Skip to main content
SpringerLink
Log in

Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo-Martin-Schwinger and Wigner-Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov-Kubo-Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the SU(2) algebra of observables in the simplest model of an ideal quantum spin paramagnet.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. N. Bogolubov, A. A. Logunov, and I. T. Todorov, Basics of Axiomatic Approach to Quantum Field Theory [in Russian], Nauka, Moscow (1969); English transl.: Introduction to Axiomatic Quantum Field Theory, (Math. Phys. Monogr. Ser., Vol. 18), Benjamin, Reading, Mass. (1975).

    Google Scholar 

  2. G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Interscience Monogr. and Texts in Phys. and Astronomy, Vol. 26), Wiley, New York (1972).

    MATH  Google Scholar 

  3. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. 1, C*- and W*-algebras, Algebras, Symmetry Groups, Decomposition of States, Springer, New York (1987).

    Google Scholar 

  4. R. D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. 1, Springer, New York (1978).

    MATH  Google Scholar 

  5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton, N. J. (1996).

    MATH  Google Scholar 

  6. S. G. Krein, ed., Functional Analysis [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  7. P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon, Oxford (1935).

    Google Scholar 

  8. N. N. Bogoliubov, “Lectures in quantum statistics,” in: Collection of Scientific Publications in 12 Volumes [in Russian], Vol. 6, Nauka, Moscow (2006), pp. 9–235; English transl.: Lectures on Quantum Statistics, Vol. 1, Quantum Statistics, Gordon and Breach, New York (1967).

    Google Scholar 

  9. L. D. Faddeev and O. A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students [in Russian], Leningrad State Univ. Press, Leningrad (1980); English transl. (Stud. Math. Libr., Vol. 47), Amer. Math. Soc., Providence, R. I. (2009).

    Google Scholar 

  10. R. Kubo, J. Phys. Soc. Japan, 12, 570–602 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  11. P. C. Martin and J. Schwinger, Phys. Rev., 115, 1342–1373 (1959).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. E. P. Wigner and M. M. Yanase, Proc. Natl. Acad. Sci. USA, 49, 910–918 (1963).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. E. H. Lieb, Adv. Math., 11, 267–288 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Bauman and R. Jost, “Remarks on a conjecture of Robinson and Ruelle concerning the quantum mechanical entropy,” in: Problems in Theoretical Physics [in Russian] (Collection of articles dedicated to Nikolai Nikolaevic Bogoljubov on the occasion of his sixtieth birthday, D. I. Blohincev, ed.), Nauka, Moscow (1969).

    Google Scholar 

  15. J. M. Luttinger, Progr. Theoret. Phys., Suppl. 37–38, 35–45 (1966).

  16. N. N. Bogoliubov, “Quasi-averages in problems of statistical mechanics [in Russian],” in: Collection of Scientific Publications in 12 Volumes, Vol. 6, Nauka, Moscow (2006), pp. 236–327; Quasi-averages in problems of statistical mechanics [in Russian], Joint Inst. Nucl. Res., Dubna (1961); Selected Papers in Statistical Physics [in Russian], MSU Publ., Moscow (1979), 193–269.

    Google Scholar 

  17. N. N. Bogoljubov and S. V. Tjablikov, Sov. Phys. Dokl., 4, 589–593 (1959).

    ADS  MathSciNet  Google Scholar 

  18. H. Mori, Progr. Theoret. Phys., 33, 423–455 (1965).

    Article  MATH  ADS  Google Scholar 

  19. R. F. Streater, Proc. Steklov Inst. Math., 228, 205–223 (2000).

    MathSciNet  Google Scholar 

  20. H. B. Callen and T. A. Welton, Phys. Rev., 83, 34–40 (1951).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. V. L. Bonch-Bruevich and S. V. Tyablikov, The Green Function Method in Statistical Mechanics [in Russian], Fizmatlit, Moscow (1961); English transl., North-Holland, Amsterdam (1962).

  22. D. N. Zubarev and Yu. A. Tserkovnikov, Proc. Steklov Inst. Math., 175, 139–185 (1988).

    MATH  Google Scholar 

  23. D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, New York (1969).

    MATH  Google Scholar 

  24. J. C. Garrison and J. Wong, Comm. Math. Phys., 26, 1–5 (1972).

    Article  ADS  MathSciNet  Google Scholar 

  25. G. Roepstorff, Comm. Math. Phys., 46, 253–262 (1976).

    Article  ADS  MathSciNet  Google Scholar 

  26. C. W. Helstrom, Quantum Detection and Estimation Theory, Acad. Press, New York (1976).

    Google Scholar 

  27. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory [in Russian], IKI, Moscow (2003); English transl. prev. ed. (North-Holland Ser. Statist. Probab., Vol. 1), North-Holland, Amsterdam (1982).

    Google Scholar 

  28. N. N. Čencov, Statistical Decision Rules and Optimal Inference [in Russian], Nauka, Moscow (1972); English transl. (Transl. Math. Monogr., Vol. 53), Amer. Math. Soc., Providence, R. I. (1982).

    Google Scholar 

  29. S.-I. Amari, Differential Geometrical Methods in Statistics (Lect. Notes Statist., Vol. 28), Springer, New York (1985).

    MATH  Google Scholar 

  30. E. A. Morozova and N. N. Chentsov, “Markov invariant geometry and manifolds of states,” J. Sov. Math., 56, 2648–2669 (1991).

    Article  MATH  Google Scholar 

  31. D. Petz, Linear Algebra Appl., 244, 81–96 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  32. H. Hasegawa and D. Petz, “Non-commutative extension of information geometry,” in: Quantum Communication, Computing, and Measurement (H. Hirota, A. S. Holevo, C. M. Caves, ed.), Plenum, New York (1997), pp. 109–118.

    Google Scholar 

  33. D. Petz and G. Toth, Lett. Math. Phys., 27, 205–216 (1993).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. S. L. Luo, Theor. Math. Phys., 143, 681–688 (2005).

    Article  Google Scholar 

  35. A. D. Sukhanov and Yu. G. Rudoy, “The geometrical aspects of the equilibrium statistical thermodynamics,” in: Proc. 5th Intl. Conf. “Bolyai-Gauss-Lobachevsky” — BGL-5 (Minsk, 10–13 Oct. 2006, Yu. Kurochkin and V. Red’kov, eds.), Minsk (2006), p. 63–75.

Download references

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia, Moscow, Russia

    Yu. G. Rudoy

Authors
  1. Yu. G. Rudoy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. G. Rudoy.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 2, pp. 352–369, August, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rudoy, Y.G. Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables. Theor Math Phys 160, 1161–1176 (2009). https://doi.org/10.1007/s11232-009-0108-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-009-0108-1

Keywords

Search

Navigation