Lorentz Abraham Dirac Equation Research Articles

Lorentz-Abraham-Dirac and Landau-Lifshitz equations of motion and the solution to a relativistic electron in a counterpropagating laser beam

Beginning with a critical examination of the Lorentz-Abraham (LA) classical equation of motion for an extended charge and the closely related Lorentz-Abraham-Dirac (LAD) equation of motion for a mass-renormalized point-charge, the Landau-Lifshitz (LL) approximate solution to the LAD equation of motion is determined for an electron subject to a counterpropagating linearly or circularly polarized plane-wave pulse with an arbitrarily shaped envelope. A convenient three-vector formulation of the LL equation is used to derive closed-form expressions for the velocities and associated powers of the electron directly in terms of the time in the laboratory frame. The three-vector formulation also reveals definitive criteria for the LL solution to be an accurate approximation to the LAD equation of motion and for the LL solution to reduce to the solution of the Lorentz force equation of motion that ignores radiation reaction. Semiclassical analyses are used to obtain simple conditions for determining the regimes where the quantum effects of either Compton electron scattering by the incident photons or electron recoil produced by the emitted photons is significant. It is proven that the LL approximation becomes an inaccurate solution to the LAD equation of motion only for large enough electron velocities and plane-wave intensities that quantum recoil effects on the electron can greatly alter the classical solution. Comparisons are made with previously published analytical and numerical solutions to the LL equation of motion for the velocity of an electron in a counterpropagating plane wave.10 MoreReceived 18 August 2021Accepted 1 November 2021DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.114002Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectromagnetismElectron beams & opticsLaser driven electron accelerationQuantum theorySingle-particle dynamicsAccelerators & BeamsGeneral PhysicsParticles & FieldsNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Stationary solutions of equations of motion for a radiating charged particle including a radiation reaction effect.

For a radiating charged particle, the radiation reaction force plays an important role in determining its motion. Several formulas have been proposed to describe the radiation reaction force. Stationary solutions of the Lorentz-Abraham-Dirac (LAD), Mo-Papas, Landau-Lifshitz, and Ford-O'Connell equations are obtained and compared for a charged particle under a static magnetic and a rotating electric field. The behaviors of the obtained solutions look quite similar. The relative differences of the Lorentz factor, calculating the values using the LAD equation compared to the other equations, are evaluated; these values are shown to be less than 10^{-6} in the regime where classical radiation reactions are applicable.

On the Schott Term in the Lorentz-Abraham-Dirac Equation

The equation of motion for a radiating charged particle is known as the Lorentz–Abraham–Dirac (LAD) equation. The radiation reaction force in the LAD equation contains a third time-derivative term, called the Schott term, which leads to a runaway solution and a pre-acceleration solution. Since the Schott energy is the field energy confined to an area close to the particle and reversibly exchanged between particle and fields, the question of how it affects particle motion is of interest. In here we have obtained solutions for the LAD equation with and without the Schott term, and have compared them quantitatively. We have shown that the relative difference between the two solutions is quite small in the classical radiation reaction dominated regime.

A comparison of the bremsstrahlung cross section in two frameworks: classical Lorentz–Abraham–Dirac dynamics and quantum field theory

In classical electromagnetic theory, the Lorentz–Abraham–Dirac (LAD) equation describes the dynamics of a charged particle, including radiation reaction. Though the LAD equation is derived from Maxwell’s equations, consistent with the conservation of energy and momentum, it admits unphysical solutions for point-like charged particles. By assuming small radiative effects, Landau and Lifshitz (LL) develop an approximation of the LAD equation that has no pathological solutions. Though the LL approximation is dynamically sound, it is the LAD equation that has firm theoretical underpinning. As such, the difficulties encountered in LAD dynamics suggest, in part, that an electrodynamic treatment of point-like particles lies outside the classical domain. Herein, we compute the cross section of a charged particle scattered by an attractive Coulombic potential using classical LAD and LL dynamics. We then compare this with the analogous cross section in quantum field theory (QFT). In particular, we compute the cross section for a charged scalar particle incident upon another extremely massive charged scalar with a single photon in the final state. We include the following -radiative corrections: scalar-photon vertex correction, infrared photon emission, vacuum polarization, and two-photon exchange. We find that the classical and quantum frameworks do not produce similar cross sections. Relative to the elastic cross section, the classical bremsstrahlung cross section vastly overestimates the relative QFT cross section; this is, in part, due to the fact that the additional radiative corrections in QFT do not have a classical analog in either the LAD or LL framework.

Erratum to: “Renormalization of the Lorentz–Abraham–Dirac equation for radiation reaction force in classical electrodynamics”

Erratum to: “Renormalization of the Lorentz–Abraham–Dirac equation for radiation reaction force in classical electrodynamics”

Explicit mass renormalization and consistent derivation of radiative response of classical electron

The radiative response of the classical electron is commonly described by the Lorentz–Abraham–Dirac (LAD) equation. Dirac’s derivation of this equation is based on energy and momentum conservation laws and on regularization of the field singularities and infinite energies of the point charge by subtraction of certain quantities: “We... shall try to get over difficulties associated with the infinite energy of the process by a process of direct omission or subtraction of unwanted terms”. To substantiate Dirac’s approach and clarify the mass renormalization, we introduce the point charge as a limit of extended charges contracting to a point; the fulfillment of conservation laws follows from the relativistic covariant Lagrangian formulation of the problem. We derive the relativistic point charge dynamics described by the LAD equation from the extended charge dynamics in a localization limit by a method which can be viewed as a refinement of Dirac’s approach in the spirit of Ehrenfest theorem. The model exhibits the mass renormalization as the cancellation of Coulomb energy with the Poincaré cohesive energy. The value of the renormalized mass is not postulated as an arbitrary constant, but is explicitly calculated. The analysis demonstrates that the local energy–momentum conservation laws yield dynamics of a point charge which involves three constants: mass, charge and radiative response coefficient θ. The value of θ depends on the composition of the adjacent potential which generates Poincaré forces. The classical value of the radiative response coefficient is singled out by the global requirement that the adjacent potential does not affect the radiated energy balance and affects only the local energy balance involved in the renormalization.

Radiation reaction in ultrarelativistic laser-spinning electron interactions

The intensity of ultra-short pulse lasers has reached 10 W/cm owing to the advancements in laser technology. The large radiation from the electron behaves something like resistance in this ultrarelativistic laser–electron interaction. This effect is called the “radiation reaction”. The equation of motion with the radiation reaction is known as the Lorentz–Abraham–Dirac equation; however, this equation does not incorporate the spin property. In laser plasmas, classical physics descriptions are preferred for simulations. This paper discusses how to describe the radiation reaction of a spinning relativistic electron in classical dynamics. Subject Index A00, A01, G03, J25

The significance of the Schott energy for energy-momentum conservation of a radiating charge obeying the Lorentz–Abraham–Dirac equation

It is shown that energy and momentum are conserved during the runaway motion of a radiating charge and during free fall of a charge in a gravitational field. The runaway motion demonstrates the consistency of classical electrodynamics and the Lorentz–Abraham–Dirac equation. The important role of the Schott (acceleration) energy in this connection is made clear, and it is shown that the Schott energy is the part of the electromagnetic field energy that is proportional to (minus) the scalar product of the velocity and acceleration of a moving accelerated charged particle. The Schott energy is negative if the acceleration is in the same direction as the velocity and positive if it is opposite. During runaway motion, the Schott energy becomes more and more negative, and for a charged particle, it is localized at the particle. It is also shown that a proton and a neutron fall with the same acceleration in a uniform gravitational field, although the proton radiates and the neutron does not. The radiation energy comes from the Schott energy.

