Book Review: The Joy of Insight

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There’s something endlessly fascinating about the early days of quantum physics. In a century, we went from a few odd, inexplicable experiments to a practically complete understanding of the fundamental constituents of matter. Along the way the new ideas ended a world war, almost fueled another, and touched almost every field of inquiry. The people lucky enough to be part of this went from familiarly dorky grad students to architects of a new reality. Victor Weisskopf was one of those people, and The Joy of Insight: Passions of a Physicist is his autobiography.

Less well-known today than his contemporaries, Weisskopf made up for it with a front-row seat to basically everything that happened in particle physics. In the late 20’s and early 30’s he went from studying in Göttingen (including a crush on Maria Göppert before a car-owning Joe Mayer snatched her up) to a series of postdoctoral positions that would exhaust even a modern-day physicist, working in Leipzig, Berlin, Copenhagen, Cambridge, Zurich, and Copenhagen again, before fleeing Europe for a faculty position in Rochester, New York. During that time he worked for, studied under, collaborated or partied with basically everyone you might have heard of from that period. As a result, this section of the autobiography was my favorite, chock-full of stories, from the well-known (Pauli’s rudeness and mythical tendency to break experimental equipment) to the less-well known (a lab in Milan planned to prank Pauli with a door that would trigger a fake explosion when opened, which worked every time they tested it…and failed when Pauli showed up), to the more personal (including an in retrospect terrifying visit to the Soviet Union, where they asked him to critique a farming collective!) That era also saw his “almost Nobel”, in his case almost discovering the Lamb Shift.

Despite an “almost Nobel”, Weisskopf was paid pretty poorly when he arrived in Rochester. His story there puts something I’d learned before about another refugee physicist, Hertha Sponer, in a new light. Sponer’s university also didn’t treat her well, and it seemed reminiscent of modern academia. Weisskopf, though, thinks his treatment was tied to his refugee status: that, aware that they had nowhere else to go, universities gave the scientists who fled Europe worse deals than they would have in a Nazi-less world, snapping up talent for cheap. I could imagine this was true for Sponer as well.

Like almost everyone with the relevant expertise, Weisskopf was swept up in the Manhattan project at Los Alamos. There he rose in importance, both in the scientific effort (becoming deputy leader of the theoretical division) and the local community (spending some time on and chairing the project’s “town council”). Like the first sections, this surreal time leads to a wealth of anecdotes, all fascinating. In his descriptions of the life there I can see the beginnings of the kinds of “hiking retreats” physicists would build in later years, like the one at Aspen, that almost seem like attempts to recreate that kind of intense collaboration in an isolated natural place.

After the war, Weisskopf worked at MIT before a stint as director of CERN. He shepherded the facility’s early days, when they were building their first accelerators and deciding what kinds of experiments to pursue. I’d always thought that the “nuclear” in CERN’s name was an artifact of the times, when “nuclear” and “particle” physics were thought of as the same field, but according to Weisskopf the fields were separate and it was already a misnomer when the place was founded. Here the book’s supply of anecdotes becomes a bit more thin, and instead he spends pages on glowing descriptions of people he befriended. The pattern continues after the directorship as his duties get more administrative, spending time as head of the physics department at MIT and working on arms control, some of the latter while a member of the Pontifical Academy of Sciences (which apparently even a Jewish atheist can join). He does work on some science, though, collaborating on the “bag of quarks” model of protons and neutrons. He lives to see the fall of the Berlin wall, and the end of the book has a bit of 90’s optimism to it, the feeling that finally the conflicts of his life would be resolved. Finally, the last chapter abandons chronology altogether, and is mostly a list of his opinions of famous composers, capped off with a Bohr-inspired musing on the complementary nature of science and the arts, humanities, and religion.

One of the things I found most interesting in this book was actually something that went unsaid. Weisskopf’s most famous student was Murray Gell-Mann, a key player in the development of the theory of quarks (including coining the name). Gell-Mann was famously cultured (in contrast to the boorish-almost-as-affectation Feynman) with wide interests in the humanities, and he seems like exactly the sort of person Weisskopf would have gotten along with. Surprisingly though, he gets no anecdotes in this book, and no glowing descriptions: just a few paragraphs, mostly emphasizing how smart he was. I have to wonder if there was some coldness between them. Maybe Weisskopf had difficulty with a student who became so famous in his own right, or maybe they just never connected. Maybe Weisskopf was just trying to be generous: the other anecdotes in that part of the book are of much less famous people, and maybe Weisskopf wanted to prioritize promoting them, feeling that they were underappreciated.

