Starting with a succinct historical account about how theoretical physics evolved since the Dirac Equation formulation in the (four-dimensional) so called Dirac Representation (being the best suited to give a tensor like appearance to mathematical expressions containing the gamma matrices), I shall report about some recent results that suggests the convenience to replace tensor language by a Weyl based two-spinor formalism. Although a personal opinion, I think that, perhaps, the time is come for adopting a new language (even at the classical level) for twenty first century physics. Recently I have shown how the tensor language (in special relativity and in electrodynamics) can be superseded by Weyl two-spinor calculus. The novelty and usefulness of this practice, already discretely mentioned by Roger Penrose and some others, is reflected in the fact that two-spinors provide, even at the classical level, a deeper insight (they also describe 1/2 spin) than classical variables. Most theoretical studies are generally developed (with the exception of the electroweak theory) in the four-component Dirac spinor representation, resembling as much as possible the tensor language stablished during the twenties of last century after the enormous success of Einstein’s Relativity.
In the new language the basic tools in classical as well as quantum dynamics are two components spinors. In the classical domain, what until now were equations describing dynamics of particles without any kind of internal structure is now turned over in expressions describing also spin 1/2. I would like to emphasize that the new master spinor equations (already presented in previous studies and having no tensor counterpart) can also be obtained via a Lagrangian formulation extended to local U(1), SU(2) and SU(3) local gauge symmetries in terms of the field strength quantities without the conventional use of the four potentials. The U(1) case has a special significance explaining in classical terms the controversial Zitterbewegung (trembling motion) phenomenon so much discussed in relation with the free particle solutions of the Dirac Equation. Finally, I shall present the geodesic equation in two-spinor language (inclusion of the electromagnetic field inducing deviation from geodesic motion). Some solutions corresponding to radial trajectories in the Schwarzschild metric will also be discussed. Something remarkable is that the Cartan-Torsion term (present in one or another disguise since almost a century in most attempts of finding a unifying theory of gravitation and electromagnetism) is now unnecessary as spin is already embedded in the spinorial variables. As this is an ongoing research project, I will not mention expectations susceptible of being considered wishful thinking.