Quantum Gravity : Mathematical Models and Experimental Bounds

2. Quantum general relativity
Quantum Gravity — A Short Overview 5
2.1. Covariant approaches
The first attempt to construct a theory of quantum theory was, of course, the
attempt to develop a perturbation theory. After all, major theories such as QED
are only understood perturbatively. However, for gravity there is a distinctive
feature: The perturbation theory is non-renormalizable, that is, infinitely many
parameters must be introduced (and experimentally determined) in order to absorb
the ensuing divergences. This seems to render the perturbation theory meaningless.
Still, even a non-renormalizable theory may be able to yield definite predictions
at low energies. This occurs on the level of effective field theories. One example is
the quantum gravitational correction to the Newtonian potential calculated in [9].
Another example concerns the application of renormalization group methods [10]:
The theory may be asymptotically safe, that is, possess a non-vanishing ultraviolet
fixed point at the non-perturbative level. This would lead to running gravitational
and cosmological ‘constants’ that could in principle provide an explanation for the
dark matter and the dark energy in the universe.
A direct ‘covariant quantization’ of general relativity would be the formulation
of a path-integral framework. This has been tried in both a Euclidean and
a Lorentzian formalism: Whereas in the former one rotates the time parameter
onto the imaginary axis, the latter leaves the Lorentzian signature of spacetime
untouched. Besides their use in the search for boundary conditions in quantum
cosmology (see below), much work has been devoted to a numerical analysis via
‘Regge calculus’ or ‘dynamical triangulation’. The latter approach has recently
provided some interesting results concerning the structure of space and time: The
Hausdorff dimension of the resulting space is consistent with the (expected) number
three, a positive cosmological constant is needed, and the universe behaves
semiclassically at large scales, cf. [11] for a recent introductory review.
2.2. Canonical approaches
The major alternative to covariant quantization is to use a Hamiltonian framework
— the framework of ‘canonical quantization’. Under the assumption of global hyperbolicity,
the classical spacetime is foliated into three-dimensional spaces each of
which is isomorphic to a given three-manifold Σ. One then chooses an appropriate
canonical variable and its momentum and turns their Poisson-bracket algebra into
a commutator algebra. According to the choice of variables one can distinguish
various subclasses of canonical quantization. The oldest is quantum geometrodynamics
in which the canonical variable is the three-dimensional metric on Σ,
and the momentum is linearly related to the extrinsic curvature. More recent approaches
employ either a connection variable or its integral along a loop in Σ. The
latter leads to what is now called ‘loop quantum gravity’.
The central equations in all these approaches are the quantum constraints.
The invariance of the classical theory under coordinate transformation leads to