On the value of geometric algebra for spacetime analyses using an investigation of the form of the self-force on an accelerating charged particle as a case study

The ability to treat vectors in classical mechanics and classical electromagnetism as single geometric objects rather than as a set of components facilitates physical understanding and theoretical analysis. To do the same in four-dimensional spacetime calculations requires a generalization of the vector cross product. Geometric algebra provides such a generalization and is much less abstract than exterior forms. It is shown that many results from geometric algebra are useful for spacetime calculations and can be presented as simple extensions of conventional vector algebra. As an example, it is shown that geometric algebra tightly constrains the possible forms of the self-force that an accelerating charged particle experiences and predicts the Lorentz–Abraham–Dirac equation of motion up to a constant of proportionality. Geometric algebra also makes the important physical content of the Lorentz–Abraham–Dirac equation more transparent than does the standard tensor form of this equation, thus allowing a proposed modification to this equation free from the problems of preacceleration and runaway motion to be easily predicted.

Dynamics of emitting electrons in strong laser fields

A new derivation of the motion of a radiating electron is given, leading to a formulation that differs from the Lorentz–Abraham–Dirac equation and its published modifications. It satisfies the proper conservation laws. Particularly, it conserves the generalized momentum, eliminating the symmetry-breaking runaway solution. The equation allows a consistent calculation of the electron current, the radiation effect on the electron momentum, and the radiation itself, for a single electron or plasma electrons in strong electromagnetic fields. The equation is then applied to a simulation of a strong laser pulse interaction with a plasma target. Some analytical solutions are also provided.

Exact Solution of the Landau-Lifshitz Equation in a Plane Wave

The Landau–Lifshitz (The Classical Theory of Fields. Elsevier, Oxford 1975) form of the Lorentz–Abraham–Dirac equation in the presence of a plane wave of arbitrary shape and polarization is solved exactly and in closed form. The explicit solution is presented in the particular, paradigmatic cases of a constant crossed field and of a monochromatic wave with circular and with linear polarization.

Maxwell–Faraday stresses in electromagnetic fields and the self-force on a uniformly accelerating point charge

The physical analysis of a uniformly accelerating point charge provides a rich problem to explore in advanced courses in electrodynamics and relativity since it brings together fundamental concepts in relation to electromagnetic radiation, Einstein's equivalence principle and the inertial mass of field energy in ways that reveal subtleties in each of these concepts. By first exploring the heuristic value of Maxwell's and Faraday's idea that the electromagnetic field is like a stressed material medium, it is shown in this paper that the problem also provides an interesting application of the electromagnetic stress–energy–momentum tensor and that an analysis using this tensor provides clear physical insight into this highly subtle and contentious problem and the so-called ‘4/3 problem’ of classical electromagnetic theory. In particular, it is shown that the stress force on a uniformly accelerating, uniformly charged spherical shell due to its own field is simply the (relativistic) inertial mass of the charge's electrostatic field times the acceleration. Since the inertial mass of the electromagnetic field forms part of the observed rest mass of a charged particle, it is argued that the results are therefore consistent with the Lorentz–Abraham–Dirac equation of motion for an accelerating point charge, which implies that for uniform acceleration, the work done by the force acting on the charge only goes into increasing the kinetic energy of the charge, none goes into the creation of radiation.

The validity limits of physical theories: response to the preceding Letter

In the preceding Letter by Marijan Ribaric and Luka Sustersic, “Qualitative properties of an equation of motion of a classical point charge”, the authors criticize a Letter of mine [1] concerning an equation that I called “the correct equation of motion of a classical point charge”. This equation was first derived by Landau and Lifshitz [2]. Later, it was proven by Spohn [3] to be the restriction of the Lorentz–Abraham–Dirac (LAD) equation to its critical surface. I shall therefore refer to that equation as the LLS equation. The criticism contained in the preceding Letter consists in the demonstration that the LLS equation has properties that are physically inadmissible. They point out that they had previously argued in support of a hypothesis that this equation (as well as the LAD equation) is only asymptotically valid when the external forces are small and slowly varying. In the following, I shall respond to their criticism of my Letter. I shall show that the validity limits of classical electrodynamics exclude from consideration the kind of external forces presented in the preceding Letter. Since the LLS equation (including its derivation) is part of classical electrodynamics, it is restricted by the same

The correct equation of motion of a classical point charge

The Lorentz–Abraham–Dirac equation is not the correct equation of motion: (1) it does not yield the law of inertia uniquely; (2) it is of third order rather than of second order; and (3) it allows a self-force also in the absence of an external force. The correct equation is derived.