Generalization of Bohmian mechanics and quantum gravity effective action

We generalize the de Broglie–Bohm (dBB) formulation of quantum mechanics to the case of quantum gravity (QG) by using the effective action (EA) for a QG theory. This is done by replacing the dBB equations of motion (EOM) with the EA EOM, which is beneficial even in the non-gravitational case, since in this way one avoids the violations of the Heisenberg uncertainty relations and the absence of the classical trajectories for stationary bound states. Another advantage of the EA formalism is that one can obtain the field configurations in the case of a quantum field theory (QFT). The proposed QG generalization is natural for Bohmiam mechanics because a dBB wavefunction is really a wavefunction of the Universe and in order to define the EA for an arbitrary initial state one needs a QG path integral. The QG EA can be constructed by using the piecewise flat QG (PFQG) theory and the PFQG EA can be approximated by the QFT EA for General Relativity coupled to matter, with a cutoff determined by the average edge length of the spacetime triangulation. One can then calculate the corresponding field configurations and from these field configurations one can obtain the trajectories for the corresponding elementary particles.