Graviton corrections to the Newtonian potential using invariant observables

We consider the effective theory of perturbative quantum gravity coupled to a point particle, quantizing fluctuations both gravitational field and particle's position around flat space. Using recent relational approach construct gauge-invariant observables, we compute one-loop graviton corrections invariant metric perturbation, whose time-time component gives Newtonian potential. The resulting correction consists two parts: first stems from loops agrees with derived by other methods, while second one is sourced energy-momentum, may be viewed as an analog "Zitterbewegung". As check on computation, also recover classical which agree expansion Schwarzschild metric.