Proper time and Minkowski structure on causal graphs

Abstract

For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition 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 a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and therey define a continuum limit. This provides an alternative to the standard coarse graining procedures. For regular causal lattice (like e.g. the 2-dim. light-cone lattice) this concept can be proven to lead to a Minkowski structure. An illustrative example of this approach is provided by the breather solutions of the Sine-Gordon model on a 2-dimensional light-cone lattice.