Tag Archives: theoretical physics

What Can Replace Space-Time?

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Nima Arkani-Hamed is famous for believing that space-time is doomed, that as physicists we will have to abandon the concepts of space and time if we want to find the ultimate theory of the universe. He’s joked that this is what motivates him to get up in the morning. He tends to bring it up often in talks, both for physicists and for the general public.

The latter especially tend to be baffled by this idea. I’ve heard a lot of questions like “if space-time is doomed, what could replace it?”

In the past, Nima and I both tended to answer this question with a shrug. (Though a more elaborate shrug in his case.) This is the honest answer: we don’t know what replaces space-time, we’re still looking for a good solution. Nima’s Amplituhedron may eventually provide an answer, but it’s still not clear what that answer will look like. I’ve recently realized, though, that this way of responding to the question misses its real thrust.

When people ask me “what could replace space-time?” they’re not asking “what will replace space-time?” Rather, they’re asking “what could possibly replace space-time?” It’s not that they want to know the answer before we’ve found it, it’s that they don’t understand how any reasonable answer could possibly exist.

I don’t think this concern has been addressed much by physicists, and it’s a pity, because it’s not very hard to answer. You don’t even need advanced physics. All you need is some fairly old philosophy. Specifically we’ll use concepts from metaphysics, the branch of philosophy that deals with categories of being.

Think about your day yesterday. Maybe you had breakfast at home, drove to work, had a meeting, then went home and watched TV.

Each of those steps can be thought of as an event. Each event is something that happened that we want to pay attention to. You having breakfast was an event, as was you arriving at work.

These events are connected by relations. Here, each relation specifies the connection between two events. There might be a relation of cause-and-effect, for example, between you arriving at work late and meeting with your boss later in the day.

Space and time, then, can be seen as additional types of relations. Your breakfast is related to you arriving at work: it is before it in time, and some distance from it in space. Before and after, distant in one direction or another, these are all relations between the two events.

Using these relations, we can infer other relations between the events. For example, if we know the distance relating your breakfast and arriving at work, we can make a decent guess at another relation, the difference in amount of gas in your car.

This way of viewing the world, events connected by relations, is already quite common in physics. With Einstein’s theory of relativity, it’s hard to say exactly when or where an event happened, but the overall relationship between two events (distance in space and time taken together) can be thought of much more precisely. As I’ve mentioned before, the curved space-time necessary for Einstein’s theory of gravity can be thought of equally well as a change in the way you measure distances between two points.

So if space and time are relations between events, what would it mean for space-time to be doomed?

The key thing to realize here is that space and time are very specific relations between events, with very specific properties. Some of those properties are what cause problems for quantum gravity, problems which prompt people to suggest that space-time is doomed.

One of those properties is the fact that, when you multiply two distances together, it doesn’t matter which order you do it in. This probably sounds obvious, because you’re used to multiplying normal numbers, for which this is always true anyway. But even slightly more complicated mathematical objects, like matrices, don’t always obey this rule. If distances were this sort of mathematical object, then multiplying them in different orders could give slightly different results. If the difference were small enough, we wouldn’t be able to tell that it was happening in everyday life: distance would have given way to some more complicated concept, but it would still act like distance for us.

That specific idea isn’t generally suggested as a solution to the problems of space and time, but it’s a useful toy model that physicists have used to solve other problems.

It’s the general principle I want to get across: if you want to replace space and time, you need a relation between events. That relation should behave like space and time on the scales we’re used to, but it can be different on very small scales (Big Bang, inside of Black Holes) and on very large scales (long-term fate of the universe).

Space-time is doomed, and we don’t know yet what’s going to replace it. But whatever it is, whatever form it takes, we do know one thing: it’s going to be a relation between events.

Research or Conference? Can’t it be both?

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“If you’re there for two months, for sure you’ll be doing research.”

I wanted to be snarky. I wanted to point out that, as a theoretical physicist, I do research wherever I go. I wanted to say that I even did research on the drive over. (This may not have been true, I think I mostly thought about Magic the Gathering cards.)

More than any of those, though, I wanted to get my travel visa. So instead I said,

“That’s fair.”

“Mmhmm, that’s fair.” Looking down at the invitation letter, she triumphantly pointed to the name of the inviting institution: “South American Institute for Fundamental Research.”

A bit of background: I’m going to Brazil this winter. Partly, this is because winter in Canada is not especially desirable, but it’s also because Sao Paulo’s International Center for Theoretical Physics is running a Program on Integrability, the arcane set of techniques that seeks to bypass the approximate perturbations we often use in particle physics and find full, exact results.

What do I mean by a Program? It’s not the sort of scientific program I’ve talked about before, though the ideas are related. When an institute holds a Program, they’re declaring a theme. For a certain length of time (generally from a few months to a whole semester), there will be a large number of talks at the institute focused on some particular scientific theme. The institute invites people from all over the world who work on that theme. Those people are there to give and attend talks, but they’re also there to share ideas with each other, to network and collaborate and do research.

This is where things get tricky. See, Brazil has multiple types of visas. A Tourist Visa can be used, among other things, for attending a scientific conference. On the other hand, someone coming to Brazil to do research uses Visa 1.

A Program is essentially a long conference…but it’s also an opportunity to do research. So are most short conferences, though! In theoretical physics we have workshops, short conferences explicitly focused on collaboration and research, but even if a conference isn’t a workshop you can bet that we’ll be doing some research there, for sure. We don’t need labs, and some of us don’t even need computers, research can happen whenever the inspiration strikes. The distinction between conferences and research, from our perspective, is an arbitrary one.

In physics, we like to cut through this sort of ambiguity by looking at what’s really important. I wanted to figure out what about research makes the Brazilian government use a different visa for it, whether it was about motivating people to enter the country for specific reasons or tracking certain sorts of activities. I wanted to understand that, because it would let me figure out whether my own research fell under those reasons, and thus figure out objectively which type of visa I ought to have.

I wanted to ask about all of this…but more than any of that, I wanted to get my travel visa. So I applied for the visa they told me to, and left.

Why I Can’t Explain Ghosts: Or, a Review of a Popular Physics Piece

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Since today is Halloween, I really wanted to write a post talking about the spookiest particles in physics, ghosts.

And their superpartners, ghost riders.

The problem is, in order to explain ghosts I’d have to explain something called gauge symmetry. And gauge symmetry is quite possibly the hardest topic in modern physics to explain to a general audience.

Deep down, gauge symmetry is the idea that irrelevant extra parts of how we represent things in physics should stay irrelevant. While that sounds obvious, it’s far from obvious how you can go from that to predicting new particles like the Higgs boson.

Explaining this is tough! Tough enough that I haven’t thought of a good way to do it yet.

Which is why I was fairly stoked when a fellow postdoc pointed out a recent popular physics article by Juan Maldacena, explaining gauge symmetry.

Juan Maldacena is a Big Deal. He’s the guy who figured out the AdS/CFT correspondence, showing that string theory (in a particular hyperbola-shaped space called AdS) and everybody’s favorite N=4 super Yang-Mills theory are secretly the same, a discovery which led to a Big Blue Dot on Paperscape. So naturally, I was excited to see what he had to say.

Big Blue Dot pictured here.

Big Blue Dot pictured here.

The core analogy he makes is with currencies in different countries. Just like gauge symmetry, currencies aren’t measuring anything “real”: they’re arbitrary conventions put in place because we don’t have a good way of just buying things based on pure “value”. However, also like gauge symmetry, then can have real-life consequences, as different currency exchange rates can lead to currency speculation, letting some people make money and others lose money. In Maldacena’s analogy the Higgs field works like a precious metal, making differences in exchange rates manifest as different prices of precious metals in different countries.

It’s a solid analogy, and one that is quite close to the real mathematics of the problem (as the paper’s Appendix goes into detail to show). However, I have some reservations, both about the paper as a whole and about the core analogy.

In general, Maldacena doesn’t do a very good job of writing something publicly accessible. There’s a lot of stilted, academic language, and a lot of use of “we” to do things other than lead the reader through a thought experiment. There’s also a sprinkling of terms that I don’t think the average person will understand; for example, I doubt the average college student knows flux as anything other than a zany card game.

Regarding the analogy itself, I think Maldacena has fallen into the common physicist trap of making an analogy that explains things really well…if you already know the math.

