Poincare Invariance in Discrete Gravity

A formulation of discrete gravity was recently proposed based on defining a lattice and a shift operator connecting the cells. Spinors on such a space will have rotational SO(d) invariance which is taken as the fundamental symmetry. Inspired by lattice QCD, discrete analogues of curvature and torsion were defined that go smoothly to the corresponding tensors in the continuous limit. In this paper, we show that the absence of diffeomorphism invariance could be replaced by requiring translational invariance in the tangent space by enlarging the tangent space from SO(d) to the inhomogeneous Lorentz group ISO(d) to include translations. We obtain the ISO(d) symmetry by taking instead the Lie group SO(d + 1) and to perform on it Inonu-Wigner contraction. We show that, just as for continuous spaces, the zero torsion constraint converts the translational parameter to a diffeomorphism parameter, thus explaining the effectiveness of this formulation.