Discrete Spaces and Geometries  

General Concepts on Discrete Spaces > s.a. cell complex; combinatorics; Continuum; forms; mathematics [finite]; spacetime [points].
* Discrete subsets of metric spaces: A subset SX of a metric space (X, d) is discrete if every point in S is isolated (S doesn't contain any accumulation points); The set S is uniformly discrete if there exists an ε > 0 such that for any two distinct x, yS, d(x, y) > ε.
@ General references: Sorkin in(83), IJTP(91); Balachandran et al NPB(94)ht/93; Immirzi NPPS(97)gq [and canonical quantum gravity]; Bianchi & Haggard PRL(11)-a1102 [from Bohr-Sommerfeld quantization]; Rovelli a1107 [reparametrization-invariant systems]; Carfora & Marzuoli 12 [moduli spaces, polyhedral manifolds, quantum geometry, etc]; Zahn & Kraus EJP(14)-a1405 [sector models, pedagogical approach to piecewise linear geometries]; Beck & Robins 15 [polyhedra and lattices]; Rovelli & Zatloukal a1902 [discrete differential calculus].
@ Discrete topological spaces: Zapatrin IJTP(93), IJTP(98)gq/97; Parfionov & Zapatrin gq/97 [and histories quantum theory]; Efremov & Mitskievich gq/03 [evolving T0 topologies].
@ Discrete manifolds: Dimakis & Müller-Hoissen JPA(94), et al JMP(95)ht/94; Williams JMP(95) [invariants]; Zapatrin JMP(97) [as polyhedra and spatial posets]; Dimakis & Müller-Hoissen JMP(03)mp/02 [differential geometry]; de Beauce & Sen ht/06-conf [discrete interior product]; > s.a. graph theory.
@ Differential geometry: Forgy & Schreiber mp/04 [including pseudo-Riemannian]; Bombelli & Lorente AIP(06)gq/05 [curvature]; Lev & Olevskii RMI-a1501 [measures with discrete support and spectrum]; Conboye et al CQG(15)-a1502 [distributed mean curvature]; > s.a. curvature; diffeomorphisms.
> Related concepts: see analysis [on discrete spaces]; connections and geodesics; Covariance [discrete model]; integration [combinatorial Stokes formula].
> Online resources: see Wikipedia page; arXiv blog [is spacetime countable?].

Discrete Spacetime > s.a. discretized or lattice gravity; modified quantum mechanics; regge calculus; sheaf; world function.
* Roots: The origin of the modern idea that nature is discrete can be traced to the ancient Greek atomism of Leucippus, Democritus and Epicurus, and to the book De rerum natura by Lucretius; In terms of space and time, Riemann suggested that a spatial metric could be explained if space was discrete, and Heisenberg noticed that infinities in quantum field theory would go away if there was a minimum spatial distance.
* Mathematical motivation: The possibility that gravity admits a discrete, combinatorial formulation in terms of triangulations is suggested by the fact that up to 6D, smooth manifolds up to diffeomorphism are characterized by their Whitehead triangulations up to PL-isomorphisms.
* Physical motivation: Desire to avoid singularities in gravity theory, and infinities in black-hole entropy; Developments in loop quantum gravity; One argument suggests that quantum gravity effects modify the Heisenberg uncertainty principle to a generalized uncertainty principle (GUP), GUP-induced corrections to the Schrödinger equation lead to the quantization of length, and corrections to the Klein-Gordon and Dirac equations give rise to length, area and volume quantizations; > s.a. quantum gravity phenomenology.
@ Books, reviews: Silberstein 36; March 50; Lorente in(95)gq/03; Regge & Williams JMP(00)gq; Hagar 14.
@ General references: Ambarzumian & Iwanenko ZP(30); March ZP(36), ZP(37), ZP(37); Heisenberg ZP(38); Schild PR(48), CJM(49); Coxeter & Whitrow PRS(50); Darling PR(50); Hill PR(55) [rational Poincaré transformations]; Coish PR(59); Das NC(60); Stiegler PPS(63); Ahmavaara JMP(65), JMP(65), JMP(66), JMP(66); Takano PTP(67), PTP(67); Bohm et al IJTP(70); Cole IJTP(72), IJTP(72) [observer-dependent cellular structure]; Lorente IJTP(76), IJTP(86); Shale AiM(79); Cobb & Smalley IJTP(82); Noyes & McGoveran PE(89); Haag CMP(90); Orland ht/93 [critical solid]; Hillman PhD(95)ht/98; El Naschie CSF(05) ['t Hooft's views]; Zhang a1003; Oriti a1107-FQXi, a1302/SHPMP; Zahedi a1501-conf [digital philosophy]; Crouse a1608; Khosravi & Saueressig a1908-GRF [spacetime cannot be discrete]; Raasakka a1908 [argument for granularity of spacetime]; Kabernik a2002 [and emergence of classicality in quantum theory].
@ Operator coordinates: Hellund & Tanaka PR(54).
@ Discrete time: Lee PLB(83), in(85); Wolf NCB(95); 't Hooft CQG(99)gq/98; Bruce PRA(01)qp [in quantum mechanics]; Khrennikov & Volovich qp/02 [particle interference], OSID(06)qp [H atom]; Budd & Loll CQG(09)-a0906 [in 2+1 quantum gravity]; He a0911 [proposed test]; Sloan SHPMP(15) [possibilities and implications]; Baez's & Unruh's opinions sa(18); Christodoulou & Rovelli a1812 [proposed test]; > s.a. time in quantum theory.
@ Continuous-discrete relationship: Das & Ghosh a1006 [based on Landau theory of liquid-solid phase transition]; Kempf NJP(10)-a1010 [continuous and discrete, information-theory approach]; Eichhorn & Koslowski PRD(14)-a1408 [phase transition]; Calcagni et al PRD(15)-a1412 [dimensional flow]; Thürigen PhD(15)-a1510 [effective dimension]; Pesci a1511, CQG(19)-a1812 [density of spacetime atoms]; Sadun a1610-in [history, spacetime is not discrete]; Kempf FP(18)-a1803-talk [neither discrete nor continuous]; > s.a. Shape Dynamics [no fundamental discreteness].
@ Other proposals: Raptis & Zapatrin IJTP(00)gq/99 [correspondence principle]; Afanas'ev ht/00; Mathur IJMPD(03)ht-GRF [bits and expansion]; Knight IJTP(03) [dislocations]; Rauch IJTP(03) [without synchronization or regularity]; 't Hooft FP(08)-a0804 [straight pieces of string]; Gudder RPMP(12)-a1108 [quantum sequential growth]; Finkelstein a1201-proc; Weinfurtner et al JPA(14)-a1210 [as a lattice gas model]; Adler a1402 [discrete space, cubic grid]; Trugenberger PRD(15)-a1501, PRE(15)-a1507 [information network, self-organization].

Other Approaches > s.a. discrete spacetime models.
@ And the GUP: Ali et al PLB(09)-a0906; Das et al PLB(10)-a1005; Abutaleb AHEP(13)-a1309; Deb et al PLB(16)-a1601 [and spacetime curvature].
@ And fermions: Diethert et al IJMPA(08)-a0710; > s.a. fermions [causal fermion systems].
> Proposals: see Non-Associative Geometry; path-integral quantum gravity; quantum spacetime and proposals.

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