## Canonical and pertubative quantum gravity

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##### Date

1993

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##### Abstract

After a review of Dirac's theory of iconstrained Hamiltonian systems and their
quantization, canonical quantum gravity is studied relying on the Arnowitt-Deser-Misner
formalism. First-class constraints of the theory are studied in some detail following De
Witt's work, and geometrical and topological properties of Wheeler's superspace are dis cussed following the mathematical work of Fisher.
Perturbative quantum gravity is then formulated in terms of amplitudes of going from
a three-metric and a matter-field configuration on a spacelike surface ~ to a three-metric
and a field configuration on a spacelike surface ~'. The Wick-rotated quantum amplitudes
are here studied under the assumption that the analytic continuation to the real Rieman- I
nian 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 mat ter fields <P about a configuration (go, <Po) 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 (go, <Po) 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 ((s) obtained
from the eigenvalues of the elliptic operator B appearing in the calculation. Such ( s) can
be analytically extended to a meromorphic function which only has poles at some finite
values of s. The values of ( and its first derivative at the origin enable one to express the
one-loop quantum amplitudes, whose scaling properties only depend on (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 (0) value can be obtained
by studying the heat equation for the elliptic operator B. The corresponding integrated
heat kernel G(T) has an asymptotic expansion as T ~ 0+ for those boundary conditions
which ensure self-adjointness of B. The (0) value is then given by the constant term in
the asymptotic form of G(T), and it also determines the one-loop divergences of physical
theories. The zeta-function technique has been recently applied to the study of one-loop
properties of supersymmetric field theories in the presence of boundaries.
Some relevant examples of gravitational background fields are then studied. These
gravitational instantons are complete, four-dimensional Riemannian manifolds whose met ric solves the Einstein equations with cosmological constant: R(X, Y) - Ag(X, Y) = o.
The possible boundary conditions are asymptotically Euclidean, asymptotically locally
Euclidean, asymptotically flat, asymptotically locally flat, compact without boundary.