Selected External References

Listed here for convenience are publications of which we are aware that may be relevant to build on in developing the Wolfram Physics Project. This is for published papers and books, or arXiv entries, except under exceptional circumstances.

Mathematical Issues

Discrete Differential Geometry

• M. Eidi and J. Jost (2019), “Ollivier Ricci Curvature of Directed Hypergraphs”. arXiv:1907.04727.
• R. Forman (2003), “Bochner’s Method for Cell Complexes and Combinatorial Ricci Curvature”, Discrete Comput Geom 29, 323–74. doi:10.1007/s00454-002-0743-x.
• Y. Ollivier (2007), “Ricci Curvature of Metric Spaces”, C R Math Acad Sci Paris 345, 643–46. doi:10.1016/j.crma.2007.10.041.
• Y. Ollivier (2013), “A Visual Introduction to Riemannian Curvatures and Some Discrete Generalizations”, in Analysis and Geometry of Metric Measure Spaces, CRM Proc Lecture Notes 56, 197–220. doi:10.1090/crmp/056/08.

Fractional Dimensional Structure

• R. Burioni and D. Cassi (1998), “The Spectral Dimension and Geometrical Universality on Graphs”, J Phys IV France 8, Pr6-81–85. doi:10.1051/jp4:1998610.
• S. Carlip (2017), “Dimension and Dimensional Reduction in Quantum Gravity”, Class Quantum Grav 34, 193001. arXiv:1705.05417.
• C. Palmer and P. N. Stavrinou (2004), “Equations of Motion in a Non-integer-dimensional Space”, J Phys A: Math Gen 37, 6987–7003. doi:10.1088/0305-4470/37/27/009.
• K. Svozil (1987), “Quantum Field Theory on Fractal Spacetime: A New Regularisation Method”, J Phys A: Math Gen 20, 3861–75. doi:10.1088/0305-4470/20/12/033.

Limits of Graphs

• D. Glasscock (2016), “What Is a Graphon?", Notices Amer Math Soc 62, 46–48. arXiv:1611.00718.
• “A Wolfram Notebook on Graphons”, develop.open.wolframcloud.com/objects/exploration/Graphons.nb.

Multiway System Features

• Y. Guiraud, P. Malbos and S. Mimram (2013), “A Homotopical Completion Procedure with Applications to Coherence of Monoids”, in 24th International Conference on Rewriting Techniques and Applications (RTA 2013), F. van Raamsdonk (ed.), Schloss Dagstuhl, 223–38. doi:10.4230/LIPIcs.RTA.2013.223.
• Y. Lafont (2019), “Homotopy and Homology of Rewriting”, presented at the 11th International School on Rewriting, Paris. isr2019.mines-paristech.fr/wp-content/uploads/2019/07/lafont.pdf.

Potentially Connected Models

Models with features that may inform or be informed by our models

Causal Sets etc.

• J. Ambjørn, M. Carfora and A. Marzuoli (1997), The Geometry of Dynamical Triangulations, Springer.
• J. Ambjørn, J. Jurkiewicz and R. Loll (2009), “Quantum Gravity, or the Art of Building Spacetime”, in Approaches to Quantum Gravity, D. Oriti (ed.), Cambridge U Press, 341–59. arXiv:hep-th/0604212.
• L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin (1987), “Space-time as a Causal Set”, Phys Rev Lett 59, 521–24. doi:10.1103/PhysRevLett.59.521.
• L. Crane (2008), “Model Categories and Quantum Gravity”. arXiv:0810.4492.
• F. Dowker (2006), “Causal Sets as Discrete Spacetime”, Contemp Phys 47, 1–9. doi:10.1080/17445760500356833.
• B. F. Dribus (2013), “On the Axioms of Causal Set Theory”. arXiv:1311.2148v3.
• R. Loll (2001), “Discrete Lorentzian Quantum Gravity”, Nucl Phys B 94, 96–107. arXiv:hep-th/0011194.
• S. Major, D. Rideout and S. Surya (2006), “Spatial Hypersurfaces in Causal Set Cosmology”, Class Quant Grav 23, 4743–52. doi:10.1088/0264-9381/23/14/011.
• S. Major, D. Rideout and S. Surya (2007), “On Recovering Continuum Topology from a Causal Set”, J Phys Math 48, 032501. doi:10.1063/1.2435599.
• S. Major, D. Rideout and S. Surya (2009), “Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory”, Class Quant Grav 26, 175008. doi:10.1088/0264-9381/26/17/175008.
• F. Markopoulou (2000), “The Internal Description of a Causal Set: What the Universe Looks Like from the Inside”, Commun Math Phys 211, 559–83. doi:10.1007/s002200050826.
• nLab authors (2010), “Discrete Causal Spaces”, The nLab, ncatlab.org/nlab/show/Discrete+causal+spaces.
• D. Rideout and P. Wallden (2009), “Emergence of Spatial Structure from Causal Sets”, J Phys Conf Ser 174, 012017. doi:10.1088/1742-6596/174/1/012017.
• D. Rideout and P. Wallden (2009), “Spacelike Distance from Discrete Causal Order”, Class Quant Grav 26, 155013. doi:10.1088/0264-9381/26/15/155013.
• R. Sorkin (2006), “Geometry from Order: Causal Sets”, Einstein Online. einstein-online.info/en/spotlight/causal_sets.

