Quantum source of entropy for black holes

We associate to any quantum field propagating in the background metric of a black hole an effective density matrix whose statistical entropy can be interpreted as a contribution to the total entropy of the black hole. By evaluating this contribution in a simplified case, we show that in general it can be expected to be finite and proportional to the area of the black hole. As a by-product of our calculation we obtain a general expression for the entropy of any real Gaussian density matrix.