Most of the example models come from a rewriting system on ordered graphs; some concepts are illustrated by examples pertaining to string rewriting systems. But at the risk of making the picture more complicated (and note that this is incredibly simple compared to the full hypergraph case), we can annotate the multiway graph by including the updating events that lead to each transition between states: But now we can ask the question: what are the causal relationships between these events? Because the math we have for understanding finite systems is robust and well-understood. And the exciting thing is that in our models, there’s an obvious resolution. But every new mathematical model pose new issues associated with itself, e.g. It has to be a language that can express computational ideas. If we just want to know what the universe does, well, then we have the universe, and we can just watch what it does. For our original hypergraphs, we imagined that we’d get something like ordinary physical space (say close to three-dimensional Euclidean space). Love from Germany. Your models are like enchanted looms. For example, there are closed timelike curves, sometimes viewed as allowing time travel. But in a curved space, you won’t: And essentially what’s happening in the uncertainty principle is that you’re doing exactly this, but in branchial space, rather than physical space. Amazing stuff. Technical Documents. Here’s another rule: {{x, y, z}, {u, y, v}} → {{w, z, x}, {z, w, u}, {x, y, w}}. But actually, in our models, the “breaking off” can be even more extreme. But in our models we don’t start from anything like this, and in fact space and time are not even at all the same kind of thing. And a pretty natural thing for observers like us to do is just to say “one set of things happens all across the universe, then another, and so on”. It also doesn’t matter what the elements are called. Your opinions are important to us. But the big recent surprise for me is that we seem to be lucking out. Perhaps the particles—like electrons—that we currently know about are the “big ones”. If you insist on a simulatable universe, it raises the bar for the completeness of a theory, and this gets you to experimentally testable ideas faster. In the path integral there’s a quantity called the action—which is a kind of relativistic analog of energy—and when one works things out more carefully, our fluxes of causal edges correspond to the action, but are also exactly what determine the rate of turning of geodesics. And every so often I’d wonder if they might be relevant for physics. 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. But let me try to give a flavor of it. In general, the “volume” of the d-dimensional analog of a sphere is a constant multiplied by rd. Do you evaluate f[9], or f[8]? The key point is to think about what an observer who is themselves part of the multiway system will conclude about the world. In a case like this where there’s a final result (as opposed to just evolving forever), causal invariance basically says: it doesn’t matter what order you do all the updates in; the result you’ll get will always be the same. It is akin to how blocks, blinkers and gliders emerge from Conway’s simple rules in his ‘Game of Life’ (R.I.P.). It’s that the same stuff that corresponds to the virtual particles is actually “making the space”, and maintaining its structure. report, by Bob Yirka , Phys.org. And what it corresponds to physically is what’s normally called a light cone (or “forward light cone”). And already there are people in the physics community expressing opinions about it ranging from disapproval to disinterest. (The constancy of ρ is in effect a reflection of the Principle of Computational Equivalence.). We reproduced, more elegantly, what I had done in the 1990s. Now we can go back and also talk about how curvature interacts with mass and energy in space. They’ve written at length about their concerns over their field drifting further and further into abstruse mathematics and further from experimental testability. We don’t know. But unless we engage and try, the bridge between discrete math and physics will remain unbuilt, and, in all likelihood, fundamental physics will stay stuck. So a “flux of causal edges” is in effect the communication of activity (i.e. We’ll discuss how this relates to quantum mechanics in our models later. In typical cosmology, it’s been quite mysterious how different parts of the early universe managed to “communicate” with each other, for example, to smooth out perturbations. But something vital is lost in this approach. In direct analogy to the case of relativity, there are many different possible choices the observer can make about how to define time—and each of them corresponds to a different foliation of the multiway graph. Once that happens, the math quickly becomes hard to handle. We don’t (yet) know an actual rule that represents our universe—and it’s almost certainly not the one we just talked about. ), To keep things tolerably simple, I’m not going to talk directly about rules that operate on hypergraphs. The Wolfram Physics Project is a project launched by computer scientist and physicist Stephen Wolfram to find the fundamental theory of physics. Yes, there are different possible paths of history. More and more it seems likely that space itself is an emergent property of some deeper non-local structure. 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 only way to keep the foliation consistent in the multiway graph above is to have it progressively expand over time. But actually—as I first discovered in the early 1980s—this kind of intrinsic, spontaneous generation of complexity turns out to be completely ubiquitous among simple rules and simple programs. (Amusingly, their basic structure can be expressed in a fraction of a line of symbolic Wolfram Language code.) But there’s more to it. As time progresses we are in effect seeing the results of more and more steps in a computation. If we based everything on the traditional methodology of mathematics, we would in effect only be able to explore what we somehow already understood. There’s quite a bit of mathematical sophistication involved (for example, we have to consider curvature in space+time, not just space), but the bottom line is that, yes, in various limits, and subject to various assumptions, our models do indeed reproduce Einstein’s equations. Let’s have a blast. Well, actually, yes there is. Take a look at this multiway graph: At each slice in the foliation, let’s draw a graph where we connect two states whenever they’re both part of the same “branch pair”, so that—like AA and ABB here—they both come from the same state on the slice before. Not only can the causal graph split; the spatial hypergraph can actually throw off disconnected pieces—each of which in effect forms a whole “separate universe”: By the way, it’s interesting to look at what happens to the foliations observers make when there’s an event horizon. And second, we have to go as far as we can in figuring out what our rules actually do. It’s just that there isn’t anything immediately useful for folk working in that field that comes from this discipline, so why attend to it? (Functions are automatically loaded from the Wolfram Function Repository), Archives of runnable notebooks from the Wolfram Physics Project, Archives of working session videos from the Wolfram Physics Project, The tools for the Wolfram Physics Project are built with the Wolfram Language, Wolfram technology is freely available through site licenses at most major universities worldwide, The Wolfram Language can be freely used in the Wolfram Cloud, Introduction to Wolfram Physics Project tools, Guide to Wolfram Physics Project functions, All code being used for the project is openly available, Contact us about education & collaboration opportunities.

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