Four-dimensional covariance of Feynman diagrams in Einstein gravity

It was previously noted that physical states in terms of the ADM formalism in the framework of four-dimensional (4D) Einstein gravity holographically reduce and can be described as three-dimensional (3D). Obviously, a problem with 4D covariance arises with such an approach; it turns out that there are two such problems with covariance. We consider methods for solving these problems. Although the unphysical character of the trace part of the fluctuation metric has long been known, it has not been considered from the standpoint of applying Feynman diagrams for computations. A proper method for treating the trace part with gauge-fixing is the key to resolving subtle covariance issues. Regarding the second problem, it turned out that a covariant renormalization can be performed to any loop order in the intermediate steps, which preserves the 4D covariance. Only at the final stage is it necessary to consider 3D physical external states. With physical external states, the one-particle-irreducible effective action becomes 3D, and renormalizability is ensured just as in the 3D case. We present the one-loop two-point renormalization with careful attention to the trace part of the fluctuation metric. In particular, we describe the one-loop renormalization of the Newton constant.