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Aharonov-Bohm effect, local field interaction, and Lorentz invariance

A field-interaction scheme is introduced for describing the Aharonov-Bohm effect, fully consistent with the principle of relativity. Our theory is based on the fact that local field interactions are present even when a particle moves only in a field-free region. The interaction Lagrangian between a charge and a flux is uniquely constructed from three principles: Lorentz covariance, linearity in the interaction strength, and a correct stationary limit of charge. Our result resolves fundamental questions raised on the standard interpretation of the Aharonov-Bohm effect, concerning its duality with the Aharonov-Casher effect and the equivalence between the potential and the field-interaction models for describing the electromagnetic interaction. Most of all, potential is eliminated in our theory, and all kind of the force-free Aharonov-Bohm effect is understood in a unified framework of the Lorentz-covariant local interaction of electromagnetic fields.


I. INTRODUCTION
Aharonov-Bohm (AB) effect [1,2] has changed our notion of electromagnetic field and potential. It is known as a milestone in our understanding of electromagnetic interactions, which describes a quantum interference of a charged particle moving in a region free of electric and magnetic fields. According to the standard interpretation, AB effect demonstrates that the classical picture of electromagnetism based on the local action of fields is invalid in quantum theory, because the interference is affected by a potential even when a charged particle moves in a field-free region. Aharonov-Casher (AC) effect [3], dual to the AB phenomenon, shows a phase shift of a fluxon moving around a charged rod. It has been shown that AC effect is also free of force [4,5], but standard view draws a clear distinction between the two phenomena in that the fluxon moves under a nonvanishing field generated by the charge in the case of AC effect [6]. Despite the fact that the observable phenomena depend only on the relative motion of a charge and a fluxon in two dimension, a unified picture, fully consistent with the principle of relativity, is lacking.
In this paper, we provide a unified theory which resolves the question of relativity, based on a Lorentz-invariant field-interaction between a charge and a localized flux. The AB effect can be understood in this fully relativistic viewpoint based on the local action of fields.
The AB phase shift is derived from the Lorentz-covariant interaction Lagrangian, and the force-free nature of the effect is also derived. In addition, we show that the AB effect should vanish under the condition of a perfect shielding of the field interactions, which has been widely overlooked. The present study suggests a fundamental change in our viewpoint of the potential: The AB effect can be described purely in terms of local interaction of fields, and we can unify the role of fields and potentials both in classical and quantum theories. We expect that our study initiates further research in this context. The role of shielding local interactions will be a key issue for understanding the underlying nature of the AB effect.

