Earman, John (2024) Explaining the Aharonov-Bohm Effect. [Preprint]
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Abstract
In an often invoked setup used to illustrate the Aharonov-Bohm (AB) effect, charged particles passing on opposite sides of an idealized infinitely long solenoid, with no flux leaks and protected from penetration of the charged particles by an infinitely high potential barrier, experience differential phase changes that result in an observable interference pattern dependent on the magnetic flux in the solenoid, despite the fact that the configuration space of the charged particles is disjoint from regions where the magnetic field B is non-zero. The philosophical literature on the AB effect focuses largely on issues of non-locality, the "reality" of electromagnetic potentials, and the like. While a thorough discussion of the effect must confront these issues, making them the focus risks underappreciating the prior and more fundamental issues about the derivation/explanation of the effect. The mainline explanation is dynamical, relying on Schrödinger evolution using a Hamiltonian operator H_{A} universally cited in standard texts. This operator is derived by following the canonical quantization procedure, starting from a classical Hamiltonian and then substituting for the classical position and momentum variables Hilbert space operators that give a representation of the Heisenberg canonical commutation relations. But H_{A} is not the unambiguous result of canonical quantization since there are representations unitarily inequivalent to the familiar Schrödinger representation used to derive H_{A}. This operator contains a gauge variable, the vector potential A of the magnetic field, and the mainline explanation relies on the gauge equivalence of Schrödinger evolution under H_{A} with free evolution multiplied by a phase factor. This equivalence is broken, and the mainline explanation is undermined by an attempt to overcome the apparent nonlocal dependence of the inference pattern on the magnetic field by regarding the vector potential A as a real physical variable. The mainline explanation can be criticized on the grounds that it attempts to explain a gauge-independent effect using gauge-dependent variables. This concern can be answered by showing that the AB phase can be obtained without using the vector potential A but only the gauge-invariant magnetic field B. The extant version of this explanation uses the path integral approach rather than canonical quantization, and it makes manifest the non-local dependence of the interference pattern on the magnetic field over all space. There are also attempts at non-dynamical (a.k.a. "topological") explanations that do not depend on the details of the Hamiltonian operator. While intriguing it is not clear that they address the relevant explanandum. In working through the thicket of issues surrounding the explanation of the AB effect it is natural to look for help from the philosophical literature on scientific explanation. But, with the exception of interventionist accounts of causal explanation, one looks in vain.
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| Item Type: | Preprint | ||||||
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| Keywords: | Aharonov-Bohm effect; non-locality; quantization of classical systems; scientific explanation; gauge symmetry; idealizations; inequivalent representations | ||||||
| Subjects: | General Issues > Causation Specific Sciences > Physics > Classical Physics General Issues > Explanation General Issues > Models and Idealization Specific Sciences > Physics > Quantum Mechanics Specific Sciences > Physics > Symmetries/Invariances |
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| Depositing User: | John Earman | ||||||
| Date Deposited: | 13 Jan 2024 22:55 | ||||||
| Last Modified: | 13 Jan 2024 22:55 | ||||||
| Item ID: | 22970 | ||||||
| Subjects: | General Issues > Causation Specific Sciences > Physics > Classical Physics General Issues > Explanation General Issues > Models and Idealization Specific Sciences > Physics > Quantum Mechanics Specific Sciences > Physics > Symmetries/Invariances |
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| Date: | 12 January 2024 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/22970 |
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