Tag Archives: quantum field theory

The Many Worlds of Condensed Matter

Physics is the science of the very big and the very small. We study the smallest scales, the fundamental particles that make up the universe, and the largest, stars on up to the universe as a whole.

We also study the world in between, though.

That’s the domain of condensed matter, the study of solids, liquids, and other medium-sized arrangements of stuff. And while it doesn’t make the news as often, it’s arguably the biggest field in physics today.

(In case you’d like some numbers, the American Physical Society has divisions dedicated to different sub-fields. Condensed Matter Physics is almost twice the size of the next biggest division, Particles & Fields. Add in other sub-fields that focus on medium-sized-stuff, like those who work on solid state physics, optics, or biophysics, and you get a majority of physicists focused on the middle of the distance scale.)

When I started grad school, I didn’t pay much attention to condensed matter and related fields. Beyond the courses in quantum field theory and string theory, my “breadth” courses were on astrophysics and particle physics. But over and over again, from people in every sub-field, I kept hearing the same recommendation:

“You should take Solid State Physics. It’s a really great course!”

At the time, I never understood why. It was only later, once I had some research under my belt, that I realized:

Condensed matter uses quantum field theory!

The same basic framework, describing the world in terms of rippling quantum fields, doesn’t just work for fundamental particles. It also works for materials. Rather than describing the material in terms of its fundamental parts, condensed matter physicists “zoom out” and talk about overall properties, like sound waves and electric currents, treating them as if they were the particles of quantum field theory.

This tends to confuse the heck out of journalists. Not used to covering condensed matter (and sometimes egged on by hype from the physicists), they mix up the metaphorical particles of these systems with the sort of particles made by the LHC, with predictably dumb results.

Once you get past the clumsy journalism, though, this kind of analogy has a lot of value.

Occasionally, you’ll see an article about string theory providing useful tools for condensed matter. This happens, but it’s less widespread than some of the articles make it out to be: condensed matter is a huge and varied field, and string theory applications tend to be of interest to only a small piece of it.

It doesn’t get talked about much, but the dominant trend is actually in the other direction: increasingly, string theorists need to have at least a basic background in condensed matter.

String theory’s curse/triumph is that it can give rise not just to one quantum field theory, but many: a vast array of different worlds obtained by twisting extra dimensions in different ways. Particle physicists tend to study a fairly small range of such theories, looking for worlds close enough to ours that they still fit the evidence.

Condensed matter, in contrast, creates its own worlds. Pick the right material, take the right slice, and you get quantum field theories of almost any sort you like. While you can’t go to higher dimensions than our usual four, you can certainly look at lower ones, at the behavior of currents on a sheet of metal or atoms arranged in a line. This has led some condensed matter theorists to examine a wide range of quantum field theories with one strange behavior or another, theories that wouldn’t have occurred to particle physicists but that, in many cases, are part of the cornucopia of theories you can get out of string theory.

So if you want to explore the many worlds of string theory, the many worlds of condensed matter offer a useful guide. Increasingly, tools from that community, like integrability and tensor networks, are migrating over to ours.

It’s gotten to the point where I genuinely regret ignoring condensed matter in grad school. Parts of it are ubiquitous enough, and useful enough, that some of it is an expected part of a string theorist’s background. The many worlds of condensed matter, as it turned out, were well worth a look.

KITP Conference Retrospective

I’m back from the conference in Santa Barbara, and I thought I’d share a few things I found interesting. (For my non-physicist readers: I know it’s been a bit more technical than usual recently, I promise I’ll get back to some general audience stuff soon!)

James Drummond talked about efforts to extend the hexagon function method I work on to amplitudes with seven (or more) particles. In general, the method involves starting with a guess for what an amplitude should look like, and honing that guess based on behavior in special cases where it’s easier to calculate. In one of those special cases (called the multi-Regge limit), I had thought it would be quite difficult to calculate for more than six particles, but James clarified for me that there’s really only one additional piece needed, and they’re pretty close to having a complete understanding of it.

There were a few talks about ways to think about amplitudes in quantum field theory as the output of a string theory-like setup. There’s been progress pushing to higher quantum-ness, and in understanding the weird web of interconnected theories this setup gives rise to. In the comments, Thoglu asked about one part of this web of theories called Z theory.

