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 
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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).zbMATHMathSciNetCrossRefADSGoogle 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
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    V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E. C. G. Sudarshan: Rep. Math. Phys. 13, 149 (1978).zbMATHMathSciNetCrossRefADSGoogle 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

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