Euclidean and canonical formulations of statistical mechanics in the presence of Killing horizons Original Research Article

The relation between the covariant Euclidean free energy FE and the canonical statistical-mechanical free energy FC in the presence of the Killing horizons is studied. FE is determined by the covariant Euclidean effective action. The definition of FC is related to the Hamiltonian, which is the generator of the evolution along the Killing time. At arbitrary temperatures FE acquires additional ultraviolet divergences because of conical singularities. The divergences of FC are different and occur since the density dn/dω 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/dω. After that the divergences of FC become identical to the divergences of FE. 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/dω 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/dω, FC and FE are presented in the most complete form.