Discrete, Causal, and Combinatorial Spacetime

Discrete quantum-spacetime programmes do not all discretize the same object. A causal set keeps order and counting; Regge calculus keeps a simplicial complex with edge lengths; causal dynamical triangulations sum causally foliated triangulations with fixed simplex geometry; tensor models and group field theories generate complexes through interaction combinatorics. Comparing them therefore requires common questions about kinematics, dynamics, continuum recovery, observables, and evidence—not a single slogan that spacetime is “made of atoms.”

Helpful background. Flat-Spacetime Quantum Field Theory supplies path integrals and renormalization; Einstein Equations supplies the continuum target; Quantum Information supplies state, correlation, and coarse-graining language; Quantum Gravity supplies common evidence standards; and Causality and Global Structure supplies causal order and global-hyperbolicity concepts.

Four ways to encode geometry

A causal set is a locally finite partially ordered set $(C,\prec)$. The order is intended to encode causal precedence, while cardinality estimates spacetime volume. A faithful continuum approximation additionally requires an order-preserving embedding whose points are distributed by a Lorentz-invariant Poisson sprinkling. Order plus number recovers Lorentzian geometry only within a manifoldlike approximation class; most finite partial orders are not known to approximate smooth spacetimes Surya 2019, §§2–3.

Regge calculus fixes a triangulation and assigns squared lengths to edges. Each simplex is flat, while curvature is concentrated on codimension-two hinges and measured by a deficit angle. It is a coordinate-free discretization of classical general relativity and a starting point for discrete path integrals, not by itself a microscopic ontology Regge 1961, §§2–5.

Causal dynamical triangulations (CDT) instead fixes a small set of Lorentzian simplex types, preserves a discrete proper-time foliation and spatial topology, sums triangulations with a Regge weight, and then performs a controlled analytic continuation for numerical sampling. Its degrees of freedom are gluings, not continuously variable edge lengths. Extended four-dimensional phases and scale-dependent spectral observables are numerical ensemble results, conditional on bare couplings, volume fixing, moves, and finite-size analysis Loll 2020, §§2–5.

Tensor models and group field theories generate labeled cellular complexes as Feynman diagrams. Tensor index contractions prescribe face identifications; group variables can add parallel transport and geometric data. A large-$N$ expansion orders combinatorial topology, while condensate and renormalization programmes seek collective continuum phases. Neither a dominant graph family nor a condensate mean field is automatically a four-dimensional Lorentzian spacetime.

Dynamics is more than a configuration space

The chapter moves from kinematics to increasingly dynamical questions:

1. Causal Set Kinematics tests order intervals, dimension estimators, and faithful embedding.
2. Causal Set Dynamics and Sequential Growth studies label-independent stochastic dynamics and the gap to a quantum measure.
3. Regge Calculus derives discrete curvature and field equations on a fixed complex.
4. Causal Dynamical Triangulations treats an ensemble computationally and asks which transition could define a continuum limit.
5. Tensor Models and Group Field Theory connects interaction vertices to gluing and large-$N$ organization.
6. Universality and Continuum Recovery asks which observables become independent of microscopic choices.

The distinctions prevent category errors. Classical sequential growth supplies probabilities over finite causal sets, not unitary quantum amplitudes. A Regge solution converging under mesh refinement validates a discretization, not a sum over geometries. A CDT phase with macroscopic volume profile is stronger than a single triangulation but still needs continuum scaling. A tensor-model $1/N$ expansion controls combinatorial complexity, not automatically signature, causality, or Einstein dynamics.

Common continuum tests

Every programme must specify a family of regulators or system sizes and a map between scales. Useful tests include:

• recovery of dimension from several independent observables rather than one fitted exponent;
• emergence of a local effective action with the Einstein term and controlled corrections;
• correct causal propagation and two long-distance gravitational polarizations;
• correlation lengths large compared with the microscopic spacing;
• stability under changes of triangulation, order estimator, measure, truncation, or coarse-graining scheme; and
• observables defined without reference to unphysical labels.

No single diagnostic is sufficient. Many nonmanifoldlike structures can share an ordering fraction with a low-dimensional sprinkling. Distinct graph ensembles can share a spectral dimension over a finite diffusion window. A second-order transition is a candidate route to a continuum limit only if physical correlation lengths diverge and renormalized observables approach regulator-independent values. Universality is an output to demonstrate, not an assumption licensed by discreteness.

