U(2,2)-invariant spinorial geometrodynamics

Some criticism of Dirac theory is presented. It is indicated, e.g., that this theory involves some hidden action-at-distance concepts incompatible with the local paradigm of gauge theories. A new approach to the conformal SU(2,2) symmetry appearing in the theory of Dirac particles is suggested. The quadruplet of scalar Klein-Gordon particles with the internal hermitian metric of signature (+ + − −) is used as a primary model. Localization of the internal symmetry leads to the concept of the SU(2,2)-ruled gauge field. After the reduction to the SL(2, C) subgroup, some subsystem of the gauge field multiplet becomes the cotetrad field, whereas another subsystem may be interpreted as the Einstein-Cartan-Weyl affine connection. Geometrodynamical degrees of freedom comprise the SU(2,2)-gauge field and the normal-hyperbolic metric. After the SL(2,C)-reduction, the wave equation for the matter field becomes the Dirac-Klein-Gordon equation; its operator is a superposition of the Klein-Gordon and Dirac operators. There is a good correspondence with the usual Dirac equation. Yang-Mills equations for the SU(2,2) gauge field correspond in a reasonable way with the equations of the Poincaré gauge theory of gravitation and with the standard general relativity.