The Inevitability of Quantum Mechanics

“But how will you ever get quantum mechanics?”, physicists would always ask me when I would describe?earlier versions of my models. In many ways, quantum mechanics is the pinnacle of existing physics. It’s always had a certain “you-are-not-expected-to-understand-this” air, though, coupled with “just-trust-the-mathematical-formalism”. And, yes, the mathematical formalism has worked well—really well—in letting us calculate things. (And it almost seems more satisfying because the calculations are?often so hard; indeed, hard enough that they’re what?first made me start using computers to do mathematics?45 years ago.)

Our usual impression of the world is that definite things happen. And before quantum mechanics, classical physics typically captured this in laws—usually equations—that would tell one what specifically a system would do. But in quantum mechanics the formalism involves any particular system doing lots of different things “in parallel”, with us just seeing samples—ultimately with certain probabilities—of these possibilities.

And as soon as one hears of a model in which there are definite rules, one might assume that it could never reproduce quantum mechanics. But, actually, in our models, quantum mechanics is not just possible; it’s absolutely inevitable. And, as we’ll see, in something I consider quite beautiful, the core of what leads to it turns out to be the same as what leads to relativity.

OK, so how does this work? Let’s go back to what we discussed when we first started talking about time. In our models there’s a definite rule for updates to make in our hypergraphs, say:

But if we’ve got a hypergraph like this:

there will usually be many places where this rule can be applied. So which update should we do first? The model doesn’t tell us. But let’s just imagine all the possibilities. The rule tells us what they all are—and we can represent them (as we discussed above) as a?multiway system—here illustrated using the simpler case of strings rather than hypergraphs:

Each node in this graph now represents a complete state of our system (a hypergraph in our actual models). And each node is joined by arrows to the state or states that one gets by applying a single update to it.

If our model had been operating “l(fā)ike classical physics” we would expect it to progress in time from one state to another, say like this:

But the crucial point is that the structure of our models leaves us no choice but to consider multiway systems. The form of the whole multiway system is completely determined by the rules. But—in a way that is already quite reminiscent of the standard formalism of quantum mechanics—the multiway system defines many different possible paths of history.

But now there is a mystery. If there are always all these different possible paths of history, how is it that we ever think that definite things happen in the world? This has been a core mystery of quantum mechanics for a century. It turns out that if one’s just using quantum mechanics to do calculations, the answer basically doesn’t matter. But if one wants to “really understand what’s going on” in quantum mechanics, it’s something that definitely does matter.

And the exciting thing is that in our models, there’s an obvious resolution. And actually it’s based on the exact same phenomenon—causal invariance—that gives us relativity.

Here’s roughly?how this works. The key point is to think about what an observer who is themselves part of the multiway system will conclude about the?world. Yes, there are different possible paths of history. But—just as in our discussion of relativity—the only aspect of them that an observer will ever be aware of is the causal relationships between the events they involve. But the point is that—even though when looked at from “outside” the paths are different—causal invariance implies that the network of relationships between causal events (which is all that’s relevant when one’s inside the system) will always be exactly the same.

In other words—much as in the case of relativity—even though from outside the system there may seem to be many possible “threads of time”, from inside the system causal invariance implies that there’s in a sense ultimately just one thread of time, or, in effect, one objective reality.

How does this all relate to the detailed standard formalism of quantum mechanics? It’s a little complicated. But let me make at least a few comments here. (There’s some more detail in my?technical document;?Jonathan Gorard has given even more.)

The states in the multiway system can be thought of as possible states of the quantum system. But how do we?characterize how observers experience them? In particular, which states is the observer aware of when? Just like in the relativity case, the observer can in a sense make a choice of how they define time. One possibility might be through a foliation of the multiway system like this:

In the formalism of quantum mechanics, one can then say that at each time, the observer experiences a superposition of possible states of the system. But now there’s a critical point. 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.

Again by analogy to relativity, we can then think of these choices as what we can call different “quantum observation frames”. Causal invariance implies that as long they respect the causal relationships in the graph, these frames can basically be set up in any way we want. In talking about relativity, it was useful to just have “tipped parallel lines” (“inertial frames”) representing observers who are moving uniformly in space.

In talking about quantum mechanics, other frames are useful. In particular, in the standard formalism of quantum mechanics, it’s common to talk about “quantum measurement”: essentially the act of taking a quantum system and determining some definite (essentially classical) outcome from it. Well, in our setup, a quantum measurement basically corresponds to a particular quantum observation frame.

Here’s an example:

The successive pink lines effectively mark off what the observer is considering to be successive moments in time. So when all the lines bunch up below the state?ABBABBwhat it means is that the observer is effectively choosing to “freeze time” for that state. In other words, the observer is saying “that’s the state I consider the system to be in, and I’m sticking to it”. Or, put another way, even though in the full multiway graph there’s all sorts of other “quantum mechanical” evolution of states going on, the observer has set up their quantum observation frame so that they pick out just a particular, definite, classical-like outcome.

OK, but can they consistently do that? Well, that depends on the actual underlying structure of the multiway graph, which ultimately depends on the actual underlying rule. In the example above, we’ve set up a foliation (i.e. a quantum observation frame) that does the best possible job in this rule at “freezing time” for the?ABBABB?state. But just how long can this “reality distortion field” be maintained?

The only way to keep the foliation consistent in the multiway graph above is to have it progressively expand over time. In other words, to keep time frozen, more and more quantum states have to be pulled into the “reality distortion field”, and so there’s less and less coherence in the system.

The picture above is for a very trivial rule. Here’s a corresponding picture for a slightly more realistic case:

And what we see here is that—even in this still incredibly simplified case—the structure of the multiway system will force the observer to construct a more and more elaborate foliation if they are to successfully freeze time. Measurement in quantum mechanics has always involved a slightly uncomfortable mathematical idealization—and this now gives us a sense of what’s really going on. (The situation is ultimately very similar to the problem of decoding “encrypted” thermodynamic initial conditions that I mentioned above.)

