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


SIGMA 5 (2009), 056, 31 pages      arXiv:0905.4491      https://doi.org/10.3842/SIGMA.2009.056
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras

Luigi Accardi a and Andreas Boukas b
a) Centro Vito Volterra, Università di Roma ''Tor Vergata'', Roma I-00133, Italy
b) Department of Mathematics, American College of Greece, Aghia Paraskevi, Athens 15342, Greece

Received November 20, 2008, in final form May 16, 2009; Published online May 27, 2009

Abstract
The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of *-representations of infinite dimensional *-Lie algebras, quantum probability, white noise and stochastic calculus and the theory of classical and quantum infinitely divisible processes.

Key words: quantum probability; quantum white noise; infinitely divisible process; quantum decomposition; Meixner classes; renormalization; infinite dimensional Lie algebra; central extension of a Lie algebra.

pdf (416 kb)   ps (247 kb)   tex (36 kb)

References

  1. Accardi L., Amosov G., Franz U., Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), 183-194.
  2. Accardi L., Arefeva I., Volovich I.V., Quadratic Fermi fields, unpublished manuscript, 2003.
  3. Accardi L., Boukas A., Central extensions of the Heisenberg algebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted, arXiv:0810.3365.
  4. Accardi L., Boukas A., Central extensions of white noise *-Lie algebras, Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted.
  5. Accardi L., Boukas A., Fock representation of the renormalized higher powers of white noise and the centerless Virasoro (or Witt)-Zamolodchikov-w *-Lie algebra, J. Phys. A: Math. Theor. 41 (2008), 304001, 12 pages, arXiv:0706.3397.
  6. Accardi L., Boukas A., Higher powers of q-deformed white noise, Methods Funct. Anal. Topology 12 (2006), 205-219.
  7. Accardi L., Boukas A., Itô calculus and quantum white noise calculus, Proceedings of ''The 2005 Abel Symposium, Stochastic Analysis and Applications'' (Symposium in Honor of Kiyosi Itô, on the occasion of his 90th birthday) (July 29 - August 4, 2005, Oslo, Norway), Editors F.E. Benth, G. Di Nunno, T. Lindstrom, B. Oksendal and T. Zhang, Springer, Berlin, 2007 Vol. 2, 7-51.
  8. Accardi L., Boukas A., Lie algebras associated with the renormalized higher powers of white noise, Commun. Stoch. Anal. 1 (2007), 57-69.
  9. Accardi L., Quantum probability: an introduction to some basic ideas and trends, in Stochastic models, II (Guanajuato, 2000), Editors D. Hernandez, J.A. Lopez-Mimbela and R. Quezada, Aportaciones Mat. Investig., Vol. 16, Soc. Mat. Mexicana, Mexico, 2001, 1-128.
  10. Accardi L., Boukas A., Recent advances in quantum white noise calculus, in Quantum Information and Computing, Quantum Probability and White Noise Analysis, Vol. 19, World Scientific, 2006, 18-27.
  11. Accardi L., Boukas A., Renormalized higher powers of white noise and the Virasoro-Zamolodchikov-w algebra, Rep. Math. Phys. 61 (2008), 1-11, hep-th/0610302.
  12. Accardi L., Boukas A., Renormalized higher powers of white noise (RHPWN) and conformal field theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 353-360, math-ph/0608047.
  13. Accardi L., Boukas A., Square of white noise unitary evolutions on boson Fock space, in Proceedings of the International Conference on Stochastic Analysis and Applications in Honor of Paul Kree (October 22-27, 2001, Hammamet, Tunisie), Editors S. Albeverio, A. Boutet de Monvel and H. Ouerdiane, Kluwer Acad. Publ., Dordrecht, 2004, 267-301.
  14. Accardi L., Boukas A., The emergence of the Virasoro and w Lie algebras through the renormalized higher powers of quantum white noise, Int. J. Math. Comput. Sci. 1 (2006), 315-342, math-ph/0607062.
