2.1 The process view

With this motivation in mind, I now set out to provide an alternative account of the temporal asymmetry of influence based on the simple requirement of avoiding contradictions. For concreteness, we will work within the context of a block-universe picture, where we take it that the universe does not come into being in some kind of temporal process but rather exists atemporally and eternally. ^{Footnote 1} I employ this conceptual framework because the block-universe approach is favored by a significant number of modern-day philosophers and physicists, in large part because it seems more hospitable to relativistic physics than other approaches to the metaphysics of time. In the context of the block-universe picture, it is reasonable to require that each variable has at most one value at a given spacetime point—that is to say, we do not allow variables whose value at a given spacetime point undergoes some change, and we do not allow variables whose value at a given spacetime point may be different relative to different observers. Note that we will exclude branching worlds from consideration, so we do not allow that values at a given spacetime point could differ across different branches.

As noted earlier, the commonsense intuition about the temporal asymmetry of influence is somewhat vaguely defined. However, it does seem clear that this intuition must be understood as having modal content: we can influence the future, but we cannot influence the past (or at least, so it seems to us). This suggests that the correct way to understand the difference between past and future is to see how it arises from some underlying modal structure. In this article, I will not make any assumptions about the nature of this modal structure: it could be understood within a metaphysically robust approach to modality that regards it as ontologically prior to the content of physical reality (Berenstain and Ladyman Reference Berenstain, Ladyman, Landry and Rickles2012; Adlam Reference Adlam2022b), or it could simply be understood within a Humean approach as the best systematization of the actual Humean mosaic (Jaag and Loew Reference Jaag and Loew2020; Lewis Reference Lewis and Richard1980). I insist only that modality is regarded as being an objective feature of reality in some sense.

The AKL approach does, in fact, proceed with an analysis of modal structure because it is grounded on an analysis of the semantics of backtracking counterfactuals. However, the relationship between this semantic approach and the way modality is modeled in physics is not always very clear, and thus in this article, the aim is to understand the modal structure of reality in a way that is more closely informed by physics. First, I model an experiment as a set of laboratories, in each of which agents may select inputs and/or receive outputs. I then define a process as a class of experiments that can be implemented anywhere in spacetime and can be composed in any desired way with other processes, such that the probabilistic relationship between inputs and outputs will always be the same if the process has been implemented correctly. That is, an experiment is a particular sequence of physical events instantiating one particular set of inputs and one particular set of outputs, whereas a process is a family of such sequences of physical events, encapsulating all possible inputs and the associated outputs and the probabilistic relationships between them. Note that for certain sorts of processes, it may be the case that there are restrictions on the way in which the laboratories can be arranged in spacetime—if these requirements are not met, then the process has not been implemented correctly, and hence the usual probabilistic relation between inputs and outputs would not be expected to hold. For example, for processes that can be regarded as transmitting signals, it is typically necessary that the output laboratory is in the future of the input laboratory, and if this requirement is not met, then the signal will not be successfully transmitted.

Evidently, the claim that some physical systems instantiate a particular process is indeed a modal one: we require not only that the actual input and output are consistent with the probabilistic relationship defining the process but also that we can make counterfactual assertions of the form, “if some other value $v$ were chosen for the input in laboratory one, then some other value $x$ would have been produced as output in laboratory two,” or probabilistic counterfactuals of the form, “if some other value $v$ were chosen for the input in laboratory one, then with probability $p$ some other value $x$ would have been produced as output in laboratory two.” Indeed, another way of thinking about a process is to think of it as an implementation of a computation, in the sense of Ladyman and Robertson (Reference Ladyman and Robertson2014): “not only … its initial and final states are those associated with the input and output states of the computation on some particular occasion, but also … had the initial state been that corresponding to one of the other input states of the computation, the resultant physical final state would have been that associated with the appropriate output state of the computation.” Thus, processes are modal in the same sense that computations are modal, and we may take it that the set of processes available in our universe is defined by the underlying objective modal structure of our universe, including the laws of nature—for example, the literature on operational formulations of quantum mechanics (Davies and Lewis Reference Davies and Lewis1970; Chiribella et al. Reference Chiribella, Mauro D’Ariano and Perinotti2011, Reference Chiribella, Mauro D’Ariano and Perinotti2010; Barrett Reference Barrett2005; Barnum et al. Reference Barnum, Barrett, Leifer, Spekkens, Stepanik, Wilce and Wilke2010; Masanes and Müller Reference Masanes and Müller2011) can be understood as characterizing the set of processes that can be constructed from quantum-mechanical systems on a fixed spacetime background.

