Causal Set Theory Resource Page

Foundations Historical Reviews Manifoldlikeness
Cosmological Constant Lorentz Invariance Black Hole Entropy Quantum Measure
Locality Dynamics Geometry Poset Theory
General Mathematics


  • D. Finkelstein, "The space-time code"Phys. Rev. 184:1261 (1969)(Foundations)

  • J. Myrheim, "Statistical geometry",  CERN preprint TH-2538 (1978) (Foundations, Historical)

  • L. Bombelli, J. Lee, D. Meyer and R.D. Sorkin, "Spacetime as a causal set",   Phys. Rev. Lett. 59:521-524 (1987); (Introduction, Foundations) 

  • C. Moore, "Comment on 'Space-time as a causal set' ",   Phys. Rev. Lett. 60, 655 (1988); (Foundations)

  • L. Bombelli, J. Lee, D. Meyer and R.D. Sorkin, "Bombelli et al. Reply," Phys. Rev. Lett. 60, 656 (1988); (Foundations)

  • L. Bombelli, "Space-time as a Causal Set",   PhD thesis (Syracuse University, 1987); (Introduction) 

  • D. Finkelstein, "Superconducting' Causal Nets" Int. J. Th. Phys 27:473(1988)(Foundations

  • R.D. Sorkin, "A Specimen of Theory Construction from Quantum Gravity", The Creation of Ideas in Physics: Studies for a Methodology of Theory Construction (Proceedings of the Thirteenth Annual Symposium in Philosophy, held Greensboro, North Carolina, March, 1989) pp. 167-179 (Number 55 in the University of Western Ontario Series in Philosophy of Science) (Kluwer Academic Publishers, Dordrecht, 1995) J. Leplin (ed.); gr-qc/9511063(Philosophical, Historical)

  • R.D. Sorkin, "Does a Discrete Order underly Spacetime and its Metric?",  Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics, (Victoria, Canada, May, 1989), edited by A. Coley, F. Cooperstock, B.Tupper, pp. 82-86, (World Scientific, 1990); (Introduction) 

  • R.D. Sorkin, "First Steps with Causal Sets",  General Relativity and Gravitational Physics, (Proceedings of the Ninth Italian Conference of the same name, held Capri, Italy, September, 1990), 68-90, (World Scientific, Singapore), (1991), R. Cianci, R. de Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (eds.);; (Introduction)

  • R.D. Sorkin, "Spacetime and Causal Sets",  Relativity and Gravitation: Classical and Quantum, (Proceedings of the SILARG VII Conference, held Cocoyoc, Mexico, December, 1990), pages 150-173, (World Scientific, Singapore, 1991), J.C. D’Olivo, E. Nahmad-Achar, M.Rosenbaum, M.P. Ryan, L.F. Urrutia and F. Zertuche (eds.);; (Introduction)

  • R.D. Sorkin, "Forks in the Road, on the Way to Quantum Gravity",  Talk given at the conference entitled “Directions in General Relativity”, held at College Park, Maryland, May, 1993, Int. J. Th. Phys. 36: 2759–2781 (1997); gr-qc/9706002; (Philosophical, Introduction)   


  • B. Riemann, ¨Uber die Hypothesen, welche der Geometrie zu Grunde liegen", The Collected Works of B. Riemann (Dover NY 1953);; (Historical)

  • A. Einstein,  "Letter to H.S. Joachim", August 14, 1954; Item 13-453 cited in J. Stachel,“Einstein and the Quantum: Fifty Years of Struggle”, in From Quarks to Quasars,Philosophical Problems of Modern Physics, edited by R.G. Colodny (U. Pittsburgh Press, 1986), pages 380-381; (Historical)

  • E.C. Zeeman, "Causality Implies the Lorentz Group",  J. Math. Phys. 5: 490-493;; (Historical, Foundations)

  • G.'t Hooft, "Quantum gravity: a fundamental problem and some radical ideas",  Recent Developments in Gravitation (Proceedings of the 1978 Cargese Summer Institute) edited by M. Levy and S. Deser (Plenum, 1979); (Introduction, Foundations, Historical)


