Branchial Motion and the Entanglement Horizon

We can think of motion in physical space as like the process of exploring new elements in the spatial hypergraph, and potentially becoming affected by them. But now that we’re talking about branchial space, it’s natural to ask whether there’s something like motion there too. And the answer is that there is. And it’s basically exactly the same kind of thing: but instead of exploring new elements in the spatial hypergraph, we’re exploring new elements in the branchial graph, and potentially becoming affected by them.

There’s a way of talking about it in the standard language of quantum mechanics: as we move in branchial space, we’re effectively getting “entangled” with more and more quantum states.

OK, so let’s take the analogy further. In physical space, there’s a maximum speed of motion—the speed of light,?c. So what about in branchial space? Well, in our models we can see that there’s also got to be a?maximum speed of motion in branchial space. Or, in other words, there’s a maximum rate at which we can entangle with new quantum states.

In physical space we talk about light cones as being the regions that can be causally affected by some event at a particular location in space. In the same way, we can talk about entanglement cones that define regions in branchial space that can be affected by events at some position in branchial space. And just as there’s a causal graph that effectively knits together elementary light cones, there’s something similar that knits together entanglement cones.

That something similar is the?multiway causal graph: a graph that represents causal relationships between all events that can happen anywhere in a multiway system. Here’s an example of a multiway causal graph for just a few steps of a very simple string substitution system—and it’s already pretty complicated:

But in a sense the multiway causal graph is the most complete description of everything that can affect the experience of observers. Some of the causal relationships it describes represent spacelike connections; some represent branchlike connections. But all of them are there. And so in a sense the multiway causal graph is where relativity and quantum mechanics come together. Slice one way and you’ll see relationships in physical space; slice another way and you’ll see relationships in branchial space, between quantum states.

To help see how this works here’s a very?toy version of a multiway causal graph:

Each point is an event that happens in some hypergraph on some branch of a multiway system. And now the graph records the causal relationship of that event to other ones. In this toy example, there are purely timelike relationships—indicated by arrows pointing down—in which basically some element of the hypergraph is affecting its future self. But then there are both spacelike and branchlike relationships, where the event affects elements that are either “spatially” separated in the hypergraph, or “branchially” separated in the multiway system.

But in all this complexity, there’s something wonderful that happens. As soon as the underlying rule has causal invariance, this implies all sorts of regularities in the multiway causal graph. And for example it tells us that all those causal graphs we get by taking different branchtime slices are actually the same when we project them into spacetime—and this is what leads to relativity.

But causal invariance has other consequences too. One of them is that there should be an analog of special relativity that applies not in spacetime but in branchtime. The reference frames of special relativity are now our quantum observation frames. And the analog of speed in physical space is the rate of entangling new quantum states.

So what about a phenomenon like relativistic time dilation? Is there an analog of that for motion in branchial space? Well, actually, yes there is. And it turns out to be what’s sometimes called the?quantum Zeno effect: if you repeatedly measure a quantum system fast enough it won’t change. It’s a phenomenon that’s implied by the add-ons to the standard formalism of quantum mechanics that describe measurement. But in our models it just comes directly from the analogy between branchial and physical space.

Doing new measurements is equivalent to getting entangled with new quantum states—or to moving in branchial space. And in direct analogy to what happens in special relativity, as you get closer to moving at the maximum speed you inevitably sample things more slowly in time—and so you get time dilation, which means that your “quantum evolution” slows down.

OK, so there are relativistic phenomena in physical space, and quantum analogs in branchial space. But in our models these are all effectively facets of one thing: the multiway causal graph. So are there situations in which the two kinds of phenomena can mix? Normally there aren’t: relativistic phenomena involve large physical scales; quantum phenomena tend to involve small ones.

But one example of an extreme situation where they can mix is black holes. I’ve mentioned several times that the formation of an event horizon around a black hole is associated with disconnection in the causal graph. But it’s more than that. It’s actually disconnection not only in the spacetime causal graph, but in the full multiway causal graph. And that means that there’s not only an ordinary causal event horizon—in physical space—but also an?“entanglement horizon” in branchial space. And just as a piece of the spatial hypergraph can get disconnected when there’s a black hole, so can a piece of the branchial graph.

What does this mean? There are a variety of consequences. One of them is that quantum information can be trapped inside the entanglement horizon even when it hasn’t crossed the causal event horizon—so that in effect the black hole is freezing quantum information “at its surface” (at least its surface in branchial space). It’s a weird phenomenon implied by our models, but what’s perhaps particularly interesting about it is that it’s very much aligned with conclusions about black holes that have emerged in some of the latest work in physics on the so-called holographic principle in quantum field theory and general relativity.

Here’s another related, weird phenomenon. If you pass the causal event horizon of a black hole, it’s an inevitable fact that you’ll eventually get infinitely physically elongated (or “spaghettified”) by tidal forces. Well, something similar happens if you pass the entanglement horizon—except now you’ll get elongated in branchial space rather than physical space. And in our models, this eventually means you won’t be able to make a quantum measurement—so in a sense as an observer you won’t be able to “form a classical thought”, or, in other words, beyond the entanglement horizon you’ll never be able to “come to a definite conclusion” about, for example, whether something fell into the black hole or didn’t.

