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