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Gupta-Bleuler condition is $$\partial^\mu A_\mu^+ | \phi \rangle=0\tag{6.54}$$ where $$A_\mu^+= \int\frac{d^3\mathbf p}{(2\pi)^3 \sqrt{2|\mathbf p|}} \sum_{\lambda=0}^3 \epsilon^\lambda_\mu a_p^\lambda e^{-ip\cdot x}.$$

David Tong's QFT lecture note claims that this implies $$(a_p^3-a_p^0)| \phi \rangle=0.\tag{6.56}$$ I can't see how to obtain that. When I apply the Gupta-Bleuler condition and substitute the above equation, I obtain $$\int\frac{d^3\mathbf p}{(2\pi)^3 \sqrt{2|\mathbf p|}} \sum_{\lambda=0}^3 (-ip^\mu)\epsilon^\lambda_\mu a_p^\lambda e^{-ip\cdot x}| \phi \rangle=0$$ Since $\epsilon^1, \epsilon^2$ are transverse photon and they are orthogonal to $p$, only $\lambda=0$ and $\lambda=3$ term contributes. Then a sufficient condition for Gupta-Bleuler condition is that $$\{(p\cdot \epsilon^0)a_p^0-(p\cdot\epsilon^3)a_p^3\}|\phi\rangle=0$$ but this is different from the lecture note!

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You are forgetting that $\epsilon^\lambda = \epsilon^\lambda(p)$.

That basis is constructed this way, referring to a fixed Minkowskian reference frame (I adopt the signature (-,+,+,+) and $\vec{a}$ indicates your ${\bf a}$)

(1) $\epsilon^3(p) = \frac{\vec{p}}{|\vec{p}|}$,

(2) $\epsilon^{1}(p), \epsilon^2(p)$ are of unit length, mutually orthogonal, and also orthogonal to $\vec{p}$.

(3) Finally, the triple of three-vectors $\epsilon^1(p),\epsilon^2(p), \epsilon^3(p)$ viewed as spacelike four-vectors, form a pseudo-ortonormal (future-oriented, positive) basis of Minkowski space when adding $\epsilon^0(p)$ to them.

Hence $\epsilon^0(p)$ must have components $(1,0,0,0)$. This way $$p\cdot \epsilon^0(p) = p^0\:,\quad p\cdot \epsilon^3(p) = \frac{\vec{p}\cdot\vec{p}}{|\vec{p}|}\:.$$ Since $p$ is light-like (we are dealing with photons) $$p^0= |\vec{p}|=\frac{\vec{p}\cdot\vec{p}}{|\vec{p}|}$$ This identity inserted in your final identity gives rise to David Tong's assertion.

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  • $\begingroup$ The problem addressed by the Gupta-Bleuler method is that gauge invariance cannot be imposed at operator level. It can only be imposed on the fields, on the solution space. The strong statement of gauge invariance known from Maxwell's equations cannot be maintained. A much weaker statement has to be accepted. This was one reason for me to publish an alternative theory, see arxiv.org/abs/physics/0106078. $\endgroup$ – my2cts Aug 8 '19 at 7:49

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