q-Gaussian processes: Non-commutative and classical aspects

Communications in Mathematical Physics

Volumn 185, Issue 1, 1997, Pages 129-154

  Bozejko, Marek ^a   Kümmerer, Burkhard ^b   Speicher, Roland ^c  

^a UNIVERSITY OF WROC AW   (Poland)

^b UNIVERSITY OF STUTTGART   (Germany)

^c UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001392502     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050084     Document Type: Article

References (57)

1. Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic processes. Publ. RIMS 18, 97-133 (1982)
2. Bhat, B.V.R., Parthasarathy, K.R.: Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory. Ann. Inst. Henri Poincaré 31, 601-651 (1995)
3. Biane, Ph.: On processes with free increments. Mfath. Z. To appear
4. Biane, Ph.: Free Brownian motion, free stochastic calculus and random matrices. In: Free probability theory (Fields Institute Communications 12), ed. D. Voiculescu, Providence: AMS, 1997, pp. 1-19
5. Biane, Ph.: Quantum Markov processes and group representations. Preprint, 1995
6. Bożejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137, 519-531 (1991)
7. Bożejko, M., Speicher, R.: An example of a generalized Brownian motion II. Quantum Probability and Related Topics VII, ed. L. Accardi. Singapore: World Scientific, 1992, pp. 219-236
8. Bożejko, M., Speicher. R.: Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces. Math. Ann. 300, 97-120 (1994)
9. Bożejko, M., Speicher, R.: Interpolations between bosonic and fermionic relations given by generalized Brownian motions. Math. Z. 222, 135-160 (1996)
10. Bressoud, D.M.: A simple proof of Mehler's formula for q-Hermite polynomials. Indiana Univ. Math. J. 29, 577-580 (1980)
11. Carlen, E.A., Lieb, E.H.: Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. Commun. Math. Phys. 155, 27-46 (1993)
12. Cuntz, J. : Simple C*-algebras generated by isometrics. Commun. Math. Phys 57, 173-185 (1977)
13. Davies, E.B.: Heat kernels and spectral theory, Cambridge: Cambridge University Press, 1989
14. Dellacherie, D., Meyer, P.A.: Probabilités et potentiel, Paris: Hermann, 1975
15. Dykema, K., Nica, A.: On the Fock representation of the q-commutation relations. J. reine angew. Math. 440, 201-212 (1993)
16. n. Publ. RIMS 16, 915-927 (1980)
17. Fivel, D.I.: Interpolation between Fermi and Bose statistics using generalized commutators. Phys. Rev. Lett. 65, 3361-3364 (1990), Erratum 69, 2020 (1992)
18. Frisch, U., Bourret, R.: Parastochastics. J. Math. Phys. 11, 364-390 (1970)
19. Gasper, G., Rahman, M.: Basic hypergeometric functions, Cambridge: Cambridge U.P., 1990
20. Greenberg, O.W.: Particles with small violations of Fermi or Bose statistics. Phys. Rev. D 43, 4111-4120 (1991)
21. Gross, L.: Existence and uniqueness of physical ground states. J. Funct. Anal. 10, 52-109 (1972)
22. Halmos, P.R.: Normal dilations and extensions of operators. Summa Brasiliensis Math. 2, 125-134 (1950)
23. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolution. Commun. Math. Phys. 93, 301-323 (1984)
24. Ismail, E.M., Stanton, D., Viennot, G.: The combinatorics of q-Hermite polynomials and Askey-Wilson integral. Europ. J. Comb. 8, 379-392 (1987)
25. Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: q-canonical commutation relations and stability of the Cuntz algebra. Pac. J. Math. 165, 131-151 (1994)
26. Jørgensen, P.E.T., Schmitt, L.M., Werner, R.F.: Positive representations of general commutation relations allowing Wick ordering. J. Funct. Anal. 134, 3-99 (1995)
