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,
?].
main page – abbreviations – journals – comments – other
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send feedback and suggestions to bombelli at olemiss.edu – modified 13
nov 2009