This process-based approach allows us to finally be more precise about the nature of the intuition that we are able to influence the future and not the past. It can be reconceived as a statement about how we can use the outputs of our own decision-making processes—specifically, it seems that in general, we can use these outputs as inputs into other processes only if those processes produce outputs that are to the future of our deliberations, and not the past. Our task now is to understand how this restriction arises from fundamental requirements on modal structure.

Whether the modal structure is understood to determine the contents of reality (as in the metaphysically robust approach) or is simply read off the actual contents of reality (as in the Humean approach), clearly, one baseline condition must be met: the modal structure of our actual universe should not give rise to contradictions. For example, if the modal structure of reality includes deterministic causation, it cannot be the case that one causal process deterministically causes the value of a certain bit to be $0$ and another causal process deterministically causes the value of the same bit to be $1$ : different parts of the modal structures of the universe must be coordinated so as to ensure that they are compatible with one another. In the Humean approach, this is automatic because the modal structure is simply the best systematization of the actual mosaic, and because the actual mosaic does not contain contradictions, a good systematization of it should not lead to any contradictions either—or at least it should predict only a negligibly small number of contradictions because any time the systematization predicts a contradiction, it fails to match the actual mosaic and hence performs worse on the criterion of strength, so it becomes less likely to be the best system. Conversely, in the metaphysically robust approach where the modal structure determines the content of reality, the modal structure must evidently be formulated in such a way that it does not give rise to contradictions because, after all, it cannot ultimately produce a reality that contains contradictions. ^{Footnote 2} Thus, in both the metaphysically robust and Humean approaches, it makes sense to ask about the conditions that the modal structure must satisfy if we are to avoid contradictions.

The demand for a consistent modal structure is somewhat similar to the “principle of self-consistency,” which was originally proposed by Novikov to deal with problems concerning closed timelike curves (CTCs) in general relativity (Novikov Reference Novikov1992). Novikov’s principle tells us that the only solutions to the laws of physics that can occur locally in the real universe are those that are globally self-consistent—all events happen only once and cannot be changed. However, whereas Novikov’s principle is a constraint on the events that actually take place, our consistency requirement applies at the level of the modal structure: not only the occurrent events themselves but also the modal structure associated with those events should be free from contradictions because the modal structure either gives rise to those events or is a systematization of those events, depending on one’s view of modality—either way, the modal structure should not give rise to contradictions because the occurrent events cannot contain contradictions.

2.2 Temporal interdicts

What does this consistency requirement entail? First, note that processes can generally be composed to form larger processes—the output of one process can be used as the input to another process, and therefore we can think of the modal structure of our universe as giving rise to a large collection of “chained” processes, forming something like an enormous directed graph. And the requirement that the modal structure should not give rise to contradictions necessarily places some constraints on the way in which processes can be composed—in particular, it must be impossible to create compositions of the kind depicted in figure 1. Bell made a similar observation in his comments on time travel in the Gödel universe (Bell Reference Belln.d.), noting that the universe must be subject to what he referred to as “temporal interdicts” preventing contradictions that could arise from time travel: he argued that “there is an important difference between the limitative principles of physics and any principles (call them ‘temporal interdicts’) invoked to block changes of the past. In the first case it is logically possible that, for example, a body’s velocity could exceed that of light or that an electron’s position and momentum could be simultaneously measured with pinpoint precision. But any violation of a temporal interdict would involve a logical contradiction.” Bell described three possibilities for these crucial temporal interdicts: time travel is impossible, time travel is possible but no “changing of the past” is allowed, or time travel is possible and the universe branches whenever we change the past. Our case is more general, but the possible temporal interdicts are quite analogous to Bell’s:

1. 1. The modal structure of the universe does not allow processes that could give rise to contradictions.

