Feynman Path Integral Formulation

5.2 Semiclassical Expansion 147
with θ ranging from 0 to π, r 0 = √ 3/λ and a < r 0 . For sufficiently small boundary
three-geometries, a ≪ r 0 , the metric interior to the boundary approaches that of
Euclidean flat space (with θ → t),
ds 2 = dt 2 + t 2 ĝ ij dφ i dφ j . (5.21)
In this limit one would expect therefore that a calculation of the quantum corrections
could be carried out in a flat background with λ = 0, with higher order corrections
then involving the ratio between the two radii in question, O(a 2 /r0 2 ). Then in the
expansion of the metric of Eq. (5.10) one just has g μν = η μν , the flat metric.
The first step in such a program is therefore a re-writing of the expansion of
the Euclidean action to quadratic order in the h field, and specifically of Î 2 [h] in
Eqs. (5.12) and (5.13), in terms of the quantities t and ĝ ij appearing in the metric in
Eq. (5.21). One finds
Î 2 [h] =− 1 ∫
d 4 x { √ gp ij ∂ t h ij − H T − H L − h 00 C − h 0i C I } , (5.22)
16πG
with background three-metric g ij = t 2 ĝ ij , √ g the square root of the determinant of
this g ij , π ij = √ gp ij the momentum conjugate to the quantum fluctuation h ij , and
H T = √ [
)]
g p ij p ij + 2t −1 p ij h ij − 1 4
(∇ k h ij ∇ k h ij + 2t −2 h ij h ij
H L = 1 √
[
2 g p i i p j j + t−1 p i ih j j
+ 1 2
(∇ k h j j ∇k h j j + 2∇ ih ij (∇ k h k j − ∇ j h k k ) − 1 2 7t−2 (h i i) 2)]
C = 1 √
[
]
2 g ∇ i ∇ j h ij − ∇ k ∇ k h i i + 2t −1 p i i − 1 2 5t−2 h i i
C i = − √ [
]
g 2∇ j p ji − t −1 ∇ i h j j
. (5.23)
The metric components h 00 and h 0i act as Lagrange multiplier, giving four constraints
C = 0 C i = 0 . (5.24)
The physical subspace is obtained by imposing a gauge condition, in the case at hand
one that leads to a substantial simplification of the problem. The gauge condition is
∇ i h ij = 0 h i i = 0 , (5.25)
and restricts the functional integration over transverse-traceless (TT) modes only.
Due the decomposition of Î 2 in Eq. (5.22) into transverse and longitudinal contributions,
H L only contains longitudinal and trace parts, and therefore vanishes.
In the physical phase space spanned by the p (π ij
TT ≡ √ gp ij
TT
) and q (hTT ij )variables
one has
∫
P[a] =
[dπ ij
TT ][dhTT ij ] exp { −Î 2 (π TT ,h TT ) } , (5.26)