Search

A quasi-mixed-potential layered medium Green’s function (QMP-LMGF) is proposed for non-Galerkin surface integral equation (SIE) formulations in the modeling of homogeneous dielectric objects in layered medium. The formulation is derived based on the pilot vector potential approach. In this method, the field-type LMGF in the L-operator is decomposed into two parts, a “vector potential” term in a dyadic form, and a “scalar potential” term in a vector form, to represent the field components from current and charge sources, respectively. Since the proposed two potential terms (vector potential and scalar potential) do not satisfy the Lorenz gauge, we term them as quasi-mixed-potentials. The property of the two potentials is investigated to reveal that the proposed QMP-LMGF is compatible with the vector and scalar potentials in free space. Moreover, due to the continuity of the integration kernels across the interfaces, undesired line integrals are absent in the proposed QMP-LMGF when the objects are straddling different layers. Three popular non-Galerkin SIEs for dielectric objects in layered medium are studied, and the corresponding numerical results are demonstrated to validate the proposed method.

This article presents the estimation of electric and magnetic fields using the Kalman filter (KF). The electric and magnetic fields in the entire space have been estimated using the scalar and vector potential. For this estimation, the measurements at a sparse discrete set of spatial pixels have been used. To implement the KF, the state space model has been obtained using the wave equations with sources satisfied by the scalar and vector potential. The proposed method has been implemented on a Hertzian dipole antenna. The fields estimated using KF have been compared with the recursive least squares (RLS) method. The KF presents better estimation than RLS, as it is an optimal estimator. This work uses the Kronecker product for compact representation of discretized fields in the form of vectors and partial differential operators in the form of matrices.

We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)

We solve the Klein–Gordon equation subject to a homogeneous magnetic field including a quantum flux under the influence of a scalar and vector potential of Coulomb types in a cosmic string space-time, and analyze the relativistic analogue of the Aharonov–Bohm effect for bound states.

In this paper, numerical calculations were made for Ar loop-type inductively coupled thermal plasma (loop-ICTP). The loop-ICTP was developed originally by the authors’ group for rapid surface modification of large areas. Loop-ICTP is sustained with a unique three-dimensional (3D) configuration inside a circular loop quartz tube and on the substrate. A 3D and two-temperature thermofluid thermal plasma model was adopted for this calculation. Mass, momentum, and energy conservation equations were solved using a Maxwell equation for vector potential, an electron energy transport equation, and Saha’s equation in the 3D space. Results indicate that Ar loop-ICTP can be sustained and formed in the loop tube and also on the substrate. Moreover, the heavy particle temperatures reaches 1800–2000 K, whereas the electron temperature is about 10,000 K. Loop size effects on the gas temperature and gas flow field were also investigated using the developed model. Results show that adoption of a larger loop tube can be expected to improve the plasma uniformity on the substrate when applied to rapid surface modification.

In this paper, the spontaneous emission rate of an excited atom between the two dispersive and dissipative dielectric slabs is determined. It is assumed that the atom has a definite transition electric dipole moment, and the dielectric function of the slabs is an arbitrarily complex function of frequency satisfying Kramers–Kronig relations. In this research, by noting to the connection between the spontaneous emission rate and the imaginary part of the vector potential Green function, only by evaluating the vector potential Green function tensor of the system of the two slabs, an expression for the spontaneous emission rate is introduced. In this formalism, it is needless to derive the vacuum electric field operator. The emission rate is obtained in terms of the dielectric function, the thickness and the distance between the slabs. By considering a Lorentz model for the dielectric function of the slabs, the typical spatial variations of the spontaneous emission rate between the two slabs are demonstrated.

We generalize our previous unification of the Schrödinger and guidance equations in a single inhomogeneous Schrödinger equation to a Riemannian manifold with an external vector potential. A special case yields the unified theory for a spin $$\frac{1}{2}$$ 1 2 rigid rotator. The theory is proved to be symmetrical under the Galileo group, the unified field that integrates the particle and guiding wave being a 2-spinor.

