Canonical quantum statistics of an isolated Schwarzschild black hole with a spectrum En=σ√nEp

Many authors - beginning with Bekenstein - have suggested that the energy levels E-n of a quantized isolated Schwarzschild black hole have the form E-n = sigma root nE(P), n = 1,2, ..., sigma = O(1), with degeneracies g(n). 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, beta = 1/k(B)T), 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, beta) in the complex g-plane from g = 1 to g --> infinity. 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 Mc(2) requires beta to have the value (in natural units) beta = 2M(ln g/sigma(2)) [1 + O(1/M-2)] (4 pi sigma(2) = ln g gives Hawking's beta(H)) and yields the entropy S = [ln g/(4 pi sigma 2)] A/4 + O(ln A), where A is the area of the horizon. (C) 1997 Elsevier Science B.V.