Abstract
This paper formulates generalized versions of the general principle of relativity and of the principle of equivalence that can be applied to general abstract spaces. It is shown that when the principles are applied to the Hilbert space of a quantum particle, its law of coupling to electromagnetic fields is obtained. It is suggested to understand the Aharonov–Bohm effect in light of these principles, and the implications for some related foundational controversies are discussed.
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Notes
From Einstein’s reply to Friedrich Kottler, Annalen der Physik 51:639–642 (1916) as, quoted in [21].
To avoid ambiguities, it is emphasized that the terms ‘passive’ and ‘active’ are used throughout this paper to distinguish between transformations of the mathematical representation (that do not alter the physical state), and transformations which replace one physical configuration with a different one.
See Footnote 2.
A comprehensive discussion of the different interpretations is given in [3].
Wallace [35] reaches a similar conclusion based on the AB effect.
Different arguments for structural realism based on the group-theoretic structure of the electromagnetic interaction are provided by Lyre [19].
See Footnote 3.
This is an approximation assuming that the wave packets are fast, such that they do not change their form during the experiment.
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Acknowledgements
I would like to thank Yemima Ben-Menahem and Daniel Rohrlich for their guidance and advice, and an anonymous reviewer for his helpful feedback. This research was supported by the Israel Science Foundation (Grant No. 1190/13) and by The Open University of Israel’s Research Fund (Grant No. 41240).
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Hetzroni, G. Relativity and Equivalence in Hilbert Space: A Principle-Theory Approach to the Aharonov–Bohm Effect. Found Phys 50, 120–135 (2020). https://doi.org/10.1007/s10701-020-00322-y
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DOI: https://doi.org/10.1007/s10701-020-00322-y