Thermodynamics of quantum fields in black hole backgrounds

We discuss the relation between the microcanonical and the canonical ensemble for black holes, and highlight some problems associated with extreme black holes already at the classical level. Then we discuss the contribution of quantum fields and demonstrate that the partition functions for scalar and Dirac (Majorana) fields in static space-time backgrounds can be expressed as functional integrals in the corresponding optical space, and point out that the difference between this and the functional integrals in the original metric is a Liouville-type action. The optical method gives both the correction to the black hole entropy and the bulk contribution to the entropy due to the radiation, while the conical singularity method just gives the divergent contribution to the black hole entropy unless one takes into account nonlocal terms. A simple derivation of a general formula for the free energy in the high-temperature approximation is given and applied to various cases. We conclude with a discussion of the second law.