Abstract
The hydrodynamical formalism for the quantum theory of a nonrelativistic particle is considered, together with a reformulation of it which makes use of the methods of kinetic theory and is based on the existence of the Wigner phase-space distribution. It is argued that this reformulation provides strong evidence in favor of the statistical interpretation of quantum mechanics, and it is suggested that this latter could be better understood as an almost classical statistical theory. Moreover, it is shown how, within this context, the Wigner and the Margenau-Hill functions are not equivalent, and that the latter is essentially unsatisfactory, as well as the associated symmetrization rule. Arguments in favor of a stochastic picture of the phenomena at the microscopic level are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Sonego, “On the semiclassical theory of gravity,” in preparation.
L. E. Ballentine,Rev. Mod. Phys. 42, 358 (1970).
T. A. Brody, inThe Concept of Probability, E. I. Bitsakis and C. A. Nicolaides, eds. (Kluwer Academic, Dordrecht, 1989).
L. E. Ballentine,Quantum Mechanics (Prentice-Hall, Englewood Cliffs, New Jersey, 1990).
T. Takabayasi,Prog. Theor. Phys. 9, 187 (1953).
C. Y. Wong,J. Math. Phys. 17, 1008 (1976).
S. K. Ghosh and B. M. Deb,Phys. Rep. 92, 1 (1982).
L. D. Landau and E. M. Lifshitz,Quantum Mechanics (Pergamon Press, New York, 1977).
P. A. M. Dirac,The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1967).
A. Messiah,Quantum Mechanics (Wiley, New York, 1966).
R. Peierls,Surprises in Theoretical Physics (Princeton University Press, Princeton, 1979).
J. A. Wheeler and W. H. Zurek, eds.,Quantum Theory and Measurement (Princeton University Press, Princeton, 1983).
E. Schrödinger,Ann. Phys. (Leipzig) 82, 265 (1926); reprinted inErwin Schrödinger Collected Papers, Vol. 3 (Österreichische Akademie der Wissenschaften, Vienna, 1984).
J. Von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955).
L. E. Ballentine,Found. Phys. 20, 1329 (1990).
K. Huang,Statistical Mechanics (Wiley, New York, 1963).
E. H. Kennard,Kinetic Theory of Gases (McGraw-Hill, New York, 1938).
L. E. Ballentine,Am. J. Phys. 54, 883 (1986).
E. P. Wigner,Phys. Rev. 40, 749 (1932).
M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner,Phys. Rep. 106, 123 (1984).
V. I. Tatarskii,Sov. Phys. Usp. 26, 311 (1983).
A. Einstein, inAlbert Einstein: Philosopher-Scientist, P. A. Schilpp, ed. (Open Court, Evanston, Illinois 1949).
A. Einstein,Ann. Phys. (Leipzig) 17, 549 (1905); see also A. Einstein,Investigations on the Theory of the Brownian Movement (Dover, New York, 1956).
D. Bohm,Phys. Rev. 85, 166, 180 (1952).
D. Bohm, B. J. Hiley, and P. N. Kaloyerou,Phys. Rep. 144, 321 (1987).
J. S. Bell,Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987).
D. Bohm,Phys. Rev. 89, 458 (1953).
A. Valentini,Phys. Lett. A 158, 1 (1991).
A. Valentini, “Subquantum statistical mechanics I,” preprint SISSA-ISAS 23/A (1991); submitted for publication.
D. Bohm and J. P. Vigier,Phys. Rev. 96, 208 (1954).
D. Bohm and B. J. Hiley,Phys. Rep. 172, 93 (1989).
E. Nelson, Phys. Rev.150, 1079 (1966); E. Nelson,Quantum Fluctuations (Princeton University Press, Princeton, 1985).
G. C. Ghirardi, C. Omero, A. Rimini, and T. Weber,Rivista del Nuovo Cimento 1, 1 (1978).
A. Valentini, “On the pilot-wave, path-integral, and stochastic formulations of quantum theory,” preprint SISSA-ISAS 19/A (1991); submitted for publication.
C. L. Mehta,J. Math. Phys. 5, 677 (1964).
L. Cohen,J. Math. Phys. 7, 781 (1966).
J. R. Shewell,Am. J. Phys. 27, 16 (1959).
L. Cohen,J. Math. Phys. 11, 3296 (1970).
F. Testa,J. Math. Phys. 12, 1471 (1971).
I. W. Mayes and J. S. Dowker,J. Math. Phys. 14, 434 (1973).
M. Mizrahi,J. Math. Phys. 16, 2201 (1975).
L. Cohen,J. Math. Phys. 17, 597 (1976).
J. S. Dowker,J. Math. Phys. 17, 1873 (1976).
S. Sonego,Phys. Rev. A 42, 3733 (1990).
H. Margenau and R. N. Hill,Prog. Theor. Phys. 26, 722 (1961).
S. Sonego, “Quasiprobabilities and explicitly covariant relativistic quantum theory,” preprint SISSA-ISAS 10/A (1990); to appear inPhys. Rev. A.
B. Sakita,Quantum Theory of Many-Particle Systems and Fields (World Scientific, Singapore, 1985).
L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, 1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sonego, S. Interpretation of the hydrodynamical formalism of quantum mechanics. Found Phys 21, 1135–1181 (1991). https://doi.org/10.1007/BF00734264
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00734264