The speed of light?c?is a fundamental physical constant that relates distance in physical space to time. In our models, there’s now a new fundamental physical constant: the maximum entanglement speed, that relates distance in branchial space to time. I call this maximum entanglement speed?ζ?(zeta) (ζ?looks a bit like a “tangled?c”). I’m not sure what its value is, but a possible estimate is that it corresponds to entangling about 10102?new quantum states per second. And in a sense the fact that this is so big is why we’re normally able to “form classical thoughts”.

Because of the relation between (multiway) causal edges and energy, it’s possible to convert?ζ?to units of energy per second, and our estimate then implies that?ζ?is about 105?solar masses per second. It’s a big value, although conceivably not irrelevant to something like a merger of galactic black holes. (And, yes, this would mean that for an intelligence to “quantum grok” our galaxy would take maybe six months.)

Finding the Ultimate Rule

I’m frankly amazed at how much we’ve been able to figure out just from the general structure of our models. But to get a final fundamental theory of physics we’ve still got to?find a specific rule. A rule that gives us 3 (or so) dimensions of space, the particular expansion rate of the universe, the particular masses and properties of elementary particles, and so on. But how should we set about finding this rule?

And actually even before that, we need to ask: if we had the right rule, would we even know it? As I mentioned earlier, there’s potentially a big problem here with?computational irreducibility. Because whatever the underlying rule is, our actual universe has applied it perhaps 10^500 times. And if there’s computational irreducibility—as there inevitably will be—then there won’t be a way to fundamentally reduce the amount of computational effort that’s needed to determine the outcome of all these rule applications.

But what we have to hope is that somehow—even though the complete evolution of the universe is computationally irreducible—there are still enough “tunnels of computational reducibility” that we’ll be able to figure out at least what’s needed to be able to compare with what we know in physics, without having to do all that computational work. And I have to say that our recent success in getting conclusions just from the general structure of our models makes me much more optimistic about this possibility.

But, OK, so what rules should we consider? The traditional approach in natural science (at least over the past few centuries) has tended to be: start from what you know about whatever system you’re studying, then try to “reverse engineer” what its rules are. But in our models there’s in a sense too much emergence for this to work. Look at something like this:

Given the overall form of this structure, would you ever figure that it could be produced just by the rule:

{{x,?y,?y}, {y,?z,?u}}?→?{{u, z,?z}, {u,?x,?v}, {y,?u,?v}}

Having myself?explored the computational universe of simple programs?for some forty years, I have to say that even now it’s amazing how often I’m humbled by the ability of extremely simple rules to?give behavior I never expected. And this is particularly common with the very structureless models we’re using here. So in the end the only real way to find out what can happen in these models is just to?enumerate possible rules, and then run them and see what they do.

But now there’s a crucial question. If we just start enumerating very simple rules, how far are we going to have to go before we find our universe? Or, put another way, just?how simple is the rule for our universe going to end up being?

It could have been that in a sense the rule for the universe would have a special case in it for every element of the universe—every particle, every position in space, etc. But the very fact that we’ve been able to find definite scientific laws—and that systematic physics has even been possible—suggests that the rule at least doesn’t have that level of complexity. But how simple might it be? We don’t know. And I have to say that I don’t think our recent discoveries shed any particular light on this—because they basically say that lots of things in physics are generic, and independent of the specifics of the underlying rule, however simple or complex it may be.

Why This Universe? The Relativity of Rules

But, OK, let’s say we find that our universe can be described by some particular rule. Then the obvious immediate question would be: why that rule, and not another? The history of science—certainly since Copernicus—has shown us over and over again evidence that we’re “not special”. But if the rule we find to describe our universe is simple, wouldn’t that simplicity be a sign of “specialness”?

I have long wondered about this. Could it for example be that the rule is only simple because of the way that we, as entities existing in our particular universe, choose to set up our ways of describing things? And that in some other universe, with some other rule, the entities that exist there would set up their ways of describing things so that the rule for their universe is simple to them, even though it might be very complex to us?

Or could it be that in some fundamental sense it doesn’t matter what the rules for the universe are: that to observers embedded in a universe, operating according to the same rules as that universe, the conclusions about how the universe works will always be the same?

Or could it be that this is a kind of question that’s just outside the realm of science?

To my considerable surprise, the paradigm that’s emerging from our recent discoveries potentially seems to suggest a definite—though at first seemingly bizarre—scientific answer.

In what we’ve discussed so far we’re imagining that there’s a particular, single rule for our universe, that gets applied over and over again, effectively in all possible ways. But what if there wasn’t just one rule that could be used? What if?all conceivable rules could be used? What if every updating event could just use any possible rule? (Notice that in a finite universe, there are only ever finitely many rules that can ever apply.)

At first it might not seem as if this setup would ever lead to anything definite. But imagine?making a multiway graph?of absolutely everything that can happen—including all events for all possible rules. This is a big, complicated object. But far from being structureless, it’s full of all kinds of structure.

