Ivancevic_Applied-Diff-Geom

1058 Applied Differential Geometry: A Modern IntroductionFig. 6.7 The time–honored way of illustrating the gravitational path integral as thepropagator from an initial to a final spatial boundary geometry (see text for explanation).integral, and is often written schematically as∫G(x ′ , t ′ ; x ′′ , t ′′ ) = Σ, D[x(t)] e iS[x(t)] , (6.101)where D[x(t)] is a functional measure on the ‘space of all paths’, and theexponential weight depends on the classical action S[x(t)] of a path. Recallalso that this procedure can be defined in a mathematically clean way ifwe Wick–rotate the time variable t to imaginary values t ↦→ τ = it, therebymaking all integrals real [Reed and Simon (1975)].Can a similar strategy work for the case of Einstein geometrodynamics?As an analogue of the particle’s position we can take the geometry [g ij (x)](i.e., an equivalence class of spatial metrics) of a constant–time slice. Canone then define a gravitational propagatorG([g ′ ij], [g ′′ij]) =∫Σ Geom(M) D[g µν ] e iSEH [g µν](6.102)from an initial geometry [g ′ ] to a final geometry [g ′′ ] (Figure 6.7) as alimit of some discrete construction analogous to that of the non-relativisticparticle (6.4)? And crucially, what would be a suitable class of ‘paths’, thatis, space–times [g µν ] to sum over? ∫Now, to be able to perform the integration ΣD[g µν ] in a meaningfulway, the strategy we will be following starts from a regularized version ofthe space Geom(M) of all geometries. A regularized path integral G(a) canbe defined which depends on an ultraviolet cutoff a and is convergent in anon–trivial region of the space of coupling constants. Taking the continuumlimit corresponds to letting a → 0. The resulting continuum theory – if itcan be shown to exist – is then investigated with regard to its geometric