Kiefer C. Quantum gravity

2
COVARIANT APPROACHES TO QUANTUM GRAVITY
2.1 The concept of a graviton
A central role in the quantization of the gravitational field is played by the
graviton—a massless particle of spin-2, which is the mediator of the gravitational
interaction. It is analogous to the photon in quantum electrodynamics. Its
definition requires, however, the presence of a background structure, at least in
an approximate sense. We shall, therefore, first review weak gravitational waves
in Minkowski space–time and the concept of helicity. It will then be explained
how gravitons are defined as spin-2 particles from representations of the Poincaré
group. Finally, the gravitational field in its linear approximation is quantized. It
is shown, in particular, how Poincaré invariance ensues the equivalence principle
and therefore the full theory of general relativity (GR) in the classical limit.
2.1.1 Weak gravitational waves
Our starting point is the decomposition of a space–time metric g µν into a fixed
(i.e. non-dynamical) background and a ‘perturbation’; see, for example, Weinberg
(1972) and Misner et al. (1973). In the following, we take for the background the
flat Minkowski space–time with the standard metric η µν = diag(−1, 1, 1, 1) and
call the perturbation f µν .Thus,
g µν = η µν + f µν . (2.1)
We assume that the perturbation is small, that is, that the components of f µν
are small in the standard cartesian coordinates. Using (2.1) without the Λ-term,
the Einstein equations (1.3) without the Λ-term read in the linear approximation
✷f µν = −16πG ( T µν − 1 2 η µνT ) , (2.2)
where T ≡ η µν T µν , and the ‘harmonic condition’ (also called the ‘de Donder
gauge’)
ν
fµν, = 1 2 f ν ν,µ (2.3)
has been used. 1 This condition is analogous to the Lorenz 2 gauge condition in
electrodynamics and is used here to partially fix the coordinates. Namely, the
invariance of the full theory under coordinate transformations
1 Indices are raised and lowered by η µν and η µν, respectively. We set c = 1 in most expressions.
2 This is not a misprint. The Lorenz condition is named after the Danish physicist Ludwig
Lorenz (1829–91).
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