Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-17T09:52:13.382Z Has data issue: false hasContentIssue false
  • English
  • Français

Logic, Probability, and Quantum Theory

Published online by Cambridge University Press:  14 March 2022

Arthur I. Fine
Arthur I. Fine*
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to present and discuss a probabilistic framework that is adequate for the formulation of quantum theory and faithful to its applications. Contrary to claims, which are examined and rebutted, that quantum theory employs a nonclassical probability theory based on a nonclassical “logic,” the probabilistic framework set out here is entirely classical and the “logic” used is Boolean. The framework consists of a set of states and a set of quantities that are interrelated in a specified manner. Each state induces a classical probability space on the values of each quantity. The quantities, so considered, become statistical variables (not random variables). Such variables need not have a “joint distribution.” For the quantum theoretic application, there is a uniform procedure that defines and determines the existence of such “joint distributions” for statistical variables. A general rule is provided and it is shown to lead to the usual compatibility-commutivity requirements of quantum theory. The paper concludes with a brief discussion of interference and the misunderstandings that are involved in the false move from interference to nonclassical probability.

Type
Research Article
Information
Philosophy of Science , Volume 35 , Issue 2 , June 1968 , pp. 101 - 111
Copyright
Copyright © 1968 The Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bohm, D. and Bub, J., Reviews of Modern Physics, Vol. 38, 1966, p. 453.10.1103/RevModPhys.38.453CrossRefGoogle Scholar
[2] Bohm, D. and Bub, J., Reviews of Modern Physics, Vol. 38, 1966, p. 470.10.1103/RevModPhys.38.470CrossRefGoogle Scholar
[3] Cohen, Leon, Philosophy of Science, Vol. 33, 1966, p. 317.10.1086/288104CrossRefGoogle Scholar
[4] Feynman, R. P., in Proc. Second Berkeley Symposium in Mathematical Statistics and Probabilities, 1951, p. 533.Google Scholar
[5] Jauch, J. M. and Piron, C., Helvetica Physica Acta, Vol. 36, 1963, p. 827.Google Scholar
[6] Kolmogorov, A. N., Foundations of the Theory of Probability (2nd English ed.), Chelsea Publishing Co., New York, 1954. First published as Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin, 1933.Google Scholar
[7] Landé, Alfred, New Foundations of Quantum Mechanics, Cambridge University Press, 1965.Google Scholar
[8] Mackey, G. W., The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin Inc., New York, 1963.Google Scholar
[9] Margenau, Henry, Annals of Physics, Vol. 23, 1963, p. 469.10.1016/0003-4916(63)90264-1CrossRefGoogle Scholar
[10] Menger, Karl, in Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probabilities, Vol. 2, 1954, p. 215.Google Scholar
[11] Menger, Karl, in Measurement: Definitions and Theories (eds. Churchman, C. W. and Ratoosh, P.), John Wiley & Sons, New York, 1959, pp. 97–128.Google Scholar
[12] von Neumann, John, Mathematical Foundations of Quantum Mechanics (trans. R. T. Beyer), Princeton University Press, Princeton, 1955, Chap. III, Sec. 5.Google Scholar
[13] Piron, C., Helvetica Physica Acta, Vol. 37, 1964, p. 439.Google Scholar
[14] Putnam, Hilary, in Beyond The Edge of Certainty (ed. Colodny, R. G.), Prentice-Hall, New Jersey, 1965, pp. 75–101.Google Scholar
[15] Reichenbach, Hans, Philosophic Foundations of Quantum Mechanics, University of California Press, Berkeley, 1944.Google Scholar
[16] Segal, I. E., Annals of Mathematics, Vol. 48, 1947, p. 930.10.2307/1969387CrossRefGoogle Scholar
[17] Schwinger, J., Proceedings of the National Academy of Sciences of the United States of America, Vol. 45, 1959, p. 1542.10.1073/pnas.45.10.1542CrossRefGoogle Scholar
[18] Schwinger, J., Proceedings of the National Academy of Sciences of the United States of America, Vol. 46, 1960, pp. 257, 570.10.1073/pnas.46.2.257CrossRefGoogle Scholar
[19] Suppes, Patrick, Philosophy of Science, Vol. 33, 1966, p. 14.10.1086/288067CrossRefGoogle Scholar
[20] Varadarajan, V. S., Communications on Pure and Applied Mathematics, Vol. 15, 1962, p. 189.10.1002/cpa.3160150207CrossRefGoogle Scholar