Effective Hamiltonian for the system of the electromagnetic field and charged particles
A classical system of point charged particles and electromagnetic field is investigated in the general formulation, in which there is no direct interaction of the particles with each other. Particles are assumed to be non-relativistic for research simplicity. The point of view is defended that gauge choice determines the physical picture of processes in the system. The idea of extended gauge, in which scalar φk and longitudinal part Alnk of the vector potentials transform separately for large and small wave numbers, is proposed. Moreover, in the new gauge φk ≠ 0 only when k ≥ k0 and Alnk ≠ 0 only when k ≤ k0 , where k0 is a certain wave number. This greatly simplifies the study of the system dynamics, in which the transverse
components Atnk of the vector potential are taken into account and do not change at the gauge transformation. Basing on the extended gauge idea, it is established that the system has a massive oscillatory mode described by a transverse potential Atnk and an oscillatory mod described by a longitudinal potential Alnk with the laws of dispersion (ωk2 + ω02 )1/2 і ω0, respectively (ωk = ck is the dispersion law for electromagnetic waves in vacuum, and ω0 is the plasma frequency). These modes can be called electromagnetic and plasma ones. At the same time, effective interaction between charged particles is introduced, which describes the screening effect. A connection of our work with Bohm and Pines investigations related to the presence of plasma modes in a Coulomb system, in which they do not use idea of gauge transformations, is analyzed.