Skip to main content
SpringerLink
Log in

Quantum equilibrium and the origin of absolute uncertainty

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

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.

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. 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. 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. J. S. Bell, On the Einstein Podolsky Rosen paradox,Physics 1:195–200 (1964) [Reprinted in refs. 58 and 10].

    Google Scholar 

  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. 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. 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. 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. J. S. Bell, On the impossible pilot wave,Found. Phys,12:989–999 (1982) [Reprinted in ref. 10].

    Google Scholar 

  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. J. S. Bell,Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987).

    Google Scholar 

  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. D. Bohm,Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951).

    Google Scholar 

  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. 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. D. Bohm, Proof that probability density approaches ¦ψ¦2 in causal interpretation of quantum theory,Phys. Rev. 89:458–166 (1953).

    Google Scholar 

  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. D. Bohm,Wholeness and the Implicate Order (Routledge & Kegan Paul, London, 1980).

    Google Scholar 

  18. D. Bohm and B. J. Hiley, Measurement understood through the quantum potential approach,Found. Phys. 14:255–274 (1984).

    Google Scholar 

  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. D. Bohm and B. Hiley,The Undivided Universe: An Ontological Interpretation of Quantum Theory (Routledge & Kegan Paul, London, 1992).

    Google Scholar 

  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. N. Bohr,Atomic Physics and Human Knowledge (Wiley, New York, 1958).

    Google Scholar 

  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. L. de Broglie, A tentative theory of light quanta,Philos. Mag. 47:446–458 (1924).

    Google Scholar 

  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. B. S. DeWitt and N. Graham, eds.,The Many-Worlds Interpretation of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1973).

    Google Scholar 

  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. 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. D. Dürr, S. Goldstein, and N. Zanghi, On the role of operators in quantum theory, in preparation.

  30. D. Dürr, S. Goldstein, and N. Zanghí, The mystery of quantization, in preparation.

  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. M. Gell-Mann and J. B. Hartle, Alternative decohering histories in quantum mechanics, preprint.

  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. J. W. Gibbs,Elementary Principles in Statistical Mechanics (Yale University Press, 1902; Dover, New York, 1960).

  35. S. Goldstein, Stochastic mechanics and quantum theory,J. Stat. Phys. 47:645–667 (1987).

    Google Scholar 

  36. R. B. Griffiths, Consistent histories and the interpretation of quantum mechanics,J. Stat. Phys. 36:219–272 (1984).

    Google Scholar 

  37. W. Heisenberg,Physics and Philosophy (Harper and Row, New York, 1958), p. 138.

    Google Scholar 

  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. N. S. Krylov,Works on the Foundations of Statistical Mechanics (Princeton University Press, Princeton, New Jersey, 1979).

    Google Scholar 

  40. L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Non-relativistic Theory (Pergamon Press, Oxford, 1958).

    Google Scholar 

  41. O. E. Lanford III and D. Ruelle,Commun. Math. Phys. 13:194–215 (1969).

    Google Scholar 

  42. A. J. Leggett, Macroscopic quantum systems and the quantum theory of measurement,Prog. Theor. Phys. Suppl. 69:80–100 (1980).

    Google Scholar 

  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. E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics,Phys. Rev. 150:1079–1085 (1966).

    Google Scholar 

  45. E. Nelson,Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, New Jersey, 1967).

    Google Scholar 

  46. E. Nelson,Quantum Fluctuations (Princeton University Press, Princeton, New Jersey, 1985).

    Google Scholar 

  47. R. Omnes, Logical reformulation of quantum mechanics I,J. Stat. Phys. 53:893–932 (1988).

    Google Scholar 

  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. R. Penrose,The Emperor's New Mind (Oxford University Press, Oxford, 1989).

    Google Scholar 

  50. P. A. Schilpp, ed.,Albert Einstein, Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois, 1949).

    Google Scholar 

  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. E. Schrödinger, Discussion of probability relations between separated systems,Proc. Camb. Philos. Soc. 31:555–563 (1935);32:446–452 (1936).

    Google Scholar 

  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. 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. 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. 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. S. Weinberg, Precision tests of quantum mechanics,Phys. Rev. Lett. 62:485–488 (1989).

    Google Scholar 

  58. J. A. Wheeler and W. H. Zurek, eds.,Quantum Theory and Measurement (Princeton University Press, Princeton, New Jersey, 1983).

    Google Scholar 

  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. E. P. Wigner, The problem of measurement,Am. J. Phys. 31:6–15 (1963) [Reprinted in refs. 61 and 58].

    Google Scholar 

  61. E. P. Wigner,Symmetries and Reflections (Indiana University Press, Bloomington, Indiana, 1967).

    Google Scholar 

  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. 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. W. H. Zurek, Environment-induced superselection rules,Phys. Rev. D 26:1862–1880 (1982).

    Google Scholar 

Download references

Author information

Author notes

  1. Detlef Dürr

    Present address: Fakultät für Mathematik, Universität München, 8000, München 2, Germany

  2. Nino Zanghí

    Present address: Istituto di Fisica, Università di Genova, INFN, 16146, Genova, Italy

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, New Jersey

    Detlef Dürr, Sheldon Goldstein & Nino Zanghí

Authors
  1. Detlef Dürr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sheldon Goldstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nino Zanghí
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper is dedicated to the memory of J. S. Bell.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dürr, D., Goldstein, S. & Zanghí, N. Quantum equilibrium and the origin of absolute uncertainty. J Stat Phys 67, 843–907 (1992). https://doi.org/10.1007/BF01049004

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01049004

Key words

Search

Navigation