Tag Archives: quantum gravity

Why I Wasn’t Bothered by the “Science” in Avengers: Endgame

Things I’d Like to Know More About

This is an accountability post, of sorts.

As a kid, I wanted to know everything. Eventually, I realized this was a little unrealistic. Doomed to know some things and not others, I picked physics as a kind of triage. Other fields I could learn as an outsider: not well enough to compete with the experts, but enough to at least appreciate what they were doing. After watching a few string theory documentaries, I realized this wasn’t the case for physics: if I was going to ever understand what those string theorists were up to, I would have to go to grad school in string theory.

Over time, this goal lost focus. I’ve become a very specialized creature, an “amplitudeologist”. I didn’t have time or energy for my old questions. In an irony that will surprise no-one, a career as a physicist doesn’t leave much time for curiosity about physics.

One of the great things about this blog is how you guys remind me of those old questions, bringing me out of my overspecialized comfort zone. In that spirit, in this post I’m going to list a few things in physics that I really want to understand better. The idea is to make a public commitment: within a year, I want to understand one of these topics at least well enough to write a decent blog post on it.

Wilsonian Quantum Field Theory:

When you first learn quantum field theory as a physicist, you learn how unsightly infinite results get covered up via an ad-hoc-looking process called renormalization. Eventually you learn a more modern perspective, that these infinite results show up because we’re ignorant of the complete theory at high energies. You learn that you can think of theories at a particular scale, and characterize them by what happens when you “zoom” in and out, in an approach codified by the physicist Kenneth Wilson.

While I understand the basics of Wilson’s approach, the courses I took in grad school skipped the deeper implications. This includes the idea of theories that are defined at all energies, “flowing” from an otherwise scale-invariant theory perturbed with extra pieces. Other physicists are much more comfortable thinking in these terms, and the topic is important for quite a few deep questions, including what it means to properly define a theory and where laws of nature “live”. If I’m going to have an informed opinion on any of those topics, I’ll need to go back and learn the Wilsonian approach properly.

Wormholes:

If you’re a fan of science fiction, you probably know that wormholes are the most realistic option for faster-than-light travel, something that is at least allowed by the equations of general relativity. “Most realistic” isn’t the same as “realistic”, though. Opening a wormhole and keeping it stable requires some kind of “exotic matter”, and that matter needs to violate a set of restrictions, called “energy conditions”, that normal matter obeys. Some of these energy conditions are just conjectures, some we even know how to violate, while others are proven to hold for certain types of theories. Some energy conditions don’t rule out wormholes, but instead restrict their usefulness: you can have non-traversable wormholes (basically, two inescapable black holes that happen to meet in the middle), or traversable wormholes where the distance through the wormhole is always longer than the distance outside.

I’ve seen a few talks on this topic, but I’m still confused about the big picture: which conditions have been proven, what assumptions were needed, and what do they all imply? I haven’t found a publicly-accessible account that covers everything. I owe it to myself as a kid, not to mention everyone who’s a kid now, to get a satisfactory answer.

Quantum Foundations:

Quantum Foundations is a field that many physicists think is a waste of time. It deals with the questions that troubled Einstein and Bohr, questions about what quantum mechanics really means, or why the rules of quantum mechanics are the way they are. These tend to be quite philosophical questions, where it’s hard to tell if people are making progress or just arguing in circles.

I’m more optimistic about philosophy than most physicists, at least when it’s pursued with enough analytic rigor. I’d like to at least understand the leading arguments for different interpretations, what the constraints on interpretations are and the main loopholes. That way, if I end up concluding the field is a waste of time at least I’d be making an informed decision.

Amplitudes in String and Field Theory at NBI

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There’s a conference at the Niels Bohr Institute this week, on Amplitudes in String and Field Theory. Like the conference a few weeks back, this one was funded by the Simons Foundation, as part of Michael Green’s visit here.

The first day featured a two-part talk by Michael Green and Congkao Wen. They are looking at the corrections that string theory adds on top of theories of supergravity. These corrections are difficult to calculate directly from string theory, but one can figure out a lot about them from the kinds of symmetry and duality properties they need to have, using the mathematics of modular forms. While Michael’s talk introduced the topic with a discussion of older work, Congkao talked about their recent progress looking at this from an amplitudes perspective.

