This is a followup to my previous question Is this proof that $a|0\rangle=0$ wrong (ladder oeprators and number operator)?
According to the comment, we should always have $a|0\rangle= 0$, because the norm of $a|0\rangle$ is 0 which I agree with. By definition of a Hilbert space, a norm of 0 is only possible for the null vector o.
However in "ulf leonhardt measuring the quantum state of light" which a copy is available here, in page 22, proposition 2.31, the author talks about the possibilty that $a|0\rangle\neq 0$ and explores it at the end of page 24. How can we even make this assumption if the norm of $a|0\rangle$ is always 0 ??
