Quantum Gravity : Mathematical Models and Experimental Bounds

Quantum Gravity
B. Fauser, J. Tolksdorf and E. Zeidler, Eds., 1–13
c○ 2006 Birkhäuser Verlag Basel/Switzerland
Quantum Gravity — A Short Overview
Claus Kiefer
Abstract. The main problems in constructing a quantum theory of gravity
as well as some recent developments are briefly discussed on a non-technical
level.
Mathematics Subject Classification (2000). Primary 83-02; Secondary 83C45;
83C47; 83F05; 83E05; 83E30.
Keywords. Quantum gravity, string theory, quantum geometrodynamics, loop
quantum gravity, black holes, quantum cosmology, experimental tests.
1. Why do we need quantum gravity?
The task to formulate a consistent quantum theory of gravity has occupied physicists
since the first attempts by Léon Rosenfeld in 1930. Despite much work it is
fair to say that this goal has not yet been reached. In this short contribution I shall
attempt to give a concise summary of the situation in 2005 from my point of view.
The questions to be addressed are: Why is this problem of interest? What are the
main difficulties? And where do we stand? A comprehensive technical treatment
can be found in my monograph [1] as well as in the Proceedings volumes [2] and
[3]. A more detailed review with focus on recent developments is presented in [4].
The reader is referred to these sources for details and references.
Why should one be interested in developing a quantum theory of the gravitational
field? The main reasons are conceptual. The famous singularity theorems
show that the classical theory of general relativity is incomplete: Under very general
conditions, singularities are unavoidable. Such singularities can be rather mild,
that is, of a purely topological nature, but they can also consist of diverging curvatures
and energy densities. In fact, the latter situation seems to be realized in
two important physical cases: The Big Bang and black holes. The presence of
the cosmic microwave background (CMB) radiation indicates that a Big Bang has
happened in the past. Curvature singularities seem to lurk inside the event horizon
of black holes. One thus needs a more comprehensive theory to understand these
I thank Bertfried Fauser and Jürgen Tolksdorf for inviting me to this stimulating workshop.