Unveiling the Role of Vector Potential in the Aharonov–Bohm Effect

Abstract

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.

1. Introduction

Since its first theoretical prediction [1,2], the Aharonov–Bohm effect has continued to provoke innumerable debates which concern the deep foundation of modern physics. (For reviews, see [3,4,5,6], for example). A central question is whether the electromagnetic potential is a physical entity or just a convenient mathematical tool [7,8,9]. Without doubt, the most popular interpretation of the AB effect is that the magnetic vector potential locally affects the complex phase of the quantum electron wave function, thereby causing a change in phase that can be verified through interference experiments [10,11]. Nonetheless, it is also true that there are many researchers who are not completely satisfied with the vector potential explanation of the AB effect [12,13,14,15]. This is because the vector potential is a gauge-variant quantity with inherent arbitrariness. For this reason, they believe that the vector potential is just a convenient mathematical tool for calculating the electromagnetic force field, as is in fact the case with classical electromagnetism. This motivates them to look for explanations which do not use the gauge-dependent vector potential [12,13,14]. One of the most influential studies along this line would be the work by Vaidman [16,17]. He proposed an explanation for the AB effect via the force between the solenoid current and the moving electron rather than via electromagnetic potential. Later, Vaidman’s paper was criticized by Aharonov, Cohen, and Rohrlich [18]. Through six thought experiments, these authors concluded that the attempt to dispense with scalar and vector potentials is at least incompatible with the attempt to interpret the AB effect as a local effect. They also pointed out a potential problem inherent in the local force explanation of the AB effect. According to this force interpretation, the change in the phase of the electron’s wave function implies a change in the electron’s velocity, which appears to be incompatible with the so-called dispersionless nature of the AB effect [19,20] (the dispersionless nature of the AB effect means that the observed AB phase shift is independent of the velocity of the electron).

Partially motivated by Vaidman’s work, several authors have focused on the interaction energy between the solenoid current and the moving electron rather than the force between them [21,22,23,24,25]. (See also a related older work by Boyer [26]). They postulate that the change in the phase of the electron wave function along a path is proportional to the change in the above interaction energy along the same path. Then, by explicitly evaluating the change in the interaction energy along a closed path of the electron, they showed that it reproduces the standard answer for the AB phase shift.

The validity or invalidity of these authors’ further claim is highly nontrivial, as follows. According to them, since the change in the above interaction energy along the path of the electron is a gauge-invariant quantity, the partial AB phase shift along a non-closed path is also a gauge-invariant quantity, which in turn implies that it can in principle be observed. However, in a recent paper [27], based on the framework of a self-contained quantum mechanical treatment of the combined system of a solenoid, an electron, and the quantized electromagnetic field, we showed that there is no evidence to support their central claim, i.e., the proportionality assumption between the interaction energy of the solenoid current and the moving electron and the corresponding partial AB phase shift. To summarize the situation to date, we feel that all attempts to dispense with the vector potential in the explanation of the AB effect have not displayed complete success. On the other hand, there already exist several works which support the physically reasonable nature of the vector potential interpretation [28,29,30]. The only drawback of this theoretical approach is the fact that the vector potential is a gauge-dependent quantity, which makes it difficult to completely dispel some researchers’ suspicion regarding its physical reality. The purpose of the present paper is to address and eliminate these doubts in the most convincing manner.

The remainder of this paper is structured as follows. First, in Section 2, we analyze the nature of the vector potential generated by an infinitely long solenoid with the specific intention of confirming its physically substantial nature. It is demonstrated that the vector potential generated by an infinitely long solenoid can be uniquely decomposed into a transverse component and a longitudinal component, provided that physically unacceptable multi-valued gauge transformation is excluded. To help our understanding of the significance of the discussion in Section 3, we provide in Section 4 a brief introduction to some past works which tried to explain the AB effect, without using the standard vector potential interpretation. Also discussed in this section is the highly nontrivial claim made in several recent studies that the partial AB phase shift corresponding to a non-closed path of the electron can in principle be observed, thereby proposing several concrete settings of measurement for verifying these authors’ claim [22,23,24,25]. Section 5 summarizes the essential points of our vector-potential-based interpretation of the AB effect improved upon in the present paper.