Z theory is weird. Most of the theories that come out of this “web” come from a consistent sort of logic: just like you can “square” Yang-Mills to get gravity, you can “square” other theories to get more unusual things. In possibly the oldest known example, you can “square” the part of string theory that looks like Yang-Mills at low energy (open strings) to get the part that looks like gravity (closed strings). Z theory asks: could the open string also come from “multiplying” two theories together? Weirdly enough, the answer is yes: it comes from “multiplying” normal Yang-Mills with a part that takes care of the “stringiness”, a part which Oliver Schlotterer is calling “Z theory”. It’s not clear whether this Z theory makes sense as a theory on its own (for the experts: it may not even be unitary) but it is somewhat surprising that you can isolate a “building block” that just takes care of stringiness.

Peter Young in the comments asked about the Correlahedron. Scattering amplitudes ask a specific sort of question: if some particles come in from very far away, what’s the chance they scatter off each other and some other particles end up very far away? Correlators ask a more general question, about the relationships of quantum fields at different places and times, of which amplitudes are a special case. Just as the Amplituhedron is a geometrical object that specifies scattering amplitudes (in a particular theory), the Correlahedron is supposed to represent correlators (in the same theory). In some sense (different from the sense above) it’s the “square” of the Amplituhedron, and the process that gets you from it to the Amplituhedron is a geometrical version of the process that gets you from the correlator to the amplitude.

For the Amplituhedron, there’s a reasonably smooth story of how to get the amplitude. News articles tended to say the amplitude was the “volume” of the Amplituhedron, but that’s not quite correct. In fact, to find the amplitude you need to add up, not the inside of the Amplituhedron, but something that goes infinite at the Amplituhedron’s boundaries. Finding this “something” can be done on a case by case basis, but it get tricky in more complicated cases.

For the Correlahedron, this part of the story is missing: they don’t know how to define this “something”, the old recipe doesn’t work. Oddly enough, this actually makes me optimistic. This part of the story is something that people working on the Amplituhedron have been trying to avoid for a while, to find a shape where they can more honestly just take the volume. The fact that the old story doesn’t work for the Correlahedron suggests that it might provide some insight into how to build the Amplituhedron in a different way, that bypasses this problem.

There were several more talks by mathematicians trying to understand various aspects of the Amplituhedron. One of them was by Hugh Thomas, who as a fun coincidence actually went to high school with Nima Arkani-Hamed, one of the Amplituhedron’s inventors. He’s now teamed up with Nima and Jaroslav Trnka to try to understand what it means to be inside the Amplituhedron. In the original setup, they had a recipe to generate points inside the Amplituhedron, but they didn’t have a fully geometrical picture of what put them “inside”. Unlike with a normal shape, with the Amplituhedron you can’t just check which side of the wall you’re on. Instead, they can flatten the Amplituhedron, and observe that for points “inside” the Amplituhedron winds around them a specific number of times (hence “Unwinding the Amplituhedron“). Flatten it down to a line and you can read this off from the list of flips over your point, an on-off sequence like binary. If you’ve ever heard the buzzword “scattering amplitudes as binary code”, this is where that comes from.

They also have a better understanding of how supersymmetry shows up in the Amplituhedron, which Song He talked about in his talk. Previously, supersymmetry looked to be quite central, part of the basic geometric shape. Now, they can instead understand it in a different way, with the supersymmetric part coming from derivatives (for the specialists: differential forms) of the part in normal space and time. The encouraging thing is that you can include these sorts of derivatives even if your theory isn’t supersymmetric, to keep track of the various types of particles, and Song provided a few examples in his talk. This is important, because it opens up the possibility that something Amplituhedron-like could be found for a non-supersymmetric theory. Along those lines, Nima talked about ways that aspects of the “nice” description of space and time we use for the Amplituhedron can be generalized to other messier theories.

While he didn’t talk about it at the conference, Jake Bourjaily has a new paper out about a refinement of the generalized unitarity technique I talked about a few weeks back. Generalized unitarity involves matching a “cut up” version of an amplitude to a guess. What Jake is proposing is that in at least some cases you can start with a guess that’s as easy to work with as possible, where each piece of the guess matches up to just one of the “cuts” that you’re checking.  Think about it like a game of twenty questions where you’ve divided all possible answers into twenty individual boxes: for each box, you can just ask “is it in this box”?

Finally, I’ve already talked about the highlight of the conference, so I can direct you to that post for more details. I’ll just mention here that there’s still a fair bit of work to do for Zvi Bern and collaborators to get their result into a form they can check, since the initial output of their setup is quite messy. It’s led to worries about whether they’ll have enough computer power at higher loops, but I’m confident that they still have a few tricks up their sleeves.

