Extending projector-based relational observables beyond leading order in G

Develop a relational-observable construction—such as the projector-based Papadodimas–Raju approach—that remains valid beyond leading order in Newton’s constant, ensuring that operators localized in a compact region commute with exterior operators to the accuracy required to define fine‑grained entropies.

Background

Relational observables are designed to recover approximate locality in gravity by dressing bulk operators relative to clock-like backgrounds, improving commutation with asymptotic observables. The projector-based construction aligns correlators of improved operators with their local counterparts and improves commutation with the Hamiltonian at leading order.

However, the authors point out explicit limitations beyond leading order, including the sensitivity to insertions involving the Hamiltonian and the emergence of nonvanishing commutators via OPEs. They explicitly state that the extension beyond leading order in G is not yet understood.