Idealizations, essential self-adjointness, and minimal model explanation in the Aharonov–Bohm effect
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Two approaches to understanding the idealizations that arise in the Aharonov–Bohm (AB) effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations.
KeywordsIdealization Aharonov–Bohm effect Representation Models Explanation Topology Emergence and reduction
I am extremely grateful to John Earman, John D. Norton, Laura Ruetsche, Robert W. Batterman, James Woodward, Mark Wilson, Giovanni Valente, Nicholaos Jones, Bryan W. Roberts, Aaron Novick, and Samuel C. Fletcher, as well as audiences in numerous workshops and conferences, for insightful conversations and many constructive comments on earlier versions of this paper going back to 2014 under the title of “Topological Idealization, Asymptotic-Minimal Model Explanation, and the Aharonov–Bohm Effect.” Also thanks to Narin Shech for help with figures, and to Michel Smith and Cesar R. de Oliveira for assistance with technical issues with this version of the paper. Special thanks to John Earman for his guidance and mentorship regarding the details and issues discussed in this paper and over the years. Needless to say, my mistakes are my own, and I refer the reader to John Earman’s own more recent and excellent contribution in his “The Role of Idealization in the Aharonov–Bohm Effect” (http://philsci-archive.pitt.edu/12696/).
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