Weisskopf keeps the physics light to try to reach a broad audience. This means he opts for short explanations, and often these are whatever is easiest to reach for. It creates some interesting contradictions: the way he describes his “almost Nobel” work in quantum electrodynamics is very much the way someone would have described it at the time, but very much not how it would be understood later, and by the time he talks about the bag of quarks model his more modern descriptions don’t cleanly link with what he said earlier. Overall, his goal isn’t really to explain the physics, but to explain the physicists. I enjoyed the book for that: people do it far too rarely, and the result was a really fun read.

Duality and Emergence: When Is Spacetime Not Spacetime?

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Spacetime is doomed! At least, so say some physicists. They don’t mean this as a warning, like some comic-book universe-destroying disaster, but rather as a research plan. These physicists believe that what we think of as space and time aren’t the full story, but that they emerge from something more fundamental, so that an ultimate theory of nature might not use space or time at all. Other, grumpier physicists are skeptical. Joined by a few philosophers, they think the “spacetime is doomed” crowd are over-excited and exaggerating the implications of their discoveries. At the heart of the argument is the distinction between two related concepts: duality and emergence.

In physics, sometimes we find that two theories are actually dual: despite seeming different, the patterns of observations they predict are the same. Some of the more popular examples are what we call holographic theories. In these situations, a theory of quantum gravity in some space-time is dual to a theory without gravity describing the edges of that space-time, sort of like how a hologram is a 2D image that looks 3D when you move it. For any question you can ask about the gravitational “bulk” space, there is a matching question on the “boundary”. No matter what you observe, neither description will fail.

If theories with gravity can be described by theories without gravity, does that mean gravity doesn’t really exist? If you’re asking that question, you’re asking whether gravity is emergent. An emergent theory is one that isn’t really fundamental, but instead a result of the interaction of more fundamental parts. For example, hydrodynamics, the theory of fluids like water, emerges from more fundamental theories that describe the motion of atoms and molecules.

(For the experts: I, like most physicists, am talking about “weak emergence” here, not “strong emergence”.)

The “spacetime is doomed” crowd think that not just gravity, but space-time itself is emergent. They expect that distances and times aren’t really fundamental, but a result of relationships that will turn out to be more fundamental, like entanglement between different parts of quantum fields. As evidence, they like to bring up dualities where the dual theories have different concepts of gravity, number of dimensions, or space-time. Using those theories, they argue that space and time might “break down”, and not be really fundamental.

(I’ve made arguments like that in the past too.)

The skeptics, though, bring up an important point. If two theories are really dual, then no observation can distinguish them: they make exactly the same predictions. In that case, say the skeptics, what right do you have to call one theory more fundamental than the other? You can say that gravity emerges from a boundary theory without gravity, but you could just as easily say that the boundary theory emerges from the gravity theory. The whole point of duality is that no theory is “more true” than the other: one might be more or less convenient, but both describe the same world. If you want to really argue for emergence, then your “more fundamental” theory needs to do something extra: to predict something that your emergent theory doesn’t predict.

Sometimes this is a fair objection. There are members of the “spacetime is doomed” crowd who are genuinely reckless about this, who’ll tell a journalist about emergence when they really mean duality. But many of these people are more careful, and have thought more deeply about the question. They tend to have some mix of these two perspectives:

First, if two descriptions give the same results, then do the descriptions matter? As physicists, we have a history of treating theories as the same if they make the same predictions. Space-time itself is a result of this policy: in the theory of relativity, two people might disagree on which one of two events happened first or second, but they will agree on the overall distance in space-time between the two. From this perspective, a duality between a bulk theory and a boundary theory isn’t evidence that the bulk theory emerges from the boundary, but it is evidence that both the bulk and boundary theories should be replaced by an “overall theory”, one that treats bulk and boundary as irrelevant descriptions of the same physical reality. This perspective is similar to an old philosophical theory called positivism: that statements are meaningless if they cannot be derived from something measurable. That theory wasn’t very useful for philosophers, which is probably part of why some philosophers are skeptics of “space-time is doomed”. The perspective has been quite useful to physicists, though, so we’re likely to stick with it.

Second, some will say that it’s true that a dual theory is not an emergent theory…but it can be the first step to discover one. In this perspective, dualities are suggestive evidence that a deeper theory is waiting in the wings. The idea would be that one would first discover a duality, then discover situations that break that duality: examples on one side that don’t correspond to anything sensible on the other. Maybe some patterns of quantum entanglement are dual to a picture of space-time, but some are not. (Closer to my sub-field, maybe there’s an object like the amplituhedron that doesn’t respect locality or unitarity.) If you’re lucky, maybe there are situations, or even experiments, that go from one to the other: where the space-time description works until a certain point, then stops working, and only the dual description survives. Some of the models of emergent space-time people study are genuinely of this type, where a dimension emerges in a theory that previously didn’t have one. (For those of you having a hard time imagining this, read my old post about “bubbles of nothing”, then think of one happening in reverse.)

