Indefinite Causal Order from Failure-to-Glue: Contextual Semantics and Parametric Time
Partha Ghose
TL;DR
The paper addresses the challenge of defining and comparing indefinite causal order (ICO) across multiple formalisms. It develops a category-theoretic gluing framework in Part I, treating definite causal orders as contexts and causal separability as the existence of a global section of a convex presheaf, complemented by a seven-valued intuitionistic contextual classifier that separates cross-context variation from true indeterminacy. In Part II, the same framework is applied to a quantum-gravity setting where time is a parametric ordering variable $\tau$, with stochastic spin-network dynamics and a Wheeler–DeWitt–type equilibrium constraining the system, so ICO becomes indeterminacy of the parametric order of coarse-grained interventions. The combined approach yields a unified language to compare ICO criteria, distinguishing genuine indeterminacy from hidden definite-order explanations and offering concrete guidelines, toy models, and a program for connecting higher-order causal structures with parametric-time descriptions of quantum gravity. This provides a clarifying bridge between established ICO strands (switches, process matrices, superpositions of causal structures) and a robust semantic–structural perspective that is adaptable to emergent geometry and equilibrium dynamics.
Abstract
Indefinite causal order (ICO) has been studied via higher-order quantum processes (e.g.\ the quantum switch), process matrices, and quantum-gravity proposals involving superposed causal structure, yet the meaning of ``indefiniteness'' and its relation to definite-order explanations often remain opaque. Part~I develops a category-theoretic formulation of definite-order explainability as a gluing problem: each definite causal ordering (a partial order/DAG type) is treated as a context, and causal separability amounts to a consistent global section (possibly after convex mixing), whereas causal nonseparability is a failure-to-glue. We also introduce a compact seven-valued contextual classifier -- an intuitionistic elaboration -- that separates variation across contexts from genuine indeterminacy. Part~II applies this framework to a quantum-gravity motivated setting where the fundamental time is a parametric ordering variable $τ$, distinct from geometric (spacetime) time. Adopting a stochastic-quantization perspective on spin-network dynamics (Hilbert space not assumed fundamental) and reading the Wheeler--DeWitt condition as an equilibrium/stationarity constraint, we interpret ICO as indeterminacy of the parametric order of coarse-grained relational interventions, even when the microscopic update process is globally ordered by $τ$. Together, the two parts provide a common language for comparing ICO criteria and for stating precisely what ``no hidden definite order'' means.