Quantum Gravity

174 QUANTUM GEOMETRODYNAMICS] [Ĥg µ , Ĥg ν =iÛµνĤg λ λ . (5.170)As shown in Barvinsky (1993b), the fact that (5.170) holds follows from theclassical gravitational constraints (5.165) by replacing the momenta p k with thefunctional covariant derivatives D k /i—covariant with respect to the Riemannconnection based on the DeWitt metric (5.168)—and by adding a purely imaginarypart (anti-Hermitian with respect to the L 2 inner product): the functionaltrace of structure functions, iUµν ν /2. With this definition of covariant derivatives,it is understood that the space of three-metrics q is regarded as a functional differentiablemanifold, and that the quantum states |Ψ(q)〉 are scalar densities ofweight 1/2. Thus, the operator realization for the full constraints including thematter parts has the formĤ ⊥ = − 12m 2 G⊥ ik D iD k + V ⊥ + iP2 U ⊥ν ν + Ĥm ⊥ , (5.171)Ĥ a = 1 i ∇i aD i + i 2 U ν aν + Ĥm a . (5.172)The imaginary parts of these operators are either formally divergent (being thecoincidence limits of delta-function type kernels) or formally zero (as in (5.171)because of vanishing structure-function components). We shall, however, keepthem in a general form, expecting that a rigorous operator regularization willexist that can consistently handle these infinities as well as the correspondingquantum anomalies (see Section 5.3.5).The highest order of the semiclassical approximation leads to a wave functionalof the form (5.153). It is sometimes convenient to consider a two-pointobject (‘propagator’) instead of a wave functional—one can, for example, easilytranslate the results into a language using Feynman diagrams. This will be donein the next subsection. We shall, therefore, consider a two-point solution K(q, q ′ )of the Wheeler–DeWitt equation. One can construct a closed expression for theone-loop pre-exponential factor of a solution which is of the semiclassical formcorresponding to (5.153) (see Barvinsky and Krykhtin (1993))K(q, q ′ )=P (q, q ′ )e im2 P S(q, q′ ) . (5.173)Here, S(q, q ′ ) is a particular solution of the Hamilton–Jacobi equations withrespect to both arguments—the classical action calculated at the extremal ofequations of motion joining the points q and q ′ ,H g µ(q, ∂S/∂q) =0. (5.174)The one-loop (O(m 0 P )partofthe1/m2 P -expansion) order of the pre-exponentialfactor P (q, q ′ ) here satisfies a set of quasi-continuity equations which follow from