Dispersionless forces and the Aharonov-Bohm effect

The independence of the Aharonov-Bohm phase shift on particle velocity is one of its defining properties. The classical counterpart to this dispersionless behavior is the absence of forces along the direction of motion of the particle. A reevaluation of the experimental demonstration that forces are absent in the AB physical system is given, including previously unpublished data. It is shown that the debate on the presence or absence of forces is not settled. Experiments that measure the influence of magnetic permeability on forces and search for dispersionless quantum forces are proposed. Copyright c © EPLA, 2015 Introduction. – Type-I Aharonov-Bohm effects [1] showcase the guiding principle of the Standard Model, local gauge invariance [2]. The Aharonov-Bohm effect is also a cornerstone phenomenon in quantum mechanics. It is thought to establish that the vector potential can cause measurable effects even when the fields are zero [3]. It is thus claimed to elevate the relevance of the vector potential from being a helpful mathematical construct to that of having direct physical reality [4]. However, Vaidman recently reconsidered this viewpoint [5]: “. . . the Aharonov-Bohm effect can be explained without the notion of potentials. It is explained by local action of the field of the electron on the source of the potential.” The passing electron is shown to exert a force on the solenoid, while the solenoid does not exert a force on the passing electron. The first part of this argument agrees with Boyer’s derivation [6]. Boyer claims that there is a force on the solenoid, but in contrast, he also claims that there is a back-action force on the electron that explains the ABphase shift. McGregor et al. have shown [7] that both viewpoints can be maintained even if they appear to be at odds with each other. If the motion of the charge carriers in the solenoid is fully constrained, the solenoid experiences a force and the passing electron does not. If the charge carriers are completely free to move, the passing electron does experience a force. This supports the generally accepted interpretation of the Aharonov-Bohm effect: “phase without a force”, in the case of fully constrained (a)E-mail: [email protected] motion. This case has been shown to be an example of a Feynman paradox [7] on conservation of momentum. Missing momentum is stored in the combined electromagnetic field of the electron and solenoid in this case where there is no back-action force. Note that for the interaction of a charged particle and a magnetic flux (due to a solenoid, for example), the existence of hidden momentum is expected to affect the equation of motion [8]. For the alternative case of fully unconstrained motion, there is a back-action force, and momentum conservation does not require field momentum. The surprise is that the backaction force is exactly the correct magnitude to explain the AB-effect [5,6]. The two limits, constrained and unconstrained motion, considered in ref. [7], are not thought to represent a detailed realistic description of a physical system. A detailed model study of the response of the solenoid has currently not been completed [9]. A definitive theoretical answer is, thus, currently not available. This leaves concerns in the interpretation of the classical part of the analysis of the Aharonov-Bohm physical system. Do the experiments belong to the constrained or unconstrained case? On the experimental side, a test showing the dispersionless nature of the Aharonov-Bohm effect with an electron wave interferometer [10,11] has never been performed. The next best approach is to rule out forces by time delay experiments. Caprez et al. have shown that an electron passing by a solenoid does not experience a force that causes a delay sufficiently large to explain the AB-effect [12,13]. It appears that this settles the issue.