27. Jørgensen, P.E.T., Werner, R.F.: Coherent states on the q-canonical commutation relations. Commun. Math. Phys. 164, 455-471 (1994)
28. Kümmerer, B.: Markov dilations on W*-algebras. J. Funct. Anal. 63, 139-177 (1985)
29. Kümmerer, B.: Survey on a theory of non-commutative stationary Markov processes. Quantum Probability and Applications III, L. Accardi, W.v. Waldenfels, ed., Berlin: Springer-Verlag, 1988, pp. 228-244
30. Kümmerer, B., Maassen, H.: Elements of Quantum Probability. Quantum Probability Communications X. To appear
31. ∞. J. Funcl. Anal. 103, 372-408 (1992)
32. van Leeuwen, H., Maassen, H.: A q-deformation of the Gauss distribution. J. Math. Phys. 36, 4743-4756 (1995)
33. van Leeuwen, H., Maassen, H.: An obstruction for q-deformation of the convolution product. J. Phys. A 29, 1-8 (1996)
34. Meyer, P.A.: Quantum probability for probabilists, Lecture Notes in Mathematics 1538, Heidelberg: Springer-Verlag, 1993
35. Møller, J.S.: Second quantization in a quon-algebra. J. Phys. A 26, 4643-4652 (1993)
36. Nelson, E.: Construction of quantum fields from Markoff fields. J. Funct. Anal. 12, 97-112 (1973)
37. Nelson, E.: The free Markoff field. J. Funct. Anal. 12, 211-227 (1973)
38. Neu, P., Speicher, R.: Spectra of Hamiltonians with generalized single-site dynamical disorder. Z. Phys. B 95, 101-111 (1994)
39. Parthasarathy, K.R.: Azéma martingales and quantum stochastic calculus. Proc. R. C. Bose Symposium, Wiley Eastern, 1990, pp. 551-569
40. Parthasarathy, K.R.: An introduction to quantum stochastic calculus. Monographs in Mathematics Vol. 85, Basel: Birkhäuser, 1992
41. Rogers, L.J.: On a three-fold symmetry in the elements of Heine series. Proc. London Math. Soc. 24, 171-179 (1893)
42. Schürmann, M.: Quantum q-white noise and a q-central limit theorem. Commun. Math. Phys. 140, 589-615 (1991)
43. Segal, I.E.: Tensor algebras over Hubert spaces. Trans. Amer. Math. Soc. 81, 106-134 (1956)
44. 2 Euclidean (Quantum) Field Theory. Princeton, NJ: Princeton University Press, 1974
45. Simon, B.: Functional Integration and Quantum Physics. New York: Academic Press, 1979
46. Speicher, R.: Generalized statistics of macroscopic fields. Lett. Math. Phys. 27, 97-104 (1993)
47. Speicher, R.: On universal products. In: "Free probability theory" (Fields Institute Communications 12), ed. D. Voiculescu. Providence: AMS, 1997, pp. 257-266
48. Stanciu, S.: The energy operator for infinite statistics. Commun. Math. Phys. 147, 211-216 (1992)
49. Szego, G.: Ein Beitrag zur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. L1 19, 242-252 (1926)
50. Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal. 9, 306-321 (1972)
51. Voiculescu, D.: Symmetries of some reduced free product C*-algebras. In: Operator Algebras and their Connection with Topology and Ergodic Theory, Lecture Notes in Mathematics Vol. 1132 Heidelberg: Springer-Verlag, 1985, pp. 556-588
52. Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. Providence, RI: AMS, 1992
53. Waldenfels, W. von: Fast positive Operatoren. Z. Wahrscheinlichkeitstheorie verw. Geb. 4, 159-174 (1965)
54. Werner, R.F.: The free quon gas suffers Gibb's paradox. Phys. Rev. D 48, 2929-2934 (1993)
55. Wilde, I.F.: The free Fermi field as a Markov field. J. Funct. Anal. 15, 12-21 (1974)
56. Yu, T., Wu, Z.-Y.: Construction of the Fock-like space for quons. Science in China (Series A) 37, 1472-1483 (1994)
57. Zagier, D.: Realizability of a model in infinite statistics. Commun. Math. Phys. 147, 199-210 (1992)