ENERGY-SPECTRUM OF A QUANTUM BLACK-HOLE

We discuss a 'minisuperspace' path integral for the partition function of a Schwarzschild black hole in thermal equilibrium within a finite spherical box. Building on a novel classical variational principle, we define and evaluate a partition function using a non-trivial complex integration contour. The partition function solves exactly the relevant differential equation related to the Wheeler-DeWitt equation, and it has the desired semiclassical behaviour indicating in particular thermodynamical stability. For a given size of the box, the density-of-states is non-vanishing only in a finite energy interval whose upper end is twice as high as would be classically expected without negative temperatures. When negative temperatures are included, this discrepancy is resolved, and the system is then analogous to certain systems in ordinary quantum statistical mechanics which admit negative temperatures. The relation to the partition function previously obtained by Whiting and York using a Hamiltonian reduction method is discussed.