Skip to main content
SpringerLink
Log in

Two Approaches to Coupling Classical and Quantum Variables

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We address the issue of coupling variables whichare essentially classical to variables that are quantum.Two approaches are discussed. In the first, continuousquantum measurement theory is used to construct a phenomenological description of theinteraction of a quasiclassical variable X with aquantum variable x, where the quasiclassical nature ofX is assumed to have come about as a result ofdecoherence. The state of the quantum subsystem evolvesaccording to the stochastic nonlinear Schrodingerequation of a continuously measured system, and theclassical system couples to a stochastic c-number\(\overline x \left( t \right)\) representing the imprecisely measured value of x. The theorygives intuitively sensible results even when the quantumsystem starts out in a superposition of well-separatedlocalized states. The second approach involves a derivation of an effective theory from theunderlying quantum theory of the combinedquasiclassical-quantum system, and uses the decoherenthistories approach to quantum theory.

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. L. Rosenfeld, Nucl. Phys. 40, 353 (1963).

    Google Scholar 

  2. C. Moller, In Les Theories Relativistes de la Gravitation, A. Lichnerowicz and M. A. Tonnelat, eds. (CNRS, Paris, 1962).

    Google Scholar 

  3. L. H. Ford, Ann. Phys. (N. Y.) 144, 238 (1982).

    Google Scholar 

  4. J. B. Hartle and G. T. Horowitz, Phys. Rev. D 24, 257 (1981).

    Google Scholar 

  5. C.-I. Kuo and L. H. Ford, Phys. Rev. D 47, 4510 (1993).

    Google Scholar 

  6. D. N. Page and C. D. Geilker, Phys. Rev. Lett. 47, 979 (1981).

    Google Scholar 

  7. T. W. B. Kibble, In Quantum Gravity 2: A Second Oxford Symposium, C. J. Isham, R. Penrose, and D. W. Sciama, ed. (Oxford University Press, New York, 1981).

    Google Scholar 

  8. A. Anderson, Phys. Rev. Lett. 74, 621 (1995); 76, 4090 (1996); in Proceedings of the Fourth Drexel Symposium on Quantum Nonintegrability, D. H. Feng, ed. (International Press, 1996).

    Google Scholar 

  9. I. V. Aleksandrov, Z. Naturforsch. 36A, 902 (1981); 1981); A. Anderson, Phys. Rev. Lett. 74, 621 (1995); Phys. Rev. Lett. 76, 4090 (1996); W. Boucher and J. Traschen, Phys. Rev. D 37, 3522 (1988); K. R. W. Jones, Phys. Rev. Lett. 76, 4087 (1996); L. Diósi, Phys. Rev. Lett. 76, 4088 (1996); I. R. Senitzky, Phys. Rev. Lett. 76, 4089 (1996).

    Google Scholar 

  10. L. Diósi, A true equation to couple classical and quantum variables, preprint quant-ph/ 9510028 (1995)

  11. A. Zoupas, Coupling of quantum to classical in the presence of a decohering environment, Imperial College preprint (1997).

  12. L. Diósi and J. J. Halliwell, Coupling classical and quantum variables using continous quantum measurement theory, Imperial College preprint 96-97/46, quant-ph/9705008 (1997); Phys. Rev. Lett.(1998).

  13. J. J. Halliwell, Phys. Rev. D 57, 2337 (1998).

    Google Scholar 

  14. M. Gell-Mann and J. B. Hartle, In Complexity, Entropy and the Physics of Information, W. Zurek, ed. (Addison-Wesley, Reading, Massachusetts, 1990); in Proceedings of the Third International Symposium on the Foundations of Quantum Mechanics in the Light of New Technology, S. Kobayashi, H. Ezawa, Y. Murayama, and S. Nomura, eds. (Physical Society of Japan, Tokyo, 1990).

    Google Scholar 

  15. M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993).

    Google Scholar 

  16. L. Diósi, Phys. Rev. A 42, 5086 (1990).

    Google Scholar 

  17. A. Barchielli, L. Lanz, and G. M. Prosperi, Nuovo Cimento 72B, 79 (1982); V. P. Belavkin and P. Staszewski, Phys. Rev. A 45, 1347 (1992).

