Причинные множества — направление исследований в квантовой гравитации, основанное на математической гипотезе о дискретной структуре пространства-времени и о частичной упорядоченности его точек, физически означающей между событиями в них с сохранением лоренцевской инвариантности.
В теории причинных множеств важную роль играет теорема Дэвида Маламента (), которая гласит, что если существует биективное отображение между двумя областями пространства-времени, которое сохраняет их причинную структуру, то это отображение является конформно изоморфным. Конформный множитель, который остается неопределенным, связан с объемом областей в пространстве-времени. Его можно найти, указав элемент объема для каждой точки пространства-времени. Общий объем области пространства-времени затем может быть найден путем подсчета количества точек в этой области. __NOTOC__ Определение Причинным множеством называется множество с отношением частичной упорядоченности , обладающее свойствами:
Рефлексивность: Для всех , выполняется . Антисимметричность: Для всех из справедливости и следует . Транзитивность: Для всех , из справедливости и следует . : Для всех , выполняется .
Если и , то используют обозначение .
Множество представляет набор событий пространства-времени, а отношение порядка представляет причинно-следственную связь между событиями (аналогичную идею в лоренцевом многообразии см. ).
Хотя в этом определении используется рефлексивное соглашение, мы могли бы выбрать нерефлексивное соглашение, в котором отношение порядка нерефлексивно.
лоренцева многообразия (без замкнутой ) удовлетворяет первым трем условиям. Это условие локальной конечности, которое вводит дискретность пространства-времени.
См. также Причинная динамическая триангуляция
Общая теория относительности Отношение порядка
Примечания
Дальнейшее чтение
Введения и обзоры L. Bombelli. Causal Set reference page (Overview) L. Bombelli. Causal Sets: Overview and Status, Talk given at Quantum Gravity in the Americas III, August 24-26, 2006; (Introduction, Overview) F. Dowker, Causal sets and the deep structure of spacetime, arXiv:gr-qc/0508109; (Introduction) F. Dowker, Causal sets as discrete spacetime, Contemporary Physics, vol. 47, Issue 1, p. 1-9; (Overview, Introduction) F. Dowker, Introduction to causal sets and their phenomenology, Gen Relativ Gravit (2013) 45:1651-1667 doi:10.1007/s10714-013-1569-y (Overview of recent research) J. Henson, The causal set approach to quantum gravity, arXiv:gr-qc/0601121; (Introduction, Overview) D.D. Reid; Introduction to causal sets: an alternate view of spacetime structure; Canadian Journal of Physics 79, 1-16 (2001); arXiv:gr-qc/9909075; (General); R.D. Sorkin; Causal set glossary and bibliography (20 November 2001); (Glossary and bibliography); 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; arXiv:gr-qc/0309009; (Introduction, Glossary)
Основы теории L. Bombelli, J. Lee, D. Meyer, 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, R.D. Sorkin, Bombelli et al. Reply, Phys. Rev. Lett. 60, 656 (1988); (Foundations) 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) D. Finkelstein, Space-time code, Phys. Rev. 184:1261 (1969); (Foundations) D. Finkelstein, «Superconducting» Causal Nets, Int. J. Th. Phys 27:473(1988); (Foundations) G. Hemion, A quantum theory of space and time; Found. Phys. 10 (1980), p. 819 (Similar proposal) J. Myrheim, Statistical geometry, CERN preprint TH-2538 (1978); (Foundations, Historical) B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, The Collected Works of B. Riemann (Dover NY 1953); ; (Historical) R.D. Sorkin; A Finitary Substitute for Continuous Topology, Int. J. Theor. Phys. 30 7: 923—947 (1991); (Foundational) 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); arXiv:gr-qc/9706002; (Philosophical, Introduction) 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) E.C. Zeeman, Causality Implies the Lorentz Group, J. Math. Phys. 5: 490—493; (Historical, Foundations)
