Rutherford scattering of quantum and classical fields

Abstract

Quantum Rutherford scattering and the scattering of classical waves by black holes have similar formal structures and can be studied using the same mathematical techniques. In both contexts, the long-range nature of the interaction leads to a divergent total cross section, which has been interpreted and regularized in various ways in the past literature. We review in detail the origin of this divergence, in both real and multipole spaces, and show that it arises from the incorrect use of approximations outside their domain of validity. We also stress that although black hole and quantum Rutherford scattering share the same formalism, the natures of the associated physical observables differ. We comment on the role of interference: while interference may be safely neglected in the context of quantum Rutherford scattering (due to the fact that the observable quantity is a flux, and the incoming flux is collimated), it should not be neglected in the context of classical waves scattered by a black hole, where one expects to see a superposition of transmitted and scattered waves in a broad region downstream from the target and a cross section is not connected to any physically observable quantity.