When one approaches the study of the quantal relativistic theory of the electron, one may be surprised by the gap which lies between the frame of the experiments, i.e. the real geometry of the space and time, and the abstraction of the complex matrices and spinors formalism employed in the presentation of the theory. This book uses a theory of the electron, introduced by David Hestenes, in which the mathematical language is the same as the one of the geometry of the space and time. Such a language not only allows one to find again the well known results concerning the one-electron atoms theory but furthermore leads easily to the resolution of problems considered for a long time without solution.

The Solutions of the Dirac Equation in Hydrogenic Atoms.- The Electromagnetic Fields Created by Time-Sinusoidal Current.- The Dirac Equation of the Electron in the Real Formalism.- The Solutions of the Dirac Equation for the Central Potential in the Real Formalism.- Fields Created by the Dirac Transition Currents Between Two States.- The Dirac Transition Currents Between Two States.- The Field at Large Distance Created by the Transition Currents.- Case of the Transitions P1/2-S1/2 and P3/2-S1/2.- Interaction with Radiation.- Interaction with an Incident Wave: The Retardation.- Relativistic Expression of the Matrix Elements.- The Photoeffect.- The Radial Functions of the Continuum.- Matrix Elements for the Transitions 1S1/2-Continuum.- Matrix Elements for the Relativistic Transitions with Retardation 1S1/2-Continuum.- The Radiative Recombination.- Interaction with a Magnetic Field.- The Zeeman Effect.- Addendum.- The Contribution of the Discrete Spectrum to the Lamb Shift of the 1S1/2 State.