At Bohr-100: Current Themes in Theoretical Physics

During the pandemic, some conferences went online. Others went dormant.

Every summer before the pandemic, the Niels Bohr International Academy hosted a conference called Current Themes in High Energy Physics and Cosmology. Current Themes is a small, cozy conference, a gathering of close friends some of whom happen to have Nobel prizes. Holding it online would be almost missing the point.

Instead, we waited. Now, at least in Denmark, the pandemic is quiet enough to hold this kind of gathering. And it’s a special year: the 100th anniversary of Niels Bohr’s Nobel, the 101st of the Niels Bohr Institute. So it seemed like the time for a particularly special Current Themes.

A particularly special Current Themes means some unusually special guests. Our guests are usually pretty special already (Gerard t’Hooft and David Gross are regulars, to just name the Nobelists), but this year we also had Alexander Polyakov. Polyakov’s talk had a magical air to it. In a quiet voice, broken by an impish grin when he surprised us with a joke, Polyakov began to lay out five unsolved problems he considered interesting. In the end, he only had time to present one, related to turbulence: when Gross asked him to name the remaining four, the second included a term most of us didn’t recognize (striction, known in a magnetic context and which he wanted to explore gravitationally), so the discussion hung while he defined that and we never did learn what the other three problems were.

At the big 100th anniversary celebration earlier in the spring, the Institute awarded a few years worth of its Niels Bohr Institute Medal of Honor. One of the recipients, Paul Steinhardt, couldn’t make it then, so he got his medal now. After the obligatory publicity photos were taken, Steinhardt entertained us all with a colloquium about his work on quasicrystals, including the many adventures involved in finding the first example “in the wild”. I can’t do the story justice in a short blog post, but if you won’t have the opportunity to watch him speak about it then I hear his book is good.

An anniversary conference should have some historical elements as well. For this one, these were ably provided by David Broadhurst, who gave an after-dinner speech cataloguing things he liked about Bohr. Some was based on public information, but the real draw were the anecdotes: his own reminiscences, and those of people he knew who knew Bohr well.

The other talks covered interesting ground: from deep approaches to quantum field theory, to new tools to understand black holes, to the implications of causality itself. One out of the ordinary talk was by Sabrina Pasterski, who advocated a new model of physics funding. I liked some elements (endowed organizations to further a subfield) and am more skeptical of others (mostly involving NFTs). Regardless it, and the rest of the conference more broadly, spurred a lot of good debate.

Einstein-Years

Scott Aaronson recently published an interesting exchange on his blog Shtetl Optimized, between him and cognitive psychologist Steven Pinker. The conversation was about AI: Aaronson is optimistic (though not insanely so) Pinker is pessimistic (again, not insanely though). While fun reading, the whole thing would normally be a bit too off-topic for this blog, except that Aaronson’s argument ended up invoking something I do know a bit about: how we make progress in theoretical physics.

Aaronson was trying to respond to an argument of Pinker’s, that super-intelligence is too vague and broad to be something we could expect an AI to have. Aaronson asks us to imagine an AI that is nothing more or less than a simulation of Einstein’s brain. Such a thing isn’t possible today, and might not even be efficient, but it has the advantage of being something concrete we can all imagine. Aarsonson then suggests imagining that AI sped up a thousandfold, so that in one year it covers a thousand years of Einstein’s thought. Such an AI couldn’t solve every problem, of course. But in theoretical physics, surely such an AI could be safely described as super-intelligent: an amazing power that would change the shape of physics as we know it.

I’m not as sure of this as Aaronson is. We don’t have a machine that generates a thousand Einstein-years to test, but we do have one piece of evidence: the 76 Einstein-years the man actually lived.

Einstein is rightly famous as a genius in theoretical physics. His annus mirabilis resulted in five papers that revolutionized the field, and the next decade saw his theory of general relativity transform our understanding of space and time. Later, he explored what general relativity was capable of and framed challenges that deepened our understanding of quantum mechanics.

After that, though…not so much. For Einstein-decades, he tried to work towards a new unified theory of physics, and as far as I’m aware made no useful progress at all. I’ve never seen someone cite work from that period of Einstein’s life.

