Interested Article - Причинные множества

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 ( ), которая гласит, что если существует биективное отображение между двумя областями пространства-времени, которое сохраняет их , 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.

Определение

Причинным множеством называется множество $C$ с отношением частичной упорядоченности $\preceq$ , обладающее свойствами:

Рефлексивность : Для всех $x\in C$ , выполняется $x\preceq x$ .
Антисимметричность : Для всех $x,y\in C$ из справедливости $x\preceq y$ и $y\preceq x$ следует $x=y$ .
Транзитивность : Для всех $x,y,z\in C$ , из справедливости $x\preceq y$ и $y\preceq z$ следует $x\preceq z$ .
: Для всех $x,z\in C$ , выполняется $|\{y\in C|x\preceq y\preceq z\}|<\aleph _{0}$ .

Если $x\preceq y$ и $x\neq y$ , то используют обозначение $x\prec y$ .

Множество $C$ представляет набор событий пространства-времени , а отношение порядка $\preceq$ 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

Примечания

Malament, David B. (July 1977). (PDF) . Journal of Mathematical Physics . 18 (7): 1399—1404. Bibcode : . doi : . (PDF) из оригинала 14 декабря 2021 . Дата обращения: 19 сентября 2021 .

Дальнейшее чтение

Введения и обзоры

L. Bombelli. (Overview)
L. Bombelli. , Talk given at Quantum Gravity in the Americas III, August 24-26, 2006; (Introduction, Overview)
, Causal sets and the deep structure of spacetime , ; (Introduction)
, , 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 , ; (Introduction, Overview)
D.D. Reid; Introduction to causal sets: an alternate view of spacetime structure ; Canadian Journal of Physics 79, 1-16 (2001); ; (General);
; (20 November 2001); (Glossary and bibliography);
, Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School) , In Proceedings of the Valdivia Summer School, edited by A. Gomberoff and D. Marolf; ; (Introduction, Glossary)

Основы теории

L. Bombelli, J. Lee, D. Meyer, , , Phys. Rev. Lett. 59:521-524 (1987); (Introduction, Foundations)
C. Moore, , Phys. Rev. Lett. 60, 655 (1988); (Foundations)
L. Bombelli, J. Lee, D. Meyer, , , 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)
, , Phys. Rev. 184:1261 (1969); (Foundations)
, , Int. J. Th. Phys 27:473(1988); (Foundations)
G. Hemion, ; Found. Phys. 10 (1980), p. 819 (Similar proposal)
J. Myrheim, , CERN preprint TH-2538 (1978); (Foundations, Historical)
, , The Collected Works of B. Riemann (Dover NY 1953); ; (Historical)
; , Int. J. Theor. Phys. 30 7: 923—947 (1991); (Foundational)
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, 30 сентября 2013 года. , 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)
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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)
, , J. Math. Phys. 5: 490—493; (Historical, Foundations)

Диссертации

L. Bombelli, , 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 ( , Trieste, 1999); (Black hole entropy)
S. Johnston, Quantum Fields on Causal Sets , PhD Thesis ( Imperial College London , 2010) (Quantum Field Theory)
D.A. Meyer, , PhD thesis ( , 1988); (Dimension theory)
L. Philpott, Causal Set Phenomenology , PhD Thesis ( Imperial College London , 2010); (Swerves, Phenomenology)
D.P. Rideout; Dynamics of Causal Sets ; PhD Thesis ( Syracuse University 2001); ; (Cosmology, Dynamics)
R.B. Salgado; ; 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); (Quantum Field Theory and Gravity)

Обсуждения

Joe Henson, ; Talk given at , 14 September 2010, Waterloo ON (Introduction)
, ; Talk given at Loops 05, 10-14 October 2005, Potsdam, (Swerves)
S. Johnston; ; Talk given at 14 April 2008 (Quantum field theory)
D.A. Meyer; Talk given at the 1997 Santa Fe workshop: ; Presented at «New Directions in Simplicial Quantum Gravity» July 28 — August 8, 1997; (Feynman diagrams, Quantum Dynamics)
D.P. Rideout; ; Talk given at Loops 05, 10-14 October 2005, Potsdam, (Spatial hyper-surfaces, Dynamics)
J. Scargle, ; Talk given at Santa Cruz Institute for Particle Physics, April 24, 2007. (Lorentz invariance, Phenomenology)
; Two Talks given at the 1997 Santa Fe workshop: and ' ; Presented at «New Directions in Simplicial Quantum Gravity» July 28 — August 8, 1997; ;;
; ; Talk given at 17/01/2007 (d’Alembertian, Locality)
, ; Talk given at Loops 05, 10-14 October 2005, Potsdam, (Overview)
Talk given at 07/09/2005
S. Surya, ; Talk given at Loops 05, 10-14 October 2005, Potsdam, (Topology)
R. Sverdlov; ; Talk given at 19/02/2008 (Quantum field theory)

