Impossibility of superluminal signalling in Minkowski space-time 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 [Vilasini and Colbeck, Phys. Rev. A. 106, 032204 (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.

Figures (2)

• Figure 1: Three examples of causal models and their compatible embeddings in (1+1)-Minkowski space-time. In each case, the operational causal structure associated with the model is given in black, circled variables are observed nodes, while uncircled ones are not and the black arrows denote causation. Space-time information is given in blue with time along the vertical and space along the horizontal axis. The solid lines represent light-like surfaces and the shaded region corresponds to the joint future of $A$ and $C$ in all cases.
• Figure 2: Calculating the observed distribution of Example 3 (a) Acyclic causal structure $\mathcal{G}^{\text{acyc}}$ obtained by splitting the node $B$ of the cyclic causal structure $\mathcal{G}^{\text{loop}}$ of Figure 1(b). (b) Table showing the conditional probability distribution $P_{\mathcal{G}^{\text{acyc}}}(ABC|B')$ of the induced causal model over (a). We then post-select on $B=B'$ to recover the original cyclic model and then ignore $B'$. But this yields an unnormalised distribution, and we must therefore renormalise the distribution to obtain the required observed distribution $P_{\mathcal{G}^{\text{loop}}}(ABC)$ as given in the last column.

Theorems & Definitions (5)

• Example 1: The one-time pad
• Example 2: A simplified jamming scenario (cf. Grunhaus1996)
• Example 3: A fine-tuned causal loop
• Definition 1.1: Blocked paths
• Definition 1.2: d-separation