  • R.D. Sorkin, "Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)", In Proceedings of the Valdivia Summer School, edited by A. Gomberoff and D. Marolf; gr-qc/0309009; (Introduction, Glossary)

  • F. Dowker,  "Causal sets and the deep structure of spacetime", gr-qc/0508109; (Introduction)

  • Joe Henson, "The causal set approach to quantum gravity", gr-qc/0601121; (Introduction, Overview)

  • L. Bombelli, "Causal Sets: Overview and Status",  Talk given at Quantum Gravity in the Americas III, August 24-26, 2006;; (Introduction, Overview)


  • R.D. Sorkin, "Posets as Lattice Topologies",  General Relativity and Gravitation, vol. I, 635-637 (Roma, Consiglio Nazionale Delle Ricerche, 1983), B. Bertotti, F. de Felice, A. Pascolini (eds.); (Topology)

  • D.A. Meyer, "The Dimension of Causal Sets"PhD thesis (M.I.T., 1988)(Dimension theory)

  • D.A. Meyer, "The Dimension of Causal Sets I: Minkowski dimension",   Syracuse University preprint (1988); (Dimension theory)

  • D.A. Meyer, "The Dimension of Causal Sets II: Hausdorff dimension",  Syracuse University preprint (1988); (Dimension theory)

  • L. Bombelli, R.D. Sorkin, " When are Two Lorentzian Metrics close?", General Relativity and Gravitation, proceedings of the 12th International Conference on General Relativity and Gravitation, held July 2-8, 1989, in Boulder, Colorado, USA, under the auspices of the International Society on General Relativity and Gravitation, 1989, p.220; (Closeness of Lorentzian manifolds)

  • R.D. Sorkin; A Finitary Substitute for Continuous Topology?Int. J. Theor. Phys. 30 7: 923-947 (1991); (Foundational)

  • L. Bombelli, "Causal sets and the closeness of Lorentzian manifolds",  Relativity in General: proceedings of the Relativity Meeting "93, held September 7-10, 1993, in Salas, Asturias, Spain. Edited by J. Diaz Alonso, M. Lorente Paramo. ISBN 2-86332-168-4. Published by Editions Frontieres, 91192 Gif-sur-Yvette Cedex, France, 1994, p. 249; (Closeness of Lorentzian manifolds)

  • D.A. Meyer, "Spherical containment and the Minkowski dimension of partial orders",  Order 10: 227-237 (1993); (Dimension theory) 

Cosmological Constant

  • Rafael D. Sorkin, "A Modified Sum-Over-Histories for Gravity”; reported in Highlights in gravitation and cosmology: Proceedings of the International Conference on Gravitation and Cosmology, Goa, India, 14-19 December, 1987, edited by B. R. Iyer, Ajit Kembhavi, Jayant V. Narlikar, and C. V. Vishveshwara, see pages 184-186 in the article by D. Brill and L. Smolin: “Workshop on quantum gravity and new directions”, pp  183-191 (Cambridge University Press, Cambridge, 1988); (PreObservation Cosmological Constant)

  • Rafael D. Sorkin; "On the Role of Time in the Sum-over-histories Framework for Gravity", paper presented to the conference on The History of Modern Gauge Theories, held Logan, Utah, July 1987; Int. J. Theor. Phys. 33 : 523-534 (1994); (PreObservation Cosmological Constant)

  • Rafael D. Sorkin, "First Steps with Causal Sets", in R. Cianci, R. de Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (eds.), General Relativity and Gravitational Physics (Proceedings of the Ninth Italian Conference of the same name, held Capri, Italy, September, 1990), pp. 68-90 (World Scientific, Singapore, 1991); (PreObservation Cosmological Constant) 

  • Rafael D. Sorkin; Forks in the Road, on the Way to Quantum Gravity, talk given at the conference entitled “Directions in General Relativity”, held at College Park, Maryland,  May, 1993; Int. J. Th. Phys. 36 : 2759–2781 (1997) ; (PreObservation Cosmological Constant) 

  • Rafael D. Sorkin, "Discrete Gravity";  a series of lectures to the First Workshop on Mathematical Physics and Gravitation, held Oaxtepec, Mexico, Dec. 1995 (unpublished); (PreObservation Cosmological Constant)  

