Feynman Path Integral Formulation

8.3 Invariant Local Gravitational Averages 277rescaling of the edge lengths. As in the continuum, they are proportional to first andsecond derivatives of Z latt with respect to k.One can contrast the behavior of the preceding averages, related to the curvature,with the corresponding quantities involving the local volumes V h (the quantity √ gdxin the continuum). Consider the average volume per siteand its fluctuation, defined as〈V 〉≡ 1 , (8.15)hχ V (k) ≡ < (∑ hV h ) 2 > − < ∑ h V h > 2< ∑ h V h >, (8.16)where V h is the volume associated with the hinge h. The last two quantities are againsimply related to derivatives of Z latt with respect to the bare cosmological constantλ 0 , as for example in ∼ ∂ lnZ latt (8.17)∂λ 0andχ V (k) ∼ ∂ 2lnZ latt . (8.18)∂λ 2 0Some useful relations and sum rules can be derived, which follow directly fromthe scaling properties of the discrete functional integral. Thus a simple scaling argument,based on neglecting the effects of curvature terms entirely (which, as willbe seen below, vanish in the vicinity of the critical point), gives an estimate of theaverage volume per edge [for example from Eqs. (7.148) and (7.149)] ∼2(1 + σ d)λ 0 d∼d=4, σ=012λ 0, (8.19)where σ is the functional measure parameter in Eqs. (2.27) and (6.76). In four dimensionsdirect numerical simulations with σ = 0 (corresponding to the lattice De-Witt measure) agree quite well with the above formula.Some exact lattice identities can be obtained from the scaling properties of theaction and measure. The bare couplings k and λ 0 in the gravitational action aredimensionful in four dimensions, but one can define the dimensionless ratio k 2 /λ 0 ,and rescale the edge lengths so as to eliminate the overall length scale √ k/λ 0 .Asaconsequence the path integral for pure gravity,∫Z latt (λ 0 ,k,a,b) = [dl 2 ] e −I(l2) , (8.20)obeys the scaling law