Quantum Gravity

34 COVARIANT APPROACHES TO QUANTUM GRAVITYD (1,1) = D (1) ⊗D (1) , (2.45)where D (1) denotes the j = 1 representation of the rotation group. (Since onehas three ‘angles’ in each of them, this yields the 3 × 3 = 9 degrees of freedom ofthe trace-free f µν .) The representation (2.45) is reducible with ‘Clebsch–Gordondecomposition’D (1) ⊗D (1) = D (2) ⊕D (1) ⊕D (0) , (2.46)corresponding to the five degrees of freedom for a massive spin-2 particle, threedegrees of freedom for a spin-1 particle, and one degree of freedom for a spin-0 particle, respectively. The latter 3 + 1 degrees of freedom can be eliminatedby the four conditions ∂ ν f µν = 0 (transversality). To obtain only two degreesof freedom one needs, however, to impose the gauge freedom (2.5). This yieldsthree additional conditions (the four ɛ µ in (2.5) have to satisfy ∂ µ ɛ µ =0—onecondition—to preserve tracelessness; the condition ✷ɛ µ = 0—needed to preservetransversality—is automatically fulfilled for plane waves). In total, one arrives at(10 − 1) − 4 − 3 = 2 degrees of freedom, corresponding to the two helicity statesof the graviton. 9The same arguments also apply of course to electrodynamics: A µ cannottransform as a Lorentz vector, since, for example, the temporal gauge A 0 =0can be chosen. Instead, one is forced to introduce gauge invariance, and thetransformation law isA µ → Λµ ν A ν + ∂ µ ɛ, (2.47)in analogy to (2.44). Therefore, a Lagrangian is needed that couples to a conservedsource, ∂ µ j µ =0. 10 The group-theoretic argument for QED goes as follows.A vector field transforms according to the D (1/2,1/2) representation of theLorentz group which, if restricted to rotations, can be decomposed asD (1/2,1/2) = D (1/2) ⊗D (1/2) = D (1) ⊕D (0) . (2.48)The D (0) describes spin-0, which is eliminated by the Lorenz condition ∂ ν A ν =0.The D (1) corresponds to the three degrees of freedom of a massive spin-1 particle.One of these is eliminated by the gauge transformation A µ → A µ + ∂ µ ɛ (with✷ɛ = 0 to preserve the Lorenz condition) to arrive at the two degrees of freedomfor the massless photon.Weinberg (1964) concluded (see also p. 537 of Weinberg 1995) that one canderive the equivalence principle (and thus GR if no other fields are present) fromthe Lorentz invariance of the spin-2 theory (plus the pole structure of the S-matrix). Similar arguments can be put forward in the electromagnetic case toshow that electric charge must be conserved. No arguments of gauge invariance9 The counting in the canonical version of the theory leads of course to the same result andis presented in Section 4.2.3.10 This is also connected with the fact that massless spins ≥ 3 are usually excluded, becauseno conserved tensor is available. Massless spins ≥ 3 cannot generate long-range forces; cf.Weinberg (1995, p. 252).