Francesca Ferrari’s talk on Tuesday also related to modular forms, while Oliver Schlotterer and Pierre Vanhove talked about a different corner of mathematics, single-valued polylogarithms. These single-valued polylogarithms are of interest to string theorists because they seem to connect two parts of string theory: the open strings that describe Yang-Mills forces and the closed strings that describe gravity. In particular, it looks like you can take a calculation in open string theory and just replace numbers and polylogarithms with their “single-valued counterparts” to get the same calculation in closed string theory. Interestingly, there is more than one way that mathematicians can define “single-valued counterparts”, but only one such definition, the one due to Francis Brown, seems to make this trick work. When I asked Pierre about this he quipped it was because “Francis Brown has good taste…either that, or String Theory has good taste.”

Wednesday saw several talks exploring interesting features of string theory. Nathan Berkovitz discussed his new paper, which makes a certain context of AdS/CFT (a duality between string theory in certain curved spaces and field theory on the boundary of those spaces) manifest particularly nicely. By writing string theory in five-dimensional AdS space in the right way, he can show that if the AdS space is small it will generate the same Feynman diagrams that one would use to do calculations in N=4 super Yang-Mills. In the afternoon, Sameer Murthy showed how localization techniques can be used in gravity theories, including to calculate the entropy of black holes in string theory, while Yvonne Geyer talked about how to combine the string theory-like CHY method for calculating amplitudes with supersymmetry, especially in higher dimensions where the relevant mathematics gets tricky.

Thursday ended up focused on field theory. Carlos Mafra was originally going to speak but he wasn’t feeling well, so instead I gave a talk about the “tardigrade” integrals I’ve been looking at. Zvi Bern talked about his work applying amplitudes techniques to make predictions for LIGO. This subject has advanced a lot in the last few years, and now Zvi and collaborators have finally done a calculation beyond what others had been able to do with older methods. They still have a way to go before they beat the traditional methods overall, but they’re off to a great start. Lance Dixon talked about two-loop five-particle non-planar amplitudes in N=4 super Yang-Mills and N=8 supergravity. These are quite a bit trickier than the planar amplitudes I’ve worked on with him in the past, in particular it’s not yet possible to do this just by guessing the answer without considering Feynman diagrams.

Today was the last day of the conference, and the emphasis was on number theory. David Broadhurst described some interesting contributions from physics to mathematics, in particular emphasizing information that the Weierstrass formulation of elliptic curves omits. Eric D’Hoker discussed how the concept of transcendentality, previously used in field theory, could be applied to string theory. A few of his speculations seemed a bit farfetched (in particular, his setup needs to treat certain rational numbers as if they were transcendental), but after his talk I’m a bit more optimistic that there could be something useful there.

How to Get a “Minimum Scale” Without Pixels

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Zoom in, and the world gets stranger. Down past atoms, past protons and neutrons, far past the smallest scales we can probe at the Large Hadron Collider, we get to the scale at which quantum gravity matters: the Planck scale.

Weird things happen at the Planck scale. Space and time stop making sense. Read certain pop science articles, and they’ll tell you the Planck scale is the smallest scale, the scale where space and time are quantized, the “pixels of the universe”.

That last sentence, by the way, is not actually how the Planck scale works. In fact, there’s pretty good evidence that the universe doesn’t have “pixels”, that space and time are not quantized in that way. Even very tiny pixels would change the speed of light, making it different for different colors. Tiny effects like that add up, and astronomers would almost certainly have noticed an effect from even Planck-scale pixels. Unless your idea of “pixels” is fairly unusual, it’s already been ruled out.

If the Planck scale isn’t the scale of the “pixels of the universe”, why do people keep saying it is?

Part of the problem is that the real story is vaguer. We don’t know what happens at the Planck scale. It’s not just that we don’t know which theory of quantum gravity is right: we don’t even know what different quantum gravity proposals predict. People are trying to figure it out, and there are some more or less viable ideas, but ultimately all we know is that at the Planck scale our description of space-time should break down.

“Our description breaks down” is unfortunately not very catchy. Certainly, it’s less catchy than “pixels of the universe”. Part of the problem is that most people don’t know what “our description breaks down” actually means.

So if that’s the part that’s puzzling you, maybe an example would help. This won’t be the full answer, though it could be part of the story. What it will be is an example of what “our description breaks down” can actually mean, how there can be a scale beyond which space-time stops making sense without there being “pixels”.

The example comes from string theory, from a concept called “T duality”. In string theory, “extra” dimensions beyond our usual three space and one time are curled up small, so that traveling along them just gets you back where you started. Instead of particles, there are strings, with length close to the Planck length.

