In this work we include the elastic scattering of longitudinal electromagnetic waves in transport theory using a medium filled with point-like, electric dipoles. The interference between longitudinal and transverse waves creates two new channels among which one allows energy transport. This picture is worked out by extending the independent scattering framework of radiative transfer to include binary dipole-dipole interactions. We calculate the diffusion constant of light in the new transport channel and investigate the role of longitudinal waves in other aspects of light diffusion by considering the density of states, equipartition, and Lorentz local field. In the strongly scattering regime, the different transport mechanisms couple and impose a minimum conductivity of electromagnetic waves, thereby preventing Anderson localization of light in the medium. We extend the self-consistent theory of localization and compare the predictions to extensive numerical simulations.

29 pages, 14 figures