The deformation quantizations of the hyperbolic plane

[en] We describe the space of (all) invariant deformation quantizations on the hyperbolic plane D as solutions of the evolution of a second order hyperbolic di erential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a uni ed method producing every quantization of D, and provides, in the 2-dimensional context, an exact solution to Weinstein's WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or `dual') to the geometry of the hyperbolic plane through the quantization process.

Disciplines :

Physics

Author, co-author :

Bieliavsky, P.

Detournay, S.

Spindel, Philippe ; Université de Mons > Faculté des Sciences > Mécanique et gravitation

Language :

English

Title :

The deformation quantizations of the hyperbolic plane

Publication date :

01 January 2009

Journal title :

Communications in Mathematical Physics

ISSN :

0010-3616

Publisher :

Springer, Germany

Volume :

289

Issue :

Pages :

529-559.

Peer reviewed :

Peer Reviewed verified by ORBi

Commentary :

e-Print : arXiv: 0806.4741 [math-ph],ISNN 1432-0916 (Online),Bibliographic Code:2009CMaPh.289..529B

Available on ORBi UMONS :

since 10 June 2010

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Bibliography

F. Bayen M. Flato C. Fronsdal A. Lichnerowicz D. Sternheimer 1978 Deformation theory and quantization I and II Ann. Phys. 111 61 151
M. De Wilde P. Lecomte 1983 Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds Lett. Math. Phys. 7 6 487 496
Fedosov, B.V.: Formal quantization. In: Some problems in modern mathematics and their applications to problems in mathematical physics (Russian), Moscow: Moskov. Fiz.-Tekhn. Inst., 1985., pp. 129-136
H. Omori Y. Maeda A. Yoshioka 1991 Weyl manifolds and deformation quantization Adv. Math. 85 2 224 255
M. Bertelson M. Cahen S. Gutt 1997 Equivalence of star products. Geometry and physics Classical Quantum Gravity 14 1A A93 A107
R. Nest R. Tsygan 1995 Algebraic index theorem Commun. Math. Phys. 172 223 262
F. Hansen 1984 Quantum mechanics in phase space Rep. Math. Phys. 19 3 361 381
M. Bertelson P. Bieliavsky S. Gutt 1998 Parametrizing equivalence classes of invariant star products Lett. Math. Phys. 46 4 339 345 (Pubitemid 128504943)
N. Seiberg E. Witten 1999 String theory and noncommutative geometry JHEP 09 032
A. Connes M.R. Douglas A.S. Schwarz 1998 Noncommutative geometry and matrix theory: Compactification on tori JHEP 9802 003
V. Schomerus 1999 D-branes and deformation quantization JHEP 9906 030
A.Y. Alekseev A. Recknagel V. Schomerus 2000 Brane dynamics in background fluxes and non-commutative geometry JHEP 0005 010
A.Y. Alekseev A. Recknagel V. Schomerus 2001 Open strings and non-commutative geometry of branes on group manifolds Mod. Phys. Lett. A16 325 336
A.Y. Alekseev A. Recknagel V. Schomerus 1999 'Non-commutative world-volume geometries: Branes on SU(2) and fuzzy spheres JHEP 9909 023
V. Schomerus 2002 Lectures on branes in curved backgrounds Class. Quant. Grav. 19 5781 5847 (Pubitemid 37028711)
P. Bieliavsky C. Jego J. Troost 2007 Nucl. Phys. B782 94 133
A. Hashimoto K. Thomas 2006 Non-commutative gauge theory on d-branes in Melvin universes JHEP 0601 083
A. Hashimoto S. Sethi 2002 Holography and string dynamics in time-dependent backgrounds Phys. Rev. Lett. 89 261601
S. Halliday R.J. Szabo 2006 Noncommutative field theory on homogeneous gravitational waves J. Phys. A39 5189 5226
W. Behr A. Sykora 2004 Construction of gauge theories on curved noncommutative spacetime Nucl. Phys. B 698 473 502 (Pubitemid 39266114)