A hybrid spatial–spectral method for rapid and broadband calculation of the vector potential normal modes and dyadic Green’s function (GF) inside the cavity of irregular shape is presented. In the broadband method of simulation, the normal modes of the irregular cavity, which are independent of operation frequency, are first computed through a linear eigenvalue problem based on a hybrid representation of the regular (rectangular) cavity. The method is based on the extraction of the singularity of the GF that results in a rapidly convergent hybrid spatial–spectral expansion of the GF. The method is applied to a V-grooved cavity to obtain the vector potential GF inside the cavity, and the results are shown over a broad range of frequencies. The results of the hybrid method are in good agreement with that of direct method of moment (MoM), while the hybrid method is much more efficient than direct MoM as the latter requires the computation at one frequency at a time.

A variant of the experiment to observe the optical Aharonov—Bohm effect has been discussed. In the experiment, a unipolar subcycle light pulse has been proposed as a source of the vector potential acting on electrons and having zero electric field strength in the region of a nonzero vector potential. Such a relation between the field and potential appears because the unipolar pulse has a nonzero electric area. An unusual situation where the fact of the passage of the pulse in one of the arms of two-beam interferometer is stored for a very long time after the passage and can be recorded in a two-beam electronic interferometer by a shift of fringes has been discussed. Detailed analysis of the phase shift under the influence of the vector potential created by the inhomogeneous unipolar pulse removes this contradiction and shows the equality of phase incursions in both arms that is unobvious at first glance.

A Feynman path integral formula for the Schr\"odinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schr\"odinger and heat equation with magnetic field., Comment: 43 pages

In this work, the interaction potential between an atom (with definite Static Polarizability) and a perfect neutral conductive wall is evaluated in two approaches. First, using, the explicit form of the singlet and doublet parts of the vacuum vector potential operator in atomic quadratic stark shift formula and taking the expectation value in vacuum state. Second, writing, the quadratic stark shift formula in terms of electric field correlation function, relating the correlation function to the imaginary parts of the vector potential Green function tensor components (via fluctuation dissipation theorem in statistical mechanics) and the evaluation of the required vector potential Green function tensor components. The consistency of the two methods is shown.

We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing for fairly general coupling functions becomes relevant when the matter-radiation system is confined to cavities with inward pointing boundary singularities., Comment: 48 pages. Second version: references updated, typos removed plus a small number of other trivial improvements

By using the vector potential operator commutation relations, for a molecule (or generally an emitter) placed between two infinite identical dielectric slabs and with the given transition frequency and electric dipole moment, the spontaneous emission rate is evaluated via Fermi golden rule. Molecules with the electric dipole moment parallel and perpendicular to the slabs are considered separately, and for each orientation, a typical variation of the emission rate in the space of the cavity is demonstrated. In the used quantization scheme, the dielectric functions of the slabs can be an arbitrary complex function of frequency (satisfying Kramers–Kronig relations) and thus, slabs generally can be dissipative and dispersive. By showing the agreement of this quantization approach with two previous green function approaches, in evaluating the spontaneous emission rate in a Fabry–Perot cavity, the consistency between field quantization and Green function approaches is shown.

A previously introduced reduction of the Dirac equation is used to study the Charmonium spectrum. A regularized vector potential that only depends on the coupling constant and on the regularization radius is adopted, considering the interacting quark as an extended source of the Chromo-electric field. A scalar interaction is also introduced with some constraints for its parameters. A good description of the structure of the Charmonium spectrum is obtained with only three free parameters., 14 pages, 2 tables

This work considers the creation of electron-positron pairs from intense electric fields in vacuum, for arbitrary temporal field variations. These processes can be useful to study quantum vacuum effects with ultra-intense lasers. We use the quantized Dirac field to explore the temporal Klein model. This model is based on a vector potential discontinuity in time, in contrast with the traditional model based on a scalar potential discontinuity in space. We also extend the model by introducing a finite time-scale for potential variations. This allows us to study the transition from a singular electric field spike, with infinitesimal duration, to the opposite case of a static field where the Schwinger formula would apply. The present results are intrinsically non-perturbative. Explicit expressions for pair-creation as a function of the potential time-scales are derived. This work explores the spacetime symmetry associated with pair creation in vacuum: the space symmetry breaking of the old Klein paradox model, in contrast with the time symmetry breaking of the temporal Klein model.

