This paper formulates generalized versions of the general principle of relativity and of the principle of equivalence that can be applied to general abstract spaces. It is shown that when the principles are applied to the Hilbert space of a quantum particle, its law of coupling to electromagnetic fields is obtained. It is suggested to understand the Aharonov-Bohm effect in light of these principles, and the implications for some related foundational controversies are discussed.
I discuss in detail the history of the Aharonov-Bohm effect in Bristol and my encounters with Akira Tonomura later on. I then propose an idea that developed following the publication of the Aharonov-Bohm effect, namely the importance of modulo momentum and Heisenberg representation in dealing with non-local quantum phenomena.
For a believer in locality of Nature, the Aharonov-Bohm effect and the Aharonov-Casher effect are paradoxes. I discuss these and other Aharonov's paradoxes and propose a local explanation of these effects. If the solenoid in the Aharonov-Bohm effect is treated quantum mechanically, the effect can be explained via local interaction between the field of the electron and the solenoid. I argue that the core of the Aharonov-Bohm and the Aharonov-Casher effects is that of quantum entanglement: the quantum wave function describes all systems together.
We obtain exact solutions to the Dirac equation and the relevant binding energies in the combined Aharonov--Bohm--Coulomb potential in 2+1 dimensions. By means of solutions obtained the quantum Aharonov--Bohm effect is studied for free and bound electron states. We show that the total scattering amplitude in the combined Aharonov--Bohm--Coulomb potential is a sum of the Aharonov--Bohm and the Coulomb scattering amplitudes. This modifies expression for the standard Aharonov--Bohm cross section due to the interference these two amplitudes with each other.
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.
We reexamine the topological and nonlocal natures of the Aharonov-Casher and scalar Aharonov-Bohm phase effects. The underlying U(1) gauge structure is exhibited explicitly. And the conditions for developing topological Aharonov-Casher and scalar Aharonov-Bohm phases are clarified. We analyse the arguments of M. Peshkin and H. J. Lipkin (Phys. Rev. Lett. 74, 2847(1995)) in detail and show that they are based on the wrong Hamiltonian which yields their conclusion incorrect.
Aharonov-Bohm (AB) and Aharonov-Casher (AC) effects are treated fully relativistically in 2+1 dimensions. The influences of the relevant geometric and topological phases on an entangled spin-1/2 system are studied. It is shown that for the AC case the correlation function of the Clauser-Horne-Shimony-Holt (CHSH) inequality for certain choices of the spin measurements depends on the AC phase explicitly.
The Schrodinger formalism of quantum mechanics is used to demonstrate the existence of the Aharonov-Bohm effect in momentum space and set-ups for experimentally demonstrating it are proposed for either free or ballistic electrons.
A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.
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.
Although the Aharonov-Bohm and related effects are familiar in solid state and high energy physics, the nonlocality of these effects has been questioned. Here we show, for the first time, that the Aharonov-Bohm effect has two very different aspects. One aspect is instantaneous and nonlocal; the other aspect, which depends on entanglement, unfolds continuously over time. While local, gauge-invariant variables may occasionally suffice for explaining the continuous aspect, we argue that they cannot explain the instantaneous aspect. Thus the Aharonov-Bohm effect is, in general, nonlocal.
A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value $< r^λ>$ $(-\infty \leq λ\leq \infty)$ is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the s…
The propagator of three-dimensional Aharonov-Bohm-Coulomb system is calculated by following the Duru-Kleinert method. It is shown that the system is reduced to two independent two dimensional Aharonov-Bohm plus harmonic oscillator systems through dimensional extension and Kustaanheimo-Stiefel transformation. The energy spectrum is deduced.
The possibility of quantum transmission of information via the induced fractional angular momentum by the Aharonov - Bohm vector potential is revealed. Its special advantage is that it is noiseless: Stray magnetic fields of environments influence the energy spectrum of the ion, but cannot contribute the fractional angular momentum to cause noise.
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…
The magnetic field due to the photon rest mass $m_{ph}$ modifies the standard results of the Aharonov-Bohm effect for electrons, and of other recent quantum effects. For the effect involving a coherent superposition of beams of particles with opposite electromagnetic properties, by means of a table-top experiment, the limit $m_{ph}x10^{-51}g$ is achievable, improving by 6 orders of magnitude that derived by Boulware and Deser for the Aharonov-Bohm effect.