2. On the Vector Potential Generated by an Infinitely Long Solenoid

In order to discuss the essence of the AB effect in the simplest possible form, it is customary to consider an idealized setting of an infinitely long solenoid with radius R directed in the z-direction. The stationary and uniform surface current distribution of the solenoid is represented as

$${\mathit{J}}_{ext}\left(\mathit{x}\right)=B\delta (\rho -R){\mathit{e}}_{\varphi }.$$

(1)

Here, we use the cylindrical coordinates $\mathit{x}=(\rho ,\varphi ,z)$ with $\rho =\sqrt{x^2+y^2}$ and $\varphi =arctan\left({\displaystyle \frac{y}{x}}\right)$. Our aim is to find the vector potential generated by the above solenoid current distribution. There are various routes to reach this goal, but probably the most instructive way is to start with the familiar Biot–Savart law represented as (note that we are basically handling magnetostatics)

$$\mathit{B}\left(\mathit{x}\right)={\displaystyle \frac{1}{4\pi }}\nabla \times \int {\displaystyle \frac{{\mathit{j}}_{ext}\left({\mathit{x}}^{\prime }\right)}{|\mathit{x}-{\mathit{x}}^{\prime }|}}{d}^3{x}^{\prime }.$$

(2)

For simplicity’s sake, we use the Heaviside–Lorentz unit combined with the natural unit $\hslash =c=1$. The importance of this formula is that the physical quantities contained in it (these are the external current distribution ${\mathit{j}}_{ext}\left(\mathit{x}\right)$ and the generated magnetic field $\mathit{B}\left(\mathit{x}\right)$) are all gauge-invariant quantities. The form of the Biot–Savart law naturally leads us to introduce the vector field $\mathit{A}\left(\mathit{x}\right)$ by the relation

$$\mathit{B}\left(\mathit{x}\right)=\nabla \times \mathit{A}\left(\mathit{x}\right).$$

(3)

This quantity $\mathit{A}\left(\mathit{x}\right)$ is nothing but what we call the (magnetic) vector potential. Naturally, the vector potential introduced in the above way is not unique. Its general form is given as

$$\mathit{A}\left(\mathit{x}\right)={\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)+\nabla \chi \left(\mathit{x}\right),$$

(4)

with the definition

$${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)\equiv {\displaystyle \frac{1}{4\pi }}\int {\displaystyle \frac{{\mathit{j}}_{ext}\left({\mathit{x}}^{\prime }\right)}{|\mathit{x}-{\mathit{x}}^{\prime }|}}{d}^3{x}^{\prime },$$

(5)

while $\chi \left(x\right)$ is an arbitrary scalar function [31]. The superscript $\left(S\right)$ on ${\mathit{A}}^{\left(S\right)}$ designates that it is a part of the vector potential $\mathit{A}\left(\mathit{x}\right)$ which is uniquely determined by the solenoid current distribution ${\mathit{j}}_{ext}\left(\mathit{x}\right)$, provided that the relevant spatial integral in (5) converges (naively, this integral diverges, but it is known to converge by using an appropriate limiting procedure [31]). The arbitrary nature of the part $\nabla \chi \left(\mathit{x}\right)$ is interpreted as gauge degrees of freedom of the vector potential. This is of course a well-known story, but we point out that there exists a physically very important constraint on the scalar function $\chi \left(\mathit{x}\right)$ given by

$$\nabla \times \nabla \chi \left(\mathit{x}\right)=0,$$

(6)

which is often forgotten when the gauge ambiguity issue of the vector potential in the AB effect is discussed. As we shall soon argue in more detail, if this condition is not satisfied, the part $\nabla \chi \left(\mathit{x}\right)$ of $\mathit{A}\left(\mathit{x}\right)$ would generate a new magnetic field distribution, which necessarily alters the original distribution $\mathit{B}\left(\mathit{x}\right)$.

