(Never) Mind your p’s and q’s: Von Neumann versus Jordan on the foundations of quantum theory

Abstract

In two papers entitled "On a new foundation [Neue Begründung] of quantum mechanics," 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. Central 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 problems 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 canonically 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 probabilities in quantum mechanics somewhat differently. Jordan (1927b) was the first to state those rules in full generality. Von Neumann (1927a) rephrased them and, in a subsequent paper , 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.

Time and quantum theory: A history and a prospectus

“…This became the modern Schrödinger picture, in which the state but not the observables varies with time and the dynamics is given by the Schrödinger equation, which provides a alternative to the Heisenberg picture suggested by matrix mechanics (in which the observables vary according the the Heisenberg equations of motion rather than the state). There is more to say here about the emergence of the modern notion of state and of measurement, some of which is covered by Duncan & Janssen (2013), but we can see that already these developments were fatal to the idea of the quantum jump as a discontinuous transition between stationary states, which relied critically on the hypothesis that a system remain in an energy eigenstate at all times.…”

Entropy from Carnot to Bekenstein

“…3 After being convinced of the mechanical equivalence of heat by James Joule's experiments, 1 A brief historical account of the work of Carnot, and its influence on the later developments of Thomson and Clausius is given in [1]. 2 This requires in particular that the two reservoirs are not in thermal contact. 3 Thomson wrote: "The characteristic property of the scale which I now propose is, that all degrees have the same value; that is, that a unit of heat descending from a body A at the temperature T • of this scale, to a body B at the temperature (T − 1) • , would give out the same mechanical effect, whatever be the number T ."…”

“…In his Nobel Prize lecture [https://www.nobelprize.org/prizes/physics/1954/born/lecture/] Max Born gives an overview of the emergence of the statistical interpretation of quantum mechanics. For an in depth historical look, focusing on the approaches of Jordan and von Neumann, see Ref [2]…”

On the Tension between Ontology and Epistemology in Quantum Probabilities

“…First, it is noteworthy that while Heisenberg was concerned with measurement "error" or "disturbance", the connections of these with the notion of standard deviation is not straightforward [46], and so, without further qualifications, (U 3 ) does not follow from, or entails, (U 2 ). Second, and even more important, Heisenberg's microscope thought experiment, and the "disturbance" view associated with it, consist of an alternative to Born's statistical interpretation of the state vector [11], or to Jordan's version of quantum theory [25], as yet another context where probabilities enter into the formalism of QM. At that time, recall, the founding fathers of QM were indecisive with respect to the statistical character of the theory.…”