# Gupta-Bleuler Condition

I am studying quantum field theory using David Tong's notes available at http://www.damtp.cam.ac.uk/user/tong/qft.html and I am stuck at page 135 eq. $$6.56$$

I fail to see how the following equation: $$(a^3_p-a^0_p)|\phi\rangle=0$$ implies that the state $$|\phi\rangle$$ must contain equal pairs of timelike and longitudinal photons.

For example, according to the above, $$|\phi\rangle=a_q^{3\dagger}a_q^{0\dagger}|0\rangle$$ should be a valid state but when I substitute this into $$(a^3_p-a^0_p)|\phi\rangle$$, I obtain $$(2\pi)^3\delta^3(p-q)[a_q^{0\dagger}+a_q^{3\dagger}]|0\rangle$$ which is different from 0?

2. The claim is not that every state that contains equal pairs of longitudinal and timelike photons is a physical state. It is that every physical state must contain such equal pairs, since if it doesn't then in $$a^3_p\lvert \phi\rangle = a^0_p\lvert \phi\rangle$$ one side will be zero while the other won't.