[en] In [Prog Theor Phys Suppl 49(3):173–196, 1999], Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [math.DG/0208171, Submitted], Bordemann proved this existence using the framework of Thomas–Whitehead connections. In [Lett Math Phys 72(3):183–196, 2005], we gave a new proof of the same theorem thanks to the Cartan connections. After these works, there was no explicit formula for the quantization. In this paper, we give this formula using the formula in terms of Cartan connections given in [Lett Math Phys 72(3):183–196, 2005]. This explicit formula constitutes the generalization to any order of the formulae at second and third orders soon published by Bouarroudj in [Lett Math Phys 51(4):265–274, 2000] and [C R Acad Sci Paris Sér I Math 333(4):343–346, 2001].
Disciplines :
Mathematics
Author, co-author :
Radoux, Fabian ; Université de Liège - ULiège > Département de mathématique > Géométrie et théorie des algorithmes
Language :
English
Title :
Explicit formula for the natural and projectively equivariant quantization
Publication date :
2006
Journal title :
Letters in Mathematical Physics
ISSN :
0377-9017
eISSN :
1573-0530
Publisher :
Springer Science & Business Media B.V., Dordrecht, Netherlands
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Bibliography
Bordemann, M.: Sur l'existence d'une prescription d'ordre naturelle projectivement invariante. Submitted for publication, math.DG/0208171
Bouarroudj, S.: Formula for the projectively invariant quantization on degree three. C. R. Acad. Sci. Paris Sér. I Math. 333(4), 343-346 (2001)
Čap, A., Slovák, J., Souček, V.: Invariant operators on manifolds with almost Hermitian symmetric structures. I. Invariant differentiation. Acta Math. Univ. Comenian. (N.S.) 66(1), 33-69 (1997)
Duval, C., Ovsienko, V.: Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions. Lett. Math. Phys. 57(1), 61-67 (2001)
Kobayashi, S.: Transformation groups in differential geometry. Springer, Berlin Heidelberg New York (1972) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70.
Lecomte, P.B.A., Ovsienko, V.Yu.: Projectively equivariant symbol calculus. Lett. Math. Phys. 49(3), 173-196 (1999)
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