  • Y. Jack Ng and H. van Dam, "A small but nonzero cosmological constant";  Int. J. Mod. Phys D. 10 : 49 (2001); (PreObservation Cosmological Constant) 

  • M. Ahmed, S. Dodelson, P.B. Greene, R.D. Sorkin,  "Everpresent lambda";  Phys. Rev. D69, 103523, (2004);   (Cosmological Constant)

  • Rafael D. Sorkin, " Big extra dimensions make Lambda too small"; Constant)

  • R.D. Sorkin,  "Is the cosmological "constant" a nonlocal quantum residue of discreteness of the causal set type?"; Proceedings of the PASCOS-07 Conference, July 2007, Imperial College London;; (Cosmological Constant)

Lorentz Invariance

  • L. Bombelli, J. Henson, R.D. Sorkin; Discreteness without symmetry breaking: a theorem;   gr-qc/0605006v1; (Lorentz invariance, Sprinkling)

  • F. Dowker, J. Henson, R.D. Sorkin, "Quantum gravity phenomenology, lorentz invariance and discreteness" ; Mod. Phys. Lett. A19, 1829–1840, (2004); (Lorentz invariance, Phenomenology)

  • J. Henson,  "Macroscopic observables and Lorentz violation in discrete quantum gravity"; gr-qc/0604040v1; (Lorentz invariance)

Black Hole Entropy

  • D. Dou, " Causal Sets, a Possible Interpretation for the Black Hole Entropy, and Related Topics";  PhD thesis (SISSA, Trieste, 1999); (Black hole entropy)

  • D. Dou, "Black Hole Entropy as Causal Links";  Fnd. of Phys, 33 2:279-296(18) (2003)(Black hole entropy) 

  • R.D. Sorkin, "On the Entropy of the Vacuum Outside a Horizon"; Tenth International Conference on General Relativity and Gravitation (held Padova, 4-9 July, 1983), Contributed Papers, vol. II, pp. 734-736 (Roma, Consiglio Nazionale Delle Ricerche, 1983), B. Bertotti, F. de Felice and A. Pascolini (eds.); (Black hole entropy) 

  • G.’t Hooft, "On the quantum structure of a black hole"; Nuclear Phys. B 256:727-745 (1985);;  (Black hole entropy) 

  • D.P. Rideout, S. Zohren, "Counting entropy in causal set quantum gravity" ; gr-qc/0612074v1; (Black hole entropy) 

  • D.P. Rideout, S. Zohren, "Evidence for an entropy bound from fundamentally discrete gravity";  Class.Quant.Grav. 23 (2006) 6195-6213; (Black hole entropy) 

Quantum Measure

  • Rafael Sorkin; Quantum mechanics as quantum measure theory;  Mod.Phys.Lett.A9:3119-3128,1994(Quantum Measure Theory)

  • Rafael D. Sorkin; Quantum measure theory and its interpretation; 4TH Drexel Symposium on Quantum Nonintegrability, 8-11 Sep 1994, Philadelphia, PA; gr-qc/9507057 ; (Quantum Measure Theory)

  • R.B. Salgado; Some Identities for the Quantum Measure and its Generalizations;; gr-qc/9903015; (Quantum Measure Theory)

  • Xavier Martin, Denjoe O'Connor , R.D. Sorkin; The Random walk in generalized quantum theory; Phys.Rev.D71:024029,2005;(Quantum Measure Theory)


  • A.R. Daughton; The Recovery of Locality for Causal Sets and Related Topics; PhD thesis (Syracuse University, 1993); ( Locality)

  • R.D. Sorkin; Does Locality Fail at Intermediate Length-Scales; Towards Quantum Gravity, Daniele Oriti (ed.) (Cambridge University Press, 2007); gr-qc/0703099v1; (d'Alembertian, Locality)


  • D.A. Meyer; Spacetime Ising models; (UCSD preprint May 1993); (Quantum Dynamics)

  • D.A. Meyer; Why do clocks tick?; General Relativity and Gravitation 25 9:893-900;; (Quantum Dynamics)

