One may regard a Regge geometry as a special case of a continuum Riemannian manifold, a so-called piecewise flat manifold, with a flat metric in the interior of its 4-simplices , and singular curvature assignments to its two-simplices (the bones or hinges).
The Einstein action with cosmological term in the Regge approach is given by have generalized (12) to the case of manifolds with boundary. The boundary contribution to the action is given by [104, 81, 18]. One is interested in the behaviour of expectation values of local observables as the simplicial complex becomes large, and the existence of critical points and long-range correlations, in a scaling limit and as the cutoffs are removed.
A first implementation of these ideas was given by Roček and Williams [181, 183]. They obtained a simplicial lattice geometry by subdividing each unit cell of a hypercubic lattice into simplices. Their main result was to rederive the continuum free propagator (see also  for related results) in the limit of weak perturbations about flat space. This calculation can be repeated for Lorentzian signature .
Some non-perturbative aspects of the path integral were investigated in  (see also ). In this work, discrete analogues of space-time diffeomorphisms are defined as the local link length transformations which leave the action invariant, and go over to translations in the flat case. It is argued that an approximate invariance should exist in 4d. One may define analogues of local conformal transformations on a simplicial complex by multiplication with a positive scale factor at each vertex, but the global group property is incompatible with the existence of the generalized triangle inequalities.
© Max Planck Society and the author(s)