Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering
Viewing 
gravitational energy-momentum PGμ as equal by 
observation, but different in essence from inertial energy-momentum PIμ 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.