Gupta-Bleuler Condition

Question

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?

Answer 1

1. The Gupta-Bleuler condition is the condition for a physical state. It is not required to hold for all states in the preliminary space of states. Your example is simply an example of a state that is unphysical because it does not fulfill the Gupta-Bleuler condition.

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.