternheimer, Déformations 1-différentiales des algèbres de Lie attachées à une variété symplectique ou de contact. Compositio Math. 31, 47–82 (1975)

    MATH  MathSciNet  Google Scholar 

  23. M. Flato, A. Lichnerowicz, D. Sternheimer, Crochets de Moyal-Vey et quantification. C. R. Acad. Sci. Paris Sér. A 283, 19–24 (1976)

    MATH  MathSciNet  Google Scholar 

  24. M. Gerstenhaber, The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Gerstenhaber, On the deformation of rings and algebras. Ann. Math. 79(1), 59–103 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Gerstenhaber, S.D. Schack, Algebraic cohomology and deformation theory, Deformation Theory of Algebras and Structures and Applications (Kluwer Academic Publishers, Dordrecht, 1988), pp. 11–264

    Chapter  Google Scholar 

  27. V.L. Ginzburg, Lectures on non commutative geometry (2005) arXiv:math/0506603

  28. H.J. Groenewold, On the principles of elementary quantum mechanics. Physics 12, 405–460 (1946)

    MATH  MathSciNet  Google Scholar 

  29. S. Gutt, Deformation quantisation of Poisson manifolds. Geom. Topol. Monogr. 17, 171–220 (2001)

    MathSciNet  Google Scholar 

  30. S. Gutt, J. Rawnsley, Equivalence of star products on a symplectic manifold: an introduction to Deligne’s Cech cohomology classes. J. Geom. Phys 29, 347–392 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. B.C. Hall, Lie Groups, Lie Algebras and Representations (Springer, Berlin, 2003)

    Book  MATH  Google Scholar 

  32. V. Hinich, V. Schechtman, Homotopy Lie algebras, in I.M.Gelfand Seminar, Advances in Soviet Mathematics, ed. by S. Gindikin, vol. 16 (1993). AMS

    Google Scholar 

  33. G. Hochschild, On the cohomology groups of an associative algebra. Ann. Math. 46(1), 58–67 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  34. G. Hochschild, B. Kostant, A. Rosenberg, Differential forms on regular affine algebras. Trans. Amer. Math. Soc. 102(3), 383–408 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Karabegov, Cohomological classification of deformation quantisations with separation of variables. Lett. Math. Phys. 43, 347–357 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. B. Keller, Deformation quantization after Kontsevich and Tamarkin. In: Déformation, quantification, théorie de Lie, vol. 20 (Panoramas et Synthèses, Société mathématique de France, 2005), pp. 19–62

    Google Scholar 

  37. M. Kontsevich, Formality conjecture, in Deformation Theory and Symplectic Geometry, ed. by D. Sternheimer, et al. (Kluwer Academic Publishers, Dordrecht, 1997), pp. 139–156

    Google Scholar 

  38. M. Kontsevich, Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. B. Kostant, Quantization and unitary representation, Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics (Springer, Berlin, 1970)

    Google Scholar 

  40. M. Manetti. Deformation Theory Via Differential Graded Lie Algebras. Seminari di Geometria Algebrica 1998–99 Scuola Normale Superiore, 1999

    Google Scholar 

  41. M. Manetti, Lectures on deformations of complex manifolds. Rendiconti di Matematica 24, 1–183 (2004)

    MATH  MathSciNet  Google Scholar 

  42. A. Messiah, Quantum Mechanics (Dover publications, Mineola, 1999)

    Google Scholar 

  43. J.E. Moyal, Quantum mechanics as a statistical theory. in Proceedings of the Cambridge Philosophical Society, 1949, vol. 45, pp. 99–124

    Google Scholar 

  44. O.M. Neroslavsky, A.T. Vlasov, Sur les deformations de l’algebre des fonctions d’une variete symplectique. C. R. Acad. Sci. 292, 71–73 (1981)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  46. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshida, Deformation quantization of Fréchet-Poisson algebras of Heisenberg type. Contemp. Math. 288, 391–395 (2001)

    Article  Google Scholar 

  47. H. Omori, Y. Maeda, A. Yoshida, Weyl manifold and deformation quantization. Adv. Math. 85(2), 224–255 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  48. M. Rieffel, Deformation quantization and operator algebra. Proceedings of Symposia in Pure Mathematics, vol. 51, (1990)

    Google Scholar 

  49. M. Rieffel, Deformation quantization for actions of \(\mathbb{R}^d\). Mem. Amer. Math. Soc. 506 (1993)

    Google Scholar 

  50. M. Rieffel, Quantization and \(C^*\)-algebras. Contemp. Math. 167, 67–97 (1994)

    MathSciNet  Google Scholar 

  51. ie segal, Quantization of nonlinear systems. J. Math. Phys. 1(6), 339–364 (1960)

    Article  Google Scholar 

  52. J-M. Souriau, Structure des systemes dymaniques. Editions Jacques Gabay, 1969

    Google Scholar 

  53. D. Sternheimer, Deformation quantization: twenty years after, in AIP Conference Proceedings (1998)

    Google Scholar 

  54. J. Vey, Déformation du crochet de Poisson sur une variété symplectique. Commentarii Mathematici Helvetici, 50(1), 1975

    Google Scholar 

  55. S. Waldmann, Poisson-Geometrie und Deformationsquantisierung (Springer, Heidelberg, 2007)

    MATH  Google Scholar 

  56. C. Weibel, An Introduction to Homological Algebras (Cambridge University Press, Cambridge, 1994)

    Book  Google Scholar 

  57. H. Weyl, Quantenmechanik und Gruppentheorie. Z. Physics 46, 1–46 (1927)

    Article  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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  1. Department of Mathematics, University of Würzburg, Würzburg, Germany

    Chiara Esposito

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Esposito, C. (2015). Deformation Quantization and Formality Theory. In: Formality Theory. SpringerBriefs in Mathematical Physics, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-09290-4_3

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