At the moment, let us go ahead by assuming that the condition (6) is satisfied. Then, if it is combined with the easily verified relation $\nabla ·{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)=0$, Equation (4) just gives the transverse–longitudinal decomposition of the vector potential [28,29,30]

$$\mathit{A}\left(\mathit{x}\right)={\mathit{A}}_{\perp }\left(\mathit{x}\right)+{\mathit{A}}_{\Vert }\left(\mathit{x}\right),$$

(7)

with the following identification

$${\mathit{A}}_{\perp }\left(\mathit{x}\right)\equiv {\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right),{\mathit{A}}_{\Vert }\left(\mathit{x}\right)\equiv \nabla \chi \left(\mathit{x}\right).$$

(8)

In fact, these two components certainly satisfy the transverse condition and the longitudinal condition, respectively.

$$\nabla ·{\mathit{A}}_{\perp }\left(\mathit{x}\right)=0,\nabla \times {\mathit{A}}_{\Vert }\left(\mathit{x}\right)=0.$$

(9)

Note that the derivation above indicates that, in our setting of an infinitely long solenoid, the transverse–longitudinal decomposition of the vector potential is unique [28,29,30]. (Note that this is equivalent to saying that the transverse part of the vector potential is unique. To avoid misunderstanding, however, we recall in Appendix A that there is one familiar physical system in which the transverse–longitudinal decomposition of the vector potential is not unique at all).

Unfortunately, the uniqueness of the transverse–longitudinal decomposition of the vector potential has been often questioned, probably because of the existence of the following gauge transformation [32]:

$${\mathit{A}}^{\prime }\left(\mathit{x}\right)=\mathit{A}\left(\mathit{x}\right)+\nabla {\chi }^{sing}\left(\mathit{x}\right),$$

(10)

which is specified by the multi-valued gauge function as follows:

$${\chi }^{sing}\left(\mathit{x}\right)=-{\displaystyle \frac{1}{2\pi }}\Phi \varphi =-{\displaystyle \frac{1}{2\pi }}\Phi arctan\left({\displaystyle \frac{y}{x}}\right).$$

(11)

Here, $\Phi =\pi R^{2}B$ is the total magnetic flux penetrating the solenoid. As can be easily verified, the rotation of ${\chi }^{sing}\left(\mathit{x}\right)$ does not vanish, i.e., $\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)\ne 0$, but it rather satisfies the transverse condition $\nabla ·\nabla {\chi }^{sing}\left(\mathit{x}\right)=0$. At first glance, this observation appears to show that the transverse–longitudinal decomposition, or equivalently, the identification of ${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$ as the transverse component, is not unique at all, once the multi-valued gauge transformation as above is permitted.

We, however, recall that our discussion of the relation (4) based on the Biot–Savart law already indicates that the scalar function $\chi \left(\mathit{x}\right)$ in this equation must satisfy the rotation-free condition $\nabla \times \nabla \chi \left(\mathit{x}\right)=0$. Otherwise, the term $\nabla \chi \left(\mathit{x}\right)$ in $\mathit{A}\left(\mathit{x}\right)$ inevitably alters the magnetic field distribution of the system. Let us look into this state of affairs in a more concrete manner. As is well-known, the vector potential ${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$ obtained from the integral (5) is given by (see page 208 of [31], for example)

$${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{\rho }{R^2}}{\mathit{e}}_{\varphi }\hfill & (\rho <R)\hfill \\ {\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{1}{\rho }}{\mathit{e}}_{\varphi }\hfill & (\rho \ge R).\hfill \end{array}\right.$$

(12)

It is an elementary exercise to obtain the explicit form of the gauge-transformed vector potential ${\mathit{A}}^{\prime }\left(\mathit{x}\right)$ given by (10). First, for $\rho \ge R$, i.e., in the outer region of the solenoid, we get

$$\nabla {\chi }^{sing}\left(\mathit{x}\right)=-{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{1}{\rho }}{\mathit{e}}_{\varphi },$$

(13)

which in turn gives

$${\mathit{A}}^{\prime }\left(\mathit{x}\right)={\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{1}{\rho }}{\mathit{e}}_{\varphi }-{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{1}{\rho }}{\mathit{e}}_{\varphi }=0(\mathrm{for}\rho \ge R).$$

(14)

This means that, by the above multi-valued gauge transformation, the vector potential outside the solenoid can be completely eliminated. At first glance, this appears to show the expulsion of the AB effect, as advocated by Bocchieri and Loisinger many years ago [32]. However, as was shown later by several researchers [33,34], the AB effect continues to exist even after such a multi-valued gauge transformation if one properly takes account of the change in the $2\pi$ periodic boundary condition of the electron wave functions (see also [35] concerning the general consideration of multi-valued wave functions).