Picture a loop of string in a small extra dimension. What can it do?

Image credit: someone who’s done a lot more work explaining string theory than I have

One thing it can do is move along the extra dimension. Since it has to end up back where it started, it can’t just move at any speed it wants. It turns out that the smaller the extra dimension, the more energy the string has when it spins around it.

The other thing it can do is wrap around the extra dimension. If it wraps around, the string has more energy if the dimension is larger, like a rubber band stretched around a pipe.

The string can do either or both of these multiple times. It can wrap many times around the extra dimension, or move in a quicker circle around it, or both at once. And if you calculate the energy of these combinations, you notice something: a string wound around a big circle has the same energy as a string moving around a small circle. In particular, you get the same energy on a circle of radius $R$ , and a circle of radius $l^2/R$ , where $l$ is the length of the string.

It turns out it’s not just the energy that’s the same: for everything that happens on a circle of radius $R$ , there’s a matching description with a circle of radius $l^2/R$ , with wrapping and moving swapped. We say that the two descriptions are dual: two seemingly different pictures that turn out to be completely physically indistinguishable.

Since the two pictures are indistinguishable, it doesn’t actually make sense to talk about dimensions smaller than the length of the string. It’s not that they can’t exist, or that they’re smaller than the “pixels of the universe”: it’s just that any description you write down of such a small dimension could just as easily have been of a larger, dual dimension. It’s that your picture, of one obvious size of the curled up dimension, broke down and stopped making sense.

As I mentioned, this isn’t the whole picture of what happens at the Planck scale, even in string theory. It is an example of a broader idea that string theorists are investigating, that in order to understand space-time at the smallest scales you need to understand many different dual descriptions. And hopefully, it’s something you can hold in your mind, a specific example of what “our description breaks down” can actually mean in practice, without pixels.

Current Themes 2018

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I’m at Current Themes in High Energy Physics and Cosmology this week, the yearly conference of the Niels Bohr International Academy. (I talked about their trademark eclectic mix of topics last year.)

This year, the “current theme” was broadly gravitational (though with plenty of exceptions!).

For example, almost getting kicked out of the Botanical Garden

There were talks on phenomena we observe gravitationally, like dark matter. There were talks on calculating amplitudes in gravity theories, both classical and quantum. There were talks about black holes, and the overall shape of the universe. Subir Sarkar talked about his suspicion that the expansion of the universe isn’t actually accelerating, and while I still think the news coverage of it was overblown I sympathize a bit more with his point. He’s got a fairly specific worry, that we’re in a region that’s moving unusually with respect to the surrounding universe, that hasn’t really been investigated in much detail before. I don’t think he’s found anything definitive yet, but it will be interesting as more data accumulates to see what happens.

Of course, current themes can’t stick to just one theme, so there were non-gravitational talks as well. Nima Arkani-Hamed’s talk covered some results he’s talked about in the past, a geometric picture for constraining various theories, but with an interesting new development: while most of the constraints he found restrict things to be positive, one type of constraint he investigated allowed for a very small negative region, around thirty orders of magnitude smaller than the positive part. The extremely small size of the negative region was the most surprising part of the story, as it’s quite hard to get that kind of extremely small scale out of the math we typically invoke in physics (a similar sense of surprise motivates the idea of “naturalness” in particle physics).

There were other interesting talks, which I might talk about later. They should have slides up online soon in case any of you want to have a look.

The Amplitudes Long View

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Occasionally, other physicists ask me what the goal of amplitudes research is. What’s it all about?

I want to give my usual answer: we’re calculating scattering amplitudes! We’re trying to compute them more efficiently, taking advantage of simplifications and using a big toolbox of different approaches, and…

Usually by this point in the conversation, it’s clear that this isn’t what they were asking.

When physicists ask me about the goal of amplitudes research, they’ve got a longer view in mind. Maybe they’ve seen a talk by Nima Arkani-Hamed, declaring that spacetime is doomed. Maybe they’ve seen papers arguing that everything we know about quantum field theory can be derived from a few simple rules. Maybe they’ve heard slogans, like “on-shell good, off-shell bad”. Maybe they’ve heard about the conjecture that N=8 supergravity is finite, or maybe they’ve just heard someone praise the field as “demoting the sacred cows like fields, Lagrangians, and gauge symmetry”.

