Possibility of causal loops without superluminal signalling -- a general framework

Abstract

Causality is fundamental to science, but it appears in several different forms. One is relativistic causality, which is tied to a space-time structure and forbids signalling outside the future. On the other hand, causality can be defined operationally using causal models by considering the flow of information within a network of physical systems and interventions on them. From both a foundational and practical viewpoint, it is useful to establish the class of causal models that can coexist with relativistic principles such as no superluminal signalling, noting that causation and signalling are not equivalent. Here we develop such a general framework that allows these different notions of causality to be independently defined and for connections between them to be established. The framework first gives an operational way to study causation that allows for cyclic, fine-tuned and non-classical causal influences. We then consider how a causal model can be embedded in a space-time structure (modelled as a partial order) and propose a mathematical condition for ensuring that the embedded causal model does not allow signalling outside the space-time future, which we call compatibility. We identify several distinct classes of causal loops that can arise in our framework, showing that compatibility with a space-time can rule out only some of them. In addition, we demonstrate the mathematical possibility of causal loops embedded in Minkowski space-time, whose existence can be operationally verified without leading to superluminal signalling. Within our framework we discuss conditions for preventing superluminal signalling within arbitrary (possibly cyclic) causal models. We also consider causation in post-quantum theories admitting so-called jamming correlations. By considering arbitrary interventions, this goes beyond previous works on jamming that focus on correlations. Finally, this work introduces a new causal modelling concept of "higher-order affects relations" and several technical results in this regard, which have applications for causal discovery in fined-tuned causal models. Contents I. Introduction II. Preliminaries: Acyclic and faithful causal models III. Motivation for analysing cyclic and fine-tuned causal models A. Friedman's thermostat and the one-time pad B. Jamming non-local correlations IV. The framework, Part 1: Causality A. Cyclic and fine-tuned causal models B. Interventions and affects relations C. Conditional and higher-order affects relations D. Relationships between concepts V. The framework, Part 2: Space-time A. Space-time structure B. Embedding of a causal model in a space-time structure C. Compatibility of a causal model with an embedding in space-time D. Necessary and sufficient conditions for compatibility VI. Causal loops and their space-time embeddings

Impossibility of Superluminal Signaling in Minkowski Spacetime Does Not Rule Out Causal Loops

“…Note that these causal mechanisms do not admit a unique solution. Nevertheless, in Section B of the Supplemental Material we apply a method proposed in [34] to uniquely determine the observed distribution based on these mechanisms, and show that the same distribution as the previous two examples is obtained. The effect of interventions in this case cannot be directly inferred from the observed correlations since none of the observed nodes are parentless.…”

“…In our associated paper [34], we have developed a general framework for causation that can describe nonclassical and cyclic causal influences as well as causation without signalling. We do so by keeping the direction of causation and the order of events in space-time distinct.…”

“…Interestingly, the same example fails to be embeddable in (3+1)-Minkowski space-time. In the associated paper [34], we have characterised a large class of operationally detectable causal loops within our framework. Whether any of these may be embeddable in (3+1) dimensions remains an interesting open question.…”

“…Causal models: correlations, interventions and finetuning-We begin by reviewing the essentials of the causal modelling framework (see [34] for details). A causal structure is modelled as a directed graph G whose nodes correspond to observed or unobserved systems and directed edges denote causation between these systems [38].…”

“…Here the observed distribution must be consistent with relationships specified by the causal mechanisms. Methods for modelling operational causal structures beyond the classical case are proposed in our associated paper [34]. In the present work, we restrict to the classical case which suffices to illustrate our main claims.…”