2. 2. The modal structure of the universe could, in principle, give rise to contradictions, but it arranges accidents or interferes with free choice to prevent contradictions from actually occurring.

3. 3. The modal structure of the universe allows “contradictions” to occur, but the universe branches so that contradictory values occur in different branches.

We have excluded branching worlds, so we can discard interdict 3. Interdict 2 remains viable, but as noted in the philosophical literature on time travel (Horwich Reference Horwich1990, chap. 7], this approach would require us to accept that every attempt to create contradictions, as in figure 1, is foiled by apparently inexplicable coincidences. This possibility seems unappealing, and more to the point, it doesn’t appear to be the way things actually work in our world—we don’t seem to observe large numbers of strange coincidences or random changes of mind, and indeed, these features would be a contraindication for the existence of agents such as ourselves, agents who, in general, seem to have a significant amount of freedom to manipulate and compose variables as we please.

That leaves interdict 1—observers locally have freedom of choice about how they compose processes, but the modal structure of the universe does not admit any processes that could be composed in such a way as to give rise to contradictions. This interdict is significantly stronger than a mere prohibition on contradictions—it is, in fact, a very general constraint on the types of processes allowed by the modal structure of the universe. That is, assuming that we have the freedom to compose processes in any way we like, this interdict asserts that we should not have access to any processes that could be used to construct contradictions (regardless of whether they are in fact used to create contradictions). Moreover, this option seems to be the one that is realized in our actual world: what stops us from creating a composition like that in figure 1 is not that things inexplicably go wrong every time we try but simply that we do not know any processes that can be composed in this way—and indeed, apart from possibly general relativity, none of our fundamental theories currently tell us that loops of this kind are possible. So there are good reasons to think that the objective modal structure of our actual world indeed limits the possible processes such that no composition of possible processes can give rise to a contradiction. We now turn to the question of what that entails about the set of possible processes.

2.3 The process matrix framework

There is, in fact, a theoretical framework in quantum foundations that has been developed specifically to study the set of all possible processes that are compatible with observers locally having free choice but that also guarantees the absence of contradictions—the process matrix framework (Oreshkov and Giarmatzi Reference Oreshkov and Giarmatzi2016; Oreshkov et al. Reference Oreshkov, Costa and Brukner2012; Araújo et al. Reference Araújo, Branciard, Costa, Feix, Giarmatzi and Brukner2015). In this framework, we consider a set of laboratories in which agents can freely choose to perform any operations permitted by quantum mechanics, and we then use a “process matrix” to encode the set of all possible dynamics between the labs—including dynamics that are not permitted by any known physics, such as dynamics in “loops,” as described here. An equivalent formulation is available for classical physics, in which case we use process functions rather than process matrices, and agents can choose from all local operations permitted by classical physics (Baumeler et al. Reference Baumeler, Costa, Ralph, Wolf and Zych2019). The process matrix or function is defined in such a way that no dynamics encoded in a valid process matrix or function can ever give rise to any contradictions. So, for example, it can be shown that the process used for the composition in figure 1 does not have a valid process function—the classical process function formalism forbids the identity operation on loops precisely because it could produce this kind of contradiction: “The problem with the argument above is that it simply assumes that the identity backwards in time is a possible solution of the dynamics, based on the intuition that such evolution would be possible if a were in the future of x, without CTCs. The studies mentioned in the introduction suggest that such an assumption is typically incorrect: The system’s evolution typically finds a way to ‘adjust itself,’ preserving the consistency of ‘free interventions’” (Baumeler et al. Reference Baumeler, Costa, Ralph, Wolf and Zych2019). Thus, if we assume that observers locally have free choice to perform any operations they like, and that the modal structure of the universe does not allow for exceptions or inexplicable coincidences, it follows that all possible processes must be associated with valid process matrices or process functions.