Let us next examine what happens with the transformed vector potential inside the solenoid. First, in the domain excluding the origin ($\rho =0$), we find that

$$\begin{array}{ccc}\hfill \nabla {\chi }^{sing}\left(\mathit{x}\right)& =& -{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{1}{\rho }}{\mathit{e}}_{\varphi }(\rho \ne 0),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\mathit{A}}^{\prime }\left(\mathit{x}\right)& =& {\displaystyle \frac{\Phi }{2\pi }}\left({\displaystyle \frac{\rho }{R^2}}-{\displaystyle \frac{1}{\rho }}\right){\mathit{e}}_{\varphi }(\rho \ne 0),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \nabla \times {\mathit{A}}^{\prime }\left(\mathit{x}\right)& =& {\displaystyle \frac{\Phi }{\pi R^2}}{\mathit{e}}_z(\rho \ne 0).\hfill \end{array}$$

(17)

The form of $\nabla {\chi }^{sing}\left(\mathit{x}\right)$ above indicates that $\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)$ has a singularity at the origin. To confirm this, let us consider a circle $C_{\epsilon }$ around the origin with an infinitesimally small radius $\epsilon (\to 0^{+})$. The area surrounded by $C_{\epsilon }$ is denoted as $S_{\epsilon }$. If we evaluate the surface integral of $\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)$ over $S_{\epsilon }$ with the use of the Stokes theorem, we obtain

$$\begin{array}{ccc}\hfill {\int }_{S_{\epsilon }}\left(\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)\right)·d\mathit{S}& =& {\oint }_{C_{\epsilon }}\nabla {\chi }^{sing}\left(\mathit{x}\right)·d\mathit{x}\hfill \\ & =& -{\displaystyle \frac{\Phi }{2\pi }}{\int }_0^{2\pi }{\displaystyle \frac{1}{\epsilon }}\epsilon d\varphi =-\Phi .\hfill \end{array}$$

(18)

This indicates that, in the vicinity of the origin, the following relation holds:

$$\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)=-{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{\delta \left(\rho \right)}{\rho }}{\mathit{e}}_z.$$

(19)

In fact, it can be verified from the following manipulation:

$${\int }_{S_{\epsilon }}\left(\nabla \times \nabla {\chi }^{sing}\left(\mathit{x}\right)\right)·d\mathit{S}=-{\displaystyle \frac{\Phi }{2\pi }}{\int }_0^{2\pi }d\varphi {\int }_0^{\epsilon }{\displaystyle \frac{\delta \left(\rho \right)}{\rho }}\rho d\rho =-\Phi .$$

(20)

To sum up, inside the solenoid, we find that

$$\begin{array}{ccc}\hfill {\mathit{B}}^{\prime }\left(\mathit{x}\right)=\nabla \times {\mathit{A}}^{\prime }\left(\mathit{x}\right)& =& {\displaystyle \frac{\Phi }{\pi R^2}}{\mathit{e}}_z-{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{\delta \left(\rho \right)}{\rho }}{\mathit{e}}_z\hfill \\ & \equiv & \mathit{B}\left(\mathit{x}\right)+{\mathit{B}}^{string}\left(\mathit{x}\right)(\rho <R).\hfill \end{array}$$

(21)

The first term of the above equation is nothing but the original uniform magnetic field $\mathit{B}\left(\mathit{x}\right)$ inside the solenoid. On the other hand, the second term shows that the multi-valued gauge transformation generates a string-like magnetic field in the direction of the negative z-axis, which is opposite to the direction of the original uniform magnetic field. In this way, as pointed out before, we confirm that that the multi-valued gauge transformation specified by (10) and (11) generates an extra magnetic field distribution which is originally absent. If we evaluate the total flux of ${\mathit{B}}^{\prime }\left(\mathit{x}\right)$ penetrating the solenoid (with the radius R), we obtain

$$\begin{array}{ccc}\hfill {\int }_{S(\rho \le R)}{\mathit{B}}^{\prime }\left(\mathit{x}\right)·\mathit{n}dS& =& {\int }_{S(\rho \le R)}{\displaystyle \frac{\Phi }{\pi R^2}}{\mathit{e}}_z·{\mathit{e}}_{z}dS-{\int }_{S(\rho \le R)}{\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{\delta \left(\rho \right)}{\rho }}{\mathit{e}}_z·{\mathit{e}}_{z}dS\hfill \\ & =& \Phi -\Phi =0,\hfill \end{array}$$

(22)

which means that the new net magnetic field penetrating the solenoid becomes precisely zero. Undoubtedly, this is the reason why the gauge-transformed vector potential ${\mathit{A}}^{\prime }\left(\mathit{x}\right)$ entirely vanishes outside the solenoid.

