Kiefer C. Quantum gravity

30 COVARIANT APPROACHES TO QUANTUM GRAVITY
action and the theory therefore GR. Minkowski space–time as a background
structure has completely disappeared. Boulanger et al. (2001) have shown that,
starting from a finite number of Fierz–Pauli Lagrangians, no consistent coupling
between the various helicity-2 fields is possible if the fields occur at most with
second derivatives—leading only to a sum of uncoupled Einstein–Hilbert actions.
Since GR follows uniquely from (2.20), the question arises whether one would
be able to construct a pure scalar, fermionic, or vector theory of gravity, cf.
Feynman et al. (1995) and Straumann (2000). As has already been known to
Maxwell, a vector theory is excluded because it would lead to repulsing forces.
Fermions are excluded because the object that emits a fermion does not remain
in the same internal state (there are also problems with a two-fermion exchange).
A scalar theory, on the other hand, would only lead to attraction. In fact, even
before the advent of GR, Nordström had tried to describe gravity by a scalar
theory, which can be defined by the Lagrangian
L = − 1 2 ηµν ∂ µ φ∂ ν φ − 4πGTφ + L matter , (2.27)
where T = η µν T µν . This leads to the field equation
✷φ =4πGT . (2.28)
The physical metric (as measured by rods and clocks) turns out to be
g µν (x) ≡ φ 2 (x)η µν .
It is thus conformally flat and possesses a vanishing Weyl tensor. A non-linear
generalization of the Nordström theory was given by Einstein and Fokker (1914);
their field equations read
R =24πGT . (2.29)
However, this theory is in contradiction with observation, since it does not implement
an interaction between gravity and the electromagnetic field (the latter
has T = 0) and the perihelion motion of Mercury comes out incorrectly. Moreover,
this theory contains an absolute structure; cf. Section 1.3: the conformal
structure (the ‘lightcone’) is given from the outset and the theory thus possesses
an invariance group (the conformal group), which in four dimensions is a finitedimensional
Lie group and which must be conceptually distinguished from the
diffeomorphism group of GR. While pure scalar fields are thus unsuitable for
a theory of the gravitational field, they can nevertheless occur in addition to
the metric of GR. In fact, this happens quite frequently in unified theories; cf.
Chapter 9.
2.1.2 Gravitons from representations of the Poincaré group
We shall now turn to the quantum theory of the linear gravitational field. The
discussion of the previous subsection suggests that it is described by the behaviour
of a massless spin-2 particle. Why massless? From the long-range nature