The causal set approach to quantum gravity

TL;DR

Causal set theory proposes that spacetime is fundamentally discrete and Lorentzian, encoded as locally finite posets where order supplies causality and element count supplies volume. The continuum is recovered via Poisson sprinkling into manifolds, with the Hauptvermutung conjecturing a near-uniqueness of the continuum corresponding to a given causal set at a fixed density. Dynamics are pursued through classical sequential growth and quantum measure formalisms, with a continuum-inspired BD action providing an effective path integral that respects Lorentz invariance and nonlocality. The framework yields concrete results on dimension estimators, topological reconstruction, scalar-field propagation (including the SJ vacuum), and cosmological implications such as Lambda fluctuations, highlighting both promising prospects and key open questions for a complete quantum gravity theory.

Abstract

The causal set theory (CST) approach to quantum gravity postulates that at the most fundamental level, spacetime is discrete, with the spacetime continuum replaced by locally finite posets or "causal sets". The partial order on a causal set represents a proto-causality relation while local finiteness encodes an intrinsic discreteness. In the continuum approximation the former corresponds to the spacetime causality relation and the latter to a fundamental spacetime atomicity, so that finite volume regions in the continuum contain only a finite number of causal set elements. CST is deeply rooted in the Lorentzian character of spacetime, where a primary role is played by the causal structure poset. Importantly, the assumption of a fundamental discreteness in CST does not violate local Lorentz invariance in the continuum approximation. On the other hand, the combination of discreteness and Lorentz invariance gives rise to a characteristic non-locality which distinguishes CST from most other approaches to quantum gravity. In this review we give a broad, semi-pedagogical introduction to CST, highlighting key results as well as some of the key open questions. This review is intended both for the beginner student in quantum gravity as well as more seasoned researchers in the field.

Figures (22)

• Figure 1: The local lightcone of a Lorentzian spacetime.
• Figure 2: An example of a signature $(-,-,+,+)$ spacetime with one spatial dimension suppressed. It is not possible to distinguish a past from a future timelike direction and hence order events, even locally.
• Figure 3: The transitivity condition $x \prec y, y \prec z \Rightarrow x\prec z$ is satisfied by the causality relation $\prec$ in any Lorentzian spacetime.
• Figure 4: The Hasse diagrams of some simple finite cardinality causal sets. Only the nearest neighbour relations or links are depicted. The remaining relations are deduced from transitivity.
• Figure 5: The lightcone lattice in $d=2$. The lattice on the left looks "regular" in a fixed frame but transforms into the "stretched" lattice on the right under a boost. The $n \sim \rho_cV$ correspondence cannot be implemented as seen from the example of the Alexandrov interval, which contains $n=7$ lattice points in the lattice in the left but is empty after a boost.

Theorems & Definitions (1)

• theorem 2