Quantum corrected geodesics

We compute the graviton-induced corrections to the trajectory of a classical test particle. We show that the motion of the test particle is governed by an effective action given by the expectation value (with respect to the graviton state) of the classical action. We analyze the quantum corrected equations of motion for the test particle in two particular backgrounds: a Robertson Walker spacetime and a $\mathrm{}(2+1)\mathrm{}$-dimensional spacetime with rotational symmetry. In both cases we show that the quantum corrected trajectory is not a geodesic of the background metric.