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Deformation theory applied to quantization and statistical mechanics

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Abstract

After a review of the deformation (star product) approach to quantization, treated in an autonomous manner as a deformation (with parameter ħ) of the algebraic composition law of classical observables on phase-space, we show how a further deformation (with parameter β) of that algebra is suitable for statistical mechanics. In this case, the phase-space is endowed with what we call a conformal symplectic (or conformal Poisson) structure, for which the bracket is the Poisson bracket modified by terms of order (1, 0) and (0, 1). As an application, one sees that the KMS states (classical or quantum) are those that vanish on the modified (Poisson or Moyal-Vey) bracket of any two observables, multiplied by a conformal factor.

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References

  1. Flato, M., Czech. J. Phys. B32, 472 (1982).

    Google Scholar 

  2. Souriau, J.M., in Proc. Journées de Géométrie Symplectique, Balaruc, 1984.

  3. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Ann. Phys. (NY) 111, 61–151 (1978); Lett. Math. Phys. 1, 521 (1977).

    Google Scholar 

  4. Basart, H., Flato, M., Lichnerowicz, A., and Sternheimer, D., C.R. Acad. Sci. Paris, 2891 (1984).

  5. Bourbaki, N., Variétés différentielles et analytiques, vols. I & II. Hermann, Paris, 1967 & 1971; Lang, S., Introduction to Differentiable Manifolds, Wiley, New York, 1962.

    Google Scholar 

  6. Schouten, J.A., Conv. Int. Geom. Diff. Italia, 1–7. Cremonese, Rome, 1954; Nijenhuis, A., Indag. Math. 17, 370–403 (1955).

    Google Scholar 

  7. Lichnerowicz, A., C.R. Acad. Sci. Paris, 296 I, 915 (1983).

    Google Scholar 

  8. Gestenhaber, M., Ann. Math. 79, 59–90 (1964).

    Google Scholar 

  9. Lichnerowicz, A., Ann. Math. Fourier 32, 157–209 (1982); and in A.O. Barut (ed.), Math. Physics Studies, vol. 4, D. Reidel, Dordrecht, 1982, pp. 3–82.

    Google Scholar 

  10. Vey, J., Comm. Math. Helv. 50, 421–454 (1975).

    Google Scholar 

  11. De Wilde, M. and Lecomte, P., Lett. Math. Phys 7, 487 (1983).

    Google Scholar 

  12. Neroslavsky, O. M. and Vlassov, A.T., C.R. Acad. Sci. Paris 292 I, 71 (1981).

    Google Scholar 

  13. Cahen, M. and Gutt, S., Lett. Math. Phys. 6, 395 (1982).

    Google Scholar 

  14. Groenewold, A., Physica 12, 405 (1946); Moyal, J., Proc. Cambridge Phil. Soc. 45, 99 (1949).

    Google Scholar 

  15. Basart, H. and Lichnerowicz, A., C.R. Acad. Sci. Paris 293 I, 347 (1981).

    Google Scholar 

  16. Flato, M., Lichnerowicz, A., and Sternheimer, D., C.R. Acad. Sci. Paris, A279, 877 (1975) Comp. Math. 71, 47–82 (1975).

    Google Scholar 

  17. Flato, M., Lichnerowicz, A., and Sternheimer, D., J. Math. Phys. 17, 1754–1762 (1976).

    Google Scholar 

  18. Fronsdal, C., Rep. Math. Phys. 15, 111 (1978); J. Math. Phys. 20, 2226 (1979).

    Google Scholar 

  19. Arnal, D., Cortet, J.C., Flato, M., and Sternheimer, D., in E.Tirapegui (ed.), Field Theory, Quantization and Statistical Physics, D. Reidel, Dordrecht, 1981.

    Google Scholar 

  20. Sternheimer, D., Lecture Notes in Physics 153, 314, Springer, Berlin, 1982; Lectures in Applied Math. 21, AMS, Providence, 1984.

    Google Scholar 

  21. Segal, I.E., Symp. Math. 14, 99 (1974).

    Google Scholar 

  22. Aizenmann, M., Gallavotti, G., Goldstein, S., and Lebowitz, J., Comm. Math. Phys. 48, 1–14 (1976); Gallavotti, G. and Verboven, E., Nuovo Cim. 28B, 274 (1975); Gallavotti, G. and Pulvirenti, M., Comm. Math. Phys 46, 1–9 (1976).

    Google Scholar 

  23. See, e.g., Feynman, R.P., Statistical Mechanics, Benjamin, Reading, Mass., 1972; Ruelle, D., Statistical Mechanics, Benjamin, New York, 1969.

    Google Scholar 

  24. Rubio, R., C.R. Acad. Sci. Paris 299 I (1984).

    Google Scholar 

  25. Kubo, R., J. Phys. Soc. Japan 12, 570–586 (1957); Martin, P.C. and Schwinger, J., Phys. Rev. 115, 1342–1373 (1959); Haag, R., Hugenholtz, N.M., and Winnik, M., Comm. Math. Phys. 16, 81–104 (1967); Haag, R., Kastler, D., and Trych-Pohlmeyer, E., Comm. Math. Phys. 38, 173–193 (1974); Araki, H., Haag, R., Kastler, D., and Takesaki, M., Comm. Math. Phys. 53, 97–134 (1977); Araki, H., Lecture Notes in Math. 650, 66–84, Springer, Berlin, 1978; Mitt. Math. Ges. DDR 1, 5–13 (1982).

    Google Scholar 

  26. Lichnerowicz, A., C.R. Acad. Sci. Paris, 299 I (1984).

    Google Scholar 

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Authors and Affiliations

  1. Physique Mathématique, Collège de France, 75231, Paris Cedex 05, France

    H. Basart, M. Flato, A. Lichnerowicz & D. Sternheimer

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  1. H. Basart
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  2. M. Flato
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  3. A. Lichnerowicz
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  4. D. Sternheimer
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Basart, H., Flato, M., Lichnerowicz, A. et al. Deformation theory applied to quantization and statistical mechanics. Lett Math Phys 8, 483–494 (1984). https://doi.org/10.1007/BF00400978

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