Euclidean and Canonical Formulations of Statistical Mechanics in the Presence of Killing Horizons

Abstract

The relation between the covariant Euclidean free-energy $F^E$ and the canonical statistical-mechanical free energy $F^C$ in the presence of the Killing horizons is studied. $F^E$ is determined by the covariant Euclidean effective action. The definition of $F^C$ is related to the Hamiltonian which is the generator of the evolution along the Killing time. At arbitrary temperatures $F^E$ acquires additional ultraviolet divergences because of conical singularities. The divergences of $F^C$ are different and occur since the density ${dn \over d\omega}$ of the energy levels of the system blows up near the horizon in an infrared way. We show that there are regularizations that make it possible to remove the infrared cutoff in ${dn \over d\omega}$. After that the divergences of $F^C$ become identical to the divergences of $F^E$. The latter property turns out to be crucial to reconcile the covariant Euclidean and the canonical formulations of the theory. The method we use is new and is based on a relation between ${dn \over d\omega}$ and heat kernels on hyperbolic-like spaces. Our analysis includes spin 0 and spin 1/2 fields on arbitrary backgrounds. For these fields the divergences of ${dn \over d\omega}$, $F^C$ and $F^E$ are presented in the most complete form.