Relativistic quantum physics with hyperbolic numbers

A representation of the quadratic Dirac equation and the Maxwell equations in terms of the three-dimensional universal complex Clifford algebra C¯3,0 is given. The investigation considers a subset of the full algebra, which is isomorphic to the Baylis algebra. The approach is based on the two Casimir operators of the Poincaré group, the mass operator and the spin operator, which is related to the Pauli-Lubanski vector. The extension to spherical symmetries is discussed briefly. The structural difference to the Baylis algebra appears in the shape of the hyperbolic unit, which plays an integral part in this formalism.