Ivancevic_Applied-Diff-Geom

1068 Applied Differential Geometry: A Modern Introductionsum (6.110) would lead to a super–exponential growth of at least ∝ N d ! ofthe number of triangulations with the volume N d . Such a divergence of thepath integral cannot be compensated by an additive renormalization of thecosmological constant of the kind outlined above.There are ways in which one can sum divergent series of this type, forexample, by performing a Borel sum. The problem with these stems fromthe fact that two different functions can share the same asymptotic expansion.Therefore, the series in itself is not sufficient to define the underlyingtheory uniquely. The non–uniqueness arises because of non–perturbativecontributions to the path integral which are not represented in the perturbativeexpansion. 16 In order to fix these uniquely, an independent, non–perturbative definition of the theory is necessary. Unfortunately, for dynamicallytriangulated models of quantum gravity, no such definitions havebeen found so far. In the context of 2D (Euclidean) quantum gravity thisdifficulty is known as the ‘absence of a physically motivated double-scalinglimit’ [Ambjørn and Kristjansen (1993)].Lastly, getting an interesting continuum limit may or may not require anadditional fine–tuning of the inverse gravitational coupling κ d−2 , dependingon the dimension d. In four dimensions, one would expect to find a secondordertransition along the critical line, corresponding to local gravitonicexcitations. The situation in d = 3 is less clear, but results get so farindicate that no fine–tuning of Newton’s constant is necessary [Ambjørnet. al. (2001b); Ambjørn et. al. (2001c)].Before delving into the details, let us summarize briefly the results thathave been get so far in the approach of Lorentzian dynamical triangulations.At the regularized level, that is, in the presence of a finite cutoff a for theedge lengths and an infrared cutoff for large space–time volume, they arewell–defined statistical models of Lorentzian random geometries in d =2, 3, 4. In particular, they obey a suitable notion of reflection-positivityand possess self–adjoint Hamiltonians.The crucial questions are then to what extent the underlying combinatorialproblems of counting all dD geometries with certain causal propertiescan be solved, whether continuum theories with non–trivial dynamics existand how their bare coupling constants get renormalized in the process.What we know about Lorentzian dynamical triangulations so far is thatthey lead to continuum theories of quantum gravity in dimension 2 and 3.In d = 2, there is a complete analytic solution, which is distinct from the16 A field–theoretic example would be instantons and renormalons in QCD.