Are There Testable Discrete Poincar\'e Invariant Physical Theories?

TL;DR

This paper investigates whether discrete models of spacetime, like random sprinklings in Minkowski space, can be empirically distinguished from continuous models, highlighting the challenges in testing discrete Poincaré invariance.

Contribution

It argues that true Poincaré invariance in discrete models requires probability distributions over entire orbits, making local violations untestable and challenging to empirically confirm or refute.

Findings

Discrete Poincaré invariance involves measure-zero sets, complicating empirical tests.

Typical sprinklings do not establish full Lorentz invariance due to possible preferred directions.

The argument against preferred timelike directions does not imply complete Lorentz invariance.

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 Poincar\'e invariant in a natural sense. We examine this possibility here. We argue that a genuinely Poincar\'e invariant model requires a probability distribution on sprinklable sets -- Poincar\'e orbits of sprinklings -- rather than individual sprinklings. The corresponding $\sigma$-algebra contains only sets of measure zero or one. This makes testing the hypothesis of discrete Poincar\'e invariance problematic, since any local violation of Poincar\'e 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.