Quantum Dynamics and the BDG Action
A discrete analogue of Einstein's action for causal sets. The Benincasa-Dowker-Glaser (BDG) action uses purely combinatorial data to approximate spacetime curvature; the quantum theory is the path integral over causal sets weighted by this action. 2D quantum gravity simulations are the modern computational test.
Placeholder for a 3D visualisation of Causal Set Theory. The interactive scene will land in Phase 3. Causal Set Theory proposes that space and time are not smooth all the way down. The fundamental structure of the universe is a discrete partial order: a collection of elementary events ordered by which ones can causally influence which others. From a distance, this discrete web of cause-and-effect looks like a smooth Lorentzian spacetime, much as a pixelated image looks smooth when zoomed out. The foundational slogan is order plus number equals geometry: if you know which events causally precede which others (the partial order) and how many elements there are (the count), you have reconstructed the shape and size of spacetime up to a Lorentz-invariant random sprinkling. Bombelli, Lee, Meyer, and Sorkin's 1987 PRL is the foundational paper. The program has developed in five overlapping threads: classical stochastic dynamics for how a causal set grows (Rideout-Sorkin 1999), quantum dynamics via a discrete action (Benincasa-Dowker 2010 BDG action), continuum reconstruction (manifold sprinkling, Myrheim-Meyer dimension estimators), and empirical handles (Lorentz-invariant swerves, Sorkin's 1991 prediction of the cosmological constant magnitude). After almost 40 years the program is technically rich and structurally distinctive among quantum-gravity proposals, but unconfirmed empirically and unresolved on its foundational inverse problem.
In one sentence
The Benincasa-Dowker action (2010) provides a discrete analogue of Einstein's action: a formula that uses counts of certain suborder patterns in a causal set to approximate the scalar curvature governing Einstein's general-relativity action. The quantum theory of causal sets is then the path integral over all possible causal sets, weighted by this action. Modern 2D causal set quantum gravity simulations (Glaser-Surya 2013-2018) test the construction by showing that quantum causal sets self-assemble into structures resembling continuous physical space.
The claim
Benincasa and Dowker's 2010 PRL provided the central technical anchor of the quantum causal-set program: a discrete analogue of Einstein's general-relativity action. The classical Einstein-Hilbert action is the integral of the scalar curvature over spacetime volume, which is the geometric quantity whose extremization gives the Einstein field equations. Benincasa and Dowker showed that the same scalar curvature can be approximated using purely combinatorial data from a causal set: specifically, counts of certain ordered-pair patterns in the partial order. The resulting BDG action depends only on the causal-set structure (no continuum metric required) and reduces to the continuum Einstein-Hilbert action in the appropriate sprinkling limit. This is the technical machinery that lets you formulate gravity quantum-mechanically on a fundamentally discrete background.
Once you have the action, the quantum theory is the sum-over-histories: a path integral over all possible causal sets, with each one weighted by exp(i times the BDG action divided by hbar). Modern computational work (Glaser-Surya 2013 on manifoldlike-locality criteria, Glaser 2014 on the closed-form BDG d'Alembertian, Glaser-Surya 2016 on 2D causal set quantum gravity) has tested this construction in low-dimensional toy models. The 2D quantum gravity simulations are the variant's modern frontier: numerical sums over 2D causal sets, weighted by the BDG action, do produce structures resembling continuous physical space rather than collapsing into pathological non-manifoldlike configurations. This is structural consistency evidence that the BDG action plays the central role the program needs it to play. The variant's editorial center is the BDG action as the unifying technical claim; 2D quantum gravity is the modern computational test; entropic dominance is the main objection.
The entropic dominance problem (the variant's central objection) is sharp. The number of causal sets with n elements grows extremely fast with n, and the overwhelming majority of large random causal sets are Kleitman-Rothschild posets: highly tangled, layered structures with no continuum manifold approximation. The BDG action's central job is to suppress these non-manifold-like configurations in the path integral, so that the dominant contributions come from manifold-like causal sets that look like real spacetime. Whether the action succeeds at this suppression is unproven. The 2D simulations provide encouraging structural evidence, but generalization to 4D and to full quantum theory remains open. This is the technical question the field is most actively working on through 2026.
The family stance
Spacetime is fundamentally discrete. The continuum of general relativity is an emergent approximation to a finite causal set of elementary events at the Planck scale. Smooth geometry, distances, dimensions, and curvature can all be reconstructed from the causal partial order plus the element count. After almost 40 years of development the program has a complete kinematic framework (the causal set), a classical dynamics (sequential growth), a candidate quantum action (BDG), several continuum-reconstruction techniques, and two empirical handles (Lorentz tests, Λ fluctuations). The 1991 Sorkin prediction that Λ should be of order 10^-120 in Planck units agreed in order of magnitude with the 1998 dark-energy discovery seven years later. No proposal in the family has been confirmed observationally.
