Van der Waals interactions: Corrections from radiation in fluids

We consider induced van der Waals interactions with corrections due to radiation in fluids consisting of polarizable hard spheres. The fluctuating polarizations are quantized while the positions of particles are treated classically. First the well known result for the induced Casimir free energy for a pair of particles is used to establish the resulting free energy at low density. The Casimir interaction includes the full effect of the quantized radiating electromagnetic field. Then the situation with electrostatic dipole-dipole interactions is considered for general density. For this situation the induced interactions are the van der Waals interactions, and we evaluate numerically the free energy based upon analytic results obtained earlier. These analytic results were obtained by extending methods of classical statistical mechanics to the path integral of quantum mechanics. We have realized that these methods can be extended to time-dependent interactions too. Thus we here also make the extension to the radiating dipole-dipole interaction between pairs of particles to obtain explicit results for more arbitrary fluid densities, and radiation corrections to the induced free energy are found both analytically and numerically.