In non-relativistic quantum mechanics, the amplitude of observing a free particle of mass $m$ at $(x_{1}, t)$ given that it was observed at $(x_{0}, 0)$ is given by:
\begin{equation}\langle x_{0}|e^{-i\hat{H}t}|x_{1}\rangle = \sqrt{\left(\frac{m}{2\pi i \hbar t}\right)}e^{-\frac{m(x - x_{1})^{2}}{2i\hbar t}}.\end{equation}
Given that QFT is taken to be fully in agreement with relativity, would the computation of this same quantity in QFT result in $0$ for measurement events such that $(x_{1} -x_{0})^{2} > c^{2}t^{2}$? If not, how can we explain this effect relativistically?
