Comments on the Aharonov-Casher effect

Quantum computation using the Aharonov-Casher set up

It is argued that the Aharonov-Casher set up could be used as the basic building block for quantum computation. We demonstrate explicitly in this scenario one- and two-qubit phase shift gates that are fault tolerant to deformations of the path when encircling two sites of the computational system around each other.

Aharonov-Casher and shielded Aharonov-Bohm effects with a quantum electromagnetic field

We use a covariant formalism that is capable of describing the electric and magnetic versions of the Aharonov-Bohm effect, as well as the Aharonov-Casher effect, through local interactions of charges and currents with the quantum electromagnetic field. By considering that only local interactions of a quantum particle with the quantum field can affect its behavior, we show that the magnetic Aharonov-Bohm effect must be present even if the solenoid generating the magnetic field is shielded by a perfect conductor, as experimentally demonstrated.

The ground state of a spin-1/2 neutral particle with anomalous magnetic moment in a Aharonov-Casher configuration

We determine the (bound) ground state of a spin 1/2 chargless particle with anomalous magnetic moment in certain Aharonov-Casher configurations. We recast the description of the system in a supersymmetric form. Then the basic physical requirements for unbroken supersymmetry are established. We comment on the possibility of neutron trapping in these systems.

Relativistic Aharonov-Casher Phase in Spin One

The Aharonov-Casher (AC) phase is calculated in relativistic wave equations of spin one. The AC phase has previously been calculated from the Dirac-Pauli equation using a gauge-like technique \cite{MK1,MK2}. In the spin-one case, we use Kemmer theory (a Dirac-like particle theory) to calculate the phase in a similar manner. However the vector formalism, the Proca theory, is more widely known and used. In the presence of an electromagnetic field, the two theories are `equivalent' and may be transformed into one another. We adapt these transformations to show that the Kemmer theory results apply…

The influence of Aharonov-Casher effect on the generalized Dirac oscillator in the cosmic string space-time

In this manuscript, we investigate the influence of the Aharonov-Casher effect on the generalized Dirac oscillator containing the Coulomb-type potential function related to a relativistic neutral particle having a permanent magnetic dipole moment interacting with the external electromagnetic fields in (1+2)-dimensional cosmic string space-time. The eigenfunctions and energy eigenvalues of such a Dirac oscillator are derived by using the Nikifornov-Uvarov method. We indicate that implementing the scenario gives the relativistic modified exact analytical solutions. In this way, we can see that t…

缺失的Aharonov-Casher几何量子相位

Missing Aharonov-Casher geometric quantum phase

本文从磁偶极矩与电场相互作用出发，发现缺失电荷导致几何量子相位，并探讨了Aharonov-Bohm型效应及持续自旋电流。

Testing Bell's inequality using Aharonov-Casher effect

We propose the Aharonov-Casher (AC) effect for four entangled spin-half particles carrying magnetic moments in the presence of impenetrable line charge. The four particle state undergoes AC phase shift in two causually disconnected region which can show up in the correlations between different spin states of distant particles. This correlation can violate Bell's inequality, thus displaying the non-locality for four particle entangled states in an objective way. Also, we have suggested how to control the AC phase shift locally at two distant locations to test Bell's inequality. We belive that a…

Testing Leggett's Inequality Using Aharonov-Casher Effect

Bell's inequality is established based on local realism. The violation of Bell's inequality by quantum mechanics implies either locality or realism or both are untenable. Leggett's inequality is derived based on nonlocal realism. The violation of Leggett's inequality implies that quantum mechanics is neither local realistic nor nonlocal realistic. The incompatibility of nonlocal realism and quantum mechanics has been urrently confirmed by photon experiments. In our work, we propose to test Leggett's inequality using the Aharonov-Casher effect. In our scheme, four entangled particles emitted fr…

(Semi)classical motion in fields of Aharonov-Bohm and Aharonov-Casher

Particle motion in the fields of Aharonov-Bohm and Aharonov-Casher is considered in framework of the classical theory to reveal conditions admitting duality of the two configurations. Important role of orientation of the magnetic dipole moment is demonstrated. Duality becomes totally destroyed by addition of electric dipole and/or higher multipole moments. Correspondence between quantum and classical considerations is also discussed.

