Sorry if this is a silly question (engineer here), but I was wondering if the math in particle physics assumes that unitarity applies even between measurements. In other words, I take it that the evolution of quantum states is governed by an operator that ensures the probability of all possible events adds up to 1 at all times. What I'm wondering -- is there an operator for which this does not apply at all times but still gives measurement results with probability between 0 and 1?
Yes, you can make a unitary asymptotic S-matrix (so asymptotic measurements) when the intermediate states do not evolve in a unitary way. This is what ghost fields do--- the intermediate states in ghost-descriptions include negative probability objects, but when you make asymptotic measurements you don't see the ghosts, you only see the positive probability objects.
In cases where you have a ghost description, there are often no-ghost formulations, like light-cone or axial gauges. In these formulations, the Hamiltonian is well defined, so that you can ask about measurements on the intermediate states and get well defined answers. These formulations have a reduced symmetry compared to the ghost formulation, but they are manifestly unitary.
In ghost formulations, you assume that every measurement is made on asymptotic states which have no ghosts. Even if it is not true that every measurement is of an S-matrix quantity, the existence of the unitary formulation guarantees that anything you build out of asymptotic states will only end up measuring a quantity which has a reasonable positive probability interpretation.