Quantum differences from direct Zerilli potential quantization

Investigate whether a direct operator representation and quantization of the polar Zerilli potential within the hybrid loop quantum cosmology framework for perturbed nonrotating black holes leads to quantum differences compared to quantizing the axial Regge–Wheeler potential and relating the polar sector via the Chandrasekhar transformation; characterize any such differences and their impact on the construction of the total Hamiltonian constraint operator.

Background

In the operator construction of the total Hamiltonian constraint for the hybrid quantization, the authors exploit the Chandrasekhar transformation to relate the polar (Zerilli) potential to the axial (Regge–Wheeler) potential. This allows them to focus on representing and quantizing the axial sector while assuming the polar sector follows by symmetry.

They note that directly treating the Zerilli potential may introduce quantum differences relative to the approach that uses the Chandrasekhar mapping. Determining whether such differences arise, and if so, characterizing them, is necessary to fully justify the equivalence of the axial-focused quantization with the polar sector in the quantum theory.