Canonical quantum statistics of an isolated Schwarzschild black hole with a spectrum En = σnEp

Many authors — beginning with Bekenstein — have suggested that the energy levels En of a quantized isolated Schwarzschild black hole have the form En = σnEp, n = 1,2,…,σ = O(1), with degeneracies gn. In the present paper properties of a system with such a spectrum, considered as a quantum canonical ensemble, are discussed: Its canonical partition function Z(g,β = 1kBT), defined as a series for g < 1, obeys the 1-dimensional heat equation. It may be extended to values g > 1 by means of an integral representation which reveals a cut of Z(g,β) in the complex g-plane from g = 1 to g → ∞. Approaching the cut from above yields a real and an imaginary part of Z. Very surprisingly, it is the (explicitly known) imaginary part which gives the expected thermodynamical properties of Schwarzschild black holes: Identifying the internal energy U with the rest energy Mc2 requires β to have the value (in natural units) β = 2M(ln g/σ2)[1+O(1/M2)] (4πσ2 = ln g gives Hawkingʹs βH) and yields the entropy S = [ln g/(4φσ2)] A/4 + O(ln A), where A is the area of the horizon.