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

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. A second is an operational notion of causation that considers the flow of information between physical systems and interventions on them. In In [Vilasini and Colbeck, Phys. Rev. A. xx (2022)], we propose a framework for characterising when a causal model can coexist with relativistic principles such as no superluminal signalling, while allowing for cyclic and non-classical causal influences and the possibility of causation without signalling. In a theory without superluminal causation, both superluminal signalling and causal loops are not possible in Minkowski space-time. Here we demonstrate that if we only forbid superluminal signalling, superluminal causation remains possible and show the mathematical possibility of causal loops that can be embedded in a Minkowski space-time without leading to superluminal signalling. The existence of such loops in the given space-time could in principle be operationally verified using interventions. This establishes that the physical principle of no superluminal signalling is not by itself sufficient to rule out causal loops between Minkowski space-time events. Interestingly, the conditions required to rule out causal loops in a space-time depend on the dimension. Whether such loops are possible in three spatial dimensions remains an important open question.Introduction.-Understanding cause-effect relations is central to the scientific method, yet there are several inequivalent notions of causality. Often, it is defined with respect to a background space-time structure, after which causal structure and space-time structure are treated synonymously. An alternative is to define causality operationally and independently of space-time. One way to do this is through causal models, which are based on intervening on physical systems and analysing the resulting correlations [1,2]. This is the approach we take here. Causal models have been extensively applied to situations involving classical variables, being used for instance for medical testing [3,4], economic predictions [1, 5], and machine learning [6][7][8].

General framework for cyclic and fine-tuned causal models and their compatibility with space-time

“…Here, we first show that the absence of Type 1 and Type 2 affects causal loops is necessary for the existence of such a non-trivial and compatible space-time embedding. The results of the associated Letter [1] show that this is no longer true for ACLs of higher types, in particular we construct an ACL of Type 4 there that does admit such a space-time embedding. This demonstrates that the absence of affects causal loops is not necessary for the existence of a non-trivial and compatible space-time embedding.…”

“…• In Section VI, we define several distinct classes of causal loops and consider theories that are consistent with the principle that signalling outside the space-time future is not possible. We show that such theories are necessarily free of certain types of causal loops, and, in the associated letter [1], we apply our framework to construct a causal model for an operationally detectable causal loop that can be embedded in Minkowski space-time without leading to superluminal signalling. We discuss this example and illustrate in Appendix B that such theories (which allow for causal loops without signalling outside the future of a partially ordered space-time) can involve further distinct classes of causal loops beyond those defined in the main text.…”

“…In the case of classical causal structures with unobserved nodes, the set of compatible observed distributions for the causal structure are obtained by marginalisation of a total distribution (over all nodes) that satisfies Equation (1).…”

“…This class of theories is quite general, it certainly includes quantum and standard GPTs and any theory that can be characterised using acyclic causal structure. In the associated Letter [1], we apply the framework developed here to construct an explicit operationally detectable causal loop that can be embedded in (1+1)-dimensional Minkowski space-time without superluminal signalling, which demonstrates that the set T can also include theories admitting causal loops. In this section, we characterise several different classes of causal loops that can arise in our framework, and we show that some of these classes can be ruled out by requiring that the causal model has a compatible spacetime embedding while the results of the associated Letter show that some other classes cannot be ruled out in this manner.…”

“…Within our framework, we distinguish these concepts. In the associated Letter [1], we apply this framework to show the mathematical possibility of causal loops between Minkowski space-time events, the existence of which can be operationally detected without leading to superluminal signalling. 1 Our framework also suggests further conditions that could be used to rule out certain types of causal loops.…”