Quantum Gravity

260 QUANTUM COSMOLOGYwaves and are gauge independent. It is possible to work exclusively with gaugeindependentvariables (Bardeen 1980), but this is not needed for the followingdiscussion.Introducing the shorthand notation {x n } for the collection of multipolesa n ,b b ,c n ,d n , the wave function is defined on an infinite-dimensional configurationspace spanned by a (or α) andφ (the ‘minisuperspace background’) andthe variables {x n }. The Wheeler–DeWitt equation can be decomposed into twoparts referring to first and second derivatives in the {x n }, respectively. The partwith the second derivatives reads (Halliwell and Hawking 1985)(H 0 +2e ∑ )3α H n (a, φ, x n ) Ψ(α, φ, {x n })=0, (8.42)nwhere H 0 denotes the minisuperspace part,( ) ( )∂2H 0 ≡∂α 2 − ∂2∂∂φ 2 +e6α m 2 φ 2 − e 4α 2≡∂α 2 − ∂2∂φ 2 + V (α, φ) , (8.43)and H n is a sum of Hamiltonians referring to the scalar, vector, and tensor partof the modes, respectively,H n = H n (S) + H n (V ) + H n (T ) .We now make the ansatzΨ(α, φ, {x n })=ψ 0 (α, φ) ∏ n>0ψ n (α, φ; x n ) (8.44)and insert this into (8.42). Following Kiefer (1987), we get− ∇2 ψ 0− 2 ∇ψ ∑0 ∇ψ n− ∑ ψ 0 ψ 0 ψn nn− ∑ ∇ψ n ∇ψ mψ n ψ mn≠m∇ 2 ψ nψ n+ V (α, φ)+2e 3α ∑ nH n ΨΨ =0,where( ) ∂∇≡∂α , ∂∂φdenotes the ‘minisuperspace gradient’. One then gets by separation of variablesthe two equations− ∇2 ψ 0+ V (α, φ) =−2f(α, φ) , (8.45)ψ 0−2 ∇ψ ∑0 ∇ψ n− ∑ ∇ 2 ψ n− ∑ ∇ψ n ∇ψ mψ 0 ψn n ψn n ψ n ψ mn≠m+2e ∑ 3α H n Ψ=2f(α, φ) , (8.46)Ψn