Results and open questions as of 2026-08-20

Programme	Strongest controlled result or structure	Limitation and competing interpretation
Causal sets	Lorentz-invariant Poisson sprinkling preserves no preferred inertial frame, and order estimators recover geometry on known sprinklings; Surya 2019, §§2–4	Manifoldlikeness and a completed quantum dynamics remain open; kinematics may be fundamental or only an approximation tool
Regge calculus	On a fixed piecewise-flat complex, deficit angles encode integrated curvature and variation of the Regge action gives discrete vacuum equations; Regge 1961, §§2–5	A suitable continuum approximation is argued, but general convergence does not follow from the original derivation; no microscopic path-integral measure is selected
CDT	Monte Carlo ensembles exhibit extended four-dimensional phases and nontrivial scale-dependent observables; Loll 2020, §§4–6	Continuum order, finite-size scaling, Wick rotation, and universality require further control
Colored tensor models	The arbitrary-rank $1/N$ expansion is ordered by Gurau degree, and the simplest leading critical sector is melonic; Gurau 2012, Eq. (16), Definition 2, and Lemma 4, Bonzom et al. 2011, §§ III–V and Propositions 3–5	Leading melonic geometry is not generic four-dimensional Einstein spacetime; double scaling and interactions change the comparison
Group field theory	Feynman amplitudes generate simplicial/spin-foam structures and condensates provide collective cosmological sectors; Oriti 2006, §§2–5	Condensate truncation, clock choice, renormalization, and connection to local observables remain conditional

The permissive view is that causal restrictions, critical phenomena, and many-body coarse graining provide complementary routes to a universal continuum phase. The conservative view accepts the exact combinatorics and finite-model results but demands a regulator-independent four-dimensional observable and the Einstein effective theory before claiming spacetime recovery. A skeptical view treats dominance by nonmanifoldlike orders, branched polymers, or melonic graphs as possible structural failures. The generic four-dimensional continuum limit remains open, and the evidence does not yet select a unique microscopic programme.

String and holographic constructions belong to their dedicated volume. For semiclassical thermodynamics and classical alternative gravity, consult their respective volumes; for generic field-theory methods, see Flat-Spacetime Quantum Field Theory. The conclusions could change after a replicated continuum critical point yields common renormalized observables, a theorem changes a manifold-reconstruction or universality claim, an adverse phase persists under enlarged ensembles, or evidence after 2026-08-20 changes the balance.

Diagnostic: order is not a drawing

Consider four elements with covering relations $a\prec b$, $a\prec c$, $b\prec d$, and $c\prec d$. Is this a causal set? What is the open interval $I(a,d)$, and does deleting a drawn direct line from $a$ to $d$ change the causal relation?

Solution

Taking the transitive closure adds $a\prec d$. The relation is irreflexive, transitive, and locally finite because every interval is finite, so it is a finite causal set. The open order interval is

$$I(a,d)=\{x\in C:a\prec x\prec d\}=\{b,c\}.$$

A Hasse diagram draws only covering relations, so it normally omits a direct $a$–$d$ edge. Deleting or adding such a redundant drawn line does not alter the partial order after transitive closure. This is why graph adjacency is not itself causal-set geometry: the physically proposed datum is the complete order, and number supplies the volume proxy.

References

• Bonzom, Valentin, Razvan Gurau, Aldo Riello, and Vincent Rivasseau. “Critical Behavior of Colored Tensor Models in the Large $N$ Limit.” Nuclear Physics B 853 (2011): 174–195. DOI. Open PDF.
• Gurau, Razvan. “The Complete $1/N$ Expansion of Colored Tensor Models in Arbitrary Dimension.” Annales Henri Poincaré 13 (2012): 399–423. DOI. Open PDF.
• Loll, Renate. “Quantum Gravity from Causal Dynamical Triangulations: A Review.” Classical and Quantum Gravity 37 (2020): 013002. Open review.
• Oriti, Daniele. “The Group Field Theory Approach to Quantum Gravity.” In Approaches to Quantum Gravity (2009). Open review.
• Regge, Tullio. “General Relativity without Coordinates.” Nuovo Cimento 19 (1961): 558–571. DOI.
• Surya, Sumati. “The Causal Set Approach to Quantum Gravity.” Living Reviews in Relativity 22 (2019): 5. Open review.