Feynman Path Integral Formulation

5.2 Semiclassical Expansion 149
logP[a] = 1 2 ζ ′ (0)+ζ (0) log(4π l 2 P μ 2 ) . (5.34)
The dependence of P[a] on a can be determined by a scaling argument. When rescaling
the original radius a appearing in the background metric g μν by a → ka
the Laplacian ∇ λ ∇ λ scales like 1/k 2 , and therefore ζ (s) by k 2s . The first term in
Eq. (5.34) therefore gives a contribution to P[a] of the form a ζ (0) .
The second contribution in Eq. (5.34) depends on μ, and therefore on the specific
details of the functional measure [dh TT
μν]. The measure factor comes in through a
local weight h p/2 ,
∫
∫
[dh TT
μν] =
∏ [deth(x)] p/2 dh TT
μν(x) , (5.35)
x
[see Eqs. (2.18) and (2.22)]. If the above measure is chosen to be scale invariant
(as in Faddeev and Popov, 1973), then the second contribution in Eq. (5.34) is scale
independent, i.e. k- ora-independent. Then only the first contribution matters, and
one has simply, to this order in the semiclassical expansion,
P[a] =N a ζ (0) , (5.36)
where N is an a-independent normalization constant. In general though one would
expect P[a], as defined here, to be sensitive to short-distance details of the theory,
and to contain some dependence on the details of the regularization procedure of
the measure. The general problem in these types of calculations seems to be the
difficulty in decoupling the short distance details of the ultraviolet cutoff μ, which
is required to make the product ∏ x in Eq. (5.35) well defined, from the other short
distance quantity appearing in this problem, namely the spatial scale for the threemetric
a.
The last point that needs to be addressed therefore is a determination of ζ (s)
from the eigenvalues of the Laplacian ∇ 2 , acting on transverse traceless tensors
vanishing on the boundary of the three-sphere. Several sub-steps are involved in this
calculation, which we will summarize here. The first is to establish a relationship
between the function ζ (s) and the short time expansion for the kernel of the heat
equation (for an accessible elementary introduction to the methods of zeta function
regularization for functional determinants see for example Ramond, 1990). One can
write for ζ (s) the integral representation
ζ (s) = 1 ∫ ∞
dτ τ s−1 G (τ) , (5.37)
Γ (s) 0
with G (τ) defined as
G (τ) =
∫
d 4 x G μν μν(x,x;τ) =∑exp(−λ n τ) . (5.38)
n
Here G μν,ρσ (x,x ′ ;τ) is a transverse-traceless Green’s function for the heat equation