The unphysical nature of such a singular gauge transformation can also be demonstrated if we evaluate the curve of ${\mathit{B}}^{\prime }\left(\mathit{x}\right)\equiv \mathit{B}\left(\mathit{x}\right)+{\mathit{B}}^{string}\left(\mathit{x}\right)$. We find that

$$\begin{array}{ccc}\hfill \nabla \times \mathit{B}\left(\mathit{x}\right)& =& B\delta (\rho -R){\mathit{e}}_{\varphi }={\mathit{J}}_{ext}\left(\mathit{x}\right)\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill \nabla \times {\mathit{B}}^{string}\left(\mathit{x}\right)& =& {\displaystyle \frac{\Phi }{2\pi }}{\displaystyle \frac{\partial }{\partial \rho }}\left({\displaystyle \frac{\delta \left(\rho \right)}{\rho }}\right){\mathit{e}}_{\varphi }\equiv {\mathit{J}}_{string}\left(\mathit{x}\right).\hfill \end{array}$$

(24)

This means that the new magnetic field ${\mathit{B}}^{\prime }\left(\mathit{x}\right)$ satisfies the following equation:

$$\nabla \times {\mathit{B}}^{\prime }\left(\mathit{x}\right)={\mathit{J}}_{ext}\left(\mathit{x}\right)+{\mathit{J}}_{string}\left(\mathit{x}\right).$$

(25)

This is clearly different from the original Maxwell equation for the magnetic field $\mathit{B}\left(\mathit{x}\right)$ given by

$$\nabla \times \mathit{B}\left(\mathit{x}\right)={\mathit{J}}_{ext}\left(\mathit{x}\right).$$

(26)

Beyond doubt, all these observations reveal the physically unacceptable nature of the above multi-valued gauge transformation. We therefore conclude that, as long as such an unphysical gauge transformation is excluded, the transverse–longitudinal decomposition of the vector potential is unique at least in our setting of an infinitely long solenoid.

3. Unveiling the Role of Vector Potential in the Aharonov–Bohm Effect

Through the discussion in the previous sections, we have verified that, at least in the setting of an infinitely long solenoid, the generated vector potential can uniquely be decomposed into a transverse component and a longitudinal component, provided that the possibility of multi-valued gauge transformation is excluded. The point is that the multi-valued gauge transformation may be mathematically allowed, but it is physically unacceptable because it alters the magnetic field distribution of the system or even the form of the basic Maxwell equation.

Most importantly, the transverse component of the vector potential, i.e., ${\mathit{A}}_{\perp }\left(\mathit{x}\right)={\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$, is unique, gauge-invariant, and it cannot be eliminated by any regular gauge transformations which leave the magnetic field distribution intact. This strongly indicates that this transverse component of the vector potential is not just a convenient mathematical tool but rather contains some definite physical entity. Nevertheless, it is also true that the entire vector potential still contains the longitudinal part, which cannot be free from gauge ambiguity. How one should confront this puzzling situation has already been discussed by several researchers [28,29,30]. Unfortunately, in any of these previous investigations, the uniqueness argument of the transverse–longitudinal decomposition has not been completed at a satisfactory level, as discussed in the present paper. This is probably the reason why such past analyses could not completely dispel the misbelief that the vector potential is just a mathematical tool with little physical substance. Now, we are ready to make more a definitive statement on this long-standing frustrating situation.