Aarsonson mentions simulating Einstein “at his peak”, and it would be tempting to assume that the unified theory came “after his peak”, when age had weakened his mind. But while that kind of thing can sometimes be an issue for older scientists, I think it’s overstated. I don’t think careers peak early because of “youthful brains”, and with the exception of genuine dementia I don’t think older physicists are that much worse-off cognitively than younger ones. The reason so many prominent older physicists go down unproductive rabbit-holes isn’t because they’re old. It’s because genius isn’t universal.

Einstein made the progress he did because he was the right person to make that progress. He had the right background, the right temperament, and the right interests to take others’ mathematics and take them seriously as physics. As he aged, he built on what he found, and that background in turn enabled him to do more great things. But eventually, the path he walked down simply wasn’t useful anymore. His story ended, driven to a theory that simply wasn’t going to work, because given his experience up to that point that was the work that interested him most.

I think genius in physics is in general like that. It can feel very broad because a good genius picks up new tricks along the way, and grows their capabilities. But throughout, you can see the links: the tools mastered at one age that turn out to be just right for a new pattern. For the greatest geniuses in my field, you can see the “signatures” in their work, hints at why they were just the right genius for one problem or another. Give one a thousand years, and I suspect the well would eventually run dry: the state of knowledge would no longer be suitable for even their breadth.

…of course, none of that really matters for Aaronson’s point.

A century of Einstein-years wouldn’t have found the Standard Model or String Theory, but a century of physicist-years absolutely did. If instead of a simulation of Einstein, your AI was a simulation of a population of scientists, generating new geniuses as the years go by, then the argument works again. Sure, such an AI would be much more expensive, much more difficult to build, but the first one might have been as well. The point of the argument is simply to show such a thing is possible.

The core of Aaronson’s point rests on two key traits of technology. Technology is replicable: once we know how to build something, we can build more of it. Technology is scalable: if we know how to build something, we can try to build a bigger one with more resources. Evolution can tap into both of these, but not reliably: just because it’s possible to build a mind a thousand times better at some task doesn’t mean it will.

That is why the possibility of AI leads to the possibility of super-intelligence. If we can make a computer that can do something, we can make it do that something faster. That something doesn’t have to be “general”, you can have programs that excel at one task or another. For each such task, with more resources you can scale things up: so anything a machine can do now, a later machine can probably do better. Your starting-point doesn’t necessarily even have to be efficient, or a good algorithm: bad algorithms will take longer to scale, but could eventually get there too.

The only question at that point is “how fast?” I don’t have the impression that’s settled. The achievements that got Pinker and Aarsonson talking, GPT-3 and DALL-E and so forth, impressed people by their speed, by how soon they got to capabilities we didn’t expect them to have. That doesn’t mean that something we might really call super-intelligence is close: that has to do with the details, with what your target is and how fast you can actually scale. And it certainly doesn’t mean that another approach might not be faster! (As a total outsider, I can’t help but wonder if current ML is in some sense trying to fit a cubic with straight lines.)

It does mean, though, that super-intelligence isn’t inconceivable, or incoherent. It’s just the recognition that technology is a master of brute force, and brute force eventually triumphs. If you want to think about what happens in that “eventually”, that’s a very important thing to keep in mind.

Carving Out the Possible

If you imagine a particle physicist, you probably picture someone spending their whole day dreaming up new particles. They figure out how to test those particles in some big particle collider, and for a lucky few their particle gets discovered and they get a Nobel prize.

Occasionally, a wiseguy asks if we can’t just cut out the middleman. Instead of dreaming up particles to test, why don’t we just write down every possible particle and test for all of them? It would save the Nobel committee a lot of money at least!

It turns out, you can sort of do this, through something called Effective Field Theory. An Effective Field Theory is a type of particle physics theory that isn’t quite true: instead, it’s “effectively” true, meaning true as long as you don’t push it too far. If you test it at low energies and don’t “zoom in” too much then it’s fine. Crank up your collider energy high enough, though, and you expect the theory to “break down”, revealing new particles. An Effective Field Theory lets you “hide” unknown particles inside new interactions between the particles we already know.

