Discrete Quantum Gravity and Its Transition in Continuous Form in Nonperturbative Regime

Abstract

This paper presents an approach, independent of the space-time background, to gravitational studies in the discrete (lattice) form enabling one to express all fundamental quantities and their variations in terms of the present momenta in the case when a theory involves the maximal momentum (energy) Pmax, Emax. The approach, in the explicit form, is applied to the Regge Calculus. In this case at low energies (far from the maximal ones) E ˝ Emax, the study covers interaction with matter fields as well as the lattice weak-field expansion and strong coupling expansion. It is demonstrated that in the latter case of the proposed pattern a theory involves no long-range order. It is shown that, almost without any changes, this approach may be extended to the case of Causal Dynamical Triangulations (CDT) and, quite naturally, to the high energies E ≈ Emax. As this takes place, a theory involves the correctly defined low-energy transition to discrete and to continuous-form gravity.