Superposition of dynamics, indefinite causal order, and quantum histories

Abstract: The process-matrix formalism describes quantum processes without assuming a fixed global causal order, but the physical meaning of indefinite causal order remains open. We address this question within histories theory, where temporal ordering and dynamical evolution are distinct structures. We show that the decoherence functional admits coherent superpositions of dynamics. In measurement settings, under suitable factorization conditions, these generate process matrices. Thus, we identify process-matrix indefinite causal order as an operationally restricted realization of the more general phenomenon of superposition of dynamics. Histories theory also admits a distinct kinematical notion of indefinite order, in which the ordering of physical events is itself a history observable. The resulting distinction between event order and intervention order clarifies the relation between indefinite causal order, quantum dynamics, and spacetime structure.