Are There Testable Discrete Poincaré Invariant Physical Theories?

Abstract

In a model of physics taking place on a discrete set of points that approximates Minkowski space, one might perhaps expect there to be an empirically identifiable preferred frame. However, the work of Dowker, Bombelli, Henson, and Sorkin might be taken to suggest that random sprinklings of points in Minkowski space define a discrete model that is provably Poincaré invariant in a natural sense. We examine this possibility here. We argue that a genuinely Poincaré invariant model requires a probability distribution on sprinklable sets -- Poincaré orbits of sprinklings -- rather than individual sprinklings. The corresponding $σ$-algebra contains only sets of measure zero or one. This makes testing the hypothesis of discrete Poincaré invariance problematic, since any local violation of Poincaré invariance, however gross and large scale, is possible, and cannot be said to be improbable. We also note that the Bombelli-Henson-Sorkin argument, which rules out constructions of preferred timelike directions for typical sprinklings, is not sufficient to establish full Lorentz invariance. For example, once a pair of timelike separated points is fixed, a preferred spacelike direction {\it can} be defined for a typical sprinkling, breaking the remaining rotational invariance.