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 $2^n$-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 $U\left(2\right)\mathrm{}\equiv U\left(1\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ symmetry, where the area operator generates the $U\left(1\right)\mathrm{}$ symmetry. The three generators of the $\mathrm{SU}\left(2\right)\mathrm{}$ 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 $2^n{/n}^{3/2}$ for large n, and therefore, the logarithmic correction term $-3/2\log A$ 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.