Reflection positivity of the Euclidean generating functional Z(F)

Ascertain whether the Euclidean generating functional Z[F] defined in equation (3.14) for the Gaussian dust–coupled canonical quantum gravity model, with integration variable Q in synchronous gauge and Euclidean action S[Q] (including the Gibbons–Hawking boundary term), satisfies time reflection positivity. Establishing reflection positivity is necessary for Osterwalder–Schrader reconstruction of a Hilbert space and Hamiltonian bounded from below for this formulation.

Background

The paper constructs a Euclidean path integral formulation for the Gaussian dust matter coupled to gravity, starting from the canonical Lorentzian Hamiltonian and performing an analytic continuation to obtain Schwinger functions. The resulting generating functional Z[F] appears in equation (3.14) and integrates over the density-weighted metric variable Q restricted to synchronous gauge, with the Euclidean Einstein–Hilbert action plus a Gibbons–Hawking boundary term.

The authors note that due to the indefinite nature of the spatial Ricci scalar and the negative conformal mode in the kinetic term (associated with the DeWitt metric signature), convergence and positivity properties of Z[F] are nontrivial. Reflection positivity is a key condition for applying Osterwalder–Schrader reconstruction to recover a Hilbert space and a Hamiltonian bounded from below, which would enable a rigorous connection back to the canonical (operator) framework. Determining whether Z[F] satisfies time reflection positivity is thus crucial for establishing the full equivalence between the Euclidean functional and the canonical theory in this setting.