The theory of the two-dimensional (2D) magnetoexcitons was enlarged in two aspects. One of them takes into account the electron-hole (e–h) exchange Coulomb interaction, which appears when the conduction and the valence electrons belong partially to both bands. The exchange Coulomb interaction gives rise to the Dirac cone dispersion law in the range of small in-plane wave vectors k|| obeying to the condition |k|||l0 < 1, where l0 is the magnetic length. Such dispersion law may change essentially the properties of the high density magnetoexcitons opening the possibility of their Bose-Einstein condensation at different from zero temperatures [1]. Another aspect concerns the high density magnetoexcitons with fractional filling factors in conditions of fractional quantum Hall effects. The Chern-Simons gauge field was introduced into the Hamiltonian of the 2D coplanar e–h system by the unitary transformation. It gives rise to the additional gauge vector potential, the influence of which leads to the anisotropic corrections to the magnetoexciton magnetic mass [2].

An increasing need for fast and reliable models has led to a continuous development of Fourier-based (FB) analytical modeling. This paper presents an overview of the techniques that are currently available in FB modeling for electric machines. By coupling that overview to the most relevant literature related to the subject, an interesting starting point is provided for anyone who wants to use or improve FB models. The following seven aspects of FB models are discussed in detail: 1) the magnetic potential (scalar or vector potential); 2) the coordinate system and the solution of the partial-differential equations for each magnetic potential and for each coordinate system; 3) the way in which time dependence is accounted for; 4) the implementation of the source terms; 5) the possibilities to account for slotted structures; 6) the modeling of eccentricity; and 7) the post-processing computation of physical quantities, such as flux density, electromotive force, torque, losses, and eddy currents in conductive objects. Furthermore, this paper gives the closed form solution of the Laplace, Poisson, and Helmholtz equations in each coordinate system. In addition, this paper tackles other important features of FB models such as computational time reduction and coupling the machine model to an electric circuit.

Vertical components of the vector potential of electromagnetic wave above two-layer strongly inductive impedance medium are determined. The solution is represented as a Sommerfeld integral. Singularities of such an integral are determined using effective parameters. The phase velocity of the surface electromagnetic wave is considered. It is shown that such a velocity is always less than the velocity of light in vacuum, so that the surface electromagnetic waves cannot be classified as the Zenneck waves for which the phase velocity is greater than the velocity of light (similarly to electromagnetic waves in waveguides).

The scattering properties of nonspherical particles can be approximately computed by equivalent spherical theory. The scattering properties of ice particles were approximately computed by Rayleigh approximation because the sizes of the ice particles are smaller than the wavelength of millimeter wave radar. Based on the above assumption, the echo fluctuation of moving particles was analyzed by computing the total backscattering field of a cirrus cloud using the classical vector potential technique. The simulation results showed that echo fluctuation influences the accuracy of retrieving the physical parameters of a cloud. To suppress the echo fluctuation of moving ice particles, a video integrator of a millimeter wave cloud radar would be used. However, video integrators lose the rapidly changing information of ice particles and reduce radar range resolution; thus, we propose the pace-diversity technique of MIMO radar to reduce the echo fluctuation, which could be validated by theoretical computation and experimental measurements.

