Some reflections on mathematicians' views of quantization

D. Sternheimer

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 199-220

Résumé

We start with a short presentation of the difference in attitude between mathematicians and physicists even in their treatment of physical reality, and look at the paradigm of quantization as an illustration. In particular we stress the differences in motivation and realization between the Berezin and deformation quantization approaches, exposing briefly Berezin's view of quantization as a functor. We continue with a schematic overview of deformation quantization and of its developments in contrast with the latter and discuss related issues, in particular the spectrality question. We end by a very short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective.

Bibliographie

[1] G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics, I, II, III”, Phys. Rev. D, 2 (1970), 2161–2186, 2187–2205, 2206–2225 | DOI | MR

[2] A. Aspect, R. Dalibard, G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers”, Phys. Rev. Lett., 49:25 (1982), 1804–1807 | DOI | MR

[3] D. Avis, H. Imai, T. Ito, Y. Sasaki, “Two-party Bell inequalities derived from combinatorics via triangular elimination”, J. Phys. A, 38:50 (2005), 10971–10987 ; arXiv: /quant-ph/0505060 | DOI | MR | Zbl

[4] H. Basart, M. Flato, A. Lichnerowicz, D. Sternheimer, “Deformation theory applied to quantization and statistical mechanics”, Lett. Math. Phys., 8 (1984), 483–494 | DOI | MR | Zbl

[5] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, “Deformation theory and quantization. I: Deformations of symplectic structures”, Ann. Physics, 111 (1978), 61–110 | DOI | MR | Zbl

[6] J. S. Bell, “On the problem of hidden variables in quantum mechanics”, Rev. Modern Phys., 38 (1966), 447–452 | DOI | MR | Zbl

[7] F. A. Berezin, “Canonical operator transformation in representation of secondary quantization”, Dokl. Akad. Nauk SSSR, 137 (1961), 311–314 | MR | Zbl

[8] F. A. Berezin, The method of second quantization, Nauka, Moscow, 1965 | MR

[9] F. A. Berezin, “Quantization”, Izv. Akad. Nauk SSSR, Ser. Mat., 38 (1974), 1116–1175 ; 39 (1975), 363–402, 472 | MR | Zbl | MR | Zbl

[10] F. A. Berezin, “General concept of quantization”, Comm. Math. Phys., 40 (1975), 153–174 | DOI | MR

[11] F. A. Berezin, “Feynman path integrals in a phase space”, Usp. Fiz. Nauk, 132 (1980), 493–548 | MR

[12] F. A. Berezin, Introduction to superanalysis, with an appendix by V. I. Ogievetsky, Mathematical Physics and Applied Mathematics, 9, ed. A. A. Kirillov, D. Reidel Publishing Co, Dordrecht, 1987 | MR | Zbl

[13] F. A. Berezin, M. A. Shubin, The Schrödinger equation, Kluwer Acad. Publ., Dordrecht, 1991 | MR | Zbl

[14] F. A. Berezin, D. A. Leites, “Supermanifolds”, Dokl. Akad. Nauk SSSR, 224:3 (1975), 505–508 | MR | Zbl

[15] 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 (1994), 125–156 | DOI | MR | Zbl

[16] P. Bonneau, M. Gerstenhaber, A. Giaquinto, D. Sternheimer, “Quantum groups and deformation quantization: explicit approaches and implicit aspects”, J. Math. Phys., 45 (2004), 3703–3741 | DOI | MR | Zbl

[17] A. S. Cattaneo, G. Felder, “A path integral approach to the Kontsevich quantization formula”, Comm. Math. Phys., 212 (2000), 591–611 ; ; “On the Globalization of Kontsevich's Star Product and the Perturbative Poisson Sigma Model”, Prog. Theor. Phys. Suppl., 144 (2001), 38–53 ; arXiv: /math.QA/9902090arXiv: /hep-th/0111028 | DOI | MR | Zbl | DOI | MR | Zbl

[18] A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994 ; “Noncommutative differential geometry”, Inst. Hautes Études Sci. Publ. Math., 1985, no. 62, 257–360 | MR | Zbl | MR

[19] A. Connes, “A short survey of noncommutative geometry”, J. Math. Phys., 41 (2000), 3832–3866 ; arXiv: /hep-th/0003006 | DOI | MR | Zbl

