Discreteness without symmetry breaking: a theorem

TL;DR

This paper proves that discrete structures derived from Poisson sprinklings in Minkowski space cannot break Lorentz symmetry, ensuring no preferred frames or Lorentz-violating effects arise from such discretizations.

Contribution

It establishes a fundamental theorem showing the impossibility of Lorentz symmetry breaking through intrinsic discrete structures in Minkowski space.

Findings

No equivariant measurable map from sprinklings to spacetime directions exists.

Discrete causal sets do not produce Lorentz-violating effects.

Finite-valency graphs cannot be Lorentz-invariantly associated with sprinklings.

Abstract

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to ``Lorentz breaking'' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.