I want to calculate the transition amplitude for a particle to start at position $q_1$ at time $t_1$ to position $q_2$ at time $t_2$ in the Heisenberg picture.
As we are in the Heisenberg picture, the time dependence is in the states, so the position operator is given by
We have instantaneous eigenstates $|q_1;t_1\rangle$ such that
Now because this is the Heisenberg picture the states don't evolve with time. They are fixed entities that tell us the entire spacetime history of a particle (if we could "solve" them, in a sense.)
The transition amplitude is given by $\langle q_2;t_2|q_1;t_1\rangle$. Now, as I said, the instantaneous eigenstates describe the entire spacetime history of the particle. The only condition on $|q_1;t_1\rangle$ is that
We have no idea how $Q(t)$ acts on $|q_1;t_1\rangle$ for times $t\neq t_1$.
So, is it then possible to view the transition amplitude as the probability amplitude that if I act on $|q_1;t_1\rangle$ with $Q(t_2)$ I will find $q_2$? i.e the probability amplitude for
$Q(t_2)|q_1;t_1\rangle=q_2|q_1;t_1\rangle$ being true?