  The known canonical quantum theory of a spherically symmetric pure
(Schwarzschild) gravitational system describes isolated black holes by plane
waves exp(-iMc^2\tau/\hbar) with respect to their continuous masses M and the
proper time \tau of obsevers 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 GM/c^3 in Euclidean time. This yields the mass spectrum
M_n=(1/2)\sqrt{n}m_P, n=1,2,... .