  15. Accardi L., Boukas A., The semi-martingale property of the square of white noise integrators, in Stochastic Partial Differential Equations and Applications (Trento, 2002), Editors G. Da Prato and L. Tubaro, Lecture Notes in Pure and Appl. Math., Vol. 227, Dekker, New York, 2002, 1-19.
  16. Accardi L., Boukas A., Unitarity conditions for the renormalized square of white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 197-222.
  17. Accardi L., Boukas A., White noise calculus and stochastic calculus, in Proceedings of the International Conference ''Stochastic Analysis: Classical and Quantum, Perspectives of White Noise Theory'' (November 1-5, 2003, Meijo University, Nagoya), Editors T. Hida and K. Saito, World Sci. Publ., Hackensack, NJ, 2005, 260-300.
  18. Accardi L., Boukas A., Franz U., Renormalized powers of quantum white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 129-147.
  19. Accardi L., Boukas A., Kuo H.-H., On the unitarity of stochastic evolutions driven by the square of white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), 579-588.
  20. Accardi L., Franz U., Skeide M., Renormalized squares of white noise and other non-Gaussian noises as Lèvy processes on real Lie algebras, Comm. Math. Phys. 228 (2002), 123-150.
  21. Accardi L., Hida T., Kuo H.H., The Itô table of the square of white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), 267-275.
  22. Accardi L., Lu Y.G., Obata N., Towards a non-linear extension of stochastic calculus, in Quantum Stochastic Analysis and Related Fields (Kyoto, 1995), Editors N. Obata, Surikaisekikenkyusho Kokyuroku, no. 957, Research Institute for Mathematical Sciences, Kyoto, 1996, 1-15.
  23. Accardi L., Lu Y.G., Volovich I., Non-linear extensions of classical and quantum stochastic calculus and essentially infinite-dimensional analysis, in Probability Towards 2000 (October 2-6, 1995, Columbia University, New York), Editors L. Accardi and C. Heyde, Lecture Notes in Statist., Vol. 128, Springer, New York, 1998, 1-33.
  24. Accardi L., Lu Y.G., Volovich I., Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002.
  25. Accardi L., Lu Y.G., Volovich I., White noise approach to classical and quantum stochastic calculi, Lecture Notes of the Volterra International School, Trento, Italy, 1999.
  26. Accardi L., Pechen A., Roschin R., Quadratic KMS states on the RSWN algebra, unpublished manuscript, 2004.
  27. Accardi L., Roschin R., Renormalized squares of Boson fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), 307-326.
  28. Accardi L., Skeide M., On the relation of the square of white noise and the finite difference algebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 185-189.
  29. Accardi L., Volovich I.V., Quantum white noise with singular non-linear interaction, in Developments of Infinite-Dimensional Noncommutative Analysis (Kyoto, 1998), Editor N. Obata, Surikaisekikenkyusho Kokyuroku, no. 1099, Research Institute for Mathematical Sciences, Kyoto, 1999, 61-69.
  30. Arveson W., Noncommutative dynamics and E-semigroups, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
  31. Ayed W., White noise approach to quantum stochastic calculus, Thesis Doctor of Mathematics, University of Tunis El Manar, Tunis, 2006.
  32. Bakas I., The structure of the W algebra, Comm. Math. Phys. 134 (1990), 487-508.
  33. Bakas I., Kiritsis E.B., Structure and representations of the W algebra, Progr. Theoret. Phys. Suppl. (1991), no. 102, 15-37.
  34. Barhoumi A., Ouerdiane H., Riahi H., Representations of the Witt and square white noise algebras through the renormalized powers of negative binomial quantum white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), 223-250.
  35. Berezansky Yu.M., Lytvynov E., Mierzejewski D.A., The Jacobi field of a Lèvy process, Ukrain. Mat. Zh. 55 (2003), 706-710 (English transl.: Ukrainian Math. J. 55 (2003), 853-858).