We consider a simplified (reduced) dynamic model of a Cosserat medium, which occupies an intermediate position between the classical dynamic theory of elasticity and the proper Cosserat medium model, which has asymmetry in the stress tensor and the presence of moment stresses. In contrast to the latter, in the simplified model, three of the six elastic constants are zero and, as a result, there is no moment stress tensor. In the two-dimensional formulation for the model of a reduced medium, the problem of the propagation of an elastic surface wave along the half-space boundary was solved. The solution of the equations was described as the sum of the scalar and vector potentials, and only one component of the vector potential is nonzero. It is shown that such a wave, in contrast to the classical surface Rayleigh wave, has a dispersion. In the plane “phase velocity-frequency” for such waves there are two dispersion branches: the lower (acoustic) and upper (optical). With increasing frequency, the phase velocity of the wave related to the lower dispersion branch decreases. The phase velocity of the wave related to the upper dispersion branch increases with increasing frequency. The phase velocity of the surface wave in the entire frequency range exceeds the phase velocity of the bulk shear wave. The stresses and displacements arising in the zone of propagation of the surface wave are calculated.

Electrons in solids constitute quantum many-body systems showing a variety of phenomena. It often happens that the eigen states of the Hamiltonian are classified into subgroups separated by energy gaps. Band structures in solids and spin polarization in Mott insulators are two representative examples. The subspace spanned by these wavefunctions belonging to each of this subgroup can be regarded as a manifold in Hilbert space, and concepts concerning differential geometry become relevant. Connection and curvature are two key quantities, which correspond to the vector potential and field strength of electromagnetism, respectively. Therefore, one can construct an effective electromagnetic field from the structure of the Hilbert space, which is called an "emergent electromagnetic field". In this article, we review the physics related to this emergent electromagnetic field in solids, including the gauge theory of strongly correlated electrons, various Hall effects, multiferroics, topological matter, magnetic texture such as skyrmions, and the shift current in noncentrosymmetric materials.

A semicircular linear motor that realizes a motion along circumference of the circle has been developed. To settle the optimal design of this motor by mathematical analysis, the theoretical equation of motor performance has to be clarified. In this paper, the mathematical modeling of semicircular linear motor is carried out from the viewpoint of Lorentz force and magnetic flux density derived by vector potential. By comparing calculated, analyzed, and measured magnetic flux density and thrust force, the validity of derived mathematical model is confirmed. In addition, the cause of difference among three results is also discussed.

A new exactly solvable case in strong-field quantum electrodynamics with a time-dependent external electric field is presented. The corresponding field is given by an analytic function, which is asymmetric (in contrast to Sauter-like electric field) with respect to the time instant, where it reaches its maximum value, that is why we call it the analytic asymmetric electric field. We managed to exactly solve the Dirac equation with such a field, which made it possible to calculate characteristics of the corresponding vacuum instability nonperturbatively. We construct the so-called in- and out-solutions and with their help calculate mean differential and total numbers of created charged particles, probability of the vacuum to remain a vacuum, vacuum mean values of current density and energy-momentum tensor of the particles. We study the vacuum instability in regimes of rapidly and slowly changing analytic asymmetric electric field, and compare the obtained results with corresponding ones obtained earlier for the case of the symmetric Sauter-like electric field. We also compare exact results in the regime of slowly changing field with corresponding results obtained within the slowly varying field approximation recently proposed by two of the authors, thus demonstrating the effectiveness of such an approximation., 27 pages, 7 figures, some minor changes introduced

Novel approach to evaluation of multilayered media Green's function for the mixed-potential integral equation (MPIE) formulation commonly used for full-wave Method of Moments (MoM) analysis of microwave circuits is proposed. Spectral Differential Equation Approximation Method (SDEAM) previously available only for the evaluation of traditional vector potential multilayered media dyadic Green's function is extended to handle the mixed-potential kernels.

