1. Introduction
The dominant paradigm in physics relies on the idea that systems evolve through time according to dynamical laws, with the state at a given time determining the entire history of the system.
General relativity challenges this view. The Einstein equations, describing the relationship between spacetime geometry and mass-energy [
1], have counterintuitive solutions containing closed time-like curves (CTCs) [
2–
17]. An event on such a curve would be both in the future and in the past of itself, preventing an ordinary formulation of dynamics according to an 'initial condition' problem. The question then arises whether some more general type of dynamics is possible.
Although it is an open question whether CTCs are possible in our Universe [
18–
22], considering dynamics beyond the ordinary temporal view is relevant to other research areas as well. In a theory that combines quantum physics with general relativity, it is expected that spacetime loses its classical properties [
23,
24], possibly leading to indefinite causal structures [
25–
27]. In a quite different direction, it has been suggested that quantum physics could be reduced to some kind of 'retrocausal' classical dynamics [
28–
39].
The main problem arising when abandoning ordinary causality is the so called 'grand father paradox' [
40]: a time traveller could kill her own grandfather and thus prevent her own birth, leading to a logical inconsistency. A popular approach holds that the grandfather paradox makes CTCs incompatible with classical physics, while appropriate modifications to quantum physics could restore consistency [
41–
56]. A common feature of the proposals within this approach is that they postulate a radical departure from ordinary physics even in regions of space-time devoid of CTCs, or in scenarios where the time traveling system does not actually interact with anything in the past [
57,
58].
A different approach is the so called 'process matrix formalism', which takes as a starting point the local validity of the ordinary laws of physics and asks what type of global processes are compatible with this assumption [
59–
74]. This framework enforces that all operations that would normally be possible in ordinary spacetime should still be available in local regions. First considered in the quantum context, this approach has been applied to classical physics too, with the remarkable discovery of classical processes that are incompatible with any causal order between events [
75–
77].
In reference [
78], a classical, deterministic version of the formalism was proposed as a possible model for CTCs. In this model, one considers a set of regions that do not contain any, but might be traversed by, CTCs. Agents in the regions receive a classical state from the past boundary, perform an arbitrary deterministic operation on it, and then send the system through the future boundary. Dynamics outside the regions determines the state each agent will observe in the past of the respective region, as a function of the states prepared by other agents. A simple characterisation was found for all processes involving up to three regions; furthermore, it was found that, for three regions, all non causally ordered processes are essentially equivalent.
In this work, we extend the characterisation of deterministic processes to an arbitrary number of regions. We provide some simple interpretation of the characterisation: when fixing the state on the future of all but two regions, the remaining two must be causally ordered, with only one directional signalling possible.
We show, by explicit examples, that there are inequivalent, non causally ordered quadripartite processes, which cannot be reduced to tripartite ones. Our results show that CTCs are not only compatible with determinism and with the local 'free choice' of operations, but also with a rich and diverse range of scenarios and dynamical processes.
(necessary cause vs. sufficient cause)
