This paper shows how to quantize certain gauge‑invariant field components on a rotating black hole background and constructs a physically reasonable quantum state for them. The authors work with Teukolsky scalars of spin 0, ±1 and ±2. These scalars are combinations of metric or electromagnetic perturbations that are well adapted to Kerr black holes. The main result is a rigorous construction of the Unruh state for those fields on a subextreme Kerr spacetime and a proof that this state satisfies the Hadamard condition on the black hole exterior and on the interior up to, but not including, the inner horizon.

The authors use the algebraic approach to quantum field theory. They start from a careful description of the classical phase space for the Teukolsky equations and then build the canonical commutation relation (CCR) algebra. A technical difficulty is that the Teukolsky fields live in a nontrivial bundle of spin‑weighted scalars that does not come with a natural positive fibre metric. To handle this, the paper introduces an extended theory that combines the spin+s and spin−s fields. The extended Teukolsky operator can be shown to be formally hermitian and Green hyperbolic, which makes the algebraic construction feasible.

To define the Unruh state the authors use a bulk‑to‑boundary construction. They embed the algebra of observables on the spacetime into an algebra on the past conformal boundary, made of the ingoing black hole horizon and past null infinity. A key technical ingredient is conservation of a charged symplectic form for solutions. Establishing that conservation depends on decay and scattering estimates for solutions of the Teukolsky equation. The paper relies on detailed analytic results cited from prior work to justify these steps.

A central technical problem is positivity of the state. Because the extended bundle’s fibre metric is not positive definite, the natural candidate two‑point functions do not automatically give a positive state. The authors resolve this by identifying a physical subspace of the extended phase space. They use the Teukolsky–Starobinsky identities, which relate solutions of the spin+s and spin−s equations, to pick out the subspace where the state is positive. The Unruh state is then defined on the corresponding physical subalgebra and shown to be positive.