Discreteness Without Symmetry Breaking: A Theorem

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.

Scalar Curvature of a Causal Set

“…Where curvature limits Lorentz symmetry, it may render the number of nearest neighbours finite but it will still be huge so long as the radius of curvature is large compared to the Planck length. Causal set theory is a discrete approach to quantum gravity which embodies Lorentz symmetry [1,2] and exhibits nonlocality of exactly this form [3,4].…”

“…These are to explain (1) how the fundamental dynamics picks out a discrete structure that is well approximated by a Lorentzian manifold and (2) why, in that case, the geometry should be a solution of the Einstein equations. This is often referred to as the problem of the continuum limit but in the context of a fundamentally discrete theory in which the discreteness scale is fixed and is not taken to zero but rather the observation scale is large, it is more accurately described as the problem of the continuum approximation.…”

Causal sets: Quantum gravity from a fundamentally discrete spacetime

“…Causal sets are constructed in such a way, that they respect Lorentz invariance at the kinematic level already 8 . These two observations are analyzed in detail in [10].…”

“…There are two ways to try and adopt the dynamics. The first 14 takes as fundamental ingredient the 10 To be more precise it is proportional and the proportionality constant in general depends on the dimension 11 Basically, it was tested that causal sets that arose from sprinkling to some continuum manifold 12 in other words take an element z such that d t (x, z) = 2n and find |V xz | 13 Adding any other element makes our set stop being an anti-chain 14 causal set itself and attempts to generate the full causal set from some basic principles that are in some sense natural to the causal set approach, in other words simply related with the partial order. The second approach 15 , is an attempt to guess some dynamics (possibly by rephrasing the continuum Einstein action, in terms of the order relation) and get "correct" result 16 .…”

Causal sets and conservation laws in tests of Lorentz symmetry

“…The discrete model we will discuss for the rest of this paper, causal sets, is singled out among discrete models as it is constructed to be Lorentz invariant by definition. Recently, Sorkin, Bombelli, and Henson [18] proved that a causal set is Lorentz invariant for an abstract operator that represents a measurement of a preferred frame for a section of the causal set (the proof is that such operators cannot exist). In this work we argue that this operator D does not quite reflect the way that we currently analyze many real Lorentz violating experiments and that such experiments (in particular astrophysical tests) may theoretically show "spurious" Lorentz violating effects if the underlying spacetime is a causal set.…”