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Electrical Engineering in Japan

The vector potential revisited

Toshimi Adachi, Shigeru Sasabe, Toshio Inagaki, Masao Ozaki

Published Jan 1, 1993• 3 citations
Publisher

Abstract

Abstract The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so‐called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov‐Bohm effect is chosen here. As for the Aharonov‐Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z‐axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov‐Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.

DOI
10.1002/eej.4391130602

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The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. We show that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is uniquely determined by the magnetic field and does not alter by a gauge transformation, on the other hand, the longitudinal part depends on the choice of special gauge. We show that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.

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Abstract The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so‐called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov‐Bohm effect is chosen here. As for the Aharonov‐Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z‐axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov‐Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. We show that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is uniquely determined by the magnetic field and does not alter by a gauge transformation, on the other hand, the longitudinal part depends on the choice of special gauge. We show that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is expressed by the gauge invariant physical current, gives the desired contribution to the Aharonov-Bohm effect. While the scalar mode of the vector potential, which changes under the gauge transformation so that it is the unphysical mode, give no contribution to the Aharonov-Bohm effect. Then Aharonov-Bohm effect really occurs by the physical longitudinal mode in the original Aharonov-Bohm’s setting. In the setting of Tonomura et al., where the magnet is shielded with the superconducting material, not only the magnetic field but also the longitudinal mode of the vector potential become massive by the Meissner effect. Then not only the magnetic field but also the physical longitudinal mode does not come out to the region where the electron travels. In such setting, only the scalar mode of the vector potential exists in the region where the electron travels, but there is no contribution to the Aharonov-Bohm effect from that mode.
The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.
It is argued that much of the recent controversy over the Aharonov-Bohm effect has been fuelled by the widespread consideration of solenoids of infinite length: in this limit, the field and the vector potential have somewhat anomalous properties and it appears superficially that the Aharonov-Bohm phase shift can be changed by a gauge transformation. This incorrect impression is here removed by considering a solenoid of finite length and by carefully distinguishing the longitudinal and transverse parts of the vector potential. It is shown that the phase shift depends only on the transverse part of the vector potential and it cannot be changed by a gauge transformation. The nature of the difficulty that occurs in the limit of a solenoid of infinite length is examined in detail. The finite-length theory given here also provides guidelines for the design of experiments on the Aharonov-Bohm effect.
In this paper, we study the Aharonov–Bohm (A–B) effect and Anderson–Higgs mechanism in Ginzburg–Landau model of superconductors from the perspective of the decomposition of U (1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A⊥ in A–B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A‖, which reflects the Anderson–Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A‖, while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.
The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.