    Google Scholar 

  18. C. M. Caves and G. J. Milburn, Phys. Rev. A 36, 5543 (1987).

    Google Scholar 

  19. L. Diósi, Phys. Lett. 129A, 419 (1988).

    Google Scholar 

  20. B. L. Hu and A. Matacz, Phys. Rev. D 51, 1577 (1995).

    Google Scholar 

  21. L. Diósi, Phys. Lett. 132A, 233 (1988); Y. Salama and N. Gisin, Phys. Lett. 181A, 269 (1993).

    Google Scholar 

  22. I. C. Percival, J. Phys. A 27, 1003 (1994).

    Google Scholar 

  23. N. Gisin and I. C. Percival, J. Phys. A 26, 2233 (1993); 26, 2245 (1993).

    Google Scholar 

  24. J. J. Halliwell and A. Zoupas, Phys. Rev. D 52, 7294 (1995); 55, 4697 (1997).

    Google Scholar 

  25. L. Diósi, Phys. Lett. 105A, 199 (1984).

    Google Scholar 

  26. T. W. B. Kibble, Comm. Math. Phys. 64, 73 (1978); T. W. B. Kibble and S. Randjbar-Daemi, J. Phys. A 13, 141 (1980).

    Google Scholar 

  27. R. Griffiths, J. Stat. Phys. 36, 219 (1984).

    Google Scholar 

  28. J. J. Halliwell, In Stochastic Evolution of Quantum States in Open Systems and Measurement Processes, L. Diósi and B. Lukács, eds. (World Scientific, Singapore, 1994).

    Google Scholar 

  29. J. J. Halliwell, In Fundamental Problems in Quantum Theory, D. Greenberger and A. Zeilinger, eds. (New York Academy of Sciences, 1994); p. 726.

  30. J. B. Hartle, In Quantum Cosmology and Baby Universes, S. Coleman, J. Hartle, T. Piran, and S. Weinberg, eds. (World Scientific, Singapore, 1991).

    Google Scholar 

  31. J. B. Hartle, In Proceedings of the 1992 Les Houches Summer School, Gravitation et Quantifications, B. Julia and J. Zinn-Justin, eds. (Elsevier, 1995).

  32. R. Omnès, The Interpretation of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1994); Rev. Mod. Phys. 64, 339 (1992), and references therein.

    Google Scholar 

  33. J. B. Hartle, In Proceedings of the Cornelius Lanczos International Centenary Confererence, J. D. Brown, M. T. Chu, D. C. Ellison, and R. J. Plemmons (SIAM, Philadelphia, 1994).

    Google Scholar 

  34. R. Feynman, Rev. Mod. Phys. 20, 367 (1948); R. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

    Google Scholar 

  35. A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983).

    Google Scholar 

  36. R. P. Feynman and F. L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963).

    Google Scholar 

  37. H. F. Dowker and J. J. Halliwell, Phys. Rev. D 46, 1580 (1992).

    Google Scholar 

  38. N. Balazs and B. K. Jennings, Phys. Rep. 104, 347 (1984); M. Hillery, R. F. O' Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984); V. I. Tatarskii, Sov. Phys. Usp. 26, 311 (1983).

    Google Scholar 

  39. J. J. Halliwell, Phys. Rev. D 48, 4785 (1993).

    Google Scholar 

  40. J. J. Halliwell, Phys. Rev. D 46, 1610 (1992).

    Google Scholar 

  41. K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940).

    Google Scholar 

  42. E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985).

    Google Scholar 

  43. J. P. Paz, S. Habib, and W. Zurek, Phys. Rev. D 47, 488 (1993).

    Google Scholar 

  44. W. Zurek, Prog. Theor. Phys. 89, 281 (1993); Physics Today 40, 36 (1991); in Physical Origins of Time Asymmetry, J. J. Halliwell, J. Perez-Mercader, and W. Zurek, eds. (Cambridge University Press, Cambridge, 1994).

    Google Scholar 

  45. T. Yu and A. Zoupas, in preparation.

  46. L. Diósi, Quant. Semiclass. Opt. 8, 309 (1996); in New Developments on Fundamental Problems in Quantum Physics, M. Ferrero and A. van der Merwe, eds. (Kluwer, Dordrecht, 1997).

    Google Scholar 

  47. E. Calzetta and B. L. Hu, Phys. Rev. D 49, 6636 (1994).

    Google Scholar 

  48. E. Calzetta and B. L. Hu, preprint hep-th/9501040, IASSNS-HEP/95/2 (1995).

  49. E. Calzetta and B. Hu, Phys. Rev. D 52, 6770 (1995).

    Google Scholar 

  50. E. Calzetta, A. Campos, and E. Verdaguer, Phys. Rev. D 56, 2163 (1997).

    Google Scholar 

Download references

Authors
  1. J. J. Halliwell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Halliwell, J.J. Two Approaches to Coupling Classical and Quantum Variables. International Journal of Theoretical Physics 38, 2969–2986 (1999). https://doi.org/10.1023/A:1026612300318

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026612300318

Keywords

Search

Navigation