Abraham–Lorentz–Dirac force

In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force or the self force.

The formula predates the theory of special relativity and is not valid at velocities of the order of the speed of light. Its relativistic generalization is called the "Abraham–Lorentz–Dirac force". Both of these are in the domain of classical physics, not quantum physics, and therefore may not be valid at distances of roughly the Compton wavelength or below. There is, however, an analogue of the formula that is both fully quantum and relativistic, called the "Abraham–Lorentz–Dirac–Langevin equation" – see Johnson and Hu.

The force is proportional to the square of the object's charge, times the jerk (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action.

It was thought that the solution of the Abraham–Lorentz force problem predicts that signals from the future affect the present, thus challenging intuition of cause and effect. For example, there are pathological solutions using the Abraham–Lorentz–Dirac equation in which a particle accelerates in advance of the application of a force, so-called pre-acceleration solutions. One resolution of this problem was discussed by Yaghjian and is further discussed by Rohrlich and Medina.

Mathematically, the Abraham–Lorentz force is given in SI units by

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