Interview with Sam Kuypers: Qubit Field Theory [9/9/2020]
On the problems with orthodox quantum field theory, how they could be solved, and the implications thereof.
Logan Chipkin: You’ve been working on an alternative to quantum field theory, and it seems that constructor theory might help you with your research. But before we get into that, what is quantum field theory in broad strokes?
Samuel Kuypers: I’ll explain quantum field theory via classical field theory. In classical field theory, a classical field is a physical system that has observables at each point in space. An example of this is a temperature field. If you consider the temperature in a room, then effectively you have a temperature field because there’s a temperature that you can associate at each point in the room. What makes it a classical field is that the temperature everywhere in the room has a sharp value. More generally, a classical field would be anything with observables at points in space with sharp values.
In quantum field theory, fields also have observables at each point in space, but these observables are not necessarily sharp. So the states of these observables aren’t certain–it could be that they are in a superposition of multiple values at the same time.
For a field to have unsharp values, it needs to have observables at a particular point in space that don’t commute with one another. In practice, this means that if you were to observe one of the field’s observables at that point in space and then another one, the outcome of the experiment would be different if you measured them in the reverse order (Measuring A and then B would give you a different result from measuring B and then A). Algebraically, it means that the multiplication of these variables (of these observables) is not the same. The order of multiplication of these variables matters.
To summarize, a quantum field is a physical system that has observables at each point in space, and these observables have noncommutative properties such that they can be unsharp—unlike with classical fields, whose observables are necessarily sharp.
Logan: What are the problems that arise in the conventional way of expressing quantum field theory? What is it about this conventional formulation that causes these problems in the first place?
Sam: I think there are several problems. Maybe the most prominent one is that these quantum fields have so-called divergences. There are observables of the field whose expectation values go up to infinity. That’s an unphysical result. I think it’s now commonly accepted that the reason for these divergences is that the interactions of a quantum field at very small scales are very violent.
The standard solution is to do quantum field theory while neglecting the interactions at very small distances. And therefore you get an effective field theory approach—you have a field theory such that you cut out the structure at very small scales. So the effective theory is such that you pretend that this structure isn’t there and still get results via something called renormalization. This is the way people deal with these divergences.
But it’s also a problem in the sense that it means your theory isn’t universally valid. At very small scales, people agree that quantum field theory cannot be valid, and that the theory tells us this because it needs renormalization. So there needs to be another theory—a universal theory that will explain the interactions at very small-length scales. So that’s one problem.
The other problem is that the fields can carry too much information in them. We know from general relativity that there is a limit on the amount of information that can be stored in a finite region of spacetime. Basically, the limit is the amount of information you can store in a black hole such that if you store too much information in a finite region of space, you automatically turn that region into a black hole.
Quantum field theory, on the other hand, seems to suggest that there is no bound on the amount of information you can store in a finite region of space. The reason for that is, again, to do with the commutativity between observables. The observables of a quantum field at different points in space necessarily commute with one another. Observables that commute can be independently prepared and measured, meaning that you can store information in them separately. And since there are infinitely many of these commuting observables in a finite region of space, it seems that you can store an arbitrary amount of information in such a region. That contradicts the notion from general relativity called the Bekenstein Bound-–that you can only store a finite amount of information in a region of the field. So that is another problem that I think exists in what I would call the orthodox quantum field theory.
Logan: So if I’m understanding correctly, two of the problems with orthodox quantum field theory are: one, at fine enough scales, it leads to unphysical quantities such as infinite expectations values.
And, two, it leads a conflict between theories—orthodox quantum field theory tells us that it’s possible to store an arbitrary amount of information that one may store in a region of the field, but general relativity tells us that there is a bound on the amount of information that one may store in a region of the field.
Sam: Yes. And I think thse problems are related. Also I should say that, nowadays, people would disagree with you that the infintiies are a problem, because they have these renormalization schemes. The tradeoff is that they can deal with the infinities but have to give up being able to provide a description of the fields at small-length scales. They basically have to cut those out. But it’s a tradeoff people are willing to make, so I think there might be disagreement about whether or not the infinities themselves are still an issue, but they definitely cause other problems even if you try to solve for them.
As for the relation between the divergences and the apparent infinite amount of information that can be stored in the fields, this idea that the field’s observables at different points in space commute with one another is actually a standard assumption in quantum theory. It’s called the canonical commutation relation, and it says that different physical systems have to have observables that commute with another such that you can independently treat and measure and interact with them.
And in the field theory case, what happens is that this canonical commutation relation-–this idea that the fields’ observables at different points in space commute with one another—actually introduces a divergence. So if you consider the field observables at two different points in space and slowly bring them together, the field observables blow up to infinity when they are at the same point in space (The technical term is that there is a Dirac delta function in the commutation relation, and that causes the fields to blow up.).
There is even a connection with the problems that existed in classical field theory and quantum field theory. Classical fields used to be problematic because they had infinite degrees of freedom, which could each store information. I think there’s a parallel: this problem of infinite information had plagued classical fields and now plagues quantum fields. In many ways, field theories have always been problematic and yet deeply fundamental—and so a nice area to study.
Logan: If this canonical commutation relation causes these problems to emerge in orthodox quantum field theory, why was it accepted in the first place?
Sam: I think it has to do with the fact that people wanted quantum theory to be local. Different physical systems have their own algebra, but no matter their specific algebra, the algebra of the lager, composite system is such that the algebra of the subsystems commute with one another.
So if you have an observable of one of the subsystems, that observable always commutes with an observable from the other subsystem. That ensures in quantum theory that you can independently prepare and measure the subsystem without affecting the other subsystems that might exist as well.
These kinds of considerations about locality, local realism, no-signaling, were the reasons why the canonical commutation relations were required in the first place. So meddling with those seems to be very tricky…people would be scared of breaking the no-signaling rules or violating locality, for example.
Logan: Have any solutions been proposed that would solve these problems in orthodox quantum field theory?
Sam: Yes. David Deutsch in 2004 or so put a paper in the archives about an alternative to quantum field theory called qubit field theory. The assumption in qubit field theory is that there are physical systems at each event in spacetime which do not necessarily obey the canonical commutation relation.
So if you evaluate the field at two very nearby points in space, the observables of those points in space would not necessarily commute with each other—which violates the canonical commutation relation.
It’s a very nicely symmetric approach to quantum field theory, for one thing because these commutation relations that are imposed on the field observables in the usual quantum field theory are asymmetric with respect to time. Field observables at nearby points in time are never assumed to commute with one another, yet they are assumed to commute with one another when they are at different points in space. And this is one of the things that qubit field theory gives up and symmetrizes the way we treat space and time.
So in qubit field theory, observables at different points in time don’t necessarily commute, and, similarly, observables at different points in space also don’t necessarily commute.
I should add that one of the nice things about qubit fields is that they are everywhere finite. The state space is finite dimensional, and there are no divergences—nothing in the theory blows up to infinity. There are many things to enjoy about qubit field theory, and one is that they are well-behaved relative to orthodox fields.
Logan: Qubit field theory drops the canonical commutation relation, and it solves the existence of infinities that have plagued orthodox quantum field theory. You’d said that one of the reasons that the canonical commutation relation was accepted in orthodox quantum field theory was so that the theory could be local. Is qubit field theory local?
Sam: Yes. Qubit fields have local equations of motion, meaning that the equations of motion of the field only depend on the field at a particular point and finitely many of its spatial derivatives. It sounds like a loophole that allows the theory to still be local without adhering to all these canonical commutation relations that are usually necessary within quantum theory to ensure that a field behaves locally.
Logan: How would constructor theory help to advance a theory like qubit field theory?
Sam: One of the reasons qubit theory was proposed was to find out about alternative quantum theories that have a limited information carrying capacity, meaning that the amount of information you could store in a finite region of the field would be limited. At the moment, there is no theory of measurement for a qubit field. And that is because the canonical commutation relation in quantum theory is deeply connected to information in determining what the information carrying degrees of the field are. And by altering that in qubit field theory, you basically have to derive a fully new theory of measurement in order to determine what the information variables of the field really are.
The way that constructor theory helps with that problem is that constructor theory provides a theory-independent way of thinking about measurements. Constructor theory is like a meta-theory of physics that tells us, ‘Given a particular theory, what should the theory adhere to for it to have things like information and measurable quantities?’
This streamlines the problem-solving for this particular issue. Instead of having to think from the ground up about what a measurement could mean for a new theory in physics like qubit field theory, you can instead immediately say, ‘There is this general framework that constructor theory provides. We just have to fit our theory within this framework.’ This is still an open problem, but it’s much simpler than having to think about the problem of what a measurement really is before you even get to solving it.
In short, we’re able to skip a step.
Logan: What is it about constructor theory—or, in this case, constructor theory of information in particular—that makes it easier to determine what will constitute a measurement in qubit field theory?
