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(Never) Mind Your P's and Q's: Von Neumann Versus Jordan on the Foundations of Quantum Theory

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(Never) Mind Your P's and Q's: Von Neumann Versus Jordan on the Foundations of Quantum Theory (Never) Mind your p’s and q’s: Von Neumann versus Jordan on the Foundations of Quantum Theory. a b, Anthony Duncan , Michel Janssen ∗ aDepartment of Physics and Astronomy, University of Pittsburgh bProgram in the History of Science, Technology, and Medicine, University of Minnesota Abstract In two papers entitled “On a new foundation [Neue Begr¨undung] of quantum me- chanics,” Pascual Jordan (1927b,g) presented his version of what came to be known as the Dirac-Jordan statistical transformation theory. As an alternative that avoids the mathematical difficulties facing the approach of Jordan and Paul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbert space formalism of quantum mechanics. In this paper, we focus on Jordan and von Neumann. Cen- tral to the formalisms of both 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 prob- lems in quantum mechanics. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canon- ically conjugate variables, i.e., p’s and q’s. For von Neumann, not constrained by the analogy to classical mechanics, it required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Jordan and von Neumann also stated the characteristic new rules for prob- abilities in quantum mechanics somewhat differently. Jordan (1927b) was the first to state those rules in full generality. Von Neumann (1927a) rephrased them and, arXiv:1204.6511v2 [physics.hist-ph] 31 Aug 2012 in a subsequent paper (von Neumann, 1927b), sought to derive them from more basic considerations. 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. Key words: Pascual Jordan, John von Neumann, transformation theory, probability amplitudes, canonical transformations, Hilbert space, spectral theorem Preprint submitted to The European Physical Journal H 3 September 2012 1 Introduction 1.1 The Dirac-Jordan statistical transformation theory On Christmas Eve 1926, Paul A. M. Dirac, on an extended visit to Niels Bohr’s institute in Copenhagen, wrote to Pascual Jordan, assistant to Max Born in G¨ottingen: 1 Dr. Heisenberg has shown me the work you sent him, and as far as I can see it is equivalent to my own work in all essential points. The way of obtaining the results may be rather different though . I hope you do not mind the fact that I have obtained the same results as you, at (I believe) the same time as you. Also, the Royal Society publishes papers more quickly than the Zeits. f. Phys., and I think my paper will appear in their January issue. I am expecting to go to G¨ottingen at the beginning of February, and I am looking forward very much to meeting you and Prof. Born there (Dirac to Jordan, December 24, 1916, AHQP). 2 Dirac’s paper, “The physical interpretation of the quantum dynamics” (Dirac, 1927), had been received by the Royal Society on December 2. Zeitschrift f¨ur Physik had received Jordan’s paper, “On a new foundation [Neue Begr¨un- dung] of quantum mechanics” (Jordan, 1927b), on December 18. 3 In both ⋆ This paper was written as part of a joint project in the history of quantum physics of the Max Planck Institut f¨ur Wissenschaftsgeschichte and the Fritz-Haber-Institut in Berlin. ∗ Corresponding author. Address: Tate Laboratory of Physics, 116 Church St. SE, Minneapolis, MN 55455, USA, Email: [email protected] 1 Reminiscing about the early days of quantum mechanics, Jordan (1955) painted an idyllic picture of G¨ottingen during this period, both of the town itself and of its academic life. J. Robert Oppenheimer, who spent a year there in 1926–1927, also remembered a darker side: “[A]lthough this society was extremely rich and warm and helpful to me, it was parked there in a very very miserable German mood which probably in Thuringia was not as horrible as a little bit further east; G¨ottingen is not in Thuringia —not as bad as in Thuringia—but it was close enough to Thuringia to be bitter, sullen, and, I would say, discontent and angry and loaded with all those ingredients which were later to produce a major disaster” (Interview with Oppenheimer for the Archive for History of Quantum Physics [AHQP], session 2, p. 5; quoted in part by Bird and Sherwin, 2005, pp. 57–58). For detailed references to material in the AHQP, see Kuhn et al. (1967). 