Towards a unified quantum theory of gravity and matter

B. Heim’s alternative quantum field theory can be considered as a candidate to consistently describe spacetime and matter since it is founded on the covariant structure of general relativity, and since the masses of the known matter particles can be calculated from it fairly well, as recently shown. Its basic nonlinear eigenvalue equation represents the quantization of spacetime in Heim’s way and merges into the Einstein field equation in the macroscopic limit. As also the relativistic equations of the Klein–Gordon and Dirac fields can be derived from it, the quantization of matter can be considered as consistently contained as well. On this basis, we show that Heim’s equation can be formulated similar to the semi-classical Einstein equation, but with the crucial difference of being linear in the matter field, and that it effectively becomes a linear field equation for both the geometric and the matter side under typical quantum scales of elementary particles. Thus, it then fulfills the quantum mechanical superposition principle. The calculated metric tensor remains regular at r = 0, t = 0 and shows oscillations whose possible similarity with BKL oscillations should be further investigated. In the weak field limit, a gravitational potential can be derived that decreases exponentially at microscopic distances, potentially shedding new light on the hierarchy problem. It can be used for an experimental test at intermediate mass-energies.