About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2011 (2011), Article ID 625978, 16 pages
http://dx.doi.org/10.1155/2011/625978
Research Article

Quantum Dynamical Semigroups and Decoherence

1Faculty of Physics, University of Bielefeld, Universitätsstraβe 25, 33615 Bielefeld, Germany
2Bundesamt für Strahlenschutz (Federal Office for Radiation Protection), Willy-Brandt-Straße 5, 38226 Salzgitter, Germany

Received 22 June 2011; Accepted 31 August 2011

Academic Editor: Christian Maes

Copyright © 2011 Mario Hellmich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Haag, Local Quantum Physics, Springer, Berlin, Germany, 2nd edition, 1996.
  2. F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific, Hackensack, NJ, USA, 2005.
  3. P. Blanchard and R. Olkiewicz, “Decoherence induced transition from quantum to classical dynamics,” Reviews in Mathematical Physics, vol. 15, no. 3, pp. 217–243, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Olkiewicz, “Environment-induced superselection rules in Markovian regime,” Communications in Mathematical Physics, vol. 208, no. 1, pp. 245–265, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. N. P. Landsman, “When champions meet: rethinking the Bohr-Einstein debate,” Studies in History and Philosophy of Science B, vol. 37, no. 1, pp. 212–242, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  6. N. P. Landsman, “Between classical and quantum,” in Handbook of the Philosophy of Science: Philosophy of Physics, J. Butterfield and J. Earman, Eds., North Holland, Amsterdam, The Netherlands, 2007.
  7. E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, UK, 1976.
  8. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, New York, NY, USA, 2002.
  9. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY, USA, 2000.
  10. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum-Statistical Mechanics I, Springer, New York, NY, USA, 2nd edition, 1987.
  11. K. Jacobs, “Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banach-Räumen,” Mathematische Zeitschrift, vol. 67, pp. 83–92, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. K. De Leeuw and I. Glicksberg, “Almost periodic compactifications,” Bulletin of the American Mathematical Society, vol. 65, pp. 134–139, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. K. De Leeuw and I. Glicksberg, “Applications of almost periodic compactifications,” Acta Mathematica, vol. 105, pp. 63–97, 1961. View at Publisher · View at Google Scholar
  14. J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser, Basel, Switzerland, 1996.
  15. W. Arendt and C. J. K. Batty, “Tauberian theorems and stability of one-parameter semigroups,” Transactions of the American Mathematical Society, vol. 306, no. 2, pp. 837–852, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. I. Lyubich and Q. P. Vû, “Asymptotic stability of linear differential equations in Banach spaces,” Studia Mathematica, vol. 88, no. 1, pp. 37–42, 1988. View at Zentralblatt MATH
  17. B. Blackadar, Operator Algebras, Springer, Berlin, Germany, 2006.
  18. B. Kümmerer and R. Nagel, “Mean ergodic semigroups on W-algebras,” Acta Scientiarum Mathematicarum, vol. 41, no. 1-2, pp. 151–159, 1979.
  19. A. Bátkai, U. Groh, C. Kunszenti-Kovács, and M. Schreiber, “Decomposition of operator semigroups on W-algebras,” http://arxiv.org/abs/1106.0287v1. View at Publisher · View at Google Scholar
  20. D. E. Evans, “Irreducible quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 54, no. 3, pp. 293–297, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. Tomiyama, “On the projection of norm one in W-algebras,” Proceedings of the Japan Academy, vol. 33, pp. 125–129, 1957.
  22. D. W. Robinson, “Strongly positive semigroups and faithful invariant states,” Communications in Mathematical Physics, vol. 85, no. 1, pp. 129–142, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Frigerio, “Stationary states of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 63, no. 3, pp. 269–276, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. Hellmich, Decoherence in infinite quantum systems, Ph.D. thesis, University of Bielefeld, Bielefeld, Germany, 2009.
  25. R. Carbone, E. Sasso, and V. Umanità, “Decoherence for positive semigroups on M2(),” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 032202, 2011.
  26. A. Dhahri, F. Fagnola, and R. Rebolledo, “The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 13, no. 3, pp. 413–433, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. R. Rebolledo, “Decoherence of quantum Markov semigroups,” Annales de l'Institut Henri Poincar é, vol. 41, no. 3, pp. 349–373, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. Rebolledo, “A view on decoherence via master equations,” Open Systems and Information Dynamics, vol. 12, no. 1, pp. 37–54, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. R. Alicki, “Controlled quantum open systems,” Lecture Notes in Physics, vol. 622, pp. 121–139, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. J. Tomiyama, “On the projection of norm one in W-algebras II,” The Tôhoku Mathematical Journal, vol. 10, pp. 125–129, 1958.
  31. P. Ługiewicz and R. Olkiewicz, “Classical properties of infinite quantum open systems,” Communications in Mathematical Physics, vol. 239, no. 1-2, pp. 241–259, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. Christensen, “Generators of semigroups of completely positive maps,” Communications in Mathematical Physics, vol. 62, no. 2, pp. 167–171, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. A. Connes, “Classification of injective factors,” Annals of Mathematics, vol. 104, no. 1, pp. 73–115, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 48, no. 2, pp. 119–130, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. H. Spohn, “An algebraic condition for the approach to equilibrium of an open N-level system,” Letters in Mathematical Physics, vol. 2, no. 1, pp. 33–38, 1977.
  36. A. Frigerio and M. Verri, “Long-time asymptotic properties of dynamical semigroups on W-algebras,” Mathematische Zeitschrift, vol. 180, no. 2, pp. 275–286, 1982. View at Publisher · View at Google Scholar