The Unruh state for bosonic Teukolsky fields on subextreme Kerr spacetimes

We perform the quantization of Teukolsky scalars of spin $0$, $\pm 1$, and $\pm 2$ within the algebraic approach to quantum field theory. We first discuss the classical phase space, from which we subsequently construct the algebra. This sheds light on which fields are conjugates of each other. Further, we construct the Unruh state for this theory on Kerr and show that it is Hadamard on the black hole exterior and the interior up to the inner horizon. This shows not only that Hadamard states exist for this theory, but also extends the existence and Hadamard property of the Unruh state to (bosonic) Teukolsky fields on Kerr, where such a result was previously missing.

💡 Research Summary

The paper presents a rigorous algebraic‑quantum‑field‑theory (AQFT) construction for Teukolsky scalar fields of spin 0, ±1, and ±2 on sub‑extremal Kerr spacetimes, i.e. the region comprising the exterior, the event horizon, and the interior up to (but not including) the inner Cauchy horizon. The authors begin by analysing the geometric setting: the Teukolsky scalars are sections of a non‑trivial bundle of spin‑weighted scalars associated with a null tetrad. Because this bundle lacks a natural Hermitian fibre metric, a direct quantisation is obstructed. To overcome this, they introduce an enlarged bundle that simultaneously carries fields of spin +s and –s. On this enlarged bundle the extended Teukolsky operator becomes formally self‑adjoint (Green‑hyperbolic), allowing the construction of a classical phase space equipped with a charged symplectic form.

A central technical ingredient is the Teukolsky‑Starobinsky identities, which relate solutions of the +s and –s equations. These identities are used to single out a physical subspace of the enlarged phase space that corresponds to the true degrees of freedom (e.g. the Hertz potential). By restricting the CCR algebra to this physical subalgebra the authors avoid the indefiniteness of the fibre metric and obtain a positive‑definite state space.

The paper then defines the Unruh state for the Teukolsky theory. The construction follows a bulk‑to‑boundary approach: the algebra of observables on the spacetime is embedded into an algebra on the past conformal boundary, consisting of past null infinity and the past event horizon. Decay estimates for solutions of the Teukolsky equation—derived from recent Fredholm theory and semiclassical flow analysis—ensure that the symplectic form is conserved under this embedding and that the two‑point function is square‑integrable on the boundary. Positivity of the state follows from the identification of the physical subspace and the positivity of the boundary symplectic form.

To prove the Hadamard property, the authors verify the microlocal spectrum condition for the Unruh two‑point function. They combine the decay results with microlocal analysis of the wave‑front set, showing that singularities are of the standard Hadamard form. The proof works both in the exterior region and in the interior up to the inner horizon, thereby establishing that the Unruh state is Hadamard throughout the entire sub‑extremal Kerr domain considered.

The main theorem (Theorem 1.1) states that for each spin s∈{0,1,2} the Teukolsky scalar theory on any sub‑extremal Kerr spacetime can be quantised via the enlarged phase space, that a physical sub‑algebra corresponding to the Hertz potential can be identified, and that the Unruh state defined on this sub‑algebra is well‑defined and Hadamard on the whole spacetime.

This work fills a gap in the literature: while Hadamard states for scalar fields and for fermions on Kerr have been constructed, no rigorous construction existed for bosonic Teukolsky fields (including linearised gravity). The paper’s methodology—extending the bundle, exploiting the Teukolsky‑Starobinsky identities, and employing recent decay and Fredholm results—provides a template for constructing physically reasonable states for other gauge‑invariant linearised theories on rotating black holes. The results are expected to be instrumental for future analytical and numerical studies of quantum effects (Hawking radiation, back‑reaction, gravitational‑wave signatures) in realistic Kerr backgrounds.