Compatibility of Cyclic Causal Structures with Spacetime in General Theories with Free Interventions

TL;DR

This paper explores the relationship between information-theoretic and relativistic causality, proposing new methods to detect causal loops and embedding causal structures into spacetime, with implications for higher-dimensional Minkowski spacetime.

Contribution

It introduces higher-order affects relations for modeling cyclic causal influences and proposes stability conditions for embedding causal structures into spacetime to rule out loops.

Findings

Detection of causal loops from higher-order affects relations

Stability conditions for spacetime embeddings to exclude loops

Conjecture that no-signalling suffices to rule out loops in higher dimensions

Abstract

By relating and ordering events, causality constitutes a pivotal feature of our world. On the one hand, there are information-theoretic notions of causality defined in terms of the information processing ability of agents and on the other hand, there are relativistic notions of causality tied to a spacetime. In this thesis, we improve upon a framework introduced by V. Vilasini and R. Colbeck in PRA, 106, 032204 (2022) and PRL, 129, 110401 (2022) for connecting these notions, where the possibility of operationally detectable causal loops embedded in (1+1)-Minkowski spacetime without superluminal signalling was demonstrated. In the first part, we take the information-theoretic point of view, where the concept of higher-order (HO) affects relations was proposed to generically model signalling in the presence of cyclic, fine-tuned and non-classical causal influences. We establish new properties of HO affects relations and apply them to infer causal structures. We then demonstrate a complete and constructive way to detect causal loops from a set of HO affects relations. In the second part, we study the embedding of information-theoretic causal structures into partially ordered spacetimes. We propose stability conditions on the spacetime embedding to rule out a class of operationally detectable loops that cannot be ruled out by the principle of no-signalling (outside the relativistic future) alone. We then propose novel order-theoretic properties that we conjecture to hold in Minkowski spacetime with $d \geq 2$ spatial dimensions. This would imply that in contrast to (1+1)-Minkowski spacetime, in higher dimensions, the no-signalling principle is indeed sufficient for ruling out this class of loops. Finally, we introduce a relation which allows for causal inference through knowledge about the absence of signalling relations and we relate it to novel order-theoretic features of spacetime.