[20] A. Connes, Cyclic cohomology, noncommutative geometry and quantum group symmetries. Noncommutative geometry, Lecture Notes in Math., 1831, Springer, Berlin, 2004 | MR | Zbl

[21] A. Connes, M. Dubois-Violette, “Moduli space and structure of noncommutative 3-spheres”, Lett. Math. Phys., 66 (2003), 99–121 ; arXiv: /math.QA/0308275 | DOI | MR

[22] A. Connes, M. Flato, D. Sternheimer, “Closed star-products and cyclic cohomology”, Lett. Math. Phys., 24 (1992), 1–12 | DOI | MR | Zbl

[23] A. Connes, D. Kreimer, “From local perturbation theory to Hopf and Lie algebras of Feynman graphs”, EuroConférence Moshé Flato 2000, Part I (Dijon), Lett. Math. Phys., 56 (2001), 3–15 | DOI | MR | Zbl

[24] E. B. Davies, “Semi-classical states for non-self-adjoint Schrödinger operators”, Comm. Math. Phys., 200 (1999), 35–41 ; “Semi-classical analysis and pseudo-spectra”, J. Differential Equations, 216 (2005), 153–187 | DOI | MR | Zbl | DOI | MR | Zbl

[25] N. Dencker, J. Sjöstrand, M. Zworski, “Pseudospectra of semiclassical (pseudo-) differential operators”, Comm. Pure Appl. Math., 57 (2004), 384–415 | DOI | MR | Zbl

[26] 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 (1983), 487–496 | DOI | MR | Zbl

[27] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Sciences Monograph Series, 2, Yeshiva University, New York, 1967 ; “Generalized Hamiltonian dynamics”, Canad. J. Math., 2 (1950), 129–148 | MR | DOI | MR | Zbl

[28] J. Dito, “Star-product approach to quantum field theory: the free scalar field”, Lett. Math. Phys., 20 (1990), 125–134 ; “Star-products and nonstandard quantization for K-G equation”, J. Math. Phys., 33 (1992), 791–801 | DOI | MR | Zbl | DOI | MR

[29] J. Dito, “An example of cancellation of infinities in star-quantization of fields”, Lett. Math. Phys., 27 (1993), 73–80 | DOI | MR | Zbl

[30] G. Dito, “Deformation quantization of covariant fields”, Deformation quantization (Strasbourg 2001), IRMA Lectures in Math. Theoret. Phys., 1, Walter de Gruyter, Berlin, 2002, 55–66 ; arXiv: /math.QA/0202271 | MR

[31] G. Dito, D. Sternheimer, “Deformation quantization: genesis, developments and metamorphoses”, Deformation quantization (Strasbourg 2001), IRMA Lect. Math. Theor. Phys., 1, Walter de Gruyter, Berlin, 2002, 9–54 ; arXiv: /math.QA/0201168 | MR

[32] M. R. Douglas, H. Liu, G. Moore, B. Zwiebach, “Open string star as a continuous Moyal product”, J. High Energy Phys., 2002, no. 22, 0204 ; arXiv: /hep-th/0202087 | MR

[33] M. R. Douglas, N. A. Nekrasov, “Noncommutative field theory”, Rev. Mod. Phys., 73 (2001), 977–1029 ; arXiv: /hep-th/0106048 | DOI | MR | Zbl

[34] V. Drinfeld, “Quantum Groups”, Proc. ICM86 Berkeley, 1, Amer. Math. Soc., Providence, 1987, 101–110 | MR

[35] B. V. Fedosov, “A simple geometrical construction of deformation quantization”, J. Diff. Geom., 40 (1994), 213–238 ; Deformation quantization and index theory, Mathematical Topics, 9, Akademie Verlag, Berlin, 1996 ; “The Atiyah-Bott-Patodi method in deformation quantization”, Comm. Math. Phys., 209 (2000), 691–728 | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[36] M. Flato, C. Fronsdal, “Composite Electrodynamics”, J. Geom. Phys., 5 (1988), 37–61 | DOI | MR | Zbl

[37] M. Flato, C. Frønsdal, D. Sternheimer, “Singletons, physics in AdS universe and oscillations of composite neutrinos”, Lett. Math. Phys., 48 (1999), 109–119 | DOI | MR | Zbl