To help you picture how this works, imagine that the pink and blue lines here represent familiar particles like electrons and quarks, while the dotted line is a new particle somebody dreamed up. (The picture is called a Feynman diagram, if you don’t know what that is check out this post.)

In an Effective Field Theory, we “zoom out”, until the diagram looks like this:

Now we’ve “hidden” the new particle. Instead, we have a new type of interaction between the particles we already know.

So instead of writing down every possible new particle we can imagine, we only have to write down every possible interaction between the particles we already know.

That’s not as hard as it sounds. In part, that’s because not every interaction actually makes sense. Some of the things you could write down break some important rules. They might screw up cause and effect, letting something happen before its cause instead of after. They might screw up probability, giving you a formula for the chance something happens that gives a number greater than 100%.

Using these rules you can play a kind of game. You start out with a space representing all of the interactions you can imagine. You begin chipping at it, carving away parts that don’t obey the rules, and you see what shape is left over. You end up with plots that look a bit like carving a ham.

People in my subfield are getting good at this kind of game. It isn’t quite our standard fare: usually, we come up with tricks to make calculations with specific theories easier. Instead, many groups are starting to look at these general, effective theories. We’ve made friends with groups in related fields, building new collaborations. There still isn’t one clear best way to do this carving, so each group manages to find a way to chip a little farther. Out of the block of every theory we could imagine, we’re carving out a space of theories that make sense, theories that could conceivably be right. Theories that are worth testing.

The Most Anthropic of All Possible Worlds

Today, we’d call Leibniz a mathematician, a physicist, and a philosopher. As a mathematician, Leibniz turned calculus into something his contemporaries could actually use. As a physicist, he championed a doomed theory of gravity. In philosophy, he seems to be most remembered for extremely cheaty arguments.

I don’t blame him for this. Faced with a tricky philosophical problem, it’s enormously tempting to just blaze through with an answer that makes every subtlety irrelevant. It’s a temptation I’ve succumbed to time and time again. Faced with a genie, I would always wish for more wishes. On my high school debate team, I once forced everyone at a tournament to switch sides with some sneaky definitions. It’s all good fun, but people usually end up pretty annoyed with you afterwards.

People were annoyed with Leibniz too, especially with his solution to the problem of evil. If you believe in a benevolent, all-powerful god, as Leibniz did, why is the world full of suffering and misery? Leibniz’s answer was that even an all-powerful god is constrained by logic, so if the world contains evil, it must be logically impossible to make the world any better: indeed, we live in the best of all possible worlds. Voltaire famously made fun of this argument in Candide, dragging a Leibniz-esque Professor Pangloss through some of the most creative miseries the eighteenth century had to offer. It’s possibly the most famous satire of a philosopher, easily beating out Aristophanes’ The Clouds (which is also great).

Physicists can also get accused of cheaty arguments, and probably the most mocked is the idea of a multiverse. While it hasn’t had its own Candide, the multiverse has been criticized by everyone from bloggers to Nobel prizewinners. Leibniz wanted to explain the existence of evil, physicists want to explain “unnaturalness”: the fact that the kinds of theories we use to explain the world can’t seem to explain the mass of the Higgs boson. To explain it, these physicists suggest that there are really many different universes, separated widely in space or built in to the interpretation of quantum mechanics. Each universe has a different Higgs mass, and ours just happens to be the one we can live in. This kind of argument is called “anthropic” reasoning. Rather than the best of all possible worlds, it says we live in the world best-suited to life like ours.

I called Leibniz’s argument “cheaty”, and you might presume I think the same of the multiverse. But “cheaty” doesn’t mean “wrong”. It all depends what you’re trying to do.

Leibniz’s argument and the multiverse both work by dodging a problem. For Leibniz, the problem of evil becomes pointless: any evil might be necessary to secure a greater good. With a multiverse, naturalness becomes pointless: with many different laws of physics in different places, the existence of one like ours needs no explanation.

In both cases, though, the dodge isn’t perfect. To really explain any given evil, Leibniz would have to show why it is secretly necessary in the face of a greater good (and Pangloss spends Candide trying to do exactly that). To explain any given law of physics, the multiverse needs to use anthropic reasoning: it needs to show that that law needs to be the way it is to support human-like life.

This sounds like a strict requirement, but in both cases it’s not actually so useful. Leibniz could (and Pangloss does) come up with an explanation for pretty much anything. The problem is that no-one actually knows which aspects of the universe are essential and which aren’t. Without a reliable way to describe the best of all possible worlds, we can’t actually test whether our world is one.

The same problem holds for anthropic reasoning. We don’t actually know what conditions are required to give rise to people like us. “People like us” is very vague, and dramatically different universes might still contain something that can perceive and observe. While it might seem that there are clear requirements, so far there hasn’t been enough for people to do very much with this type of reasoning.

However, for both Leibniz and most of the physicists who believe anthropic arguments, none of this really matters. That’s because the “best of all possible worlds” and “most anthropic of all possible worlds” aren’t really meant to be predictive theories. They’re meant to say that, once you are convinced of certain things, certain problems don’t matter anymore.

Leibniz, in particular, wasn’t trying to argue for the existence of his god. He began the argument convinced that a particular sort of god existed: one that was all-powerful and benevolent, and set in motion a deterministic universe bound by logic. His argument is meant to show that, if you believe in such a god, then the problem of evil can be ignored: no matter how bad the universe seems, it may still be the best possible world.

Similarly, the physicists convinced of the multiverse aren’t really getting there through naturalness. Rather, they’ve become convinced of a few key claims: that the universe is rapidly expanding, leading to a proliferating multiverse, and that the laws of physics in such a multiverse can vary from place to place, due to the huge landscape of possible laws of physics in string theory. If you already believe those things, then the naturalness problem can be ignored: we live in some randomly chosen part of the landscape hospitable to life, which can be anywhere it needs to be.

So despite their cheaty feel, both arguments are fine…provided you agree with their assumptions. Personally, I don’t agree with Leibniz. For the multiverse, I’m less sure. I’m not confident the universe expands fast enough to create a multiverse, I’m not even confident it’s speeding up its expansion now. I know there’s a lot of controversy about the math behind the string theory landscape, about whether the vast set of possible laws of physics are as consistent as they’re supposed to be…and of course, as anyone must admit, we don’t know whether string theory itself is true! I don’t think it’s impossible that the right argument comes around and convinces me of one or both claims, though. These kinds of arguments, “if assumptions, then conclusion” are the kind of thing that seems useless for a while…until someone convinces you of the conclusion, and they matter once again.

So in the end, despite the similarity, I’m not sure the multiverse deserves its own Candide. I’m not even sure Leibniz deserved Candide. But hopefully by understanding one, you can understand the other just a bit better.

I saw a discussion on twitter recently, about PhD programs in the US. Apparently universities are putting more and more weight whether prospective students published a paper during their Bachelor’s degree. For some, it’s even an informal requirement. Some of those in the discussion were skeptical that the students were really contributing to these papers much, and thought that most of the work must have been done by the papers’ other authors. If so, this would mean universities are relying more and more on a metric that depends on whether students can charm their professors enough to be “included” in this way, rather than their own abilities.

I won’t say all that much about the admissions situation in the US. (Except to say that if you find yourself making up new criteria to carefully sift out a few from a group of already qualified-enough candidates, maybe you should consider not doing that.) What I did want to say a bit about is what undergraduates can typically actually do, when it comes to research in my field.

First, I should clarify that I’m talking about students in the US system here. Undergraduate degrees in Europe follow a different path. Students typically take three years to get a Bachelor’s degree, often with a project at the end, followed by a two-year Master’s degree capped with a Master’s thesis. A European Master’s thesis doesn’t have to result in a paper, but is often at least on that level, while a European Bachelor project typically isn’t. US Bachelor’s degrees are four years, so one might expect a Bachelor’s thesis to be in between a European Bachelor’s project and Master’s thesis. In practice, it’s a bit different: courses for Master’s students in Europe will generally cover material taught to PhD students in the US, so a typical US Bachelor’s student won’t have had some courses that have a big role in research in my field, like Quantum Field Theory. On the other hand, the US system is generally much more flexible, with students choosing more of their courses and having more opportunities to advance ahead of the default path. So while US Bachelor’s students don’t typically take Quantum Field Theory, the more advanced students can and do.

Because of that, how advanced a given US Bachelor’s student is varies. A small number are almost already PhD students, and do research to match. Most aren’t, though. Despite that, it’s still possible for such a student to complete a real research project in theoretical physics, one that results in a real paper. What does that look like?

Sometimes, it’s because the student is working with a toy model. The problems we care about in theoretical physics can be big and messy, involving a lot of details that only an experienced researcher will know. If we’re lucky, we can make a simpler version of the problem, one that’s easier to work with. Toy models like this are often self-contained, the kind of thing a student can learn without all of the background we expect. The models may be simpler than the real world, but they can still be interesting, suggesting new behavior that hadn’t been considered before. As such, with a good choice of toy model an undergraduate can write something that’s worthy of a real physics paper.

Other times, the student is doing something concrete in a bigger collaboration. This isn’t quite the same as the “real scientists” doing all the work, because the student has a real task to do, just one that is limited in scope. Maybe there is particular computer code they need to get working, or a particular numerical calculation they need to do. The calculation may be comparatively straightforward, but in combination with other results it can still merit a paper. My first project as a PhD student was a little like that, tackling one part of a larger calculation. Once again, the task can be quite self-contained, the kind of thing you can teach a student over a summer project.

Undergraduate projects in the US won’t always result in a paper, and I don’t think anyone should expect, or demand, that they do. But a nontrivial number do, and not because the student is “cheating”. With luck, a good toy model or a well-defined sub-problem can lead a Bachelor’s student to make a real contribution to physics, and get a paper in the bargain.

At Mikefest

I’m at a conference this week of a very particular type: a birthday conference. When folks in my field turn 60, their students and friends organize a special conference for them, celebrating their research legacy. With COVID restrictions just loosening, my advisor Michael Douglas is getting a last-minute conference. And as one of the last couple students he graduated at Stony Brook, I naturally showed up.

The conference, Mikefest, is at the Institut des Hautes Études Scientifiques, just outside of Paris. Mike was a big supporter of the IHES, putting in a lot of fundraising work for them. Another big supporter, James Simons, was Mike’s employer for a little while after his time at Stony Brook. The conference center we’re meeting in is named for him.

I wasn’t involved in organizing the conference, so it was interesting seeing differences between this and other birthday conferences. Other conferences focus on the birthday prof’s “family tree”: their advisor, their students, and some of their postdocs. We’ve had several talks from Mike’s postdocs, and one from his advisor, but only one from a student. Including him and me, three of Mike’s students are here: another two have had their work mentioned but aren’t speaking or attending.

Most of the speakers have collaborated with Mike, but only for a few papers each. All of them emphasized a broader debt though, for discussions and inspiration outside of direct collaboration. The message, again and again, is that Mike’s work has been broad enough to touch a wide range of people. He’s worked on branes and the landscape of different string theory universes, pure mathematics and computation, neuroscience and recently even machine learning. The talks generally begin with a few anecdotes about Mike, before pivoting into research talks on the speakers’ recent work. The recent-ness of the work is perhaps another difference from some birthday conferences: as one speaker said, this wasn’t just a celebration of Mike’s past, but a “welcome back” after his return from the finance world.

One thing I don’t know is how much this conference might have been limited by coming together on short notice. For other birthday conferences impacted by COVID (and I’m thinking of one in particular), it might be nice to have enough time to have most of the birthday prof’s friends and “academic family” there in person. As-is, though, Mike seems to be having fun regardless.

Happy Birthday Mike!

Things Which Are Fluids

For overambitious apes like us, adding integers is the easiest thing in the world. Take one berry, add another, and you have two. Each remains separate, you can lay them in a row and count them one by one, each distinct thing adding up to a group of distinct things.

Other things in math are less like berries. Add two real numbers, like pi and the square root of two, and you get another real number, bigger than the first two, something you can write in an infinite messy decimal. You know in principle you can separate it out again (subtract pi, get the square root of two), but you can’t just stare at it and see the parts. This is less like adding berries, and more like adding fluids. Pour some water in to some other water, and you certainly have more water. You don’t have “two waters”, though, and you can’t tell which part started as which.

Some things in math look like berries, but are really like fluids. Take a polynomial, say $5 x^2 + 6 x + 8$. It looks like three types of things, like three berries: five $x^2$, six $x$, and eight $1$. Add another polynomial, and the illusion continues: add $x^2 + 3 x + 2$ and you get $6 x^2+9 x+10$. You’ve just added more $x^2$, more $x$, more $1$, like adding more strawberries, blueberries, and raspberries.

But those berries were a choice you made, and not the only one. You can rewrite that first polynomial, for example saying $5(x^2+2x+1) - 4 (x+1) + 7$. That’s the same thing, you can check. But now it looks like five $x^2+2x+1$, negative four $x+1$, and seven $1$. It’s different numbers of different things, blackberries or gooseberries or something. And you can do this in many ways, infinitely many in fact. The polynomial isn’t really a collection of berries, for all it looked like one. It’s much more like a fluid, a big sloshing mess you can pour into buckets of different sizes. (Technically, it’s a vector space. Your berries were a basis.)

Even smart, advanced students can get tripped up on this. You can be used to treating polynomials as a fluid, and forget that directions in space are a fluid, one you can rotate as you please. If you’re used to directions in space, you’ll get tripped up by something else. You’ll find that types of particles can be more fluid than berry, the question of which quark is which not as simple as how many strawberries and blueberries you have. The laws of physics themselves are much more like a fluid, which should make sense if you take a moment, because they are made of equations, and equations are like a fluid.

So my fellow overambitious apes, do be careful. Not many things are like berries in the end. A whole lot are like fluids.

W is for Why???

Have you heard the news about the W boson?

The W boson is a fundamental particle, part of the Standard Model of particle physics. It is what we call a “force-carrying boson”, a particle related to the weak nuclear force in the same way photons are related to electromagnetism. Unlike photons, W bosons are “heavy”: they have a mass. We can’t usually predict masses of particles, but the W boson is a bit different, because its mass comes from the Higgs boson in a special way, one that ties it to the masses of other particles like the Z boson. The upshot is that if you know the mass of a few other particles, you can predict the mass of the W.

And according to a recent publication, that prediction is wrong. A team analyzed results from an old experiment called the Tevatron, the biggest predecessor of today’s Large Hadron Collider. They treated the data with groundbreaking care, mindbogglingly even taking into account the shape of the machine’s wires. And after all that analysis, they found that the W bosons detected by the Tevatron had a different mass than the mass predicted by the Standard Model.

How different? Here’s where precision comes in. In physics, we decide whether to trust a measurement with a statistical tool. We calculate how likely the measurement would be, if it was an accident. In this case: how likely it would be that, if the Standard Model was correct, the measurement would still come out this way? To discover a new particle, we require this chance to be about one in 3.5 million, or in our jargon, five sigma. That was the requirement for discovering the Higgs boson. This super-precise measurement of the W boson doesn’t have five sigma…it has seven sigma. That means, if we trust the analysis team, then a measurement like this could come accidentally out of the Standard Model only about one in a trillion times.

Ok, should we trust the analysis team?

If you want to know that, I’m the wrong physicist to ask. The right physicists are experimental particle physicists. They do analyses like that one, and they know what can go wrong. Everyone I’ve heard from in that field emphasized that this was a very careful group, who did a lot of things impressively right…but there is still room for mistakes. One pointed out that the new measurement isn’t just inconsistent with the Standard Model, but with many previous measurements too. Those measurements are less precise, but still precise enough that we should be a bit skeptical. Another went into more detail about specific clues as to what might have gone wrong.

If you can’t find an particle experimentalist, the next best choice is a particle phenomenologist. These are the people who try to make predictions for new experiments, who use theoretical physics to propose new models that future experiments can test. Here’s one giving a first impression, and discussing some ways to edit the Standard Model to agree with the new measurement. Here’s another discussing what to me is an even more interesting question: if we take these measurements seriously, both the new one and the old ones, then what do we believe?

I’m not an experimentalist or a phenomenologist. I’m an “amplitudeologist”. I work not on the data, or the predictions, but the calculational tools used to make those predictions, called “scattering amplitudes”. And that gives me a different view on the situation.

See in my field, precision is one of our biggest selling-points. If you want theoretical predictions to match precise experiments, you need our tricks to compute them. We believe (and argue to grant agencies) that this precision will be important: if a precise experiment and a precise prediction disagree, it could be the first clue to something truly new. New solid evidence of something beyond the Standard Model would revitalize all of particle physics, giving us a concrete goal and killing fruitless speculation.

This result shakes my faith in that a little. Probably, the analysis team got something wrong. Possibly, all previous analyses got something wrong. Either way, a lot of very careful smart people tried to estimate their precision, got very confident…and got it wrong.

(There’s one more alternative: maybe million-to-one chances really do crop up nine times out of ten.)

If some future analysis digs down deep in precision, and finds another deviation from the Standard Model, should we trust it? What if it’s measuring something new, and we don’t have the prior experiments to compare to?

(This would happen if we build a new even higher-energy collider. There are things the collider could measure, like the chance one Higgs boson splits into two, that we could not measure with any earlier machine. If we measured that, we couldn’t compare it to the Tevatron or the LHC, we’d have only the new collider to go on.)

Statistics are supposed to tell us whether to trust a result. Here, they’re not doing their job. And that creates the scary possibility that some anomaly shows up, some real deviation deep in the sigmas that hints at a whole new path for the field…and we just end up bickering about who screwed it up. Or the equally scary possibility that we find a seven-sigma signal of some amazing new physics, build decades of new theories on it…and it isn’t actually real.

We don’t just trust statistics. We also trust the things normal people trust. Do other teams find the same result? (I hope that they’re trying to get to this same precision here, and see what went wrong!) Does the result match other experiments? Does it make predictions, which then get tested in future experiments?

All of those are heuristics of course. Nothing can guarantee that we measure the truth. Each trick just corrects for some of our biases, some of the ways we make mistakes. We have to hope that’s good enough, that if there’s something to see we’ll see it, and if there’s nothing to see we won’t. Precision, my field’s raison d’être, can’t be enough to convince us by itself. But it can help.

The Undefinable

If I can teach one lesson to all of you, it’s this: be precise. In physics, we try to state what we mean as precisely as we can. If we can’t state something precisely, that’s a clue: maybe what we’re trying to state doesn’t actually make sense.

Someone recently reached out to me with a question about black holes. He was confused about how they were described, about what would happen when you fall in to one versus what we could see from outside. Part of his confusion boiled down to a question: “is the center really an infinitely small point?”

I remembered a commenter a while back who had something interesting to say about this. Trying to remind myself of the details, I dug up this question on Physics Stack Exchange. user4552 has a detailed, well-referenced answer, with subtleties of General Relativity that go significantly beyond what I learned in grad school.

According to user4552, the reason this question is confusing is that the usual setup of general relativity cannot answer it. In general relativity, singularities like the singularity in the middle of a black hole aren’t treated as points, or collections of points: they’re not part of space-time at all. So you can’t count their dimensions, you can’t see whether they’re “really” infinitely small points, or surfaces, or lines…

This might surprise people (like me) who have experience with simpler equations for these things, like the Schwarzchild metric. The Schwarzchild metric describes space-time around a black hole, and in the usual coordinates it sure looks like the singularity is at a single point where r=0, just like the point where r=0 is a single point in polar coordinates in flat space. The thing is, though, that’s just one sort of coordinates. You can re-write a metric in many different sorts of coordinates, and the singularity in the center of a black hole might look very different in those coordinates. In general relativity, you need to stick to things you can say independent of coordinates.