Predictions
- The BDG action can be defined for any causal set using purely combinatorial data (counts of certain ordered-pair patterns), with no continuum metric required
- The BDG action reduces to the continuum Einstein-Hilbert action when evaluated on causal sets that are sprinkled into a smooth manifold; this is the discrete-to-continuum correspondence that anchors the construction
- The quantum sum-over-histories with BDG-weighted path integral suppresses non-manifold-like configurations (Kleitman-Rothschild posets), at least in 2D toy models; whether this generalizes to 4D is the central open question
- 2D causal set quantum gravity, as computed via path-integral simulations, reproduces structures resembling 2D physical space; this is structural consistency evidence for the construction
Evidence
- Benincasa-Dowker 2010 (133 cites) established the BDG action as a well-defined discrete analogue of Einstein's action, with explicit demonstration that it reduces to the Einstein-Hilbert action in the continuum limit
- Glaser-Surya 2013 (47 cites) developed manifoldlike-locality criteria that give concrete tests for whether a causal set looks like it came from a smooth manifold; the framework is used throughout subsequent computational work
- Glaser 2014 (45 cites) provided the closed-form expression for the causal set d'Alembertian (the discrete analogue of the wave operator), enabling explicit calculations on specific causal sets
- Glaser-Surya 2016 (30 cites) demonstrated finite-size scaling in 2D causal set quantum gravity: the path-integral simulations show non-trivial structure consistent with continuum physical space, providing the first computational evidence that the BDG quantum theory has the correct continuum limit in low dimensions
Counterpoints
- Entropic dominance (Kleitman-Rothschild posets, see family-level objection 3). Whether the BDG action successfully suppresses non-manifold-like causal sets in the full quantum theory is unproven; if it does not, the quantum theory's predictions are dominated by structures with no continuum interpretation
- Generalization beyond 2D. All existing path-integral simulations are in 2D. Whether the construction succeeds in 4D (the physically relevant case) is the central open computational question
- The BDG action is constructed to reproduce the continuum limit, not derived from a deeper principle. Whether it captures the full quantum theory beyond the continuum limit, or is only an effective construction good for one purpose, is contested
- Boundary terms. The original BDG action was defined for causal sets without spatial boundaries. Extensions to finite causal sets with boundaries have been worked out in follow-up papers; whether the boundary terms preserve the discrete-to-continuum correspondence is technically delicate
Variants in this family
▸Go deeperTechnical detail with proper terminology
BDG action structure: the discrete action is a sum over the causal set of terms that count specific local ordered-pair patterns. For each element, the action counts certain configurations in its causal past (intervals of given size; pairs with specific relative positions). The total is a discrete approximation to the integral of the scalar curvature over spacetime volume. Benincasa-Dowker 2010 showed that for sprinklings of a flat Minkowski region the discrete action equals zero (matching continuum flat-space Einstein-Hilbert), and that for curved sprinklings the discrete value approximates the continuum curvature integral.
Kleitman-Rothschild posets: random causal sets with n elements (uniformly distributed over all possible partial orders) are overwhelmingly dominated by 3-layered structures with about half the elements in a middle layer. These structures have no continuum manifold approximation and represent the entropic-dominance problem. The BDG action's role is to suppress them in the path integral. Whether the suppression is strong enough depends on the ratio of the BDG action's effective scale to the natural growth rate of Kleitman-Rothschild numbers.
2D simulations: the canonical computational setup uses 2D causal sets with hundreds to thousands of elements, with the path integral over the BDG action computed via Markov-chain Monte Carlo (MCMC) or related techniques. Results show that for appropriate values of the BDG action's parameters, the path integral is dominated by manifoldlike causal sets that approximate continuous 2D spacetime. The Glaser-Surya 2016 paper is the canonical computational reference.
Connection to other discrete-spacetime programs: the BDG approach is structurally different from Loop Quantum Gravity (which discretizes spatial geometry directly) and Causal Dynamical Triangulations (which sums over geometrically-discrete piecewise-flat spacetimes). The CST approach is more parsimonious (causal order is the only input) but technically less mature than CDT or LQG in the sense that the 4D version is not yet computationally feasible.
References
- EstablishedBenincasa & Dowker (2010). The Scalar Curvature of a Causal Set. Phys. Rev. Lett. 104, 181301
- EstablishedGlaser & Surya (2013). Towards a Definition of Locality in a Manifoldlike Causal Set. Phys. Rev. D 88, 124026
- EstablishedGlaser (2014). A closed form expression for the causal set d'Alembertian. Class. Quant. Grav. 31, 095007
- EstablishedGlaser & Surya (2016). Finite size scaling in 2D causal set quantum gravity. Class. Quant. Grav. 35, 045006
Last reviewed May 19, 2026
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