Causal Set Theory

Causal set theory, pioneered by Bombelli, Lee, Meyer, and Sorkin, rests on a simple but profound idea: the deep structure of spacetime is a locally finite partial order. A causal set $(\mathcal{C}, \preceq)$ is a set equipped with a partial order that is:

• Reflexive: $x \preceq x$ for all $x \in \mathcal{C}$
• Antisymmetric: $x \preceq y$ and $y \preceq x$ implies $x = y$
• Transitive: $x \preceq y$ and $y \preceq z$ implies $x \preceq z$
• Locally finite: $|\{z : x \preceq z \preceq y\}| < \infty$ for all $x, y$

The local finiteness condition is the key departure from the continuum: it implies a fundamental discreteness at the Planck scale. The number of causal set elements in a spacetime region determines its volume:

where $N(\mathcal{R})$ is the number of elements in region $\mathcal{R}$ and $d$ is the spacetime dimension. This is the “order plus number equals geometry” motto.

The central conjecture (Hauptvermutung) of causal set theory states that the causal set faithfully encodes the continuum geometry. More precisely, a theorem by Malament shows that for distinguishing spacetimes, the causal order determines the conformal geometry:

The causal structure determines the metric up to a conformal factor $\Omega^2$. The missing volume information is then supplied by counting elements. The sprinkling process generates a causal set from a Lorentzian manifold $(M, g)$ by placing points via a Poisson process with density $\rho = 1/\ell_P^d$:

The Hauptvermutung asserts that the resulting causal set uniquely determines $(M, g)$ up to isometry, with high probability for large enough sprinklings. This has been proven in various special cases.

A critical test of causal sets is recovering the spacetime dimension from purely order-theoretic data. The Myrheim-Meyer estimator uses the ratio of causal relations to elements. For $N$ elements sprinkled into a $d$-dimensional Alexandrov interval:

where $R$ is the number of related pairs. For $d = 2$, this ratio is $1/4$; for $d = 4$, it is $4!/\sqrt{\pi} \cdot \Gamma(2)/(4\,\Gamma(6)) \approx 1/3$. Inverting this relation gives a dimension estimator purely from counting causal relations.

To formulate dynamics, one needs a discrete action. The Benincasa-Dowker (BD) action is a causal set analogue of the Einstein-Hilbert action. For a causal set $\mathcal{C}$ in $d = 2$ dimensions:

where $N_k(x)$ is the number of $k$-element chains in the past of $x$ (intervals of length $k$). In $d = 4$ dimensions, the action becomes:

with specific coefficients $\alpha_k$ chosen so that $\langle S_{\rm BD}^{(4)} \rangle \to \int \sqrt{-g}\,R\,d^4x$ when averaged over sprinklings of a curved spacetime. The key result of Benincasa and Dowker is:

Sorkin and Rideout proposed a stochastic dynamics for causal set growth: elements are “born” one at a time, each new element choosing its causal past according to a probability distribution. The classical sequential growth (CSG) models satisfy general covariance (label invariance) and Bell causality. The transition probability for adding element $n+1$ to an $n$-element causet with past set $p$ is:

where $|p|$ is the cardinality of the past and $t_k$ are coupling constants. The full quantum dynamics would use a path sum over causets weighted by $e^{iS_{\rm BD}}$:

A remarkable prediction of causal set theory is a small but nonzero cosmological constant. Sorkin argued that spacetime discreteness at scale $\ell_P$ implies fluctuations in $\Lambda$ of order $\Lambda \sim 1/\sqrt{V_4}$, where $V_4$ is the 4-volume in Planck units. This gave a prediction of $\Lambda \sim 10^{-120}\,\ell_P^{-2}$ before the observational discovery of dark energy.

We generate a random causal set by sprinkling $N = 80$ points in a 1+1D Minkowski diamond, compute the causal relations, extract chain length statistics for dimension estimation, and evaluate the Benincasa-Dowker action: