Quantum Gravity

32 COVARIANT APPROACHES TO QUANTUM GRAVITYOne-particle states are classified according to their behaviour with respect toPoincaré transformations. Since the components P µ of the four-momentum commutewith each other, we shall choose their eigenstates,P µ ψ p,σ = p µ ψ p,σ , (2.37)where σ stands symbolically for all other variables. Application of the unitaryoperator then yieldsU(1,a)ψ p,σ =e −ipµ a µψ p,σ . (2.38)How do these states transform under Lorentz transformations (we only considerorthochronous proper transformations)? According to the method of inducedrepresentations, it is sufficient to find the representations of the little group.This group is characterized by the fact that it leaves a ‘standard’ vector k µinvariant (within each class of given p 2 ≤ 0 7 and given sign of p 0 ). For positivep 0 , one can distinguish between the following two cases. The first possibilityis p 2 = −m 2 < 0. Here one can choose k µ =(m, 0, 0, 0), and the little group isSO(3), since these are the only Lorentz transformations that leave a particle withk = 0 at rest. The second possibility is p 2 =0.Onechoosesk µ =(1, 0, 0, 1), andthe little group is ISO(2), the invariance group of Euclidean geometry (rotationsand translations in two dimensions). Any p µ within a given class can be obtainedfrom the corresponding k µ by a Lorentz transformation. The normalization ischosen such that〈ψ p′ ,σ ′,ψ p,σ〉 = δ σσ ′δ(p − p ′ ) . (2.39)Consider first the case m ≠ 0 where the little group is SO(3). As is well knownfrom quantum mechanics, its unitary representations are a direct sum of irreducibleunitary representations D (j)σσ with dimensions 2j +1 (j =0, 1 ′ 2 , 1 ...).Denoting the angular momentum with respect to the z-axis by J (j)3 ≡ J (j)12 ,onehas (J (j)3 ) σσ ′ = σδ σσ′ with σ = −j,...,+j.On the other hand, for m = 0, the little group is ISO(2). This is the case ofinterest here. It turns out that the quantum-mechanical states are only distinguishedby the eigenvalue of J 3 , the component of the angular momentum in thedirection of motion (recall k µ =(1, 0, 0, 1) from above),The eigenvalue σ is called the helicity. One then getsJ 3 ψ k,σ = σψ k,σ . (2.40)U(Λ, 0)ψ p,σ = Ne iσθ(Λ,p) ψ Λp,σ , (2.41)where θ denotes the angle contained in the rotation part of Λ. Since masslessparticles are not at rest in any inertial system, helicity is a Lorentz-invariantproperty and may be used to characterize a particle with m = 0. There remain7 This restriction is imposed in order to avoid tachyons (particles with m 2 < 0).