Теория многообразий

L. Bombelli, D.A. Meyer; ; Phys. Lett. A 141:226-228 (1989); (Manifoldness)
L. Bombelli, , 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. ISBN 2-86332-168-4 . 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); (Closeness of Lorentzian manifolds, Manifoldness)
A.R. Daughton, , 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; ; (Continuum limit, Sprinkling)
S. Major, D.P. Rideout, S. Surya, On Recovering Continuum Topology from a Causal Set , J.Math.Phys.48:032501,2007; (Continuum Topology)
S. Major, D.P. Rideout, S. Surya; Spatial Hypersurfaces in Causal Set Cosmology ; Class. Quantum Grav. 23 (2006) 4743-4752; ; (Observables, Continuum topology)
S. Major, D.P. Rideout, S. Surya, Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory , (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, , 10: 227—237 (1993); (Dimension theory)
J. Noldus, , Class. Quant. Grav 19: 6075-6107 (2002); (Closeness of Lorentzian manifolds)
J. Noldus, , 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; (Dimension theory)
S. Surya, Causal Set Topology ;

Геометрия

E. Bachmat; Discrete spacetime and its applications ; ; (Geodesics, Antichains)
G. Brightwell, R. Gregory; ; Phys. Rev. Lett. 66:260-263 (1991); (Geodesic Length)
, S. N. Solodukhin; The Geometry of Small Causal Diamonds (Causal intervals)
S.W. Hawking , A.R. King, P.J. McCarthy; ; J. Math. Phys. 17 2:174-181 (1976); (Geometry, )
S. He, D.P. Rideout; A Causal Set Black Hole ; (Causal structure of Schwarzschild spacetime, Sprinklings)
R. Ilie, G.B. Thompson, D.D. Reid; ; 2006 Class. Quantum Grav. 23 3275-3285 (Geodesic length)
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, )
Malament, David B. (July 1977). (PDF) . Journal of Mathematical Physics . 18 (7): 1399—1404. Bibcode : . doi : .
D.P. Rideout, P. Wallden; Spacelike distance from discrete causal order ; (Spatial distances)

Вычисление космологической постоянной

M. Ahmed, S. Dodelson, P.B. Greene, , Everpresent lambda ; Phys. Rev. D69, 103523, (2004) ; (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)
Y. Kuznetsov, On cosmological constant in Causal Set theory ;
, 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, , and , 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)
; , 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)
, 30 сентября 2013 года. , 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)
; 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)
, Discrete Gravity ; a series of lectures to the First Workshop on Mathematical Physics and Gravitation, held Oaxtepec, Mexico, Dec. 1995 (unpublished); (PreObservation Cosmological Constant)
, Big extra dimensions make Lambda too small ; ; (Cosmological Constant)
, 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)
J. Zuntz, The CMB in a Causal Set Universe , (CMB)

Лоренцевская инвариантность

L. Bombelli, J. Henson, ; Discreteness without symmetry breaking: a theorem ; ; (Lorentz invariance, Sprinkling)
, J. Henson, , Quantum gravity phenomenology, Lorentz invariance and discreteness ; Mod. Phys. Lett. A19, 1829—1840, (2004) ; (Lorentz invariance, Phenomenology, Swerves)
, J. Henson, , Discreteness and the transmission of light from distant sources ; (Coherence of light, Phenomenology)
J. Henson, Macroscopic observables and Lorentz violation in discrete quantum gravity ; ; (Lorentz invariance, Phenomenology)
N. Kaloper, D. Mattingly, Low energy bounds on Poincaré violation in causal set theory ; Phys. Rev. D 74, 106001 (2006) (Poincaré invariance, Phenomenology)
D. Mattingly, Causal sets and conservation laws in tests of Lorentz symmetry ; Phys. Rev. D 77, 125021 (2008) (Lorentz invariance, Phenomenology)
L. Philpott, , , Energy-momentum diffusion from spacetime discreteness ; (Phenomenology, Swerves)

Энтропия черных дыр

D. Dou, Black Hole Entropy as Causal Links ; Fnd. of Phys, 33 2:279-296(18) (2003); (Black hole entropy)
D.P. Rideout, S. Zohren, Counting entropy in causal set quantum gravity ; ; (Black hole entropy)
D.P. Rideout, S. Zohren, Evidence for an entropy bound from fundamentally discrete gravity ; Class. Quantum Grav. 23 (2006) 6195-6213; (Black hole entropy)