Other Mathematical Structures

• K. Khanna (2019), “Aggregation Systems: A Stochastic Approach to CA”, Wolfram Summer School, community.wolfram.com/groups/-/m/t/1728662.
• V. R. Pratt (1994), “Chu Spaces: Automata with Quantum Aspects”, Proceedings Workshop on Physics and Computation. PhysComp ’94, IEEE, 186–95. doi:10.1109/PHYCMP.1994.363682.
• D. Scott and P. Suppes (1958), “Foundational Aspects of Theories of Measurement”, J Symbolic Logic 23, 113–28. doi:10.2307/2964389.
• B. Zilber (2010), “On Model Theory, Non-Commutative Geometry and Physics”, survey draft.

Physical Formalisms

Spacetime Structure

• R. Arnowitt and S. Deser (1959), “Dynamical Structure and Definition of Energy in General Relativity”, Phys Rev 116, 1322–30. doi:10.1103/PhysRev.116.1322.
• N. Bao, C. Cao, et al. (2017), “Quantum Circuit Cosmology: The Expansion of the Universe Since the First Qubit”. arXiv:1702.06959.
• J. L. F. Barbón (2009), “Black Holes, Information and Holography”, J Phys: Conf Ser 171, 012009. doi:10.1088/1742-6596/171/1/012009.
• R. de Mello Koch and J. Murugan (2009), “Emergent Spacetime”. arXiv:0911.4817. In Foundations of Space and Time: Reflections on Quantum Gravity (2012), J. Murugan, A. Weltman and G. F. R. Ellis (eds.), Cambridge U Press, 164–84.
• A. Dragan and A. Ekert (2020), “Quantum Principle of Relativity”, New J Phys 22, 033038. doi:10.1088/1367-2630/ab76f7.
• J. Frauendiener and H. Friedrich (eds.) (2002), The Conformal Structure of Space-Times, Springer.
• C. Kelly and C. A. Trugenberger (2019), “Combinatorial Quantum Gravity: Emergence of Geometric Space from Random Graphs”, J Phys Conf Ser 1275, 012016. doi:10.1088/1742-6596/1275/1/012016.
• C. Kelly, C. A. Trugenberger and F. Biancalana (2019), “Self-Assembly of Geometric Space from Random Graphs”, Class Quantum Grav 36, 125012. doi:10.1088/1361-6382/ab1c7d.
• A. Mallios and I. Raptis (2003), “Finitary, Causal, and Quantal Vacuum Einstein Gravity”, Int J Theor Phys 42, 1479–619. doi:10.1023/A:1025732112916.
• K. Martin and P. Panangaden (2004), “A Domain of Spacetime Intervals for General Relativity”. Slides presented at Dagstuhl Seminar 04351, Spatial Representation: Discrete vs. Continuous Computational Models, Wadern, Germany, August 2004.
• J. W. Moffat (2002), “Variable Speed of Light Cosmology: An Alternative to Inflation”. arXiv:hep-th/0208122.
• Y. C. Ong (2020), “Spacetime Singularities and Cosmic Censorship Conjecture: A Review with Some Thoughts”, Int J Mod Phys A 35, 2030007. doi:10.1142/S0217751X20300070.
• E. P. Verlinde (2017), “Emergent Gravity and the Dark Universe”, SciPost Phys 2, 1–41. doi:10.21468/SciPostPhys.2.3.016.
• E. P. Verlinde (2011), “On the Origin of Gravity and the Laws of Newton”, J High Energ Phys 2011. doi:10.1007/JHEP04(2011)029.

Quantum Structure

• S. D. Bartlett, T. Rudolph and R. W. Spekkens (2007), “Reference Frames, Superselection Rules, and Quantum Information”, Rev Mod Phys 79, 555–609. doi:10.1103/RevModPhys.79.555.
• D. C. Brody and L. P. Hughston (2001), “Geometric Quantum Mechanics”, J Geom Phys 38, 19–53. doi:10.1016/S0393-0440(00)00052-8.
• B. Coecke and A. Kissinger (2017), Picturing Quantum Processes, Cambridge U Press. doi:10.1017/9781316219317.
• L. Crane (1995), “Clock and Category: Is Quantum Gravity Algebraic?”, J Math Phys 36, 6180–93. doi:10.1063/1.531240.
• P. Facchi, R. Kulkarni, et al. (2010), “Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics”, Phys Lett A 374, 4801–3. doi:10.1016/j.physleta.2010.10.005.
• H. W. Hamber, R. Toriumi and R. M. Williams (2012), “Wheeler–DeWitt Equation in 2 + 1 Dimensions”, Phys Rev D 86, 084010. doi:10.1103/PhysRevD.86.084010.
• J. B. Hartle (1991), “The Quantum Mechanics of Cosmology”, in Quantum Cosmology and Baby Universes: Proceedings of the 7th Jerusalem Winter School for Theoretical Physics, S. Coleman, J. B. Hartle, et al. (eds.), World Scientific, 65–157. doi:10.1142/9789814503501_0002.
• P. Leifer (1997), “Superrelativity as an Element of a Final Theory”, Found Phys 27, 261–85. doi:10.1007/BF02550454.
• L. Smolin (2014), “Time, Laws, and the Future of Cosmology”, Phys Today 67, 38–43. doi:10.1063/PT.3.2310.
• R. W. Spekkens (2008), “In Defense of the Epistemic View of Quantum States: A Toy Theory”. arXiv:quant-ph/0401052. Updated from Phys Rev A 75 (2007), 032110. doi:10.1103/PhysRevA.75.032110.
• G. ’t Hooft (2010), “Classical Cellular Automata and Quantum Field Theory”, Int J Mod Phys A 35, 4385–96. doi:10.1142/S0217751X10050469.
• G. ’t Hooft (2016), The Cellular Automaton Interpretation of Quantum Mechanics, Springer. doi:10.1007/978-3-319-41285-6.

Physical Implications

Physical Scales

• R. J. Adler (2010), “Six Easy Roads to the Planck Scale”, Am J Phys 78, 925–32. arXiv:1001.1205.
• Y. J. Ng (2003), “Selected Topics in Planck-scale Physics”, Mod Phys Lett A 18, 1073–97. arXiv:gr-qc/0305019.

Potential Types of Quantum Experiments

• A. Choua, H. Glass, et al. (2017), “The Holometer: An Instrument to Probe Planckian Quantum Geometry”, Class Quantum Grav 34, 065005. doi:10.1088/1361-6382/aa5e5c.
• D.-C. Dai, D. Minic, et al. (2020), “Testing ER=EPR”. arXiv:2002.08178.
• S. Deffner and S. Campbell (2017), “Quantum Speed Limits: From Heisenberg’s Uncertainty Principle to Optimal Quantum Control”, J Phys A: Math Gen 50, 453001. doi:10.1088/1751-8121/aa86c6.
• C. J. Hogan and O. Kwon (2017), “Statistical Measures of Planck Scale Signal Correlations in Interferometers”, Class Quant Grav 34, 075006. doi:10.1088/1361-6382/aa601e.
• W. M. Itano (2006), “Perspectives on the Quantum Zeno Paradox”, J Phys: Conf Ser 196, 012018. doi:10.1088/1742-6596/196/1/012018.
• E. P. Verlinde and K. M. Zurek (2019), “Observational Signatures of Quantum Gravity in Interferometers”. arXiv:1902.08207.

Computational Formalisms

Distributed Computing

• L. Lamport (1978), “Time, Clocks, and the Ordering of Events in a Distributed System”, Commun ACM 21, 558–65. doi:10.1145/359545.359563.

Practical Computation

Graph Rewriting

• N. Behr, V. Danos and I. Garnier (2016), “Stochastic Mechanics of Graph Rewriting”, in LICS ’16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, Association for Computing Machinery, 46–55. doi:10.1145/2933575.2934537.
• S. Luke and L. Spector (1996), “Evolving Graphs and Networks with Edge Encoding: Preliminary Report”, in Late Breaking Papers at the Genetic Programming 1996 Conference, J. R. Koza (ed.), Stanford U Bookstore, 117–24.