And there’s one very important thing about it: it’s basically guaranteed to have causal invariance (basically because if there’s a rule that does something, there’s always another rule somewhere that can undo it).

So now we can make a rule-space multiway causal graph—which will show a rule-space analog of relativity. And what this means is that in the rule-space multiway graph, we can expect to make different foliations, but have them all give consistent results.

It’s a remarkable conceptual unification. We’ve got physical space, branchial space, and now also what we can call rulial space (or just rule space). And the same overall ideas and principles apply to all of them. And just as we defined reference frames in physical space and branchial space, so also we can define reference frames in rulial space.

But what kinds of reference frames might observers set up in rulial space? In a typical case we can think of different reference frames in rulial space as corresponding to different description languages in which an observer can describe their experience of the universe.

In the abstract, it’s a familiar idea that given any particular description language, we can always explicitly program any?universal computer?to translate it to another description language. But what we’re saying here is that in rulial space it just takes choosing a different reference frame to have our representation of the universe use a different description language.

And roughly the reason this works is that different foliations of rulial space correspond to different choices of sequences of rules in the rule-space multiway graph—which can in effect be set up to “compute” the output that would be obtained with any given description language. That this can work ultimately depends on the fact that sequences of our rules can support universal computation (which the?Principle of Computational Equivalence implies they ubiquitously will)—which is in effect why it only takes “choosing a different reference frame in rule space” to “run a different program” and get a different description of the observed behavior of the universe.

It’s a strange but rather appealing picture. The universe is effectively using all possible rules. But as entities embedded in the universe, we’re picking a particular foliation (or sequence of reference frames) to make sense of what’s happening. And that choice of foliation corresponds to a description language which gives us our particular way of describing the universe.

But what is there to say definitely about the universe—independent of the foliation? There’s one immediate thing: that the universe, whatever foliation one uses to describe it, is just a?universal computer, and nothing more. And that?hypercomputation?is never possible in the universe.

But given the structure of our models, there’s more. Just like there’s a maximum speed in physical space (the speed of lightc), and a maximum speed in branchial space (the maximum entanglement speed?ζ), so also there must be a maximum speed in rulial space, which we can call?ρ—that’s effectively another fundamental constant of nature. (The constancy of?ρ?is in effect a reflection of the Principle of Computational Equivalence.)

But what does moving in rulial space correspond to? Basically it’s a change of rule. And to say that this can only happen at a finite speed is to say that there’s?computational irreducibility: that one rule cannot emulate another infinitely fast. And given this finite “speed of emulation” there are “emulation cones” that are the analog of light cones, and that define how far one can get in rulial space in a certain amount of time.

What are the units of?ρ? Essentially they are program length divided by time. But whereas in the theory of computation one typically imagines that program length can be scaled almost arbitrarily by different models of computation, here this is a measure of program length that’s somehow fundamentally anchored to the structure of the rule-space multiway system, and of physics. (By the way, there’ll be an analog of curvature and Einstein’s equations in rulial space too—and it probably corresponds to a geometrization of computational complexity theory and?questions like P?=NP.)

There’s more to say about the structure of rulial space. For example, let’s imagine we try to make a foliation in which we freeze time somewhere in rulial space. That’ll correspond to trying to describe the universe using some computationally reducible model—and over time it’ll get more and more difficult to maintain this as emulation cones effectively deliver more and more computational irreducibility.

So what does all this mean for our original goal—of finding a rule to describe our universe? Basically it’s saying that any (computation universal) rule will do—if we’re prepared to craft the appropriate description language. But the point is that we’ve basically already defined at least some elements of our description language: they are the?kinds of things our senses detect, our measuring devices measure, and our existing physics describes. So now our challenge is to find a rule that successfully describes our universe within this framework.

For me this is a very satisfactory solution to the mystery of why some particular rule would be picked for our universe. The answer is that there isn’t ultimately ever a particular rule; basically any rule capable of universal computation will do. It’s just that—with some particular mode of description that we choose to use—there will be some definite rule that describes our universe. And in a sense whatever specialness there is to this rule is just a reflection of the specialness of our mode of description. In effect, the only thing special about the universe to us is us ourselves.

And this suggests a definite answer to another longstanding question: could there be other universes? The answer in our setup is basically no. We can’t just “pick another rule and get another universe”. Because in a sense our universe already contains all possible rules, so there can only be one of it. (There could still be other universes that do various levels of hypercomputation.)

But there is something perhaps more bizarre that is possible. While we view our universe—and reality—through our particular type of description language, there are endless other possible description languages which can lead to descriptions of reality that will seem coherent (and even in some appropriate definition “meaningful”) within themselves, but which will seem to us to correspond to utterly incoherent and meaningless aspects of our universe.

I’ve always assumed that any entity that exists in our universe must at least “experience the same physics as us”. But now I realize that this isn’t true. There’s actually an almost infinite diversity of different ways to describe and experience our universe, or in effect an almost infinite diversity of different “planes of existence” for entities in the universe—corresponding to different possible reference frames in rulial space, all ultimately connected by universal computation and rule-space relativity.