BOHM EFFECT
Take, for instance, the interaction of a charge, e, and a localized fluxon with its flux value Φ, in two dimension [3,5] (Figure 1(a)). The observable phenomena here depend only on the relative motion of the two entities, in that (1) a phase shift is acquired for one-loop rotation of one particle moving around the other, and, (2) the interaction between the particles is free of force in both cases. In spite of this equivalence, the standard view draws a clear distinction between the two cases, namely "Type I" and "Type II" (Figure 1(b)) [6]. In Type I, the charge is moving in a field-free region. The observed AB phase is interpreted as a pure topological effect where the charge is interacting with the flux only in a nonlocal way.
In the case of Type II, the fluxon moves under the influence of the electric field generated by the charge [3]. Although the net force applied to the fluxon vanishes [4], the observed phase shift can be fully understood in terms of local interaction of the fluxon with the external electric field [7]. This distinction of the types depending on the reference frame should not be inherent, because the AB and the AC effect in two dimension is in fact the same problem in that only a relative motion of the two objects matters. A unified view, fully consistent with the principle of relativity, is desirable for understanding both types of the AB effect.
We also clarify a critical issue concerning a possible shielding of local field interaction.
It is a common notion that the AB effect survives in a condition of shielding any kind of local field interactions. This viewpoint has been strengthened due to the seminal experiment by Tonomura et al. [8,9]. However, this notion is misleading, as we show in the following analysis.
Let us consider a moving electron and a stationary flux confined in an ideal Faraday cage made of a hollow superconductor ( Figure 2). Here the superconductor is introduced only as an ideal Faraday cage, without taking into account its material properties. For simplicity, the problem is analysed for a two dimension with a circular hollow cage, with the condition that no electric/magnetic field penetrates inside the hollow. In this condition, the electromagnetic interaction has three different contributions, namely, L int = L eΦ + L sΦ + L es , where L eΦ , L sΦ , and L es are Lagrangians representing electron-flux, superconductor-flux, and electron-superconductor interactions, respectively. L es is independent of Φ, and therefore irrelevant to the AB effect. The electron-flux interaction is where ω is the angular velocity of the electron and the vector potential A is given by a symmetric gauge. The moving electron induces a charge density, δσ(φ), on the outer surface of the superconductor with radius R, which depends on the angle φ on the conductor, where a (> R) is the distance of the electron from the center. Accordingly a counter-flowing is generated, whereφ denotes the azimuthal unit vector. It is straightforward to find that this induced current gives the interaction term thereby the Φ-dependent interaction Lagrangian vanishes: L eΦ + L sΦ = 0. The chargeflux interaction is completely shielded by the counter-flowing current in the superconductor.
Therefore, the flux Φ does not contribute any phase shift for an arbitrary evolution of the electron-conductor composite system. This implies that, although the vector potential is nonzero in the electron's path, the AB effect vanishes under the condition of a perfect shielding of field interaction. This effect has been widely ignored, and is crucial in our understanding the role of field and potential.
Note that the quantization of charge (by 2e×integer n) in the superconductor does not affect the analysis given above. By taking the charge quantization into account, the ideal shielding condition of the induced charge density (Eq. (2)) is uniquely determined by where δσ(φ), given by Eq. 2, is responsible for the image current: j c (φ) = Rω δσ(φ)φ, while the background charge σ 0 = (2n + 1)e/2πR does not contribute to the electric/magnetic field inside the cage.
The result of our analysis invokes another important question: Is the local field interaction picture really irrelevant to describe the Aharonov-Bohm effect? In fact, equivalence of two different approaches for describing the electromagnetic interaction energy is well established in classical electrodynamics [10]. For a current distribution (j 1 ) in the presence of a magnetic field B 2 (= ∇ × A 2 ) from another source, the interaction energy of the two independent sources is given by 1 c j 1 · A 2 dx, coupling of the current and the vector potential (A 2 ).
Alternatively, it can be expressed as 1 where B 1 is the magnetic field generated by the current j 1 . When it is applied to the AB effect, the general consensus is that only the potential picture is valid. It is hard to see why and how the equivalence is broken in a quantum mechanical treatment of the interaction. If one tries to maintain the standard viewpoint that only the potential-based model is relevant, it should be explicitly shown how the equivalence of the two model is broken, which has never been done. In the following, we argue that this is not the case, and the equivalence of the two approaches is maintained also in quantum theory.

III. LORENTZ-INVARIANT FIELD-INTERACTION APPROACH
In order to resolve the questions raised above, we develop a field-interaction scheme which is fully consistent with the special theory of relativity. We treat the local field interaction of two particles (charge and fluxon) in two dimension, on an equal footing (Figure 1(a)).
For moving charge and fluxon (with their velocitiesṙ andṘ, respectively), there are two sources of field interactions, that is, (1) between the magnetic field of the fluxon and that of the moving charge, and, (2) between the electric field of the charge and that of the moving fluxon. Each interaction term was treated for a classical force explanation of the AB [11][12][13], and of the Aharonov-Casher (AC) [14] phase shifts, respectively. Although the classical force based on the field interactions have been refuted both theoretically [4,5] and experimentally [15], we point out that the field-interaction framework itself is not erroneous.
The main drawback of the previous field-interaction approaches is that the interaction term depends on the reference frame, violating the principle of relativity. Below, we provide a Lorentz-covariant field interaction scheme to get rid of this problem. L int is reduced to −eV , the correct non-relativistic limit for a stationary charge, with the electric scalar potential V generated by the moving fluxon. The interaction term is uniquely determined from these constraints as where F (e) µν and F µν(Φ) are the electromagnetic field tensors generated by the charge and the fluxon, respectively. Eq. (5) can also be written in terms of more familiar electric and magnetic fields as where B (e) (E (e) ) and B (Φ) (E (Φ) ) represent the magnetic (electric) fields of the charge and the fluxon, respectively. L int can be simplified by adopting the relations where is the field momentum produced by the two particles. Note that Eq. (7) is equivalent to the interaction Lagrangian of [3,5] based on the vector potential A, except that in the former case (Eq. (7)) gauge dependence is absent.
From now on we focus on the nonrelativistic limit (|ṙ|, |Ṙ| ≪ c), although the Lorentzcovariant interaction Lagrangian of Eq. (7) should generally be valid. In this limit, the Lagrangian of the system is reduced to where the field momentum is withφ the azimuthal unit vector of r − R. m and M denote the masses of the charge and the fluxon, respectively. It is instructive to note that this Lagrangian is transformed to a where p and P are the canonical momenta of the variables r and R, respectively.

