Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1067The Euclideanized path sum after the Wick rotation has the formZ eu (κ d−2 , κ d ) = ∑ T1C Te −κ dN d (T )+κ d−2 N d−2 (T )= ∑ N de −κ dN d∑T | Nd1C Te κ d−2N d−2 (T )= ∑ N de −κ dN de κcrit d (κ d−2)N d× subleading(N d ). (6.110)In the last equality we have used that the number of Lorentzian triangulationsof discrete volume N d to leading order scales exponentially with N d forlarge volumes. This can be shown explicitly in space–time dimension 2 and3. For d = 4, there is strong (numerical) evidence for such an exponentialbound for Euclidean triangulations, from which the desired result for theLorentzian case follows (since W maps to a strict subset of all Euclideansimplicial manifolds).From the functional form of the last line of (6.110) one can immediatelyread off some qualitative features of the phase diagram, an example of whichappeared already earlier in Figure 6.8. Namely, the sum over geometriesZ eu converges for values κ d > κ critdof the bare cosmological constant, anddiverges (ie. is not defined) below this critical line. Generically, for allmodels of dynamical triangulations the infinite–volume limit is attained byapproaching the critical line κ critd(κ d−2 ) from above, ie. from inside theregion of convergence of Z eu . In the process of taking N d → ∞ and thecutoff a → 0, one gets a renormalized cosmological constant Λ through(κ d − κ critd ) = a µ Λ + O(a µ+1 ). (6.111)If the scaling is canonical (which means that the dimensionality of the renormalizedcoupling constant is the one expected from the classical theory),the exponent is given by µ = d. Note that this construction requires a positivebare cosmological constant in order to make the state sum converge.Moreover, by virtue of relation (6.111) also the renormalized cosmologicalconstant must be positive. Other than that, its numerical value is not determinedby this argument, but by comparing observables of the theory whichdepend on Λ with actual physical measurements. 15 Another interestingobservation is that the inclusion of a sum over topologies in the discretized15 The non–negativity of the renormalized cosmological coupling may be taken as afirst ‘prediction’ of our construction, which in the physical case of four dimensions isindeed in agreement with current observations.