4D covariance of holographic quantization of Einstein gravity

It has been observed in [Park 2014] that the physical states of the ADM formulation of 4D Einstein gravity holographically reduce and can be described by a 3D language. Obviously the approach poses the 4D covariance issue; it turns out that there are two covariance issues whose address is the main theme of the present work. Although the unphysical character of the trace piece of the fluctuation metric has been long known, it has not been taken care of in a manner suitable for the Feynman diagram computations; a proper handling of the trace piece through gauge-fixing is the key to more subtler of the covariance issues. As for the second covariance issue, a renormalization program can be carried out covariantly to any loop order at intermediate steps, thereby maintaining the 4D covariance; it is only at the final stage that one should consider the 3D physical external states. With the physical external states, the 1PI effective action reduces to 3D and renormalizability is restored just as in the entirely-3D approach of [Park 2014]. We revisit the one-loop two-point renormalization with careful attention to the trace piece of the fluctuation metric and in particular outline one-loop renormalization of the Newton's constant.