We present a canonical quantization framework for static spherically symmetric spacetimes described by the Einstein–Hilbert action with a cosmological constant. In addition to recovering the classical Schwarzschild–(Anti)-de Sitter solutions via the Ehrenfest theorem, we investigate the quantum uncertainty relations that arise among the geometric operators in this setup. Our analysis uncovers an intriguing relation to black hole thermodynamics and opens a new angle towards generalized uncertainty relations. We further obtain an upper and a lower limit of the mass that is allowed in our model, for a given value of the cosmological constant. Both limits, when evaluated for the known value of the cosmological constant, have a stunning relation to observed bounds. These findings open a promising avenue for deeper insights into how quantum effects manifest in spacetime geometry and gravitational systems.
Koch et al. (Tue,) studied this question.