Feynman Path Integral Formulation

7.4 Strong Coupling Expansion 245
At the next order one has
c 3 = Nh 3 [〈(δA)4 〉−4〈δA〉〈(δA) 3 〉−3〈(δA) 2 〉 2 + 12〈(δA) 2 〉〈δA〉 2 − 6〈δA〉 4 ]/6 ,
(7.74)
and so on. Note that the expressions in square parentheses become rapidly quite
small, O(1/Nh n ) with increasing order n, as a result of large cancellations that must
arise eventually between individual terms inside the square parentheses. In principle,
a careful and systematic numerical evaluation of the above integrals (which is quite
feasible in practice) would allow the determination of the expansion coefficients in
k for the average curvature < δA > to rather high order.
As an example, consider a non-analyticity in the average scalar curvature
R(k) = < ∫ dx √ g(x)R(x) >
< ∫ dx √ g(x) > ≈ < ∑ h δ h A h >
< ∑ h V h > , (7.75)
assumed for concreteness to be of the form of an algebraic singularity at k c , namely
R(k)
∼
k→k c
A R (k c − k) δ , (7.76)
with δ some exponent. It will lead to a behavior, for the general term in the series
in k, of the type
(−1) n (δ − n + 1)(δ − n + 2)...δ
A R
n!kc
n−δ k n . (7.77)
Given enough terms in the series, the singularity structure can then be investigated
using a variety of increasingly sophisticated series analysis methods.
It can be advantageous to isolate in the above expressions the local fluctuation
term, from those terms that involve correlations between different hinges. To see
this, one needs to go back, for example, to the first order expression in Eq. (7.72)
and isolate in the sum ∑ h the contribution which contains the selected hinge with
value δA, namely
∑
h
δ h A h = δ A + ∑
′ δ h A h , (7.78)
h
where the primed sum indicates that the term containing δA is not included. The
result is
∫
( ∫ 2
dμ(l 2 )(δ A) 2 dμ(l 2 )δ A)
c 1 = ∫
− ( ∫
dμ(l 2 2
)
dμ(l )) 2
∫
( ∫
dμ(l 2 )δ A ∑
′ δ h A h dμ(l 2 )δ A) ( )
∫
dμ(l 2 ) ∑
′ δ h A h
h
h
+ ∫
−
( ∫
dμ(l 2 2
.
)
dμ(l )) 2
(7.79)