Quantum Equations in Cosmological Spaces

The Dirac equations for a free electron in a cosmological space are solved by means of separation of variables. It is shown that the wave functions depend on the angles $\theta$ and $\varphi$ in the same manner as those of a free electron in flat space time. The radial functions are obtained and it is shown that they go over into the usual ones in the limit. The explicit form of the time dependence of the wave functions cannot be obtained until an arbitrary function $R\left(t\right)\mathrm{}$ is specified. Three different cases are discussed. The energy of the free electron is then determined for each of these. Finally the connection between the equation used here and that proposed by Dirac for the DeSitter space is discussed. It is shown that they are similar and that the imaginary part of the complex mass that he was forced to introduce has a geometrical origin.