Feynman Path Integral Formulation

8.3 Invariant Local Gravitational Averages 277
rescaling of the edge lengths. As in the continuum, they are proportional to first and
second 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 √ gdx
in the continuum). Consider the average volume per site
and its fluctuation, defined as
〈V 〉≡ 1 <
N 0
∑V h > , (8.15)
h
χ 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 again
simply related to derivatives of Z latt with respect to the bare cosmological constant
λ 0 , as for example in
∼ ∂ lnZ latt (8.17)
∂λ 0
and
χ V (k) ∼ ∂ 2
lnZ latt . (8.18)
∂λ 2 0
Some useful relations and sum rules can be derived, which follow directly from
the scaling properties of the discrete functional integral. Thus a simple scaling argument,
based on neglecting the effects of curvature terms entirely (which, as will
be seen below, vanish in the vicinity of the critical point), gives an estimate of the
average volume per edge [for example from Eqs. (7.148) and (7.149)]
∼
2(1 + σ d)
λ 0 d
∼
d=4, σ=0
1
2λ 0
, (8.19)
where σ is the functional measure parameter in Eqs. (2.27) and (6.76). In four dimensions
direct 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 the
action and measure. The bare couplings k and λ 0 in the gravitational action are
dimensionful 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 .Asa
consequence the path integral for pure gravity,
∫
Z latt (λ 0 ,k,a,b) = [dl 2 ] e −I(l2) , (8.20)
obeys the scaling law