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Relativity and Equivalence in Hilbert Space: A Principle-Theory Approach to the Aharonov–Bohm Effect

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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

  1. Expositions of the principle along similar lines are found in Einstein’s more formal review of the theory such as [9]. The conceptual relations between general covariance, relativity and symmetry are a matter of an enduring controversy [22]

  2. From Einstein’s reply to Friedrich Kottler, Annalen der Physik 51:639–642 (1916) as, quoted in [21].

  3. 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.

  4. See Footnote 2.

  5. A comprehensive discussion of the different interpretations is given in [3].

  6. Wallace [35] reaches a similar conclusion based on the AB effect.

  7. Different arguments for structural realism based on the group-theoretic structure of the electromagnetic interaction are provided by Lyre [19].

  8. The Aharonov–Bohm effect (like ’tHooft’s double slit experiment with a phase shifter [33]), is thus considered as an observation of (global) gauge symmetry in analogy to the observation of boost-symmetry in Newtonian mechanics using the Galilean ship experiment. See detailed discussion in [12].

  9. This issue is the subject of the lion’s of the contemporary philosophical discourse about the topological approach to the effect, see for example [23, 30] and references therein.

  10. See Footnote 3.

  11. 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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  1. Edelstein Center for the History and Philosophy of Science, The Hebrew University of Jerusalem, Jerusalem, Israel

    Guy Hetzroni

  2. Department of Natural Sciences, The Open University of Israel, Ra’anana, Israel

    Guy Hetzroni

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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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