Algebraic solution and coherent states for the Dirac oscillator interacting with the Aharonov-Casher system in the cosmic string background

We introduce an $SU(1,1)$ algebraic approach to study the $(2+1)$-Dirac oscillator in the presence of the Aharonov-Casher effect coupled to an external electromagnetic field in the Minkowski spacetime and the cosmic string spacetime. This approach is based on a quantum mechanics factorization method that allows us to obtain the $su(1,1)$ algebra generators, the energy spectrum and the eigenfunctions. We obtain the coherent states and their temporal evolution for each spinor component of this problem. Finally, for these problems, we calculate some matrix elements and the Schrödinger uncertainty…

Reply to "Comment on "Aharonov-Casher and Scalar Aharonov-Bohm Topological Effects""

In this Reply we argue that (i) the Hamiltonian, Eq. (17) in our paper (Phys. Rev. Lett. 108, 070405 (2012)), is definitely Lorentz invariant; (ii) the conditions of generating topological Aharonov-Casher(AC) and Scalar Aharonov-Bohm (SAB) effects are essential and physically meaningful; (iii) the Hamiltonians both in Phys. Rev. Lett. 74, 2847 (1995) and arXiv:1311.4011 are not suitable to describe the polarized spinor particles.

Comment on `Energy spectrum of a Dirac particle with position-dependent mass under the influence of the Aharonov-Casher effect'

It is shown that the paper `Energy spectrum of a Dirac particle with position-dependent mass under the influence of the Aharonov-Casher effect', by Oliveira, Borges and Sousa [Braz. J. Phys. 49, 801 (2019)], is based on a series of ingredients clearly incorrect.

Precession and interference in the Aharonov-Casher and scalar Aharonov-Bohm effect

The ideal scalar Aharonov-Bohm (SAB) and Aharonov-Casher (AC) effect involve a magnetic dipole pointing in a certain fixed direction: along a purely time dependent magnetic field in the SAB case and perpendicular to a planar static electric field in the AC case. We extend these effects to arbitrary direction of the magnetic dipole. The precise conditions for having nondispersive precession and interference effects in these generalized set ups are delineated both classically and quantally. Under these conditions the dipole is affected by a nonvanishing torque that causes pure precession around …

On the interaction of the Dirac oscillator with the Aharonov-Casher system in topological defect backgrounds

In this paper, we study the influence of the Aharonov-Casher effect [Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984).] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov-Casher geometric phase. We also discuss the influence of curvature and torsion on the relativist…

The Aharonov Casher Effect: The Case of g not equal to 2

The Aharonov Casher effect predicts the existence in two dimensions of ceil(Phi/2pi) -1 bounded zero modes associated with a magnetic flux Phi. Aharonov and Casher discussed the case of gyromagnetic factor equals 2, we will discuss the general case of any gyromagnetic factor. As a simple model, we study the case where the magnetic field lies in a thin annulus. First we examine the wavefunctions of the zero-energy bounded states, predicted by the Aharonov Casher Effect for electrons with gyromagnetic ratio equal 2. We then calculate the wave function and energies for a gyromagnetic ratio g not …

Aharonov-Bohm-Casher效应的定域性

Locality of the Aharonov-Bohm-Casher effect

本文通过分析理想屏蔽电磁场重叠的各种构型，证明Aharonov-Bohm-Casher干涉可由局域场作用解释，支持其定域性观点。

Comment on "Aharonov-Casher and Scalar Aharonov-Bohm Topological Effects"

In this Comment we point out (i) that the Hamiltonian, Eq. (17) in the Letter(Phys. Rev. Lett. 108, 070405 (2012)), is not a relativistic Hamiltonian, (ii) then that the conditions in the Letter are irrelevant for a topological AC and SAB effects, and (iii) conclusively that the non-relativistic Hamiltonian employed by Peshkin and Lipkin (Phys. Rev. Lett. 74, 2847 (1995)) has the same $U(1)_{mm}$ gauge structure for a fixed spin and then is not wrong, but their incorrect interpretation of the spin autocorrelations led to the incorrect conclusion.