In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is written by the gauge invariant physical current $(-e)\barψ{\boldsymbol γ} ψ$, 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 {\it et al.}, wh…
An experiment is proposed in which the Aharonov-Bohm effect can be veryfied through a spectroscopic measurement. The apparatus consists of gaseous hydrochloric acid (HCl) immersed in the vector potential ${\bf A}$ present in the center of a toroidal coil, where ${\bf B}=0$. Changes due to ${\bf A}$ in the absorption spectrum of the gas are investigated.
In this note we present a simplified derivation of the fact that the moduli space of flat connections in the abelian Aharonov-Bohm effect is isomorphic to the circle. The length of this circle is the electric charge.
An experiment that shows the modulation of the Aharonov-Bohm oscillations of magneto-resistance in a mesoscopic ring is described. Possible theoretical explanations of this modulation due to the interaction of the electron spin with the magnetic and electric fields are considered.
It is shown that the phase of the hidden momentum in Aharonov-Bohm (AB) solenoid effect is equal in magnitude to the phase of the electron but with opposite sign. The phase of the hidden momentum is equal to that obtained by the energy of interference calculated in our previous paper (J.Opt.Soc.Am. B 17, 2052 2000).
When the magnetic vector potential is expressed in terms of the magnetic field it, is found to be explicitly non-local in space. This gives support to the conclusions of Aharonov et al. in a recent comment, that the Aharonov-Bohm effect may be interpreted as being either due to a local gauge potential or else due to non-local gauge-invariant fields but not due to local gauge-invariant fields.
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.
In the Landau problem on the two-dimensional plane, magnetic translation of a quantum wave can be induced by an in-plane electric field. The geometric phase accompanying such magnetic translation around a closed path differs from the topological phase of Aharonov and Bohm in two essential aspects: The wave is in direct contact with the magnetic flux and the geometric phase has an opposite sign from the Aharonov-Bohm phase. We show that magnetic translation on the two-dimensional cylinder implemented by the Schrödinger time evolution truly leads to the Aharonov-Bohm effect. The magnetic field n…
We suggest as gedanken experiment a generalization of the Aharonov-Bohm experiment, based on an array of solenoids. This experiment allows in principle to measure the decomposition into homotopy classes of the quantum mechanical propagator. This yields information on the geometry of the average path of propagation and allows to determine its Hausdorff dimension.
A phase space representation of the Aharonov-Bohm effect is presented. It shows that the shift of interference fringes is associated to the interference term of the Wigner distribution function of the total wavefunction, whereas the interference pattern is defined by the common projections of the Wigner distribution functions of the interfering beams
A basic analysis is provided for the differential cross section characterizing Aharonov--Bohm effect with non standard (non regular) boundary conditions imposed on a wave function at the potential barrier. If compared with the standard case two new features can occur: a violation of rotational symmetry and a more significant backward scattering.
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.
Using the theory of self-adjoint extensions, we construct all the possible hamiltonians describing the non relativistic Aharonov-Bohm effect. In general the resulting hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.
The role of the inert magnetic source used in the Tonomura experiment that has confirmed the magnetic Aharonov-Bohm effect is discussed. For this purpose, an analysis of a thought experiment is carried out. Here the permanent magnet is replaced by a classical source which is made of an ideal coil. A detailed calculation of this noninert source proves that in this case the effect disappears. This outcome provides another support for the crucial role of an inert source in the Aharonov-Bohm effect. A new aspect of quantum nonlocality is pointed out.
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.
There is a consensus today that the the main lesson of the Aharonov-Bohm effect is that a picture of electromagnetism based on the local action of the field strengths is not possible in quantum mechanics. Contrary to this statement it is argued here that when the source of the electromagnetic potential is treated in the framework of quantum theory, the Aharonov-Bohm effect can be explained without the notion of potentials. It is explained by local action of the field of the electron on the source of the potential. The core of the Aharonov-Bohm effect is the same as the core of quantum entangle…
We investigate the scattering of neutral polarizable atoms from an electrically charged wire placed in a homogeneous magnetic field. The atoms carry an induced electric dipole. The reflecting wire is discussed. We calculate the scattering amplitude and cross section the practically more important case that atoms are totally absorbed at the surface of the wire. If the magnetic field is present, there is a dominating Aharonov-Bohm peak in the forward direction followed by decreasing oscillations for larger angles. An experimental realization of this modulated Aharonov-Bohm scattering should be p…
We apply the recently generalized Levinson theorem for potentials with inverse square singularities [Sheka et al, Phys.Rev.A, v.68, 012707 (2003)] to Aharonov-Bohm systems in two-dimensions. By this theorem, the number of bound states in a given m-th partial wave is related to the phase shift and the magnetic flux. The results are applied to 2D soliton-magnon scattering.
