The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Cli?ord algebra Cl(M) associated with the Minkwoski space¿time M, that is, the euclidean 4 R space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an importantplaceintheresolutionoftheDiracequationforthecentralpotential problem). The clarity comes from the real form given by D. Hestenes to the electron wavefunctionthatreplaces,inastrictequivalence,theDiracspinor.Thisform is directly inscribed in the frame of the geometry of the Minkwoski space in which the experiments are necessarily placed. The simplicity derives from the uni?cation of the language used to describe the mathematical objects of the theory and the data of the experiments. The mathematics concerning the de?nition and the use of the algebra Cl(M) are not very complicated. Anyone who knows what a vector space is will be able to understand the geometrical implications of this algebra. The lecture will be perhaps more di?cult for the readers already acquainted with the complex formalism of the matrices and spinors, to the extent that the new language will appear di?erent from the one that they have used. But the correspondence between the two formalisms is ensured in the text at each stage of the theory.

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.