On the Born-Infeld Field Theory of the Electron

Abstract

Since the field equations of the Born-Infeld field theory impose no essential restrictions on the world lines of the singularities, a dynamical condition, equivalent to equations of motion, must be added to complete the theory. The variation problem from which Born and Infeld obtain equations of motion by varying the world lines of the singularities does not appear to be susceptible of generalization to include radiation reaction and, moreover, the equations of motion are not invariant in form under a Lorentz transformation. The definition of a magnetic charge and current vector (which vanishes for an isolated singularity in uniform rectilinear motion) makes possible the derivation from the energy-momentum conservation laws of a relation (Eq. (32)) which is the formal analogue of the dynamical assumption, in the classical theory of electromagnetic mass, that the total force on the electron, including the self-reaction, vanishes. This result is not, however, equivalent to equations of motion because the non-classical part of the field arising from the non-linear character of the field equations always adjusts itself, for arbitrary motion of the singularity, to maintain the relation. In the coordinate system in which the singularity is momentarily at rest the relation simplifies notably and points to a new dynamical condition which appears to be singled out from all other possible conditions by its compelling simplicity. The resulting equations of motion contain the usual radiation resistance term and also higher order terms in the self-interaction.