Quantum Gravity

THE CONCEPT OF A GRAVITON 27two independent linear polarization states are usually called the + polarizationand the × polarization (Misner et al. 1973).Consider, for example, a plane wave moving in x 1 ≡ x direction. In thetransversal (y and z) directions, a ring of test particles will be deformed intoa pulsating ellipse, with the axis of the + polarization being rotated by 45 ◦compared to the × polarization. One has explicitly(f µν =2Re e µν e −iω(t−x)) , (2.10)with x 0 ≡ t, k 0 = k 1 ≡ ω>0, k 2 = k 3 = 0. Denoting with e y and e z the unitvector in y and z direction, respectively, one has for the + and the × polarization,the expressionse 22 e + = e 22 (e y ⊗ e y − e z ⊗ e z ) (2.11)ande 23 e × = e 23 (e y ⊗ e z + e z ⊗ e y ) (2.12)for the polarization tensor, respectively (e 22 and e 23 are numbers giving theamplitude of the wave.) General solutions of the wave equation can be found byperforming superpositions of the linear polarization states. In particular,e R = √ 1 (e + +ie × ), e L = √ 1 (e + − ie × ) (2.13)2 2are the right and the left circular polarization states, respectively. The generalcase of an elliptic polarization also changes the shape of the ellipse.Of special interest is the behaviour of the waves with respect to a rotationaround the axis of propagation (here: the x-axis). Rotating counterclockwise withan angle θ, the polarization states transform according toFor (2.13), this corresponds toe ′ + = e + cos 2θ + e × sin 2θ ,e ′ × = e × cos 2θ − e + sin 2θ . (2.14)e ′ R =e−2iθ e R , e ′ L =e2iθ e L . (2.15)The polarization tensors thus rotate with an angle 2θ. This corresponds to asymmetry with respect to a rotation by 180 ◦ .If a plane wave ϕ transforms as ϕ → e ihθ under a rotation around the directionof propagation, one calls h its helicity. The left (right) circular polarizedgravitational wave thus has helicity 2 (−2). In the quantum theory, these stateswill become the states of the ‘graviton’, see Section 2.1.2. For plane waves withhelicity h, the axes of linear polarization are inclined towards each other by anangle 90 ◦ /h. For a spin-1/2 particle, for example, this is 180 ◦ ,whichiswhyinvariance for them is only reached after a rotation by 720 ◦ .