Feynman Path Integral Formulation

4.6 Matter Source Terms 113
One still has the same definitions as before for the (Lagrange multiplier) lapse and
shift function, namely N =(− 4 g 00 ) −1/2 and N i = g ij 4 g 0 j . The gravitational constraints
are modified as well, since now one defines
T ≡ 1
16πG H(g ij,π ij )+H φ (g ij ,π ij ,φ,π φ )
T i ≡ 1
16πG H i(g ij ,π ij )+H φ i (g ij,π ij ,φ,π φ ) , (4.61)
with the first part describing the gravitational part already given in Eq. (4.51),
H(g ij ,π ij )=G ij,kl π ij π kl − √ g 3 R + 2λ √ g
H i (g ij ,π ij )=−2g ij ∇ k π jk , (4.62)
here conveniently re-written using the (inverse of) the DeWitt supermetric of
Eq. (2.14),
G ij,kl = 1 2 g−1/2 ( g ik g jl + g il g jk + α g ij g kl
)
, (4.63)
with parameter α = −1. In the previous expression a cosmological term (proportional
to λ) has been added as well, for future reference. For the matter part one
has
H φ (g ij ,π ij ,φ,π φ )= √ gT 00 (g ij ,π ij ,φ,π φ )
H φ i (g ij,π ij ,φ,π φ )=− √ gT 0i (g ij ,π ij ,φ,π φ ) . (4.64)
We note here that the (inverse of the) DeWitt supermetric in Eq. (4.63) can be used
to define a distance in the space of three-metrics g ij (x). Consider an infinitesimal
displacement of such a metric g ij → g ij + δg ij , and define the natural metric G on
such deformations by considering a distance in function space
∫
‖δg‖ 2 = d 3 xN(x) G ij,kl (x) δg ij (x)δg kl (x) . (4.65)
Here the lapse N(x) is an essentially arbitrary but positive function, to be taken
equal to one in the following. The quantity G ij,kl (x) is the three-dimensional version
version of the DeWitt supermetric,
G ij,kl = 1 √
(
2 g g ik g jl + g il g jk + ᾱ g ij g kl) , (4.66)
with the parameter α of Eq. (4.63) related to ᾱ in Eq. (4.66) by ᾱ = −2α/(2+3α),
so that α = −1givesᾱ = −2. One can then verify that for any choice of α ≠ −2/3
(
)
G ij,ab G kl,ab = 1 2
δk i δ j
l
+ δl i δ j
k
. (4.67)
As shown originally by DeWitt, in three dimensions the supermetric has signature
(+,+,+,+,+,−), implying infinitely many negative signs, one for each spatial
point x. The negative directions in function space can be shown to correspond to