Quantum Gravity

THE CONCEPT OF A GRAVITON 31of the gravitational interaction, it is clear that the graviton must have a smallmass. Since the presence of a non-vanishing mass, however small, affects thedeflection of light discontinuously, one may conclude that the graviton mass isstrictly zero; see van Dam and Veltman (1970) and Carrera and Giulini (2001).Such an argument cannot be put forward for the photon.In the following, we shall give a brief derivation of the spin-2 nature in theframework of representation theory (see e.g. Weinberg 1995 or Sexl and Urbantke2001). In this subsection, we shall only deal with one-particle states (‘quantummechanics’), while field-theoretic aspects will be discussed in Section 2.1.3. Theimportant ingredient is the presence of flat Minkowski space–time with metricη µν as an absolute background structure and the ensuing Poincaré symmetry.The use of the Poincaré group, not available beyond the linearized level, alreadyindicates the approximate nature of the graviton concept. We shall set =1inmost of the following expressions.According to Wigner, ‘particles’ are classified by irreducible representationsof the Poincaré group. We describe a Poincaré transformationasx ′µ =Λ µ ν xν + a µ , (2.30)where the Λ µ ν denote Lorentz transformations and the a µ denote space–timetranslations. According to Wigner’s theorem, (2.30) induces a unitary transformation5 in the Hilbert space of the theory,ψ → U(Λ,a)ψ . (2.31)This ensures that probabilities remain unchanged under the Poincaré group.Since this group is a Lie group, it is of advantage to study group elements closeto the identity,Λ µ ν = δ µ ν + ω µ ν , a µ = ɛ µ , (2.32)where ω µν = −ω νµ . This corresponds to the unitary transformation 6U(1 + ω, ɛ) =1+ 1 2 iω µνJ µν − iɛ µ P µ + ... , (2.33)where J µν and P µ denote the 10 Hermitian generators of the Poincaré group,which are the boost generators, the angular momentum and the four-momentum,respectively. They obey the following Lie-algebra relations,[P µ ,P ρ ]=0, (2.34)i[J µν ,J λρ ]=η νλ J µρ − η µλ J νρ − η ρµ J λν + η ρν J λµ , (2.35)i[P µ ,J λρ ]=η µλ P ρ − η µρ P λ . (2.36)5 The theorem also allows anti-unitary transformations, but these are relevant only for discretesymmetries.6 The plus sign on the right-hand side is enforced by the commutation relations [J i ,J k ]=iɛ ikl J l (where J 3 ≡ J 12 , etc.), whereas the minus sign in front of the second term is pureconvention.