This is a problem I see pretty frequently. I keep picking on this article, and I apologize for doing so, but it’s got a great example of this when it describes supersymmetry as involving “a whole new class of number that can be thought of as the square roots of zero”. That’s a really great analogy…if you’re a student learning about the math behind supersymmetry. If you’re not, it doesn’t tell you anything about what supersymmetry does, or how it works, or why anyone might study it. It relates something unfamiliar to something unfamiliar.

I’m worried that Maldacena is doing that in this paper. His setup is mathematically rigorous, but doesn’t say much about the why of things: why do physicists use something like this economic model to understand these forces? How does this lead to what we observe around us in the real world? What’s actually going on, physically? What do particles have to do with dimensionless constants? (If you’re curious about that last one, I like to think I have a good explanation here.)

It’s not that Maldacena ignores these questions, he definitely puts effort into answering them. The problem is that his analogy itself doesn’t really address them. They’re the trickiest part, the part that people need help picturing and framing, the part that would benefit the most from a good analogy. Instead, the core imagery of the piece is wasted on details that don’t really do much for a non-expert.

Maybe I’m wrong about this, and I welcome comments from non-physicists. Do you feel like Maldacena’s account gives you a satisfying idea of what gauge symmetry is?

The Near and the Far: Motivations for Physics

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When I introduce myself, I often describe my job like this:

“I develop mathematical tools to make calculations in particle physics easier and more efficient.”

However, I could equally well describe my job like this:

“I’m looking for a radical new way to reformulate particle physics in order to solve fundamental problems in space and time.”

These may sound very different, but they’re both correct. That’s because in theoretical physics, like in many branches of science, we have two types of goals: near-term and far-term.

In the near-term, I develop mathematical tools and tricks, which let me calculate things I (and others) couldn’t calculate before. Pushing the tricks to their limits gives me more proficiency, making the tools I develop more robust. In the future, I can imagine applying the tools to more types of calculations, and specifically to more “important” calculations.

All of that still involves relatively near-term goals, though. Develop a new trick, and you can already envision what it might be used for. The far-term goals are generally deeper.

End of the road, not just the next tree.

In the far term, the new techniques that I and others develop might lead to fundamentally new ways to understand particle physics. That’s because a central feature of most of the tricks we develop is that they rephrase the calculation in a way that leaves out something that used to be thought of as fundamental. They’re “revolutions”, overthrowing some basic principle of how we do things. The hope is that the right “revolution” will help us solve problems that our current understanding of physics seems incapable of solving.

Most scientists have both sorts of goals. Someone who studies quantum mechanics might talk about developing a quantum computer, but in the near-term be interested in perfecting some algorithm. A biologist might study how information is stored in a cell, but introduce themself as someone trying to cure cancer.

For some people, the far-term goals are a big component of how they view themselves. Nima Arkani-Hamed, for example, has joked that believing that “spacetime is doomed” is what allows him to get out of bed in the morning. (For a transcript of the relevant parts, see here.) There are plenty of others with similar perspectives, people who need a “big” goal to feel motivated.

Myself, I find it harder to identify with these kinds of goals, because the payoff is so uncertain. Rephrasing particle physics in a new way might be the solution to a fundamental problem…but it could also just be another way to say the same thing. There’s no guarantee that any one project will be that one magical solution. In contrast, for me, near term goals are something I can feel confident I’m making real progress on. I can envision each step along the way, and see the part my work plays in a larger picture, led along by the satisfaction of solving each puzzle as it comes.

Neither way is better than the other, and both are important parts of science. Some people do better with one, some do better with the other, and in the end, everyone can view themselves as accomplishing something they care about.

What’s an Amplitude? Just about everything.

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I am an Amplitudeologist. In other words, I study scattering amplitudes. I’ve explained bits and pieces of what scattering amplitudes are in other posts, but I ought to give a short definition here so everyone’s on the same page:

A scattering amplitude is the formula used to calculate the probability that some collection of particles will “scatter”, emerging as some (possibly different) collection of particles.

