Entropy of the quantum scalar field in static black holes

The quantum corrections to the entropy of static black holes are investigated by two methods: the brick wall method of 't Hooft and the Euclidean path integral approach of Gibbons and Hawking. Two general formulas for the entropy are obtained and some examples are considered. It is shown that if the contribution from the vacuum surrounding the system is ignored, then the two approaches give the same results. It is found that the entropy of the quantum scalar field in a general static black hole consists of two parts: a quadratically divergent term which takes a geometric character and a logarithmically divergent term which is not proportional to the horizon area. The logarithmically divergent term, in general, depends on the black hole characteristics (in particular, the whole entropy is determined only by this term for some extreme cases) and therefore cannot be neglected as a nonessential additive constant. The renormalization of the entropy is also discussed briefly.