Hamilton-Jacobi approach to photon wave mechanics: near-field aspects

After having briefly reviewed the Hamilton-Jacobi theory of classical point-particle mechanics, its extension to the quantum regime and the formal identity between the Hamilton-Jacobi equation for Hamilton's characteristic function and the eikonal equation of geometrical optics, an eikonal theory for free photons is established. The space-time dynamics of the photon is described on the basis of the six-component Riemann-Silberstein energy wave function. Form-identical eikonal equations are obtained for the positive and negative helicity dynamics. Microscopic response theory is used to describe the linear photon-matter interaction. In the presence of matter the free-photon concept is replaced by a quasi-photon concept, and there is a quasi-photon for each of the two helicity states. After having established integro-differential equations for the wave functions of the two quasi-photons, the eikonal conditions for the quasi-photons are determined. It appears that the eikonal condition contains complicated space integrals of the gradient of the eikonal over volumes of near-field domain size. In these space integrals the dynamics of the electrons (matter particles) appears via transverse transition current densities between pairs of many-body states. Generalized microscopic polarization and magnetization fields are introduced to establish the connection between the quasi-photon and macroscopic eikonal theories.