Abstract
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 nonsimply 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 selfadjoint 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.
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Notes
An astute reader will notice that, as defined, the AB effect cannot be manifested in the laboratory, but the effect is by now almost universally accepted as empirically confirmed. The tension between the AB effect as a concrete, phenomenological effect that has been confirmed, on the one hand, and its conventional definition, on the other, will be the main focus of this paper. Compare the situation with phase transitions (Shech 2013): Boiling kettles and magnetized iron are finite systems that we interact with regularly. Yet, according to our best theories of phase transitions “... the existence of a phase transition requires an infinite system. No phase transitions occur in systems with a finite number of degrees of freedom” (Kadanoff 2000, p. 238). In addition, it should be added that whether one considers the “AB effect” to have been confirmed will depend on controversial issues such as the local or nonlocal nature and physical basis (i.e., reality of vector potential) of the effect. E.g., see recent Magni (1995, p. 186):
Finally, the reader might wish that a definite answer be given to the vexed question about the existence or nonexistence of the Aharonov–Bohm effect. In order to answer it we have to split the question in two:

(a)
If, under the name of the Aharonov–Bohm effect a shifting of interference pattern is meant, then our answer is that, at any rate, the Schrödinger equation leads exactly to this prediction.

(b)
If, on the other hand, what is meant is a shifting due to any independent effect of the vector potential then our answer is no, there is no such thing coming from the Schrödinger equation.

(a)
This paper is a development of my Shech (2015, Section 5). See Earman (2016) for similar claims, a masterful philosophical analysis of the AB effect controversy in the physics literature, and an identification of the connection between the AB effect idealizations and the existence of unitarily inequivalent representations of the canonical commutation relations.
This is the magnetic AB effect. There is an analogue electric AB effect, implied by Lorentz covariance, which arises from electric (instead of magnetic) fields. See Peshkin and Tonomura (1989) and Tonomura (1999, 2010) for a review of the effect and its experimental confirmation, along with a selective review of the debate that arose with respect to the reality of the effect in the physics literature. Ehrenberg and Siday (1949) are usually credited with first noting the effect and Chambers (1960) with being the first experimental confirmation. See Hiley (2013) for the early history of the AB effect. Tonomura et al. (1982, 1986) are considered the first definitive confirmation of the effect and Caprez et al. (2007) further confirms the effect’s purely quantum mechanical origin. As Tonomura (1999, Ch. 6) explains, the definitive confirmation of the AB effect by Tonomura et al. (1982, 1986) did not make use of infinite and impenetrable solenoids. Instead, superconducting toroidal solenoids where used. Such solenoids are prepared so as to minimize magnetic field leakage and maximally shield the solenoid from an electron beam with a copper and niobium coating. A shift in interference pattern was observed. Although part of the shift could be explained in terms of the unideal conditions and other sources of error, these aspects could not account for the entire shift in the pattern. For example, it was calculated that the magnetic field leakage can affect the phase of an incident electron beam by at most \(\pi \), but the observed relative phase shift was about \(12\,\,\pi \). For these reasons there is little doubt now that the (concrete) AB effect is a real physical effect and not an artifact of idealizations.
Roberts (2016) notes that there is a generalized Stone’s theorem (see Cooper 1947, 1948) for maximal symmetric operators. An operator H is symmetric if it has the property that \(H\Psi =H^{*}\Psi \) for all \(\Psi \) in the common domain of H and its adjoint \(H^{*}\). An operator is maximal symmetric if it does not admit any selfadjoint extensions. Since, in the context discussed here, \(H_{AB}^I\) does have selfadjoint extensions, the generalized Stone’s theorem is inapplicable.
