On the geometric and algebraic foundation of the spinor formalism and the application to relativistic field equations

Abstract

The Hilbert calculus of segments plays an important role in the axiomatic foundation of the Euclidean geometry, as the relationship to some fundamental agebraic structures can be made more apparent. An extension of the Hilbert calculus to the field of the quaternionsU2 or biquaternionsU4 leads to some new aspects on the spinor formalism. By that, a geometrical interpretation of the Dirac equation is obtained. Including the torsion of the Minkowski space (Cartan geometry), the affine connection of the spinor spaceU4 also can be interpreted with the help of a generalized Hilbert calculus. These considerations lead to a simple geometrical access to the nonlinear spinor theory, proposed by Ivanenko, Heisenberg, Dürr, etc.