I encountered the following passage in Quantum Field Theory: Lectures of Sidney Coleman, page 280:
Since Lorentz transformations don't change $\phi'(0)$, or change any one-meson state to any other one-meson state, the coefficient $\langle k\rvert\phi'(0)\lvert 0\rangle$ of $e^{ik\cdot x}$ must be Lorentz invariant, and so can depend only on $k^2$. Then $k^2=\mu^2$, and$\langle k\rvert\phi'(0)\lvert 0\rangle$ is a constant.
Then the author goes on defining the field renormalisation constant $Z_3$ from this quantity.
My problem is, I thought that one-meson states $\lvert k\rangle$ were not invariant, since they should transform to $\lvert \Lambda k\rangle$, $\Lambda$ being the Lorentz transform.The other quantity, $\phi'(0)$, is surely constant since $\phi$ is a scalar field and $\Lambda 0=0$.I checked in the previous chapters on scalar fields, and indeed I found this transformation law.What am I missing here? Is that quantity really Lorentz-invariant, and if so, how can it be explained?
