Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 4 / e

Using ${\alpha }_0^2=-2$, tracelessness and transversality of ${ϵ}_{\mu \nu }$ givesand $v\cdot {\alpha }_0=-t^{\lambda }{}_{\lambda }/2$, so the constraints hold.

On the momentum vacuum,andThereforewhereThe vacuum has $L_0|0,p⟩={\alpha }_0^2|0,p⟩/2=-|0,p⟩$ and is annihilated by positive modes, so part c proves that $|n⟩$ is null. The physical content is consequently the transverse traceless tensor ${ϵ}_{\mu \nu }$.