LANCZOS - EINSTEIN - PETIAU: From Dirac’s equation to nonlinear wave mechanics

In 1929 Lanczos showed how to derive Dirac’s equation from a more fundamental system that predicted that spin 1 2 particles should come in pairs. Today, these pairs can unambiguously be interpreted as isospin doublets. From the same fundamental equation Lanczos derived also the correct form of the wave equation of massive spin 1 particles that would be rediscovered by Proca in 1936. Lanczos’s fundamental system was put in Lagrangian form and generalized by Einstein and Mayer in 1933. Although they not did study all possible solutions, Einstein and Mayer showed that the doublets consisted of particles with different mass and charge. In fact, there are two main classes of doublets: proton/neutron and electron/neutrino pairs. In trying to use Proca’s equation for the electromagnetic field of the electron, Lanczos reached the conclusion that the mass term could not be a constant, but had to be a function of space-time. On very general grounds he then proposed that the elementary solutions of a fundamental nonlinear field theory should be eigensolutions. In complete independence of Lanczos, Gérard Petiau discovered in 1957 a nonlinear generalization of quantum mechanics which is very close to Lanczos ideas. In such a theory the Hamiltionian scales with the 4th power of the proper frequency. Postulating a fundamental length equal to αre, with re the classical radius of the electron, we find that the Lanczos-EinsteinPetiau model is applicable to the problem of the mass of the quarks and electrons. The compatibility of the existence of this fundamental length with quantum theory and special relativity limits the number of quarks to five and the number of electrons to three. Published in W.R. Davis, et al., Cornelius Lanczos Collected Published Papers With Commentaries (North Carolina State University, Raleigh, 1998) Vol. III, p. 2-1248 to 2-1277.