Note that I’m using some weasel words here. The scattering amplitude is not a probability itself, but “the formula used to calculate the probability”. For those familiar with the mathematics of waves, the scattering amplitude gives the amplitude of a “probability wave” that must be squared to get the probability. (Those familiar with waves might also ask: “If this is the amplitude, what about the period?” The truth is that because scattering amplitudes are calculated using complex numbers, what we call the “amplitude” also contains information about the wave’s “period”. It may seem like an inconsistent way to name things from the perspective of a beginning student, but it is actually consistent with the terminology in a large chunk of physics.)

In some of the simplest scattering amplitudes particles literally “scatter”, with two particles “colliding” and emerging traveling in different directions.

A scattering amplitude can also describe a more complicated situation, though. At particle colliders like the Large Hadron Collider, two particles (a pair of protons for the LHC) are accelerated fast enough that when they collide they release a whole slew of new particles. Since it still fits the “some particles go in, some particles go out” template, this is still described by a scattering amplitude.

It goes even further than that, though, because “some particles” could also just be “one particle”. If you’re dealing with something unstable (the particle equivalent of radioactive, essentially) then one particle can decay into two or more particles. There’s a whole slew of questions that require that sort of calculation. For example, if unstable particles were produced in the early universe, how many of them would be left around today? If dark matter is unstable (and some possible candidates are), when it decays it might release particles we could detect. In general, this sort of scattering amplitude is often of interest to astrophysicists when they happen to get involved in particle physics.

You can even use scattering amplitudes to describe situations that, at first glance, don’t sound like collisions of particles at all. If you want to find the effect of a magnetic field on an electron to high accuracy, the calculation also involves a scattering amplitude. A magnetic field can be thought of in terms of photons, particles of light, because light is a vibration in the electro-magnetic field. This means that the effect of a magnetic field on an electron can be calculated by “scattering” an electron and a photon.

4gravanom

If this looks familiar, check the handbook section.

In fact, doing the calculation in this way leads to what is possibly the most accurately predicted number in all of science.

Scattering amplitudes show up all over the place, from particle physics at the Large Hadron Collider to astrophysics to delicate experiments on electrons in magnetic fields. That said, there are plenty of things people calculate in theoretical physics that don’t use scattering amplitudes, either because they involve questions that are difficult to answer from the scattering amplitude point of view, or because they invoke different formulas altogether. Still, scattering amplitudes are central to the work of a large number of physicists. They really do cover just about everything.

Am I a String Theorist?

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Perimeter, like most institutes of theoretical physics, divides their researchers into semi-informal groups. At Perimeter, these are:

  • Condensed Matter
  • Cosmology
  • Mathematical Physics
  • Particle Physics
  • Quantum Fields and Strings
  • Quantum Foundations
  • Quantum Gravity
  • Quantum Information
  • Strong Gravity

I’m in the Quantum Fields and Strings group, which many people seem to refer to simply as the String Theory group. So for the past week or so, I’ve been introducing myself as a String Theorist. As I briefly mention in my Who Am I? post, this isn’t completely accurate.

Am I a String Theorist?

The theories that I study do derive from string theory. They were first framed by string theorists, and research into them is still deeply intertwined with string theory research. I’ve definitely had occasion to compare my results to those of string theorists, or to bring in calculations by string theorists to advance my work.

And if you’re the kind of person who views the world as a competition between string theory and its rivals (like Loop Quantum Gravity) then I suppose I’m on the string theory “side”. I’m optimistic, at least, that the reason why string theory research is so much more common than any other approach to quantum gravity is simply because string theory provides many more interesting and viable projects for researchers.

On the other hand, though, there’s the basic fact that the theories I work with are not, themselves, string theories. They’re quantum field theories, the broader class that encompasses the modern synthesis of quantum mechanics and special relativity. The theories I work with are often reasonably close to the well-tested theories of the real world, close enough that the calculations are more “particle physics” than the they are “string theory”.

Of course, all of that could change. One of the great things about string theory is the way it connects lots of different interesting quantum field theories together. There’s a “string”, the “GKP string”, involved in the work of Basso, Sever, and Vieira, work that I will probably get involved with here at Perimeter. The (2,0) theory is a quantum field theory, but it’s much closer to string theory than to particle physics, so if I get more involved with the (2,0) theory would that make me a string theorist?

The fact is, these days string theory is so ubiquitous that the question “Am I a String Theorist?” doesn’t actually mean anything. String theory is there, lurking in the background, able to get involved at any time even if it’s not directly involved at present. Theoretical physicists don’t fall into neat categories.

