Perturbation methods in quantum gravity: The multiple-scattering expansion

An alternative to the conventional approach to perturbation theory in quantum gravity is proposed, based on techniques developed by Balian and Bloch for semiclassical expansions in nonrelativistic quantum mechanics. The resulting methods involve the properties of classical general relativity in an essential way, and allow the convergence of the integrals appearing in perturbation theory to be investigated from a new standpoint. The basic tool is the multiple-scattering expansion for the amplitude to go from a given initial three-dimensional metric $g_{\mathrm{ij}}^{\prime }$ to a final metric $g_{\mathrm{ij}}$, which constructs all higher-order terms from the one-loop amplitude $A\exp \left(\frac{i{S}_c}{\hslash }\right)$. Such terms are found by considering broken classical paths from $g_{\mathrm{ij}}^{\prime }$ to $g_{\mathrm{ij}}$ which bounce at a number of intermediate three-geometries $g_{\mathrm{ij}}^{\prime \prime }$, each segment of the path is a solution of the classical Einstein equations weighted by its one-loop amplitude, and a discontinuity in the second fundamental form occurs at each bounce, where a suitable vertex must be used before functional integrations are carried out over the intermediate three-metrics. This expansion is developed from the canonical formulation of quantum gravity and leads to an expression for the amplitude as the Fourier transform of a generating functional $\Omega \mathrm{}\left(s\right)\mathrm{}$ which depends only on a complex action parameter $s$ and the classical paths, thus providing a description of quantum gravity without $\hslash$. Since the expansion includes the contributions of metrics $g_{\mathrm{ij}}^{\prime \prime }$ very distant from the direct classical path, the nature of such metrics and some properties of the corresponding amplitudes are examined, with a view to understanding their possible role in improving the convergence of the theory.