[38] M. Flato, L. K. Hadjiivanov, I. T. Todorov, “Quantum deformations of singletons and of free zero-mass fields”, Found. Phys., 23 (1993), 571–586 | DOI | MR

[39] M. Flato, P. Hillion, D. Sternheimer, “Un groupe de type Poincaré associé à l'équation de Dirac-Weyl”, C. R. Acad. Sci. Paris, 264 (1967), 85–88; M. Flato, P. Hillion, “Poincaré-like group associated with neutrino physics, and some applications”, Phys. Rev. D, 1 (1970), 1667–1673 | DOI | MR | Zbl

[40] M. Flato, C. Piron, J. Gréa, D. Sternheimer, J. P. Vigier, “Are Bell's inequalities concerning hidden variables really conclusive?”, Helv. Phys. Acta, 48:2 (1975), 219–225 | MR

[41] M. Flato, D. Sternheimer, “Topological Quantum Groups, star products and their relations”, St. Petersburg Mathematical Journal (Algebra and Analysis), 6 (1994), 242–251 | MR | Zbl

[42] A. Froelicher, A. Nijenhuis, “A theorem on stability of complex structures”, Proc. Nat. Acad. Sci USA, 43 (1957), 239–241 | DOI | MR | Zbl

[43] J. Fröhlich,, Atomism and quantization, , 2006 http://math.caltech.edu/SimonFest/ Abstracts/frohlich_abst.pdf

[44] C. Fronsdal, “Some ideas about quantization”, Rep. Math. Phys., 15 (1978), 111–145 ; F. Bayen, C. Fronsdal, “Quantization on the sphere”, J. Math. Phys., 22 (1981), 1345–1349 | DOI | MR | DOI | MR | Zbl

[45] C. Frønsdal, “Quantization on Curves. Appendix by M. Kontsevich”, Lett. Math. Phys., to be published | MR

[46] I. Gelfand, V. Retakh, M. Shubin, “Fedosov manifolds”, Adv. Math., 136 (1998), 104–140 | DOI | MR | Zbl

[47] M. Gerstenhaber “On the deformation of rings and algebras”, Ann. Math., 79 (1964), 59–103 ; II 84 (1966), 1–19 ; IV 99 (1974), 257–276 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[48] M. Gerstenhaber, “On the deformation of sheaves of rings”, Global Analysis, Papers in Honor of K. Kodaira, Univ. Tokyo Press, Tokyo, 1969, 149–157 | MR

[49] M. Gerstenhaber, S. D. Schack, “Algebraic cohomology and deformation theory”, Deformation Theory of Algebras and Structures and Applications, NATO ASI Ser. C, 247, eds. M. Hazewinkel, M. Gerstenhaber, Kluwer Acad. Publ., Dordrecht, 1988, 11–264 | MR

[50] J. W. v. Goethe,, Maximen und Reflexionen, http://www.wissen-im-netz.info/ literatur/goethe/maximen/1-16.htm#1005

[51] A. Groenewold, “On the principles of elementary quantum mechanics”, Physica, 12 (1946), 405–460 | DOI | MR | Zbl

[52] G. 't Hooft, “How does God play dice? (Pre-)determinism at the Planck scale”, Quantum [un]speakables (Vienna, 2000), Springer, Berlin, 2002, 307–316 ; ; Springer, Berlin, 2005 ; arXiv: /hep-th/0104219arXiv: /quant-ph/0212095 | MR | MR | Zbl

[53] E. Ínönü, E. P. Wigner, “On the contraction of groups and their representations”, Proc. Nat. Acad. Sci. USA, 39 (1953), 510–524 | DOI | MR | Zbl

[54] M. Jimbo, “A $q$-difference algebra of $\mathcal{U}(\mathfrak{g})$ and the Yang–Baxter equation”, Lett. Math. Phys., 10 (1985), 63–69 | DOI | MR | Zbl

[55] K. Kodaira, D. C. Spencer, “On deformations of complex analytic structures”, Ann. Math., 67 (1958), 328–466 | DOI | MR | Zbl

[56] M. Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys., 66 (2003), 157–216 ; ; “Operads and motives in deformation quantization”, Lett. Math. Phys., 48 (1999), 35–72 ; arXiv: /q-alg/9709040arXiv: /math.QA/9904055 | DOI | MR | Zbl | DOI | MR | Zbl

