Causal Sets  

In General > s.a. approaches to quantum gravity / posets.
* Causal set: A (locally) finite partially ordered set, in which the order is causally interpreted.
* Causal history: A causal set with additional structure, e.g., the SUq(2) intertwiners of spin networks, and/or other fields.
* Quantum causal history: A covariant functor from the poset of antichains to the category of Hilbert spaces.
@ Overviews: Sorkin in(90), in(91), in(95)gq, in(05)gq/03; Reid CJP(01)gq/99; in Markopoulou in(04)gq/02; Dowker in(05)gq, CP(06); Henson in(09)gq/06; Sorkin EO(06); Wallden JPCS(10)-a1001; Henson a1003-proc; Surya a1103-in.
@ Proposals: Kronheimer & Penrose PCPS(67); Myrheim CERN(78); 't Hooft in(78); Bombelli et al PRL(87); Bombelli PhD(87); Raptis gq/02 [algebraic version]; Sverdlov a0910 [reinterpretation]; Krugly a1004, a1008; Bolognesi a1004 [and the computational universe]; Dribus a1311.
@ Related topics and phenomenology: Ambjørn & Loll NPB(98)ht [2D]; Dou PhD(99)gq/01 [and black-hole entropy]; Blute et al gq/01 [decoherent histories on causal sets]; Kaloper & Mattingly PRD(06)ap [momentum space diffusion]; Zuntz PRD(08)-a0711 [and cmb]; Mattingly PRD(08)-a0708 [energy-momentum non-conservation]; Zohren PhD(08)-a0905 [and dynamical triangulations]; Sorkin JPCS(09)-a0910 [field of moving charge]; Knuth & Bahreyni a1005 [and special relativity]; Philpott PhD-a1009; Surya CQG(12)-a1110 [2D, phase transition]; Wüthrich JGPS(12)-a1207 [meaning]; Dowker GRG(13) [rev]; > s.a. black-hole entropy; causality [probabilistic]; entropy bound; information.

Kinematics > s.a. Alexandrov Sets; causal structure and spacetime; spin-foam models.
* Hauptvermutung: (Original version) If a causal set can be faithfully embedded in two Lorentzian manifolds (M, g) and (M', g'), then those two manifolds are close down to scales of the order of the embedding density.
* Coarse graining: A random coarse-graining procedure consists in starting with a causal set C and removing each point with probability p.
* Feature: Can implement the notion that spacetime topology may be scale-dependent; No known continuum approach can do this.
@ And posets: Low JMP(00); Droste JMP(05)gq [universal past-finite causal set]; Vatandoost & Bahrampour JMP(11) [and sphere orders].
@ Dimension: Meyer PhD(88), Ord(93); Reid PRD(03)gq/02; Eichhorn & Mizera a1311 [spectral dimension].
@ And continuum: Bombelli & Meyer PLA(89); Daughton CQG(98) [symmetric case]; Brightwell & Gregory PRL(91); Filk CQG(01)gq [time]; Requardt JMP(03)gq/01 [renormalization group]; Ilie et al CQG(06)gq/05 [longest paths and geodesics]; Henson CQG(06)gq [manifoldlike causal sets]; Brightwell et al CQG(08)-a0706, JPCS(09) [2D model]; Surya TCS(08)-a0712 [topology]; Rideout & Wallden CQG(09)-a0810, JPCS(09)-a0811 [lengths]; Benincasa & Dowker PRL(10)-a1001 [scalar curvature]; Krugly a1006 [unfaithful embeddings and matter]; Glaser & Surya a1309 [proposed definition of locality]; Saravani & Aslanbeigi a1403.
@ Thickened spatial hypersurfaces: Major et al CQG(06)gq/05; Major et al JMP(07)gq/06; Major et al CQG(09)-a0902 [stable homology and manifoldlikeness].
@ Specific types of metrics: He & Rideout CQG(09)-a0811 [Schwarzschild]; D'Ariano & Tosini a1008, a1109/SHPMP [Minkowski].