Correspondences and Quantum Description of Aharonov-Bohm and Aharonov-Casher Effects

We establish systematic consolidation of the Aharonov-Bohm and Aharonov-Casher effects including their scalar counterparts. Their formal correspondences in acquiring topological phases are revealed on the basis of the gauge symmetry in non-simply connected spaces and the adiabatic condition for the state of magnetic dipoles. In addition, investigation of basic two-body interactions between an electric charge and a magnetic dipole clarifies their appropriate relative motions and discloses physical interrelations between the effects. Based on the two-body interaction, we also construct an exact …

阿哈罗诺夫 - 卡什尔定理中的规范自由度：二维和一维问题

The gauge freedom in the Aharonov-Casher theorem: The problem in two and one dimensions

研究阿哈罗诺夫 - 卡什尔定理中规范自由度在一维和二维的作用，分析其渐近行为，指出二维规范场唯一确定，一维存在规范自由度，由常数k决定，k受限以保证可归一化零模存在。

The time-dependent Aharonov-Casher effect

In this paper we give a covariant expression for Aharonov-Casher phase. This expression is a combination of the canonical electric field, Aharonov-Casher phase plus a magnetic field phase shift. We use this covariant expression for the Aharonov-Casher phase to investigate the case of a neutral particle with a non-zero magnetic moment moving in the {\it time dependent} electric and magnetic fields of a plane electromagnetic wave background. We focus on the case where the magnetic moment of the particle is oriented so that both the electric and magnetic field lead to non-zero phases, and we look…

Exact solutions of the Dirac oscillator under the influence of the Aharonov-Casher effect in the cosmic string background

In this work, we study the Aharonov-Casher effect in the $(2+1)$-dimensional Dirac oscillator coupled to an external electromagnetic field. We set up our system in two different scenarios: in the Minkowski spacetime and the cosmic string spacetime. In both cases, we solve exactly the Dirac oscillator and we determine the energy spectrum and the eigenfunctions for the bound states. We verify that in the Minkowski spacetime, the Dirac oscillator spectrum depends linearly on the strength of the magnetic field $B$, and on the Aharonov-Casher phase. In addition, we explicitly obtain the corrections…

Remarks on the Aharonov-Casher dynamics in a CPT-odd Lorentz-violating background

The Aharonov-Casher problem in the presence of a Lorentz-violating background nonminimally coupled to a spinor and a gauge field is examined. Using an approach based on the self-adjoint extension method, an expression for the bound state energies is obtained in terms of the physics of the problem by determining the self-adjoint extension parameter.

On Aharonov-Casher bound states

In this work bound states for the Aharonov-Casher problem are considered. According to Hagen's work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $\boldsymbol{\nabla}\cdot\mathbf{E}$ term cannot be neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Ahar…

阿哈罗诺夫-卡谢尔（Aharonov-Casher）相位：关于力、时间依赖性与贝里（Berry）相位的思考

The Aharonov-Casher Phase: Considerations Regarding Force, Time-Dependence, and Berry Phase

该研究通过Foldy-Wouthuysen变换推导AC哈密顿量一般形式，考察相位解析几何构型及与Berry相位的关联，得出不同电场强度下AC相位与场强的对应关系，明确了AC效应与磁矩粒子受力的联系。

An angular frequency dependence on the Aharonov-Casher geometric phase

A quantum effect characterized by a dependence of the angular frequency associated with the confinement of a neutral particle to a quantum ring on the quantum numbers of the system and the Aharonov-Casher geometric phase is discussed. Then, it is shown that persistent spin currents can arise in a two-dimensional quantum ring in the presence of a Coulomb-type potential. A particular contribution to the persistent spin currents arises from the dependence of the angular frequency on the geometric quantum phase.

