Starting from recent observations [1,2] about quasi-normal modes, we use semi-classical arguments to derive the Bekenstein-Hawking entropy spectrum for d-dimensional spherically symmetric black holes . We find that, as first suggested by Bekenstein, the entropy spectrum is equally spaced: SBH = k ln(m0)n, where m0 is a fixed integer that must be derived from the microscopic theory. As shown in [2],4-d loop quantum gravity yields precisely such a spectrum with m0 = 3 providing the Immirzi parameter is chosen appropriately. For d-dimensional black holes of radius RH (M ), our analysis predicts the existence of a unique quasinormal mode frequency in the large damping limit ω (d) 4π ln(m0), where m0 is an integer and Γ (d−2) is the volume of the unit d − 2 sphere.

We consider the most general dilaton gravity theory l + l dimensions. By suitably parametrizing the metric and scalar field we find a simple expression that relates the energy of a generic solution to the magnitude of the corresponding Killing vector. In theories that admit black hole solutions, this relationship leads directly to an expression for the entropy S = ~~T o / G , where TO is the value of the scalar field (in this parametrization) at the event horizon. This result agrees with the one obtained using the more general method of Wald. Finally, we point out an intriguing connection between the black hole entropy and the imaginary part of the "phase" of the exact Dirac quantum wave functionals for the theory. PACS number(s): 04.70.D~

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