The most straightforward type of process matrix is one that is causally ordered; that is, it is compatible with the existence of a strict partial order such that the probabilities for the outcome obtained in some laboratory depend only on the settings in experiments that precede it in the partial order. We can also imagine process matrices that are not causally ordered because the order for some subset of the laboratories may depend on the settings in other laboratories but that are still causal in the sense that we can write down a well-defined probability distribution over possible strict partial orders such that in each possible case, the choice of setting in a local experiment does not affect the occurrence of events that are earlier in the order or the strict partial order on the set of the events and the experiment in question (Oreshkov and Giarmatzi Reference Oreshkov and Giarmatzi2016)—that is to say, at each individual implementation, we will be able to put the experiments into a strict partial order. Causal processes cannot violate “causal inequalities” (Branciard et al. Reference Branciard, Araújo, Feix, Costa and Brukner2015) because the correlations they produce can always be embedded into a global strict partial order. And thus far, all processes known to be realized in nature (without postselection) are causal.

I will return to the topic of noncausal processes in section 5, but for now, I simply follow Feix et al. (Reference Feix, Araújo and Brukner2016) in postulating that, at least in the regimes with which we are familiar, the way the modal structure of the universe prevents contradictions is by limiting the allowed processes to those which are causal. Moreover, because composing processes gives rise to another process, it follows that in the regimes with which we are familiar, processes can only be composed in such a way that the resulting composite process is causal—that is, compatible with a strict partial order. Evidently, as long as all processes and compositions of processes are compatible with an underlying strict partial order, contradictions of the kind depicted in figure 1 cannot occur.

As noted by Jia and Sakharwade (Reference Jia and Sakharwade2018), defining a general composition rule for process matrices is not straightforward—simply taking the tensor product of two process matrices can sometimes produce something that is not a valid process matrix, so it’s necessary to impose some restrictions to ensure that composing processes yields another valid process. And presumably, we will need some further restrictions if we also want to ensure that composing two causal processes yields another causal process. But to get a picture of what these restrictions will look like, let’s focus on the case where we are composing a set of bipartite signaling causal processes—bipartite meaning the process takes an input in one lab and produces an output in another lab, and signaling meaning the output depends nontrivially on the inputs; that is, for some possible output $O$ and input $I$ , we have $p(O|I) \ne p\left( O \right)$ . Evidently, if we chain processes of this kind together by using the output of one process as the input to the next, the resulting composed process will be compatible with an underlying strict partial order, provided that there are no loops in the chain, because then we can obtain a strict partial order for the composed process by simply concatenating the strict partial orders for all of the individual causal processes. That is to say, the composed process is causal, provided that the directed graph corresponding to the way in which the processes are chained together is acyclic.

Moreover, in the regions and regimes we have so far probed, the “strict partial order” to which processes conform is simply temporal order: when we perform a causal bipartite signaling process whose input is in the past lightcone of its output, the output of that process can only be used as the input into another signaling process if the input of that process is also in the past lightcone of its output. Provided that we are in a spacetime that does not contain CTCs, this will ensure that all the resulting composed processes are compatible with a strict partial order, which is simply given by the temporal order of the experiments in any valid reference frame. I will use the term consistent chaining to refer to the requirement that the output of a signaling process that is oriented forward in time can only be used as input to other signaling processes that are also oriented forward in time, thus giving rise to a process graph that is acyclic. I emphasize that consistent chaining arises not merely from a prohibition on contradictions, but a prohibition on the existence of processes that could be used to create contradictions by agents with complete freedom to create arbitrary compositions of processes. Recall that the justification for taking this approach is ultimately empirical: as discussed in section 2.2, temporal interdict 2 remains a logically possible alternative to consistent chaining, but as far as we can tell from our best current physics, all known physical processes do indeed obey consistent chaining.