Scattering Amplitudes at KITP

I’ve been visiting the Kavli Institute for Theoretical Physics in Santa Barbara for a program on scattering amplitudes. This week they’re having a conference, so I don’t have time to say very much.


The conference logo, on the other hand, seems to be saying quite a lot

We’ve had talks from a variety of corners of amplitudes, with major themes including the web of theories that can sort of be described by string theory-esque models, the amplituhedron, and theories you can “square” to get other theories. I’m excited about Zvi Bern’s talk at the end of the conference, which will describe the progress I talked about last week. There’s also been recent progress on understanding the amplituhedron, which I will likely post about in the near future.

We also got an early look at Whispers of String Theory, a cute short documentary filmed at the IGST conference.

The Road to Seven-Loop Supergravity

There’s an obvious way to put together a theory of quantum gravity. And it doesn’t work.

Do the same thing you would with any other theory, and you get infinity. You get repeated infinities, an infinity of infinities. And while you could fix one or two infinities, fixing an infinite number requires giving up an infinity of possible predictions, so in the end your theory predicts nothing.

String theory fixes this with its own infinity, the infinite number of ways a string can vibrate. Because this infinity is organized and structured and well-understood, you’re left with a theory that is still at least capable of making predictions.

(Note that this is an independent question from whether string theory can make predictions for experiments in the real world. This is a much more “in-principle” statement: if we knew everything we might want to about physics, all the fields and particles and shapes of the extra dimensions, we could use string theory to make predictions. Even if we knew all of that, we still couldn’t make predictions from naive quantum gravity.)

Are there ways to fix the problem that don’t involve an infinity of vibrations? Or at least, to fix part of the problem?

That’s what Zvi Bern, John Joseph Carrasco, Henrik Johansson, and a growing cast of collaborators have been trying to find out.

They’re investigating N=8 supergravity, a theory that takes gravity and adds on a host of related particles. It’s one of the easiest theories to get from string theory, by curling up extra dimensions in a particularly simple way and ignoring higher-energy vibrations.

Bern, along with Lance Dixon and David Kosower, invented the generalized unitarity technique I talked about last week. Along with Carrasco and Johansson, he figured out another important trick: the idea that you can do calculations in gravity by squaring the appropriate part of calculations in Yang-Mills theory. For N=8 supergravity, the theory you need to square is my favorite theory, N=4 super Yang-Mills.

Using this, they started pushing forward, calculating approximations to greater and greater precision (more and more loops).

What they found, at each step, was that N=8 supergravity behaved better than expected. In fact, it behaved like N=4 super Yang-Mills.

N=4 super Yang-Mills is special, because in four dimensions (three space and one time, the dimensions we’re used to in daily life) there are no infinities to fix. In a world with more dimensions, though, you start getting infinities, and with more and more loops you need fewer and fewer dimensions to see them.

N=8 supergravity, unexpectedly, was giving infinities in the same dimensions that N=4 super Yang-Mills did (and no earlier). If it kept doing that, you might guess that it also had no infinities in four dimensions. You might wonder if, at least loop by loop, N=8 supergravity could be a way to fix quantum gravity without string theory.

Of course, you’d only really know if you could check in four dimensions.

If you want to check in four dimensions, though, you run into a problem. The fewer dimensions you’re looking at, the more loops you need before N=8 supergravity could possibly give infinity. In four dimensions, you need a forbidding seven loops of precision.

(To compare, the highest precision of things we’ve actually tested in the real world is four loops.)

Still, Bern, Carrasco, and Johansson were up to the challenge. Along with Lance Dixon, David Kosower, and Radu Roiban, they looked at three loops, calculating an interaction of four gravitons, and the pattern continued. Four loops, and it was still going strong.

At around this time, I had just started grad school. My first project was a cumbersome numerical calculation. To keep me motivated, my advisor mentioned that the work I was doing would be good preparation for a much grander project: the calculation of whether the four-graviton interaction in N=8 supergravity diverges at seven loops. All I’d have to do was wait for Bern and collaborators to get there.

I named this blog “4 gravitons and a grad student”, and hoped I would get a chance to contribute.

And then something unexpected happened. They got stuck at five loops.

The method they were using, generalized unitarity, is an ansatz-based method. You start with a guess, then refine it. As such, the method is ultimately only as good as your guess.

