Dynamical Triangulation Approach to Regge Calculus  

In General
* Idea: A path-integral approach to spacetime geometry in which one sums over piecewise-flat geometries with fixed edge lengths, and varies the triangulation by adding/removing simplices with well-defined moves.
* Motivation: Supposed to be ergodic in the space of geometries, 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 N2 – 10 αN4, with α = arcos(1/5), Nn = number of n-simplices.
* 2D: Dynamical triangulations are equivalent to matrix models.
* Phase transition: As one changes the curvature (Newton) coupling, 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 CQG(95) [2D]; Ambjørn et al LNP(97)ht/96; Schleich & Witt gq/96-proc NPPS(98)gq/97.
@ 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]; Ambjørn & Budd APPB(14)-a1310-ln [2D, coupled to matter]; Coumbe & Laiho JHEP(15)-a1401 [non-trivial measure term]; Laiho et al a1604 [and asymptotic safety]; Laiho et al a1701-proc [recent results].

blue bullet Related subjects: see causal dynamical triangulations; spin-foam models [spincube models].


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