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(-iMc(2) tau/h) with respect to their continuous masses M and the proper time tau 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 Delta in real time with the period Delta(H) = 8 pi G M/c(3) in Euclidean time. This yields the mass spectrum M(n) = (1/2) root n mp, n = 1,2,....