Note that the consistent-chaining requirement does not require us to postulate an objective distinction between the inputs and outputs of processes. We are always free to rename inputs as outputs, and vice versa, and indeed, Baumeler et al. (Reference Baumeler, Costa, Ralph, Wolf and Zych2019) demonstrate that (in the case of classical deterministic dynamics) every process function can be extended to an invertible one, which ensures that it is always possible to reverse the description so that inputs become outputs, and vice versa. However, if we are going to exchange the inputs and outputs for one process, then we will also have to do so for all the other processes to which it is linked; otherwise, we will be trying to set the output of one process equal to the output of another process. So, assuming that the process graph has no disconnected parts, this transformation will amount to changing the directions of all of the edges of the graph: clearly, if it is acyclic under the original ordering, it will remain acyclic under such a transformation. Thus, we can equally well describe the consistent-chaining requirement by saying that the output of a signaling process that is oriented “backward in time” can only be used as input to signaling processes that are also oriented “backward in time”—from the block-universe perspective, these descriptions are equivalent. Thus, a consistent-chaining mechanism does not have to rely on the existence of an objective preferred direction of time: we can make sense of this chaining requirement no matter which way we imagine the processes running. Indeed, as Ismael (Reference Ismael2016) notes, “nomological relationships do not have an intrinsic direction of determination…. Dynamical laws … are constraints on the relationships between states at different times, but … there is nothing in the law itself to say that either determines the other,” and this feature of our dynamical laws gives us good reason to suppose that all of the relevant modal relations are perfectly symmetric—so the processes are not actually running in either direction, but nonetheless, due to the consistent-chaining requirement, once we choose a temporal orientation for one process, that will induce an orientation for all the other processes.

2.4 Agency

Now let us try to add agents such as ourselves into this picture. Evidently, we can think of a decision made by an agent as a kind of signaling process: the agent takes an input (i.e., a set of memories and/or present perceptions) and produces an output (the final decision). The memories are themselves the output of a set of signaling processes that produce records of earlier events, and clearly, the agent will regard the temporal direction recorded by their memories as the past and the other direction as the future; hence, the agent will perceive their deliberative process as being temporally oriented from the past into the future—that is, the process of making a decision will always be perceived as having an output that is later than its input relative to the direction of time perceived by the agent. A decision then typically leads the agent to manipulate some external variables (e.g., writing and posting a letter), so the output of the decision-making process is used as the input to some other process, typically a signaling process (e.g., the process of sending a message through the post). Decisions are therefore part of the process graph, so they must be “chained” in a way that obeys the consistent-chaining condition. It follows that the output of a decision can only be used as the input to a signaling process if that process has the same temporal orientation as the deliberative process—which is to say, it has an output that is later than its input and therefore later than the decision. Thus, if we take it that “influence” necessarily requires nontrivial dependence of the event in question on the actions that supposedly influence it, so “influence” must be mediated by signaling processes, it follows that agents will find themselves only able to influence events occurring later than their deliberation, according to their own perceptual direction of time. And this is, of course, exactly how agency does work: in the words of Price and Weslake (Reference Price, Weslake, Beebee, Menzies and Hitchcock2008), “we can only wiggle handles which lie in the immediate future, with respect to our own deliberations on the matter,” or in Frisch’s words, “It is a striking fact about experimental interactions that we can only intervene into a system ‘from the past,’ as it were” (Frisch Reference Frisch, Suárez, Dorato and Rédei2010).

Furthermore, part of what it is to be an “agent” persisting over time is that the inputs to our decisions include information about our past decisions. Thus, from the process point of view, an agent is simply a set of chained decisions, that is, an ordered set $S$ of edges of the process graph, with the input of each edge connected either directly or indirectly to the outputs of all of the previous ones. Because these processes are signaling ones, consistent chaining then mandates that all of the edges in the set $S$ have the same orientation in time, so the agent’s perceived “past” and “future” will be consistent throughout the lifetime of the agent. In particular, because the agent can’t make decisions with outputs earlier than the time of the decision relative to the agent’s perceptual direction of time, the agent will be unable to “go backward” or “travel back in time” unless they encounter any CTCs.

