Toy Model of Electrodynamics in (1+1) Dimensions

Here I describe a toy model of electrodynamics in (1+1) dimensions that I developed to try to help clarify various problems in the foundations of electrodynamics. The toy model shares much of the conceptual structure of ordinary electrodynamics but is mathematically much simpler.

Overview of the toy model

Here are some of the features of the model:
  • There are fields E(x,t) and B(x,t), which can be thought of as the analogs to the electric and magnetic fields of ordinary electrodynamics. These fields mediate forces between charged particles and support freely propagating radiation.

  • The fields E(x,t) and B(x,t) obey equations of motion that are closely analogous to Maxwell's equations.

  • The coupling of a charged particle to its own field leads to phenomena such as radiation, radiation damping, and mass renormalization.

  • There is a direct analog to the Lorentz-Dirac equation of motion.

  • One can derive expressions for the energy density u(x,t) and momentum density s(x,t) of the field:
    u(x,t) = (1/2)(E2(x,t) + B2(x,t)),             s(x,t) = -E(x,t) B(x,t).
    These expressions are closely analogous to the corresponding expressions for ordinary electrodynamics. One can show that the total energy and momentum of the coupled particle-field system is conserved, and that the energy lost by a particle due to radiation damping shows up in the radiated field.

There are also some important differences between the toy model and ordinary electrodynamics:

  • For ordinary electrodynamics charges of the same sign feel a repulsive force, but for the toy model they feel an attractive force.

  • Ordinary electrodynamics is Lorentz invariant, but the toy model is neither Lorentz nor Galilean invariant: there is a preferred frame in which the equations of motion for the model are valid, and a particle moving with respect to this preferred frame feels a drag force that arises from radiation damping.

  • The Lorentz-Dirac equation is third-order in time, and this leads to conceptual problems such as runaway and preaccelerated solutions. The analogous equation of motion for the toy model is only second-order in time, and thus is free from these problems.

  • For both ordinary electrodynamics and for the toy model, a charged particle has a self-energy that arises from the coupling of the particle to its own static field. For ordinary electrodynamics the self-energy scales like 1/r, where r is the size of the particle, and diverges in the limit of a point particle. For the toy model the self-energy scales linearly with r, and vanishes in the limit of a point particle. Thus, unlike ordinary electrodynamics, the toy model is well-defined in the point-particle limit.

Movies of the toy model

Here are several movies of the toy model that illustrate various radiative processes. The movies were made by simulating the toy model on a computer as described in reference [3].

Movie: harmonically bound particle

This movie illustrates radiation from a charged particle that is harmonically bound to the origin. The figures below are two frames from the movie. In each figure the red curve is the electric field E(x,t) and the green curve is the magnetic field B(x,t).

       

The figure on the left shows the initial state of the system: the particle is held stationary at a fixed position to the right of the origin, and the E and B fields are just the static fields of the stationary charge. At time t=0 the particle is released and begins to oscillate about the origin and emit radiation. The figure on the right shows the state of the system at time t=1320.

[download .avi (2.1 mb)]

Movie: two particles

This movie illustrates radiation from two particles of equal charge that are symmetrically displaced from the origin. The figures below are two frames from the movie. In each figure the red curve is the electric field E(x,t) and the green curve is the magnetic field B(x,t).

       

The figure on the left shows the initial state of the system: the particles are held stationary at a finite distance from the origin, and the E and B fields are just the static fields of the stationary charges. At time t=0 the particles are released. The mutual attraction of the two particles causes them to oscillate about the origin and emit radiation. The figure on the right shows the state of the system at time t=1016.

[download .avi (2.1 mb)]

Movie: braking radiation

This movie illustrates braking radiation from a single charged particle. The figures below are two frames from the movie. In each figure the red curve is the electric field E(x,t) and the green curve is the magnetic field B(x,t).

       

The figure on the left shows the initial state of the system: the particle is held stationary at a the origin, and the E and B fields are just the static fields of the stationary charge. At time t=0 the particle is given an impulsive momentum kick and it begins to move the right. The motion is subsequently damped and pulses of emitted radiation propagate outward to the left and right. The figure on the right shows the state of the system at time t=420.

[download .avi (980 kb)]

Movie: scattering

This movie illustrates the scattering of radiation by a single particle that is harmonically bound to the origin. The figures below are two frames from the movie. Each figure consists of two panels. In the top panel, the red curve is the electric field E(x,t) and the green curve is the magnetic field B(x,t). In the bottom panel, the red and green curves are the energy density and energy flux of the radiation field, as computed by subtracting the instantaneous static field of the charge from the total field.

       

[download .avi (3.2 mb)]

The figure on the left shows the state of the system at time t=360: the particle is stationary at a the origin and incoming radiation approaches the particle from the left. The frequency of the incoming radiation is chosen to be resonant with the harmonic frequency of the particle. The incoming radiation drives the particle, causing it to oscillate and emit outgoing radiation. At around time t=2000 the energy of the particle saturates; that is, the rate at which it gains energy by absorbing radiation balances the rate at which it loses energy by emitting outgoing radiation. At this point almost all of the incident radiation is back-reflected by the particle: to the right of the particle the incoming and outgoing radiation interfere destructively and cancel, and to the left of the particle the incoming and outgoing radiation interfere constructively and form a standing wave. The figure on the right shows the state of the system at t=2480, after the energy of the particle has saturated.

Computer programs

Here are several computer programs for simulating the toy model on a computer. These programs were used to create the figures for references [3] and [5] and to create the movies shown above.
electrodynamics.c

Numerically integrate the equations of motion for the toy model for the case of a single particle. Used damped boundary conditions for the field.

electrodynamics-two-particles.c

Numerically integrate the equations of motion for the toy model for the case of two particles of equal charge that are symmetrically displaced about the origin. Used damped boundary conditions for the field.

electrodynamics-scattering.c

Numerically integrate the equations of motion for the toy model for the case of a single particle. Use boundary conditions that describe incoming radiation that approaches the particle from the left.

finite-size.c

Numerically integrate the equation of motion for a spatially extended charged particle.

two-particle.c

Numerically integrate the equation of motion for two point particles of equal charge that are symmetrically displaced about the origin.

Papers that discuss the toy model

  1. A. D. Boozer, "A toy model of electrodynamics in (1+1) dimensions," Eur. J. Phys. 28 447--464 (2007). [pdf]

    This paper describes the classical version of the toy model and compares the toy model with ordinary electrodynamics.

  2. A. D. Boozer, "A toy model of quantum electrodynamics in (1+1) dimensions," Eur. J. Phys. 29 815--830 (2008). [pdf]

    This paper discusses the quantized version of the toy model.

  3. A. D. Boozer, "Simulating a toy model of electrodynamics in (1+1) dimensions," Am. J. Phys. 77 (3), 262--269 (2009). [pdf]

    This paper describes how the toy model can be simulated on a computer and discusses several numerical experiments.

  4. A. D. Boozer, "Retarded potentials and the radiative arrow of time," Eur. J. Phys. 28 1131--1143 (2007). [pdf]

    This paper uses the toy model to discuss time-reversal invariance and time-asymmetry in classical electrodynamics.

  5. A. D. Boozer, "Advanced action in classical electrodynamics," J. Phys. A: Math. Theor. 41 425202 (2008). [pdf]

    This paper discusses the physical meaning of the preaccelerated solutions in classical electrodynamics and explains why such solutions do not occur in the toy model.

David Boozer
Last modified 1 March 2009
boozer at caltech dot edu