This question concerns the quantisation of the EM gauge potential $A_\mu$. When the Gupta-Bleuler formalism is introduced, it is usually stated that the creation/annihilation operators satisfy $\langle 0|a_\mu (p)a_\nu^\dagger(p') |0\rangle \propto \delta(p-p') \eta_{\mu\nu}$ due to Lorentz covariance. Since $\eta_{\mu\nu}$ is indefinite, this implies the existence of negative norm states.
But why exactly must $\langle 0|a_\mu (p)a_\nu^\dagger(p') |0\rangle \propto \delta(p-p') \eta_{\mu\nu}$ hold? Specifically, why the factor of $\eta_{\mu\nu}$? (I'm not worried about the $\delta(p-p')$ factor). Here's the way I've always thought about it (which I now believe is flawed):
Define the array $T_{\mu\nu} = \langle 0|a_\mu (0)a_\nu^\dagger(0) |0\rangle $. Then, since $a_\mu$ transforms as a 4-vector under a boost $\Lambda$ with unitary rep $U(\Lambda)$ -- that is, $U(\Lambda)^\dagger a_\mu(p) U(\Lambda)=\Lambda_\mu^{\space\space\rho} A_\rho(\Lambda^{-1}p)$ -- and the vacuum is invariant under $U(\Lambda)$, we find by inserting unitaries that:
\begin{align} T_{\mu\nu}&= \langle 0| U(\Lambda)^\dagger a_\mu(0) U(\Lambda) U(\Lambda)^\dagger a_\nu(0)^\dagger U(\Lambda) |0\rangle\\ &= \langle0| \Lambda_\mu^{\space\space\rho} a_\rho(0) \Lambda_\nu^{\space\space \sigma}a_\sigma^\dagger(0)|0\rangle\\ &= \Lambda_\mu^{\space\space\rho} \Lambda_\nu^{\space\space\sigma} T_{\rho\sigma} \end{align} In other words, $T=\Lambda T \Lambda^T$ for all boosts $\Lambda$, and I'm willing to believe that the only matrices satisfying this are scalar multiples of $\eta_{\mu\nu}$, concluding the proof.
The issue is that, as I've defined it, $T_{\mu\nu}$ is infinite: it contains a $\delta(0)$ factor. I'd always assumed that this infinity could somehow be dealt with rigorously, making the proof valid. But I recently posted this argument, which uses very similar reasoning to the above to reach the absurd conclusion that any spin-$\frac{1}{2}$ field must identically annihilate the vacuum. The flaw, I was told, is that my argument contains infinities. So now I'm worried about the above reasoning for $A_\mu$. Is there a way to make it rigorous? Why does the infinity cause a problem in the spin-$\frac{1}{2}$ case but not in the spin-1 case?