First, let us consider the most fundamental AB phase shift measured through the interference of the two electron beams. (See the schematic picture illustrated in Figure 1). According to the standard analysis, the phase change of the electron wave function along the path $C_1$ is given by

$$\Delta {\varphi }_{AB}\left(C_1\right)=e{\int }_{C_1}\mathit{A}\left(\mathit{x}\right)·d\mathit{x},$$

(27)

while the phase change along the path $C_2$ is given by

$$\Delta {\varphi }_{AB}\left(C_2\right)=e{\int }_{C_2}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}.$$

(28)

Since the observable AB phase shift corresponds to the difference between the above two phase shifts, it is eventually given by the following expression, which is proportional to the closed line integral of the vector potential $\mathit{A}\left(\mathit{x}\right)$ represented as

$${\varphi }_{AB}=e{\int }_{C_1}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}-e{\int }_{C_2}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}=e{\oint }_{C_1-C_2}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}.$$

(29)

By now, we know that the vector potential is generally given as $\mathit{A}\left(\mathit{x}\right)={\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)+\nabla \chi \left(\mathit{x}\right)$. The above closed line integral of the vector potential is then given by

$${\oint }_{C_1-C_2}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}={\oint }_{C_1-C_2}{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)·d\mathit{x}+{\oint }_{C_1-C_2}\nabla \chi \left(\mathit{x}\right)·d\mathit{x}.$$

(30)

Here, we have

$${\oint }_{C_1-C_2}{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)·d\mathit{x}={\int }_S\mathit{B}\left(\mathit{x}\right)·d\mathit{S}=\Phi ,$$

(31)

and

$${\oint }_{C_1-C_2}\nabla \chi \left(\mathit{x}\right)·d\mathit{x}={\int }_S\left(\nabla \times \nabla \chi \left(\mathit{x}\right)\right)·d\mathit{S}=0,$$

(32)

since the gauge function $\chi \left(\mathit{x}\right)$ is demanded to satisfy the constraint $\nabla \times \nabla \chi \left(\mathit{x}\right)=0$. This means that the transverse component ${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$ solely explains the Aharonov–Bohm phase shift, and the gauge-dependent longitudinal component never contributes to it.

Importantly, however, if we consider the phase change corresponding to a non-closed path connecting the two spatial points ${\mathit{x}}_i$ and ${\mathit{x}}_f$, it is given by

$$\Delta {\varphi }_{AB}=e{\int }_{{\mathit{x}}_i}^{{\mathit{x}}_f}\mathit{A}\left(\mathit{x}\right)·d\mathit{x}.$$

(33)

This quantity is also divided into two pieces as

$$\begin{array}{ccc}\hfill \Delta {\varphi }_{AB}& =& e{\int }_{{\mathit{x}}_i}^{{\mathit{x}}_f}\left({\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)+\nabla \chi \left(\mathit{x}\right)\right)·d\mathit{x}\hfill \\ & =& e{\int }_{{\mathit{x}}_i}^{{\mathit{x}}_f}{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)·d\mathit{x}+e\left(\chi \left({\mathit{x}}_f\right)-\chi \left({\mathit{x}}_i\right)\right).\hfill \end{array}$$

(34)

Although the first term of the above equation is gauge-invariant, the second term is not, because of the arbitrariness of the gauge function $\chi \left(\mathit{x}\right)$ contained in the longitudinal part. This means that such a partial AB phase shift is a gauge-dependent quantity. Hence, as long as we believes the widely accepted gauge principle, we must conclude that it would not correspond to any measurable quantity. The statement above may sound self-evident to some researchers. We, however, recall that this conclusion contradicts the recent claims by several authors that such a partial Aharonov–Bohm phase shift can in principle be observed. In the next section, we briefly introduce and comment on these challenging claims.

4. On Some Attempts to Explain the AB Effect Without Using the Gauge-Variant Electromagnetic Potential

It is widely accepted that the vital importance of the vector potential in quantum mechanics was established in the paper by Aharonov and Bohm on the quantum phenomenon associated with their names [2]. However, it appears that, because of the gauge-dependent nature of the vector potential, Aharonov himself is not completely satisfied with the vector-potential-based interpretation of the AB effect. This motivates him and his collaborators to look for explanations which do not use the gauge-dependent vector potential [13,14]. Along with this line of exploration, Vaidman argued that when the source of the electromagnetic potential is treated in the framework of quantum theory, the AB effect can be explained without the notion of vector potential [16]. To be more concrete, he considered the following setup. The solenoid consists of two cylinders and the opposite charges Q and $-Q$ spread on their surfaces. The cylinders rotate in opposite directions with a certain surface velocity. The electron is supposed to encircle the solenoid with some velocity in a superposition of being in the left and in the right sides of the circular trajectory. When the electron enters one arm of the circle, it changes the magnetic flux through a cross-section of the solenoid and then induces an electromagnetic force acting on the solenoid. Vaidman explicitly calculated the shift of the wave packet of each cylinder during the motion of the electron. Combining this result with the information on the relevant wavelength of the de Broglie wave of each cylinder, he eventually showed that this analysis precisely reproduces the familiar AB phase shift.

