Tag Archives: quantum gravity

How to Get a “Minimum Scale” Without Pixels

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.

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Current Themes 2018

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!).

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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

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

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.

Bubbles of Nothing

I recently learned about a very cool concept, called a bubble of nothing.

Read about physics long enough, and you’ll hear all sorts of cosmic disaster scenarios. If the Higgs vacuum decays, and the Higgs field switches to a different value, then the masses of most fundamental particles would change. It would be the end of physics, and life, as we know it.

A bubble of nothing is even more extreme. In a bubble of nothing, space itself ceases to exist.

The idea was first explored by Witten in 1982. Witten started with a simple model, a world with our four familiar dimensions of space and time, plus one curled-up extra dimension. What he found was that this simple world is unstable: quantum mechanics (and, as was later found, thermodynamics) lets it “tunnel” to another world, one that contains a small “bubble”, a sphere in which nothing at all exists.

giphy

Except perhaps the Nowhere Man

A bubble of nothing might sound like a black hole, but it’s quite different. Throw a particle into a black hole and it will fall in, never to return. Throw it into a bubble of nothing, though, and something more interesting happens. As you get closer, the extra dimension of space gets smaller and smaller. Eventually, it stops, smoothly closing off. The particle you threw in will just bounce back, smoothly, off the outside of the bubble. Essentially, it reached the edge of the universe.

The bubble starts out small, comparable to the size of the curled-up dimension. But it doesn’t stay that way. In Witten’s setup, the bubble grows, faster and faster, until it’s moving at the speed of light, erasing the rest of the universe from existence.

You probably shouldn’t worry about this happening to us. As far as I’m aware, nobody has written down a realistic model that can transform into a bubble of nothing.

Still, it’s an evocative concept, and one I’m surprised isn’t used more often in science fiction. I could see writers using a bubble of nothing as a risk from an experimental FTL drive, or using a stable (or slowly growing) bubble as the relic of some catastrophic alien war. The idea of a bubble of literal nothing is haunting enough that it ought to be put to good use.

We Didn’t Deserve Hawking

I don’t usually do obituaries. I didn’t do one when Joseph Polchinksi died, though his textbook is sitting an arm’s reach from me right now. I never collaborated with Polchinski, I never met him, and others were much better at telling his story.

I never met Stephen Hawking, either. When I was at Perimeter, I’d often get asked if I had. Visitors would see his name on the Perimeter website, and I’d have to disappoint them by explaining that he hadn’t visited the institute in quite some time. His health, while exceptional for a septuagenarian with ALS, wasn’t up to the travel.

Was his work especially relevant to mine? Only because of its relevance to everyone who does gravitational physics. The universality of singularities in general relativity, black hole thermodynamics and Hawking radiation, these sharpened the questions around quantum gravity. Without his work, string theory wouldn’t have tried to answer the questions Hawking posed, and it wouldn’t have become the field it is today.

Hawking was unique, though, not necessarily because of his work, but because of his recognizability. Those visitors to Perimeter were a cross-section of the Canadian public. Some of them didn’t know the name of the speaker for the lecture they came to see. Some, arriving after reading Lee Smolin’s book, could only refer to him as “that older fellow who thinks about quantum gravity”. But Hawking? They knew Hawking. Without exception, they knew Hawking.

Who was the last physicist the public knew, like that? Feynman, at the height of his popularity, might have been close. You’d have to go back to Einstein to find someone who was really solidly known like that, who you could mention in homes across the world and expect recognition. And who else has that kind of status? Bohr might have it in Denmark. Go further back, and you’ll find people know Newton, they know Gaileo.

Einstein changed our picture of space and time irrevocably. Newton invented physics as we know it. Galileo and Copernicus pointed up to the sky and shouted that the Earth moves!

Hawking asked questions. He told us what did and didn’t make sense, he showed what we had to take into account. He laid the rules of engagement, and the rest of quantum gravity came and asked alongside him.

We live in an age of questions now. We’re starting to glimpse the answers, we have candidates and frameworks and tools, and if we’re feeling very optimistic we might already be sitting on a theory of everything. But we haven’t turned that corner yet, from asking questions to changing the world.

These ages don’t usually get a household name. Normally, you need an Einstein, a Newton, a Galileo, you need to shake the foundations of the world.

Somehow, Hawking gave us one anyway. Somehow, in our age of questions, we put a face in everyone’s mind, a figure huddled in a wheelchair with a snarky, computer-generated voice. Somehow Hawking reached out and reminded the world that there were people out there asking, that there was a big beautiful puzzle that our field was trying to solve.

