Deformation quantization by Moyal star-product and Stratonovich chaos
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We make a deformation quantization by Moyal star-product on a space of functions endowed with the normalized Wick product and where Stratonovich chaos are well defined.
Keywords: Moyal product, Stratonovich chaos, white noise analysis.
Mots-clés : Connes algebra
@article{SIGMA_2012_8_a10,
     author = {R\'emi L\'eandre and Maurice Obame Nguema},
     title = {Deformation quantization by {Moyal} star-product and {Stratonovich} chaos},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a10/}
}
TY  - JOUR
AU  - Rémi Léandre
AU  - Maurice Obame Nguema
TI  - Deformation quantization by Moyal star-product and Stratonovich chaos
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a10/
LA  - en
ID  - SIGMA_2012_8_a10
ER  - 
%0 Journal Article
%A Rémi Léandre
%A Maurice Obame Nguema
%T Deformation quantization by Moyal star-product and Stratonovich chaos
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a10/
%G en
%F SIGMA_2012_8_a10
Rémi Léandre; Maurice Obame Nguema. Deformation quantization by Moyal star-product and Stratonovich chaos. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a10/

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

[2] Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., “Deformation theory and quantization. II. Physical applications”, Ann. Physics, 111 (1978), 111–151 | DOI | MR | Zbl

[3] Dito G., “Deformation quantization on a Hilbert space”, Noncommutative Geometry and Physics, World Sci. Publ., Hackensack, NJ, 2005, 139–157 ; arXiv: math.QA/0406583 | DOI | MR

[4] Dito J., “Star-products and nonstandard quantization for Klein–Gordon equation”, J. Math. Phys., 33 (1992), 791–801 | DOI | MR

[5] Dito J., “Star-product approach to quantum field theory: the free scalar field”, Lett. Math. Phys., 20 (1990), 125–134 | DOI | MR | Zbl

[6] Dito G., Léandre R., “Stochastic Moyal product on the Wiener space”, J. Math. Phys., 48 (2007), 023509, 8 pp. | DOI | MR | Zbl

[7] Dito G., Sternheimer D., “Deformation quantization: genesis, developments and metamorphoses”, Deformation Quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys., 1, 9–54 | MR | Zbl

[8] Dütsch M., Fredenhagen K., “Perturbative algebraic field theory, and deformation quantization”, Mathematical Physics in Mathematics and Physics (Siena, 2000), Fields Inst. Commun., 30, Amer. Math. Soc., Providence, RI, 2001, 151–160 | MR

[9] Gerstenhaber M., “On the deformation of rings and algebras”, Ann. of Math. (2), 79 (1964), 59–103 | DOI | MR | Zbl

[10] Getzler E., Cyclic cohomology and the path integral of Dirac operator, unpublished

[11] Hida T., Kuo H.H., Potthoff J., Streit L., White noise, Mathematics and its Applications, 253, Kluwer Academic Publishers Group, Dordrecht, 1993 | MR | Zbl

[12] Hu Y.Z., Meyer P.A., “Sur les intégrales multiples de Stratonovitch”, Séminaire de Probabilités, XXII, Lecture Notes in Math., 1321, Springer, Berlin, 1988, 72–81 | DOI | MR

[13] Léandre R., “Connes–Hida calculus in index theory”, in XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005, 493–497 | MR

[14] Léandre R., “Deformation quantization in infinite dimensional analysis”, Trends in Stochastic Analysis, London Math. Soc. Lecture Note Ser., 353, Cambridge Univ. Press, Cambridge, 2009, 303–325 | MR

[15] Léandre R., “Deformation quantization in white noise analysis”, SIGMA, 3 (2007), 027, 8 pp. ; arXiv: math.QA/0702624 | DOI | MR

[16] Léandre R., “Fedosov quantization in white noise analysis”, J. Nonlinear Math. Phys., 15:3 (2008), 251–263 | DOI | MR | Zbl

[17] Léandre R., “Wiener analysis and cyclic homology”, Stochastic Analysis and Mathematical Physics (SAMP/ANESTOC 2002), World Sci. Publ., River Edge, NJ, 2004, 115–127 | DOI | MR

[18] Maeda Y., “Deformation quantization and noncommutative differential geometry”, Sugaku Expositions, 16 (2003), 1–23 | MR

[19] Malliavin P., Stochastic analysis, Grundlehren der Mathematischen Wissenschaften, 313, Springer-Verlag, Berlin, 1997 | MR | Zbl

[20] Malliavin P., “Stochastic calculus of variation and hypoelliptic operators”, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Wiley, New York, 1978, 195–263 | MR

[21] Meyer P.A., Quantum probability for probabilists, Lecture Notes in Mathematics, 1538, Berlin, Springer-Verlag, 1993 | MR | Zbl

[22] Nualart D., The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995 | MR

[23] Obata N., White noise calculus and Fock space, Lecture Notes in Mathematics, 1577, Springer-Verlag, Berlin, 1994 | MR | Zbl

[24] Solé J.L., Utzet F., “Intégrale multiple de Stratonovich pour le processus de Poisson”, Séminaire de Probabilités, XXV, Lecture Notes in Math., 1485, Springer, Berlin, 1991, 270–283 | DOI | MR

[25] Üstünel A.S., An introduction to analysis on Wiener space, Lecture Notes in Mathematics, 1610, Springer-Verlag, Berlin, 1995 | MR

[26] Weinstein A., “Deformation quantization”, Astérisque, no. 227, Exp. No. 789, 5, 1995, 389–409 | MR | Zbl

[27] Witten E., “Noncommutative geometry and string field theory”, Nuclear Phys. B, 268 (1986), 253–294 | DOI | MR