Causal set theory models spacetime by locally finite partially ordered sets (posets). Minz (2024) proved that Poisson sprinkles into Minkowski spacetime are, with probability 1, 'total locally unsymmetric' — no deletion of finitely many elements can create a local symmetry — while combinatorially generic finite posets (Kleitman–Rothschild orders, which dominate the enumeration of posets) can carry such symmetries and have only a few 'layers'. This poses an unresolved program with three linked questions: (1) whether total local unsymmetry is necessary, sufficient, or characteristic for a causal set to be 'manifold-like' (a faithful discrete approximation of a spacetime manifold), so that it can serve as a structural diagnostic selecting continuum-approximating causets from generic orders; (2) whether total local unsymmetry rules out infinite posets with only finitely many layers, a conjecture stated without proof; and (3) how far the probabilistic nonexistence of local symmetries extends from Minkowski spacetime to other physically relevant spacetimes such as cosmological models and black-hole backgrounds. All three questions remain open.

Background

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete: a causal set (or causet) is a locally finite partially ordered set (poset) whose order relation represents the causal relations between events (Bombelli, Lee, Meyer and Sorkin, 1987). Locally finite means every interval — the set of elements between two comparable elements — is finite. A central conjecture of the approach is that a suitable ensemble of causets corresponds to a spacetime manifold, with all continuum properties, including the spacetime symmetries, recoverable from the partial orders of the ensemble's members. The causets that represent a given spacetime manifold MM are usually generated by sprinkling, a Poisson point process at density ρ\rho on MM that keeps the selected points and their causal relations: the probability of finding exactly nn sprinkled points in a compact region KK of volume ν(K)\nu(K) is e−ρν(K)(ρν(K))n/n!e^{-\rho\nu(K)}(\rho\nu(K))^n/n! (Fewster, Hawkins, Minz and Rejzner, 2021). A well-known tension is that each individual sprinkle manifestly fails to carry the continuum's global symmetries (for example, no random point set in Minkowski spacetime is invariant under a nontrivial Poincaré transformation); symmetry is recovered only statistically, at the level of the ensemble. In this setting one may ask which combinatorial properties of individual causal sets are compatible with being a faithful discrete model of a continuum spacetime — such causets are called manifold-like (formally, a causet that admits a faithful embedding into a Lorentzian manifold, i.e. an order-preserving injection whose image is a Poisson sprinkling up to the discreteness scale). Symmetries of a poset are its automorphisms, i.e. bijections of the element set that preserve the order. Minz (2024) developed a systematic framework for the symmetries of finite and infinite posets. Two elements of a poset are singleton-symmetric if their sets of immediate predecessors and successors (their link sets) coincide; more generally, a poset is locally (Q,r)(Q,r)-symmetric if it admits a nontrivial symmetry quotient, a retract obtained by collapsing rr occurrences of a fixed pattern poset QQ (for instance a two-element chain), and locally unsymmetric if it admits no such retract. For finite posets, being locally unsymmetric is equivalent to having a trivial automorphism group. Two stability refinements are needed for infinite posets: a poset is kk-stable locally unsymmetric if deleting any subset of at most kk elements leaves a locally unsymmetric poset, and total locally unsymmetric if this holds for every finite kk — that is, no finite deletion can ever create a local symmetry. The physical contrast is with generic posets. A classical enumeration theorem of Kleitman and Rothschild (1975) implies that asymptotically almost all finite posets have only three layers, where the number of layers of a poset is the length of its longest chain — informally, the number of 'instances of time'. Such Kleitman–Rothschild orders do not resemble sprinkles into any large spacetime region, which have many layers, and they are known to be dynamically suppressed by the actions used in the causal-set path integral (Loomis and Carlip, 2018; Carlip, Carlip and Surya, 2023, 2024). Minz (2024) adds a structural counterpart on the sprinkled side: with probability 1, a Poisson sprinkle into dd-dimensional Minkowski spacetime is total locally unsymmetric. The proof is a volume-divergence argument: for any two distinct sprinkled points xx and yy whose futures could be made to agree, the causal interval ItI_t from xx toward the relevant future direction grows without bound in volume as t→∞t \to \infty, hence contains infinitely many sprinkled points with probability 1, so the set that would have to be deleted to make xx and yy locally symmetric cannot be finite. What the proof does not deliver — and what the source explicitly leaves open — is whether this absence of local symmetries can be turned around into a criterion for being manifold-like, whether it constrains the layer structure of infinite posets on purely combinatorial grounds, and whether it survives the transition from flat to curved spacetimes.

