Covariant perturbation theory and the Randall–Sundrum picture

The effective action for quantum fields on a d-dimensional spacetime can be computed using a non-local expansion in powers of the curvature. We show explicitly that, for conformal fields and up to quadratic order in the curvature, the non-local effective action is equivalent to the d+1 action for classical gravity in AdSd+1 restricted to a (d−1)-brane. This generalizes previous results about quantum corrections to the Newtonian potential and provides an alternative method for making local a non-local effective action. The equivalence can be easily understood by comparing the Kallen–Lehmann decomposition of the classical propagator with the spectral representation of the non-local form factors in the quantum effective action.