Sam: The constructor theory of information has provided us with a definition of information, which is an attribute of a system that can be used to perform classical computations and that can be copied given a suitable substrate. Using this definition, we can extract from qubit field theory what it is within the theory that constitutes information.
So, again, we’re using this meta-theory of physics to get to the answer quicker—via the constructor theory of information, we have an understanding of what a measurement is in general. That is no longer a problem that we have to think about.
Logan: Can you give us a brief example of what making a measurement looks like in orthodox quantum field theory?
Sam: I can give a sketch of what that means more broadly. If you have two quantum systems, then because they commute with one another, you can measure the observables of one of the systems without affecting the observables of the other system.
But you can’t measure the observables of the same system without affecting those. For example, say you have a single qubit. It will have three observables-–the qubit descriptors. You can measure one of them, but if you subsequently measure another one, then your previous measurement result will have affected the second measurement. This is what noncommutativity means—measuring A and then B is not the same as measuring B and then A.
You treat the field observables of the quantum field like separate physical systems, each with observables. Measuring and interacting with those can be done independently from the other field observables. In practice, it’s always to an approximation, but the theory gives no bound to how well you can independently interact with one part of the field.
It’s because you’re dropping this assumption in qubit field theory that interacting with the observables at two different points in space affects the measurements of those observables. It seems like if you were to observe the field at point A and then the field at point B, they would affect one another, because the field observables don’t commute with one another. It’s like they are part of the same physical system, which is not the case in orthodox quantum theory. I don’t think there is currently a solution to that problem—we don’t yet know what that means for the information in the qubit field.
This notion of measurement in orthodox quantum field theory is not carried over to the notion of a measurement in qubit field theory, because the canonical commutation relation is very important for defining information in orthodox quantum field theory. And you simply drop this relation, so you have this new problem to solve. And again I think that’s where constructor theory will come in and be very useful.
Logan: It sounds like if qubit field theory turns out to be more correct than orthodox quantum field theory, then the world is even more ‘quantum’ than we had previously thought.
Sam: Yes, that’s how I think about it. The property that things commute is a very classical property. We are very used to being able to interact with things such that our measurements commute. In fact, noncommutativity is very characteristic of quantum systems. And yet we’ve treated the combining of quantum systems rather classically, since we’ve introduced—on assumption—the idea that the observables of separate quantum systems must commute with one another.
So I agree with you that if you drop that assumption, then things will become even more quantum mechanical. You have even more of these observables that don’t commute with one another. It adds another layer of strangeness to your description of reality.
Logan: What other open problems in quantum mechanics do you think researchers could solve by dropping the canonical commutation relation?
Sam: If qubit field theory turns out to be true, then it will just be the theory of quantum fields. But of course there are many tests that it would have to go through. For example, before it can really stand up as a theory, I think it should solve this problem of information. This was one of the reasons for its invention in the first place! And again, constructor theory will be very vital for this.
There are other problems, like: how should the fields interact with other fields?
If qubit field theory manages to solve these problems, then it will be a reformulation of field theory as we currently know it. For example, if qubit field theory is true, then there are no fields that adhere to the canonical commutation relation, since commutativity can spread to other fields. That means that you’d have to reformulate all of quantum field theory in terms of qubit field theory, which would then mean that you’d have to solve the Standard Model not with orthodox quantum fields but instead with qubit fields. And that would be a very exciting project. If qubit field theory takes off, then that would definitely be one of the things it would be applied to.
Another relevant area is fields in a black hole. Quantum fields inside black hole cannot adhere to canonical commutation relations, either. I think there will be a link to general relativity. What might qubit fields tell us about the relationship between general relativity and quantum mechanics? It’s a very exciting and open problem.
Logan: You don’t hear much about noncommutativity in popular science media, despite its apparent potential to solve a number of problems in physics.
Sam: One reason is that it’s a very tentative solution. Qubit field theory is in its infancy at the moment. It’s somewhat unsurprising that people haven’t noticed it more, but at the same time, more and more people are noticing problems with quantum field theory.
For example, Sean Carroll wrote a paper a while back about why the state space of any physical system should be finite, including for quantum fields. This is a very related problem. I don’t think he concluded anything about how quantum field theory should change, but in this paper he noted that there is definitely a problem with the size of the state space of a quantum field. And this is related to how much information you can store in a field, which was something he also noted.
If qubit field theory were to make progress, it would appeal to many people’s intuitions about how spacetime and quantum fields should interface with each other.
Note — watch interview here.