I am studying relativistic quantum mechanics and I can't really understand how propagators arise from the theory.
They are generally defined as the Wightman function $$ W_F (t',\vec{x}', t,\vec{x}) \equiv \langle{0| \phi(t, \vec x) \phi(t', \vec x') |0}\rangle. $$
I understand that the state $\phi(t, \vec x)| 0\rangle $ is the state representing an elementary particle of the field. The propagator gives us information about how two points $\vec x, \vec x' $ are related and also is a kernel so it can be solved using Green's functions. My question is: why are propagators Wightman functions? Are they supposed to be interpreted as the probability of transition between two points $\langle t, \vec x|t', \vec x' \rangle$? In what way is the Wightman function related to the probability of transition between two points?
