Discretization and Continuum Limit of Quantum Gravity on a Four-Dimensional Space-Time Lattice

The Regge Calculus is a powerful method to approximate a continuous manifold by a simplicial lattice, keeping the connectivities of the underlying lattice fixed and taking the edge lengths as degrees of freedom. The Discrete Regge Model limits the choice of the link lengths to a finite number. We examine the phase structure of Standard Regge Calculus in four dimensions and compare our Monte Carlo results with those of the $Z_2$-Regge Model as well as with another formulation of lattice gravity derived from group theoretical considerations. Within all of the three models of quantum gravity we find an extension of the well-defined phase to negative gravitational couplings and a new phase transition. We calculate two-point functions between geometrical quantities at the corresponding critical point and estimate the masses of the respective interaction particles. A main concern in lattice field theories is the existence of a continuum limit which requires the existence of a continuous phase transition. The recently conjectured second-order transition of the four-dimensional Regge skeleton at negative gravity coupling could be such a candidate. We examine this regime with Monte Carlo simulations and critically discuss its behavior.