Path-dependent Lagrangians in relativistic electrodynamics

The interaction of an electromagnetic field and a set of point charges is governed by the Maxwell-Lorentz and Minkowski equations. These relate variables-the electromagnetic field tensor and the charge-current density vector-that have a direct physical significance and are uniquely defined (once a convention for their behaviour under improper Lorentz transformations has been adopted). The Lagrangian formalism, however, requires the interaction to be expressed in terms of one variable that is uniquely defined together with one that is capable of being altered by a gauge transformation. It is demonstrated here in an explicitly Lorentz covariant manner that there exists a class of path-dependent Lagrangians with the property that the so-called 'minimal-coupling' and 'multipolar' interactions are equal. In such Lagrangians, whether the path dependence (which is a manifestation of non-uniqueness or gauge dependence) is to be carried by the electromagnetic potential or by the polarisation-magnetisation tensor of the system of particles is entirely a question of convenience. (author)