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.
We determine the energy spectrum and the corresponding eigenfunctions of a 2D Dirac oscillator in the presence of Aharonov-Bohm (AB) effect . It is shown that the energy spectrum depends on the spin of particle and the AB magnetic flux parameter. Finally, when the irregular solution occurs it is shown that the energy takes particular values. The nonrelativistic limit is also considered.
Decades ago, Aharonov and Bohm showed that electrons are affected by electromagnetic potentials in the absence of forces due to fields. Zeilinger's theorem describes this absence of classical force in quantum terms as the "dispersionless" nature of the Aharonov-Bohm effect. Shelankov predicted the presence of a quantum "force" for the same Aharonov-Bohm physical system as elucidated by Berry. Here, we report an experiment designed to test Shelankov's prediction and we provide a theoretical analysis that is intended to elucidate the relation between Shelankov's prediction and Zeilinger's theore…
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
Partial wave theory of a two dimensional scattering problem for an arbitray short range potential and a nonlocal Aharonov-Bohm magnetic flux is established. The scattering process of a ``hard disk'' like potential and the magnetic flux is examined. Since the nonlocal influence of magnetic flux on the charged particles is universal, the nonlocal effect in hard disk case is expected to appear in quite general potential system and will be useful in understanding some phenomena in mesoscopic phyiscs.
The Aharonov-Bohm effect is considered by most authors as a quantum effect, but a generally accepted explanation does not seem to be available. The phenomenon is studied here under the assumption that hypothetical electric dipole distributions configured by moving charges in the solenoid act on the electrons as test particles. The relative motions of the interacting charged particles introduce relativistic time dilations. The massless dipoles are postulated as part of an impact model that has recently been proposed to account for the far-reaching electrostatic forces between charged particles …
We suggest a method for trapping photons in quasi one-dimensional waveguide or coupled-resonator lattices, which is based on an optical analogue of the Aharonov-Bohm cages for charged particles. Light trapping results from a destructive interference of Aharonov-Bohm type induced by a synthetic magnetic field, which is realized by periodic modulation of the waveguide/resonator propagation constants/resonances.
We analyze a Hanbury Brown Twiss geometry in which particles are injected from two independent sources into a mesoscopic electrical conductor. The set-up has the property that all partial waves end in different reservoirs without generating any single particle interference. There is no single particle Aharonov-Bohm effect. However, exchange effects lead to two-particle Aharonov-Bohm oscillations in current correlations. We demonstrate that the two-particle Aharonov-Bohm effect is connected to orbital entanglement which can be used for violation of a Bell Inequality.
The topological calculation of Aharonov-Bohm phases associated with D-branes in the absence of a Neveu-Schwarz B-field is explored. The K-theoretic classification of Ramond-Ramond fields in Type II and Type I theories is used to produce formulae for the Aharonov-Bohm phase associated with a torsion flux. A topological construction shows that K-theoretic pairings to calculate such phases exist and are well-defined. An analytic perspective is then taken, obtaining a means for determining Aharonov-Bohm phases by way of the reduced eta-invariant. This perspective is used to calculate the phase for…
In the framework of QED we evaluate the cross section for electron-positron pair production by a single photon in the presence of the external Aharonov-Bohm potential in first order of perturbation theory. We analyse energy, angular and polarization distributions at different energy regimes: near the threshold and at high photon energies.
The phase shifts of the Aharonov-Bohm effect are generally determined by means of the partial wave decomposition of the underlying Schrodinger equation. It is shown here that they readily emerge from an o(2,1) calculation of the energy levels employing an added harmonic oscillator potential which discretizes the spectrum.
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 …
The Aharonov-Bohm scattering of a localized wave packet is considered. A careful analysis of the forward direction points out new results: according to the time-dependent solution obtained by means of the asymptotic representation for the propagator (kernel), a phenomenon of auto-interference occurs along the forward direction, where, also, the probability density current is evaluated and found finite.
A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymmetric structure is shown to be intimately related with the scale symmetry. As a result, a bosonized superconformal symmetry is identified in the system. Different self-adjoint extensions of the Aharonov-Bohm problem are studied in the light of this superconformal structure and interacting anyons. Scattering problem of the original Aharonov-Bohm model is discussed in the context of the revealed supersymmetry.
