Einstein-Dirac system in semiclassical gravity

We study the Dirac equation minimally coupled to general relativity using quantum field theory and semiclassical gravity approximation. Previous studies of Einstein-Dirac system did not quantize required multiple independent fields preserve spherical symmetry. canonically a single in static spherically symmetric curved spacetime background. Using approximation, which Einstein equations are sourced by expectation value stress-energy-momentum tensor, we derive whose solutions describe self-gravitating configurations identical spin-$1/2$ particles. self-consistently solve these present example configurations. Although limiting cases our reproduce multifield found literature, is derived from excitations field.