Kolmogorov systems and Anosov systems in quantum theory

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Volumn 4, Issue 1, 2001, Pages 85-119

  Narnhofer, H ^a  

^a UNIVERSITY OF VIENNA   (Austria)

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EID: 0347081592     PISSN: 02190257     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219025701000401     Document Type: Article

References (79)

1. P. Walters, An Introduction to Ergodic Theory (Springer, 1982); I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory (Springer, 1982).
2. P. Walters, An Introduction to Ergodic Theory (Springer, 1982); I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory (Springer, 1982).
3. K. Schmidt, Dynamical Systems of Algebraic Origin (Birkhäuser, 1995).
4. G. Gallavotti, Statistical Mechanics, A Short Treatise (Springer, 1999).
5. G. Landi and R. J. Szabo, "From large N matrices to the noncommutative torus", hep-th/9912130.
6. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II (Springer, 1981).
7. F. Benatti, Deterministic Chaos in Infinite Quantum Systems, SISSA Notes in Physics (Springer, 1993).
8. A. N. Kolmogorov, A new metric invariant of transitive systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk 119 (1958) 861-864.
9. A. Connes, H. Narnhofer and W. Thirring, Dynamical entropy of C*-algebras and von Neumann algebras, Commun. Math. Phys. 112 (1987) 691-719.
10. H. Narnhofer, E. Størmer and W. Thirring, C*-dynamical systems for which the tensor product formula for entropy fails, Ergodic Theory and Dynamical Systems 15 (1995) 961-968.
11. H. Narnhofer and W. Thirring, Lett. Math. Phys. 135 (1995) 145-154.
12. R. Alicki and H. Fannes, Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994) 75-82.
13. D. V. Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras 2, Commun. Math. Phys. 170 (1995) 249-282.
14. H. v. Beijeren, J. R. Dorfmann, H. A. Posch and Ch. Dellago, Kolmogorov-Sinai entropy for dilute gases in equilbrium, Phys. Rev. E56/5 (1997) 5272-5277.
15. D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95 (1999) 393-468.
16. G. G. Emch, H. Narnhofer, G. L. Sewell and W. Thirring, Anosov actions on non-commutative algebras, J. Math. Phys. 35/11 (1994) 5582-5599.
17. H. J. Borchers, On modular inclusion and spectrum condition, Lett. Math. Phys. 27 (1993) 311-324; 184 (1997) 683-685; "On the revolutionization of quantum field theory by Tomita's modular theory", preprint ESI, 160 pages, 148 references.
18. H. J. Borchers, On modular inclusion and spectrum condition, Lett. Math. Phys. 27 (1993) 311-324; 184 (1997) 683-685; "On the revolutionization of quantum field theory by Tomita's modular theory", preprint ESI, 160 pages, 148 references.
19. H. J. Borchers, On modular inclusion and spectrum condition, Lett. Math. Phys. 27 (1993) 311-324; 184 (1997) 683-685; "On the revolutionization of quantum field theory by Tomita's modular theory", preprint ESI, 160 pages, 148 references.
20. H. J. Borchers, On modular inclusion and spectrum condition, Lett. Math. Phys. 27 (1993) 311-324; 184 (1997) 683-685; "On the revolutionization of quantum field theory by Tomita's modular theory", preprint ESI, 160 pages, 148 references.
21. H. W. Wiesbrock, Half-sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 157 (1993) 83-92; 184 (1997) 683-685; Symmetries and modular intersections of von Neumann algebras, Lett. Math. Phys. 39 (1997) 203-212; Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys. 193 (1998) 269-285.
22. H. W. Wiesbrock, Half-sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 157 (1993) 83-92; 184 (1997) 683-685; Symmetries and modular intersections of von Neumann algebras, Lett. Math. Phys. 39 (1997) 203-212; Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys. 193 (1998) 269-285.
