Contrary Inferences for Classical Histories within the Consistent Histories Formulation of Quantum Theory

Abstract

In the histories formulation of quantum theory, sets of coarse-grained histories that are consistent obey the classical probability rules. It has been argued that these sets can describe the quasi-classical behaviour of closed quantum systems, e.g. Omnès (Rev. Mod. Phys. 64(2), 339, 1992) and Hartle (Les Houches1992). Most physical scenarios admit multiple different consistent sets and one can view each of these as a separate context. Using propositions from different consistent sets to make inferences leads to paradoxes such as contrary inferences, first noted by Kent (Phys. Rev. Lett. 78(15), 2874, 1997). In this contribution, we use the consistent histories to describe a quasi-classical and macroscopic system to show that paradoxes involving contextuality persist even in the quasi-classical limit. This is distinctively different from the contextuality of standard quantum theory, where the contextuality paradoxes do not persist in the quasi-classical limit. Specifically, we consider different consistent sets for the arrival time problem of a (quasi-classical) ball in an infinite square well. For this setting, we construct two different consistent sets. We find the probabilities that each consistent set assigns to the simple question of whether the ball ever crossed the middle of the interval. We show that one consistent set concludes with certainty that the ball crossed it while the other consistent set concludes with certainty that it did not. Our results point to the need for constraints on the histories sets, additional to the consistency condition, to recover the correct quasi-classical limit in this formalism and lead to the motto ‘all consistent sets are equal’, but ‘some consistent sets are more equal than others’.