10:25 PM
Oh, that is actually very surprising
and actually raises an even bigger problem for me then, because in the operator product expansion at some point we have a "sum over primaries", implying that there is more than one conformal primary operator that produces the conformal primary states from the vacuum. On the other hand in (say) scalar field theory we only have one operator at each point in spacetime, so the existence of more than one such conformal primary operator is confusing
What I would have said was sure, you just evaluate the field away from the origin, but I've also been told that $\phi_I(x\neq0)|0\rangle$ is not a conformal primary state