Often, they’ve heard a little bit of all of these. Sometimes they’re excited, sometimes they’re skeptical, but either way, they’re usually more than a little confused. They’re asking how all of these statements fit into a larger story.

The glib answer is that they don’t. Amplitudes has always been a grab-bag of methods: different people with different backgrounds, united by their interest in a particular kind of calculation.

With that said, I think there is a shared philosophy, even if each of us approaches it a little differently. There is an overall principle that unites the amplituhedron and color-kinematics duality, the CHY string and bootstrap methods, BCFW and generalized unitarity.

If I had to describe that principle in one word, I’d call it minimality. Quantum field theory involves hugely complicated mathematical machinery: Lagrangians and path integrals, Feynman diagrams and gauge fixing. At the end of the day, if you want to answer a concrete question, you’re computing a few specific kinds of things: mostly, scattering amplitudes and correlation functions. Amplitudes tries to start from the other end, and ask what outputs of this process are allowed. The idea is to search for something minimal: a few principles that, when applied to a final answer in a particular form, specify it uniquely. The form in question varies: it can be a geometric picture like the amplituhedron, or a string-like worldsheet, or a constructive approach built up from three-particle amplitudes. The goal, in each case, is the same: to skip the usual machinery, and understand the allowed form for the answer.

From this principle, where do the slogans come from? How could minimality replace spacetime, or solve quantum gravity?

It can’t…if we stick to only matching quantum field theory. As long as each calculation matches one someone else could do with known theories, even if we’re more efficient, these minimal descriptions won’t really solve these kinds of big-picture mysteries.

The hope (and for the most part, it’s a long-term hope) is that we can go beyond that. By exploring minimal descriptions, the hope is that we will find not only known theories, but unknown ones as well, theories that weren’t expected in the old understanding of quantum field theory. The amplituhedron doesn’t need space-time, it might lead the way to a theory that doesn’t have space-time. If N=8 supergravity is finite, it could suggest new theories that are finite. The story repeats, with variations, whenever amplitudeologists explore the outlook of our field. If we know the minimal requirements for an amplitude, we could find amplitudes that nobody expected.

I’m not claiming we’re the only field like this: I feel like the conformal bootstrap could tell a similar story. And I’m not saying everyone thinks about our field this way: there’s a lot of deep mathematics in just calculating amplitudes, and it fascinated people long before the field caught on with the Princeton set.

But if you’re asking what the story is for amplitudes, the weird buzz you catch bits and pieces of and can’t quite put together…well, if there’s any unifying story, I think it’s this one.

The State of Four Gravitons

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This blog is named for a question: does the four-graviton amplitude in N=8 supergravity diverge?

Over the years, Zvi Bern and a growing cast of collaborators have been trying to answer that question. They worked their way up, loop by loop, until they stalled at five loops. Last year, they finally broke the stall, and last week, they published the result of the five-loop calculation. They find that N=8 supergravity does not diverge at five loops in four dimensions, but does diverge in 24/5 dimensions. I thought I’d write a brief FAQ about the status so far.

Q: Wait a minute, 24/5 dimensions? What does that mean? Are you talking about fractals, or…

Nothing so exotic. The number 24/5 comes from a regularization trick. When we’re calculating an amplitude that might be divergent, one way to deal with it is to treat the dimension like a free variable. You can then see what happens as you vary the dimension, and see when the amplitude starts diverging. If the dimension is an integer, then this ends up matching a more physics-based picture, where you start with a theory in eleven dimensions and curl up the extra ones until you get to the dimension you’re looking for. For fractional dimensions, it’s not clear that there’s any physical picture like this: it’s just a way to talk about how close something is to diverging.

Q: I’m really confused. What’s a graviton? What is supergravity? What’s a divergence?

I don’t have enough space to explain these things here, but that’s why I write handbooks. Here are explanations of gravitons, supersymmetry, and (N=8) supergravity, loops, and divergences. Please let me know if anything in those explanations is unclear, or if you have any more questions.

Q: Why do people think that N=8 supergravity will diverge at seven loops?

There’s a useful rule of thumb in quantum field theory: anything that can happen, will happen. In this case, that means if there’s a way for a theory to diverge that’s consistent with the symmetries of the theory, then it almost always does diverge. In the past, that meant that people expected N=8 supergravity to diverge at five loops. However, researchers found a previously unknown symmetry that looked like it would forbid the five-loop divergence, and only allow a divergence at seven loops (in four dimensions). Zvi and co.’s calculation confirms that the five-loop divergence doesn’t show up.

