Search for the Geometrodynamical Gauge Group: Hypotheses and Some Results

Abstract

Discussed is the problem of the mutual interaction between spinor and gravitational fields. The special stress is laid on the problem of the proper choice of the gauge group responsible for the spinorial geometrodynamics. According to some standard views this is to be the local, i.e., $x$-dependent, group SL$(2,\mathbb{C})$, the covering group of the Lorentz group which rules the internal degrees of freedom of gravitational cotetrad. Our idea is that this group should be replaced by SU$(2,2)$, i.e., the covering group of the Lorentz group in four dimensions. This leads to the idea of Klein-Gordon-Dirac equation which in a slightly different context was discovered by Barut and coworkers. The idea seems to explain the strange phenomenon of appearing leptons and quarks in characteristic, mysterious doublets in the electroweak interaction.