Probing the quantum nature of curvature in the deep quantum regime of a black hole

In this Letter, we probe the quantum nature of the Kretschmann invariant $K$ in the deep quantum regime that dominates in the vicinity of the black hole singularity. To overcome challenges in formulating a quantum mechanical operator $\hat{K}$ corresponding to the Kretschmann invariant, we employ Bohmian quantum interpretation for the Wheeler-DeWitt equation that emerges from a Kantowski-Sachs representation of the black hole interior. We find that the expectation value $〈\hat{K}〉$ of the Kretschmann operator with respect to regular wave functions obtained from solutions of the Wheeler-DeWitt equation is well-behaved in the deep quantum regime in restricted subregions parametrized by the eigenvalues.