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*Slobodan Perovic*
Why Were Logically Distinct Theories Deemed Equivalent in Early Quantum Mechanics?

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## Summary

Recently, several philosophers have carefully scrutinized the key arguments pursued by physicists at the beginning of the Quantum Revolution. Based on their analysis of the physicists’ arguments, these philosophers have characterized some of the essential agreements between physicists as unsubstantiated and unjustified. Most notably, the supposed equivalence of the two competing accounts of quantum phenomena, namely V. Heisenberg’s Matrix Mechanics (MM) and E. Schrödinger’s Wave Mechanics (WM), was recently debunked by the rethinking of the history of the debate over the foundations of quantum theory.

As MM and WM differed substantially both in terms of the mathematical techniques they employed and in terms of ontological assumptions about the microphysical systems at stake, and were successful in accounting for two distinct sets of experimental results, the supposed equivalence of the two was perceived as a major breakthrough when first conceptualized.

F. A. Muller (1997), however, argues, that the agreement concerning equivalence was unjustified and that only later developments in the late 1920s and early 1930s, and especially the work of a mathematician John von Neumman (1932) provided the sound proof of equivalence, as opposed to the famous proof provided by Schrödinger (1927) and the attempts by others (Eckart; Dirac; Pauli). If this re-evaluation is true, it would imply that the wide agreement among physicists on the equivalence of two formalisms in the mid-1920s, on which further developments of the theory were critically predicated, was an unjustified, indeed an irrational, act of faith (“a myth” – Muller) on the part of the physics community.

Thus, given that even in its best moments, the practice of physics does not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, one might well question the very foundations of rationality in physics.

In response, I will argue that rationality in physics appears elusive even in its key moments only if we premise our analysis of actual scientific practice on narrow models of scientific knowledge. These models, such as that of P. Suppes (1957, 1960) used in the above-outlined analysis of the equivalence case, focus their conceptual and historical analysis on the aspects of scientific knowledge committed to the mathematical-logical analysis of the formalisms (such as MM and WM), which, although indispensable in scientific practice, may not be even the most important mark of its rationality. Such a narrowly-focused analysis is bound to miss some key aspects of the physicists’ arguments, embedded as they are in philosophical and historical contexts, contexts which must be unraveled if one is to do justice to the physicists’ thinking.

More specifically, with respect to the case of equivalence, the kind of equivalence that was pursued at a later stage by Von Neumann, and which allegedly represents a moment of lucidity in the overwhelming messiness of the development of QM, was only a one possible refinement of the previous agreement on the preliminary, yet both experimentally and conceptually sufficiently substantiated, concept of equivalence. Some of the key early experimental results demonstrated that the N. Bohr’s (1985) model of atom provides a satisfactory common ground, unifying the opposed formalisms of MM and WM in very tangible terms and setting the groundwork for further exploration of their relationship. The formal proofs were only a part of this overall effort. Only at a later stage of development were the proofs worked out in precisely the terms that the above-mentioned historical and conceptual analyses that I criticize, takes to be the subject of the agreement on equivalence of the mid-1920s. Indeed, the claim of equivalence was based on the postulates of Bohr’s model concerning the so-called stationary states (i.e., the permitted energy states) of the atom that proved to be indispensable in both formalisms, which, while formulated in different ways, turned out to be reconcilable within the framework of Bohr’s model. Thus, the 1920s agreement on equivalence appears to be a myth only if we leave out the ontological (and focus exclusively on a purely formal goal) which was shared by the physics community of that time, of providing a coherent overall model of the atom. Although the equivalence of the 1920s was provisional, it was justified in virtue of its ontological aim.