The domain of \({\bar{H}}_{AB}^I\) is \(Dom({\bar{H}}_{AB}^I)=\mathcal{H}^{2}( {S_{out} })\cap \mathcal{H}_0^1( {S_{out} })\) where \(\mathcal{H}^{d}\) denotes the usual Sobolev space of square integrable functions so that \(\mathcal{H}^{2}\) is the domain of the free Hamiltonian (i.e., the negative Laplacian \(\Delta )\) in \(L^{2}( {S_{out} })\) and \(\mathcal{H}_0^1( {S_{out} } )\) is a subspace of \(\mathcal{H}^{1}( {S_{out} })\) with elements vanishing (in the sense of Sobolev traces) at the solenoid boundary of \(S_{in} \). Note that the Dirichlet boundary conditions are selected because of the space \(\mathcal{H}_0^1( {S_{out} })\) in the limit operator domain. More precisely, \({\bar{H}}\) is the Friedrichs extensions of the formal operator \(\mathop \sum \nolimits _{i=1}^3( {i\frac{\partial }{\partial x_i }+\frac{q}{c}A_i })^{2}\) with homogeneous Dirichlet conditions at the boundary of \(S_{in} \). Also notice that \(Dom({\bar{H}}_{AB}^I)\supset C_0^\infty ( {S_{out} } )\). See Magni and ValzGris (1995) and de Oliveira and Pereira (2008, 2011) for details.
See Healey (2007, Ch. 1–2) for an introduction.
Sentiments of this sort arise in, among others, Aharonov and Bohm (1959), Batterman (2003), Belot (1998, p. 544), Lyre (2001, 2009), Nounou (2003), Peshkin and Tonomura (1989), Ryder (1996), and Wu and Yang (1975, p. 3845). Healey (2007), while agreeing that topological considerations may be important, disagrees with Batterman’s (2003) and Nounou’s (2003) claim that the AB effect occurs because of the topology of the base manifold.
The idea that “different selfadjoint extensions of a symmetric operator may lead to very different physics” was studied in detail first in Ruijsenaars (1983, p. 3).
Butterfield (2011, Section 3) discusses similar distinctions. In particular, he makes a distinction between a system\(\sigma \left( N \right) \) that depends on some parameter N (let \(\{\sigma \left( N \right) \)} denote a sequence of such systems), a quantity defined on the system \(f\left( {\sigma \left( N \right) } \right) \) (let \(\left\{ {f\left( {\sigma \left( N \right) } \right) } \right\} \) denote a sequence of quantities on successive systems), and a (real number) value\(v\left( {f\left( {\sigma \left( N \right) } \right) } \right) \) of quantities on successive systems (where a sequence of states on \(\sigma \left( N \right) \) is implicitly understood; let \(\left\{ {v\left( {f\left( {\sigma \left( N \right) } \right) } \right) } \right\} \) denote a sequence of values on successive systems). A limit system\(\sigma \left( \infty \right) \) arises when \(\mathop {\lim }\nolimits _{N\rightarrow \infty } \left\{ {\sigma \left( N \right) } \right\} \) is welldefinedotherwise there is no limit system. A property of a limit system refers to the value \(v(f\left( {\sigma \left( \infty \right) } \right) \) of the (natural) limit quantity \(f\left( {\sigma \left( \infty \right) } \right) \) (in the natural limit state) on \(\sigma \left( \infty \right) \). A limit property\(v(f\left( {\sigma \left( N \right) } \right) \) is a limit of a sequence of values of quantities on successive systems (or, values on the systems on the way to the limit) and is welldefined when \(\mathop {\lim }\nolimits _{N\rightarrow \infty } \left\{ {v\left( {f\left( {\sigma \left( N \right) } \right) } \right) } \right\} \) exists. The question of whether a property of a limit system equals the corresponding limit property asks if the following holds: \(v(f\left( {\sigma \left( \infty \right) } \right) =\mathop {\lim }\nolimits _{N\rightarrow \infty } \left\{ {v\left( {f\left( {\sigma \left( N \right) } \right) } \right) } \right\} \) (assuming both are welldefined).
In discussing limiting procedures I’m mostly concentrating on the cylindrical solenoid case as in Magni and ValzGris (1995) and de Oliveira and Pereira (2008, 2010, 2011). See Fig. 4 for the toroidal solenoid scenario where the limit taken is that of infinite impenetrability as in Ballesteros and Weder (2009, 2011) and de Oliveira and Pereira (2011).
Notwithstanding the work of de Oliveira and Pereira (2010) which surely puts in question our right to assume a limit system in which Dirichlet boundary conditions hold, my point here is that evenif we allow for such a limit system, the claim that an idealized solenoid wellapproximates actual systems is left without justification.
