Solving the Effective Field Equations for the Newtonian Potential

Loop corrections to the gravitational potential are usually inferred from scattering amplitudes, which seems quite different from how the linearized Einstein equations are solved with a static, point mass to give the classical potential. In this study we show how the Schwinger-Keldysh effective field equations can be used to compute loop corrections to the potential in a way which parallels the classical treatment. We derive explicit results for the one loop correction from the graviton self-energy induced by a massless, minimally coupled scalar.