Quantum Gravity

THE CONCEPT OF A GRAVITON 33only the two possibilities of having the angular momentum in direction along oropposite to the direction of propagation.Comparison with (2.17) exhibits that σ = ±1 characterizes the photon. 8 Becauseof the helicity-2 nature of weak gravitational waves in a flat background,see (2.15), we attribute the particle with σ = ±2 with the gravitational interactionand call it the graviton. Since for a massless particle, |σ| is called its spin, werecognize that the graviton has spin 2. The helicity eigenstates (2.41) correspondto circular polarization (see Section 2.1.1), while their superpositions correspondin the generic case to elliptic polarization or (for equal absolute values of theamplitudes) to linear polarization.2.1.3 Quantization of the linear field theoryWe now turn to field theory. One starts from a superposition of plane-wavesolutions (2.8) and formally turns this into an operator,f µν (x) = ∑ ∫d 3 k [√ a(k,σ)eµν (k,σ)e ikx + a † (k,σ)e ∗ µν (k,σ)e−ikx] .2|k|σ=±2(2.42)As in the usual interpretation of free quantum field theory, a(k,σ)(a † (k,σ)) isinterpreted as the annihilation (creation) operator for a graviton of momentumk and helicity σ (see e.g. Weinberg 1995). They obey[a(k,σ),a † (k ′ ,σ ′ )] = δ σσ ′δ(k − k ′ ) , (2.43)with all other commutators vanishing. The quantization of the linearized gravitationalfield was already discussed by Bronstein (1936).Since we only want the presence of helicities ±2, f µν cannot be a true tensorwith respect to Lorentz transformations (note that the TT-gauge condition is notLorentz invariant). As a consequence, one is forced to introduce gauge invarianceand demand that f µν transform under a Lorentz transformation according tof µν → Λ λµ Λ ρν f λρ − ∂ ν ɛ µ − ∂ µ ɛ ν , (2.44)in order to stay within the TT-gauge; cf. (2.5). Therefore, the coupling in theLagrangian (2.20) must be to a conserved source, ∂ ν T µν = 0, because otherwisethe coupling is not gauge invariant.The occurrence of gauge invariance can also be understood in a grouptheoreticway. We start with a symmetric tensor field f µν with vanishing trace(nine degrees of freedom). This field transforms according to the irreducibleD (1,1) representation of the Lorentz group; see, for example, section 5.6 in Weinberg(1995). Its restriction to the subgroup of rotations yields8 Because of space inversion symmetry, σ = 1 and σ = −1 describe the same particle.This holds also for the graviton, but due to parity violation not, for example, for (massless)neutrinos.