Renormalization in the Classical Equation of Motion of a Point Charge

Recent attempts to obtain the force of radiative reaction in the classical equations of motion of point charges have proceeded from two different viewpoints. Each of these has to introduce one basic assumption in addition to Maxwell's equations, namely the conservation law for the electromagnetic energy-momentum tensor in field theory and the relation between the Lorentz force and the momentum of the particle in action-at-a-distance theory. In previous field-theoretical derivations the Lorentz-Dirac equations including radiation damping are obtained only if one takes the field produced by the particle to be the retarded field; but equations of motion without the damping term are obtained if one uses half the sum of retarded and advanced fields. On the other hand, the theory of action at a distance as developed by Wheeler and Feynman was able to obtain the radiation damping using fields symmetric in time. It is noted that the need for the exclusive use of retarded fields arose only in the field-theoretical derivations for the one-particle problem. The considerations of Wheeler and Feynman on the total field due to all particles of the universe are, however, applicable to field theory in the symmetric form as well as to action-at-a-distance theory. The acceptance of their condition of "complete absorption" again leads to the radiation damping term in the equations of motion of the many-body problem. Some implications of this result are discussed.

During the past century, there has been considerable discussion and analysis of the motion of a point charge in an external electromagnetic field in special relativity, taking into account ``self-force'' effects due to the particle's own electromagnetic field. We analyze the issue of ``particle motion'' in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density ${J}^{a}(\ensuremath{\lambda})$ and stress-energy tensor ${T}_{ab}(\ensuremath{\lambda})$ scale to zero size in an asymptotically self-similar manner about a worldline $\ensuremath{\gamma}$ as $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. In this limit, the charge, $q$, and total mass, $m$, of the body go to zero, and $q/m$ goes to a well-defined limit. The Maxwell field ${F}_{ab}(\ensuremath{\lambda})$ is assumed to be the retarded solution associated with ${J}^{a}(\ensuremath{\lambda})$ plus a homogeneous solution (the ``external field'') that varies smoothly with $\ensuremath{\lambda}$. We prove that the worldline $\ensuremath{\gamma}$ must be a solution to the Lorentz force equations of motion in the external field ${F}_{ab}(\ensuremath{\lambda}=0)$. We then obtain self-force, dipole forces, and spin force as first-order perturbative corrections to the center-of-mass motion of the body. We believe that this is the first rigorous derivation of the complete first-order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the ``reduction of order'' procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. (There is no corresponding justification for the non-reduced-order equations.) We restrict consideration in this paper to classical electrodynamics in flat spacetime, but there should be no difficulty in extending our results to the motion of a charged body in an arbitrary globally hyperbolic curved spacetime.

The motion of non-relativistic test charges is studied for a class of fields (arising, for example, in cavity resonators) which oscillate with a high frequency ω comparable with or greater than the local peak gyromagnetic frequency ωg of the charges. Thus the recently developed methods based upon ω « ωg are supplemented whenever the (variable) magnetic field so confines the charges that the total field may be regarded as locally homogeneous. Rapidly converging expansions involving integral order Bessel functions Jn(ωg/ω) as coefficients are used to obtain asymptotic expressions for the kinetic energy limited by a simplified collision term in Lorentz's equations of motion. Kinetic energy resonances, reminiscent of the action of the betatron, arise for certain values of the critical ratio ωg/ω, notably near the peaks of [(ωg/ω)J1(ωg/ω)]2.

Most non-Abelian gauge theories admit the existence of conserved and quantized topological charges as generalizations of the Dirac monopole. Their interactions are dictated by topology. In this paper, previous work in deriving classical equations of motion for these charges is extended to quantized particles described by Dirac wave functions. The resulting equations show intriguing similarities to the Yang-Mills theory. Further, although the system is not dual symmetric, its gauge symmetry is nevertheless doubled as in the Abelian case from {ital G} to {ital G}{times}{ital G}, where the second {ital G} has opposite parity to the first but is mediated instead by an antisymmetric second-rank tensor potential.

