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(Never) Mind your p's and q's: Von Neumann versus Jordan on the Foundations of Quantum Theory

Duncan, Anthony and Janssen, Michel (2012) (Never) Mind your p's and q's: Von Neumann versus Jordan on the Foundations of Quantum Theory. [Preprint]

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In early 1927, Pascual Jordan (1927b) published his version of what came to be known as the Dirac-Jordan statistical transformation theory. Later that year and partly in response to Jordan, John von Neumann (1927a) published the modern Hilbert space formalism of quantum mechanics. Central to both formalisms are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems in the new theory. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canonically conjugate variables, i.e., p’s and q’s. Jordan (1927e) ran into serious difficulties when he tried to extend his approach from quantities with fully continuous spectra to those with wholly or partly discrete spectra. For von Neumann, not constrained by the analogy to classical physics and aware of the daunting mathematical difficulties facing the approach of Jordan (and, for that matter, Dirac (1927)), the solution of a problem in the new quantum mechanics required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Related to their disagreement about the appropriate general formalism for the new theory, Jordan and von Neumann stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan (1927b) was the first to state those rules in full generality, von Neumann (1927a) rephrased them and then sought to derive them from more basic considerations (von Neumann, 1927b). In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and Nordheim (1928). We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics.

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Item Type: Preprint
Duncan, Anthony
Janssen, Michel
Additional Information: The main difference between this new version and the old one is that the introduction has been completely rewritten. Sec. 1 (pp. 2–12) of the old version has been replaced by Secs. 1.1-1.4 (pp. 2–31) in the new version. This has resulted in a number of additional entries in the bibliography (e.g., Born 1926c and Heisenberg 1927a). The rest of the paper has stayed the same except for the occasional footnote (see, e.g., notes 71, 94, 107, 133, 134, and 137).
Keywords: Pascual Jordan, John von Neumann, transformation theory, probability amplitudes, canonical transformations, Hilbert space, spectral theorem
Subjects: Specific Sciences > Physics > Quantum Mechanics
General Issues > Theory Change
Depositing User: Michel Janssen
Date Deposited: 23 Aug 2012 15:59
Last Modified: 23 Aug 2012 15:59
Item ID: 9283
Subjects: Specific Sciences > Physics > Quantum Mechanics
General Issues > Theory Change
Date: 29 April 2012

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