Proper time and Minkowski structureon causal graphs

Abstract

For causal graphs we propose a definition of proper time which for smallscales is based on the concept of volume, while for large scales the usualdefinition of length is applied. The scale where the change from `volume' to`length' occurs is related to the size of a dynamical clock and defines anatural cut-off for this type of clock. By changing the cut-off volume wemay probe the geometry of the causal graph on different scales andthereby define a continuum limit. This provides an alternative to thestandard coarse-graining procedures. For regular causal lattices (such as,for example, the two-dimensional lightcone lattice) this concept can be proven to lead to aMinkowski structure. An illustrative example of this approach is providedby the breather solutions of the sine-Gordon model on a two-dimensionallightcone lattice.