  36. Bogoliubov N.N., Shirkov D.V., Introduction to the theory of quantized fields, John Wiley & Sons, 1980.
  37. Boukas A., An example of a quantum exponential process, Monatsh. Math. 112 (1991), 209-215.
  38. El Kinani E.H., Akhoumach K., Generalized Clifford algebras and certain infinite dimensional Lie algebras, Adv. Appl. Clifford Algebras 10 (2000), 1-6.
  39. Feinsilver P., Discrete analogues of the Heisenberg-Weyl algebra, Monatsh. Math. 104 (1987), 89-108.
  40. Feinsilver P.J., Schott R., Algebraic structures and operator calculus, Vols. I and III, Kluwer Academic Publishers Group, Dordrecht, 1993.
  41. Feinsilver P.J., Schott R., Differential relations and recurrence formulas for representations of Lie groups, Stud. Appl. Math. 96 (1996), 387-406.
  42. Fuchs J., Schweigert C., Symmetries, Lie algebras and representations. A graduate course for physicists, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997.
  43. Green M.B., Schwarz J.H., Witten E., Superstring theory, Vol. I, Cambridge University Press, Cambridge, 1988.
  44. Guichardet A., Symmetric Hilbert spaces and related topics. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes, Lecture Notes in Mathematics, Vol. 261, Springer-Verlag, Berlin - New York, 1972.
  45. Hida T., Selected papers of Takeyuki Hida, Editors L. Accardi, H.H. Kuo, N. Obata, K. Saitô, Si Si and L. Streit, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  46. Ivanov V.K., The algebra of elementary generalized functions, Dokl. Akad. Nauk SSSR 246 (1979), 805-808 (English transl.: Soviet Math. Dokl. 20 (1979), 553-556).
  47. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
  48. Kac V.G., Highest weight representations of infinite-dimensional Lie algebras, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 299-304.
  49. Kac V.G., Contravariant form for infinite-dimensional Lie algebras and superalgebras, in Proceedings of the Seventh International Colloquium and Integrative Conference on Group Theory and Mathematical Physics (September 11-16, 1978, University of Texas, Austin), Editors W. Beigelböck, A. Böhm and E. Takasugi, Springer Lecture Notes in Physics, Vol. 94, Springer-Verlag, Berlin - New York, 1979, 441-445.
  50. Ketov S.V., Conformal field theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  51. Kruchkovich G.I., Classification of three-dimensional Riemannian spaces according to groups of motions, Uspehi Matem. Nauk 9 (1954), no. 1, 3-40 (in Russian).
  52. Lytvynov E., The square of white noise as a Jacobi field, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), 619-630, math.PR/0401370.
  53. Meixner J., Orthogonale Polynomsysteme mit einen besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.
  54. Ovando G., Four dimensional symplectic Lie algebras, Beiträge Algebra Geom. 47 (2006), 419-434, math.DG/0407501.
  55. Parthasarathy K.R., Schmidt K., Positive definite kernels continuous tensor products and central limit theorems of probability theory, Springer Lecture Notes in Mathematics, Vol. 272, Springer-Verlag, Berlin - New York, 1972.
  56. Pope C.N., Lectures on W algebras and W gravity, Lectures given at the Trieste Summer School in High-Energy Physics, August 1991.
  57. Prokhorenko D.V., Squares of white noise, SL(2,C}) and Kubo-Martin--Schwinger states, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 491-511.
  58. Sniady P., Quadratic bosonic and free white noises, Comm. Math. Phys. 211 (2000), 615-628, math-ph/0303048.
  59. Wanglai L., Wilson R., Central extensions of some Lie algebras, Proc. Amer. Math. Soc. 126 (1998), 2569-2577.
  60. Zamolodchikov A.B., Infinite additional symmetries in two-dimensional conformal quantum field theory, Teoret. Mat. Fiz. 65 (1985), 347-359.


Previous article   Next article   Contents of Volume 5 (2009)