When a system is expressed in terms of creation and annihilation operators for bosonic/fermionic modes, what exactly is the physical meaning of the order in which the operators act?
For example, for a fermionic system with states $i$ and $j$, $c_i c_j^\dagger$ is different from $c_j^\dagger c_i$ by a sign change, due to anticommutativity. I understand the mathematics of this, but what does it mean intuitively?
The former would be described as destroying a particle in state $j$ "before" creating one in state $i$, but what does "before" actually mean in this context, since there's no notion of time?
As another (bosonic) example, $a_i^\dagger a_i$ is clearly different from $a_ia_i^\dagger$, since acting the former on a vacuum state $|0\rangle$ gives zero while for the latter, $|0\rangle$ is an eigenstate, but again, what is the physical interpretation?
My normal interpretation of commutativity as a statement regarding the effect of a measurement on a state fails here since creation/annihilation are obviously not observables.
I hope the question makes sense and isn't too abstract!