Advertisement

Journal of Statistical Physics

, Volume 67, Issue 5–6, pp 843–907 | Cite as

Quantum equilibrium and the origin of absolute uncertainty

  • Detlef Dürr
  • Sheldon Goldstein
  • Nino Zanghí
Articles

Abstract

The quantum formalism is a “measurement” formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that “particles” means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by “ρ = ¦ψ¦2”. A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.

Key words

Quantum randomness quantum uncertainty hidden variables effective wave function collapse of the wave function the measurement problem Bohm's causal interpretation of quantum theory pilot wave foundations of quantum mechanics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Time symmetry in the quantum process of measurement,Phys. Rev. B 134:1410–1416 (1964) [Reprinted in ref. 58].Google Scholar
  2. 2.
    J. S. Bell, On the problem of hidden variables in quantum mechanics,Rev. Mod. Phys. 38:447–452 (1966) [Reprinted in refs. 58 and 10].Google Scholar
  3. 3.
    J. S. Bell, On the Einstein Podolsky Rosen paradox,Physics 1:195–200 (1964) [Reprinted in refs. 58 and 10].Google Scholar
  4. 4.
    J. S. Bell, The measurement theory of Everett and de Broglie's pilot wave, inQuantum Mechanics, Determinism, Causality, and Particles, L. de Broglie and M. Flato, eds. (D. Reidel, Boston, 1976), pp. 11–17) [Reprinted in ref. 10].Google Scholar
  5. 5.
    J. S. Bell, De Broglie-Bohm, delayed-choice double-slit experiment, and density matrix,Int. J. Quantum Chem. Symp. 14:155–159 (1980) [Reprinted in ref. 10].Google Scholar
  6. 6.
    J. S. Bell, Bertlmann's socks and the nature of reality,J. Phys. (Paris)C2-42:41–61 (1981) [Reprinted in ref. 10].Google Scholar
  7. 7.
    J. S. Bell, Quantum mechanics for cosmologists, inQuantum Gravity 2, C. Isham, R. Penrose, and D. Sciama, eds. (Oxford University Press, New York, 1981), pp. 611–637 [Reprinted in ref. 10].Google Scholar
  8. 8.
    J. S. Bell, On the impossible pilot wave,Found. Phys,12:989–999 (1982) [Reprinted in ref. 10].Google Scholar
  9. 9.
    J. S. Bell, Are there quantum jumps?, inSchrödinger. Centenary Celebration of a Polymath, C.W. Kilmister, ed. (Cambridge University Press, Cambridge, 1987) [Reprinted in ref. 10].Google Scholar
  10. 10.
    J. S. Bell,Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987).Google Scholar
  11. 11.
    J. S. Bell, Against “measurement”,Phys. World 3:33–40 (1990); also inSixty-two Years of Uncertainty: Historical. Philosophical, and Physical Inquiries into the Foundations of Quantum Mechanics, A. I. Miller, ed. (Plenum Press, New York, 1990), pp. 17–31.Google Scholar
  12. 12.
    D. Bohm,Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951).Google Scholar
  13. 13.
    D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden variables”: Part I,Phys. Rev. 85:166–179 (1952) [Reprinted in ref. 58].Google Scholar
  14. 14.
    D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden variables”: Part II,Phys. Rev. 85:180–193 (1952) [Reprinted in ref. 58].Google Scholar
  15. 15.
    D. Bohm, Proof that probability density approaches ¦ψ¦2 in causal interpretation of quantum theory,Phys. Rev. 89:458–166 (1953).Google Scholar
  16. 16.
    D. Bohm and B. J. Hiley, On the intuitive understanding of non-locality as implied by quantum theory,Found. Phys. 5:93–109 (1975).Google Scholar
  17. 17.
    D. Bohm,Wholeness and the Implicate Order (Routledge & Kegan Paul, London, 1980).Google Scholar
  18. 18.
    D. Bohm and B. J. Hiley, Measurement understood through the quantum potential approach,Found. Phys. 14:255–274 (1984).Google Scholar
  19. 19.
    D. Bohm and B. J. Hiley, An ontological basis for the quantum theory I: Non-relativistic particle systems,Phys. Rep. 144:323–348 (1987).Google Scholar
