Communications in Mathematical Physics

, Volume 35, Issue 4, pp 265–277 | Cite as

The classical limit for quantum mechanical correlation functions

  • Klaus Hepp


For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space aroundħ−1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,ħ−1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.


Correlation Function Phase Space Quantum System Mechanical System Coherent State 
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.


  1. 1.
    Schrödinger, E.: Ann. d. Phys.79, 489 (1926)Google Scholar
  2. 2.
    Heisenberg, W.: Die physikalischen Prinzipien der Quantentheorie. Leipzig: Hirzel 1930Google Scholar
  3. 3.
    Pauli, W.: Die allgemeinen Prinzipien der Wellenmechanik, Handbuch der Physik, V. 1, Berlin-Göttingen-Heidelberg: Springer 1958Google Scholar
  4. 4.
    Andrié, M.: Comment. Phys. Math.41, 333 (1971)Google Scholar
  5. 5.
    Maslov, V.P.: Uspekhi Mat. Nauk,15, 213 (1960); Théorie des perturbations et méthodes asymptotiques. Paris: Dunod 1972Google Scholar
  6. 6.
    Feynman, R.P.: Rev. Mod. Phys.20, 367 (1948)Google Scholar
  7. 7.
    Nelson, E.: J. Math. Phys.5, 332 (1964)Google Scholar
  8. 7a.
    Berezin, F.A., Šubin, M.A.: Coll. Math. Soc. J. Bolyai,5 (1970)Google Scholar
  9. 8.
    Ehrenfest, P.: Z. Physik45, 455 (1927)Google Scholar
  10. 9.
    Klauder, J.R.: J. Math. Phys.4, 1058 (1963);5, 177 (1964);8, 2392 (1967)Google Scholar
  11. 10.
    Glauber, R.J.: Phys. Rev.131, 2766 (1963)Google Scholar
  12. 11.
    Goldstone, J.: Nuovo Cim.19, 154 (1961)Google Scholar
  13. 12.
    Gross, E.P.: Phys. Rev.100, 1571 (1955);106, 161 (1957); Ann. Phys.4, 57 (1958);9, 292 (1960)Google Scholar
  14. 13.
    Hepp, K., Lieb, E.H.: Ann. Phys.76, 360 (1973); Helv. Phys. Acta46 (1973).Google Scholar
  15. 13a.
    Constructive quantum field theory. Velo, G., Wightman, A.S., eds., Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer 1973Google Scholar
  16. 14.
    Hepp, K.: Helv. Phys. Acta45, 237 (1972)Google Scholar
  17. 15.
    Kato, T.: Perturbation theory of linear operators. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  18. 16.
    Wigner, E.P.: Phys. Rev.40, 749 (1932)Google Scholar
  19. 17.
    Kirkwood, J.G.: Phys. Rev.44, 31 (1933);45, 116 (1934)Google Scholar
  20. 18.
    Berezin, F.A.: Math. USSR Sbornik15, 577 (1971);17, 269 (1972); Izv. Akad. Nauk Ser. Mat.37, 1134 (1972)Google Scholar
  21. 19.
    Hepp, K., Lieb, E.H.: Phys. Rev.A8, 2517 (1973)Google Scholar
  22. 20.
    Perelomov, A.M.: Commun. math. Phys.26, 222 (1972)Google Scholar
  23. 21.
    Kostant, B.: In: Group representations in mathematics and physics, Bargmann, V., ed., Berlin-Heidelberg-New York: Springer 1970Google Scholar
  24. 22.
    Souriau, J.M.: Structure des systèmes dynamiques. Paris: Dunod 1970Google Scholar
  25. 23.
    Kirillov, A.A.: Elements of representation theory. Moscow: Nauka 1972Google Scholar
  26. 24.
    Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Phys. Rev.A6, 2211 (1972)Google Scholar
  27. 25.
    Lieb, E.H.: Commun. math. Phys.31, 327 (1973)Google Scholar
  28. 26.
    Glimm, J., Jaffe, A.M.: In: Mathematics of contemporary physics, Streater, R.F., ed., London: Academic P. 1972Google Scholar
  29. 27.
    Jörgens, K.: Math. Z.77, 295 (1961)Google Scholar
  30. 28.
    Browder, F.E.: Math. Z.80, 249 (1962)Google Scholar
  31. 29.
    Segal, I.E.: Ann. Math.78, 339 (1963)Google Scholar
  32. 30.
    Coleman, S., Weinberg, E.: Phys. Rev. D7, 1888 (1973)Google Scholar
  33. 31.
    Glimm, J.: Commun. math. Phys.10, 1 (1968)Google Scholar
  34. 32.
    Osterwalder, K.: Fortschr. Physik19, 43 (1971)Google Scholar
  35. 33.
    Ruelle, D.: Statistical mechanics: Rigorous results, New York: Benjamin 1969Google Scholar
  36. 34.
    Ginibre, J.: In: Statistical mechanics and quantum field theory, de Witt, C., Stora, R., eds. Paris: Gordon & Breach 1971Google Scholar
  37. 35.
    Bloch, C., de Dominicis, C.: Nucl. Phys.10, 181 (1959)Google Scholar
  38. 36.
    Bialynicki-Birula, I.: Ann. Phys.67, 252 (1971)Google Scholar
  39. 37.
    Martin-Löf, A.: Skand. Aktuarietidskr.1967, 70Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • Klaus Hepp
    • 1
  1. 1.Physics DepartmentETHZürichSchweiz

Personalised recommendations