Notice that from the external, objective point of view, there is still no preferred direction here. An agent is just a signaling process, and the process can equivalently be described in either direction: in reverse, a decision would look like a map from a final decision to the set of memories and/or outputs of past decisions that produce this decision, and the memories in turn become inputs to other past-oriented processes that ultimately produce the events that the memories are of. It is not objectively the case that the memories are the inputs and the decisions are the outputs; that is simply how we experience the process because, of course, we remember the content of the memories. So the asymmetry comes solely from the fact that the agent has a perspective defined by the fact that they have memories of one temporal direction and not the other: that direction is necessarily perceived as the past, and the other direction, the direction toward which agents produce decisions as outputs, is therefore perceived as the future. There are, of course, further questions about how there comes to be an agent with memories and thus a particular temporal perspective in the first place, and it seems likely those questions must be answered by appeal to thermodynamics. But conditional on the existence of such an agent with memories of the past, the fact that the agent can only output decisions that affect the “future” relative to their own orientation is an objective fact about the modal structure that arises from the consistent-chaining requirement. Thus, for such an agent, there is a very concrete distinction between past and future that has nothing to do with thermodynamics: given that the agent is oriented in a certain direction, they can only use their actions as inputs to processes that are oriented in the same direction, and therefore there is nothing epistemic or approximate about the temporal asymmetry of influence.

2.5 Example

One way to clarify the distinction between our approach to the temporal asymmetry of influence and the usual thermodynamical account is to consider a case where the thermodynamic asymmetry fails to hold in the usual way. For example, consider Price’s example of the “Stargate Doughnut,” involving a microscopic gate that may block the path of a photon or allow the photon to pass through (Price Reference Price, Price and Corry2005). Because we are dealing with only microscopic entities, it seems that thermodynamics is not relevant to this scenario, and yet we tend to think that the gate’s being closed will prevent the photon from arriving at its destination but will not prevent it from being emitted in the first place, so there appears to be some asymmetry in our description of the situation. Price argues that “what we bring to the case, in imagination, is the typical perspective we have as deliberating agents … according to the perspectival view, it is this asymmetric perspective on our part that grounds the intuitive asymmetry.” That is to say, our intuitions about this case are essentially based on an analogy—we are importing a causal asymmetry extrapolated from our experience even though there is no real asymmetry here.

However, the consistent-chaining account entails that even though this scenario is microscopic, the temporal asymmetry of influence still applies. For example, suppose some microscopic device is oriented with respect to the Stargate so as to accept information as input from some temporal direction ${t_1}$ , which we will refer to as the “past.” Then that device can gain information on whether or not the photon is present in the “past” before determining if the gate should be open or closed (assuming that the gate can be opened or closed instantaneously). So if this device’s action in closing the gate can bring it about that the photon is not present in the “past,” then this device could create a paradox by checking to see if the photon is present on the left side and closing the gate if it is. However, if this device’s action in closing the gate can only bring it about that the photon is not present in the opposite temporal direction ${t_2}$ , which we will refer to as the “future,” then no paradox can arise because the device cannot get information from the future before deciding whether or not to open the gate. (Obviously, exactly the same argument could be made in reverse about a different device configured so as to accept information as input from the temporal direction ${t_2}$ instead.) So the difference between the past and future in this case doesn’t just come from making analogies to more standard thermodynamic cases: it is an objective fact that if some process can gain information about the passage of the photon from one temporal direction, then the output of that process in terms of opening or closing the gate can only influence what happens to the photon in the other temporal direction, or else logical paradoxes would ensue. The consistent-chaining account therefore entails that in such scenarios, it will always be the case for any given agent or device that one direction of time is inaccessible and the other direction is accessible, so in a sense, there is indeed an objective asymmetry present in this scenario, although the direction of the asymmetry is not fixed, and differently oriented agents or devices will be subject to different temporal asymmetries.

