Imprint of Operator Ordering on the Quantum Nature of DeWitt-regular Black Holes

Abstract

In making a transition from classical to quantum mechanics in the canonical framework, operator ordering in the Hamiltonian of a non-trivial system plays a crucial role. Such is the case in formulating a Wheeler-DeWitt equation from the classical Hamiltonian of a black hole. In this paper, we therefore study the critical role of operator ordering controlled by a parameter while obtaining the Wheeler-DeWitt equation from a Kantowski-Sachs representation of the black hole interior. Upon solving the Wheeler-DeWitt equation, we find that the solutions reveal the critical role played by operator ordering. Notably, the solutions uncover the following important consequences: (1) The black hole interior wave function categorizes itself into three different classes, with wide domains for the operator ordering parameter in consistency with the DeWitt criterion. (2) The DeWitt-regular, asymptotically safe, wave functions are intimately related with the operator ordering as they carry imprints of the parameter controlling operator ordering. (3) The DeWitt-regular wave functions give finite and well-behaved expectation values of the Kretschmann curvature in the vicinity of the black hole singularity with reduced but still infinite extent for the operator ordering parameter. We thus find that the black hole singularity can be resolved with infinite ways of choosing operator ordering.