calculated. The reason for this is twofold: Neglect of the "self-fields" usually leads to answers that are accurate enough for many applications, and Inclusion of self-fields leads to problems in physics such as renormalization, some of which still unsolved, that relate to the very nature of matter and energy. This conceptual problems created by self-fields are highlighted in a standard graduate text. [Jackson] The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~1948–50) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain. The magnetic radiation reaction force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating non-relativistic particles with associated magnetic moment emit radiation. The Abraham–Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of quantum effects leads one to quantum electrodynamics. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization. This has led to a theory that is able to make the most accurate predictions that humans have made to date. See precision tests of QED. The renormalization process fails, however, when applied to the gravitational force. The infinities in that case are infinite in number, which causes the failure of renormalization. Therefore general relativity has unsolved self-field problems. String theory is a current attempt to resolve these problems for all forces. Derivation We begin with the Larmor formula for radiation of the second derivative of a magnetic moment with respect to time: In the case that the magnetic moment is produced by an electric charge moving along a circular path is where