We predict that a charge under the influence of a quantum electrodynamic potential exhibits a force-free Aharonov-Bohm effect. The specific system considered in this study is a superconducting charge qubit nonlocally interacting with a cavity electromagnetic field. We find that this nonlocal interaction gives rise to remarkable quantum electrodynamic phenomena such as vacuum Rabi splitting and oscillation, and Lamb shift, under the condition that the qubit is located in an electromagnetic-field-free region. Our result can be verified in a realistic experimental setup, and provides a new approa…
Hamilton-Jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials (A,ϕ), similar to Schroedinger equation. We derived the Aharonov-Bohm effect from Hamilton-Jacobi equation thereby having proved that this effect is of classical origin. These facts enable us to arrive at the following conclusions: a) the very idea of special role of potentials (A,ϕ) in quantum mechanics (different from their role in classical physics) lost the ground, and becomes dubious, as this idea is based on the Aharonov-Bohm effect, b) failure to find any s…
We investigate Faraday's law of induction manifested in the quantum state of Aharonov-Bohm loops. In particular, we propose a flux-switching experiment for a double-dot AB ring to verify the phase shift induced by Faraday's law. We show that the induced {\em Faraday phase} is geometric and nontopological. Our study demonstrates that the relation between the local phases of a ring at different fluxes is not arbitrary but is instead determined by Faraday's inductive law, which is in strong contrast to the arbitrary local phase of an Aharonov-Bohm ring for a given flux.
Bound states of massive fermions in the Aharonov-Bohm like fields have analytically been studied. The Hamiltonians with the Aharonov--Bohm like potentials are essentially singular and therefore require specification of a one-parameter self-adjoint extension. We construct self-adjoint Dirac Hamiltonians with the Aharonov-Bohm (AB) potential in 2+1 dimensions that are specified by boundary conditions at the origin. It is of interest that for some range of extension parameter the AB potential can bind relativistic charged massive fermions. The bound-state energy is determined by the AB magnetic f…
The minimal coupling rule is "derived" starting from Landau's relativistically invariant classical action for a charge in the presence of classical electromagnetic fields. Experiments are then proposed to see the resulting electromagnetic angular momentum of a classical, "lumpy" charged ring enclosing a solenoid. These classical, macroscopic experiments are similar in spirit to those proposed by Aharonov and Bohm at the quantum level.
We consider the phase-coherent transport of electrons passing through an Aharonov-Bohm ring while interacting with a tunnel charge in a double quantum dot (representing a charge qubit) which couples symmetrically to both arms of the ring. For Aharonov-Bohm flux Phi_AB=h/2e we find that electrons can only be transmitted when they flip the charge qubit's pseudospin parity an odd number of times. The perfect correlations of the dynamics of the pseudospin and individual electronic transmission and reflection events can be used to entangle the charge qubit with an individual passing electron.
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.
The canonical Aharonov-Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov-Bohm phase acquired by a charged particle moving in {\it time-dependent} potentials . In particular, we focus on the case of a charged particle moving in the time varying field of a plane electromagnetic wave. We work out the Aharonov-Bohm phase using both the potential ({\it i.e.} $\oint A_μdx ^μ$) and field ({\it i.e.} $\frac{1}{2}\int F_{μν} d σ^{μν}$) forms of the Aharanov-Bohm phase. We give conditions in terms of the parameters of the system (frequency of the …
Another manifestation of the Aharonov-Bohm effect is introduced to chemistry, in fact to nuclear dynamics and high-resolution molecular spectroscopy. As demonstrated, the overall rotation of a symmetric-top molecule influences the dynamics of an internal vibrational motion in a way that is analogous to the presence of a solenoid carrying magnetic flux. To a good approximation, the low-energy rovibrational energy-level structure of the quasistructural molecular ion H5+ can be understood entirely in terms of this effect.
The role of Aharonov-Bohm effect in quantum tunneling is examined when a potential is defined on the $S^1$ and has $N$-fold symmetry. We show that the low-lying energy levels split from the $N$-fold degenerate ground state oscillate as a function of the Aharonov-Bohm phase, from which general degeneracy conditions depending on the magnetic flux is obtained. We apply these results to the spin tunneling in a spin system with $N$-fold rotational symmetry around a hard axis.
