Kiefer C. Quantum gravity

42 COVARIANT APPROACHES TO QUANTUM GRAVITY
(HerewehavetransformedK → iK according to the convention used in Section
4.2.1.) One recognizes that the action can be made arbitrarily negative by
choosing a highly varying conformal factor Ω. The presence of such metrics in the
path integral then leads to its divergence. This is known as the conformal-factor
problem. There are, however, strong indications for a solution of this problem. As
Dasgupta and Loll (2001) have argued, the conformal divergence can cancel with
a similar term of opposite sign arising from the measure in the path integral (cf.
Section 2.2.3 for a discussion of the measure); see Hartle and Schleich (1987) for
a similar result in the context of linearized gravity. Euclidean path integrals are
often used in quantum cosmology, being related to boundary conditions of the
universe (see Section 8.3), so a clarification of these issues is of central interest.
Since the gravitational path integral is of a highly complicated nature, the
question arises whether it can be evaluated by discretization and performing the
continuum limit. In fact, among others, the following two methods have been
employed (see Section 2.2.6 for details):
1. Regge calculus: Originally conceived by Tullio Regge as a method for classical
numerical relativity, it was applied to the Euclidean path integral from
the 1980s on. The central idea is to decompose four-dimensional space into
a set of simplices and treat the edge lengths as dynamical entities.
2. Dynamical triangulation: In contrast to Regge calculus, all edge lengths
are kept fixed, and the sum in the path integral is instead taken over all
possible manifold-gluings of equilateral simplices. The evaluation is thus
reduced to a combinatorical problem. In contrast to Regge calculus, this
method is applied to Lorentzian geometries, emphasizing the importance
of the lightcone structure already at the level of geometries in the path
integral.
The discussion of path integrals will be continued in Section 2.2.3, where
emphasis is put on the integration measure and the derivation of Feynman rules
for gravity. In the next subsection, we shall give an introduction into the use of
perturbation theory in quantum gravity.
2.2.2 The perturbative non-renormalizability
In Section 2.1, we have treated the concept of a graviton similar to the photon—
within the representation theory of the Poincaré group. We have, in particular,
discussed the Fierz–Pauli Lagrangian (2.20) which is (up to a total derivative)
gauge invariant and which at the classical level inevitably leads to GR. The
question thus arises whether this Lagrangian can be quantized in a way similar
to electrodynamics where one arrives at the very successful theory of QED. More
generally then, why should one not perform a quantum perturbation theory of
the Einstein–Hilbert action (1.1)?
The typical situation for applications of perturbation theory in quantum field
theory addresses ‘scattering’ situations in which asymptotically free quantum
states (representing ingoing and outgoing particles) are connected by a region of