Quantum correction of the Dirac field to the entropy of a static black hole

The one-loop effective action for the quantum Dirac field is regulated by Pauli–Villars scheme, and the renormalized action for the coupled gravitational-quantum Dirac field is obtained. The quantum entropy of the massive Dirac field outside a static black hole is calculated by Rindler approximation. The divergencies of the free energy and entropy are regulated by the large mass of some fictitious regulators. The divergent entropy is renormalizable and yields the standard Bekenstein–Hawking formula, in terms of the Newtonʹs constant.