Causal
Sets| 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, gq/03-in;
Reid CJP(01)gq/99;
in Markopoulou gq/02-in;
Dowker gq/05-in,
CP(06);
Henson gq/06-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].
@ 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]; > s.a. black hole
entropy, entropy
bound.
Kinematics > s.a. causal
structure and spacetime; spin foam.
* 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].
@ Dimension: Meyer PhD(88), Ord(93); Reid PRD(03)gq/02.
@ 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 [2D
model]; Surya a0712-JTCS
[topology].
@ Thickened spatial hypersurfaces: Major et al CQG(06)gq/05;
Major et al JMP(07)gq/06.
Dynamics
* 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 gq/02-PhD;
Varadarajan & Rideout PRD(06)gq/05 [solution];
Georgiou RSA(05)
[random binary 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].
@ 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;
Sverdlov & Bombelli a0801 [action
in causal set terms, + scalar]; Johnston a0806 [particle
propagators].
@ Dynamics of matter: Sverdlov a0807 [gauge
theory].
@ From spin networks: Markopoulou gq/97, & Smolin NPB(97)gq, & Smolin PRD(98)gq/97 [surfaces].
@ Observables:
Brightwell et al gq/02-in, PRD(03)gq/02;
Dowker & Surya CQG(06)gq/05.
@ And the cosmological constant: Ahmed et al PRD(04)ap/02 [unimodular
relativity], comment
Barrow PRD(07)gq/06;
Kuznetsov a0706; > s.a. cosmological
constant.
Similar Proposals > s.a. 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; > 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 ht/06-in;
Livine & Terno PRD(07)
[and information theory].
@ 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
[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].
Online Resources > see Einstein Online page; Sumati Surya's resource page; Wikipedia page.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
28 jul 2008