  In this work, with the help of the quantum hydrodynamic formalism, the
gravitational equation associated to the Dirac field is derived. The
hydrodynamic representation of the Dirac equation have been generalizaed to the
curved space-time in the covariant form. Thence, the metric of the spacetime
has been defined by imposing the minimum action principle. The derived gravity
shows the spontaneous emergence of the cosmological gravity tensor (CGT) as a
part of the energy-impulse tensor density (EITD) that in the classical limit
leads to the cosmological constant (CC). Even if the classical cosmological
constant is set to zero, the CGT is non zero, allowing to have a stable quantum
vacuum (out of the collapsed branched polymer phase). The theory shows that in
the classical limit, the gravity equation leads to the general relativity
equation. In the perturbative approach, the CGT leads to a second order
correction to the Newtonian gravity that takes contribution from the space
where the mass is localized (and the spacetime is curvilinear) while tends to
zero as the spacetime approaches to the flat vacuum leading, as a mean, to an
overall cosmological cosmological constant that may possibly be compatible with
the astronomical observations. The Dirac field gravity shows analogies with the
modified Brans-Dicke gravity where each spinor term brings an effective gravity
constant G divided by its field squared.