Decoherence induced by a fluctuating Aharonov-Casher phase

Dipoles interference is studied when atomic systems are coupled to classical electromagnetic fields. The interaction between the dipoles and the classical fields induces a time-varying Aharonov-Casher phase. Averaging over the phase generates a suppression of fringe visibility in the interference pattern. We show that, for suitable experimental conditions, the loss of contrast for dipoles can be observable and almost as large as the corresponding one for coherent electrons. We analyze different trajectories in order to show the dependence of the decoherence factor with the velocity of the part…

Spin-selective Aharonov-Casher caging in a topological quantum network

A periodic network of connected rhombii, mimicking a spintronic device, is shown to exhibit an intriguing spin selective extreme localization, when submerged in a uniform out of plane electric field. The topological Aharonov Casher phase acquired by a travelling spin is seen to induce a complete caging, triggered at a special strength of the spin orbit coupling, for half odd integer spins s \ge n\hbar/2, with n odd, sparing the integer spins. The observation finds exciting experimental parallels in recent literature on caged, extreme localized modes in analogous photonic lattices. Our results …

穿越量子方形环的双体纠缠对的Aharonov-Casher相位

The Aharonov Casher phase of a bipartite entanglement pair traversing a quantum square ring

本文提出一种量子方形环，利用自旋纠缠粒子对产生、湮灭和提取非阿贝尔Aharonov-Casher相位，并证明纠缠可简化几何相位实验研究。

Aharonov-Casher-Effect Suppression of Macroscopic Tunneling of Magnetic Flux

We suggest a system in which the amplitude of macroscopic flux tunneling can be modulated via the Aharonov-Casher effect. The system is an rf-SQUID with the Josephson junction replaced by a Bloch transistor -- two junctions separated by a small superconducting island on which the charge can be induced by an external gate voltage. When the Josephson coupling energies of the junctions are equal and the induced charge is q=e, destructive interference between tunneling paths brings the flux tunneling rate to zero. The device may also be useful as a qubit for quantum computation.

Single atom-scale diamond defect allows large Aharonov-Casher phase

We propose an experiment that would produce and measure a large Aharonov-Casher (A-C) phase in a solid-state system under macroscopic motion. A diamond crystal is mounted on a spinning disk in the presence of a uniform electric field. Internal magnetic states of a single NV defect, replacing interferometer trajectories, are coherently controlled by microwave pulses. The A-C phase shift is manifested as a relative phase, of up to 17 radians, between components of a superposition of magnetic substates, which is two orders of magnitude larger than that measured in any other atom-scale quantum sys…

利用拓扑Aharonov-Casher相位与He-McKellar-Wilkens相位测试Bell-CHSH不等式

Testing Bell-CHSH Inequalities Using topological Aharonov-Casher and He-McKellar-Wilkens Phases

本研究利用带相位延迟器的Mach-Zehnder型干涉仪与线性电荷分布，分析拓扑AC-HMW相位对纠缠自旋1/2系统Bell角及Bell-CHSH不等式最大违反的影响，以自旋和路径为可观测量检验非语境隐变量理论，并探讨方案的物理实现。

The Aharonov-Casher Effect for Particles of Arbitrary Spin

The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two specia…

Aharonov-Casher 相位是几何的而非拓扑的

The Aharonov-Casher phase is geometrical and not topological

本文通过反例证明 Aharonov-Casher 相位是几何相位而非拓扑相位，其依赖于路径细节，且非阿贝尔情形下路径依赖更明显。

On the Aharonov-Casher scattering in a CPT-odd Lorentz-violating background

The effects of a Lorentz symmetry violating background vector on the Aharonov-Casher scattering in the nonrelativistic limit is considered. By using the self-adjoint extension method we found that there is an additional scattering for any value of the self-adjoint extension parameter and non-zero energy bound states for negative values of this parameter. Expressions for the energy bound states, phase-shift and the scattering matrix are explicitly determined in terms of the self-adjoint extension parameter. The expression obtained for the scattering amplitude reveals that the helicity is not co…

Locality and topology in the molecular Aharonov-Bohm effect

It is shown that the molecular Aharonov-Bohm effect is neither nonlocal nor topological in the sense of the standard magnetic Aharonov-Bohm effect. It is further argued that there is a close relationship between the molecular Aharonov-Bohm effect and the Aharonov-Casher effect for an electrically neutral spin$-{1/2}$ particle encircling a line of charge.

On the alleged nonlocal and topological nature of the molecular Aharonov-Bohm effect

The nonlocal and topological nature of the molecular Aharonov-Bohm (MAB) effect is examined for real electronic Hamiltonians. A notion of preferred gauge for MAB is suggested. The MAB effect in the linear + quadratic $E\otimes ε$ Jahn-Teller system is shown to be essentially analogues to an anisotropic Aharonov-Casher effect for an electrically neutral spin$-{1/2}$ particle encircling a certain configuration of lines of charge.

