Spectrum of charged black holes - the big fix mechanism revisited

Following an earlier suggestion of the authors (Barvinsky A and Kunstatter G 1997 Mass spectrum for black holes in generic 2-D dilaton gravity Proc. 2nd International A D Sakharov Conference on Physics ed I M Dremin and A M Seminkhatov (Singapore: World Scientific) pp 210-15), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner-Nordstrom black hole we get A/4Gh = pi (2n + p + 1), where the integer p = 0, 1, 2.... gives the charge gives spectrum, with Q = +/- root hp. The quantity pi (2n + 1), n = 0, 1.... gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: Q = +/- me, in = 0, 1, 2,.... As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole-a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is e(2)/h = p/m(2) and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously.