Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1063‘scans’ the space of geometries by letting the l i ’s run continuously over allvalues compatible with the triangular inequalities.In a nutshell, Lorentzian dynamical triangulations give a definite meaningto the ‘integral over geometries’, namely, as a sum over inequivalentLorentzian gluings T over any number N d of d−simplices,∫Σ Geom(M) D[g µν ] e iS[gµν]LDT−→∑ T ∈T1C Te iSReg (T ) , (6.105)where the symmetry factor C T = |Aut(T )| on the r.h.s. is the order ofthe automorphism group of the triangulation, consisting of all maps of Tonto itself which preserve the connectivity of the simplicial lattice. We willspecify below what precise class T of triangulations should appear in thesummation.It follows from the above that in this formulation all curvatures andvolumes contributing to the Regge simplicial action come in discrete units.This can be illustrated by the case of a 2D triangulation with Euclideansignature, which according to the prescription of dynamical triangulationsconsists of equilateral triangles with squared edge lengths +a 2 . All interiorangles of such a triangle are equal to π/3, which implies that the deficitangle at any vertex v can take the values 2π − k v π/3, where k v is thenumber of triangles meeting at v. As a consequence, the Einstein–Reggeaction S Reg assumes the simple formS Reg (T ) = κ d−2 N d−2 − κ d N d , (6.106)where the coupling constants κ i = κ i (λ, G N ) are simple functions of thebare cosmological and Newton constants in d dimensions. Substituting thisinto the path sum in (6.105) leads toZ(κ d−2 , κ d ) = ∑ N de −iκ dN d∑N d−2e iκ d−2N d−2∑T | Nd ,N d−21C T, (6.107)The point of taking separate sums over the numbers of d− and (d −2)−simplices in (6.107) is to make explicit that ‘doing the sum’ is tantamountto the combinatorial problem of counting triangulations of a givenvolume and number of simplices of codimension 2 (corresponding to thelast summation in (6.107)). 12 It turns out that at least in two space–timedimensions the counting of geometries can be done completely explicitly,12 The symmetry factor C T is almost always equal to 1 for large triangulations.