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An argument, based on Lorentz invariance for the number of discrete objects and Lorentz non-invariance for continuous physical quantities, is used to arrive at an uncertainty relation involving dipole moment and mass. Applied to a photon, a virtual dipole moment is defined and the photon itself is described as an electromagnetic wave. The small distance singularity in the Coulomb potential is removed by using a complex number for distance.

A modified Robertson-Walker metric is used to obtain solutions for matter density in the time interval −∞

Abstract: A complex length is introduced into the Coulomb and Yukawa potential energy equations. It is shown how this complex length can be used to remove the small-distance singularities in the above potentials. In addition, the Yukawa potential now exhibits a hard core.

Approximate analytical self-consistent field wave functions were calculated for Pr3+. For the purpose of this work we combined the minimum and the saturated basis set approximations. All the orbital exponents were optimized.