Abstract
The standard mathematical formulation of quantum mechanics is specified. Bohm's ontological interpretation of quantum mechanics is then shown to be incapable of providing a suitable interpretation of that formulation. It is also shown that Bohm's interpretation may well be viable for two alternative mathematical formulations of quantum mechanics, meaning that the negative result is a significant though not a devastating criticism of Bohm's interpretation. A preliminary case is made for preferring one alternative formulation over the other.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Akhiezer, N. I. and I. M. Glazman: 1993, Theory of Linear Operators in Hilbert Space, Dover Publications, New York: Originally published in New York by F. Ungar Pub. Co., 1961, 1963.
Amrein, W. O., J. M. Jauch, and K. B. Sinha: 1977, Scattering Theory in Quantum Mechanics: Physical Principles and Mathematical Methods, W. A. Benjamin, Reading, Mass.
Antoine, J. P.: 1969, 'Dirac Formalism and Symmetry Problems in Quantum Mechanics I and II', Journal of Mathematical Physics 10, 53-69, 2276–2290.
Antoniou, I. and S. Tasaki: 1993, 'Generalized Spectral Decompositions of Mixing Dynamical Systems', International Journal of Quantum Chemistry 46, 425-474.
Berndl, K.: 1996, 'Global Existence of Bohmian Trajectories', in Bohmian Mechanics and Quantum Theory: An Appraisal, Kluwer, Boston.
Blank, J., P. Exner, and M. Havlicek: 1994, Hilbert Space Operators in Quantum Physics, American Institute of Physics, New York.
Bogolubov, N. N., A. A. Logunov, and I. T. Todorov: 1975, Introduction to Axiomatic Quantum Field Theory, W. A. Benjamin, Reading, Mass.
Bohm, D.: 1951, Quantum Theory, Prentice-Hall, New York.
Bohm, D.: 1952, 'A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden' Variables, I and II', Physical Review 85, 166-179, 180–193.
Bohm, A.: 1966, 'Rigged Hilbert Space and Mathematical Description of Physical Systems', in Lectures in Theoretical Physics 9A: Mathematical Methods of Theoretical Physics, Wiley, New York.
Bohm, A. and M. Gadella: 1989, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Springer-Verlag, New York.
Bohm, A., S. Maxon, M. Loewe, and M. Gadella: 1997, 'Quantum Mechanical Irreversibility', Physica A 236, 485-549.
Bratteli, O. and D. W. Robinson: 1987, Operator Algebras and Quantum Statistical Mechanics 1, Springer-Verlag, New York.
Cohen-Tannoudji, C., B. Diu, and F. Laloë: 1977, Quantum Mechanics, Wiley, New York.
Constantinescu, F.: 1980, Distributions and Their Applications in Physics, Pergamon Press, New York.
Cushing, J. T.: 1994, Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony, University of Chicago Press, Chicago.
Dubin, D. A. and M. A. Henning: 1990, Quantum Mechanics, Algebras, and Distributions, Longman Scientific & Technical, Harlow, Essex, England.
Gel'fand, I. M. and G. E. Shilov: 1964, Generalized Functions II: Fundamental and Generalized Functions, Academic Press, New York. [Translation of the 1958 Russian edition.]
Gel'fand, I. M. and N. Y. Vilenkin: 1964, Generalized Functions IV: Applications of Harmonic Analysis, Academic Press, New York. [Translation of the 1961 Russian edition.]
Gottfried, K.: 1974, Quantum Mechanics, W. A. Benjamin, New York.
Haag, R.: 1992, Local Quantum Physics, Springer-Verlag, New York.
Holland, P.R.: 1993, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, New York.
Horvath, J.: 1966, Topological Vector Spaces and Distributions, Addison-Wesley, Reading, MA.
Jammer, M.: 1966, The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York.
Jauch, J. M.: 1968, Foundations of Quantum Mechanics, Addison-Wesley, Reading, MA.
Jordan, T. F.: 1969, Linear Operators for Quantum Mechanics, Wiley, New York.
Melsheimer, O.: 1972, 'Rigged Hilbert Space Formalism as an Extended Mathematical Formalism for Quantum Systems I and II', Journal of Mathematical Physics 15, 902-926.
Merzbacher, E.: 1970, Quantum Mechanics, Second edition, Wiley, New York.
Messiah, A.: 1961, Quantum Mechanics, Wiley, New York.
Nagel, B.: 1989, 'Introduction to Rigged Spaces', in Resonances, Springer-Verlag, New York.
Pauling, L. and E. B. Wilson: 1935, Introduction to Quantum Mechanics, with Applications to Chemistry, McGraw-Hill, New York. Republished by Dover Publications, 1985.
Petrosky, T. and I. Prigogine: 1997, 'The Liouville Space Extension of Quantum Mechanics', Advances in Chemical Physics 99, 1-120.
Prugovecki, E.: 1981, Quantum Mechanics in Hilbert Space, Second edition, Academic Press, New York.
Reed, M. and B. Simon: 1980, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, New York.
Roberts, J. E.: 1966, 'The Dirac Bra and Ket Formalism', Journal of Mathematical Physics 7, 1097-1104.
Roberts, J. E.: 1966, 'Rigged Hilbert Space in Quantum Mechanics', Communications in Mathematical Physics 3, 98-119.
Schiff, L. I.: 1968, Quantum Mechanics, Third edition, McGraw-Hill, New York.
Schwartz, L.: 1966, Theorie des Distributions, New edition, revised and augmented, Hermann, Paris. [First published in 1950–1951.]
Streater, R. F. and A. S. Wightman: 1964, PCT, Spin and Statistics, and All That, W. A. Benjamin, New York.
Suchanecki, Z., I. Antoniou, S. Tasaki, and O. F. Bandtlow: 1996, 'Rigged Hilbert Spaces for Chaotic Dynamical Systems', Journal of Mathematical Physics 37, 5837-5847.
Sudbery, A.: 1986, Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians, Cambridge University Press, New York.
Treves, F.: 1967, Topological Vector Spaces, Distributions and Kernals, Academic Press, New York.
Van Eijndhoven, S. J. L. and J. de Graaf: 1986, A Mathematical Introduction to Dirac's Formalism, North-Holland, New York.
Von Neumann, J.: 1955, Mathematical Foundations of Quantum Mechanics, Princeton University Press, N.J.[First published in German in 1932.]
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kronz, F.M. Bohm's Ontological Interpretation and its Relations to Three Formulations of Quantum Mechanics. Synthese 117, 31–52 (1998). https://doi.org/10.1023/A:1005098623123
Issue Date:
DOI: https://doi.org/10.1023/A:1005098623123