Recently, I am reading the Tong's note of QFT. I have 2 question.
In the note, $\hat{\phi}(x)|0\rangle:=|x\rangle$ being interpreted as creating a particle in spacetime $(x^0,\vec{x})$. It is quite abrupt for me about the interpretation. Specifically, I do not understand the logic why we could be sure such an interpretation is correct. To my knowledge, I think we need to compare all the properties of $|x\rangle$ in classical quantum mechanics with the definition above and hopefully find that all those properties match. However, even though we have all the same properties, we still cannot say they are the same.
Suppose above is true. Then under the SI unit, the interpretation of $\langle0|\hat{\phi}(x)|\vec{k}\rangle=\langle x|\vec{k}\rangle$ is the non-normalized coefficient (here, I say it is the coefficient because its norm square is the probability distribution) of the probability distribution of a free particle with momentum $\vec{k}\hbar$ being observed at $(x^0/c,\vec{x})$.(Here, the momentum has a coefficient $\hbar$ because I have the commutation relation $[\hat{a}(\vec{k}),a^\dagger(\vec{k'})]=(2\pi)^3\,2\hbar\, E_{\vec{k}}\,\delta^{(3)}(\vec{k}-\vec{k'})$) ). I want to know whether I am correct.