Comment on `Aharonov--Casher-Effect Suppression of Macroscopic Tunneling of Magnetic Flux'

A recent Letter [1] has proposed a device, consisting of an rf SQUID with the junction replaced by a double one, i.e. a Bloch transistor. For symmetric Josephson couplings, and for specific flux bias and gate voltage applied to the transistor's island, the tunneling of encircled flux between its two local potential minima is claimed to be suppressed completely due to destructive interference. This comment purports to show that this claim in general is not valid. Namely, the very Hamiltonian (1) in [1], from which the effect is derived, describes the circuit only under an additional condition. …

Magnetic Properties of Scalar Particles --The Scalar Aharonov-Casher Effect and Supersymmetry

The original topological Aharonov-Casher (AC) effect is due to the interaction of the anomalous magnetic dipole moment (MDM) with certain configurations of electric field. Naively one would not expect an AC effect for a scalar particle for which no anomalous MDM can be defined in the usual sense. In this letter we study the AC effect in supersymmetric systems. In this framework there is the possibility of deducing the AC effect of a scalar particle from the corresponding effect for a spinor particle. In 3+1 dimensions such a connection is not possible because the anomalous MDM is zero if super…

Persistent spin currents in an elastic Landau system

We consider a neutral particle with permanent magnetic dipole moment in an elastic medium with the presence of a uniform distribution of screw dislocations interacting with a radial electric field. We show that the uniform distribution of dislocations plays the role of an effective uniform magnetic field, and obtain a spectrum of energy which depends on the Aharonov-Casher geometric phase [Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984)]. Moreover, from the dependence of energy levels on the Aharonov-Casher geometric phase, we calculate the persistent spin currents in this elastic L…

Quantum effects of Aharonov-Bohm type and noncommutative quantum mechanics

Quantum mechanics in noncommutative space modifies the standard result of the Aharonov-Bohm effect for electrons and other recent quantum effects. Here we obtain the phase in noncommutative space for the Spavieri effect, a generalization of Aharonov-bohm effect which involving a coherent superposition of particles with opposite charges moving along single open interferometric path. By means of the experimental considerations a limit $\sqrtθ\simeq\left( 0,13T\text{eV}\right) ^{-1}$ is achieved, improving by 10 orders of magnitude the derived by Chaichian et. al. for the Aharonov-Bohm effect. It…

超冷暗态原子的Aharonov-Casher Chern能带

Aharonov-Casher Chern bands for ultracold dark state atoms

本文研究了Λ型原子-光耦合方案中超冷原子绝热跟随暗态时的Aharonov-Casher条件，推导了其适用要求，并展示了通过适当组合非精细调谐和有限耦合强度可实现完全平坦的最低能带，为模拟分数量子霍尔态提供完美拓扑。

Quantum gates with topological phases

We investigate two models for performing topological quantum gates with the Aharonov-Bohm (AB) and Aharonov-Casher (AC) effects. Topological one- and two-qubit Abelian phases can be enacted with the AB effect using charge qubits, whereas the AC effect can be used to perform all single-qubit gates (Abelian and non-Abelian) for spin qubits. Possible experimental setups suitable for a solid state implementation are briefly discussed.

阿哈罗诺夫-卡舍相位是几何相位还是动力学相位？

Is the Aharonov-Casher phase geometrical or dynamical?

该研究针对含拉什巴自旋-轨道相互作用的两类二维电子系统，证明狄拉克体系的阿哈罗诺夫-卡舍相位为动力学相位，薛定谔体系的该相位为几何相位。

Reply to the comment of Y. Aharonov and L. Vaidman on ``Time asymmetry in quantum mechanics: a retrodiction paradox''

In the standard physical interpretation of quantum theory, prediction and retrodiction are not symmetric. The opposite assertion by some authors results from their use of non-standard interpretations of the theory.

Comment on "On the Electric Charge Quantization from the Aharonov-Bohm Potential"

In the paper quant-ph/0503212, Barone and Halayel-Neto (BH) claim that charge quantization in quantum mechanics can be proven without the need for the existence of magnetic monopoles. In this paper it is argued that their claim is untrue.

