The Dirac wave function is represented in a form where all its components have obvious geometrical and physical interpretations. Six components compose a Lorentz transformation determining the electron velocity are spin directions. This provides the basis for a rigorous connection between relativistic rigid body dynamics and the time evolution of the wave function. The scattering matrix is given a new form as a spinor-valued operator rather than a complex function. The approach reveals a geometric structure of the scattering matrix and simpli¯es scattering calculations. This claim is supported by an explicit calculation of the di®erential crosssection and polarization change in Coulomb scattering. Implications for the structure and interpretation of relativistic quantum theory are discussed.