* Idea: The formulation of dynamics must ultimately be done in the context of a quantum theory, the most promising approach being a sum-over-histories one, for example with amplitudes of the type U(A, B) = ∑ paths exp{i S/}; Until a quantum framework can be developed, classical models can provide useful insights.
* Sequential growth dynamics: A classical stochastic evolution scheme in which posets are sequentially grown, with covariance and causality restrictions; Each new element is assigned a probability of being related to each existing one.
* Other examples: 2000, An amplitude exp{–bR} has been tested by Reid & Sorkin, but no published results.
@ Sequential growth: Sorkin IJTP(97)gq, IJTP(00)gq; Rideout & Sorkin PRD(00)gq/99, PRD(01)gq/00; Martin et al PRD(01)gq/00 [cosmology]; Rideout PhD(01)gq/02; Varadarajan & Rideout PRD(06)gq/05 [solution]; Georgiou RSA(05) [random binary growth]; Krugly a1106, a1112; Gudder a1108, a1108 [quantum]; Krugly & Stepanian a1111-conf, Krugly a1201-conf [directed dyadic acyclic graph]; Gudder a1204, a1305 [quantum sequential growth].
@ Sequential growth, mathematical properties: Alon et al AAP(94) [transitive percolation]; Ash & McDonald JMP(03)gq/02 [characterization], JMP(05) [Markov chains and posts]; Gudder a1208 [the causal poset is directed but not lattice ordered].
@ Other proposals and matter: Criscuolo & Waelbroeck CQG(99)gq/98 [percolation]; Raptis IJTP(00)gq/99; Blute et al IJTP(03)gq/01 [framework]; Zizzi gq/02; Foster & Jacobson JHEP(04)ht [2D growing lattice]; Sverdlov & Bombelli CQG(09)-a0801 [action in causal set terms, + scalar], JPCS(09)-a0905 [+ scalar + gauge field]; Bolognesi a1004 [deterministic]; Gudder a1204, a1303.
@ d'Alambertians: Dowker & Glaser CQG(13)-a1305, Glaser a1311; Aslanbeigi et al a1403 [generalized].
@ Particles and fields: Johnston CQG(08)-a0806 [particle propagators], PRL(09)-a0909 [Feynman propagator]; Sverdlov a0807 [gauge theory], a0807 [bosonic fields], a0808 [spinors], PhD(09)-a0905; Philpott et al PRD(09)-a0810 [particle energy-momentum diffusion]; Philpott CQG(10)-a0911 [simulations]; Scargle & Simić eConf-a0912 [photon dispersion]; Dowker et al PRD(10)-a1009 [scalar field propagation]; Johnston PhD(10)-a1010, Sorkin JPCS(11)-a1107 [quantum fields on causal set backgrounds]; Sverdlov 12-a1201 [corrections to bosonic-field Lagrangians]; Knuth a1212-proc, Noldus a1305 [Fermions and the Dirac equation].
@ From spin networks: Markopoulou gq/97, & Smolin NPB(97)gq, & Smolin PRD(98)gq/97 [surfaces].
@ Observables: Brightwell et al gq/02-proc, PRD(03)gq/02; Dowker & Surya CQG(06)gq/05.
@ And cosmology: Ahmed et al PRD(04)ap/02 [unimodular relativity], comment Barrow PRD(07)gq/06; Kuznetsov a0706; Ahmed & Rideout PRD(10)-a0909 [de Sitter space]; Krioukov et al NatSR(12)-a1203 [and the structure of complex networks]; > s.a. cosmological constant.
@ Related topics: Benincasa et al CQG(11)-a1011 [discrete action for a 2D Lorentzian manifold].

Similar Proposals > s.a. discrete geometries; models of spacetime; quantum spacetime and proposals [branching spacetime].
@ Quantum sets, causal nets: Finkelstein PR(69), PRD(72), PRD(72), PRD(74), IJTP(88), IJTP(89), IJTP(89), et al PRD(74), CQG(97)qp/96, qp/96; Finkelstein & Gibbs IJTP(93) [and groups]; Selesnick JMP(94) [Dirac fields], JMP(95) [gauge fields]; Hitchcock qp/00; Mallios & Raptis IJTP(01)gq [sheaves], IJTP(03)gq/02; Dukovski PRD(13); > s.a. observable algebras.
@ Causal histories: Markopoulou CMP(00)gq/98, CQG(00)ht/99, NPPS(00)ht/99; Hawkins et al CQG(03)ht; Markopoulou in(09)ht/06; Livine & Terno PRD(07) [and information theory]; Knuth & Bahreyni a1209.
@ Other proposals: Hemion FP(80), IJTP(88); Rylov JMP(90) [based on world function]; Raptis gq/99, gq/01 [based on topos]; Krugly IJTP(00) [with Grassmann variables], IJTP(02) [special types, and particles]; Zizzi gq/01/GRG-conf [qubit network]; Christensen & Crane JMP(05)gq/04 [causal sites]; Stavraki G&C(06) ["causal virtual algebraic structure"]; Brout gq/06 [with energy exchange and vacuum fluctuations]; Sen 10; Elze a1001-FQXi [dual partial orders]; Marin et al a1201 [gravity from order]; Mayburov PPN-a1205 [fuzzy points/posets, and massive particle interactions]; Cortês & Smolin a1307, a1308 [energetic causal sets]; Kalogeropoulos a1309-proc [asymptotic cones]; Gudder a1311 [covariant causal sets].

Online Resources > see Sumati Surya's resource page; Wikipedia page.

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