Extend covariant POVM results to curved spacetime

Generalize Theorem 3.20 from transitive group actions on homogeneous spaces to suitable B-compatible measurable projections on the Poincaré groupoid of a curved spacetime, thereby establishing L1 Radon–Nikodym densities for the corresponding spacetime marginal frame measures.

Background

In the flat case, covariance guarantees absolute continuity of the relevant Born measures and produces L1 smearing functions. For curved spacetime, the authors expect an analogous result only for certain bisection-compatible projections, such as those associated with isometries, but leave its proof and precise hypotheses unresolved.

References

It may be that Thm. 3.20 can be generalised to curved spacetime for certain B-compatible π (e.g. when B = BisIsom(Poin(M, g))) so that we can sometimes write dμFxR,πωx (y) = f FxR,πωx (y)dμg (y) with f FxR,πωx ∈L1(M, μg ). In such cases, we can write the relational semilocal quantum fields as ˆΦRx (ω) =∫MˆϕR,πx,ω (y)f FxR,πωx (y)dμg (y) (262) as a direct generalisation of Eqn. (234). We keep such a generalisation of Thm. 3.20 to curved spacetimes to future work.

— A groupoidal approach to quantum reference frames  (2608.14133 - Fedida et al., 14 Aug 2026) in Remark 6.10, Section 6.1, p. 47

It is an interesting open question to understand whether given two B-compatible measurable projections π1 and π2 for B ≤ Bis(Poin(M, g)), OS ⊂ A(HS ) is (SR, σ, π1)-causal (resp. weakly microcausal, weakly semi-causal) if and only if OS is (SR, σ, π2)-causal (resp. weakly microcausal, weakly semi-causal). That is, whether the notion of causality is stable under changes of notions of the frame’s localization in spacetime.

— A groupoidal approach to quantum reference frames  (2608.14133 - Fedida et al., 14 Aug 2026) in Remark 6.26, Section 6.3, p. 51