Perturbative Quantum Gravity
Perturbative quantum gravity can be formulated in terms of amplitudes of going from a three-metric and a matter-field configuration on a spacelike surface Σ to a three-metric and a field configuration on a spacelike surface Σ′. The Wick-rotated quantum amplitudes are here studied under the assumption that the analytic continuation to the real Riemannian section of the complexified space-time is possible, but this is not a generic property. Within the background-field method, one then expands both the four-metric g and matter fields ø about a configuration (g 0, ø 0) which is a solution of the classical equations of motion. If the one-loop approximation holds, the part of the action quadratic in the fluctuations about (g 0, ø 0) gives the dominant contribution to the quantum amplitudes. This leads to Gaussian integrals and to formally divergent amplitudes, since the one-loop result involves the determinant of second-order elliptic operators.
The corresponding divergences are regularized using the zeta-function method. For this purpose, following Hawking, one first defines a generalized zeta-function ζ(s) obtained from the eigenvalues of the elliptic operator B appearing in the calculation. Such ζ(s) can be analytically extended to a meromorphic function which only has poles at some finite values of s. The values of ζ and its first derivative at the origin enable one to express the one-loop quantum amplitudes, whose scaling properties only depend on ζ(0) under suitable assumptions on the measure in the path integral. Although it frequently happens that the eigenvalues of B cannot be computed exactly, the regularized ζ(0) value can be obtained by studying the heat equation for the elliptic operator B. The corresponding integrated heat kernel G(τ) has an asymptotic expansion as τ → 0+ for those boundary conditions which ensure self-adjointness of B. The ζ(0) value is then given by the constant term in the asymptotic form of G(τ), and it also determines the one-loop divergences of physical theories.
Some relevant examples of gravitational background fields are then studied. These gravitational instantons are complete, four-dimensional Riemannian manifolds whose metric solves the Einstein equations with cosmological constant: R(X,Y) − Λg(X,Y) = 0. The possible boundary conditions are: asymptotically Euclidean, asymptotically locally Euclidean, asymptotically flat, asymptotically locally flat, compact without boundary.
Perturbative renormalization, asymptotic expansions and summability methods in quantum field theory are finally described, and the standard argument showing the nonrenormalizability of quantized general relativity is presented.
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