Realism and causality. II. Retrocausality in energetic causal sets

We describe a new form of retrocausality, which is found in the behavior of a class of causal set theories, called energetic causal sets (ECS). These are discrete sets of events, connected by causal relations. They have three orders: (1) a birth order, which is the order in which events are generated; this is a total order which is the true causal order, (2) a dynamical partial order, which prescribes the flows of energy and momentum among events, (3) an emergent causal order, which is defined by the geometry of an emergent Minkowski spacetime, in which the events of the causal sets are embedded. However, the embedding of the events in the emergent Minkowski spacetime may preserve neither the true causal order in (1) nor correspond completely with the microscopic partial order in (2). We call this disordered causality, and we here demonstrate its occurrence in specific ECS models. This is the second in a series of papers centered around the question: should we accept violations of causality as a lesser price to pay in order to keep realist formulations of quantum theory? We begin to address this in the first paper [E. Cohen, M. Cortês, A. C. Elitzur, and L. Smolin, preceding paper, Realism and causality. I. Pilot wave and retrocausal models as possible facilitators, Phys. Rev. D 102, 124027 (2020).] and continue here by giving an explicit example of an ECS model in the classical regime, in which causality is disordered.

Figure 1

Simulation of a typical energetic causal set model, with ($1+1$) dimensions and a cyclic space coordinate. This model has 20 different families and total event number ${10}^4$. The figure depicts events as single dots in the emergent ($1+1$)-dimensional Minkowski spacetime. For the purposes of the current work, the main aspect to highlight in the simulation is the emergence, at late times, of regular lines which we call quasiparticle trajectories. This marks the transition of the time irreversible to the time reversible phase.

Figure 2

Events generated in a ($1+1$)-dimensional energetic causal set model with 20 different families and total duration of ${10}^4$ events. This panel shows the same events as in Fig. 1 but now with the underlying causal links of the network uncovered. That is, this figure links the events one by one in the order that they were generated. For comparison purposes with Fig. 1, we have left the scale of this plot unchanged, though this means density of causal links is high and the individual links are indistinguishable. In red, we show the causal links which are highly discausal, i.e., events which move against the direction of time in Minkowski spacetime. We display in red only large discausal jumps, with a backward jump which is more than $1/20$ of the full time range. See text for explanation of this individual choice of $1/20$th jump size. In Fig. 4, we will zoom-in and reveal distinct regions of the causal set for different stages of evolution.

Figure 3

Minkowski coordinate time versus birth order. A causal set sequence is plotted in order to trace the frequency of discausal jumps in the sequence. If causal jumps were always aligned with emergent time, the plot would be monotonically increasing. Downward moves represent causal jumps opposing the arrow of time. In red, we highlight large discausal jumps: we choose these to be moves that jump backward in time with amplitude larger than a threshold of $1/20$ of the full time range of the plot. See text for choice of threshold value.

Figure 4

Zoom-in on the full sequence of the causal set structure, represented in the emergent Minkowski spacetime. The panels depict (a) the first ${10}^3$ events, (b) ${10}^3$ events in middle sequence, and (c) the last ${10}^3$ events in the sequence. The blue lines represent the causal links. Red displays highly discausal jumps: moves that jump backward in time with amplitude larger than $1/20$ of the time scale of each individual plot. Although the amplitude of discausal jumps increases with the direction of emergent time, their frequency decreases as the causal set evolves. This is in compliance with the overall evolution toward time symmetry of the structure.

Figure 5

(a) Difference between consecutive pairs in the causal set sequence plotted to investigate which jumps are retrocausal. A zero value signals moves which are causally ordered and a nonzero value signals retrocausal moves. The frequency of occurring zeros in this plot indicates then that the arrows of causality and emergent Minkowski time are aligned. The system starts off strongly discausal in the time-asymmetric phase, as noted by the low occurrence of zeros, and progressively evolves toward domination by causal moves at late times, in the symmetric phase. In (b), we show the frequency of zeros, computed in bins of 40 elements, in the evolution of the causal set. This figure confirms the indication of the left panel of a rapid increase in the frequency of zeros toward late times, denoting the decrease in discausal jumps.

Figure 6

The three basic stages of Becoming.

Figure 7

Simulation I: robustness of results to variation of initial conditions. Simulation with the same model parameters and different 20 initial spatial positions of each family. Panels show (a) events in emergent spacetime. (b) Evolution of Minkowski time with event number. (c) Difference between two consecutive elements for detecting the presence of limit cycles—when the difference is zero. (d) Evolution of the amount of limit cycles given by the amount of zeros of the previous plot.

Figure 8

Simulation I: robustness of results to variation of initial conditions. Simulation with the same model parameters and different 20 initial spatial positions of each family. Panels show the network of events in the emergent spacetime connected in order of sequence in the causal set. (a) Full evolution with ${10}^4$ events. (b) Zoom-in of first ${10}^3$ events. (c) Zoom-in of ${10}^3$ events from the middle of sequence. (d) Last ${10}^3$ events.

Figure 9

As Fig. 7, for Simulation II.

Figure 10

As Fig. 8, for Simulation II.

Figure 11

As Fig. 7, for Simulation III.

Figure 12

As Fig. 8, for Simulation III.