Ultraviolet-Finite Perturbative Expansion of Quantum Gravity at Null Infinity
We present a perturbative formulation of quantum gravity for asymptotically flat vacuum spacetimes based on the Null Surface Formulation (NSF), in which the expansion is ultraviolet-finite term by term up to the orders computed, without the need for renormalization. The outgoing Bondi shear operators are constructed explicitly up to fourth order, with interaction kernels determined recursively from on-shell gravitational data at null infinity. Ultraviolet finiteness at each order follows from the on-shell structure of the construction and the restriction of all integrations to the compact celestial sphere, eliminating off-shell propagators. The map between the in and out states admits a perturbative construction, and unitarity is verified explicitly up to fourth order. The outgoing operators satisfy the same commutation relations as the incoming ones, indicating that the transformation is canonical and consistent with the unitary implementation. Collinear configurations give rise to infrared singularities, as expected in massless quantum field theories, but do not affect the ultraviolet behavior established here. In coherent states, the expectation value of the shear reproduces the known finite classical graviton scattering at lowest nontrivial order. These results provide a perturbative framework for quantum gravity with improved ultraviolet behavior relative to the covariant approach.