Their guesses, in general, were pretty good. The trick they were using, squaring N=4 to get N=8, requires a certain type of guess: one in which the pieces they square have similar relationships to the different types of charge in Yang-Mills theory. There’s still an infinite number of guesses that can obey this, so they applied more restrictions, expectations based on other calculations, to get something more manageable. This worked at three loops, and worked (with a little extra thought) at four loops.

But at five loops they were stuck. They couldn’t find anything, with their restrictions, that gave the correct answer when “cut up” by generalized unitarity. And while they could drop some restrictions, if they dropped too many they’d end up with far too general a guess, something that could take months of computer time to solve.

So they stopped.

They did quite a bit of interesting work in the meantime. They found more theories they could square to get gravity theories, of more and more unusual types. They calculated infinities in other theories, and found surprises there too, other cases where infinities didn’t show up when they were “supposed” to. But for some time, the N=8 supergravity calculation was stalled.

And in the meantime, I went off in another direction, which long-time readers of this blog already know about.

Recently, though, they’ve broken the stall.

What they realized is that the condition on their guess, that the parts they square be related like Yang-Mills charges, wasn’t entirely necessary. Instead, they could start with a “bad” guess, and modify it, using the failure of those relations to fill in the missing pieces.

It looks like this is going to work.

We’re all at an amplitudes program right now in Santa Barbara. Walking through the halls of the KITP, I overhear conversations about five loops. They’re paring things down, honing their code, getting rid of the last few bugs, and checking their results.

They’re almost there, and it’s exciting. It looks like finally things are moving again, like the train to seven loops has once again left the station.

Increasingly, they’re beginning to understand the absent infinities, to see that they really are due to something unexpected and new.

N=8 supergravity isn’t going to be the next theory of everything. (For one, you can’t get chiral fermions out of it.) But if it really has no infinities at any loop, that tells us something about what a theory of quantum gravity is allowed to be, about the minimum necessary to at least make sense on a loop-by-loop level.

And that, I think, is worth being excited about.

Generalized Unitarity: The Frankenstein Method for Amplitudes

This is going to be a bit more technical than my usual, but you were warned.

There are a few things you’ll need to know to understand this post.

First, you should know that when we calculate probabilities of things happening in particle physics, we can do it by drawing Feynman diagrams, pictures of particles traveling and interacting. These diagrams can have loops, and the particle in the loop can have any momentum, from zero on up to infinity: you have to add up all the possibilities to get whatever you’re trying to calculate.

Second, you should understand that the “particles” in these loops aren’t really particles. They’re “virtual particles”, better understood as disturbances in quantum fields. Matt Strassler has a very nice article about this. In particular, these “particles” don’t have to obey E=mc^2 (or rather, if we include kinetic energy, E^2=p^2 c^2+m^2 c^4, where p is the momentum).

You can imagine a space that the momentum and energy “live in”. It’s got three dimensions for the three directions momentum can have, and one more dimension for the energy. Virtual particles can live anywhere in this four-dimensional space, but real particles have to live on a “shell” of points that obey E^2=p^2 c^2+m^2 c^4. If you’ve heard physicists say “on-shell” or “off-shell”, they’re referring to whether a particle is virtual, a quantum mechanical disturbance (and thus lives anywhere in the space) or a real classical particle (living on this “shell”).

Third, you should appreciate that in quantum physics, in Scott Aaronson’s words, we put complex numbers in our ontologies. Often, quantum weirdness shows itself when we look at our calculations as functions of complex numbers.

Let’s say I’m calculating an amplitude with one loop, and I draw a diagram like this:


Unitarity is how particle physicists say “all probabilities have to add up to one”. Since we have complex numbers in our ontologies, this statement is more complicated than it looks. One thing it ends up implying is that if I calculate an amplitude from the one-loop diagram above, its imaginary part will be given by multiplying together two simpler amplitudes:


Here you can imagine that I took a pair of scissors and “cut” the diagram in two along the dashed line. Now that the diagram has been “cut”, the particles I cut through are no longer part of a loop, so they’re no longer virtual: they’re real, on-shell particles.

If I wanted, I could keep “cutting” the diagram, generalizing this implication of unitarity. (For those who know some complex analysis, this involves taking residues.) I could cut all of the lines in the loop, like this:


Now something interesting happens. Here I’ve forced all four of the particles in the loop to be “on-shell”, to obey E^2=p^2 c^2+m^2 c^4. Previously, the momentum and energy in the loop was entirely free, living in its four-dimensional space. Now, though, it must obey four equations. And for those who’ve seen some algebra, four independent equations and four unknowns gives us one solution. By cutting all of these particles, we’ve killed all of the freedom that the loop momentum had. Instead of the living, quantum amplitude we had, we’ve cut it up into a bunch of dead, classical parts.

