Minimum length scale from graviton propagator

Many approaches to quantum gravity predict the existence of a fundamental minimum length. In this work we distinguish between the minimum geometrical length and the minimum length scale, and explore how they arise from graviton propagators subject to different boundary conditions. We show that in-in propagators allow one to calculate the minimum geometrical length as the expectation value of the proper distance, and that it vanishes to all orders of metric perturbations in a non-interacting theory. After incorporating a one-loop correction to the graviton propagator, this outcome remains unchanged at second order in perturbation theory. In contrast, we find that the minimum length scale, defined by in-out expectation values and interpreted as the scale of scattering processes, is non-vanishing. Treating general relativity as an effective field theory, we establish that this scale is of Planckian order both in the free theory and after including the one-loop correction to the graviton propagator.