ar X iv : h ep - p h / 01 12 31 7 v 2 1 8 O ct 2 00 8 Lanczos ’ s equation to replace Dirac ’ s equation ?

Lanczos’s quaternionic interpretation of Dirac’s equation provides a unified description for all elementary particles of spin 0, 1 2 , 1, and 3 2 . The Lagrangian formulation given by Einstein and Mayer in 1933 predicts two main classes of solutions. (1) Point like partons which come in two families, quarks and leptons. The correct fractional or integral electric and baryonic charges, and zero mass for the neutrino and the u-quark, are set by eigenvalue equations. The electro-weak interaction of the partons is the same as with the Standard model, with the same two free parameters: e and sin θ. There is no need for a Higgs symmetry breaking mechanism. (2) Extended hadrons for which there is no simple eigenvalue equation for the mass. The strong interaction is essentially non-local. The pion mass and pion-nucleon coupling constant determine to first order the nucleon size, mass and anomalous magnetic moment. Since 1928, Dirac’s relativistic wave-equation has been rewritten in various forms in order to facilitate its interpretation. For example, in 1930, Sauter [1] and Proca [2] rewrote it using Clifford numbers. However, the most direct road was taken by Lanczos in 1929 [3]. He showed how to derive Dirac’s equation from the coupled biquaternion system: ∇A = mB, ∇B = mA. (1)