  • D.A. Meyer; Talk given at the 1997 Santa Fe workshop: "Causal Sets and Feynman diagrams"; Presented at "New Directions in Simplicial Quantum Gravity" July 28 - August 8, 1997;;Feynman diagrams; (Quantum Dynamics)

  • R.D. Sorkin; Two Talks given at the 1997 Santa Fe workshop: “A Review of the Causal Set Approach to Quantum Gravity” “A Growth Dynamics for Causal Sets”;  Presented at ”New Directions in Simplicial Quantum Gravity” July 28 - August 8, 1997;;;

  • A. Criscuolo, H. Waelbroeck; Causal Set Dynamics: A Toy Model; Class. Quant. Grav.16:1817-1832 (1999); gr-qc/9811088; (Quantum Dynamics, Toy Model)

  • D.P.Rideout and R.D. Sorkin, A classical sequential growth dynamics for causal sets, Phys. Rev D, 6, 024002 (2000); (Cosmology, Dynamics)

  • D.P. Rideout, R.D. Sorkin,  Evidence for a continuum limit in causal set dynamics.  Phys.Rev.D63:104011,2001; (Cosmology, Dynamics)

  • R.D. Sorkin; Indications of causal set cosmology; Int. J. Theor. Ph. 39(7):1731-1736 (2000); gr-qc/0003043; (Cosmology, Dynamics)

  • X. Martin, D. O'Connor, D.P. Rideout and R.D. Sorkin; On the “renormalization” transformations induced by cycles of expansion and contraction in causal set cosmology; Phys. Rev. D 63, 084026 (2001) (Cosmology, Dynamics)

  • D.P. Rideout; Dynamics of Causal Sets; PhD Thesis (Syracuse University 2001); gr-qc/0212064v1;  (Cosmology, Dynamics) 

  • G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; General covariance and the `problem of time'in a discrete cosmology; In ed. K. Bowden, Correlations:Proceedings of the ANPA 23 conference, August 16-21, 2001, Cambridge, England, pp. 1-17. Alternative Natural Philosophy Association, (2002).;gr-qc/0202097; (Cosmology, Dynamics, Observables)

  • G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; 'Observables' in causal set cosmology; Phys. Rev. D67, 084031, (2003); gr-qc/0210061 (Cosmology, Dynamics, Observables)

  • A.L. Krugly; Causal Set Dynamics and Elementary Particles; Int. J. Theo. Phys 41 1:1-37(2004);; (Quantum Dynamics)

  • M. Varadarajan, D.P. Rideout; A general solution for classical sequential growth dynamics of Causal Sets; Phys.Rev. D73 (2006) 104021; gr-qc/0504066v3;   (Cosmology, Dynamics)

  • F. Dowker, S. Surya; Observables in extended percolation models of causal set cosmology;Class. Quant. Grav. 23, 1381-1390 (2006); gr-qc/0504069v1; (Cosmology, Dynamics,  Observables)

  • S. Major, D.P. Rideout, S. Surya; Spatial Hypersurfaces in Causal Set Cosmology; Class.Quant.Grav. 23 (2006) 4743-4752; gr-qc/0506133v2; (Observables, Continuum topology)

  • J. Henson; Comparing causality principles; Stud.Hist.Philos.Mod.Phys. 36 (2005) 519-543; quant-ph/0410051v3; (Quantum Dynamics, Philosophy)

  • R.D. Sorkin; Relativity theory does not imply that the future already exists: a counterexample; Relativity and the Dimensionality of the World, Vesselin Petkov (ed.) (Springer 2007, in press); gr-qc/0703098v1; (Dynamics, Philosophy)

  • G. Brightwell, J. Henson, S. Surya; A 2D model of Causal Set Quantum Gravity: The emergence of the continuum;; (Quantum Dynamics, Toy Model)

  • G. Brightwell, R. Gregory; The Structure of Random Discrete Spacetime; Phys. Rev. Lett. 66:260-263 (1991) (Geodesic Length)

  • R.D. Sorkin, E. Woolgar; A Causal Order for Spacetimes with C^0 Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves; Classical & Quantum Gravity 13: 1971-1994 (1996); (Causal Structure)

  • A.A. Robb; Geometry of Time and Space; Cambridge University Press, 1936; (Geometry, Causal Structure)