23. H. W. Wiesbrock, Half-sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 157 (1993) 83-92; 184 (1997) 683-685; Symmetries and modular intersections of von Neumann algebras, Lett. Math. Phys. 39 (1997) 203-212; Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys. 193 (1998) 269-285.
24. H. W. Wiesbrock, Half-sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 157 (1993) 83-92; 184 (1997) 683-685; Symmetries and modular intersections of von Neumann algebras, Lett. Math. Phys. 39 (1997) 203-212; Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys. 193 (1998) 269-285.
25. H. Narnhofer and W. Thirring, Equivalence of modular K and Anosov dynamical systems, Rev. Math. Phys. 12 (2000) 445-459.
26. H. Narnhofer and W. Thirring, Quantum K-systems, Commun. Math. Phys. 125 (1989) 565-577.
27. H. Narnhofer and W. Thirring, Algebraic K-systems, Lett. Math. Phys. 20 (1990) 231-250.
28. H. Narnhofer and W. Thirring, Clustering for algebraic K-systems, Lett. Math. Phys. 30 (1994) 307-316.
29. H. Narnhofer and W. Thirring, Ergodic properties of quantum fields, in Mathematical Physics Towards the 21st Century, eds. R. N. Sea and A. Gersten, BeerSheva, Israel (1994), pp. 16-32.
30. B. Kümmerer, Survey on a theory of noncommutative stationary Markov processes, in Quantum Probability and Applications III, eds. L. Accardi and W. v. Waldenfeld (Springer, 1988), pp. 228-244.
31. L. Accardi, A. Frigerio and J. T. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto 18 (1982) 97-133.
32. 1 factors, ibid 39 (1987) 492-511.
33. 1 factors, ibid 39 (1987) 492-511.
34. V. Ya. Golodets and E. Størmer, Entropy of C*-dynamical systems defined by bitstreams, Ergodic Theory Dynamical Systems 18 (1998) 1-16.
35. V. Ya. Golodets and S. V. Neshveyev, Non Bernoullian quantum K-systems, Commun. Math. Phys. 195 (1998) 213-232.
36. V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1-25.
37. R. Longo, Simple injective subfactors, Adv. Math. 63 (1987) 152-171; Index of subfactors and statistics of quantum fields, Commun. Math. Phys. 130 (1990) 285-309.
38. R. Longo, Simple injective subfactors, Adv. Math. 63 (1987) 152-171; Index of subfactors and statistics of quantum fields, Commun. Math. Phys. 130 (1990) 285-309.
39. H. Kosaki, Extension of Jones theory on index to arbitrary factors, J. Funct. Anal. 66 (1986) 123-140.
40. M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986) 57-106; Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988) 127-133.
41. M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986) 57-106; Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988) 127-133.
42. S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995) 427-445.
43. H. Narnhofer and W. Thirring, Realizations of two-sided quantum K-systems, ESI 717, Rep. Math. Phys. 45 (2000) 239-256.
44. D. Shlyakhtenko, Free quasifree states, Pac. J. Math. 177 (1997) 329-368.
45. V. I. Arnold and A. Avez, Problèmes Ergodique de la Mecanique Classique (Gauthier-Villars, 1967).
46. G. G. Emch and I. Peter, Quantum Anosov flows: A new family of examples, J. Math. Phys. 39/9 (1998) 4513-4539.
47. H. Reeh and S. Schlieder, Eine Bemerkung zur Unitäräquivalenz von Lorentz-invarianten Feldern, Nuovo Cimemto 22 (1961) 1051-1068.
48. H. Narnhofer, "Time reversibility for modular K-systems", Rep. Math. Phys. 45 (2000) 107-120.
49. J. Bisognano and E. H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys. 16 (1975) 985-1007; On the duality condition for quantum fields, ibid 17 (1976) 303-312.
50. J. Bisognano and E. H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys. 16 (1975) 985-1007; On the duality condition for quantum fields, ibid 17 (1976) 303-312.