Quantum measurement is really about what an observer perceives. But if you are for example trying to construct a quantum computer, it’s not just a question of having a qubit be perceived as being maintained in a particular state; it actually has to be maintained in that state. And for this to be the case we actually have to freeze time for that qubit. But here’s a very simplified example of how that can happen in a multiway graph:

All this discussion of “freezing time” might seem weird, and not like anything one usually talks about in physics. But actually, there’s a wonderful connection: the freezing of time we’re talking about here can be thought of as happening because we’ve got the analog in the space of quantum states of a black hole in physical space.

The picture above makes it plausible that we’ve got something where things can go in, but if they do, they always get stuck. But there’s more to it. If you’re an observer far from a black hole, then you’ll never actually see anything fall into the black hole in finite time (that’s why black holes are called “frozen stars” in Russian). And the reason for this is precisely because (according to the mathematics) time is frozen at the event horizon of the black hole. In other words, to successfully make a qubit, you effectively have to isolate it in quantum space like things get isolated in physical space by the presence of the event horizon of a black hole.

General Relativity and Quantum Mechanics Are the Same Idea!

General relativity and quantum mechanics are the two great foundational theories of current physics. And in the past it’s often been a struggle to reconcile them. But one of the beautiful outcomes of our project so far has been the realization that at some deep level general relativity and quantum mechanics are?actually the same idea. It’s something that (at least so far) is only clear in the context of our models. But the basic point is that both theories are consequences of causal invariance—just applied in different situations.

Recall our discussion of causal graphs in the context of relativity above. We drew foliations and said that if we looked at a particular slice, it would tell us the arrangement of the system in space at what we consider to be a particular time. So now let’s look at multiway graphs. We saw in the previous section that in quantum mechanics we’re interested in foliations of these. But if we look at a particular slice in one of these foliations, what does it represent? The foliation has got a bunch of states in it. And it turns out that we can think of them as being laid out in an abstract kind of space that we’re calling “branchial space”.

To make sense of this space, we have to have a way to say what’s near what. But actually the multiway graph gives us that. 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. Here are the graphs we get by doing this for successive slices:

We call these branchial graphs. And we can think of them as?representing the correlation—or entanglement—of quantum states. Two states that are nearby in the graph are highly entangled; those further away, less so. And we can imagine that as our system evolves, we’ll get larger and larger branchial graphs, until eventually, just like for our original hypergraphs, we can think of these graphs as limiting to something like a continuous space.

But what is this space like? For our original hypergraphs, we imagined that we’d get something like ordinary physical space (say close to three-dimensional Euclidean space). But?branchial space?is something more abstract—and much wilder. And typically it won’t even be finite-dimensional. (It might?approximate a projective Hilbert space.) But we can still think of it mathematically as some kind of space.

OK, things are getting fairly complicated here. But let me try to give at least a flavor of how things work. Here’s an example of a wonderful correspondence: curvature in physical space is like the uncertainty principle of quantum mechanics. Why do these have anything to do with each other?

The uncertainty principle says that if you measure, say, the position of something, then its momentum, you’ll get a different answer than if you do it in the opposite order. But now think about what happens when you try to make a rectangle in physical space by going in direction?x?first, then?y, and then you do these in the opposite order. In a flat space, you’ll get to the same place. 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. And it’s because branchial space is wild—and effectively very curved—that you?get the uncertainty principle.

Alright, so the next question might be: what’s the analog of the Einstein equations in branchial space? And again, it’s quite wonderful: at least in some sense, the answer is that it’s the?path integral—the fundamental mathematical construct of modern quantum mechanics and quantum field theory.

This is again somewhat complicated. But let me try to give a flavor of it. Just as we discussed geodesics as describing paths traversed through physical space in the course of time, so also we can discuss?geodesics as describing paths traversed through branchial space?in the course of time. In both cases these geodesics are determined by curvature in the corresponding space. In the case of physical space, we argued (roughly) that the presence of excess causal edges—corresponding to energy—would lead to what amounts to curvature in the spatial hypergraph, as described by Einstein’s equations.

OK, so what about branchial space? Just like for the spatial hypergraph, we can think about the causal connections between the updating events that define the branchial graph. And we can once again imagine identifying the flux of causal edges—now not through spacelike hypersurfaces, but through branchlike ones—as corresponding to energy. And—much like in the spatial hypergraph case—an excess of these causal edges will have the effect of producing what amounts to curvature in branchial space (or, more strictly, in branchtime—the analog of spacetime). But this curvature will then affect the geodesics that traverse branchial space.

In general relativity, the presence of mass (or energy) causes curvature in space which causes the paths of geodesics to turn—which is what is normally interpreted as the action of the force of gravity. But now we have an analog in quantum mechanics, in our branchial space. The presence of energy effectively causes curvature in branchial space which causes the paths of geodesics through branchial space to turn.

What does turning correspond to? Basically it’s exactly what the path integral talks about. The path integral (and the usual formalism of quantum mechanics) is set up in terms of complex numbers. But it can just as well be thought of in terms of turning through an angle. And that’s exactly what’s happening with our geodesics in branchial space. 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.

It all fits together beautifully. In physical space we have Einstein’s equations—the core of general relativity. And in branchial space (or, more accurately, multiway space) we have Feynman’s path integral—the core of modern quantum mechanics. And in the context of our models they’re just different facets of the same idea. It’s an amazing unification that I have to say I didn’t see coming; it’s something that just emerged as an inevitable consequence of our simple models of applying rules to collections of relations, or hypergraphs.