World quantum gravity: quantum test objects and Synge’s world function
function Ding Jia (贾丁)1, 2, 3, ∗ 1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada 3Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Abstract A new path integral approach of quantum gravity based on relational variables and quantum test objects is presented. We take as a basic variables the squared invariant distance. This invariant quantity could be technically simpler to work with than variant quantities such as the metric tensor. It also could facilitate the studies of matter coupling and quantum spacetime causal structures. In contrast to approaches based on piecewise linear geometries, here gravity is captured by its effects on quantum test particles and fields. By an observation of Parker, under a Feynman sum a gravitational phase can be traded into a Van Vleck-Morette determinant term. This leads to a new candidate path integral for gravity, which in certain special cases can be computed efficient in the Lorentzian signature. We discuss some ambiguities left in the path integral measure, which invite further clarifications.