[57] M. Kontsevich, “Deformation quantization of algebraic varieties”, Lett. Math. Phys., 56 (2001), 271–294 ; arXiv: /math.AG/0106006 | DOI | MR | Zbl

[58] B. Kostant, “Quantization and unitary representations”, Lecture Notes in Math., 170, Springer-Verlag, Berlin, 1970, 87–208 ; “On the definition of quantization”, Géométrie symplectique et physique mathématique, Colloq. Int. CNRS, 237, Éds. CNRS, Paris, 1975, 187–210 ; “Graded manifolds, graded Lie theory, and prequantization”, Differential geometrical methods in mathematical physics, Lecture Notes in Math., 570, Springer, Berlin, 1977, 177–306 | MR | MR | MR

[59] V. Mathai, R. B. Melrose, I. M. Singer, “Fractional analytic index”, J. Diff. Geom., in press

[60] R. A. Minlos, “Felix Alexandrovich Berezin (A Brief Scientific Biography)”, Contemporary Mathematical Physics, F. A. Berezin Memorial Volume, American Mathematical Society Translations, Ser. 2, 175, eds. R. L. Dobrushin, R. A. Minlos, M. A. Shubin, A. M. Vershik, 1996, 1–13 ; Lett. Math. Phys., 74 (2005), 5–19 | MR | Zbl | DOI | MR | Zbl

[61] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Phil. Soc., 45 (1949), 99–124 | DOI | MR | Zbl

[62] F. Nadaud, “Generalized deformations, Koszul resolutions, Moyal Products”, Reviews Math. Phys., 10:5 (1998), 685–704 ; Thèse, Dijon (janvier 2000) | DOI | MR | Zbl

[63] R. Nest, B. Tsygan, “Algebraic index theorem”, Comm. Math. Phys., 172 (1995), 223–262 ; “Algebraic index theorem for families”, Adv. Math., 113 (1995), 151–205 ; “Formal deformations of symplectic manifolds with boundary”, J. Reine Angew. Math., 481 (1996), 27–54 | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl

[64] H. Omori, Y. Maeda, A. Yoshioka, “Weyl manifolds and deformation quantization”, Adv. Math., 85 (1991), 225–255 ; “Existence of a closed star product”, Lett. Math. Phys., 26 (1992), 285–294 | DOI | MR | DOI | MR | Zbl

[65] Pankaj Sharan, “Star-product representation of path integrals”, Phys. Rev. D, 20 (1979), 414–418 | DOI | MR

[66] M. Rieffel, “Matrix algebras converge to the sphere for quantum Gromov–Hausdorff distance”, Mem. Amer. Math. Soc., 168:796 (2004), 67–91 ; arXiv: /math.OA/0108005 | MR

[67] I. E. Segal, “A class of operator algebras which are determined by groups”, Duke Math. J., 18 (1951), 221–265 | DOI | MR | Zbl

[68] D. Sternheimer, “Deformation quantization: Twenty years after”, Particles, Fields, and Gravitation (Lodz 1998), ed. J. Rembieliński, AIP Press, New York, 1998, 107–145 ; ; “A personal view on Moshé Flato's scientific legacy”, Conférence Moshé Flato 1999, Math. Phys. Studies, 21, Kluwer Acad. Publ., Dordrecht, 2000, 31–41 arXiv: /math.QA/9809056 | MR | Zbl | MR

[69] D. Sternheimer, “Quantization is deformation”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemporary Mathematics, 391, eds. J. Fuchs et al., Amer. Math. Soc., 2005, 331–352 | MR | Zbl

[70] D. Sternheimer, “Quantization: deformation and/or functor?”, Lett. Math. Phys., 74 (2005), 293–309 | DOI | MR | Zbl

[71] L. N. Trefethen, M. Embree, Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, NJ, 2005 | MR | Zbl

[72] H. Weyl, The theory of groups and quantum mechanics, Dover, New-York, 1931; “Quantenmechanik und Gruppentheorie”, Z. Physik, 46 (1927), 1–46 | DOI | Zbl

[73] E. P. Wigner, “Quantum corrections for thermodynamic equilibrium”, Phys. Rev., 40 (1932), 749–759 | DOI

[74] N. M. J. Woodhouse, Geometric quantization. 2nd edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1992 | MR