  20. 20.
    D. Bohm and B. Hiley,The Undivided Universe: An Ontological Interpretation of Quantum Theory (Routledge & Kegan Paul, London, 1992).Google Scholar
  21. 21.
    N. Bohr, Discussion with Einstein on epistemological problems in atomic physics, inAlbert Einstein, Philosopher-Scientist, P. A. Schilpp, ed. (Library of Living Philosophers, Evanston, Illinois, 1949), pp. 199–244 [Reprinted in refs. 22 and 58].Google Scholar
  22. 22.
    N. Bohr,Atomic Physics and Human Knowledge (Wiley, New York, 1958).Google Scholar
  23. 23.
    M. Born, Quantenmechanik der Stossvorgänge,Z. Phys. 38:803–827 (1926) [English translation: Quantum mechanics of collision processes, inWave Mechanics, G. Ludwig, eds. (Pergamon Press, Oxford, 1968)].Google Scholar
  24. 24.
    L. de Broglie, A tentative theory of light quanta,Philos. Mag. 47:446–458 (1924).Google Scholar
  25. 25.
    L. de Broglie, La nouvelle dynamique des quanta, inÉlectrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique tenu à Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de l'Institut International de Physique Solvay (Gauthier-Villars, Paris, 1928), pp. 105–132.Google Scholar
  26. 26.
    B. S. DeWitt and N. Graham, eds.,The Many-Worlds Interpretation of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1973).Google Scholar
  27. 27.
    R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity,Theory Prob. Appl. 13:197–224 (1968).Google Scholar
  28. 28.
    D. Dürr, S. Goldstein, and N. Zanghí, On a realistic theory for quantum physics, inStochastic Processes, Geometry and Physics, S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, and R. Mortesi, eds. (World Scientific, Singapore, 1990), pp. 374–391.Google Scholar
  29. 29.
    D. Dürr, S. Goldstein, and N. Zanghi, On the role of operators in quantum theory, in preparation.Google Scholar
  30. 30.
    D. Dürr, S. Goldstein, and N. Zanghí, The mystery of quantization, in preparation.Google Scholar
  31. 31.
    M. Gell-Mann and J. B. Hartle, Quantum mechanics in the light of quantum cosmology, inComplexity, Entropy, and the Physics of Information, W. Zurek, ed. (Addison-Wesley, Reading, Massachusetts, 1990), pp. 425–458; also inProceedings of the 3rd International Symposium on Quantum Mechanics in the Light of New Technology, S. Kobayashi, H. Ezawa, Y. Murayama, and S. Nomura, eds. (Physical Society of Japan, 1990).Google Scholar
  32. 32.
    M. Gell-Mann and J. B. Hartle, Alternative decohering histories in quantum mechanics, preprint.Google Scholar
  33. 33.
    G. C. Ghirardi, A. Rimini, and T. Weber, Unified dynamics for microscopic and macroscopic systems,Phys. Rev. D 34:470–491 (1986).Google Scholar
  34. 34.
    J. W. Gibbs,Elementary Principles in Statistical Mechanics (Yale University Press, 1902; Dover, New York, 1960).Google Scholar
  35. 35.
    S. Goldstein, Stochastic mechanics and quantum theory,J. Stat. Phys. 47:645–667 (1987).Google Scholar
  36. 36.
    R. B. Griffiths, Consistent histories and the interpretation of quantum mechanics,J. Stat. Phys. 36:219–272 (1984).Google Scholar
  37. 37.
    W. Heisenberg,Physics and Philosophy (Harper and Row, New York, 1958), p. 138.Google Scholar
  38. 38.
    E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environment,Z. Phys. B 59:223–243 (1985).Google Scholar
  39. 39.
    N. S. Krylov,Works on the Foundations of Statistical Mechanics (Princeton University Press, Princeton, New Jersey, 1979).Google Scholar
  40. 40.
    L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Non-relativistic Theory (Pergamon Press, Oxford, 1958).Google Scholar
  41. 41.
    O. E. Lanford III and D. Ruelle,Commun. Math. Phys. 13:194–215 (1969).Google Scholar
  42. 42.
    A. J. Leggett, Macroscopic quantum systems and the quantum theory of measurement,Prog. Theor. Phys. Suppl. 69:80–100 (1980).Google Scholar