Bloch oscillations (BOs) were initially predicted for electrons in a solid lattice to which a static electric field is applied. The observation of BOs in solids remains challenging due to the collision scattering and barrier tunnelling of electrons. Nevertheless, analogies of electron BOs for photons, acoustic phonons and cold atoms have been experimentally demonstrated in various lattice systems. Recently, BOs in the frequency dimension have been proposed and studied by using an optical micro-resonator, which provides a unique approach to controlling the light frequency. However, the finite resonator lifetime and intrinsic loss hinder the effect from being observed practically. Here, we experimentally demonstrate BOs in a synthetic frequency lattice by employing a fibre-loop circuit with detuned phase modulation. We show that a detuning between the modulation period and the fibre-loop roundtrip time acts as an effective vector potential and hence a constant effective force that can yield BOs in the modulation-induced frequency lattices. With a dispersive Fourier transformation, the pulse spectrum can be mapped into the time dimension, and its transient evolution can be precisely measured. This study offers a promising approach to realising BOs in synthetic dimensions and may find applications in frequency manipulations in optical fibre communication systems.

Using a method based on the Magnus expansion, the survival of Rydberg atoms in intense laser fields is studied. Usually, for atoms in intense laser fields, effects beyond the electric dipole approximation need to be considered. In the past few years, researchers have primarily paid attention to the nondipole effects induced by the electric quadrupole term of the optical field. The quadrupole term plays a more important role than the electric dipole term in the calculation of the survival probability of Rydberg atoms in intense laser fields even though the magnitude of the former is smaller than that of the latter in terms of the multipole expansion of the optical vector potential. Meanwhile, the effects of higher-order corrections beyond the quadrupole term remain ambiguous. In this paper, we confirm that corrections containing higher-order terms, except for the quadrupole term, can be neglected in the calculation of the transitions of Rydberg atoms in external fields. We then use the second-order time evolution operator containing the quadrupole term to study the scaling law of the transition probability of Rydberg He in laser fields with different parameters.

This paper deals with the so-called AB effect — AB stands for the names of the two physicists Y. Aharonov and D. Bohm — which has received new attention during the last decade, especially concerning its physical meaning. This effect should be referred by the acronym ESAB in order to recognize the pioneering work of W. Ehrenberg and R. Siday. Here we review the role of the electromagnetic potential in the interference phenomenon with charged particles and we recall that such a potential is never unambiguously defined since it depends on the choice of an arbitrary gauge. Therefore, it cannot play a primary role with respect to the electromagnetic field from which it is derived. However this does not rule out its great interest in theoretical physics. Indeed, it appears naturally when defining important quantities, such as the Lagrangian, the linear momentum, and the Hamiltonian, both in classical and quantum physics. By referring to a large panel of published experimental tests of the ESAB effect, we argue that only a pantopical point of view, i.e. rejecting the hypothesis of locality, can account for the behavior of an electron (or any charge) in an interferometer and its interaction with an electromagnetic field.

The radial and tangential force components acting on each sub-conductor of a Roebel bar in a large generator lead to mechanical vibration of conductors. This double-frequency vibration results in insulation deterioration and looseness of stator bars. Understanding the origin of these vibrations and developing models to anticipate them is essential to increase the reliability and expected lifetime of these systems. This chapter presents a comprehensive study on the origin and behaviors of Roebel bar vibrations under different operation modes. A 3D-FEM model of a generator is developed and the vector potential values in the air gap are calculated. These vector potential values are applied as boundary condition to a detailed 2D-FEM model to calculate the sub-conductor forces. The results obtained from Maxwell stress tensor and Lorentz force methods are compared to each other. Moreover, the Roebel bar force behavior for different power factors and load angles is investigated.

