Quantum Gravity

264 QUANTUM COSMOLOGYExcept in simple minisuperspace models, the path integral in (8.58) cannotbe evaluated exactly. It is therefore usually being calculated in a semiclassical(‘saddle point’) approximation. Since there exist in general several saddle points,one must address the issue of which contour of integration has to be chosen.Depending on the contour only part of the saddle points may contribute to thepath integral.The integral over the four-metric in (8.58) splits into integrals over threemetric,lapse function, and shift vector; cf. Section 5.3.4. In a Friedmann model,only the integral over the lapse function remains and turns out to be an ordinaryintegral (cf. Halliwell (1988)),∫ ∫ψ(a, φ) = dN Dφ e −I[a(τ ),φ(τ ),N] , (8.59)where we have denoted the Euclidean minisuperspace action by I instead of S E .For notational simplicity, we have denoted the arguments of the wave functionby the same letters than the corresponding functions which are integrated overin the path integral. For the Friedmann model containing a scalar field, theEuclidean action readsI = 1 2∫ τ2τ 1dτN[− a ( daN 2 dτ) 2 ( ) 2+ a3 dφN 2 − a + a 3 V (φ)]dτ, (8.60)where V (φ) may denote just a mass term, V (φ) ∝ m 2 φ 2 , or include a selfinteractionsuch as V ∝ φ 4 .How is the no-boundary proposal being implemented in minisuperspace? Theimprint of the restriction in the class of contours in (8.58) is to integrate overEuclidean paths a(τ) with the boundary condition a(0) = 0; cf. the discussionin Halliwell (1991). This is supposed to implement the idea of integration overregular four-geometries with no ‘boundary’ at a = 0. (The point a =0hasto be viewed like the pole of a sphere, which is completely regular.) One isoften interested in discussing quantum-cosmological models in the context ofinflationary cosmology. Therefore, in evaluating (8.59), one might restrict oneselfto the region where the scalar field φ is slowly varying (‘slow-roll approximation’of inflation). One can then neglect the kinetic term of φ and integrate overEuclidean paths with φ(τ) ≈ constant. For a 2 V < 1, one gets the following twosaddle point actions (Hawking 1984; Halliwell 1991):I ± = − 1 [1 ± (1 − a 2 V (φ)) 3/2] . (8.61)3V (φ)The action I − is obtained for a three-sphere being closed off by less than half thefour-sphere, while in evaluating I + , the three-sphere is closed off by more thanhalf the four-sphere.There exist various arguments in favour of which of the two extremal actionsare distinguished by the no-boundary proposal. This can in general only be