Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 1 / d / Solution

For a null momentum $q^{\mu }$, impose transversalityThis removes one of four vector components. The remaining equivalenceremoves a second component, leaving two transverse polarizations. For momentum along the third axis they may be chosen aswhich carry helicity $\pm 1$.

The equivalence is a gauge redundancy: vectors that differ by a multiple of $q^{\mu }$ describe the same physical state rather than distinct measurable configurations. A Lorentz transformation of a chosen transverse representative can require a compensating gauge transformation. Therefore the amplitudemust be invariant under ${ϵ}_{\mu }\mapsto {ϵ}_{\mu }+\alpha q_{\mu }$. For arbitrary $\alpha$, this is exactly the Ward identity