Why do this?

Well, suppose we have a guess for what the full amplitude should be. We’ve still got some uncertainty in our guess: it’s an ansatz.

If we wanted to check our guess, to fix the uncertainty in our ansatz, we could compare it to the full amplitude. But then we’d have to calculate the full quantum amplitude, and that’s hard.

It’s a lot easier, though, to calculate those “dead” classical amplitudes.

That’s the method we call “generalized unitarity”. We stitch together these easier-to-calculate, “dead” amplitudes. Enough different stitching patterns, and we can fix all the uncertainty in our ansatz, ending up with a unique correct answer without ever doing the full quantum calculation. Like Frankenstein, from dead parts we’ve assembled a living thing.


It’s off-shell!

How well does this work?

That depends on how good the ansatz is. The ansatze for one loop are very well understood, and for two loops the community is getting there. For higher loops, you have to be either smart or lucky. I happen to know some people who are both, I’ll be talking about them next week.

Poll Results, and What’s Next

I’ll leave last week’s poll up a while longer as more votes trickle in, but the overall pattern (beyond “Zipflike“) is pretty clear.

From pretty early on, most requests were for more explanations of QFT, gravity, and string theory concepts, with amplitudes content a clear second. This is something I can definitely do more of: I haven’t had much inspiration for interesting pieces of this sort recently, but it’s something I can ramp up in future.

I suspect that many of the people voting for more QFT and more amplitudes content were also interested in something else, though: more physics news. Xezlec mentioned that with Résonaances and Of Particular Significance quiet, there’s an open niche for vaguely reasonable people blogging about physics.

The truth is, I didn’t think of adding a “more physics news” option to the poll. I’m not a great source of news: not being a phenomenologist, I don’t keep up with the latest experimental results, and since my sub-field is small and insular I’m not always aware of the latest thing Witten or Maldacena is working on.

For an example of the former: recently, various LHC teams presented results at the Moriond and Aspen conferences, with no new evidence of supersymmetry in the data they’ve gathered thus far. This triggered concessions on several bets about SUSY (including an amusingly awkward conversation about how to pay one of them).

And I only know about that because other bloggers talked about it.

So I’m not going to be a reliable source of physics news.

With that said, knowing there’s a sizable number of people interested in this kind of thing is helpful. I’ve definitely had times when I saw something I found interesting, but wasn’t sure if my audience would care. (For example, recently there’s been some substantial progress on the problem that gave this blog its name.) Now that I know some of you are interested, I’ll err on the side of posting about these kinds of things.

“What’s it like to be a physicist” and science popularization were both consistently third and fourth in the poll, switching back and forth as more votes came in. This tells me that while many of you want more technical content, there are still people interested in pieces aimed to a broader audience, so I won’t abandon those.

The other topics were fairly close together, with the more “news-y” ones (astrophysics/cosmology and criticism of bad science coverage) beating the less “news-y” ones. This also supports my guess that people were looking for a “more physics news” option. A few people even voted for “more arguments”, which was really more of a joke topic: getting into arguments with other bloggers tends to bring in readers, but it’s not something I ever plan to do intentionally.

So, what’s next? I’ll explain more quantum field theory, talk more about interesting progress in amplitudes, and mention news when I come across it, trusting you guys to find it interesting. I’ll keep up with the low-level stuff, and with trying to humanize physics, to get the public to understand what being a physicist is all about. And I’ll think about some of the specific suggestions you gave: I’m always looking for good post ideas.

The Way to a Mathematician’s Heart Is through a Pi

Want to win over a mathematician? Bake them a pi.

Of course, presentation counts. You can’t just pour a spew of digits.


If you have to, at least season it with 9’s

Ideally, you’ve baked your pi at home, in a comfortable physical theory. You lay out a graph to give it structure, then wrap it in algebraic curves before baking under an integration.

(Sometimes you can skip this part. My mathematician will happily eat graphs and ignore the pi.)

At this point, if your motives are pure (or at least mixed Tate), you have your pi. To make it more interesting, be sure to pair with a well-aged Riemann zeta value. With the right preparation, you can achieve a truly cosmic pi.


Fine, that last joke was a bit of a stretch. Hope you had a fun pi day!