Quantum mechanics notoriously doesn’t play by the rules of the classical world. One of the most famous and fundamental examples of quantum weirdness is the idea that two observable quantities, such as a particle’s position and momentum, can be incompatible, or not simultaneously measurable. If you want to measure them both, the order in which you make the measurements matters. In mathematical terms, their commutator—momentum times position minus position times momentum—is nonzero.
When considering a time-evolving system, it can be helpful to talk about the so-called out-of-time-order (OTO) commutator between two quantities measured at different times—say, position at time 0 and momentum at time t > 0. From a theoretical perspective, there’s nothing paradoxical about the idea of making a later measurement first and an earlier measurement second. One need only write down a factor of eiHt in between, where H is the system’s Hamiltonian. And perhaps surprisingly, experimenters too have tricks for turning back the clock on a quantum system—although they can be challenging to implement—so OTO commutators make both mathematical and physical sense.
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