Feynman Path Integral Formulation

146 5 Semiclassical GravityAn alternative way of doing the calculation is to use a hybrid Hamiltonian formalismwith an explicit choice of gauge, and later integrate (with some suitablefunctional measure) over the physical degrees only (Schleich, 1985). The procedurerelies on establishing a correspondence between the covariant path integral approachand some aspects of the machinery, notably the constraint equations, of canonicalquantum gravity. Such a procedure is possible because the Feynman path integral,already for a non-relativistic particle, can be written in the equivalent formG(q f ,t f ;q i ,t i )=〈q f ,t f |q i ,t i 〉∫ q[ ]f (t f ) dq(t)dp(t)=2π ¯hq i (t i ){ ∫ ī t}fexp dt [p ˙q − H(p,q)]h t i,(5.18)which gives the expression in Eq. (5.4) after integrating out the p’s (see, for example,Abers and Lee, 1973). Some of the earliest discussions on the connection betweenthe canonical and functional-integral approaches to quantum gravity, and of some ofthe subtleties that arise in such a correspondence, can be found in (Leutwyler, 1964;Faddeev and Popov, 1973; Fradkin and Vilkovisky, 1973).5.2 Semiclassical ExpansionTo proceed further it will help to be a bit more specific about the boundary geometryin Eq. (5.6). If one wants to investigate the quantum mechanical behavior ofclosed cosmologies in the vicinity of classical singularities, one is naturally led toconsider the behavior of the wave functional Ψ at small volume, where correctionsto the classical solution can be large and new effects can arise. In the semiclassicalexpansion of quantum gravity it is therefore natural to consider as saddle points ofthe Euclidean action solutions whose boundary geometry has the shape of a threesphere.Indeed in the case of a positive cosmological constant λ any regular Euclideansolution of the field equations is necessarily compact, with the solution ofgreatest symmetry corresponding to the four-sphere or radius √ 3/λ.In the following it will be assumed (Schleich, 1985) that the boundary geometryis a three-sphere with radius a, such thatds 2 = a 2 ĝ ij dφ i dφ j , (5.19)where a is the radius of the three-sphere, and ĝ ij is the metric on the unit threesphere.With such a boundary condition, the compact extrema of the action inEq. (5.7) are sections of Euclidean de Sitter space, withds 2 = dθ 2 + r 2 0 sin 2 (θ/r 0 ) ĝ ij dφ i dφ j , (5.20)