Reply to a Comment on "Role of Potentials in the Aharonov-Bohm Effect"

Preceding Comment challenged my claim that potentials might be just auxiliary mathematical tools and that they are not necessary for explaining physical phenomena. The Comment did not confront my explanation without potentials of the Aharonov-Bohm effects appeared in the original paper, but stated that I cannot apply this explanation for seven other examples. In my reply I provide explanations using my method, of one of the examples, show that two other example are not relevant, and agree that the remaining examples require further analysis. However, I argue that none of the examples provide r…

Comments on ``Differential cross section for Aharonov-Bohm effect with nonstandard boundary conditions''

We show that the violation of rotational symmetry for differential cross section for Aharonov-Bohm effect with nonstandard boundary conditions has been known for some time. Moreover, the results were applied to discuss the Hall effect and persistent currents of fermions in a plane pierced by a flux tube.

Quantum states of a neutral massive fermion with an anomalous magnetic moment in an external electric field

The planar non-relativistic quantum dynamics of a neutral massive fermion with an anomalous magnetic moment (AMM) in the presence of the electric field of infinitely long and thin thread with a charge density distributed uniformly along it (an Aharonov--Casher field) is examined. The relevant Hamiltonian is singular and requires additional specification of a one-parameter self-adjoint extension, which can be given in terms of physically acceptable (self-adjoint) boundary conditions. We find all possible self-adjoint Hamiltonians with an Aharonov--Casher field (ACF) by constructing the correspo…

Towards a quantum Hall effect for atoms using electric fields

An atomic analogue of Landau quantization based on the Aharonov-Casher (AC) interaction is developed. The effect provides a first step towards an atomic quantum Hall system using electric fields, which may be realized in a Bose-Einstein condensate.

Comment on "Protective measurements and Bohm's trajectories" by Y.Aharonov, B.G.Englert, and M.O.Scully

In the above paper [1], it is claimed that Bohm's mechanics can be challenged by a simple gedanken experiment involving a protecting measurement. Here we show that this gedanken experiment can be justified without problem and without contradicting the axiom attributing a physical meaning to the Bohmian trajectorie

Comment on Macroscopic Test of the Aharonov-Bohm Effect

In this Comment it is shown that it cannot be argued that in the magnetic AB effect there is no force acting on the particle, i.e., that the observed phase shift is entirely due to nonzero vector potential. In stationary resistive conductors carrying constant currents there are quasistatic surface charges, which generate not only the electric field inside the wire driving the current, but also a static electric field outside it. These external static electric fields have nothing to do with Boyer's force picture and with his result for the existence of a time delay.

2+1维阿哈罗诺夫-玻姆型配置中的奇偶异常

Parity anomalies in 2+1-dimensional Aharonov-Bohm type configurations

该研究通过Fujikawa方法揭示了2+1维阿哈罗诺夫-玻姆型配置中狄拉克费米子的奇偶异常，并展示了其拓扑相的特征。

Comment on: "Aharonov-Bohm effect for bound states from the interaction of the magnetic quadrupole moment of a neutral particle with axial fields". Phys. Rev. A 101, 032102 (2020)

We analyze recently published results about the Aharonov-Bohm effect for bound states from the interaction of the magnetic quadrupole moment of a neutral particle with axial fields. We show that the eigenvalues obtained by the authors from an arbitrary truncation of the Frobenius power series correspond to more than one model and, consequently, are unsuitable for drawing any sound physical conclusion. Besides, the prediction of allowed oscillator frequencies is a mere consequence of the truncation condition just mentioned.

Comment on "Role of Potentials in the Aharonov-Bohm Effect"

Are the electromagnetic scalar and vector potentials dispensable? Lev Vaidman has suggested that local interactions of gauge-invariant quantities, e.g. magnetic torques, suffice for the description of all quantum electromagnetic phenomena. We analyze six thought experiments that challenge this suggestion. All of them have explanations in terms of $local$ interactions of gauge-dependent quantities, in addition, some have explanations in terms of nonlocal interactions of gauge-invariant quantities. We claim, however, that two of our examples have no gauge-invariant formal description and that, i…