Quantum dynamical semigroups and complete positivity. An application to isotropic spin relaxation

  • Vittorio Gorini
  • Maurizio Verri
  • E. C. G. Sudarshan
Canonical Transformation and Quantum Mechanics
Part of the Lecture Notes in Physics book series (LNP, volume 135)


Density Matrix Relaxation Rate Hyperfine Splitting Dynamical Semi Complete Positivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Footnotes and References

  1. 1.
    Here the derivative at the l.h.s of (1.1) is defined as where is the trace norm on (we denote by B* the adjoint of an operator B). The domain D(L) is the set of all for which dρ/dt exists.Google Scholar
  2. 2.
    K. Kraus: Ann. Phys. (N.Y.) 64, 311 (1971).MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    V. Gorini, A. Kossakowski and E. C. G. Sudarshan: J. Math. Phys. 17, 821 (1976).MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    G. Lindblad: Commun. Math. Phys. 48, 119 (1976).MATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    For a partial result when L is unbounded see E. B. Davies, Generators of dynamical groups, semigroups, preprint (1977) For the classification of dynamical semigroups on arbitrary Von Neumann algebras and with bounded L see E. Christensen, Commun. Math. Phys. 62, 167 (1978).Google Scholar
  6. 6.
    W. Happer: Rev. Mod. Phys. 44, 169 (1972) and references contained therein.CrossRefADSGoogle Scholar
  7. 7.
    A. Omont: Progr. Quantum Electronics 5, 69 (1977) and references contained therein.CrossRefADSGoogle Scholar
  8. 8.
    See, e.g., Ref. 7 and J.F. Papp and F.A. Franz, Phys Rev. A5, 1763 (1972).ADSGoogle Scholar
  9. 9.
    V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E. C. G. Sudarshan: Rep. Math. Phys. 13, 149 (1978).MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    M. Verri and V. Gorini: Quantum dynamical semigroups and isotropic relaxation of two coupled spins, in preparation.Google Scholar
  11. 11.
    V. Gorini, G. Parravicini, E.C.G. Sudarshan and M. Verri, Positive and completely positive SU(2) — invariant dynamical semigroups, in preparation.Google Scholar
  12. 12.
    A superscript bar denotes complex conjugation.Google Scholar
  13. 13.
    U. Fano and G. Racah: Irreducible tensorial sets, Academic Press, New York (1957).Google Scholar
  14. 14.
    A. Omont: J. Phys. 26, 26 (1965).Google Scholar
  15. 15.
    V. Gorini and A. Kossakowski: J. Math. Phys. 17, 1298 (1976).MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    W. Happer: Phys. Rev. B1, 2203 (1970).ADSGoogle Scholar
  17. 17.
    M. Verri and V. Gorini: J. Math. Phys. 19, 1803 (1978)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    The statement in [17] that for isotropic relaxation of a single spin positivity and complete positivity are equivalent is false. Actually, the argument given there allows only to prove that positivity implies λ2J⩾0.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Vittorio Gorini
    • 1
  • Maurizio Verri
    • 2
  • E. C. G. Sudarshan
    • 3
  1. 1.Istituto di Fisica dell'UniversitàMilanoItaly
  2. 2.Informatica e Sistemistica dell' UniversitàIstituto di MatematicaUdineItaly
  3. 3.Department of Physics, CPTThe University of Texas at AustinAustinUSA

Personalised recommendations