Is it possible to measure the time of annihilation of two particles? Suppose we have two delocalized particles, one is a particle and the other its antiparticle. These two particles can annihilate each other at some time, or perhaps not. 
Question: Is it possible to measure the time of annihilation? (I understand that time itself is not an observable, but I am not sure if that means that the time of an event cannot be measured)
If it can be measured: How is the information about annihilation time represented in Hilbert space? Is there an infinite number of superposed states, each one representing a different annihilation time? 
EDIT
Is the system in a superposition of states related to annihilation vs no annihilation? of so, is a superposition of two states (annihilated vs no annihilated), or of an infinite number of states (one for each time in which the particles might have annihilated)? 
 A: To answer the first part: 
When the particles annihilate each other, the energy is not lost and will instead be transformed to something else. In the case of electron-positron annihilation, you'll get two gamma photons. I'm not sure if you'll always get light, or if sometimes a particle-antiparticle pair annihilates into new particles. 
Anyway, you can set up a detector for the gamma photons that show up in the annihilation and that will tell you when the pair annihilated.
In terms of Hilbert space: If you don't check, then the state will be a superposition of states where the pair has annihilated and states where it hasn't. If you measure, though, it'll collapse to a defined state.
Time not being an observable means that there's no hermitian operator whose expectation value corresponds to some concept of "time", but it doesn't mean that we can't measure the time of events.

To explain a bit more the superposition thing: Let's look at a simpler example of an electron in an excited state $|1\rangle$ eventually decaying into the ground state $|0\rangle$.
At the beginning, the wavefunction will be just the excited state, $|\psi\rangle = |1\rangle$. As time goes on, it will become a superposition of $|1\rangle$ and $|0\rangle$ and $|0\rangle$ and will look like
$$|\psi(t)\rangle = A(t) |1\rangle + B(t) |0\rangle$$
where $A(0) = 1$, $B(0) = 0$, and as time goes on $A$ will shrink and $B$ will grow until, at $t \rightarrow \infty$ we have $A = 0$ and $B = 1$.
So in this case it's always a superposition of the two possible states, and the weight of the states within that superposition will change over time. It is not a superposition of all the possible decay-times!