More generally, string theory not only avoids divergences but clears up other phenomena, like black holes. These two things seem tied together: string theory cleans up problems in quantum gravity in a consistent, unified way. There isn’t a clear way for N=8 supergravity on its own to clean up these kinds of problems, which makes some people skeptical that it can match string theory’s advantages. Either way N=8 supergravity, unlike string theory, isn’t a candidate theory of nature by itself: it would need to be modified in order to describe our world, and no-one has suggested a way to do that.

Q: Why do people think that N=8 supergravity won’t diverge at seven loops?

There’s a useful rule of thumb in amplitudes: amplitudes are weird. In studying amplitudes we often notice unexpected simplifications, patterns that uncover new principles that weren’t obvious before.

Gravity in general seems to have a lot of these kinds of simplifications. Even without any loops, its behavior is surprisingly tame: it’s a theory that we can build up piece by piece from the three-particle interaction, even though naively we shouldn’t be able to (for the experts: I’m talking about large-z behavior in BCFW). This behavior seems to have an effect on one-loop amplitudes as well. There are other ways in which gravity seems better-behaved than expected, overall this suggests that we still have a fair ways to go before we understand all of the symmetries of gravity theories.

Supersymmetric gravity in particular also seems unusually well-behaved. N=5 supergravity was expected to diverge at four loops, but doesn’t. N=4 supergravity does diverge at four loops, but that seems to be due to an effect that is specific to that case (for the experts: an anomaly).

For N=8 specifically, a suggestive hint came from varying the dimension. If you checked the dimension in which the theory diverged at each loop, you’d find it matched the divergences of another theory, N=4 super Yang-Mills. At $l$ loops, N=4 super Yang-Mills diverges in dimension $4+6/l$ . From that formula, you can see that no matter how much you increase $l$ , you’ll never get to four dimensions: in four dimensions, N=4 super Yang-Mills doesn’t diverge.

At five loops, N=4 super Yang-Mills diverges in 26/5 dimensions. Zvi Bern made a bet with supergravity expert Kelly Stelle that the dimension would be the same for N=8 supergravity: a bottle of California wine from Bern versus English wine from Stelle. Now that they’ve found a divergence in 24/5 dimensions instead, Stelle will likely be getting his wine soon.

Q: It sounds like the calculation was pretty tough. Can they still make it to seven loops?

I think so, yes. Doing the five-loop calculation they noticed simplifications, clever tricks uncovered by even more clever grad students. The end result is that if they just want to find out whether the theory diverges then they don’t have to do the “whole calculation”, just part of it. This simplifies things a lot. They’ll probably have to find a few more simplifications to make seven loops viable, but I’m optimistic that they’ll find them, and in the meantime the new tricks should have some applications in other theories.

Q: What do you think? Will the theory diverge?

I’m not sure.

To be honest, I’m a bit less optimistic than I used to be. The agreement of divergence dimensions between N=8 supergravity and N=4 super Yang-Mills wasn’t the strongest argument (there’s a reason why, though Stelle accepted the bet on five loops, string theorist Michael Green is waiting on seven loops for his bet). Fractional dimensions don’t obviously mean anything physically, and many of the simplifications in gravity seem specific to four dimensions. Still, it was suggestive, the kind of “motivation” that gets a conjecture started.

Without that motivation, none of the remaining arguments are specific to N=8. I still think unexpected simplifications are likely, that gravity overall behaves better than we yet appreciate. I still would bet on seven loops being finite. But I’m less confident about what it would mean for the theory overall. That’s going to take more serious analysis, digging in to the anomaly in N=4 supergravity and seeing what generalizes. It does at least seem like Zvi and co. are prepared to undertake that analysis.

Regardless, it’s still worth pushing for seven loops. Having that kind of heavy-duty calculation in our sub-field forces us to improve our mathematical technology, in the same way that space programs and particle colliders drive technology in the wider world. If you think your new amplitudes method is more efficient than the alternatives, the push to seven loops is the ideal stress test. Jacob Bourjaily likes to tell me how his prescriptive unitarity technique is better than what Zvi and co. are doing, this is our chance to find out!

Overall, I still stand by what I say in my blog’s sidebar. I’m interested in N=8 supergravity, I’d love to find out whether the four-graviton amplitude diverges…and now that the calculation is once again making progress, I expect that I will.

4 gravitons

The trials and tribulations of four gravitons and a postdoc