The Dirac field and the possible origin of gravity
  The spin connections of the Dirac field have three ingredients that are connected with the Ricci rotations, the Maxwell field, and an axial field which minimally interacts with the axial current. I demonstrate that the axial field provides an effective mechanism of auto-localization of the Dirac field into compact objects. The condition that these objects are stable (the energy-momentum is self-adjoint) leads to Einstein's field equations. The Dirac field with its spin connection seem to be a natural material carrier of the space-time continuum in which compact objects are moving along geodesic lines. The long distance effect of the axial field is indistinguishable from Newton's gravity, which reveals the microscopic nature of gravity and the origin of the gravitational mass. 
Alexander Makhlin