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  • $\begingroup$ I think you need to show that $\alpha = f(p)e^{ipx}$ generates the proposed ansatz (which you have verified is also a solution to the vacuum equation). $\endgroup$
    – ShKol
    Commented Mar 3 at 15:13
  • $\begingroup$ I'm sorry, I don't think I'm quite understanding, do you mean that $\alpha = f(p)p_{\mu}$ generates the proposed ansatz? $\endgroup$
    – Sophie Schot
    Commented Mar 3 at 16:54
  • $\begingroup$ You already have $A_{\mu} = \epsilon_{\mu}(p)e^{ipx}$. Choose $\alpha = -if(p)e^{ipx}$. This means the gauge transformed field is $A_{\mu} = \epsilon_{\mu}(p)e^{ipx} + \partial_{\mu}(-if(p)e^{ipx}) = ( \epsilon_{\mu}(p) +f(p)p_{\mu})e^{ipx}$ which is the new solution that has been proposed. $\endgroup$
    – ShKol
    Commented Mar 4 at 3:08