Hermiticity/factor-ordering duality in quantum black holes

As the property of duality exists in various disciplines of physics since early times, it is important to look for the existence of this symmetry in the physics of black holes described by quantum gravity. While canonically quantizing gravity, it is well-known that ambiguity in operator ordering and the requirement of Hermiticity play their crucial roles in determining the wave function of the black hole interior. Consequently, it is important to look for how Hermiticity and operator ordering could be interrelated, presumably by means of some duality between the two. In order to study this feature in a concrete way, we consider in this paper two distinct Kantowski–Sachs minisuperspace models for the black hole interior. In the process of canonical quantization in the first model, we maintain Hermiticity in the ensuing Wheeler–DeWitt equation. On the other hand, in the second model, we do not impose Hermiticity, but consider a general operator ordering in the Wheeler–DeWitt equation. Despite these crucial differences in the two frameworks, we find that there exists an exact duality between the wave functions occurring as solutions of the Wheeler–DeWitt equations in the two scenarios, called herein Hermiticity/factor-ordering duality.