Feynman Path Integral Formulation

150 5 Semiclassical Gravity[ ]ddτ − ∇λ ∇ λ G μν,ρσ (x,x ′ ;τ) =δ μν,ρσ (x,x ′ )δ(τ) , (5.39)whose expansion in a complete orthonormal basis of TT eigenfunction of the Laplacian∇ 2 readsG μν,ρσ (x,x ′ ;τ) =∑nφ (n)∗μν (x)φ (n)ρσ (x ′ ) exp(−λ n τ) , (5.40)(similar to the usual quantum-mechanical completeness, here in imaginary time τ).Using completeness it is then easy to see that the above relations indeed hold, andthat G (τ) vanishes exponentially for large τ, so that the integral in Eq. (5.37) isconvergent for large argument.If G (τ) has a short time asymptotic expansion of the formG (τ) = 1τ 2∑ g i/2 τ i/2 , (5.41)i=0then a determination of ζ (0) in Eq. (5.37) requires (since 1/Γ (s) has a simple zeroat s = 0) the knowledge of the simple pole term at s = 0intheτ integral, whichcomes from the g 2 (constant) term.After transforming from the imaginary time variable τ to the Laplace transform“energy” variable E[E 2 − ∇ λ ∇ λ]G μν,ρσ (x,x ′ )=δ μν,ρσ (x,x ′ ) , (5.42)and expanding the solution out in a complete set of transverse-traceless hypersphericalharmonics, one can solve the radial equation (in the original metric coordinatet) and extract form it (by inverse Laplace transform) the small τ behavior for G (τ).In the end one finds (Schleich, 1985) g 2 = −γ with γ = 278/45 ≈ 6.18 in Eq. (5.41),and thus ζ (0)=−γ in Eq. (5.36), and finallyP[a] =N a −γ . (5.43)One concludes therefore that at least in the semi-classical approximation the amplitudeΨ diverges at small volume. In general the probability P is related to thesquare of the amplitude,d P(a) =|Ψ(a)| 2 dμ(a) , (5.44)where now dμ(a) is a suitable measure, induced from a functional measure on theoriginal space on which Ψ(h) in Eq. (5.2) is defined, and here Ψ(a) ≡ P[a].We conclude this section with a brief discussion of regularization issues. Theproblems that arise in attempting to regulate the determinant in Eq. (5.29) can, inour opinion, be illustrated through the following simple example. Consider the prototypeintegral over a spinless field h(x)