Quantum Gravity

36 COVARIANT APPROACHES TO QUANTUM GRAVITY∑η n g n p ν n =0,nwhich is equivalent to the statement that ∑ n g np ν n is conserved. But for nontrivialprocesses, the only linear combination of momenta that is conserved isthe total momentum ∑ n η np ν n. Consequently, the couplings g n must all be equal,andonecansetg n ≡ √ 8πG. Therefore, all low-energy particles with spin 2 andm = 0 couple to all forms of energy in an equal way. As in Section 2.1.1, thisshows the ‘equivalence’ of a spin-2 theory with GR. From this point of view,GR is a consequence of quantum theory. Weinberg (1964) also showed that theeffective gravitational mass m g is given bym g =2E − m2 iE ,where m i denotes as in Section 1.1.4 the inertial mass, and E is the energy. ForE → m i , this leads to the usual equivalence of inertial and gravitational mass. Onthe other hand, one has m g =2E for m i → 0. How can this be interpreted? In his1911-calculation of the deflection of light, Einstein found from the equivalenceprinciple alone (setting m g = E), the Newtonian expression for the deflectionangle. The full theory of GR, however, yields twice this value, corresponding tom g =2E.Weinberg’s arguments, as well as the approaches presented in Section 2.1.1,are important for unified theories such as string theory (see Chapter 9) in whicha massless spin-2 particle emerges with necessity, leading to the claim that suchtheories contain GR in an appropriate limit.Upon discussing linear quantum gravity, the question arises whether thisframework leads to observable effects in the laboratory. A brief estimate suggeststhat such effects would be too tiny: comparing in atomic physics, the quantumgravitationaldecay rate Γ g with its electromagnetic counterpart Γ e , one wouldexpect for dimensional reasonsΓ gΓ e∼ α n (mem P) 2(2.50)with some power n of the fine-structure constant α, andm e being the electronmass. The square of the mass ratio already yields the tiny number 10 −45 . Still,it is instructive to discuss some example in detail. Following Weinberg (1972),the transition rate from the 3d level to the 1s level in the hydrogen atom dueto the emission of a graviton will be calculated. One needs at least the 3d level,since ∆l = 2 is needed for the emission of a spin-2 particle.One starts from the classical formula for gravitational radiation and interpretsit as the emission rate of gravitons with energy ω,Γ g = P ω , (2.51)