Establish ultraweak continuity of relativization maps

Determine whether suitable ultraweak continuity assumptions on groupoid frame POVMs imply that the relativization map sends ultraweakly continuous operator fields to ultraweakly continuous invariant operator fields and that the restricted relativization map preserves ultraweak continuity.

Background

The relativization results establish measurability and boundedness but do not assume ultraweak continuity of the field of POVMs. The authors identify as unresolved whether stronger continuity properties of the relativization maps can be obtained under corresponding continuity assumptions on the frame observables.

References

Note that we do not necessarily require the field of POVMs to be ultraweakly continuous here, though to prove results of the form €R(A(HS )uw) ⊂ A(HS ⊠ HR)Γuw and €Rω (A(HS )uw) ⊂ A(HS )uw this might become necessary. We leave these considerations to future work.

— A groupoidal approach to quantum reference frames  (2608.14133 - Fedida et al., 14 Aug 2026) in Remark 4.31, Section 4.3, p. 29

Whether it is possible, in relational quantum field theory, to retrieve an analogous version of Wightmannian quantum fields defined instead over L1(M, C) and described using a localizable QRF is an interesting open question.

— A groupoidal approach to quantum reference frames  (2608.14133 - Fedida et al., 14 Aug 2026) in Remark 5.20, Section 5.4, p. 43