51. G. G. Emch, Generalized K-flows, Commun. Math. Phys. 49 (1976) 191-215.
52. R. Haag, Local Quantum Physics (Springer, 1992).
53. P. D. Hislop and R. Longo, Modular structure of the local algebra associated with a free massless scalar field theory, Commun. Math. Phys. 84 (1982) 71-85.
54. D. Buchholz and K. Fredenhagen, Dilatations and interactions, J. Math. Phys. 18 (1977) 1107-1111.
55. H. J. Borchers, "On thermal states of (1+1)-dimensional quantum systems", preprint (1999).
56. H. Narnhofer, A. Pflug and W. Thirring, Mixing and entropy increase in quantum systems, Scuola Normale Sup. Pisa (1989), pp. 597-626.
57. D. Buchholz and E. Wichmann, Causal independence and the energy-level density of states in local quantum field theory, Commun. Math. Phys. 106 (1986) 321-344.
58. C. Jäkel, "The Reeh-Schlieder property for thermal fields", hep-th/9904049; P. Junglas, Thesis, Hamburg (1989).
59. C. Jäkel, "The Reeh-Schlieder property for thermal fields", hep-th/9904049; P. Junglas, Thesis, Hamburg (1989).
60. H. J. Borchers and J. Yngvason, Modular groups of quantum fields in thermal states, J. Math. Phys. 40 (1999) 601-624
61. D. Buchholz and S. J. Summers, An algebraic characterization of vacuum states in Minkowski space, Commun. Math. Phys. 155 (1993) 449-458.
62. D. Buchholz, O. Dreyer, M. Florig and S. Summers, "Geometric modular action and spacetime symmetry groups", preprint (1998).
63. J. Bros, H. Epstein and U. Moschella, Analyticity properties and thermal effects for general quantum field theory on de Sitter space, Commun. Math. Phys. 196 (1998) 535-570.
64. R. Figari, R. Hoegh-Krohn and C. R. Nappi, Interacting relativistic boson fields in the de Sitter universe with two space-time-dimensions, Commun. Math. Phys. 44 (1975) 265-278.
65. H. J. Borchers and D. Buchholz, Global properties of vacuum states in de Sitter space, Ann. Inst. H. Poincaré 70 (1999) 23-40.
66. H. Narnhofer, I. Peter and W. Thirring, How hot is the de Sitter space, Internat. J. Mod. Phys. B10 (1996) 1507-1520.
67. H. Narnhofer and W. Thirring, in preparation.
68. R. Haag, H. Narnhofer and U. Stein, On quantum field theory in gravitational background, Commun. Math. Phys. 94 (1984) 219-238.
69. M. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996) 529-553.
70. M. Bozejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Commun. Math. Phys. 185 (1997) 129-154.
71. M. Choda, Entropy on Cuntz's canonical endomorphism, Pac. J. Math. 190 (1999) 235-245.
72. F. Acerbi, G. Morchio and F. Strocchi, Algebraic fermion bosonization, Lett. Math. Phys. 26 (1992) 13-22; Thetra vacua, charge confinement and charged sectors from nonregular representations of CCR algebras, Lett. Math. Phys. 27 (1993) 1-12.
73. F. Acerbi, G. Morchio and F. Strocchi, Algebraic fermion bosonization, Lett. Math. Phys. 26 (1992) 13-22; Thetra vacua, charge confinement and charged sectors from nonregular representations of CCR algebras, Lett. Math. Phys. 27 (1993) 1-12.
74. N. Ilieva and H. Narnhofer, A Fermi field algebra as crossed product, Osterr. Akad. Wiss., Sitzungsber. Ab. II (1996) 13-33.
75. N. Ilieva and W. Thirring, Anyons and the Bose-Fermi duality in the finite temperature Thirring model, Theor. Math. Phys. 121 (1999) 40-65 (in Russian); ESI 720.
76. N. Ilieva and W. Thirring, Anyons and the Bose-Fermi duality in the finite temperature Thirring model, Theor. Math. Phys. 121 (1999) 40-65 (in Russian); ESI 720.
77. N. Ilieva, H. Narnhofer and W. Thirring, "Thermal correlators of anyons in two dimensions", math-ph/0004006.
78. J. M. Luttinger, An exactly soluble model of a many fermion system, J. Math. Phys. 4 (1963) 1154-1162.
79. D. C. Mattis and E. Lieb, Exact solution of a many fermion system and its associated boson field, J. Math. Phys. 6 (1965) 304-312.