In this work, we investigate the quantum Brownian motion of a point charge arising as a consequence of two fluctuating point-like boundaries. The study considers Dirichlet, Neumann, and mixed boundary conditions imposed on a real massless scalar field. Additionally, we analyze the effects of a fluctuating compactification length on the random motion of the point charge, induced by the imposition of a quasi-periodic condition on the scalar field. By associating a wave function with the length scale of each system, we demonstrate that typical divergences, which commonly appear in scenarios with fixed boundaries and compactification size, are effectively smoothed out. This approach generalizes and extends previous results found in the literature, offering new insights into the regularization of divergences appearing in idealized systems.

The classical Ramo-Shockley theorem (RS) [1] , [2] gives the current induced on perfect conductors by the motion of nearby charges. It is a foundational result in vacuum electronics, discharge physics, and semiconductor devices. The original theorem was derived by assuming quasi-static fields, ignoring both relativistic and radiative effects. This paper illustrates, for the first time, how relativistic and electromagnetic effects modify RS in some simple examples and examines the magnitude of the corrections. Specifically, we present explicit, closed form analytic solutions of Maxwell’s equations for the induced current distribution on perfectly conducting plates due to the motion of a line charge moving either parallel or perpendicular to the plates. Our exact closed form solutions have been verified by several methods, including particle-in-cell simulations. Several novel features were revealed that are absent in the classical RS: electromagnetic transients due to the sudden acceleration of charge, multiple reflections of electromagnetic waves by conductors, and generation of an electromagnetic shock when the charge strikes a conductor and is subsequently removed. We have found that the classical RS works rather well otherwise, even for a highly relativistic beam, except near the wavefront(s) where the transients dominate. This partially explains the mystery as to why the non-relativistic RS has been very successful in the theory and numerical simulations of vacuum electronic devices, in which only the quasi-static, sinusoidal steady solutions are included. The electromagnetic shock could be a source of contamination on the quality of signals in satellite communication, by the electrons in a multipactor discharge [3] . These issues are being examined. The induced current due to motion of a point charge is also under investigation.

The gross charge distribution in an electrified cloud has already been estimated by polarity distribution of the electrostatic field on the ground surface. While either a dipole or a tripole charge structure is commonly accepted, the increase–decrease and motion of each point charge in those models are both still unclear. This paper presents a new network of electric field mills for multipoint electrostatic measurement to evaluate the temporal variations of a simple cloud charge model with second-scale resolution. Details of our newly developed equipment are described, with an emphasis on its advantages. This network was deployed in the north Kanto area of Japan and operated during the summer season in 2020. In order to simplify the relationship between cloud charge positions and the horizontal distribution of the measured electrostatic field, an isolated thundercloud is focused on. As an initial analysis, a negative point charge model is applied to an isolated cloud observed on 27 August 2020. The quantity and height of the point charge were estimated as being approximately −20 C and 7 km, respectively. The calculated charge location is generally coincident with the C-band radar echo regions. Significant correspondence is demonstrated between the intensity distribution of the electrostatic fields measured at seven sites and that calculated with estimated point charge. This result indicates the possibility to determine the amounts and positions of cloud charges inside the dipole charge structure based on multipoint measurement of the electrostatic field.

The two-particle system with the time-asymmetric retarded-advanced electromagnetic interaction known as the Staruszkiewicz–Rudd–Hill model is considered in the de Sitter space-time. The manifestly covariant descriptions of the model within the Lagrangian and Hamiltonian formalisms with constraints are proposed. It is shown that the model is de Sitter-invariant and integrable. An explicit solution of the equations of motion is derived. We use the covariant electromagnetic Green function in the de Sitter space in order to derive the equation of motion of a point charge in an external electromagnetic field, where the radiation reaction is taken into account.

The Lorentz–Dirac theory of radiation reaction on the motion of point charges is beset by the well known problems of runaway solutions and preacceleration. We examine the classical theory of extended charged particles and obtain a differential-difference type equation of motion. Analysis of this equation reveals that the theory is internally consistent (i.e., no runaways or acausality) whenever the size of the particle exceeds the classical radius (defined as the radius for which the electrostatic self-energy equals the mass of the particle). A specific example is presented which explicitly shows the different character of the extended and point charge solutions.

