Algebraic approach to quantum black holes: Logarithmic corrections to black hole entropy

The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As shown previously, for a neutral nonrotating black hole, such eigenvalues must be 2ⁿ-fold degenerate if one constructs the black hole stationary states by means of a pair of creation operators subject to a specific algebra. We show that the algebra of these two building blocks exhibits U(2)≡U(1)×SU(2) symmetry, where the area operator generates the U(1) symmetry. The three generators of the SU(2) symmetry represent a global quantum number (hyperspin) of the black hole, and we show that this hyperspin must be zero. As a result, the degeneracy of the n-th area eigenvalue is reduced to 2ⁿ/n^3/2 for large n, and therefore, the logarithmic correction term -3/2logA should be added to the Bekenstein-Hawking entropy. We also provide a heuristic approach explaining this result, and evidence for the existence of two building blocks.