Discrete Spaces and Geometries  

General Concepts on Discrete Spaces > s.a. cell complex; combinatorics; Continuum; forms; mathematics [finite]; spacetime [points].
* Models: One possible way to group them into different types is:
- Combinatorial, non-topological versions, such as posets or graphs;
- Topological versions, such as simplicial or cell complexes;
- Piecewise-flat versions, such as Regge calculus, dynamical triangulations or twisted geometries;
To establish a connection with continuum geometry, each proposal of one of the above types seems to need the ones below as an intermediate step.
* 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 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]; Beck & Robins 15 [polyhedra and lattices].
@ 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. diffeomorphisms [discrete].
> 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.
* 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.
@ 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]; > 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 a1510 [effective dimension]; Pesci a1511 [density of spacetime atoms]; Sadun a1610-in [history, spacetime is not discrete]; > s.a. Shape Dynamics [no fundamental discreteness].
@ 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].
@ Causal fermion systems: Finster in(06)gq [variational principle]; Finster a1605-book [continuum limit]; > s.a. emergence; Initial-Value Problem.
@ 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]; Diethert et al IJMPA(08)-a0710 [from fermion system]; '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].

Approaches / Proposals > see discrete models; Non-Associative Geometry; path-integral quantum gravity; quantum spacetime and proposals.


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