B. Understanding the force-free Aharonov-Bohm effect
Several noticeable features can be found from the Lagrangian of Eq. (10) (or from the Hamiltonian of Eq. (11)). First, this Lagrangian is equivalent to the one represented by a vector potential [3,5], except that there is no freedom to choose a gauge in Eq. (10). The equivalence of the "potential" and the "field-interaction" approaches is restored. Second, the Euler-Lagrange equation for the classical dynamics leads to an equation of motion of There is no mutual classical force between the two particles, contrary to the previous classical explanation of the AB effect based on the field interactions [11][12][13][14]. This indicates that the classical force claimed in the previous schemes is an artefact which arises from neglect of Lorentz invariance. For instance, if −Ṙ · Π term (which corresponds to the electric field interaction) is neglected in Eq. (10), we find that MR = 0, which is the basis of the claim in Ref. [12]. Apparently, this force is absent if the Lorentz invariance is taken into account in the Lagrangian. Third, on encircling around the fluxon, the charge acquires an AB phase The same is true for the fluxon encircling around the charge. Recall that, in obtaining this, we have not relied on the vector potential. Force-free AB phase can be explained in a general way without the notion of potential, in contrast to the widely accepted viewpoint.
Our findings are not limited to the two point particles in two dimension, but can be applied to more realistic case with distributed flux (charge). Consider, for example, a charge (e) and distributed magnetic flux schematically drawn in Figure 3. The interaction Lagrangian in this case is given in the same form of Eq. (10) with the field momentum replaced by The integral is over the cross sectional area S of the flux lines, andφ is the azimuthal unit vector of r − R. The force-free nature of the interaction is maintained. Also, it is straightforward to show that the one-loop integral of Π gives the AB phase of φ AB = eΦ/(hc), where Φ is the net magnetic flux.

IV. QUESTIONS ABOUT SHIELDING
An interesting question has been raised by Erlichson [16] already in 1970 concerning a shielding of local field interactions. In his suggestion, a superconducting barrier is placed between a charge and a magnetic flux in order to prevent the penetration of the charge's magnetic field into the flux. Whether the local action of fields play a major role or not would be revealed with this kind of experiment. It is widely believed that this matter has been settled in a seminal experiment by Tonomura et al. [8,9]. In this experiment, a phase shift is reported even when the flux is confined in a superconducting shield, and their result is interpreted in terms of the potential (gauge field). However, as we have shown in Eqs. (1,3), the AB effect should vanish in a condition of perfect shielding of field interaction. This point should be clarified. Absence of the AB effect can be more directly understood in our field interaction framework. For a moving electron and a stationary flux confined in an ideal Faraday cage (Figure 2), the magnetic field (B e ) generated by the moving electron does not penetrate into the hollow. Therefore, the local field interactions vanish inside the cage.
To be precise, the moving electron's magnetic field is compensated by a field generated in with the localized flux, due to a complete shielding. The widely accepted notion -that the AB effect survives even though the field interaction is absent -is misleading, as we have shown in our analysis (both in terms of the potential-based and of the field-interaction framework). Instead, the interference pattern will be independent of the localized flux if the field interactions are perfectly shielded: That is, the AB effect vanishes under the condition of perfect shielding of local actions.

V. CONCLUSION
Since the discovery of the AB effect, it has become a common notion that a framework based on the local action of fields is impossible in quantum mechanics. Here we have provided an alternative, unified framework based on the Lorentz-covariant field-interaction, which shows that the AB effect can be universally described in terms of local interaction of fields. Our result does not reduce the significance of the AB effect, nor that of its various applications, such as the concept of gauge field. It suggests, however, the following crucial change in our understanding of the electromagnetic interaction in quantum theory. First, vector potential is eliminated in our scheme, and thus, the force-free AB effect can be explained purely in terms of local field interactions. This possibility was recently addressed with some specific examples by Vaidman [18]. Our result shows, in a general way, that there is no physical effect in the absence of a local overlap of fields. Second, our study restores the equivalence of the potential-based and the field-interaction-based schemes in quantum theory of electromagnetic interactions. The equivalence is already present in the Lagrangian of the system, and therefore, there is no reason to discard the field-interaction approach for a quantum mechanical treatment of the interaction. Third, the AB effect can be described in a unified framework independent of the reference frame, without a need for distinction of its types [6]. With this unified viewpoint, the principle of relativity is fully restored. Fourth, there is no freedom to choose a gauge for a potential in our scheme, simply because the Lagrangian is uniquely determined by the field strengths. In fact, this is the underlying reason why the local phase can be uniquely determined in an Aharonov-Bohm loop, as shown recently by the present author [19]. Finally, it is straightforward to apply the Lorentz-covariant field-interaction framework for the electric AB effect. This implies that all kind of the AB effect can be described in a unified approach. Together with the issue of shielding, further experimental efforts are necessary to verify the validity of the local action of fields.