Discrete Geometries and Spacetime Models

General Concepts on Discrete Spaces > s.a. cell complex; combinatorics; Continuum; forms.
@ General references: Sorkin in(83), IJTP(91); Balachandran et al NPB(94)ht/93; Immirzi NPPS(97)gq [and canonical quantum gravity].
@ 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-in [discrete interior product]; > s.a. graph theory.
@ Differential geometry: Forgy & Schreiber mp/04 [including pseudo-Riemannian]; Bombelli & Lorente gq/05-in [curvature]; > s.a. diffeomorphisms [distributional].
> Related concepts: see Covariance [discrete model]; analysis [on discrete spaces]; integration [combinatorial Stokes formula].

Discrete Spacetime > s.a. modified quantum mechanics; 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.
@ Reviews: Lorente in(95)gq/03; Regge & Williams JMP(00)gq;
@ General references: Ambarzumian & Iwanenko ZP(30); Silberstein 36; March ZP(36), ZP(37), ZP(37), 50; 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 ht/98-PhD; El Naschie CSF(05) ['t Hooft's views].
@ 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.
@ Discretizations of continuum gravity: de Albuquerque et al PRL(03)ht, MPLA(03)ht-in [Euclidean, non-commutative spectral principle dynamics]; Gambini & Pullin in(05)gq, IJMPD(06)gq/05-in; > s.a. lattice gravity.
@ Snyder proposal: Snyder PR(47) [coordinates as operators]; Guo et al FPC(07)ht/06, Guo ht/06-in [and de Sitter invariance]; Romero & Zamora PLB(08)-a0802 [quantum of area].
@ 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]; Finster in(06)gq [variational principle]; Diethert et al IJMPA(08)-a0710 [from fermion system]; 't Hooft FP(08)-a0804 [straight pieces of string]; Ali et al PLB(09)-a0906 [from gup].
> Approaches, proposals: see path-integral quantum gravity; proposals for quantum spacetime; regge calculus.

Models > s.a. 3D quantum gravity; canonical quantum gravity; causal sets; physics [ultimate theory proposals]; renormalization.
@ Spin networks and lqg: in Penrose 66, pr(66), in(70), in(71), in(72), in(79); Hasslacher & Perry PLB(81); Moussouris PhD(83); Kauffman IJMPA(90); LaFave gq/93; Zapata GRG(98)gq/97; > s.a. spin foam.
@ Other graphs / networks: Antonsen PhD(92), IJTP(94); Requardt ht/96, ht/96, ht/96-in [random graphs], JPA(98)ht, gq/99, CQG(00)gq/99, & Roy CQG(01)gq/00 [and random metric space]; Requardt gq/03; Markopoulou & Smolin PRD(04)gq/03; Konopka et al ht/06 [complete lattice, phase transition]; Efremov et al a0706 [BF theory on graph cobordisms]; Kan PTPS-a0712 [and cosmology]; Konopka et al PRD(08)-a0801 [quantum graphity]; Konopka PRD(08)-a0805 [statistical mechanics of graphity].
@ Cellular automaton: Feynman; Finkelstein PR(69); 't Hooft NPB(90), et al NPB(92), gq/93; Requardt ht/95 [varying connections]; Zizzi gq/01-in, gq/01, in(04)gq/03 [computational network]; Eakins & Jaroszkiewicz gq/03.
@ Fuzzy: Francis phy/99-wd, phy/99 [and particles]; Zizzi MPLA(05)gq/04-in [N-qubits on quantum space].
@ Condensed-matter-inspired models: Tahim et al MPLA(09)-a0705 [deformable solid]; 't Hooft IJMPA(09) [4D crystal with defects].
@ Other models: Noyes 01; Galiautdinov IJTP(02)ht-PhD [and spin-1/2 particles]; Buniy et al PLB(05)ht [and discrete Hilbert space]; Tanaka FPL(06)ht/05 [Buniy-Hsu-Zee model and Yang's quantized spacetime algebra]; Zachos IJMPA(80)-a0710 [umbral deformation as discretization of continuum physics]; > s.a. approaches to quantum gravity; non-commutative geometry; quantum spacetime models [categorical].

Particles and Fields > s.a. Continuous Media; quantum particles.
@ General references: Dimakis et al JMP(95)ht/94 [fields]; Bojowald & Date CQG(04)gq/03 [consistency conditions on differential equations]; Batsin & Kilmister 09; Elze a0906-in.
@ Quantum field theory: Yamamoto PRD(85) [bosons and fermions]; Finster ATMP(07)mp/06, JPCS(07)ht/06, in(09)-a0712 [fermions].
> Theories in discrete settings: see dirac fields; electromagnetism in curved spacetime; field theory [discretizations]; graphs in physics; modified quantum field theories.

Consequences > s.a. quantum-gravity phenomenology; singularities; ultra-high-energy cosmic rays.
@ Observability: Amelino-Camelia CzJP(98)gq-in [discrete lengths and areas]; Rivero gq/06 [consequences]; Dreyer a0805 [cosmological perturbation spectrum].
@ Compatible with Lorentz symmetry: Rovelli & Speziale PRD(03)gq/02; Livine & Oriti JHEP(04)gq [toy model + DSR quantum geometry]; Bombelli et al MPLA(09)gq/06 [no breaking in random sprinklings]; Bonder a0801-in [model, and Newtonian limit].
@ Lorentz symmetry, other: Sidharth IJTP(04); Dowker et al MPLA(04)gq/03 [not necessary]; Vergeles NPB(06)ht/05; Henson gq/06; > s.a. lorentz symmetry and modifications.
@ Continuum limit: Ling MPLA(05)gq/03 [and singularities]; Requardt IJGMP(06)mp/05 [and distance between metric spaces].
@ Related topics: Badiali JPA(05) [and statistical thermodynamics]; Bachmat gq/07 [applications, ?].

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