Although Vaidman emphasized the quantum mechanical nature of his analysis, there appear to be close connections between his analysis and Boyer’s semiclassical analysis of the AB effect [36], especially as to the basic interaction dynamics of the solenoid and electron. Boyer considered a solenoid as a stack of electric current loops. He calculated Lorentz force due to the electron acting on charge carriers flowing in each current loop. This Lorentz force was shown to generate the change in velocity and the electron paths. By calculating the difference in path length for electron paths passing on either side of the solenoid, Boyer demonstrated that the resultant path length difference leads to a semiclassical phase shift which reproduces the known AB phase shift.

In any case, a common ingredient in the explanations of Vaidman and of Boyer is the presence of force acting on the solenoid induced by the motion of the electron. However, we recall that such an explanation of the AB phase shift due to force has been believed to be incompatible with the dispersionless nature of the AB effect, which means that the magnitude of the AB phase shift is independent of the electron velocity [19,20]. Unfortunately, no decisive experiment to verify the dispersionless nature of the AB effect has been carried out for a long time. Some years ago, however, Caprez, Barwick, and Batelaan carried out a crucial time delay measurement of the electron beam and verified that no time delay was observed, thereby concluding that all force explanations of the AB phase shift are ruled out [37].

Also worthy of mention is the existence of still another explanation of the AB phase shift. The basic postulation of this approach is that the AB phase shift is proportional to the change in the interaction energy between the charged particle and the solenoid along the path of the moving charge. This idea can be traced back to Boyer’s older work [26], which is based on the framework of classical electrodynamics. He assumed that the AB phase shift for a given path of the moving charge is proportional to the change in the interaction energy between the magnetic field ${\mathit{B}}^s$ generated by the current of an infinitely long solenoid and the magnetic field ${\mathit{B}}^{\prime }$ generated by a moving charge with a constant velocity $\mathit{v}$ as

$$\Delta {\varphi }_{AB}\propto \Delta ϵ\left(\mathrm{Boyer}\right)=\int {\mathit{B}}^s\left({\mathit{x}}^{\prime }\right)·{\mathit{B}}^{\prime }({\mathit{x}}^{\prime },t)d^3{x}^{\prime },$$

(35)

where ${\mathit{B}}^s\left(\mathit{x}\right)$ the magnetic field generated by the surface current ${\mathit{j}}_{ext}\left(\mathit{x}\right)$ of the solenoid according to the Maxwell equation,

$$\nabla \times {\mathit{B}}^s\left(\mathit{x}\right)={\mathit{j}}_{ext}\left(\mathit{x}\right).$$

(36)

After transforming the above expression by making full use of the knowledge of classical electrodynamics, Boyer arrived at a remarkable relation

$$\Delta \epsilon \left(\mathrm{Boyer}\right)=e\mathit{v}·{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right),$$

(37)

with

$${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)={\displaystyle \frac{1}{4\pi }}\int {\displaystyle \frac{{\mathit{j}}_{ext}\left({\mathit{x}}^{\prime }\right)}{|\mathit{x}-{\mathit{x}}^{\prime }|}}{d}^3{x}^{\prime }.$$

(38)

As emphasized by Boyer, the above $\Delta \epsilon \left(\mathrm{Boyer}\right)$ is free from the gauge choice. This is because, in the above expression, the quantity ${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$ is uniquely determined by the surface current ${\mathit{j}}_{ext}\left(\mathit{x}\right)$ of the solenoid, which is gauge-invariant. This claim sounds reasonable, because the interaction energy is likely to be a gauge-invariant quantity.