In the context of intra-cluster medium turbulence, it is essential to be able to split the turbulent velocity field in a compressive and a solenoidal component. We describe and implement a new method for this aim, i.e., performing a Helmholtz-Hodge decomposition, in multi-grid, multi-resolution descriptions, focusing on (but not being restricted to) the outputs of AMR cosmological simulations. The method is based on solving elliptic equations for a scalar and a vector potential, from which the compressive and the solenoidal velocity fields, respectively, are derived through differentiation. These equations are addressed using a combination of Fourier (for the base grid) and iterative (for the refinement grids) methods. We present several idealised tests for our implementation, reporting typical median errors in the order of $1\unicode{x2030}$-$1\%$, and with 95-percentile errors below a few percents. Additionally, we also apply the code to the outcomes of a cosmological simulation, achieving similar accuracy at all resolutions, even in the case of highly non-linear velocity fields. We finally take a closer look to the decomposition of the velocity field around a massive galaxy cluster., Comment: 12 pages, 8 figures; accepted for publication in Computer Physics Communications

Based on the Dyson-Schwinger equation, we compute the resummed gluon propagator in a holonomous plasma that is described by introducing a constant background field for the vector potential $A_{0}$. Due to the transversality of the holonomous Hard-Thermal-Loop in gluon self-energy, the resummed propagator has a similar Lorentz structure as that in the perturbative Quark-Gluon Plasma where the holonomy vanishes. As for the color structures, since diagonal gluons are mixed in the over-complete double line basis, only the propagators for off-diagonal gluons can be obtained unambiguously. On the other hand, multiplied by a projection operator, the propagators for diagonal gluons, which exhibit a highly non-trivial dependence on the background field, are uniquely determined after summing over the color indices. As an application of these results, we consider the Debye screening effect on the in-medium binding of quarkonium states by analyzing the static limit of the resummed gluon propagator. In general, introducing non-zero holonomy merely amounts to modifications on the perturbative screening mass $m_D$ and the resulting heavy-quark potential, which remains the standard Debye screened form, is always deeper than the screened potential in the perturbative Quark-Gluon Plasma. Therefore, a weaker screening, thus a more tightly bounded quarkonium state can be expected in a holonomous plasma. In addition, both the diagonal and off-diagonal gluons become distinguishable by their modified screening masses ${\cal M}_D$ and the temperature dependence of the ratio ${\cal M}_D/T$ shows a very similar behavior as that found in lattice simulations., final version, published in PRD

The postulation of Flowline design method, in the frame of nonimaging optics, was a fundamental step in the development of Field Theory of Geometrical Optics. During lasts years there have been new advances in the application of Field Theory elements to Flowline analysis, vector potential, interface factors, contour integrals. In this paper we starts a new natural step in the development of Flowline theory, we study Hamiltonian formulation of Flowline, as a first step for Wigner formalism analysis. We study two different perspectives in our approach to the problem, first the classical ray optics approach and second a new approach from Radiative force point of view. Radiative force approach seems to be more advantageous than ray optics, but further studies will be needed to confirm that hypothesis.

Recent experiments with the Aharonov-Bohm geometry have shown that in addition to an electron-interference fringe shift, there is also a lateral displacement of the electron diffraction envelope. In this paper, we derive an electron displacement force based on a second-order expansion of the magnetic vector potential. The analysis illustrates the conservation of canonical angular momentum, where the mechanical angular momentum and field angular momentum sum to a constant of the motion; the azimuthal force required to change the mechanical momentum is thus supplied by changes in field momentum associated with the second-order vector potential term. Our results are consistent with all known Aharonov-Bohm experiments, including interference fringe shifts, lateral displacements, and the absence of longitudinal time delays.

In this paper, we present a method by which it is possible to describe a dissipative system (that is modeled by a linear differential equation) in Lagrangian formalism, without the trouble of finding the proper way to model the environment. The concept of the presented method is to create a function that generates the measurable physical quantity, similarly to electrodynamics, where the scalar potential and vector potential generate the electric and magnetic fields. The method is examined in the classical case, the question of quantization is unanswered.