2 This letter is quoted almost in its entirety (cf. note 42) and with extensive com- mentary by Mehra and Rechenberg (2000–2001, p. 72, pp. 83–87). 3 We will refer to this paper as Neue Begr¨undung I to distinguish it from Neue Begr¨undung II, submitted June 3, 1927 to the same journal, a sequel in which Jordan tried both to simplify and generalize his theory (Jordan, 1927g). A short 2 cases it took a month for the paper to get published: Dirac’s appeared Jan- uary 1, Jordan’s January 18. 4 What Dirac and Jordan, independently of one another, had worked out and presented in these papers has come to be known as the Dirac-Jordan (statistical) transformation theory. 5 As Jordan wrote in a volume in honor of Dirac’s 70th birthday: After Schr¨odinger’s beautiful papers [Schr¨odinger, 1926a], I formulated what I like to call the statistical transformation theory of quantum mechanical systems, answering generally the question concerning the probability of find- ing by measurement of the observable b the eigenvalue b′, if a former mea- surement of another observable a had given the eigenvalue a′. The same answer in the same generality was developed in a wonderful manner by Dirac (Jordan, 1973, p. 296; emphasis in original). In our paper, we focus on Jordan’s version of the theory and discuss Dirac’s version only to indicate how it differs from Jordan’s. 6 Exactly one month before Dirac’s letter to Jordan, Werner Heisenberg, Bohr’s assistant in the years 1926–1927, had already warned Jordan that he was about to be scooped. In reply to a letter, apparently no longer extant, in which Jordan must have given a preview of Neue Begr¨undung I, he wrote: I hope that what’s in your paper isn’t exactly the same as what’s in a paper Dirac did here. Dirac’s basic idea is that the physical meaning of Sαβ, given in my note on fluctuations, can greatly be generalized, so much so that it covers all physical applications of quantum mechanics there have been so far, and, according to Dirac, all there ever will be (Heisenberg to Jordan, November 24, 1926, AHQP). 7 The “note on fluctuations” is a short paper received by Zeitschrift f¨ur Physik on November 6 but not published until December 20. In this note, Heisenberg (1926b), drawing on the technical apparatus of an earlier paper on resonance version of Neue Begr¨undung I was presented to the G¨ottingen Academy by Born on behalf of Jordan in the session of January 14, 1927 (Jordan, 1927c). 4 The extensive bibliography of Mehra and Rechenberg (2000–2001) gives the dates papers were received and published for all the primary literature it lists. 5 The term ‘transformation theory’—and even the term ‘statistical transformation theory’—is sometimes used more broadly (see cf. note 50). 6 For discussions of Dirac’s version, see, e.g., Jammer (1966, pp. 302–305), Kragh (1990, pp. 39–43), Darrigol (1992, pp. 337–345), Mehra and Rechenberg (2000–2001, 72–89), and Rechenberg (2010, 543–548). 7 Mehra and Rechenberg (2000–2001, p. 72) quote more extensively from this let- ter. They also quote and discuss (ibid., p. 78) a similar remark in a letter from Heisenberg to Wolfgang Pauli, November 23, 1926 (Pauli, 1979, Doc. 148, p. 357). In this letter, Heisenberg talks about Dirac’s “extraordinary grandiose [großz¨ugige] generalization” (quoted by Kragh, 1990, p. 39). 3 phenomena (Heisenberg, 1926a), analyzed a simple system of a pair of iden- tical two-state atoms perturbed by a small interaction to allow for the flow- ing back and forth of the amount of energy separating the two states of the 8 atoms. The Sαβ’s are the elements of a matrix S implementing a canonical 1 transformation, f ′ = S− fS, from a quantum-mechanical quantity f for the unperturbed atom to the corresponding quantity f ′ for the perturbed one. The discrete two-valued indices α and β label the perturbed and unperturbed states, respectively. Heisenberg proposed the following interpretation of these matrix elements: 2 If the perturbed system is in a state α, then Sαβ gives the probability that (because of collision processes, because the perturbation| | suddenly stops, etc.) the system is found to be in state β (Heisenberg, 1926b, p. 505; emphasis in the original). Heisenberg (1926b, pp. 504–505) emphasized that the same analysis applies to fluctuations of other quantities (e.g., angular momentum) and to other quantum systems with two or more not necessarily identical components as long as their spectra all share the same energy gap. In the introduction of his paper on transformation theory, Dirac (1927, p.
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