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Make precise a state-dependent dressing to radiation for island operators

Develop a precise construction of gauge-invariant operators localized in an entanglement island by dressing to features of the quantum state of the Hawking radiation ("dressing to the radiation"), including a concrete procedure for identifying partner modes and localizing operator insertions, and demonstrate its validity within perturbative gravity.

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Background

In backgrounds with exact time-translation symmetry (e.g., eternal black holes in equilibrium), classical symmetry breaking is absent, obstructing standard relational dressings. The authors propose dressing to the quantum state by using the entanglement pattern between Hawking quanta and their interior partners to localize operators within the island.

They do not provide a full construction and explicitly defer the detailed development of this state-dependent dressing, identifying it as a problem for future work.

References

Since the breaking of the symmetry is provided not by classical background fields but by a quantum state, this dressing goes beyond the discussion above. We will not attempt to make it more precise here, leaving the details of this construction as an interesting problem for the future.

— An apologia for islands (2506.04311 - Antonini et al., 4 Jun 2025) in Section 4.4 (Subtleties), subsubsection “Dressing to the quantum state?”