A generalization of the method of separating longitudinal and transverse waves in electrodynamics

Abstract

A generalization of the method of separating longitudinal and transverse waves in electrodynamics is proposed in this paper. It consists in splitting up each Fourier-component of the wave-field 4-vector with respect to two null-vectors k and l k , where k is the Fourier vector of propagation, and l k is an arbitrary (real) function of k satisfying (l k , k) = 1 and l -k = — l k . This amounts to referring each Fourier-component of the field to a different time-axis in the usual method of splitting up. l k may also depend on the co-ordinates of the particles of the system. The longitudinal field variables are eliminated from the Hamiltonian formulation of electrodynamics by a contact transformation; the term which then replaces the longitudinal field depends on the co-ordinates of the particles, and in general also on the transverse field variables, except when l k is independent of the co-ordinates of the particles. The work holds both in the classical and in the quantum theory. For the problem of an electron moving in the field of a nucleus a particular form is chosen for the l k , which depends on the initial velocities of both the particles. The new longitudinal waves are eliminated, and the classical deflexion formula is derived on the assumption that the interaction with the new transverse waves can be neglected. The new method leads to half the deflexion derived by the usual method of working only with the Coulomb interaction force (and so neglecting radiation damping in a certain way), when the nuclear recoil is neglected and the deflexion angle is small. For a low-velocity electron, the effect of the new transverse waves cannot be small, but for a very high energy electron the new method may be more suitable than the usual one.