Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 066, 13 pages      arXiv:0809.4790      https://doi.org/10.3842/SIGMA.2008.066
Contribution to the Special Issue on Deformation Quantization

Hochschild Cohomology Theories in White Noise Analysis

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

Received June 18, 2008, in final form September 08, 2008; Published online September 27, 2008

Abstract
We show that the continuous Hochschild cohomology and the differential Hochschild cohomology of the Hida test algebra endowed with the normalized Wick product are the same.

Key words: white noise analysis; Hochschild cohomology.

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References

  1. Albeverio S., Høegh-Krohn R., Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), 1-57.
  2. 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.
  3. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  4. Berezansky Y.M., Kondratiev Y.G., Spectral methods in infinite-dimensional analysis, Vols. I, II, Kluwer Academic Publishers, Dordrecht, 1995.
  5. Berezin F.A., The method of second quantization, Academic Press, New York, 1966.
  6. Berezin F.A., Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86(128) (1971), 578-610.
  7. Chung D.M., Ji U.C., Obata N., Higher powers of quantum white noises in terms of integral kernel operators, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 533-559.
  8. Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. (1985), no. 62, 257-360.
  9. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Springer-Verlag, New York, 1997.
  10. Dito G., Star-product approach to quantum field theory: the free scalar field, Lett. Math. Phys. 20 (1990), 125-134.
  11. Dito G., Star-products and nonstandard quantization for Klein-Gordon equation, J. Math. Phys. 33 (1992), 791-801.
  12. Dito G., Deformation quantization on a Hilbert space, in Noncommutative Geometry and Physics (Yokohama, 2004), Editors Y. Maeda and et al., World Sci. Publ., Hackensack, NJ, 2005, 139-157, math.QA/0406583.
  13. Dito G., Léandre R., Stochastic Moyal product on the Wiener space, J. Math. Phys. 48 (2007), 023509, 8 pages.
  14. Dütsch M., Fredenhagen K., Perturbative algebraic field theory and deformation quantization, in Mathematical Physics in Mathematics and Physics (Siena, 2000), Fields Inst. Commun., Vol. 30, Amer. Math. Soc., Providence, RI, 2001, 151-160, hep-th/0101079.
  15. Gerstenhaber M., Schack S.D., Algebraic cohomology and deformation theory, in Deformation Theory of Algebras and Structures and Application (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 247, Editors M. Hazewinkel and M. Gerstenhaber, Kluwer Acad. Publ., Dordrecht, 1988, 11-264.
  16. Glimm J., Jaffe A., Boson quantum fields, in Collected Papers. I, Birkhäuser, Basel, 1985, 125-199.
  17. Haag R., On quantum field theory, Danske Vid. Selsk. Mat.-Fys. Medd. 29 (1955), no. 12, 37 pages.
  18. Hida T., Analysis of Brownian functionals, Carleton Mathematical Lecture Notes, no. 13, Carleton Univ., Ottawa, Ont., 1975.
  19. Hida T., Kuo H.H., Potthoff J., Streit L., White noise: an infinite-dimensional calculus, Kluwer Academic Publishers, Dordrecht, 1993.
  20. Ji U.C., Obata N., A unified characterization theorem in white noise analysis, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 167-178.
  21. Ji U.C., Obata N., Quantum white noise calculus, in Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads, Quantum Probab. White Noise Anal., Vol. 16, Editors N. Obata, T. Matsui and A. Hora, World Sci. Publ., River Edge, NJ, 2002, 143-191.
  22. Léandre R., Wiener analysis and cyclic cohomology, in Stochastic Analysis and mathematical physics (Santiago, 2001), Editors R. Rebolledo and J.C. Zambrini, World Sci. Publ., River Edge, NJ, 2004, 115-127.
  23. Léandre R., Deformation quantization in white noise analysis, in Geometric Aspects of Integrable Systems (Coimbra, 2006), Editor J.P. Françoise et al., SIGMA 3 (2007), 027, 8 pages, math.QA/0702624.
  24. Léandre R., Fedosov quantization in white noise analysis, in NEEDS 2007 (Almetla del Mar, 2007), Editors J. Puig et al., J. Nonlinear Math. Phys., to appear.
  25. Léandre R., Deformation quantization in infinite-dimensional analysis, in Festchrift in Honour of H.V. Weizsaecker (Kaiserslautern, 2007), Editors M. Scheutzow et al., to appear.
  26. Léandre R., Rogers A., Equivariant cohomology, Fock space and loop groups, J. Phys. A: Math. Gen. 39 (2006), 11929-11946.
  27. Maassen H., Quantum Markov processes on Fock space described by integral kernels, in Quantum Probability and Applications, II (Heidelberg, 1984), Lecture Notes in Math., Vol. 1136, Springer, Berlin, 1985, 361-374.
  28. Malliavin P., Stochastic calculus of variations and hypoelliptic operators, in Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Editor K. Itô, Wiley, New York - Chichester - Brisbane, 1978, 195-263.
  29. Meyer P.-A., Elements de probabilites quantiques. I-V, in Séminaire de Probabilites, XX, 1984/85, Lecture Notes in Math., Vol. 1204, Editors J. Azéma and M. Yor, Springer, Berlin, 1986, 186-312.
  30. Meyer P.-A., Distributions, noyaux, symboles d'apres Krée, in Seminaire de Probabilites, XXII, Lecture Notes in Math., Vol. 1321, Editors J. Azéma and M. Yor, Springer, Berlin, 1988, 467-476.
  31. Meyer P.-A., Quantum probability for probabilists, Lecture Notes in Mathematics, Vol. 1538, Springer-Verlag, Berlin, 1993.
  32. Nadaud F., On continuous and differential Hochschild cohomology, Lett. Math. Phys. 47 (1999), 85-95.
  33. Nadaud F., Déformations et déformations généralisées, These, Université de Bourgogne, 2000.
  34. Obata N., An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), 421-445.
  35. Obata N., White noise analysis and Fock space, Lecture Notes in Mathematics, Vol. 1577, Springer-Verlag, Berlin, 1994.
  36. Pflaum M., On continuous Hochschild homology and cohomology groups, Lect. Math. Phys. 44 (1998), 43-51.
  37. Pinczon G., On the equivalence between continuous and differential deformation theories, Lett. Math. Phys. 89 (1997), 143-156.
  38. Pinczon G., Ushirobira R., Supertrace and superquadratic Lie structure on the Weyl algebra, and applications to formal inverse Weyl transform, Lett. Math. Phys. 74 (2005), 263-291, math.RT/0507092.
  39. Streit L., An introduction to white noise analysis, in Stochastic Analysis and Applications in Physics (Funchal, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 449, Editors A.I. Cardoso et al., Kluwer Acad. Publ., Dordrecht, 1994, 415-439.
  40. Witten E., Noncommutative geometry and string field theory, Nuclear Phys. B 268 (1986), 253-294.

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