An anonymous referee notes that a finite solenoid topology gives \({\mathbb {R}}^{3}\) minus a point or minus a closed ball, and is topologically \(S^{2}\times {\mathbb {R}}\), which is simply connected by not homeomorphic to \({\mathbb {R}}^{3}\). Also, whether or not the limiting procedures advertised in Sect. 4 will succeed or fail may well depend on the choice of configuration space that we begin with. From my perspective, if we take my instances of referring to \({\mathbb {R}}^{3}\) to implicitly mean \({\mathbb {R}}^{3}\) minus a point or minus a closed ball so that the limiting procedures work, my arguments would still hold since both spaces are simply connected and the dispensability of an idealized nonsimply connected space succeeds.
Compare with Shech (2013) for the paradox of phase transitions.
Since \(\{ {H_{L,n} }\}\) are unbounded operators, there is no obvious way of defining a limiting procedure. Magni and ValzGris (1995, p. 180) note that, at the very least, \(\mathop {\lim }\nolimits _{n,L\rightarrow \infty } \{ {H_{L,n} }\}=H_{\infty ,\infty } \) should be the generator of the limit time evolution, and it is for this reason that I discuss the strong resolvent limit in the text. In other words, and roughly, if our system is initially in the state \(u_0 \), then the selfadjointness of \(H_{L,n} \) means that the time evolution \(u( t)=e^{iH_{L,n} t}u_0 \) is well defined and we can consider the associated one parameter groups \(\{ {U( t )} \}\) such that \(u( t)=U( t)u_0\) . \(\mathop {\lim }\nolimits _{n,L\rightarrow \infty } \{ {H_{L,n} }\}\) is then defined to be the generator of the limit evolution group \(\{ {U( t)}\}\) so that \(u( t)=e^{iH_{\infty ,\infty } t}u_0 \). The limiting procedure that Magni and ValzGris (1995, pp. 181–182) discuss is one in which \(H_n =\mathop \sum \nolimits _{i=1}^3 ( {i\frac{\partial }{\partial x_i }+\frac{q}{c}A_i } )^{2}+nV\) has domain \(Dom( {H_n })=\mathcal{H}^{2}( {{\mathbb {R}}^{3}})\forall n\) and \(\mathop {\lim }\nolimits _{n\rightarrow \infty }\{ {H_n }\}=H_\infty \) has domain \(Dom( {H_\infty })=\mathcal{H}^{2}( {S_{out} })\cap \mathcal{H}_0^1 ( {S_{out} })\), where \(\mathcal{H}^{d}\) is the usual Sobolev space of square summable functions with square summable derivatives up to order d (although it seems we can equally consider the space of square integrable functions), and \(\mathcal{H}_0^1 ( {S_{out} })\) is the subspace of \(\mathcal{H}^{1}( {S_{out} })\) with elements vanishing (in the sense of Sobolev traces) at the boundary of \(S_{in} \). They then explain that \(H_\infty \) is nothing but a precise form of the operator \({\bar{H}}_{AB}^I\) introduced by Aharonov and Bohm (1959) with homogenous Dirichlet conditions at the boundary of \(S_{in} \), which has the domain \(Dom({\bar{H}}_{AB}^I )=\mathcal{H}^{2}( {S_{out} } )\cap \mathcal{H}_0^1 ( {S_{out} })\). de Oliveira and Pereira (2008)’s discussion builds upon the results of Magni and ValzGris (1995) and is more general. \(S_{in}\) is defined as \(S_{in} =\{( {x_1 ,x_2 ,x_3 }):x_1^2 +x_2^2<r_0 ^{2}, {x_3 }<L\}\), where \((x_1 ,x_2 ,x_3 )\) denote Cartesian coordinates in \({\mathbb {R}}^{3}\), \(r_0 >0\) is radius of solenoid S with finite length \(2L>0\), and \(S_{out} \) is the exterior region. Note that, whereas in the main text I assume \(S_{in} \) includes the boundary \(S=\partial S_{in} \), here \(S_{out} ={\mathbb {R}}^{3}\backslash S\cup S_{in}\). The Hamiltonians \(\{ {H_{L,n} }\}\) have a domain \(Dom( {H_{L,n} } )=\mathcal{H}^{2}( {{\mathbb {R}}^{3}})\), where \(\mathcal{H}^{2}\) denotes the usual Sobolev space domain of the free Hamiltonian (i.e., the negative Laplacian \(\Delta \)) in \(L^{2}( {S_{out} })\). de Oliveira and Pereira (2011) note that \(C_0^\infty ( {{\mathbb {R}}^{3}} )\) is the core of \(H_{L,n} \) (i.e., \(H_{L,n} \) restricted to \(C_0^\infty ( {{\mathbb {R}}^{3}})\) has just one selfadjoint extension). \({\varvec{A}}_{\varvec{L}} \) is chosen such that as \(L\rightarrow \infty \), there is a pointwise convergence, i.e., a convergence in the pointwise topology, of \({\varvec{A}}_{\varvec{L}}\) to \({\varvec{A}}\). de Oliveira and Pereira (2008) show that the domain of \(\mathop {\lim }\nolimits _{n,L\rightarrow \infty } \{ {H_{L,n} }\}=H_{\infty ,\infty } \) is \(Dom( {H_{\infty ,\infty } } )=\mathcal{H}^{2}( {S_{out} })\cap \mathcal{H}_0^1 ( {S_{out} })\) so that \(\mathop {\lim }\nolimits _{n,L\rightarrow \infty }\{ {H_{L,n} } \}={\bar{H}}_{AB}^I\) holds in the strong resolvent sense. They note that, except in the case of the \(L\rightarrow \infty \) in \({\mathbb {R}}^{3}\) (since they assume that the border region of the solenoid is a bounded set), \(H_{L,n} \) uniformly converges to \({\bar{H}}_{AB}^I\), i.e., with the uniform topology, and this is why de Oliveira and Pereira (2011) are able to extend said results to the norm resolvent sense in the \(n\rightarrow \infty \) case. An important point in the work of Magni and ValzGris (1995) and de Oliveira and Pereira (2008, 2011) is that the selection of Dirichlet boundary conditions arises through the modeling of the impenetrability process of \(n\rightarrow \infty \) in which the repulsive barrier becomes infinitely impenetrable and creates a hole in the space; they are not just chosen without justification. Last, for completeness, note that an additional idealization that I do not discuss was studied by Weisskopf (1961) via a limiting process and concerns the fact that the electromagnetic field associated with the solenoid will interact with surrounding particles when the solenoid is being turned on and off.
More generally, we may think of the AB effect as a dependency of the behavior of electrons, such as in scattering experiments or in doubleslit experiments, on magnetic flux in regions where the wavefunction and flux do not overlap (at least not to the extent that the overlapping can account for the dependency).
See Kadanoff (2000) for accounts of phase transitions and critical phenomena, and renormalization group techniques.
Trigger Warning: notions such as “fixed point” and “universality class” are being used loosely here, extended well beyond their usual scope.
See Wayne (2009) for a criticism of the idea that singular limits are a mark of emergence.
The name is due to Nobel laureate Wilczek (1982). Unfortunately, it is beyond the scope of this paper to discuss anyons (which are not paraparticles) or the FQHE. See Wilczek (1990) for a classic collection of papers on anyons, Ezawa (2013) (and references therein) for the FQHE, and Shech (2015) and Bain (2016) for a philosophical analysis.
Note that the purely quantum mechanical origin of the AB effect, although by now a common view, is still contested among some, e.g., Boyer (2008) and Lyre (2009). Lyre (2009) specifically notes that the AB effect is not quantum mechanical in origin because of the analogue classical gravitational AB effect. Moreover, notice that I’m not saying that the issue of nonlocality or nonseparability has been settled—see Healey (2007), especially Ch. 1–2, for a review of the state of the literature on this matter and Wallace (2014) for a recent local account. Instead, I only claim that part of what is at stake in the debate about nonlocality/nonseparability becomes clearer once it is realized that quantum and classical theories make strikingly different predictions about highly idealized systems (viz., an infinitely long and absolutely impenetrable cylindrical solenoid or an absolutely impenetrable toroidal solenoid).
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Acknowledgements
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, AsymptoticMinimal 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://philsciarchive.pitt.edu/12696/).
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Shech, E. Idealizations, essential selfadjointness, and minimal model explanation in the Aharonov–Bohm effect. Synthese 195, 4839–4863 (2018). https://doi.org/10.1007/s1122901714286
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DOI: https://doi.org/10.1007/s1122901714286
Keywords
 Idealization
 Aharonov–Bohm effect
 Representation
 Models
 Explanation
 Topology
 Emergence and reduction