Canonical Quantization, Conformal Fields and the Statistical Entropy of the Schwarzschild Black Hole

The Wheeler-DeWitt wave functional of a Schwarzschild black hole obeys the two dimensional, massless Klein-Gordon equation and is supported only in the interior of the Kruskal manifold. The horizon acts as a perfectly reflecting surface. Energy eigenstates of definite parity exhibit the Bekenstein mass spectrum, M ∼ nMp. The wave function can be thought of as describing a particle in a one dimensional box whose length is proportional to the particle energy. Alternatively, one can think in terms of a conformal scalar field in two dimensions with appropriate boundary conditions on the horizon. We exploit this analogy to obtain the entropy of the Schwarzschild black hole by explicitly counting the microstates corresponding to a macrostate of fixed energy. 1 On leave of absence from the Universidade do Algarve, Faro, Portugal 2 Email: [email protected] 1 It has been recognized for some time that black holes behave as thermodynamic objects with a characteristic temperature and entropy and that these quantities are inherently quantum mechanical in nature.1,2 This makes a clear understanding of the origins of black hole thermodynamics in terms of statistical principles one of the more interesting problems in physics today. The earliest attempt at a microscopic theory of black holes was due to Bekenstein,3 who proposed that the mass spectrum of the black hole is discrete in such a way that the eigenvalues of the horizon area operator of the black hole are equally spaced. Bekenstein’s hypothesis has led to many interesting attempts to derive the black hole entropy from first principles. These attempts have taken either the traditional quantum gravity4 approach or the string theory5. The interest in a microscopic description of the black hole entropy owes to the fact that such a description requires a quantum theory of gravity, therefore detailed first-principle investigations of the black hole entropy should contribute toward a construction of such a theory. Alternatively, the ability to reproduce the Bekenstein entropy from a microscopic counting of states may be considered a measure of the success of a candidate quantum theory of gravity. Thus, the successful derivation of the Bekenstein-Hawking entropy, for example, from a microcanonical ensemble of D−brane states6 is generally considered a triumph for string theory and good evidence that string theory is a convincing candidate for a theory of quantum gravity. An approach to the entropy problem that makes explicit reference neither to string theory nor to canonical quantum gravity is to count the number of states for a conformal field theory corresponding to the asymptotic symmetries of the black hole or to the symmetries of its horizon. This mechanism for calculating the number of microstates was recently initiated by Strominger7 for the BTZ black hole.8 Strominger made use of a discovery by Brown and Henneaux9 that the asymptotic symmetry group of AdS3 is generated by two copies of the Virasoro algebra with central charge 3l/2G, and then used Cardy’s formula10 to compute the asymptotic growth of states for this conformal field theory. More recently, Carlip11 has argued that the result is more generic than originally believed as the algebra