Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering

Abstract

Viewing gravitational energy-momentum <i>P_G<sup style="margin-left:-7px;"> μ </sup></i> as equal by observation, but different in essence from inertial energy-momentum <i>P_I<sup style="margin-left:-7px;"> μ </sup></i> naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S -matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity.