Source Your Common Sense

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When I wrote that post on crackpots, one of my inspirations was a particularly annoying Twitter conversation. The guy I was talking to had convinced himself that general relativity was a mistake. He was especially pissed off by the fact that, in GR, energy is not always conserved. Screw Einstein, energy conservation is just common sense! Right?

Think a little bit about why you believe in energy conservation. Is it because you run into a lot of energy in your day-to-day life, and it’s always been conserved? Did you grow up around something that was obviously energy? Or maybe someone had to explain it to you?

Teacher Pointing at Map of World

Maybe you learned about it…from a physics teacher?

A lot of the time, things that seem obvious only got that way because you were taught them. “Energy” isn’t an intuitive concept, however much it’s misused that way. It’s something defined by physicists because it solves a particular role, a consequence of symmetries in nature. When you learn about energy conservation in school, that’s because it’s one of the simpler ways to explain a much bigger concept, so you shouldn’t be surprised if there are some inaccuracies. If you know where your “common sense” is coming from, you can anticipate when and how it might go awry.

Similarly, if, like one of the commenters on my crackpot post, you’re uncomfortable with countable and uncountable infinities, remember that infinity isn’t “common sense” either. It’s something you learned about in a math class, from a math teacher. And just like energy conservation, it’s a simplification of a more precise concept, with epsilons and deltas and all that jazz.

It’s not possible to teach all the nuances of every topic, so naturally most people will hear a partial story. What’s important is to recognize that you heard a partial story, and not enshrine it as “common sense” when the real story comes knocking.

Don’t physicists use common sense, though? What about “physical intuition”?

Physical intuition has a lot of mystique behind it, and is often described as what separates us from the mathematicians. As such, different people mean different things by it…but under no circumstances should it be confused with pure “common sense”. Physical intuition uses analogy and experience. It involves seeing a system and anticipating the sorts of things you can do with it, like playing a game and assuming there’ll be a save button. In order for these sorts of analogies to work, they generally aren’t built around everyday objects or experiences. Instead, they use physical systems that are “similar” to the one under scrutiny in important ways, while being better understood in others. Crucially, physical intuition involves working in context. It’s not just uncritical acceptance of what one would naively expect.

So when your common sense is tingling, see if you can provide a source. Is that source relevant, experience with a similar situation? Or is it in fact a half-remembered class from high school?

Starshot: The Right Kind of Longshot

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On Tuesday, Yuri Milner and Stephen Hawking announced Starshot, a $100 million dollar research initiative. The goal is to lay the groundwork for a very ambitious, but surprisingly plausible project: sending probes to the nearest star, Alpha Centauri. Their idea is to have hundreds of ultra-light probes, each with a reflective sail a few meters in diameter. By aiming an extremely powerful laser at these sails, it should be possible to accelerate the probes up to around a fifth of the speed of light, enough to make the trip in twenty years. Here’s the most complete article I’ve found on the topic.

I can’t comment on the engineering side of the project. The impression I get is that nothing they’re proposing is known to be impossible, but there are a lot of “ifs” along the way that might scupper things. What I can comment on is the story.

Milner and Hawking have both put quite a bit of effort recently into what essentially amounts to telling stories. Milner’s Breakthrough Prizes involve giving awards of $3 million to prominent theoretical physicists (and, more recently, mathematicians). Quite a few of my fellow theorists have criticized these prizes, arguing that the money would be better spent in a grant program like that of the Simons Foundation. While that would likely be better for science, the Breakthrough Prize isn’t really about that. Instead, it’s about telling a story: a story in which progress in theoretical physics is exalted in a public, Nobel-sized way.

Similarly, Hawking’s occasional pronouncements about aliens or AI aren’t science per se, and the media has a tendency to talk about his contributions to ongoing scientific debates out of proportion to their importance. Both of these things, though, contribute to the story of Hawking: a mascot for physics, someone to carry Einstein’s role of the most recognizable genius in the world. Hawking Inc. is about a role as much as it is about a man.

In calling Hawking and Milner’s activity “stories”, I’m not dismissing them. Stories can be important. And the story told by Starshot is a particularly important one.

Cosmology isn’t just a scientific subject, it contributes to how people see themselves. Here I don’t just mean cosmology the field, but cosmology in the broader sense of our understanding of the universe and our place in it.

A while back, I read a book called The View from the Center of the Universe. The book starts by describing the worldviews of the ancients, cosmologies in which they really did think of themselves as the center of the universe. It then suggests that this played an important role: that this kind of view of the world, in which humans have a place in the cosmos, is important to how we view ourselves. The rest of the book then attempts to construct this sort of mythological understanding out of the modern cosmological picture, with some success.

One thing the book doesn’t discuss very much, though, is the future. We care about our place in the universe not just because we want to know where we came from, but because we want to have some idea of where we’re going. We want to contribute to a greater goal, to see ourselves making progress towards something important and vast and different. That’s why so many religions have not just cosmologies, but eschatologies, why people envision armageddons and raptures.

Starshot places the future in our sight in a way that few other things do. Humanity’s spread among the stars seems like something so far distant that nothing we do now could matter to it. What Starshot does is give us something concrete, a conceptual stepping-stone that can link people in to the broader narrative. Right now, people can work on advanced laser technology and optics, work on making smaller chips and lighter materials, work that would be useful and worth funding regardless of whether it was going to lead to Alpha Centauri. But because of Starshot, we can view that work as the near-term embodiment of humanity’s interstellar destiny.

That combination, bridging the gap between the distant future and our concrete present, is the kind of story people need right now. And so for once, I think Milner’s storytelling is doing exactly what it should.

GUTs vs ToEs: What Are We Unifying Here?

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“Grand Unified Theory” and “Theory of Everything” may sound like meaningless grandiose titles, but they mean very different things.

In particular, Grand Unified Theory, or GUT, is a technical term, referring to a specific way to unify three of the fundamental interactions: electromagnetism, the weak force, and the strong force.

In contrast, guts unify the two fundamental intestines.

Those three forces are called Yang-Mills forces, and they can all be described in the same basic way. In particular, each has a strength (the coupling constant) and a mathematical structure that determines how it interacts with itself, called a group.

The core idea of a GUT, then, is pretty simple: to unite the three Yang-Mills forces, they need to have the same strength (the same coupling constant) and be part of the same group.

But wait! (You say, still annoyed at the pun in the above caption.) These forces don’t have the same strength at all! One of them’s strong, one of them’s weak, and one of them is electromagnetic!

As it turns out, this isn’t as much of a problem as it seems. While the three Yang-Mills forces seem to have very different strengths on an everyday scale, that’s not true at very high energies. Let’s steal a plot from Sweden’s Royal Institute of Technology:

Why Sweden? Why not!

What’s going on in this plot?

Here, each $\alpha$ represents the strength of a fundamental force. As the force gets stronger, $\alpha$ gets bigger (and so $\alpha^{-1}$ gets smaller). The variable on the x-axis is the energy scale. The grey lines represent a world without supersymmetry, while the black lines show the world in a supersymmetric model.

So based on this plot, it looks like the strengths of the fundamental forces change based on the energy scale. That’s true, but if you find that confusing there’s another, mathematically equivalent way to think about it.

You can think about each force as having some sort of ultimate strength, the strength it would have if the world weren’t quantum. Without quantum mechanics, each force would interact with particles in only the simplest of ways, corresponding to the simplest diagram here.

However, our world is quantum mechanical. Because of that, when we try to measure the strength of a force, we’re not really measuring its “ultimate strength”. Rather, we’re measuring it alongside a whole mess of other interactions, corresponding to the other diagrams in that post. These extra contributions mean that what looks like the strength of the force gets stronger or weaker depending on the energy of the particles involved.

(I’m sweeping several things under the rug here, including a few infinities and electroweak unification. But if you just want a general understanding of what’s going on, this should be a good starting point.)

If you look at the plot, you’ll see the forces meet up somewhere around 10^16 GeV. They miss each-other for the faint, non-supersymmetric lines, but they meet fairly cleanly for the supersymmetric ones.

So (at least if supersymmetry is true), making the Yang-Mills forces have the same strength is not so hard. Putting them in the same mathematical group is where things get trickier. This is because any group that contains the groups of the fundamental forces will be “bigger” than just the sum of those forces: it will contain “extra forces” that we haven’t observed yet, and these forces can do unexpected things.

In particular, the “extra forces” predicted by GUTs usually make protons unstable. As far as we can tell, protons are very long-lasting: if protons decayed too fast, we wouldn’t have stars. So if protons decay, they must do it only very rarely, detectable only with very precise experiments. These experiments are powerful enough to rule out most of the simplest GUTs. The more complicated GUTs still haven’t been ruled out, but it’s enough to make fewer people interested in GUTs as a research topic.

What about Theories of Everything, or ToEs?

While GUT is a technical term, ToE is very much not. Instead, it’s a phrase that journalists have latched onto because it sounds cool. As such, it doesn’t really have a clear definition. Usually it means uniting gravity with the other fundamental forces, but occasionally people use it to refer to a theory that also unifies the various Standard Model particles into some sort of “final theory”.

Gravity is very different from the other fundamental forces, different enough that it’s kind of silly to group them as “fundamental forces” in the first place. Thus, while GUT models are the kind of thing one can cook up and tinker with, any ToE has to be based on some novel insight, one that lets you express gravity and Yang-Mills forces as part of the same structure.

So far, string theory is the only such insight we have access to. This isn’t just me being arrogant: while there are other attempts at theories of quantum gravity, aside from some rather dubious claims none of them are even interested in unifying gravity with other forces.

This doesn’t mean that string theory is necessarily right. But it does mean that if you want a different “theory of everything”, telling physicists to go out and find a new one isn’t going to be very productive. “Find a theory of everything” is a hope, not a research program, especially if you want people to throw out the one structure we have that even looks like it can do the job.

I Don’t Get Crackpots

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[Note: not an April fool’s post. Now I’m wishing I wrote one though.]

After the MHV@30 conference, I spent a few days visiting my sister. I hadn’t seen her in a while, and she noticed something new about me.

“You’re not sure about anything. It’s always ‘I get the impression’ or ‘I believe so’ or ‘that seems good’.”

On reflection, she’s right.

It’s a habit I’ve picked up from spending time around scientists. When you’re surrounded by people who are likely to know more than you do about something, it’s usually good to qualify your statements. A little intellectual humility keeps simple corrections from growing into pointless arguments, and makes it easier to learn from your mistakes.

With that kind of mindset, though, I really really don’t get crackpots.

For example, why do they always wear funnels on their heads?

The thing about genuine crackpots (as opposed to just scientists with weird ideas) is that they tend to have almost none of the relevant background for a given field, but nevertheless have extremely strong opinions about it. That basic first step, of assuming that there are people who probably know a lot more about whatever you’re talking about? Typically, they don’t bother with that. The qualifiers, the “typically” and “as far as I know” just don’t show up. And I have a lot of trouble understanding how a person can work that way.

Is some of it the Dunning-Kruger effect? Sure. If you don’t know much about something, you don’t know the limits of your own knowledge, so you think you know more than you really do. But I don’t think it’s just that…there’s a baseline level of doubt, of humility in general, that just isn’t there for most crackpots.

I wonder if some fraction of crackpots are genuinely mentally ill, but if so I’m not sure what the illness would be. Mania is an ok fit some of the time, and the word salad and “everyone but me is crazy” attitude almost seem schizophrenic, but I doubt either is really what’s going on in most cases.

All of this adds up to me just being completely unable to relate to people who display a sufficient level of crackpottery.

The thing is, there are crackpots out there who I kind of wish I could talk to, because if I could maybe I could help them. There are crackpots who seem genuinely willing to be corrected, to be told what they’re doing wrong. But that core of implicit arrogance, the central assumption that it’s possible to make breakthroughs in a field while knowing almost nothing about it, that’s still there, and it makes it impossible for me to deal with them.

I kind of wish there was a website I could link, dedicated to walking crackpots through their mistakes. There used to be something like that for supernatural crackpots, in the form of the James Randi Educational Foundation‘s Million Dollar Prize, complete with forums where (basically) helpful people would patiently walk applicants through how to set up a test of their claims. There’s never been anything like that for science, as far as I’m aware, and it seems like it would take a lot more work. Still, it would be nice if there were people out there patient enough to do it.

Four Gravitons and Some Wildly Irresponsible Amplitudes Predictions

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My post on the “physics of decimals” a couple of weeks back caught physics blogger Luboš Motl’s attention, with predictable results. Mostly, this led to a rather unproductive debate about semantics, but he did bring up one thing that I think deserves some further clarification.

In my post, I asked you to imagine asking a genie for the full consequences of quantum field theory. Short of genie-based magic, is this the sort of thing I think it’s at all possible to know?

A Candle of Invocation? Sure, why not.

In a word, no.

The world is messy, not the sort of thing that tends to be described by neat exact solutions. That’s why we use approximations, and it’s why physicists can’t just step in and solve biology or psychology by deriving everything from first principles.

That said, the nice thing about approximations is that there’s often room for improvement. Sometimes this is quantitative, literally pushing to the next order of decimals, while sometimes it’s qualitative, viewing problems from a new perspective and attacking them from a new approach.

I’d like to give you some idea of the sorts of improvements I think are possible. I’ll focus on scattering amplitudes, since they’re my field. In order to be precise, I’ll be using technical terms here without much explanation; if you’re curious about something specific go ahead and ask in the comments. Finally, there are no implied time-scales here: I’ll be rating things based on whether I think they’re likely to eventually be understood, not on how long it will take us to get there.

Let’s begin with the most likely category:

Probably going to happen:

Mathematicians characterize the set of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms.

The seven-loop N=8 supergravity integrand is found, and the coefficient of its potential divergence is evaluated.

The dual Amplituhedron is found.

A general procedure is described for re-summing the L-loop coefficient of the Pentagon OPE for any L into a polylogarithmic form, at least at six points.

We figure out what the heck is up with the MHV-NMHV relation we found here.

Likely to happen, but there may be unforeseen complications:

N=8 supergravity is found to be finite at seven loops.

A symbol bootstrap becomes workable for QCD amplitudes at two or three loops, perhaps involving Landau singularities.

Something like a symbol bootstrap becomes workable for elliptic integrals, though it may only pass a “physicist” level of rigor.

Analogues to all of the work up to the actual Amplituhedron itself are performed for non-planar N=4 super Yang-Mills.

Quite possible, but I’m likely overoptimistic:

The space of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms that also obey the first entry condition and some number of final entry conditions turns out to be well-constrained enough that some all-loop all-point statements can be made, at least for MHV.

The enhanced cancellations observed in supergravity theories are understood, and used to provide a strong argument that N=8 supergravity is perturbatively finite.

All-multiplicity analytic QCD results at two loops, for at least the simpler helicity configurations.

The volume of the dual Amplituhedron is characterized by mathematicians and the connection to cluster polylogarithms is fully explored.

A non-planar Amplituhedron is found.

Less likely, but if all of the above happens I would not be all that surprised:

A way is found to double-copy the non-planar Amplituhedron to get an N=8 supergravity Amplituhedron.

The enhanced cancellations in N=8 supergravity turn out to be something “deep”: perhaps they are derivable from string theory, or provide a novel constraint on quantum gravity theories.

Various all-loop statements about the polylogarithms present in N=4 are used to make more restricted all-loop statements about QCD.

The Pentagon OPE is re-summed for finite coupling, if not into known functions than into a form that admits good numerics and various analytic manipulations. Alternatively, the sorts of functions that the Pentagon OPE can sum to are characterized and a bootstrap procedure becomes viable for them.

Irresponsible speculations, suited to public talks or grant applications:

The N=8 Amplituhedron leads to some sort of reformulation of space-time in a way that solves various quantum gravity paradoxes.

The sorts of mathematical objects found in the finite-coupling resummation of the Pentagon OPE lead to a revival of the original analytic S-matrix program, now with an actual chance to succeed.

Extremely unlikely:

Analytic all-loop QCD results.

Magical genie land:

Analytic finite coupling QCD results.

It Was Thirty Years Ago Yesterday, Parke and Taylor Taught the Band to Play…

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Just a short post this week. I’m at MHV@30, a conference at Fermilab in honor of Parke and Taylor’s landmark paper from March 17, 1986. I don’t have time to write up an explanation of their work’s importance, but luckily I already have.

It’s my first time visiting Fermilab. They took us on a tour of their neutrino detectors 100m underground. Since we theorists don’t visit experiments very often, it was an unusual experience.

In case you wanted to know what a neutrino beam looks like, look at the target.

The fun thing about these kinds of national labs is the sheer variety of research, from the most abstract theory to the most grounded experiments, that spring from the same core goals. Physics almost always involves a diversity of viewpoints and interests, and that’s nowhere more obvious than here.

In Defense of Lord Kelvin, Michelson, and the Physics of Decimals

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William Thompson, Lord Kelvin, was a towering genius of 19th century physics. He is often quoted as saying,

There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

Certainly sounds like something I would say!

As it happens, he never actually said this. It’s a paraphrase of a quote from Albert Michelson, of the Michelson-Morley Experiment:

While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice. It is here that the science of measurement shows its importance — where quantitative work is more to be desired than qualitative work. An eminent physicist remarked that the future truths of physical science are to be looked for in the sixth place of decimals.

Now that’s more like it!

In hindsight, this quote looks pretty silly. When Michelson said that “it seems probable that most of the grand underlying principles have been firmly established” he was leaving out special relativity, general relativity, and quantum mechanics. From our perspective, the grandest underlying principles had yet to be discovered!

And yet, I think we should give Michelson some slack.

Someone asked me on twitter recently what I would choose if given the opportunity to unravel one of the secrets of the universe. At the time, I went for the wishing-for-more-wishes answer: I’d ask for a procedure to discover all of the other secrets.

I was cheating, to some extent. But I do think that the biggest and most important mystery isn’t black holes or the big bang, isn’t asking what will replace space-time or what determines the constants in the Standard Model. The most critical, most important question in physics, rather, is to find the consequences of the principles we actually know!

We know our world is described fairly well by quantum field theory. We’ve tested it, not just to the sixth decimal place, but to the tenth. And while we suspect it’s not the full story, it should still describe the vast majority of our everyday world.

If we knew not just the underlying principles, but the full consequences of quantum field theory, we’d understand almost everything we care about. But we don’t. Instead, we’re forced to calculate with approximations. When those approximations break down, we fall back on experiment, trying to propose models that describe the data without precisely explaining it. This is true even for something as “simple” as the distribution of quarks inside a proton. Once you start trying to describe materials, or chemistry or biology, all bets are off.

This is what the vast majority of physics is about. Even more, it’s what the vast majority of science is about. And that’s true even back to Michelson’s day. Quantum mechanics and relativity were revelations…but there are still large corners of physics in which neither matters very much, and even larger parts of the more nebulous “physical science”.

New fundamental principles get a lot of press, but you shouldn’t discount the physics of “the sixth place of decimals”. Most of the big mysteries don’t ask us to challenge our fundamental paradigm: rather, they’re challenges to calculate or measure better, to get more precision out of rules we already know. If a genie gave me the solution to any of physics’ mysteries I’d choose to understand the full consequences of quantum field theory, or even of the physics of Michelson’s day, long before I’d look for the answer to a trendy question like quantum gravity.

Things You Don’t Know about the Power of the Dark Side

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Last Wednesday, Katherine Freese gave a Public Lecture at Perimeter on the topic of Dark Matter and Dark Energy. The talk should be on Perimeter’s YouTube page by the time this post is up.

Answering twitter questions during the talk made me realize that there’s a lot the average person finds confusing about Dark Matter and Dark Energy. Freese addressed much of this pretty well in her talk, but I felt like there was room for improvement. Rather than try to tackle it myself, I decided to interview an expert on the Dark Side of the universe.

Twitter doesn’t know the power of the dark side!

Lord Vader, some people have a hard time distinguishing Dark Matter and Dark Energy. What do you have to say to them?

Fools! Light side astronomers call “dark” that which they cannot observe and cannot understand. “Fear” and “anger” are different heights of emotion, but to the Jedi they are only the path to the Dark Side. Dark Energy and Dark Matter are much the same: both distinct, both essential to the universe, and both “dark” to the telescopes of the light.

Let’s start with Dark Matter. Is it really matter?

You ask an empty question. “Matter” has been defined in many ways. When we on the Dark Side refer to Dark Matter, we merely mean to state that it behaves much like the matter you know: it is drawn to and fro by gravity, sloshing about.

It is distinct from your ordinary matter in that two of the forces of nature, the strong nuclear force and electromagnetism, do not concern it. Ordinary matter is bound together in the nuclei of atoms by the strong force, or woven into atoms and molecules by electromagnetism. This makes it subject to all manner of messy collisions.

Dark Matter, in contrast, is pure, partaking neither of nuclear nor chemical reactions. It passes through each of us with no notice. Only the weak nuclear force and gravity affect it. The latter has brought it slowly into clumps and threads through the universe, each one a vast nest for groupings of stars. Truly, Dark Matter surrounds us, penetrates us, and binds the galaxy together.

Could Dark Matter be something we’re more familiar with, like neutrinos or black holes? What about a modification of gravity?

Many wondered as much, when the study of the Dark Side was young. They were wrong.

The matter you are accustomed to composes merely a twentieth of the universe, while Dark Matter is more than a quarter. There is simply not enough of these minor contributions, neutrinos and black holes, to account for the vast darkness that surrounds the galaxy, and with each astronomer’s investigation we grow more assured.

As for modifying gravity, do you seek to modify a fundamental Force?

If so, you should be wary. Forces, by their nature, are accompanied by particles, and gravity is no exception. Take care that your tinkering does not result in a new sort of particle. If so, you may be unknowingly walking the path of the Dark Side, for your modification may be just another form of Dark Matter.

What sort of things could Dark Matter be? Can Dark Matter decay into ordinary matter? Could there be anti-Dark Matter?

As of yet, your scientists are still baffled by the nature of Dark Matter. Still, there are limits. Since only rare events could produce it from ordinary matter, the universe’s supply of Dark Matter must be ancient, dating back to the dawn of the cosmos. In that case, it must decay only slowly, if at all. Similarly, if Dark Matter had antimatter forms then its interactions must be so weak that it has not simply annihilated with its antimatter half across the universe. So while either is possible, it may be simpler for your theorists if Dark Matter did not decay, and was its own antimatter counterpart. On the other hand, if Dark Matter did undergo such reactions, your kind may one day be able to detect it.

Of course, as a master of the Dark Side I know the true nature of Dark Matter. However, I could only impart it to a loyal apprentice…

Yeah, I think I’ll pass on that. They say you can only get a job in academia when someone dies, but unlike the Sith they don’t mean it literally.

Let’s move on to Dark Energy. What can you tell us about it?

Dark “Energy”, like Dark Matter, is named for what people on your Earth cannot comprehend. Nothing, not even Dark Energy, is “made of energy”. Dark Energy is “energy” merely because it behaves unlike matter.

Matter, even Dark Matter, is drawn together by the force of gravity. Under its yoke, the universe would slow down in its expansion and eventually collapse into a crunch, like the throat of an incompetent officer.

However, the universe is not collapsing, but accelerating, galaxies torn away from each other by a force that must compose more than two thirds of the universe. It is rather like the Yuuzhan Vong, a mysterious force from outside the galaxy that scouts persistently under- or over-estimate.

Umm, I’m pretty sure the Yuuzhan Vong don’t exist anymore, since Disney got rid of the Expanded Universe.

That perfidious Mouse!

Well folks, Vader is now on a rampage of revenge in the Disney offices, so I guess we’ll have to end the interview. Tune in next week, and until then, may the Force be with you!

Symbology 101

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I work with functions called polylogarithms. There’s a whole field of techniques out there for manipulating these functions, and for better or worse people often refer to them as symbology.

My plan for this post is to give a general feel for how symbology works: what we know how to do, and why. It’s going to be a lot more technical than my usual posts, so the lay reader may want to skip this one. At the same time, I’m not planning to go through anything rigorously. If you want that sort of thing there are plenty of good papers on the subject, here’s one of mine that covers the basics. Rather, I’m going to draw what I hope is an illuminating sketch of what it is we do.

Still here? Let’s start with an easy question.

What’s a log?

Ok, besides one of these.

For our purposes, a log is what happens when you integrate dx/x.

$\log x=\int \frac{dx}{x}$

Schematically, a polylog is then what happens when you iterate these integrations:

$G=\int \frac{dx_1}{x_1} \int \frac{dx_2}{x_2}\ldots$

The simplest thing you can get from this is of course just a product of logs. The next most simple thing is one of the classical polylogarithms. But in general, this is a much wider class of functions, known as multiple, or Goncharov, polylogarithms.

The number of integrations is the transcendental weight. Naively, you’d expect an L-loop Feynman integral in four dimensions to give you something with transcendental weight 4L. In practice, that’s not the case: some of the momentum integrations end up just giving delta functions, so in the end an L-loop amplitude has transcendental weight 2L.

