The quantum levels of isolated spherically symmetric gravitational systems

The known canonical quantum theory of a spherically symmetric pure (Schwarzschild) gravitational system describes isolated black holes by plane waves exp(−iMc2τh̷) with respect to their continuous masses M and the proper time τ of observers at spatial infinity. On the other hand Bekenstein and Mukhanov postulated discrete mass levels for such black holes in the spirit of the Bohr-Sommerfeld quantisation in atomic physics. The two approaches can be related by postulating periodic boundary conditions in time for the plane waves and by identifying the period Δ in real time with the period ΔH = 8πGMc3 in Euclidean time. This yields the mass spectrum MN=(12)nmP, n = 1, 2,….