  43. 43.
    F. W. London and E. Bauer,La Théorie de l'Observation en Mécanique Quantique (Hermann, Paris, 1939). [English translation in ref. 58].Google Scholar
  44. 44.
    E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,Phys. Rev. 150:1079–1085 (1966).Google Scholar
  45. 45.
    E. Nelson,Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, New Jersey, 1967).Google Scholar
  46. 46.
    E. Nelson,Quantum Fluctuations (Princeton University Press, Princeton, New Jersey, 1985).Google Scholar
  47. 47.
    R. Omnes, Logical reformulation of quantum mechanics I,J. Stat. Phys. 53:893–932 (1988).Google Scholar
  48. 48.
    R. Penrose, Quantum gravity and state-vector reduction, inQuantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Oxford University Press, Oxford, 1985).Google Scholar
  49. 49.
    R. Penrose,The Emperor's New Mind (Oxford University Press, Oxford, 1989).Google Scholar
  50. 50.
    P. A. Schilpp, ed.,Albert Einstein, Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois, 1949).Google Scholar
  51. 51.
    E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik,Naturwissenschaften 23:844–849 (1935) [English translation, The present situation in quantum mechanics: A translation of Schrödinger's “cat paradox” paper,Proc. Am. Philos. Soc. 124:323–338 (1980) [Reprinted in ref. 58].Google Scholar
  52. 52.
    E. Schrödinger, Discussion of probability relations between separated systems,Proc. Camb. Philos. Soc. 31:555–563 (1935);32:446–452 (1936).Google Scholar
  53. 53.
    J. T. Schwarz, The pernicious influence of mathematics on science, inDiscrete Thoughts: Essays on Mathematics, Science, and Philosophy, M. Kac, G. Rota, and J. T. Schwartz, eds. (Birkhauser, Boston, 1986), p. 23.Google Scholar
  54. 54.
    M. O. Scully and H. Walther, Quantum optical test of observation and complementarity in quantum mechanics,Phys. Rev. A 39:5229–5236 (1989).Google Scholar
  55. 55.
    H. P. Stapp, Light as foundation of being, inQuantum Implications: Essays in Honor of David Bohm, B.J. Hiley and F. D. Peat, eds. (Routledge & Kegan Paul, London, 1987).Google Scholar
  56. 56.
    J. von Neumann,Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932) [English translation,Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1955)].Google Scholar
  57. 57.
    S. Weinberg, Precision tests of quantum mechanics,Phys. Rev. Lett. 62:485–488 (1989).Google Scholar
  58. 58.
    J. A. Wheeler and W. H. Zurek, eds.,Quantum Theory and Measurement (Princeton University Press, Princeton, New Jersey, 1983).Google Scholar
  59. 59.
    E. P. Wigner, Remarks on the mind-body question, inThe Scientist Speculates, I. J. Good, ed. (Basic Books, New York, 1961) [Reprinted in refs. 61 and 58].Google Scholar
  60. 60.
    E. P. Wigner, The problem of measurement,Am. J. Phys. 31:6–15 (1963) [Reprinted in refs. 61 and 58].Google Scholar
  61. 61.
    E. P. Wigner,Symmetries and Reflections (Indiana University Press, Bloomington, Indiana, 1967).Google Scholar
  62. 62.
    E. P. Wigner, Interpretation of quantum mechanics, inQuantum Theory and Measurement, J. A. Wheeler and W. H. Zurek, eds. (Princeton University Press, Princeton, New Jersey, 1983).Google Scholar
  63. 63.
    E. P. Wigner, Review of the quantum mechanical measurement problem, inQuantum Optics, Experimental Gravity and Measurement Theory, P. Meystre and M. O. Scully, eds. (Plenum Press, New York, 1983), pp. 43–63.Google Scholar
  64. 64.
    W. H. Zurek, Environment-induced superselection rules,Phys. Rev. D 26:1862–1880 (1982).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Detlef Dürr
    • 1
  • Sheldon Goldstein
    • 1
  • Nino Zanghí
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew Brunswick

Personalised recommendations