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In quantum gravity, we perform a Wick rotation and work with 4 dimensional Riemannian manifolds instead of pseudo Riemannian manifolds. We also assume the manifolds we are working with are compact, connected and boundaryless (i.e. no singularities or anything of that sort! although since the measure is concentrated on "distributional" metrics...). Then, we do a functional integration over it.
- <math>\int \mathcal{D}\bold{g}\, \mathcal{D}\phi\, \exp\left(-\int d^4x \sqrt{|\bold{g}|}(R+\mathcal{L}_\mathrm{matter})\right)<math>
where φ denotes all the matter fields. See Einstein-Hilbert action.
But what functional measure <math>\mathcal{D}\bold{g}<math> do we use? It makes no sense to insist upon something like translational invariance of the measure for something as nonlinear as the metric g.
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