2.6 Indeterministic processes

The reasoning that led us to the need for consistent-chaining requirements was concerned specifically with deterministic signaling processes. But reality also appears to contain some signaling processes that are not deterministic; indeed, if everything is ultimately made up of quantum fields undergoing scattering processes, then one might worry that, really, no processes are deterministic. And clearly, the earlier argument will not go through if we compose two indeterministic signaling processes. Returning to figure 1, suppose each of the processes has a probability $0.99$ of producing an output equal to its input: then we can always have one of the channels produce an output different from its input, rendering the loop consistent. This is admittedly an unlikely turn of events, but there is no contradiction here.

However, suppose that we create a large ensemble of such loops. In order to avoid inconsistencies, in every case, one of the channels must produce an output different from its input, and therefore in the context of this composition, the relative frequency with which these channels produce outputs different from their inputs will be $0.5$ , which is very far from the expected relative frequency of $0.99$ . For a large ensemble, the probability of such a large mismatch between the expected and predicted relative frequencies becomes extremely small, and it can be made arbitrarily close to zero by making the ensemble large enough. Moreover, this mismatch will necessarily occur across all cases of this kind of composition, and therefore there will be no meaningful sense in which it is true that the probability for these processes to produce an output matching their input remains $0.99$ under this composition—regardless of one’s preferred account of probability, the only reasonable conclusion would be that the probability is somewhere close to $0.5$ under this composition. But we have specified that for a process, the probabilistic relationship between inputs and outputs must remain the same under this composition, provided that the process is being implemented correctly; so under these circumstances, we would have to conclude that we have not succeeded in reproducing the same process because the probabilities do not match the usual ones under this sort of composition.

Note that as in the deterministic case, such inconsistencies can only arise if the composed process is not causal, and because all processes known to occur in nature are causal, it seems that at least in the regimes with which we are familiar, the way the modal structure of the universe prevents these sorts of inconsistencies is by limiting the allowed processes to those that are causal. So even when we have access to indeterministic signaling processes, agents can only output decisions that influence the “future” relative to their own orientation, and thus neither deterministic nor indeterministic signaling processes allow agents to influence the past in a spacetime without CTCs.

2.7 Nonlocality

We have so far considered only signaling processes, that is, processes such that something can be inferred about the input of the process from the output. However, when we move into quantum regimes, we encounter novel processes that are not signaling but where, nonetheless, the relationship between the input and the output is not trivial. For example, using quantum mechanics, we can construct a process composed of two laboratories, each containing half of an entangled pair of particles, with the measurement directions as the “inputs” and the measurement outcomes as the “outputs.” Evidently, this is indeed a process in the sense that there exists a consistent probabilistic relationship between the inputs and outputs, and moreover, Bell’s theorem (Bell Reference Bell1990) shows that the relationship between the laboratories is nontrivial: the output in one laboratory really does depend on the input in the other laboratory in a way that can’t be explained by a common cause in the past lightcone. So should consistent-chaining requirements apply to such processes?

Well, note that if we use nonsignaling processes of this kind to create the composition shown in figure 1, we will not obtain either a contradiction or a probability inconsistency—in fact, the resulting process will still be causal because the correlations will be compatible with a strict partial order (although the choice of order is not unique—different reference frames will lead to different quantum descriptions and thus different orders). Thus intuitively, it would seem that consistent chaining need not place any restrictions on the way in which nonlocal, nonsignaling processes can be arranged in spacetime. And indeed, it transpires that this is the case: if a process is constructed from measurements on entangled particles, the probabilistic relationships between inputs and outputs will be the same regardless of where the laboratories are placed in spacetime, so it doesn’t matter whether they are spacelike, lightlike, or timelike separated. It has been observed that the existence of quantum processes of this kind that are nonlocal but also nonsignaling is a striking feature of reality that seems in need of explanation: for example, if we try to represent such a process by a causal model, it can be shown that the parameters of the model must be very carefully fine-tuned (Wood and Spekkens Reference Wood and Spekkens2015).The fact that processes of this kind must be nonsignaling in order to avoid contradictions might be regarded as furnishing the desired explanation: from this point of view, the universe has very carefully fine-tuned the “toolbox” of processes available to us to ensure that we are not provided with any processes that could give rise to contradictions.