  • A.V. Levichev; Prescribing the conformal geometry of a lorentz manifold by means of its causal structure; Soviet Math. Dokl. 35:452-455, (1987); (Geometry, Causal Structure)

  • D. Malament; The class of continuous timelike curves determines the topology of spacetime; J. Math. Phys. 18 7:1399-1404 (1977) (Geometry, Causal Structure)

  • S.W. Hawking, A.R. King, P.J. McCarthy; A new topology for curved space–time which incorporates the causal, differential, and conformal structures; J. Math. Phys. 17 2:174-181 (1976);  (Geometry, Causal Structure)

Poset Theory

  • R.D. Sorkin; Causal set glossary and bibliography (20 November 2001);;;Glossary and bibliography;

  • D.D. Reid; Discrete Quantum Gravity and Causal Sets; Canadian Journal of Physics 79, 1-16 (2001); gr-qc/9909075; General;

  • H. Reichenbach; ;Physikal. Zeitschr. 22:683 (1921);;

  • H. Reichenbach; Axiomatik der relativistische Raum-Zeit-Lehre (translated into English as Axiomatization of the theory of relativity); Berkeley, University of California Press, 1969;;

  • I. Raptis; Algebraic Quantization of Causal Sets; Int.J.Theor.Phys 39:1233-1240,2000; gr-qc/9906103; Algebraic quantization;


  • G. Brightwell; Models of Random Partial Orders; Surveys in Combinatorics, 1993, London Math. Soc. Lecture Notes Series 187:53-83, ed. Keith;; Random partial orders;

  • B. Bollobas, G. Brightwell; Box spaces and random partial orders; Trans. Amer. Math. Soc 324(1): 59-72 (1991); Random partial orders;

  • B. Bollobas, G. Brightwell; The Height of a Random Partial Order: Concentration of Measure; Annals of Applied Probability 2: 1009-1018 (1992); Geodesics;

  • N. Alon, B. Bollobas, G. Brightwell, and S. Janson; Linear extensions of a random partial order; Ann. Applied Prob. 4: 108-123 (1994); Random partial orders;

  • B. Bollobas, G. Brightwell; The structure of random graph orders; SIAM J. Discrete Math.10: 318-335 (1997); Random partial orders;

  • B. Bollobas, G. Brightwell; The dimension of random graph orders; The Mathematics of Paul Erdos II, R.L. Graham and J. Nesetril, eds. (Springer-Verlag, 1996), pp. 51-69; Random partial orders;

  • B. Bollobas, G. Brightwell; The width of random graph orders; Math. Scientist 20: 69-90 (1995); Random partial orders;

  • B. Pittel, R. Tungol; A Phase Transition Phenomenon in a Random Directed Acyclic Graph; Combinatorics, Probability and Computing 18 2 p164; ; Phase Transition, Random partial orders; 

  • J.H. Kim, B. Pittel; On tail distribution of interpost distance; J. Combinatorial Theory, B 80 1:49-56 (2000); Geodesics;

  • D. Crippa, K. Simon and P. Trunz; Markov Processes Involving q-Stirling Numbers; Combinatorics, Probability and Computing 6: 165-178 (1997);;;

  • K. Simon, D. Crippa, F. Collenberg; On the Distribution of the Transitive Closure in a Random Acyclic Digraph; Lecture Notes in Computer Science 726: 345-356 (1993); ;

  • K. Simon; Improved Algorithm for Transitive Closure on Acyclic Digraphs; Theoretical Computer Science 58 (1988);; Transitive closure, Graph theory;

  • C. M. Newman; Chain Lengths in Certain Random Directed Graphs; Random Structures and Algorithms 3: 243-253 (1992); ; Chain lengths;

  • S. Felsner, P.C. Fishburn, W.T. Trotter; Finite three dimensional partial orders which are not sphere orders; Discrete Math. 201: 101-132 (1999);;

  • R.D. Sorkin; Indecomposable Ideals in Incidence Algebras;Mod.Phys.Lett. A18 (2003) 2491-2500;math/0309126v1; Order theory;


Thanks to Steven Johnston and Rafael Sorkin for sending me extensive lists.