Quantum Gravity

30 COVARIANT APPROACHES TO QUANTUM GRAVITYaction and the theory therefore GR. Minkowski space–time as a backgroundstructure has completely disappeared. Boulanger et al. (2001) have shown that,starting from a finite number of Fierz–Pauli Lagrangians, no consistent couplingbetween the various helicity-2 fields is possible if the fields occur at most withsecond derivatives—leading only to a sum of uncoupled Einstein–Hilbert actions.Since GR follows uniquely from (2.20), the question arises whether one wouldbe 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 toMaxwell, a vector theory is excluded because it would lead to repulsing forces.Fermions are excluded because the object that emits a fermion does not remainin 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, evenbefore the advent of GR, Nordström had tried to describe gravity by a scalartheory, which can be defined by the LagrangianL = − 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 beg µν (x) ≡ φ 2 (x)η µν .It is thus conformally flat and possesses a vanishing Weyl tensor. A non-lineargeneralization of the Nordström theory was given by Einstein and Fokker (1914);their field equations readR =24πGT . (2.29)However, this theory is in contradiction with observation, since it does not implementan interaction between gravity and the electromagnetic field (the latterhas T = 0) and the perihelion motion of Mercury comes out incorrectly. Moreover,this theory contains an absolute structure; cf. Section 1.3: the conformalstructure (the ‘lightcone’) is given from the outset and the theory thus possessesan invariance group (the conformal group), which in four dimensions is a finitedimensionalLie group and which must be conceptually distinguished from thediffeomorphism group of GR. While pure scalar fields are thus unsuitable fora theory of the gravitational field, they can nevertheless occur in addition tothe metric of GR. In fact, this happens quite frequently in unified theories; cf.Chapter 9.2.1.2 Gravitons from representations of the Poincaré groupWe shall now turn to the quantum theory of the linear gravitational field. Thediscussion of the previous subsection suggests that it is described by the behaviourof a massless spin-2 particle. Why massless? From the long-range nature