Conditioning on the Future: A Filtration-Theoretic Formalization of Wheeler's Participatory Universe

Abstract

Wheeler's delayed-choice experiments and the time-symmetric formalisms of quantum measurement have long fueled a debate over whether time symmetry in quantum theory entails retrocausality. We argue that the debate conflates two distinct structures, and that the framework for separating them is the classical theory of conditioning: Doob $h$-transforms, Markov bridges, and enlargement of filtrations on the probabilistic side, and the forward-state/backward-effect (two-state-vector) formalism of pre- and post-selected systems on the quantum side. We sharpen Wheeler's participatory universe, delayed choice, and ``It from Bit'' into three theses and build a correspondence dictionary that maps each to a precise statement in one or both formalisms. The dictionary is anchored by two structurally isomorphic results: an operational proposition that a future measurement choice sorts the past ensemble into subensembles without disturbing any earlier marginal, and its classical counterpart, the disintegration of an unconditioned law into future-conditioned bridges -- both a single fact: marginalizing over the final measurement leaves earlier marginals fixed, by completeness of the POVM together with trace preservation on the quantum side and by the tower property of conditional expectation on the classical. The conditional calculus is time-symmetric; its causal structure is not. Delayed choice is thereby a selection effect rather than retrocausation, and the apparent ``pull from the future'' is a bridge drift. We state explicitly what the framework does not do -- it neither derives the Born rule nor solves the measurement problem -- and delineate the legitimate scope of participatory language in quantum mechanics.