euclidean quantum gravity

Euclidean quantum gravity refers to a Wick rotated version of quantum gravity, formulated as a quantum field theory. The manifolds that are used in this formulation are 4 dimensional Riemannian manifolds instead of pseudo Riemannian manifolds. It is also assumed that the manifolds are compact, connected and boundaryless (i.e. no singularities). Following the usual quantum field-theoretic formulation, the vacuum to vacuum amplitude is written as a functional integral over the metric tensor, which is now the quantum field under consideration.

\int \mathcal{D}\bold{g}\, \mathcal{D}\phi\, \exp\left(-\int d^4x \sqrt{|\bold{g}|}(R+\mathcal{L}_\mathrm{matter})\right)

where φ denotes all the matter fields. See Einstein-Hilbert action.

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