I am a String Theorist. Also, I am not.

Perimeter!

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I’m moving in at Perimeter this week, so I don’t have time to write a long post. For those who aren’t familiar with it, the Perimeter Institute for Theoretical Physics is an independent research institute, not affiliated with any university. Instead, it’s funded by a combination of government and private sources (for why private sources might fund theoretical physics, read my discussion here). Because it’s not a university they have budgets to do things like hire people to make the transition process easier, so everything has been really nice and well-organized.

The postdoc offices are really nice, with a view of the nearby park, shown below.

On the Perimeter...of Waterloo Park

On the Perimeter…of Waterloo Park

Hexagon Functions II: Lost in (super)Space

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My new paper went up last night.

It’s on a very similar topic to my last paper, actually. That paper dealt with a specific process involving six particles in my favorite theory, N=4 super Yang-Mills. Two particles collide, and after the metaphorical dust settles four particles emerge. That means six “total” particles, if you add the two in with the four out, for a “hexagon” of variables. To understand situations like that, my collaborators and I created “hexagon functions”, formulas that depended on the states of the six particles.

One thing I didn’t emphasize then was that that calculation only applied to one specific choice of particles, one in which all of the particles are Yang-Mills bosons, particles (like photons) created by the fundamental forces. There are lots of other particles in N=4 super Yang-Mills, though. What happens when they collide?

That question is answered by my new paper. Though it may sound surprising, all of the other particles can be taken into account with a single formula. In order to explain why, I have to tell you about something called superspace.

A while back I complained about a blog post by George Musser about the (2,0) theory. One of the things that irked me about that post was his attempt to explain superspace:

Supersymmetry is the idea that spacetime, in addition to its usual dimensions of space and time, has an entirely different type of dimension—a quantum dimension, whose coordinates are not ordinary real numbers but a whole new class of number that can be thought of as the square roots of zero.

This is actually a great way to think about superspace…if you’re already a physicist. If you’re not, it’s not very informative. Here’s a better way to think about it:

As I’ve talked about before, supersymmetry is a relationship between different types of particles. Two particles related by supersymmetry have the same mass, and the same charge. While they can be very different in other ways (specifically, having different spin), supersymmetric particles are described by many of the same equations as each-other. Rather than writing out those equations multiple times, it’s often nicer to write them all in a unified way, and that’s where superspace comes in.

At its simplest, superspace is just a trick used to write equations in a simpler way. Instead of writing down a different equation for each particle we write one equation with an extra variable, representing a “dimension” of supersymmetry. Traveling in that dimension takes you from particle to particle, in the same way that “turning” the theory (as I phrase it here) does, but it does it within the space of a single equation.

That, essentially, is the trick that we use. With four “superspace dimensions”, we can include the four supersymmetries of N=4 super Yang-Mills, showing how the formulas vary when you go beyond the equation from our first paper.

So far, you may be wondering why I’m calling superspace a “dimension”, when it probably sounds like more of a label. I’ve mentioned before that, just because something is a variable, doesn’t mean it counts as a real dimension.

The key difference is that superspace dimensions are related to regular dimensions in a precise way. In a sense, they’re the square roots of regular dimensions. (Though independently, as George Musser described, they’re the square roots of zero: go in the same direction twice in supersymmetry, and you get back where you’re started, going zero distance.) The coexistence of these two seemingly contradictory statements isn’t some sort of quantum mystery, it’s just a consequence of the fact that, mathematically, I’m saying two very different things. I just can’t think of a way to explain them differently without math.

Superspace isn’t a real place…but it can often be useful to think of it that way. In theories with supersymmetry, it can unify the world, putting disparate particles together into a single equation.

Stop! Impostor!

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Ever felt like you don’t belong? Like you don’t deserve to be where you are, that you’re just faking competence you don’t really have?

If not, it may surprise you to learn that this is a very common feeling among successful young academics. It’s called impostor syndrome, and it happens to some very talented people.

It’s surprisingly easy to rationalize success as luck, to assume praise comes from people who don’t know the full story. In science, we’re surrounded by people who seem to come up with brilliant insights on a regular basis. We see others’ successes far more often than we see their failures, and often we forget that science is at its heart a process of throwing ideas against a wall until something sticks. Hyper-aware of our own failures, when we present ourselves as successful we can feel like we’re putting on a paper-thin disguise, constantly at risk that someone will see through it.

As paper-thin disguises go, I prefer the classics.

In my experience, theoretical physics is especially heavy on impostor syndrome, for a number of reasons.

First, there’s the fact that beginning grad students really don’t know all they need to. Theoretical physics requires a lot of specialized knowledge, and most grad students just have the bare bones basics of a physics undergrad degree. On the strength of those basics, you’re somehow supposed to convince a potential advisor, an established, successful scientist, that you’re worth paying attention to.

Throw in the fact that many people have a little more than the basics, whether from undergrad research projects or grad-level courses taken early, and you have a group where everyone is trying to seem more advanced than they are. There’s a very real element of fake it till you make it, of going to talks and picking up just enough of the lingo to bluff your way through a conversation.

And the thing is, even after you make it, you’ll probably still feel like you’re faking it.

As I’ve mentioned before, there’s an enormous amount of jury-rigging that goes into physics research. There are a huge number of side-disciplines that show up at one point or another, from numerical methods to programming to graphic design. We can’t hire a professional to handle these things, we have to learn them ourselves. As such, we become minor dabblers in a whole mess of different fields. Work on something enough and others will start looking to you for help. It won’t feel like you’re an expert, though, because you know in the back of your mind that the real experts know so much more.

In the end, the best approach I’ve found is simply to keep saying yes. Keep using what you know, going to talks and trying new things. The more you “pretend” to know what you’re doing, the more experience you’ll get, until you really do know what you’re doing. There’s always going to be more to learn, but chances are if you’re feeling impostor syndrome you’ve already learned a lot. Take others’ opinions of you at face value, and see just how far you can go.

Feeling Perturbed?

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You might think of physics as the science of certainties and exact statements: action and reaction, F=ma, and all that. However, most calculations in physics aren’t exact, they’re approximations. This is especially true today, but it’s been true almost since the dawn of physics. In particular, approximations are performed via a method known as perturbation theory.

Perturbation theory is a trick used to solve problems that, for one reason or another, are too difficult to solve all in one go. It works by solving a simpler problem, then perturbing that solution, adjusting it closer to the target.

To give an analogy: let’s say you want to find the area of a circle, but you only know how to draw straight lines. You could start by drawing a square: it’s easy to find the area, and you get close to the area of the circle. But you’re still a long ways away from the total you’re aiming for. So you add more straight lines, getting an octagon. Now it’s harder to find the area, but you’re closer to the full circle. You can keep adding lines, each step getting closer and closer.

And so on.

And so on.

This, broadly speaking, is what’s going on when particle physicists talk about loops. The calculation with no loops (or “tree-level” result) is the easier problem to solve, omitting quantum effects. Each loop then is the next stage, more complicated but closer to the real total.

There are, as usual, holes in this analogy. One is that it leaves out an important aspect of perturbation theory, namely that it involves perturbing with a parameter. When that parameter is small, perturbation theory works, but as it gets larger the approximation gets worse and worse. In the case of particle physics, the parameter is the strength of the forces involves, with weaker forces (like the weak nuclear force, or electromagnetism) having better approximations than stronger forces (like the strong nuclear force). If you squint, this can still fit the analogy: different shapes might be harder to approximate than the circle, taking more sets of lines to get acceptably close.

Where the analogy fails completely, though, is when you start approaching infinity. Keep adding more lines, and you should be getting closer and closer to the circle each time. In quantum field theory, though, this frequently is not the case. As I’ve mentioned before, while lower loops keep getting closer to the true (and experimentally verified) results, going all the way out to infinite loops results not in the full circle, but in an infinite result instead. There’s an understanding of why this happens, but it does mean that perturbation theory can’t be thought of in the most intuitive way.

Almost every calculation in particle physics uses perturbation theory, which means almost always we are just approximating the real result, trying to draw a circle using straight lines. There are only a few theories where we can bypass this process and look at the full circle. These are known as integrable theories. N=4 super Yang-Mills may be among them, one of many reasons why studying it offers hope for a deeper understanding of particle physics.