R.-G. Cai J.-X. Lu N. Ohta 2003 Ncos and d-branes in time-dependent backgrounds Phys. Lett. B551 178 186
R.-G. Cai N. Ohta 2006 Holography and d3-branes in Melvin universes Phys. Rev. D73 106009
R.-G. Cai N. Ohta 2000 On the thermodynamics of large N non-commutative super Yang-Mills theory Phys. Rev. D61 124012
S. Halliday R.J. Szabo 2005 Isometric embeddings and noncommutative branes in homogeneous gravitational waves Class. Quant. Grav. 22 1945 1990 (Pubitemid 40900832)
R.J. Szabo 2006 Symmetry, gravity and noncommutativity Class. Quant. Grav. 23 R199 R242
J.M. Maldacena H. Ooguri 2001 Strings in AdS(3) and SL(2,R) WZW model. I J. Math. Phys. 42 2929 2960
J. Teschner 1999 On structure constants and fusion rules in the SL(2,C)/SU(2) WZNW model Nucl. Phys. B546 390 422
C. Bachas M. Petropoulos 2001 JHEP 0102 025
P. Bieliavsky S. Detournay P. Spindel M. Rooman 2004 Star products on extended massive non-rotating BTZ black holes JHEP 06 031
P. Bieliavsky 2008 Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces J. Phys.: Conf. Ser. 103 012001
A. et J. Unterberger 1988 Quantification et analyse pseudo- différentielle Ann. Scient. Ec. Norm. Sup. 4E série t. 21 133 158
A. et J. Unterberger 1984 La série discrète de SL(2,R) et les opérateurs pseudo-différentiels sur une demi-droite Ann. Scient. Ec. Norm. Sup. 4E série t.17 83 116
A. Weinstein 1994 Traces and triangles in symmetric symplectic spaces Contemp. Math. 179 262 270
S. Klimek A. Lesniewski 1992 Quantum Riemann surfaces. I. The unit disc Commun. Math. Phys. 146 1 103 122
M. Cahen S. Gutt J. Rawnsley 1993 Quantization of Kähler manifolds. II Trans. Amer. Math. Soc. 337 1 73 98
S. Klimek A. Lesniewski 1992 Quantum Riemann surfaces. II. The discrete series Lett. Math. Phys. 24 2 125 139
E. Hawkins 2005 Quantization of multiply connected manifolds Commun. Math. Phys. 255 3 513 575 (Pubitemid 40526412)
T. Natsume R. Nest 1999 Topological approach to quantum surfaces Commun. Math. Phys. 202 1 65 87
A. Carey K. Hannabuss V. Mathai P. McCann 1998 Quantum Hall effect on the hyperbolic plane Commun. Math. Phys. 190 3 629 673 (Pubitemid 128360181)
Bieliavsky, P.: Espace symétriques symplectiques. PhD. thesis, ULB, 1995
Kirillov, A.A.: Elementy teorii predstavlenǐ. (Russian) [Elements of the theory of representations] Moscow: Izdat. "Nauka", 1972
Kostant, B.: Quantization and unitary representations. In: Lectures in modern analysis and applications, III, Lecture Notes in Math., vol. 170, Berlin: Springer, 1970, pp. 87-208
Helgason S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, Londen-NewYork (1978)
P. Bieliavsky 2002 Strict quantization of solvable symmetric spaces J. Sympl. Geom. 1 2 269 320
P. Bieliavsky Ph. Bonneau Y. Maeda 2007 Universal deformation formulae, symplectic Lie groups and symmetric spaces Pacific J. Math. 230 1 41 57
P. Bieliavsky Ph. Bonneau 2003 On the geometry of the characteristic class of a star product on a symplectic manifold Rev. Math. Phys. 15 2 199 215 (Pubitemid 36594182)
A. Connes M. Flato D. Sternheimer 1992 Closed star products and cyclic cohomology Lett. Math. Phys. 24 1 1 12
Watson G.N.: A treatise on the Theory of Bessel Function. Second edition. Cambridge Univ. Press, Cambridge (1966)
A. Connes H. Moscovici 2004 Rankin-Cohen brackets and the Hopf algebra of transverse geometry Mosc. Math. J. 4 1 111 130
Zagier, D.: Modular forms and differential operators. In: K. G. Ramanathan memorial issue, Proc. Indian Acad. Sci. Math. Sci. 104(1), 57-75 (1994)
P. Bieliavsky M. Massar 2001 Oscillatory integral formulae for left-invariant star products on a class of Lie groups Lett. Math. Phys. 58 115 128 (Pubitemid 33613273)