Before rejecting the assumptions that lead to this conclusion, we need to be clear about what they amount to. Saying that a “green” atom went through L cannot mean that this very atom would also have gone through L if the cavity containing the photon emitted by it had not been ascertained. For if this cavity had not been ascertained, the experimenters could have checked how the atom went through both slits (in phase or out of phase), and regardless of the outcome they would have found that it went through both slits.
Rather, a “green” atom went through L only because this is what was indicated by a measurement. An atom goes through a particular slit only if the appropriate measurement is made, and it goes through L only if this is the outcome. It goes through both slits with a particular phase relation only if a different measurement is made, and it goes through both slits in phase only if that is the outcome. If neither measurement is made, it doesn’t go through the left slit and it doesn’t go through the right slit and it doesn’t go through both slits in phase and it doesn’t go through both slits out of phase. To paraphrase a well-known dictum by John Wheeler — a towering figure of 20th Century physics — no property (or behavior) is a possessed (occurrent) property (behavior) unless it is a measured (or indicated) property (behavior). Measurements do not merely reveal pre-existent properties or behaviors that occurred independently of measurements; they create their outcomes.
Seen in this light, the possibility of partially determining an atom’s past behavior no longer seems preposterous. If (as suggested by some of the experiments previously discussed) the properties of the quantum world exist only if, and only to the extent that, they are measured, then the behavior of a quantum system at an earlier time can depend on a measurement performed at a later time, and thus also on a decision taken at a later time. Nothing like the backward-in-time causation of some fantasy novels, which changes the past, is suggested. By choosing to perform one of the two experiments, the experimenters cause no change in the atom’s past behavior. Rather, they contribute to create or determine its past behavior.
The world has exactly one history. Its state at any given time is what it is; it cannot be changed. But it can be what it is because of a measurement performed at a later time. (In fact, events indicating the possession of a property occur quite generally after the time of possession of the indicated property, albeit usually a fairly short time after.)
But still, if no behavior is an occurrent behavior unless it is a measured or indicated behavior, we have reasons to doubt that the “green” atoms actually went through L. For, strictly speaking, the conclusion that is warranted by the detection of a photon in the left cavity is only this counterfactual (contrary-to-fact) statement: if a direct measurement of the slit taken by the corresponding atom had been made, then this would have been found taking the left slit. But here we need to take into account that no measurement is direct. Measurement outcomes are inferred from pointer positions, digital displays, detector clicks, computer printouts, and what not. If we had no right to infer the position of a particle from such an indicator — if we could only conclude ad infinitum that a particle would be found if the appropriate measurement were made — then it would be impossible to measure the position of a particle, or anything else for that matter. But if we do have the right to infer the position of a particle from the click of a detector, then it appears that we also have the right to infer the slit taken by an atom from the click of a photosensor (if one is placed in each cavity).
1. [↑] “No elementary phenomenon is a phenomenon until it is a registered (observed) phenomenon.” — Wheeler, J.A. (1983). Law without law, in Wheeler and Zurek (eds.), Quantum Theory and Measurement, Princeton University Press, 182–213.