Introduction.- Comparison between Complex and Real Algebraic Languages.- The Clifford Algebra Associated with the Minkowski Space-Time M.- Comparison between Real and Complex Languages.- The U(1) Gauge in Complex and Real Languages - Geometrical Properties and Relation with the Spin and the Energy of a Particle of Spin 1/2.- Geometrical Properties of the U(1) Gauge.- Relation between the U(1) Gauge, the Spin and the Energyof a Particle of Spin 1/2.-...