In most theories, you get a mix of functions: some with weight 2L, some with weight 2L-1, etc., all the way down to rational functions. N=4 super Yang-Mills is special: there, everything is at the maximum transcendental weight. In either case, though, being able to manipulate transcendental functions is very useful, and the symbol is one of the simplest ways to do so.

The core idea of the symbol is pretty easy to state, though it takes a bit more technology to state it rigorously. Essentially, we take our schematic polylog from above, and just list the logs:

$\mathcal{S}(G)=\ldots\otimes x_2\otimes x_1$

(Here I have switched the order in order to agree with standard conventions.)

What does that do? Well, it reminds us that these aren’t just some weird functions we don’t understand: they’re collections of logs, and we can treat them like collections of logs.

In particular, we can do this with logs,

$\log (x y)=\log x+\log y$

so we can do it with symbols as well:

$x_1\otimes x y\otimes x_3=x_1\otimes x \otimes x_3+x_1\otimes y\otimes x_3$

Similarly, we can always get rid of unwelcome exponents, like so:

$\log (x^n)=n\log x$

$x_1\otimes x^n\otimes x_3=n( x_1\otimes x \otimes x_3)$

This means that, in general, we can always factorize any polynomial or rational function that appears in a symbol. As such, we often express symbols in terms of some fixed symbol alphabet, a basis of rational functions that can be multiplied to get any symbol entry in the function we’re working with. In general, it’s a lot easier to calculate amplitudes when we know the symbol alphabet beforehand. For six-particle amplitudes in N=4 super Yang-Mills, the symbol alphabet contains just nine “letters”, which makes it particularly easy to work with.

That’s arguably the core of symbol methods. It’s how Spradlin and Volovich managed to get a seventeen-page expression down to two lines. Express a symbol in the right alphabet, and it tends to look a lot more simple. And once you know the right alphabet, it’s pretty straightforward to build an ansatz with it and constrain it until you get a candidate function for whatever you’re interested in.

There’s more technical detail I could give here: how to tell whether a symbol actually corresponds to a function, how to take limits and do series expansions and take derivatives and discontinuities…but I’m not sure whether anyone reading this would be interested.

As-is, I’ll just mention that the symbol is only part of the story. In particular, it’s a special case of something called a coproduct, which breaks up polylogarithms into various chunks. Break them down fully until each chunk is just an individual log, and you get the symbol. Break them into larger chunks, and you get other components of the coproduct, consisting of tensor products of polylogarithms with lower transcendental weight. These larger chunks mean we can capture as much of a function’s behavior as we like, while still taking advantage of these sorts of tricks. While in older papers you might have seen mention of “beyond-the-symbol” terms that the symbol couldn’t capture, this doesn’t tend to be a problem these days.

You Go, LIGO!

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Well folks, they did it. LIGO has detected gravitational waves!

FAQ:

What’s a gravitational wave?

Gravitational waves are ripples in space and time. As Einstein figured out a century ago, masses bend space and time, which causes gravity. Wiggle masses in the right way and you get a gravity wave, like a ripple on a pond.

Ok, but what is actually rippling? It’s some stuff, right? Dust or something?

In a word, no. Not everything has to be “stuff”. Energy isn’t “stuff”, and space-time isn’t either, but space-time is really what vibrates when a gravitational wave passes by. Distances themselves are changing, in a way that is described by the same math and physics as a ripple in a pond.

What’s LIGO?

LIGO is the Laser Interferometer Gravitational-Wave Observatory. In simple terms, it’s an observatory (or rather, a pair of observatories in Washington and Louisiana) that can detect gravitational waves. It does this using beams of laser light four kilometers long. Gravitational waves change the length of these beams when they pass through, causing small but measurable changes in the laser light observed.

Are there other gravitational wave observatories?

Not currently in operation. LIGO originally ran from 2002 to 2010, and during that time there were other gravitational wave observatories also in operation (VIRGO in Italy and GEO600 in Germany). All of them (including LIGO) failed to detect anything, and so LIGO and VIRGO were shut down in order for them to be upgraded to more sensitive, advanced versions. Advanced LIGO went into operation first, and made the detection. VIRGO is still under construction, as is KAGRA, a detector in Japan. There are also plans for a detector in India.

