Quantum mechanics of the Schwarzschild–de Sitter black hole

In the present paper, we apply the quantum general relativity theory to the Schwarzschild–de Sitter black hole. We start by writing down the super-Hamiltonian and supermomentum constraints for general, neutral, spherically symmetric, space-times, with a positive cosmological constant. Then we solve the constraints for the canonical momenta and canonically quantize the space-time using Dirac’s formalism for constrained systems. The resulting operatorial equations are exactly solved in the WKB approximation, giving rise to a wave function. Finally, for a particular ansatz of the canonical variables, we compute the quantum mechanical density probability and show how it depends on the mass of the hole and on the cosmological constant.