ABSTRACT
We describe some mathematical models based on classical and quantum field theory and statistical field theory for explaining the refractive indices of materials. The first model proposes to describe the electromagnetic field interacting with the Dirac field of electrons and positrons by replacing the value of the electronic charge with a functional of the electromagnetic field. This idea is based on the fact that the nature of the singularity of the electromagnetic field completely describes the nature, ie location and value of the point charges in it. The solution to the resulting Dirac equation in a background electromanetic field will then give us the probability density function for the spatial location of the electron and by averaging the electron’s electric and magnetic dipole moment operator with respect to this probability distribution, we can obtain formulas for the quantum averaged polarization and magnetization or equivalently the permittivity and permeability of the medium without having to describe the electronic field directly in terms of the electronic charge. This philosophy is in conformity with what many physicists believe today [Hans Van Leunen] that all properties of electrons should be derivable from the electromagnetic field itself. The second model describes a direct approach to computing the RI of a material based on Dirac’s quantum mechanics for a system of N interacting particles in an external electromagnetic field. If we solve the Dirac equation using perturbation theory for a single particle in an electromagnetic field, we could then calculate the quantum averaged electric and magnetic dipole moment of the electron which would in turn enable us to determine the permittivity and permeability of the medium in terms of the electric and magnetic fields. However, this analysis does not show how the RI depends upon the temperature of 186the material. In order to obtain temperature dependence, we consider Dirac’s quantum mechanics for an N particle system taking interparticle interactions into account apart from interaction of the particles with an external electromagnetic field and by partial tracing the mixed state Dirac equation over the other particles and then making some approximations we derive a quantum Boltzmann equation for the quantum density operator and if this equation is solved using perturbation theory with the intial state as the Gibbs state (which has temperature dependence), then the final equilibrium state in the presence of a static electromagnetic field and interparticle interactions will also depend upon temperature. When this final density matrix is used to compute quantum averages of the electric and magnetic dipole moment, we are able to explain the dependence of the RI on both the electromagnetic field and temperature. The wavelength dependence of the RI can be explained by assuming the background electromagnetic field to be black-body radiation which has the energy density of the electromagnetic field dependent upon both frequency/wavelength and temperature. The final model described in this manuscript takes into account cosmological and background gravitational effects on the refractive index of the material. Gravity affects quantum mechanics via the spinor connection of the gravitational field which has to be introduced into Dirac’s equation in order to make it invariant under local Loretnz transformations and arbitrary diffeomorphisms of space-time. Thus, this general relativistic generalization of Dirac’s equation gives us the dependence of the wave function on the background metric tensor of curved space-time. If this background metric is taken to the Schwarzchild metric, the wave function would depend upon the mass of the blackhole and the gravitational constant while if it is taken to be Robertson-Walker metric for an expanding homogeneous and isotropic universe, then the wave function will also depend on the radius of the universe and hence on Hubble’s constant. Calculating the average electric and magnetic dipole moments w.r.t such a wave function would then yield the dependence of the RI on the radius of the expanding universe and on its curvature. By taking fine measurements of the RI, we would then in principle be able to measure Hubble’s constant and hence the radius of the universe at the present epoch.
