Generalize the quadrupole–volume proportionality beyond tetrahedra

Determine whether, within the U(N)-invariant sector of the two-vertex model, the proportionality between the quadrupole-based approximate volume and the exact polyhedron volume observed for tetrahedra (N=4) holds for polyhedra with any other number of faces.

Background

The authors analytically paper the volume dynamics of polyhedra dual to the two-vertex graph using the geometric quadrupole as an approximation to volume. In the U(N)-symmetric sector, they show that the approximate volume has the same evolution as the exact volume for tetrahedra, leading to proportionality.

They explicitly leave open whether this proportionality extends to polyhedra with more than four faces, identifying a concrete question for future investigation.

References

We have found that the approximation to the volume in terms of quadrupoles is proportional to the exact volume for the case of the tetrahedron ($N=4$). It is left for future work to check whether this behavior is true for any other number of faces.

— From loop quantum gravity to cosmology: the two-vertex model (2403.15320 - Cendal et al., 22 Mar 2024) in Section 6 (Conclusions)