SIGMA 3 (2007), 027, 8 pages      math.QA/0702624      https://doi.org/10.3842/SIGMA.2007.027
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

Deformation Quantization in White Noise Analysis

Rémi Léandre
Institut de Mathématiques, Université de Bourgogne, 21000 Dijon, France

Received August 02, 2006, in final form January 30, 2007; Published online February 21, 2007

Abstract
We define and present an example of a deformation quantization product on a Hida space of test functions endowed with a Wick product.

Key words: Moyal product; white noise analysis.

pdf (200 kb)   ps (150 kb)   tex (11 kb)

References

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.
2. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
3. Berezanskii Yu., Kondratiev Yu., Spectral methods in infinite-dimensional analysis, Vols. 1, 2, Kluwer, Dordrecht, 1995.
4. Dito J., Star-product approach to quantum field theory: the free scalar field, Lett. Math. Phys. 20 (1990), 125-134.
5. Dito J., Star-products and nonstandard quantization for Klein-Gordon equation, J. Math. Phys. 33 (1992), 791-801.
6. Dito J., Deformation quantization on a Hilbert space, in Noncommutative Geometry and Physics, Editors Y. Maeda et al., World Scientific, Singapore, 2005, 139-157, math.QA/0406583.
7. Dito G., Léandre R., A stochastic Moyal product on the Wiener space, J. Math. Phys., to appear.
8. Dito J., Sternheimer D., Deformation quantization: genesis, developments and metamorphoses, in Deformation Quantization, Editor G. Halbout, IRMA Lectures Maths. Theor. Phys., Walter de Gruyter, Berlin, 2002, 9-54, math.QA/0201168.
9. Duetsch M., Fredenhagen K., Perturbative algebraic field theory and deformation quantization, Fields Inst. Commun. 30 (2001), 151-160, hep-th/0101079.
10. Garding L., Wightman A., Representations of the commutation relations, Proc. Natl. Acad. Sci. USA 40 (1954), 622-626.
11. Hida T., Analysis of Brownian functionals, Carleton. Maths. Lect. Notes, Vol. 13, Ottawa, 1975.
12. Hida T., Kuo H.H., Potthoff J., Streit L., White noise: an infinite dimensional calculus, Kluwer, Dordrecht, 1993.
13. Huang Z., Luo S., Quantum white noises and free fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998) 68-82.
14. Huang Z., Rang G., White noise approach to Φ₄⁴ quantum fields, Acta. Appl. Math. 77 (2003), 299-318.
15. Ikeda N., Watanabe S., Stochastic differential equations and diffusion processes, 2nd ed., North-Holland, Amsterdam, 1989.
16. Léandre R., Rogers A., Equivariant cohomology, Fock space and loop groups, J. Phys. A: Math. Gen. 39 (2006), 11929-11946.
17. Maeda Y., Deformation quantization and non commutative differential geometry, Sugaku Expositions 16 (1991), 224-255.
18. Malliavin P., Stochastic calculus of variations and hypoelliptic operators, in Stochastic Analysis, Editor K. Itô, Kinokuyina, Tokyo, 1978, 155-263.
19. Malliavin P., Stochastic analysis, Springer, Berlin, 1997.
20. Nualart D., Malliavin calculus and related topics, Springer, Berlin, 1995.
21. Obata N., White noise analysis and Fock space, Lect. Notes. Math., Vol. 1577, Springer, Berlin, 1994.
22. Ustunel A.S., An introduction to analysis on Wiener space, Lect. Notes. Math., Vol. 1610, Springer, Berlin, 1995.
23. Weinstein A., Deformation quantization. Séminaire Bourbaki, Astérisque 227. S.M.F., Paris (1994) 389-409.
24. Witten E., Noncommutative geometry and string field theory, Nuclear Phys. B. 268 (1986), 253-294.