Construct causal relational fields on curved spacetimes

Construct explicit nontrivial examples of causal relational quantum fields on non-homogeneous curved Lorentzian spacetimes, including examples on Schwarzschild spacetime, and develop the corresponding curved-spacetime causality theory.

Background

The paper proves implications among several proposed causality notions for relational quantum fields on curved spacetime, but does not provide nontrivial curved-spacetime models realizing them. It specifically identifies non-homogeneous backgrounds such as Schwarzschild spacetime as target examples.

References

We can therefore talk about causality in relational quantum field theory in curved spacetime. We keep as future work the task of constructing explicit nontrivial examples of causal relational quantum fields, as was done in Minkowski spacetime in [22] using similar notions of causality.

— A groupoidal approach to quantum reference frames  (2608.14133 - Fedida et al., 14 Aug 2026) in Remark 6.25, Section 6.3, p. 51; Conclusions and Outlook, pp. 60–61

With $V_{l}(x)$, the situation turns out to be much more difficult --- at the moment I have no rigorous mathematical proof that the integrals in CCRKS2c vanish for $|t-t'|<|x-x'|$.

— Remarks on quantum scalar field theory for a uniformly accelerated reference frame and a Schwarzschild black hole  (2608.23108 - Smolyakov, 24 Aug 2026) in Section 5.3, “Is there a consistent quantum scalar field theory in Kruskal-Szekeres spacetime?”