Deep down, I’m not sure we deserved that. I hope we deserve it soon.

Epistemology, Not Metaphysics, Justifies Experiments

While I was visiting the IAS a few weeks back, they had a workshop on Quantum Information and Black Holes. I didn’t see many of the talks, but I did get to see Leonard Susskind talk about his new slogan, GR=QM.

For some time now, researchers have been uncovering deep connections between gravity and quantum mechanics. Juan Maldacena jump-started the field with the discovery of AdS/CFT, showing that theories that describe gravity in a particular curved space (Anti-de Sitter, or AdS) are equivalent to non-gravity quantum theories describing the boundary of that space (specifically, Conformal Field Theories, or CFTs). The two theories contain the same information and, with the right “dictionary”, describe the same physics: in our field’s vernacular, they’re dual. Since then, physicists have found broader similarities, situations where properties of quantum mechanics, like entanglement, are closely linked to properties of gravity theories. Maldacena’s ER=EPR may be the most publicized of these, a conjectured equivalence between Einstein-Rosen bridges (colloquially known as wormholes) and entangled pairs of particles (famously characterized by Einstein, Podolsky, and Rosen).

GR=QM is clearly a riff on ER=EPR, but Susskind is making a more radical claim. Based on these developments, including his own work on quantum complexity, Susskind is arguing that the right kind of quantum mechanical system automatically gives rise to quantum gravity. What’s more, he claims that these systems will be available, using quantum computers, within roughly a decade. Within ten years or so, we’ll be able to do quantum gravity experiments.

That sounds ridiculous, until you realize he’s talking about dual theories. What he’s imagining is not an experiment at the absurdly high energies necessary to test quantum gravity, but rather a low-energy quantum mechanics experiment that is equivalent, by something like AdS/CFT, to a quantum gravity experiment.

Most people would think of that as a simulation, not an actual test of quantum gravity. Susskind, though, spends quite a bit of time defending the claim that it really is gravity, that literally GR=QM. His description of clever experiments and overarching physical principles is aimed at piling on evidence for that particular claim.

What do I think? I don’t think it matters much.

The claim Susskind is making is one of metaphysics: the philosophy of which things do and do not “really” exist. Unlike many physicists, I think metaphysics is worth discussing, that there are philosophers who make real progress with it.

But ultimately, Susskind is proposing a set of experiments. And what justifies experiments isn’t metaphysics, it’s epistemology: not what’s “really there”, but what we can learn.

What can we learn from the sorts of experiments Susskind is proposing?

Let’s get this out of the way first: we can’t learn which theory describes quantum gravity in our own world.

That’s because every one of these experiments relies on setting up a quantum system with particular properties. Every time, you’re choosing the “boundary theory”, the quantum mechanical side of GR=QM. Either you choose a theory with a known gravity partner, and you know how the inside should behave, or you choose a theory with an unknown partner. Either way, you have no reason to expect the gravity side to resemble the world we live in.

Plenty of people would get suspicious of Susskind here, and accuse him of trying to mislead people. They’re imagining headlines, “Experiment Proves String Theory”, based on a system intentionally set up to have a string theory dual, a system that can’t actually tell us whether string theory describes the real world.

That’s not where I’m going with this.

The experiments that Susskind is describing can’t prove string theory. But we could still learn something from them.

For one, we could learn whether these pairs of theories really are equivalent. AdS/CFT, ER=EPR, these are conjectures. In some cases, they’re conjectures with very good evidence. But they haven’t been proven, so it’s still possible there’s a problem people overlooked. One of the nice things about experiments and simulations is that they’re very good at exposing problems that were overlooked.

For another, we could get a better idea of how gravity behaves in general. By simulating a wide range of theories, we could look for overarching traits, properties that are common to most gravitational theories. We wouldn’t be sure that those properties hold in our world…but with enough examples, we could get pretty confident. Hopefully, we’d stumble on things that gravity has to do, in order to be gravity.

Susskind is quite capable of making these kinds of arguments, vastly more so than I. So it frustrates me that every time I’ve seen him talk or write about this, he hasn’t. Instead, he keeps framing things in terms of metaphysics, whether quantum mechanics “really is” gravity, whether the experiment “really” explores a wormhole. If he wants to usher in a new age of quantum gravity experiments, not just as a buzzword but as real, useful research, then eventually he’s going to have to stop harping on metaphysics and start talking epistemology. I look forward to when that happens.