Usually when talking about material particles exceeding the speed of light they talk about being able to travel to the past and that breaking causality. There's another particular case which they don't talk about (at least in the literature used in my course).
Let's consider a stationary reference frame $\mathsf S$ and an inertial one $\mathsf S'$ travelling at velocity $v=\beta c$ wrt $\mathsf S$. Say there's an event $(t,x)=(0,0)$ and by a physical effect it causes $(\Delta t>0, \Delta x>0)$ with $U=\frac{\Delta x}{\Delta t}>c$. If we apply a boost in the $x$ direction, $\Lambda_v:\mathsf S\to\mathsf S'$, then the time interval in $\mathsf S'$ would be $$\Delta t'=\gamma_v\left[\Delta t-\beta\frac{\Delta x}{c}\right]=\gamma_v\Delta t\left[1-\beta\frac{U}{c}\right]$$ At this point they say that if $\frac{c}{U}<\beta<1$, then $\Delta t'<0$ and so, $(\Delta x,\Delta t)$ in frame of reference $\mathsf S'$ happens in the past, thus violating causality. The case they don't mention is when $U=\frac{c}{\beta}$, namely, when $\Delta t'=0$ ! Does this mean the observer in $\mathsf S'$ would see a superposition of all states of the particle travelling at speed $U$? Is there any illustrative example to understand what is going on in this case and what implications it has?
