Dynamical Triangulation Approach to Quantum Gravity |

**In General** > s.a. regge calculus.

* __Idea__: A path-integral approach to
spacetime and gravity in which one sums over piecewise-flat geometries with fixed
edge lengths, and varies the triangulation by adding or removing simplices with
well-defined moves; The approach is discrete, but the fixed edge length is seen as a
regulator, the idea being that the true theory is recovered in the limit in which it
approaches zero, so the discreteness is not considered as fundamental; To recover the
continuum limit, in the zero-edge-length limit one looks for critical points where the
correlation length diverges, signaling the long-range order of a smooth geometry.

* __Motivation__: One of the strengths of this approach, as for Regge calculus, is that one does not need coordinates and one bypasses the whole issue of gauge freedom; The procedure is supposed
to be ergodic in the space of geometries (is there a proof?), and one expects to be
able to get many more.

* __Drawbacks__: Recovery of the
Einstein-Hilbert action is more problematic, since diffeomorphism invariance is lost.

* __Action (4D)__: Given by *I* = 2π
*N*_{2} − 10 *αN*_{4},
with *α* = arcos(1/5), *N*_{n}
= number of *n*-simplices.

* __2D__: Dynamical triangulations are equivalent to matrix models.

* __Phase structure__: As one changes the curvature (Newton) coupling,
there is a phase transition between an elongated and a crumpled phase.

**References**

@ __General__:
Godfrey & Gross PRD(91) [more than 2D];
Ambjørn & Jurkiewicz PLB(92);
Nabutovsky & Ben-Av CMP(93) [4D, non-computability];
Ambjørn et al LNP(97)ht/96;
Schleich & Witt gq/96-proc [quantum];
Bialas et al NPPS(98)gq/97;
Loll LRR(98)gq [rev].

@ __2D__: Ambjørn CQG(95);
Ambjørn & Budd APPB(14)-a1310-ln [coupled to matter].

@ __3D__: Carfora & Marzuoli IJMPA(93) [and Reidemeister torsion];
Egawa & Tsuda PLB(98) [random surfaces].

@ __Random surfaces__:
David et al NPB(87) [critical exponents];
Migdal JGP(88).

@ __Related topics__: Renken NPB(97)hl/96 [renormalization group];
Catterall et al PLB(98) [singular geometries];
Henson CQG(09)-a0907 [coarse-graining].

@ __Phase structure, transitions__:
Agishtein & Migdal NPB(92),
MPLA(92);
Varsted NPB(94) [and continuum limit];
Brügmann & Marinari JMP(95) [no exponential bound];
de Bakker PLB(96) [phase transition, first-order];
Renken et al NPB(98),
Warner et al PLB(98) [3D];
Warner & Catterall PLB(00)hl [4D, with boundary];
Laiho & Coumbe PRL(11)-a1104,
Coumbe & Laiho a1201-PoS [Euclidean, asymptotic safety and spectral dimension];
Rindlisbacher & de Forcrand a1311-conf,
JHEP-a1503
[4D Euclidean, the phase transition is 1st order].

@ __Euclidean__: Brügmann PRD(93),
& Marinari PRL(95) [4D, measure];
Ambjørn et al JMP(95)ht [2D];
Veselov & Zubkov PLB(04) [10D];
Coumbe & Laiho JHEP(15)-a1401 [non-trivial measure term];
Laiho et al PRD(17)-a1604 [and asymptotic safety];
Laiho et al a1701-proc [recent results].

@ __With matter__: in Loll LRR(98)gq; in Ambjørn et al PRP(12)-a1203.

__Related subjects__:
see causal dynamical triangulations;
spin-foam models [spincube models].

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send feedback and suggestions to bombelli at olemiss.edu – modified 28 jul 2019