Локальность и квантовая теория поля

G. Hemion, ; Int. J. Theor. Phys. 27 (1988), p. 1145 (Classical electrodynamics)
S. Johnston; Particle propagators on discrete spacetime ; 2008 Class. Quantum Grav. 25 202001; (Quantum Field Theory)
S. Johnston; The Feynman propagator for a Free Scalar Field on a Causal Set ; Phys. Rev. Lett. 103, 180401 (2009); (Quantum Field Theory)
; Does Locality Fail at Intermediate Length-Scales ; Towards Quantum Gravity, Daniele Oriti (ed.) (Cambridge University Press, 2007); ; (d’Alembertian, Locality)
R. Sverdlov, L. Bombelli; Gravity and Matter in Causal Set Theory ;
R. Sverdlov; A Geometrical Description of Spinor Fields ;
R. Sverdlov; Bosonic Fields in Causal Set Theory ;
R. Sverdlov; Gauge Fields in Causal Set Theory ;
R. Sverdlov; Spinor fields in Causal Set Theory ;

Динамика

M. Ahmed, D. Rideout, Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets ;
A.Ash, P. McDonald, Moment Problems and the Causal Set Approach to Quantum Gravity ; J.Math.Phys. 44 (2003) 1666—1678;
A.Ash, P. McDonald, ; J.Math.Phys. 46 (2005) 062502 (Analysis of number of posts in growth processes)
D.M.T. Benincasa, , The Scalar Curvature of a Causal Set ; ; (Scalar curvature, actions)
G. Brightwell; ; Order-invariant Measures on Causal Sets ; ; (Measures on causal sets)
G. Brightwell; ; Order-invariant Measures on Fixed Causal Sets ; ; (Measures on causal sets)
G. Brightwell, , R.S. Garcia, J. Henson, ; 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).; ; (Cosmology, Dynamics, Observables)
G. Brightwell, , R.S. Garcia, J. Henson, ; «Observables» in causal set cosmology ; Phys. Rev. D67, 084031, (2003); ; (Cosmology, Dynamics, Observables)
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, N. Georgiou; University of Bristol preprint. (Dynamics)
A. Criscuolo, H. Waelbroeck; Causal Set Dynamics: A Toy Model ; Class. Quantum Grav.16:1817-1832 (1999); ; (Quantum Dynamics, Toy Model)
, S. Surya; Observables in extended percolation models of causal set cosmology ;Class. Quantum Grav. 23, 1381—1390 (2006); ; (Cosmology, Dynamics, Observables)
M. Droste, Universal homogeneous causal sets , J. Math. Phys. 46, 122503 (2005); ; (Past-finite causal sets)
J. Henson, D. Rideout, , S. Surya; Onset of the Asymptotic Regime for (Uniformly Random) Finite Orders ; Experimental Mathematics 26, 3:253-266 (2017); (Cosmology, Dynamics)
A.L. Krugly; ; Int. J. Theo. Phys 41 1:1-37(2004);; (Quantum Dynamics)
X. Martin, D. O’Connor, D.P. Rideout, ; On the «renormalization» transformations induced by cycles of expansion and contraction in causal set cosmology ; Phys. Rev. D 63, 084026 (2001); (Cosmology, Dynamics)
D.A. Meyer; Spacetime Ising models ; (UCSD preprint May 1993); (Quantum Dynamics)
D.A. Meyer; ; General Relativity and Gravitation 25 9:893-900;; (Quantum Dynamics)
I. Raptis; Quantum Space-Time as a Quantum Causal Set ,
D.P. Rideout, ; A classical sequential growth dynamics for causal sets , Phys. Rev. D, 6, 024002 (2000); (Cosmology, Dynamics)
D.P. Rideout, ; Evidence for a continuum limit in causal set dynamics Phys. Rev. D 63:104011,2001; (Cosmology, Dynamics)
; Indications of causal set cosmology ; Int. J. Theor. Ph. 39(7):1731-1736 (2000); ; (Cosmology, Dynamics)
; 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); ; (Dynamics, Philosophy)
M. Varadarajan, D.P. Rideout; A general solution for classical sequential growth dynamics of Causal Sets ; Phys. Rev. D 73 (2006) 104021; ; (Cosmology, Dynamics)
M.R., Khoshbin-e-Khoshnazar (2013). "Binding Energy of the Very Early Universe: Abandoning Einstein for a Discretized Three–Torus Poset.A Proposal on the Origin of Dark Energy". Gravitation and Cosmology . 19 (2): 106—113. Bibcode : . doi : . S2CID . ;(Dynamics, Poset)

Ссылки

- a review article by Joe Henson on causal sets
— one of the first papers by Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. Sorkin
— non-technical article by Rafael D. Sorkin on