Motivated by the work of Boyer, several researchers have investigated the interaction energy between the solenoid and a moving charge within the framework of quantum electrodynamics [21,22,23,24]. (Also noteworthy is a related but slightly different approach discussed in [38]). They evaluated the interaction energy between the solenoid current and the charged particle mediated by the exchange of a virtual photon within the framework of the quantum electrodynamics, thereby arriving at the following answer:

$$\Delta \epsilon \left(\mathrm{virtual}\mathrm{photon}\mathrm{exchange}\right)=-e\mathit{v}·{\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right),$$

(39)

where ${\mathit{A}}^{\left(S\right)}\left(\mathit{x}\right)$ is the same quantity as appearing in the corresponding interaction energy obtained by Boyer [26].

Important messages from the authors of the above investigations are as follows. The change in interaction energy between the solenoid and the charged particle along the path of the moving charge is a gauge-invariant quantity. Therefore, if one accepts the above-mentioned postulation that the phase change of the electron wave function is proportional to the change in interaction energy along the path of the moving charge, the AB phase shift for a non-closed path is also a gauge-invariant quantity. This appears to indicate that it can in principle be observed. In fact, based on this belief, several authors proposed some concrete measurements for extracting the partial AB phase shift corresponding to a non-closed path [22,23,24].

This claim was, however, criticized in a recent paper by ourselves [27]. It was pointed out that, very strangely, the expressions of the interaction energy of Boyer and that due to the virtual-photon exchange are identical with opposite signs (this remarkable fact was never noticed before, since the above researchers paid attention only to the absolute magnitude of the predicted AB phase shift). It was further shown that, within the framework of a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic field, the interaction energy of Boyer and that due to virtual-photon exchange exactly cancel each other out. (Since this demonstration requires fairly careful preparation, interested readers are recommended to read the original paper [27]). The analysis there rather shows that the origin of the AB phase shift can be traced back to other part of the self-contained treatment above, which is, after all, identical to the standard mechanism as explained in Section 3 of the present paper. This means that the AB phase shift corresponding to a non-closed path is not a gauge-invariant quantity, meaning that its observation is most likely to contradict the celebrated gauge principle. In any case, it seems to us that all attempts at explaining the AB phase shift without using the notion of the vector potential have been unsuccessful up to the present. At this point in time, the vector potential interpretation seems to be the simplest and most reasonable physical explanation of the AB effect.

5. Summary and Conclusions

The vector-potential-based interpretation of the AB effect is not universally accepted because of the gauge-variant nature of the vector potential. Even now, multiple researchers seem to believe that the vector potential is just a convenient tool for obtaining the electromagnetic field, and they are searching for an explanation of the AB effect without using the vector potential concept. In the present paper, we tried to demonstrate that the vector potential is not just a convenient mathematical tool with little physical substance. The argument proceeds as follows. Employing the simplest setting of the system, i.e., an infinitely long solenoid, we have shown the following facts:

• The vector potential generated by the infinitely long solenoid is given as a sum of the transverse part and the longitudinal part.
• The above decomposition is unique as long as the multi-valued gauge transformation is excluded. In particular, the transverse part of the vector potential is uniquely determined by the surface electric current distribution of the solenoid.
• The multi-valued (and singular) gauge transformation is not allowed from the physical point of view, because it inevitably generates a new or extra magnetic field distribution which is originally absent in the system.
• The transverse part of the vector potential solely explains the standard Aharonov–Bohm effect corresponding to a closed path of the electron’s trajectory.
• Nevertheless, one should not forget the fact that the vector potential still contains the longitudinal part which has inherent gauge arbitrariness. It seems to us that this gauge arbitrariness forbids the observation of the partial AB phase shift corresponding to a non-closed path, which was recently claimed to be possible by several researchers.

To sum up, we conclude that the vector potential contains in it a piece which is unique, gauge-invariant, and cannot be eliminated by any regular gauge transformations. This part of the vector potential solely explains the standard Aharonov–Bohm effect. However, the remaining ambiguity of the longitudinal part of the vector potential is thought to forbid the observability of the partial Aharonov–Bohm phase shift corresponding to a non-closed path, because it is gauge-dependent and its observation contradicts the celebrated gauge principle. Conversely speaking, if the AB phase shift corresponding to a non-closed path were observed, it would give us the first counterexample to the validity of the gauge principle. Undoubtedly, this last statement is closely related to the authenticity of the vector-potential-based interpretation of the Aharonov–Bohm effect considerably refined in the present paper.