The non-ideal theory of the Aharonov–Bohm effect

Abstract

Elay Shech and John Earman have recently argued that the common topological interpretation of the Aharonov–Bohm (AB) effect is unsatisfactory because it fails to justify idealizations that it presupposes. In particular, they argue that an adequate account of the AB effect must address the role of boundary conditions in certain ideal cases of the effect. In this paper I defend the topological interpretation against their criticisms. I consider three types of idealization that might arise in treatments of the effect. First, Shech takes the AB effect to involve an idealization in the form of a singular limit, analogous to the thermodynamic limit in statistical mechanics. But, I argue, the AB effect itself features no singular limits, so it doesn’t involve idealizations in this sense. Second, I argue that Shech and Earman’s emphasis on the role of boundary conditions in the AB effect is misplaced. The idealizations that are useful in connecting the theoretical description of the AB effect to experiment do interact with facts about boundary conditions, but none of these idealizations are presupposed by the topological interpretation of the effect. Indeed, the boundary conditions for which Shech and demands justification are incompatible with some instances of the AB effect, so the topological interpretation ought not justify them. Finally, I address the role of the non-relativistic approximation usually presumed in discussions of the AB effect. This approximation is essential if—as the topological interpretation supposes—the AB effect constrains and justifies a relativistic theory of the electromagnetic interaction. In this case the ends justify the means. So the topological view presupposes no unjustified idealizations.

The Aharonov-Bohm effect in a closed flux line

“…This debate is currently unsettled. Many papers have addressed this debate [43,44,45,46,47,48,49,50,51,52] and proposed other models that intend to explain the AB effect [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76]. This debate may be naturally translated to the case of the AB effect involving a vector potential outside a closed flux line and a magnetic field confined along this closed flux line.…”

How to Choose a Gauge? The Case of Hamiltonian Electromagnetism

“…For recent controversy about such idealizations, cf Shech (2018),Earman (2019). andDougherty (2020).…”

Scientific understanding in the Aharonov‐Bohm effect

“…The debate concerned both the nature and reality of (what has come to be known as) the (magnetic) "AB effect", as well as various foundational issues. Philosophers of physics have been attentive to the AB effect (e.g., Batterman, 2003;Belot, 1998;Daugherty, 2021;Healey, 1997Healey, , 1999Healey, , 2007Mattingly, 2006;Maudlin, 1998;Wallace, 2014), and some recent contributions (viz., Shech, 2018;Earman, 2019) have focused specifically on the role that idealisations play, suggesting that they provide genuine understanding. For instance, Earman (2019Earman ( , p. 1991 holds that the point of studying the idealised AB effect "is to understand the foundations of quantum mechanics and how its predictions depart from those of classical mechanics".…”