3.1 Retrocausality

The consistent-chaining requirement would seem to imply that retrocausality is impossible because it prohibits the existence of past-oriented processes that can take as inputs the outputs of our deliberations. However, I have noted that the consistent-chaining approach is closely related to the perspectival account of causation, and Price himself uses the perspectival account as a way of arguing that retrocausality is at least conceptually possible (Price Reference Price, Price and Corry2005), so it seems that there is some tension here. But in fact, I have only argued that in all regimes so far probed, the universe seems to avoid contradictions by limiting the processes available to us in accordance with the consistent-chaining requirement—that is, temporal interdict 1. It is in these regimes that consistent chaining rules out the possibility of using the outputs of our deliberations as inputs to past-directed processes. But this empirical observation does not entail that the consistent-chaining requirement holds in all physical regimes: it remains possible that in some special regimes (e.g., regimes that are difficult to access, so that agents have limited freedom to compose processes defined in those regimes), temporal interdict 2 applies instead. So we could still find that in some special cases, we are able to use the outputs of our deliberations as inputs to past-oriented processes, and a perspectival analysis in the style of Price would presumably yield the conclusion that these sorts of cases do count as genuine retrocausality. Thus, the consistent-chaining approach offers an important caveat to Price’s discussion of retrocausality because it explains why we don’t see retrocausality in most ordinary regimes, but it does not contradict Price’s conclusion that retrocausality is at least conceptually possible within the perspectival account.

Moreover, even if the consistent-chaining requirement were genuinely universal, this would not entirely rule out retrocausality, for it is important to keep in mind that, as detailed by Adlam (Reference Adlam2022c), there are two very different notions of retrocausality. Some approaches to retrocausality postulate two distinct directions of dynamical causality that together determine intermediate events by forward and backward evolution, respectively, from separate and independent initial and final states—for example, the forward-evolving state and the backward-evolving state in the two-state vector interpretation (Aharonov and Vaidman Reference Aharonov, Vaidman, Muga, Mayato and Egusquiza2002). Other approaches postulate an “all-at-once” picture where the laws of nature apply atemporally to the whole of history, such as, for example, in Wharton’s all-at-once Lagrangian models (Wharton Reference Wharton2018); in such a picture, the past and the future have a reciprocal effect on one another, so there is definitely some kind of influence from the future to the past at play, but these effects can’t be separated out into separate forward and backward evolution. ^{Footnote 3}

If we had access to dynamical retrocausality, then we would presumably be able to use the outputs of our deliberations as inputs to these past-directed processes, so processes exemplifying dynamical retrocausality would seem to violate the consistent-chaining requirement. Indeed, Maudlin (Reference Maudlin2002) and Bracken et al. (Reference Bracken, Hance and Hossenfelder2021) describe several quantum-mechanical experiments where attempting to give an account in terms of dynamical retrocausality leads to paradoxes of a very similar kind to the one depicted in figure 1, precisely exemplifying the problems that arise in physical theories that do not obey consistent chaining. But as explained by Adlam (Reference Adlam2022c), all-at-once retrocausality does not lead to these kinds of paradoxes. And evidently, all-at-once retrocausality is, in any case, better suited for the picture I advocate here, as I have reinforced that the relationships underlying the process picture need not be thought of as asymmetric causal relationships; there are good reasons to think that they are really symmetrical, reciprocal relationships that appear asymmetric to us only because we happen to be instantiated in some particular direction along the chain of processes. And in a way, any such symmetric processes are automatically “retrocausal” in the all-at-once sense because a symmetric modal relation entails that the future influences the past, and simultaneously, the past influences the future; that is, past and future are on a par in terms of causal influence. Thus, consistent-chaining requirements do not rule out retrocausality in the more sophisticated all-at-once sense.