A fields only version of the Lorentz Force Law: Particles replaced by   their fields
  We show that the Lorentz force law, F^L_1=q_1(E+v_1xB) being the charge on particle 1 interacting with the electromagnetic fields due to all other particles, can be written in a pure field form F^L_1=-\nabla_1 U^{EM}. In this expression U^{EM} is the total electromagnetic energy of the system of particle 1 and all other particles. In deriving this result we review the old but not widely known results that Maxwell's equations follow uniquely from Special Relativity, and that the Lorentz force law follows from applying Hamilton's variational principle to this result.   For a two particle system, the standard view is that the electromagnetic force on particle 1 is the result of the charge of particle 1 interacting locally with the field of particle 2, and conversely. Both charges 1 and 2, and fields 1 and 2 are needed. In our approach, the fields of all particles contribute to the electromagnetic interaction everywhere, over all of space. The charges of the particles do not enter the theory except incidentally, via Gauss's law. This has novel interpretational consequences. In particular, it allows a charged particle to be replaced by its electric and magnetic fields, much as a particle in quantum mechanics is replaced by its complex valued wavefunction. 
Philip H. Butler, Niels G. Gresnigt, Martin B. van der Mark, Peter F.   Renaud
physics.gen-ph physics.class-ph