Causal-Set Kinematics and Dynamics

A causal set is a locally finite partial order intended to retain Lorentzian causal order and spacetime volume. Poisson sprinkling into a continuum spacetime gives Lorentz-invariant kinematics and testable dimension estimators. It does not supply the dynamics: a successful theory must assign amplitudes or probabilities that favor manifoldlike orders and recover causal propagation.

Required background. Microcausality and Relativistic Compatibility supplies the causal target. Relational, Boundary, and Asymptotic Observables supplies the observable standard.

Helpful background. Limits, Completeness, and Modes of Convergence and Green Operators, Causal Propagators, and State-Dependent Two-Point Functions supply continuum comparisons.

Order, volume, and faithful sprinkling

A causal set $(C,\prec)$ is transitive, irreflexive, and locally finite: every interval

$$I(x,y)=\{z\in C:x\prec z\prec y\}$$

has finite cardinality. A Poisson sprinkling of density $\rho$ into region $V$ has

$$\Pr(N=n)=e^{-\rho V}\frac{(\rho V)^n}{n!}.$$

The order is inherited from continuum causality, while counting estimates volume as $V\simeq N/\rho$. The Poisson process introduces no preferred inertial lattice frame Bombelli et al. 1987.

Application: dimension and wave-operator tests

Sprinkle $N$ points into a flat Alexandrov interval and count related unordered pairs $R$. Define the ordering fraction

$$r=\frac{2R}{N(N-1)}.$$

Its continuum expectation in dimension $d$ is

$$r_d=\frac{\Gamma(d+1)\Gamma(d/2)} {2\Gamma(3d/2)}.$$

Thus $r_2=1/2$ and $r_4=1/10$. Invert $r_d$ to obtain the Myrheim–Meyer dimension, repeat over nested intervals, and quote Poisson and boundary errors.

A retarded discrete wave operator uses order layers $L_k(x)$:

$$(B_\rho\phi)(x)=\rho^{2/d} \left[a_d\phi(x)+\sum_{k=1}^{K}b_{k,d} \sum_{y\in L_k(x)}\phi(y)\right].$$

Coefficients are fixed so that its sprinkling expectation approaches $\Box\phi$ plus curvature terms as $\rho\to\infty$ Benincasa and Dowker 2010. Test constants, linear fields, and a retarded Green response, not only the dimension estimator.

Dynamics and manifoldlikeness

Classical sequential growth models assign label-independent transition probabilities subject to causal conditions Rideout and Sorkin 2000. Path-integral proposals instead weight causal sets by discrete actions. Generic finite partial orders are not manifoldlike, so dynamics must suppress the enormous nonmanifoldlike entropy while retaining an appropriate continuum phase.

Adversarial tests

Change the sprinkling region while holding density fixed, remove boundary layers, and vary the nonlocality scale of $B_\rho$. Dimension and propagator estimates should converge. Then sample from the proposed dynamics rather than from an imposed continuum sprinkling. If manifoldlikeness disappears, the success was kinematic input, not dynamical recovery.

Scope

Causal order plus number determine continuum conformal geometry and volume under suitable manifold assumptions, but do not prove those assumptions for a random causal set. Cross-program continuum criteria are applied on Emergence and Continuum-Limit Tests.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Benincasa, D. M. T., and F. Dowker. “The Scalar Curvature of a Causal Set.” Physical Review Letters 104 (2010): 181301. DOI.
• Bombelli, L., J. Lee, D. Meyer, and R. D. Sorkin. “Space-Time as a Causal Set.” Physical Review Letters 59 (1987): 521–524. DOI.
• Rideout, D. P., and R. D. Sorkin. “A Classical Sequential Growth Dynamics for Causal Sets.” Physical Review D 61 (2000): 024002. DOI.