Dark soliton is one of most interesting nonlinear excitations in physical systems, manifesting a spatially localized density "dip" on a uniform background accompanied with a phase jump across the dip. However, the topological properties of the dark solitons are far from fully understood. Our investigation for the first time uncover a vector potential underlying the nonlinear excitation whose line integral gives the striking phase jump. More importantly, we find that the vector potential field has a topological configuration in analogous to the Wess-Zumino term in a Lagrangian representation. It can induce some point-like magnetic fields scattered periodically on a complex plane, each of them has a quantized magnetic flux of elementary $\pi$. We then calculate the Euler characteristic of the topological manifold of the vector potential field and classify all known dark solitions according to the index., Comment: 6 pages, 3 figures

A semi-analytical approach for rapid calculation of the vector potential normal modes and dyadic Green's function inside the cavity of irregular shape over a broad range of frequency is presented. The method is based on the extraction of the Green's function at an imaginary wave number from itself to obtain a rapidly convergent hybrid spatial-spectral expansion of the Green's function. The extracted term encompasses the singularity of the Green's function and is computed using surface integral equation. The method is applied to a V-grooved cavity to obtain the vector potential Green's function inside the cavity.

The broadband Green's function method is applied to characterize the band diagram and modal fields of 3D periodic PEC scatterers, and it converts the band characterization problem into a linear eigenvalue problem. This is the first time that the broadband Green's function method is used to deal with 3D scattering problem of periodic scatterers. Distinct from the conventional surface integral equation (SIE) formulation using electric field E and the magnetic field H, we formulate the SIE using the vector potential A and its dyadic Green's function. The periodic dyadic Green's function of vector potential A is represented in a fast convergent Floquet plane wave series and an exponentially convergent spatial series evaluated at an imaginary wavenumber. The Galerkin's method is applied to discretize the SIE into a matrix form, where RWG basis functions and rooftop basis functions are used to represent the vector and scalar unknowns on the surface, respectively. This matrix equation is then converted into a linear eigenvalue problem of a moderate size with the help of the hybrid representation of the periodic dyadic Green's function. The solution to the linear eigenvalue problem then yields all the band solutions and modal fields of the periodic scatters simultaneously.

In this work we investigate theoretically the influence of a Fermi velocity modulation in the electronic and transport properties of magnetic graphene superlattices. We solve the effective Dirac equation for graphene with a position dependent vector potential and Fermi velocity and use the transfer matrix method to obtain the transmission coefficient for the finite cases and the dispersion relation for a periodic superlattice. Our results reveals that the Fermi velocity modulation can control the resonance peaks of the transmittance and also works as a switch, turning on/off the transmission through the magnetic barriers. The results obtained her e can be used for the fabrication of graphene-based electronic devices.

Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract

We theoretically investigate the phenomenon of modulation instability for systems obeying the nonlinear Schr\"odinger equation, which are under the influence of an external homogeneous synthetic magnetic field. For an initial condition, the instability is detected numerically by comparing dynamics with and without a small initial perturbation; the perturbations are characterized in a standard fashion by wave vectors in momentum space. We demonstrate that the region of (in)stability in momentum space, as well as time evolution in real space, for identical initial conditions, depend on the choice of the gauge (i.e., vector potential) used to describe the homogeneous synthetic magnetic field. This superficially appears as if the gauge invariance is broken, but this is not true. When the system is evolved from an identical initial condition in two different gauges, it is equivalent to suddenly turning on the synthetic magnetic field at $t=0$. This gives rise, via Faraday's law, to an initial instantaneous kick of a synthetic electric field to the wave packet, which can differ for gauges yielding an identical uniform magnetic field for $tg0$.

