Feynman Path Integral Formulation

7.4 Strong Coupling Expansion 243where, as customary, the lattice ultraviolet cutoff is set equal to one (i.e. all lengthscales are measured in units of the lattice cutoff). For definiteness the measure willbe of the form ∫ ∫ ∞[dl 2 ]= ∏ [V d (s)] σ ∏ dlij 2 Θ[lij] 2 . (7.65)0 sijThe lattice partition function Z latt should be compared to the continuum EuclideanFeynman path integral of Eq. (2.34),∫Z cont =[dg μν ] e −λ ∫ dx √ g+ 16πG1 ∫ √ dx gR. (7.66)When doing an expansion in the kinetic term proportional to k, it will be convenientto include the λ-term in the measure. We will set therefore in this Section as inEq. (6.93)dμ(l 2 ) ≡ [dl 2 ]e −λ 0 ∑ h V h. (7.67)It should be clear that this last expression represents a fairly non-trivial quantity,both in view of the relative complexity of the expression for the volume of a simplex,Eq. (6.5), and because of the generalized triangle inequality constraints alreadyimplicit in [dl 2 ]. But, like the continuum functional measure, it is certainly local,to the extent that each edge length appears only in the expression for the volume ofthose simplices which explicitly contain it. Also, we note that in general the integral∫ dμ can only be evaluated numerically; nevertheless this can be done, at leastin principle, to arbitrary precision. Furthermore, λ 0 sets the overall scale and cantherefore be set equal to one without any loss of generality.Thus the effective strong coupling measure of Eq. (7.67) has the properties that(a) it is local in the lattice metric of Eq. (6.3), to the same extent that the continuummeasure is ultra-local, (b) it restricts all edge lengths to be positive, and (c)it imposes a soft cutoff on large simplices due to the λ 0 -term and the generalizedtriangle inequalities. Apart from these constraints, it does not significantly restrictthe fluctuations in the lattice metric field at short distances. It will be the effect ofthe curvature term to restrict such fluctuation, by coupling the metric field betweensimplices, in the same way as the derivatives appearing in the continuum Einsteinterm couple the metric between infinitesimally close space-time points.As a next step, Z latt is expanded in powers of k,∫Z latt (k) = dμ(l 2 ) e k ∑ h δ h A h=∞∑n=0∫1n! kn( ) ndμ(l 2 ) ∑δ h A h . (7.68)hIt is easy to show that Z(k) =∑ ∞ n=0 a n k n is analytic at k = 0, so this expansion shouldbe well defined up to the nearest singularity in the complex k plane. A quantitativeestimate for the expected location of such a singularity in the large-d limit will begiven later in Sect. 7.6. Beyond this singularity Z(k) can sometimes be extended, for