Wheeler-DeWitt canonical quantum gravity in hydrogenoid atoms according to de Broglie-Bohm and the geodesic hypothesis. Einstein’s Quantum Field Equation

Abstract

We explore the geodesic hypothesis of orbital trajectories of the electrons in hydrogenoid atoms, in the frame of de Broglie-Bohm quantum theory. It is intended that the space-time can be curved, at very short distances, by the effect of the joint action of the energy content of the atomic system and the contribution of the electric and quantum potentials. The geodesic hypothesis would explain the non-lose of energy in the electron orbital trajectories. So we explore a conception where particles and waves interact in a closed system: the waves guide the particles and the particles generate the spacetime perturbation that acts as a wave, beyond the pilot-wave theory.

We establish the equivalence, in a local neighborhood, between the electron trajectory of an hydrogenoid atom in the Minkowskian space where the de Broglie-Bohm can be cast with its movement in a Lorentzian manifold, according to the concept of metric tangent. Through the geodesic condition and the invariance of the elemental length, we establish a relationship between some components of the metrics. But as the particles in microphysics do not follow the Einstein’s field equation, we consider the 3+1 decomposition according to ADM and the quantization in the Wheeler De Witt theory and with a so-called quantum Einstein field equation, with a decomposition of spacetime into three-dimensional sheets of a spatial character. Then we derive from the moment-energy tensor a further equation between the components of the metrics. It opens the avenue to characterize the metric by an exact solution of the Einstein’s field equations.