Feynman Path Integral Formulation

4.10 Minisuperspace 119
is the inverse of the deWitt supermetric G ij,kl .
Due to ambiguities in the choice of operator ordering in the Wheeler-DeWitt
equations, not all terms and their coefficients can be fixed in a unique way. General
symmetry requirements (here covariance in superspace) restrict the Hamiltonian
H = ∫ d 3 xN(x)H(x) in Eqs. (4.62) and (4.72) to be of the following form (Hawking
and Page, 1986 a,b)
∫
H = − 1 2 ∇2 + β · 16πG
∫
V =
d 3 xNg −1/2 g ij
δ
δg ij
+ ε + V (4.92)
d 3 xN √ g [ 1
16πG (−3 R + 2λ)+U(φ) ] (4.93)
where ∇ 2 the covariant Laplacian in the function space W with metric Γ (N) [see
Eq. (4.90)],
∫
ε = ξ R(g)+η d 3 x √ g , (4.94)
is a scalar term involving the scalar curvature on the function space W, ξ and η two
constants, and
U = T 00 − 1 2 π2 φ . (4.95)
Since in general the β-term violates the self-adjointness requirement on H, one
sets β = 0. The most natural (and simplest choice) for ξ , the coefficient of the
scalar curvature term R, is zero. The η term can be re-absorbed into a shift of the
cosmological term λ. We shall not dwell here on the rather technical point that in
general the function N enters non-linearly in the superspace connection on W, and
therefore in H, which then spoils the interpretation of N as a Lagrange multiplier.
In general the wavefunction for all the dynamical variables of the gravitational
field in the universe is difficult to calculate, since an infinite number of degrees
of freedom are involved: the infinitely many values of the metric at all spacetime
points, and the infinitely many values of the matter field φ at the same points. One
option is to restrict the choice of variable to a finite number of suitable degrees of
freedom (Blyth and Isham, 1975; Hartle and Hawking, 1983). As a result the overall
quantum fluctuations are severely restricted, since these are now only allowed to be
nonzero along the surviving dynamical directions. If the truncation is severe enough,
the transverse-traceless nature of the graviton fluctuation is lost as well. Also, since
one is not expanding the quantum solution in a small parameter, it can be difficult
to estimate corrections. In a cosmological context, it seems natural to consider initially
a homogeneous and isotropic model, and restrict the function space to two
variables, the scale factor a(t) and a minimally coupled homogeneous scalar field
φ(t) (Hawking and Page, 1986a,b). The space-time metric is given by
dτ 2 = N 2 (t)dt 2 − g ij dx i dx j . (4.96)
The three-metric g ij is then determined entirely by the scale factor a(t),
g ij = a 2 (t) ˜g ij , (4.97)