Abstract
In this paper, we explore various aspects of the Cosmological Special Relativity theory. Our investigations include Maxwell equations, electromagnetic wave equations and functions, energy-momentum relations, the Klein-Gordon equation and wave function, the Yukawa potential in the Cosmological Inertial Frame (CIF), the Schrdinger equation, the Dirac equation, and their respective solutions. We also analyze the interaction between A-type wave function and B-type extended distance.Our particular focus is on the time evolution of the Yukawa potential in the CIF and the quantification of the Klein-Gordon scalar field. Through our analysis, we have established a connection between the newly modified Friedman-Lematre-Robertson-Walker metric, the Dirac equation, and the Klein-Gordon equation.