Feynman Path Integral Formulation

4.10 Minisuperspace 119is the inverse of the deWitt supermetric G ij,kl .Due to ambiguities in the choice of operator ordering in the Wheeler-DeWittequations, not all terms and their coefficients can be fixed in a unique way. Generalsymmetry requirements (here covariance in superspace) restrict the HamiltonianH = ∫ d 3 xN(x)H(x) in Eqs. (4.62) and (4.72) to be of the following form (Hawkingand Page, 1986 a,b)∫H = − 1 2 ∇2 + β · 16πG∫V =d 3 xNg −1/2 g ijδδg ij+ ε + V (4.92)d 3 xN √ g [ 116πG (−3 R + 2λ)+U(φ) ] (4.93)where ∇ 2 the covariant Laplacian in the function space W with metric Γ (N) [seeEq. (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 twoconstants, andU = T 00 − 1 2 π2 φ . (4.95)Since in general the β-term violates the self-adjointness requirement on H, onesets β = 0. The most natural (and simplest choice) for ξ , the coefficient of thescalar curvature term R, is zero. The η term can be re-absorbed into a shift of thecosmological term λ. We shall not dwell here on the rather technical point that ingeneral the function N enters non-linearly in the superspace connection on W, andtherefore in H, which then spoils the interpretation of N as a Lagrange multiplier.In general the wavefunction for all the dynamical variables of the gravitationalfield in the universe is difficult to calculate, since an infinite number of degreesof freedom are involved: the infinitely many values of the metric at all spacetimepoints, and the infinitely many values of the matter field φ at the same points. Oneoption is to restrict the choice of variable to a finite number of suitable degrees offreedom (Blyth and Isham, 1975; Hartle and Hawking, 1983). As a result the overallquantum fluctuations are severely restricted, since these are now only allowed to benonzero along the surviving dynamical directions. If the truncation is severe enough,the transverse-traceless nature of the graviton fluctuation is lost as well. Also, sinceone is not expanding the quantum solution in a small parameter, it can be difficultto estimate corrections. In a cosmological context, it seems natural to consider initiallya homogeneous and isotropic model, and restrict the function space to twovariables, the scale factor a(t) and a minimally coupled homogeneous scalar fieldφ(t) (Hawking and Page, 1986a,b). The space-time metric is given bydτ 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)