Abstract
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.
Highlights
IntroductionImagine a world in which physicists would be forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory—a language consisting of terms such as state vectors in Fock spaces, causal quantum fields, operators, probability amplitudes, observables, propagators, conserved quantities
Imagine a world in which physicists would be forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory—a language consisting of terms such as state vectors in Fock spaces, causal quantum fields, operators, probability amplitudes, observables, propagators, conserved quantitiesHow to cite this paper: Wiesendanger, C. (2015) Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering
Wiesendanger such as the electric charge, energy-momentum and the like. Within that framework they might try to answer questions such as “Given a certain number of incoming particles described by free state vectors with given inertial energy-momenta piμ and other quantum numbers what is the probability—after they have interacted gravitationally—to observe a certain number of outgoing particles described by free state vectors with measured inertial energy-momenta p μ f and other quantum numbers?”—or to construct the S-matrix for quantum gravity
Summary
Imagine a world in which physicists would be forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory—a language consisting of terms such as state vectors in Fock spaces, causal quantum fields, operators, probability amplitudes, observables, propagators, conserved quantities. Our physicists would assure the observed equality of inertial and gravitational energy-momentum for on-shell observable physical objects in this approach by taking the gravitational limit, i.e. equating both types of momenta They would note—being forced from the outset to think about gravitation in terms and in the language of relativistic quantum field theory—that the language of classical physics with its reference to spacetime trajectories of particles makes no sense in the context of constructing such a theory—as would the principle of equivalence. This is the basis of our claim that the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space is a viable, renormalizable theory of quantum gravity This being the case we can directly analyze physical situations for which we can compare predictions both within the framework of the theory presented as well as within the standard framework of Newtonian gravity dealt with quantum-mechanically such as the gravitational scattering of two particles with different masses. We calculate the scattering cross-section of two Dirac particles with different masses and compare it in an appropriate limit with the cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity—and determine the numerical value of the coupling constant of our theory in the process
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