Диссертации L. Bombelli, Space-time as a Causal Set, PhD thesis (Syracuse University, 1987); (Introduction, Kinematics) A.R. Daughton; The Recovery of Locality for Causal Sets and Related Topics; PhD thesis (Syracuse University, 1993); (Locality) D. Dou, Causal Sets, a Possible Interpretation for the Black Hole Entropy, and Related Topics; PhD thesis (SISSA, Trieste, 1999); arXiv:gr-qc/0106024 (Black hole entropy) S. Johnston, Quantum Fields on Causal Sets, PhD Thesis (Imperial College London, 2010) arXiv:1010.5514 (Quantum Field Theory) D.A. Meyer, The Dimension of Causal Sets, PhD thesis (M.I.T., 1988); (Dimension theory) L. Philpott, Causal Set Phenomenology, PhD Thesis (Imperial College London, 2010); arXiv:1009.1593 (Swerves, Phenomenology) D.P. Rideout; Dynamics of Causal Sets; PhD Thesis (Syracuse University 2001); arXiv:gr-qc/0212064; (Cosmology, Dynamics) R.B. Salgado; Toward a Quantum Dynamics for Causal Sets; PhD Thesis (Syracuse University 2008); (Scalar field theory, Quantum Measure Theory) R. Sverdlov; Quantum Field Theory and Gravity in Causal Sets; PhD Thesis (University of Michigan 2009); arXiv: 0905.2263 (Quantum Field Theory and Gravity)
Обсуждения Joe Henson, An Invitation to Causal Sets; Talk given at Perimeter Institute, 14 September 2010, Waterloo ON (Introduction) F. Dowker, Causal Set Phenomenology; Talk given at Loops 05, 10-14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics (Swerves) S. Johnston; Particle Propagators from Discrete Spacetime; Talk given at Perimeter Institute 14 April 2008 (Quantum field theory) 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) D.P. Rideout; Spatial Hypersurfaces in Causal Set Cosmology; Talk given at Loops 05, 10-14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics (Spatial hyper-surfaces, Dynamics) J. Scargle, Testing Quantum Gravity Theories with GLAST; Talk given at Santa Cruz Institute for Particle Physics, April 24, 2007. (Lorentz invariance, Phenomenology) R.D. Sorkin; Two Talks given at the 1997 Santa Fe workshop: A Review of the Causal Set Approach to Quantum Gravity and ''; Presented at «New Directions in Simplicial Quantum Gravity» July 28 — August 8, 1997; ;; R.D. Sorkin; Does quantum gravity give rise to an observable nonlocality?; Talk given at Perimeter Institute 17/01/2007 (d’Alembertian, Locality) R.D. Sorkin, Some Insights for Quantum Gravity Derived from Work on Causal Sets; Talk given at Loops 05, 10-14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics (Overview) R.D. Sorkin Is a past-finite causal order the inner basis of spacetime? Talk given at Perimeter Institute 07/09/2005 S. Surya, Recovering spacetime topology from a causet; Talk given at Loops 05, 10-14 October 2005, Potsdam, Max Planck Institute for Gravitational Physics (Topology) R. Sverdlov; Introduction of bosonic fields into causal set theory; Talk given at Perimeter Institute 19/02/2008 (Quantum field theory)
Теория многообразий L. Bombelli, D.A. Meyer; The origin of Lorentzian geometry; Phys. Lett. A 141:226-228 (1989); (Manifoldness) 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) 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. . Published by Editions Frontieres, 91192 Gif-sur-Yvette Cedex, France, 1994, p. 249; (Closeness of Lorentzian manifolds) L. Bombelli, Statistical Lorentzian geometry and the closeness of Lorentzian manifolds, J. Math. Phys.41:6944-6958 (2000); arXiv:gr-qc/0002053 (Closeness of Lorentzian manifolds, Manifoldness) A.R. Daughton, An investigation of the symmetric case of when causal sets can embed into manifolds, Class. Quantum Grav.15(11):3427-3434 (Nov., 1998) (Manifoldness) J. Henson, Constructing an interval of Minkowski space from a causal set, Class. Quantum Grav. 23 (2006) L29-L35; arXiv:gr-qc/0601069; (Continuum limit, Sprinkling) S. Major, D.P. Rideout, S. Surya, On Recovering Continuum Topology from a Causal Set, J.Math.Phys.48:032501,2007; arXiv:gr-qc/0604124 (Continuum Topology) S. Major, D.P. Rideout, S. Surya; Spatial Hypersurfaces in Causal Set Cosmology; Class. Quantum Grav. 23 (2006) 4743-4752; arXiv:gr-qc/0506133v2; (Observables, Continuum topology) S. Major, D.P. Rideout, S. Surya, Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory, arXiv:0902.0434 (Continuum topology and homology) 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) D.A. Meyer, Spherical containment and the Minkowski dimension of partial orders, Order 10: 227—237 (1993); (Dimension theory) J. Noldus, A new topology on the space of Lorentzian metrics on a fixed manifold, Class. Quant. Grav 19: 6075-6107 (2002); (Closeness of Lorentzian manifolds) J. Noldus, A Lorentzian Gromov-Hausdorff notion of distance, Class. Quantum Grav. 21, 839—850, (2004); (Closeness of Lorentzian manifolds) D.D. Reid, Manifold dimension of a causal set: Tests in conformally flat spacetimes, Phys. Rev. D67 (2003) 024034; arXiv:gr-qc/0207103v2 (Dimension theory) S. Surya, Causal Set Topology; arXiv:0712.1648
Геометрия E. Bachmat; Discrete spacetime and its applications; arXiv:gr-qc/0702140; (Geodesics, Antichains)