On the Locality Principle Keeping in Aharonov-Bohm Effect

Foundations of Physics, 2021

I address the view that the classical electromagnetic potentials are shown by the Aharonov-Bohm effect to be physically real (which I dub: 'the potentials view'). I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into 'gauge-underdetermination': different gauge choices represent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature (or in classrooms), neither by proponents nor by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellectually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e. will prefer some elements of such an equivalence class over others. I consider various formulations of locality, such as Bell locality, local interaction Hamiltonians, and signal locality. I show that Bell locality can only be evaluated if one fixes the gauge freedom completely. Yet, an explanation in terms of signal locality can be accommodated by the Lorenz gauge: the potentials propagate in waves at finite speed. I therefore suggest the Lorenz gauge potentials theory -- an even narrower gauge equivalence relation -- as the ontology of electrodynamics.

Foundations of Physics, 2020

We propose a simple situation in which the magnetic Aharonov-Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov-Bohm effect.

2021

In this paper, we present a novel semi-classical theory of the electrostatic and magnetostatic fields and explain the nonlocality problem in the context of the Aharonov-Bohm effect [1]. Specifically, we show that the electrostatic and the magnetostatic fields possess a quantum nature that manifests if certain conditions are met. In particular, the wave amplitudes of the fields are seen to exist even in the regions where the classical fields vanish and they operate on the electron wave functions locally as unitary phases. This formulation also sheds light on the quantisation of electric charges and magnetic flux.

Eprint Arxiv 1007 2538, 2010

In this work we consider a quantum variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. On the other hand we consider a classical analogy and variation of the usual Aharonov-Bohm effect in which Aharonov-Bohm solenoid is used for the primary coil inside secondary large coil in the remarkable classical Faraday experiment of the electromagnetic induction.

Journal of Mathematical Physics, 2011

The seminal paper of Aharonov and Bohm [Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485-491 ] is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, on spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been extensively studied. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Observation of Aharonov-Bohm effect by electron holography, Phys. Rev. Lett. 48 (1982) 1443-1446 , Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56 (1986) 792-795] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Macroscopic test of the Aharonov-Bohm effect, Phys. Rev. Lett. 99 (2007) 210401] that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. The theoretical issues were settled Ballesteros and Weder [High-velocity estimates for the scattering operator and Aharonov-Bohm effect in three dimensions,

arXiv (Cornell University), 2021

An experiment to observe the Aharonov-Bohm effect is discussed. A solenoid which consists of a large number of point magnetic dipoles is considered as the source of a vector potential, which acts on a charged particle, and such potential has an electromagnetic field of zero strength in the region of a nonzero vector potential. A detailed microscopic analysis of the change in the quantum state phase of the entire system, namely, a particle and a set of dipoles, reveals the origin of the apparent nonlocality of the action of the vector potential, and shows the locality of the phase change mechanism. An analysis of an experiment with a solenoid shielded by a superconducting shell is given.

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2007

I display, by explicit construction, an account of the Aharonov-Bohm effect that employs only locally operative electrodynamical field strengths. The terms in the account are the components of the magnetic field of the solenoid at the location of electron, and even though the total field vanishes there, the components do not. That such a construction can be carried out demonstrates at least that whatever virtues they have for understanding and constructing new field theories, gauge fields in general make no metaphysical demands, and commit us to no novel ontology. I reflect on the significance of this for our understanding of quantum time evolution and conclude that we should think of quantized matter as interacting individually with the other matter in the systems of which it is a part.

The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper argues that gauge-invariant explanations, which attribute the phase to measurable quantities like magnetic flux Φ, fail to account for its continuous accumulation along the electron's path-a process evident in the generalized AB effect, where a time-varying magnetic flux Φ(t) induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a critical analysis spanning quantum mechanics and quantum electrodynamics, I demonstrate that these explanations rely on nonlocal and discontinuous mechanisms, rendering them inadequate. Instead, the electromagnetic potentials A µ , fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase's generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials g µν may anchor spacetime's substantival reality via a generalized gravitational AB effect. Experimental proposals test this continuous accumulation, while a speculative massive photon scenario via the Proca equation deepens the potentials' ontological role. Reframing the AB effect as a linchpin for quantum reality, this study advocates a potential-centric ontology, challenging gauge symmetry's primacy and reshaping the foundations of quantum theory, particle physics, and gravitation with profound philosophical and empirical consequences.

Europhysics Letters (EPL), 1999

A fully quantum theory of the Aharonov-Bohm effect is presented. It is based on the interaction, mediated by virtual photon exchange, between the traveling electron and the atomic magnetic dipoles of an infinitely long permanent magnet. The calculation involves second-order time-dependent perturbation theory. As expected, the relative phase between the two states-corresponding to the two alternative paths-interfering on the screen agrees with that predicted in the usual theory. However our method may allow to get corrections, for instance due to the size and shape of the electron wave packet, the possible soft-photon emission or the excitation of the atoms in the magnet.

We derive the phase shift for the Aharonov-Bohm (AB) effect under time-dependent magnetic vector potential. It turns out that the phase shift is proportional to the time average of the enclosed magnetic flux. This result offers a new perspective on the AB effect, strongly suggesting that the AB phase is locally and continuously generated via the action of gauge-dependent potentials.