Ok, you might say, so the usual mathematics can’t answer the question. Can we use more unusual mathematics? If our definition of dimensions doesn’t tell us whether the singularity is a point, maybe we just need a new definition!

According to user4552, people have tried this…and it only sort of works. There are several different ways you could define the dimension of a singularity. They all seem reasonable in one way or another. But they give different answers! Some say they’re points, some say they’re three-dimensional. And crucially, there’s no obvious reason why one definition is “right”. The question we started with, “is the center really an infinitely small point?”, looked like a perfectly reasonable question, but it actually wasn’t: the question wasn’t precise enough.

This is the real problem. The problem isn’t that our question was undefined, after all, we can always add new definitions. The problem was that our question didn’t specify well enough the definitions we needed. That is why the question doesn’t have an answer.

Once you understand the difference, you see these kinds of questions everywhere. If you’re baffled by how mass could have come out of the Big Bang, or how black holes could radiate particles in Hawking radiation, maybe you’ve heard a physicist say that energy isn’t always conserved. Energy conservation is a consequence of symmetry, specifically, symmetry in time. If your space-time itself isn’t symmetric (the expanding universe making the past different from the future, a collapsing star making a black hole), then you shouldn’t expect energy to be conserved.

I sometimes hear people object to this. They ask, is it really true that energy isn’t conserved when space-time isn’t symmetric? Shouldn’t we just say that space-time itself contains energy?

And well yes, you can say that, if you want. It isn’t part of the usual definition, but you can make a new definition, one that gives energy to space-time. In fact, you can make more than one new definition…and like the situation with the singularity, these definitions don’t always agree! Once again, you asked a question you thought was sensible, but it wasn’t precise enough to have a definite answer.

Keep your eye out for these kinds of questions. If scientists seem to avoid answering the question you want, and keep answering a different question instead…it might be their question is the only one with a precise answer. You can define a method to answer your question, sure…but it won’t be the only way. You need to ask precise enough questions to get good answers.

Of Snowmass and SAGEX

arXiv-watchers might have noticed an avalanche of papers with the word Snowmass in the title. (I contributed to one of them.)

Snowmass is a place, an area in Colorado known for its skiing. It’s also an event in that place, the Snowmass Community Planning Exercise for the American Physical Society’s Division of Particles and Fields. In plain terms, it’s what happens when particle physicists from across the US get together in a ski resort to plan their future.

Usually someone like me wouldn’t be involved in that. (And not because it’s a ski resort.) In the past, these meetings focused on plans for new colliders and detectors. They got contributions from experimentalists, and a few theorists heavily focused on their work, but not the more “formal” theorists beyond.

This Snowmass is different. It’s different because of Corona, which changed it from a big meeting in a resort to a spread-out series of meetings and online activities. It’s also different because they invited theorists to contribute, and not just those interested in particle colliders. The theorists involved study everything from black holes and quantum gravity to supersymmetry and the mathematics of quantum field theory. Groups focused on each topic submit “white papers” summarizing the state of their area. These white papers in turn get organized and summarized into a few subfields, which in turn contribute to the planning exercise. No-one I’ve talked to is entirely clear on how this works, how much the white papers will actually be taken into account or by whom. But it seems like a good chance to influence US funding agencies, like the Department of Energy, and see if we can get them to prioritize our type of research.

Europe has something similar to Snowmass, called the European Strategy for Particle Physics. It also has smaller-scale groups, with their own purposes, goals, and funding sources. One such group is called SAGEX: Scattering Amplitudes: from Geometry to EXperiment. SAGEX is an Innovative Training Network, an organization funded by the EU to train young researchers, in this case in scattering amplitudes. Its fifteen students are finishing their PhDs and ready to take the field by storm. Along the way, they spent a little time in industry internships (mostly at Maple and Mathematica), and quite a bit of time working on outreach.

They have now summed up that outreach work in an online exhibition. I’ve had fun exploring it over the last couple days. They’ve got a lot of good content there, from basic explanations of relativity and quantum mechanics, to detailed games involving Feynman diagrams and associahedra, to a section that uses solitons as a gentle introduction to integrability. If you’re in the target audience, you should check it out!