Einstein-Dirac system in semiclassical gravity

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