The anti-damping effect of radiation reaction, which means the radiation reaction does non-negative work on a radiating charge, is investigated at length by using the Lorentz–Dirac equation (LDE) for the motion of a point charge respectively acted on by (a) a pure electric field, (b) a pure magnetic field and (c) the fields of an electromagnetic wave. We found that the curvature of the charge's trajectory plays an important role in the radiation reaction force, and the anti-damping effect cannot take place for the real macroscopic motions of a point charge. The condition for this anti-damping effect to take place is that the gradient of the external force field must exceed a certain value over the region of magnitude of the classical radius of massive charges (∼10−15 m). Our results are potentially helpful to lessen the controversy on LDE and justify it as the correct classical equation describing the radiating charge's motion. If this anti-damping effect of LDE were a real existing physical process, it could serve as a mechanism within the context of classical electrodynamics for the stability of hydrogen atoms. Using the picture of an electron in quantum electrodynamics, namely the negative bare charge surrounded by the polarized positive charges of vacuum, we can obtain a reasonable explanation for the energy transferred to the electron during the occurrence of the anti-damping effect, on which the venerable work of Wheeler and Feynman has thrown some light.

The Holst action containing the Immirzi parameter for pure gravity is generalized to supergravity theories. Supergravity equations of motion are not modified by such generalizations, thus preserving supersymmetry. Dependence on the Immirzi parameter does not emerge in the classical equations of motion. This is in contrast with the recent observation of Perez and Rovelli for gravity action containing the original Holst term and a minimally coupled Dirac fermion, where the classical equations of motion do develop a dependence on the Immirzi parameter.

The equation of motion in ℝ d of n generalized point charges interacting via the s -dimensional Coulomb potential, which contains for d = 2 a constant magnetic field, is considered. Planar exact solutions of the equation are found if either negative n − 1 > 2 charges and their masses are equal or n = 3 and the charges are different. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the infinite time for the considered equation without a magnetic field are also obtained. A condition permitting the existence of such solutions is proposed.

Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure

Assuming the initial streamer development of the cloud discharge be represented by the movement of a point charge with constant velocity, the parameters of the initial streamer, such as height of starting point, charge amount, direction and velocity of propagation, were determined by using the data of electric field-change measured simul-taneously at three to seven stations. The following characteristics of the initial streamer of the cloud discharge were deduced from the analyses of thirteen discharges in the winter thunderstorms at the Hokuriku coast: (1) Cloud discharges originate in the region of the temperature from -6°to -10°, which is the same temperature range as that in the summer thunderclouds. (2) The electric charge carried by an initial streamer is about 30 to 130C, and the average value is 63C. (3) The direction of streamer progress is upward, and the zenith angle of the progress is less than 30°. (4) The streamer velocity is of the order of 104m/s, and depends on the altitude of its starting point.

The paper is concerned with the motion of a point electric charge in de Sitter spacetime. A point particle of mass m and charge q moving on a geodesic curve produces electromagnetic field that diverges at a particle’s position. The field is determined by the electromagnetic Green’s function by Higuchi and Lee (2008 Phys. Rev. D 78 084031). The self-force contains both divergent and finite terms, and the latter are responsible for the radiation reaction. Our derivation of an effective equations of motion is based on conservation laws corresponding to the group of isometry of de Sitter space. The Nöther quantities consist of particle’s individual characteristics and energy, momentum, and angular momentum carried by particle’s electromagnetic field. Following the Detweiler–Whiting concept that a charge’s motion should only be enforced by the regular component of its own field, we ignore the divergent terms in conservation laws. We assume that the divergencies are absorbed by particle’s individual characteristics within the renormalization procedure. Finite radiative terms together with kinematic particle’s characteristics constitute ten conserved quantities of closed particle plus field system. Their differential consequences yield the effective equations of motion of radiating charge in an external electromagnetic field and gravitation. Contributions to already renormalized particle’s four-momentum and its inertial mass originated from electromagnetic field and background gravity are also derived from ten balance equations.