Problem Statement

The common objective is to determine whether the absence of local symmetries — specifically the property of total local unsymmetry, meaning that no local symmetry (in the retract-based sense: a singleton-symmetry or, more generally, a nontrivial (Q,r)(Q,r)-symmetry quotient) can be created in a locally finite poset by deleting any finite set of elements — can serve as a structural diagnostic that separates continuum-approximating (manifold-like) causal sets from combinatorially generic posets, and to establish the order-theoretic and spacetime-geometric limits of this diagnostic. Within this objective, the source poses three linked questions, all open: (1) Characterization: is total local unsymmetry necessary, sufficient, characteristic, or neither for a causal set to be manifold-like in the sense of faithfully approximating a spacetime manifold, as sprinkles do — i.e. does the property select continuum-approximating causets from generic orders? (2) Layer structure: does total local unsymmetry rule out infinite posets with a finite number of layers, where the number of layers is the length of the poset's longest chain (the author conjectures that it may, but provides no proof)? (3) Scope of the probabilistic theorem: to which other physically relevant spacetimes — such as cosmological models or black-hole spacetimes — does the theorem that a Poisson sprinkle is total locally unsymmetric with probability 1 generalize? Each of (1)–(3) can be addressed independently: resolving any one of them is a partial result, and the full task requires addressing all three. Within (1), its four alternatives are separately resolvable: establishing or refuting any one of necessity, sufficiency, or characteristic is itself a partial result, and (1) is resolved once the status of total local unsymmetry among the four alternatives is determined.

References

  1. Christoph Minz, 'Do causal sets have symmetries?', arXiv:2410.02862 (2024), https://arxiv.org/abs/2410.02862 (contribution to the 17th Marcel Grossmann Meeting)
  2. Christoph Minz, 'Local symmetries in partially ordered sets', arXiv:2406.14533 (2024), https://arxiv.org/abs/2406.14533
  3. Christopher J. Fewster, Eli Hawkins, Christoph Minz, Kasia Rejzner, 'Local structure of sprinkled causal sets', Physical Review D 103, 086020 (2021), DOI 10.1103/PhysRevD.103.086020, https://doi.org/10.1103/PhysRevD.103.086020
  4. Peter Carlip, Steve Carlip, Sumati Surya, 'Path integral suppression of badly behaved causal sets', Classical and Quantum Gravity 40, 095004 (2023), DOI 10.1088/1361-6382/acc50c, https://doi.org/10.1088/1361-6382/acc50c
  5. Peter Carlip, Steve Carlip, Sumati Surya, 'The Einstein–Hilbert action for entropically dominant causal sets', Classical and Quantum Gravity 41, 145005 (2024), DOI 10.1088/1361-6382/ad506e, https://doi.org/10.1088/1361-6382/ad506e
  6. Luca Bombelli, Joohan Lee, David Meyer, Rafael D. Sorkin, 'Space-time as a causal set', Physical Review Letters 59, 521 (1987), DOI 10.1103/PhysRevLett.59.521, https://doi.org/10.1103/PhysRevLett.59.521
  7. D. J. Kleitman, B. L. Rothschild, 'Asymptotic enumeration of partial orders on a finite set', Transactions of the American Mathematical Society 205, 205–220 (1975), DOI 10.1090/S0002-9947-1975-0369090-9, https://doi.org/10.1090/S0002-9947-1975-0369090-9
  8. S. P. Loomis, S. Carlip, 'Suppression of non-manifold-like sets in the causal set path integral', Classical and Quantum Gravity 35, 024002 (2018), DOI 10.1088/1361-6382/aa980b, https://doi.org/10.1088/1361-6382/aa980b