Other sorts of experiments can detect gravitational waves on different scales. eLISA is a planned space-based gravitational wave observatory, while Pulsar Timing Arrays could use distant neutron stars as an impromptu detector.

What did they detect? What could they detect?

The gravitational waves that LIGO detected came from a pair of black holes merging. In general, gravitational waves come from a pair of masses, or one mass with an uneven and rapidly changing shape. As such, LIGO and future detectors might be able to observe binary stars, supernovas, weird-shaped neutron stars, colliding galaxies…pretty much any astrophysical event involving large things moving comparatively fast.

What does this say about string theory?

Basically nothing. There are gravity waves in string theory, sure (and they play a fairly important role), but there were gravity waves in Einstein’s general relativity. As far as I’m aware, no-one at this point seriously thought that gravitational waves didn’t exist. Nothing that LIGO observed has any bearing on the quantum properties of gravity.

But what about cosmic strings? They mentioned those in the announcement!

Cosmic strings, despite the name, aren’t a unique prediction of string theory. They’re big, string-shaped wrinkles in space and time, possible results of the rapid expansion of space during cosmic inflation. You can think of them a bit like the cracks that form in an over-inflated balloon right before it bursts.

Cosmic strings, if they exist, should produce gravitational waves. This means that in the future we may have concrete evidence of whether or not they exist. This wouldn’t say all that much about string theory: while string theory does have its own explanations for cosmic strings, it’s unclear whether it actually has unique predictions about them. It would say a lot about cosmic inflation, though, and would presumably help distinguish it from proposed alternatives. So keep your eyes open: in the next few years, gravitational wave observatories may well have something important to say about the overall history of the universe.

Why is this discovery important, though? If we already knew that gravitational waves existed, why does discovering them matter?

LIGO didn’t discover that gravitational waves exist. LIGO discovered that we can detect them.

The existence of gravitational waves is no discovery. But the fact that we now have observatories sensitive enough to detect them is huge. It opens up a whole new type of astronomy: we can now observe the universe not just by the light it sheds (and neutrinos), but through a whole new lens. And every time we get another observational tool like this, we notice new things, things we couldn’t have seen without it. It’s the dawn of a new era in astronomy, and LIGO was right to announce it with all the pomp and circumstance they could muster.

My impressions from the announcement:

Speaking of pomp and circumstance, I was impressed by just how well put-together LIGO’s announcement was.

As the US presidential election heats up, I’ve seen a few articles about the various candidates’ (well, usually Trump’s) use of the language of political propaganda. The idea is that there are certain visual symbols at political events for which people have strong associations, whether with historical events or specific ideas or the like, and that using these symbols makes propaganda more powerful.

What I haven’t seen is much discussion of a language of scientific propaganda. Still, the overwhelming impression I got from LIGO’s announcement is that it was shaped by a master in the use of such a language. They tapped in to a wide variety of powerful images: from the documentary-style interviews at the beginning, to Weiss’s tweed jacket and handmade demos, to the American flag in the background, that tied LIGO’s result to the history of scientific accomplishment.

Perimeter’s presentations tend to have a slicker look, my friends at Stony Brook are probably better at avoiding jargon. But neither is quite as good at propaganda, at saying “we are part of history” and doing so without a hitch, as the folks at LIGO have shown themselves to be with this announcement.

I was also fairly impressed that they kept this under wraps for so long. While there were leaks, I don’t think many people had a complete grasp of what was going to be announced until the week before. Somehow, LIGO made sure a collaboration of thousands was able to (mostly) keep their mouths shut!

Beyond the organizational and stylistic notes, my main thought was “What’s next?” They’ve announced the detection of one event. I’ve heard others rattle off estimates, that they should be detecting anywhere from one black hole merger per year to a few hundred. Are we going to see more events soon, or should we settle into a long wait? Could they already have detected more, with the evidence buried in their data, to be revealed by careful analysis? (The waves from this black hole merger were clear enough for them to detect them in real-time, but more subtle events might not make things so easy!) Should we be seeing more events already, and does not seeing them tell us something important about the universe?

Most of the reason I delayed my post till this week was to see if anyone had an answer to these questions. So far, I haven’t seen one, besides the “one to a few hundred” estimate mentioned. As more people weigh in and more of LIGO’s run is analyzed, it will be interesting to see where that side of the story goes.

4 gravitons

Stories about physics from someone who's been there