Operator calculus: the lost formulation of quantum mechanics
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Operator calculus: the lost formulation of quantum mechanics A mathematical reconstruction Gonzalo Gimeno1
· Mercedes Xipell2
· Marià Baig2
Received: 27 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in historical accounts of quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions. Communicated by Tilman Sauer. We are particularly grateful to Arianna Borrelli, whose previous (unpublished) work on the topic has inspired several sections of this paper; we also thank the members of the department of History of Science of the Technische Universität Berlin for their encouragement and suggestions. We also thank Anthony Duncan for sharing his points of view with us. This paper has additionally benefitted from the debates in the framework of the Fourth Conference on History of Quantum Physics (HQ-4), 2015, in Spain. We thank Pablo Soler and Miguel Gimeno for their careful review and comments. We thank Jaume Navarro and two anonymous referees for their suggestions, which have decidedly contributed to the readability and understandability of this work. We acknowledge the support of the Spanish Ministry of Science and Innovation (PID2019-105131GB-I00).
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Gonzalo Gimeno [email protected] Mercedes Xipell [email protected] Marià Baig [email protected]
1
Centre for the History of Science, Research Module C, Carrer de Can Magrans s/n, 08193 Bellaterra, Barcelona, Spain
2
Centre for the History of Science, Universitat Autònoma de Barcelona, Barcelona, Spain
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G. Gimeno et al.
1 Introduction In the first quarter of the twentieth century, soon after the appearance of the seminal papers on the new quantum mechanics, the physicist Pascual Jordan enumerated the four different formulations of the theory that existed at the time in a paper: Heisenberg’s matrix theory, Schrödinger’s wave mechanics, Dirac’s theory of q-numbers and Born and Wiener’s theory. Nearly eighty years later, in an account of nine formulations of quantum mechanics published in the American Journal of Physics (Styer et al. 2002), the last of these formulations had disappeared: the theory of Born and Wiener was not even mentioned. What exactly was this last formulation? Was there any reason to neglect it? In January 1926, shortly after the publication of the ea
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