In the Wikipedia article it says "the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time".
Let us write the propagator going from state $A$ to state $B$ as $K_t(A,B)$.
According to the definition this amplitude should correspond in some sense to the probability $P(A|B)$. But this can't be so since: $$\sum_AP(A|B) = 1$$ but $$\sum_A |K_t(A,B)|^2 \neq 1$$ instead: $$\sum_A K_t(A,B) =1$$
However we do have $\sum\limits_B K_t(A,B)\Psi_0(B) = \Psi_t(A)$ which corresponds to the probabilities $P(A) = \sum\limits_B P(A|B)P(B)$. But we have $|\sum\limits_B K_t(A,B)\Psi(B)|^2 = |\Psi_0(A)|^2$. Hence the amplitude $K_t(A,B)$ can't be associated with any probability as the equation cannot be factored. Wave functions, of course, satisfy $\sum\limits_A|\Psi_t(A)|^2=1$
Hence, what right have we to call $K_t(A,B)$ a "probability amplitude" when it's absolute square does not correspond to a probability?
It is acceptable to call $K_t(A,B)$ a "time evolution operator" as we can interpret it as a matrix $K(t)^{AB}$ where $A$ and $B$ can be thought of as indices of a matrix.
Is this just being semantically picky? Or is there a fundamental difference between the complex amplitude given by a wave function $\Psi$ and the complex "amplitude" given by the propagator $K$?Because it seems to me that $K$ does not satisfy the definition of amplitude given by a complex number whose modulus squared represented a probability.