Can a discrete lattice really be Lorentz invariant?
The `usual’ discrete structures which we encounter, e.g. as discrete approximations to spatial geometry, have a `mean valence’ of order 1. e.g. each `node’ of a Cartesian lattice in three dimensions has six nearest neighbors. Random spatial lattices, such as a Voronoi complex, will similarly have valences of order 1 (or perhaps more properly of order of the spatial dimension). Such discrete structures cannot hope to capture the noncompact Lorentz symmetry of spacetime. Causal sets, however, have a `mean valence’ which grows with some finite power of the number of elements in the causet set. It is this `hyper-connectivity’ that allows them to maintain Lorentz invariance in the presence of discreteness. Below is a demonstration of the Lorentz invariant character of causal sets. The top left image is a square region of 1+1 Minkowski space, into which has been sprinkled 4096 points. To the right is a blow up of a small region of the original region. The bottom left image shows the same poi
