Existence of a kinematical normalised observable reproducing the sector-wise position reduction map

Determine whether there exists a kinematical, normalised observable on the system Hilbert space whose associated Page–Wootters reduction map, when restricted via the Hamiltonian constraint for a free particle, reduces to the sector-wise position reduction states |x0,σ>_S constructed from momentum components with weight sqrt(σ p/m) and phase e^{-i x0 p} (i.e., the states used to define the map in Eq. (ePositionReductonStates)), thereby providing a marginal kinematical interpretation without violating the imposed spatial translation covariance and time-covariant POVM requirements.

Background

The paper constructs a time-of-arrival distribution within the Page–Wootters formalism by defining a sector-wise reduction map that yields the state of a clock conditioned on the particle’s position. The resulting distribution matches Kijowski’s distribution and crucially relies on a sector decomposition enforced by the Hamiltonian constraint.

The authors emphasize three guiding requirements: (1) compatibility with the Hamiltonian constraint (working within the physical Hilbert space), (2) appropriate covariance under spatial translations for the position parameter, and (3) a clock-time POVM covariant with respect to the clock Hamiltonian. These requirements lead to the sector-wise position reduction states |x0,σ>_S with weights sqrt(σ p/m) and phase e^{-i x0 p}.

They note that this reduction map is not associated with any normalised kinematical observable, which complicates giving a marginal kinematical interpretation. An attempt to "restore" such a marginal interpretation by replacing the reduction states with those of a kinematically normalised observable would either violate one of the three conditions or would need to reduce to the same sector-wise states when restricted via the constraint. Whether such a normalised observable exists is left unresolved, as they were unable to find an example.