In his recently discovered handwritten notes on "An alternate way to handle electrodynamics" dated on 1963, Richard P. Feynman speculated with the idea of getting the inhomogeneous Maxwell's equations for the electric and magnetic fields from the wave equation for the vector potential. With the aim of implementing this pedagogically interesting idea, we develop in this paper the approach of introducing the scalar and vector potentials before the electric and magnetic fields. We consider the charge conservation expressed through the continuity equation as a basic axiom and make a heuristic handle of this equation to obtain the retarded scalar and vector potentials, whose wave equations yield the homogeneous and inhomogeneous Maxwell's equations. We also show how this axiomatic-heuristic procedure to obtain Maxwell's equations can be formulated covariantly in the Minkowski spacetime., Comment: The paper includes an extract of Feynman's first handwritten note on his "Alternate Way to Handle Electrodynamics."

We give here the field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This derivation however illustrates the proper use of the Euler–Lagrange equations and the canonical formalism that fails if one tries to quantize the classical theory of point-like particles interacting with the electromagnetic fields. In the same time it confirms the 1∕c2 result of the preceding chapter on states without photons.

In this chapter we solve the Laplace equation for the magnetic vector potential of two fundamental current distributions and plot the result. The solution for the line current is relatively straight forward. For the current loop we use the method of eigenfunction expansion. The mathematical preliminaries such as the Dirac Delta distribution and Green’s function are succinctly defined. We uncover how the geometry of the magnetic vector potential follows the geometry of the current distribution and how the magnetic field lines are equipotential lines of the vector potential field.

A study is undertaken to derive the exact solutions of the Klein–Gordon with the Mie-type potential in the presence of the nonzero space component of a vector potential field. The wave functions and the energy spectrum are derived via the Laplace transformation method and by assuming equal scalar and time component of the vector potential. The Mie-type potential describes four different types of potential at the same time. The energy spectrum separately is obtained for each potential, and an estimation of the most general case is introduced.

We consider a linear magnetic response to a point electric charge embedded in the background of parallel constant electric and magnetic fields in the framework of nonlinear electrodynamics. We find two types of responses. One is given by a vector potential free of any string singularity. The corresponding magnetic field may be thought of as two magnetic poles of opposite polarity coexisting at one point. The other response is given by a vector potential singular on a half-axis directed along the background fields. Its magnetic field is a magnetic monopole plus a field confined to an infinitely thin solenoid, whose role is the same as that of the Dirac string. The value of the magnetic charge is determined by the electric charge and the background fields and is expressed in terms of derivatives of the nonlinear local Lagrangian. Once the potential is singular, the nonlinear Maxwell equations written for potentials and field intensities are not equivalent. We argue why the preference should be given to potentials.

A two-dimensional quantum system of a charged quantum particle of charge \(q\in \mathbb {R}\setminus \{0\}\) with a singular vector potential A is considered. In this system, there appears an irreducible self-adjoint representation πA of the CCR with two degrees of freedom. A necessary and sufficient condition is formulated for πA to be inequivalent to the Schrodinger representation \(\pi _{\mathrm {S}}^{(2)}\) in terms of the magnetic flux of the system. It is shown that the case where πA is inequivalent to \(\pi _{\mathrm {S}}^{(2)}\) corresponds to the occurrence of the so-called Aharonov–Bohm effect in the present context. Moreover, inequivalence between two representations πA and \(\pi _{\boldsymbol {A}'}\) for different vector potentials A and A′ is discussed. For a vector potential A fixed, πq := πA may be regarded as a representation depending on the charge q. A necessary and sufficient condition for πq to be inequivalent to \(\pi _{q'}\) (\(q'\in \mathbb {R}\setminus \{0\}\)) is given.

In this article, we discuss how a combination of electrodynamics and quantum mechanics makes interference measurements of the quantum phase possible in terms of the vector potential, neither of which is detectable independently. This effect, predicted by Aharonov and Bohm, is of great significance in the contemporary interesting topic of nanoscopic physics. We also indicate how the effect can be incorporated in a solid state device by employing the tight-binding (TB) model. The TB model can be realized in a mesoscopic ring which allows the measurement of the bond current and the associated diamagnetism. An exactly solvable case of a threesite ring is presented that serves as a pedagogic example providing further insights into the phenomenon.