Building blocks of a black hole

What is the nature of the energy spectrum of a black hole? The algebraic approach to black hole quantization requires the horizon area eigenvalues to be equally spaced. As stressed long ago by Mukhanov, such eigenvalues must be exponentially degenerate with respect to the area quantum number if one is to understand black hole entropy as reflecting degeneracy of the observable states. Here we construct the black hole stationary states by means of a pair of “creation operators” subject to a particular simple algebra, a slight generalization of that for a pair of harmonic oscillators. This algebra reproduces the main features of the algebraic approach, in particular the equally spaced area spectrum. We then prove rigorously that the $n\mathrm{th}$ area eigenvalue is exactly $2^n$-fold degenerate. Thus black hole entropy qua logarithm of the number of states for a fixed horizon area is indeed proportional to that area.