Feynman Path Integral Formulation

5.1 Cosmological Wavefunctions 143Physically the above expression means that the total quantum-mechanical amplitudefor a particle to arrive at (x,t) is a sum over all possible values x ′ of the total amplitudeto arrive at the (x ′ ,t ′ ) (which is given by Ψ(x ′ ,t ′ ), multiplied by the amplitudeto go from x to x ′ , which is given by the propagator G(x,t;x ′ ,t ′ ). The propagatoritself corresponds to a special situation: the amplitude where the particle started outat precisely (x ′ ,t ′ ).Let us recall that in Feynman’s formulation of quantum mechanics the propagatorG is expressed as a sum over all paths connecting initial and final points, weightedby an action I,∫ x{ }f (t f )īG(x f ,t f ;x i ,t i ) ≡〈x f ,t f |x i ,t i 〉 = [dx(t)] exph I[x(t)] . (5.4)x i (t i )The paths x(t) contributing to the integral are known to be continuous, but not necessarilydifferentiable (one can give arguments in support of the statement that differentiablepaths have measure zero), which requires in general that the above integralbe carefully defined on a lattice of N points with spacing a, with the limit a → 0,N → ∞ taken at the end.Returning to the gravitational case, the question arises then of how to computethe path integral in Eq. (5.2), even in the absence of matter, and what boundaryconditions need to be imposed. In gravity the analogue of Eq. (5.4) is the quantummechanical amplitude∫ ( j) g〈g ( f )ij,φ ( f ) |g (i)ij ,φ(i) 〉 = [dgg (i)ij ,φ (i)ij ,φ ( f ){μν ][dφ] exp − 1¯h }Î(g μν,φ), (5.5)where the functional integral is over all four-geometries that match the initial (i)and final ( f ) field configurations on the two spacelike surfaces. One noteworthyaspect of such gravitational amplitudes is the fact that, since all intervening fourgeometriesare summed over, there is no notion of unique intervening proper timeinterval: the proper time distance between the two hypersurfaces will depend on thespecific choice of interpolating four-geometry in the ensemble.As mentioned previously, in computing the ground state wave functional Ψ ofEq. (5.2) the proposal has been put forward to functionally integrate over all metricsassociated with compact Euclidean four-geometries specified by g μν , with a giventhree-metric g ij on the boundary. For obvious reasons this is usually referred to asthe “no-boundary” proposal. It elegantly bypasses the issue of having to specify aboundary or continuity condition on cosmological singularities, by suitably restrictingthe choice of geometries at “initial” times. In this approach the wave functionalfor pure gravity is given by (from now on we set again ¯h = 1)∫Ψ[g ij ]= [dg μν ] exp { −Î(g μν ) } , (5.6)Mwith an Euclidean action containing both volume (M) and boundary (∂M) terms,