New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

4 Mathematical Tools for Calculation of the Effective Action 205The determinants of the operators F and γ are usually represented as a result of theintegration over some auxiliary Grassmannian variables, so-called ghost fields.This equation can be used to construct the semi-classical perturbation theory inpowers of the Planck constant (loop expansion), which gives the effective action interms of the bare propagators and the vertex functions. In particular, one finds theone-loop effective actionƔ (1) =− 1 2i log Det ˆL + 1 i log Det F + 1 log Det γ, (45)2iwhere ˆL is an operator defined byd 2 {dε 2 S(ϕ + εh) + 1 } ∣∣∣ε=02 〈χ(ϕ + εh), γ χ(ϕ + εh)〉 =−(h, ˆLh). (46)In DeWitt notation it readsˆL k j = E −1ki L ij , (47)whereL ij =−δ2 Sδϕ i δϕ j − δχαδϕ i γ δχ βαβδϕ j . (48)4.2.3 Quantum General RelativityEinstein’s theory of general relativity is an example of a gauge theory with the gaugegroup G being the group of all diffeomorphisms of the spacetime manifold M andthe configuration space M being the space of all pseudo-Riemannian metrics on M.The physical configuration space M/G of all orbits of the gauge group is then thespace of all geometries on the spacetime.The gravitational field can be parametrized by the metric tensor of the spacetimeAn invariant fiber metric is defined byϕ i = g μν (x), i ≡ (μν, x). (49)E μναβ = g μ(α g β)ν − ϰg μν g αβ , (50)where ϰ ̸= 1/n is a real parameter. The inverse metric is thenEμναβ −1 = g μ(αg β)ν −ϰnϰ − 1 g μνg αβ . (51)