Relate group-based principal-bundle and groupoid approaches
Establish the links between the groupoid-based approach to quantum reference frames and observables and the group-based extension to principal bundles that relativizes operator-valued functions, in order to clarify how the two formalisms compare.
References
Understanding the links between the approach of the present paper that deals with groupoids and observables (and recovers a notion of quantum field as a necessary emergent feature of the relativistic structure of the theory), and that explored by Głowacki that uses groups and operator-valued functions (where one is relativising already-existing quantum fields rather than observables), is kept for future work.
One may also want to generalise the notion of groupoid QRFs to measurable groupoids, and relax the continuity assumptions of the field of Hilbert spaces and of the unitary representation. One challenge to overcome this would be to define the countable additivity of POVMs ExR (∞⊔n=1∆n)=∞∑n=1ExR(∆n), ∆n ∈ Bor(Γx) (133) in some suitable topology; we leave this for future work.
It is not yet a locally covariant quantum field theory in the functorial sense: that would require assigning compatible algebras and QRF data to a category of spacetimes and causality-preserving embeddings. We keep this for future work.