It’s premature to say space-time is doomed, at least as a definite statement. But it is looking like, one way or another, space-time won’t be the right picture for fundamental physics. Maybe that’s because it’s equivalent to another description, redundant embellishment on an essential theoretical core. Maybe instead it breaks down, and a more fundamental theory could describe more situations. We don’t know yet. But physicists are trying to figure it out.

What Are Students? We Just Don’t Know

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I’m taking a pedagogy course at the moment, a term-long follow-up to the one-week intro course I took in the spring. The course begins with yet another pedagogical innovation, a “pre-project”. Before the course has really properly started, we get assembled into groups and told to investigate our students. We are supposed to do interviews on a few chosen themes, all with the objective of getting to know our students better. I’m guessing the point is to sharpen our goals, so that when we start learning pedagogy we’ll have a clearer idea of what problems we’d like to solve.

The more I think about this the more I’m looking forward to it. When I TAed in the past, some of the students were always a bit of a mystery. They sat in the back, skipped assignments, and gradually I saw less and less of them. They didn’t go to office hours or the help room, and I always wondered what happened. When in the course did they “turn off”, when did we lose them? They seemed like a kind of pedagogical dark matter, observable only by their presence on the rosters. I’m hoping to detect a little of that dark matter here.

As it’s a group project, we came up with a theme as a group, and questions to support that theme (in particular, we’re focusing on the different experiences between Danes and international students). Since the topic is on my mind in general though, I thought it would be fun to reach out to you guys. Educators in the comments: if you could ask your students one question, what would it be? Students, what is one thing you think your teachers are missing?

The Unpublishable Dirty Tricks of Theoretical Physics

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As the saying goes, it is better not to see laws or sausages being made. You’d prefer to see the clean package on the outside than the mess behind the scenes.

The same is true of science. A good paper tells a nice, clean story: a logical argument from beginning to end, with no extra baggage to slow it down. That story isn’t a lie: for any decent paper in theoretical physics, the conclusions will follow from the premises. Most of the time, though, it isn’t how the physicist actually did it.

The way we actually make discoveries is messy. It involves looking for inspiration in all the wrong places: pieces of old computer code and old problems, trying to reproduce this or that calculation with this or that method. In the end, once we find something interesting enough, we can reconstruct a clearer, cleaner, story, something actually fit to publish. We hide the original mess partly for career reasons (easier to get hired if you tell a clean, heroic story), partly to be understood (a paper that embraced the mess of discovery would be a mess to read), and partly just due to that deep human instinct to not let others see us that way.

The trouble is, some of that “mess” is useful, even essential. And because it’s never published or put into textbooks, the only way to learn it is word of mouth.

A lot of these messy tricks involve numerics. Many theoretical physics papers derive things analytically, writing out equations in symbols. It’s easy to make a mistake in that kind of calculation, either writing something wrong on paper or as a bug in computer code. To correct mistakes, many things are checked numerically: we plug in numbers to make sure everything still works. Sometimes this means using an approximation, trying to make sure two things cancel to some large enough number of decimal places. Sometimes instead it’s exact: we plug in prime numbers, and can much more easily see if two things are equal, or if something is rational or contains a square root. Sometimes numerics aren’t just used to check something, but to find a solution: exploring many options in an easier numerical calculation, finding one that works, and doing it again analytically.

“Ansatze” are also common: our fancy word for an educated guess. These we sometimes admit, when they’re at the core of a new scientific idea. But the more minor examples go un-mentioned. If a paper shows a nice clean formula and proves it’s correct, but doesn’t explain how the authors got it…probably, they used an ansatz. This trick can go hand-in-hand with numerics as well: make a guess, check it matches the right numbers, then try to see why it’s true.

The messy tricks can also involve the code itself. In my field we often use “computer algebra” systems, programs to do our calculations for us. These systems are programming languages in their own right, and we need to write computer code for them. That code gets passed around informally, but almost never standardized. Mathematical concepts that come up again and again can be implemented very differently by different people, some much more efficiently than others.

I don’t think it’s unreasonable that we leave “the mess” out of our papers. They would certainly be hard to understand otherwise! But it’s a shame we don’t publish our dirty tricks somewhere, even in special “dirty tricks” papers. Students often start out assuming everything is done the clean way, and start doubting themselves when they notice it’s much too slow to make progress. Learning the tricks is a big part of learning to be a physicist. We should find a better way to teach them.

The arXiv SciComm Challenge

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Fellow science communicators, think you can explain everything that goes on in your field? If so, I have a challenge for you. Pick a day, and go through all the new papers on arXiv.org in a single area. For each one, try to give a general-audience explanation of what the paper is about. To make it easier, you can ignore cross-listed papers. If your field doesn’t use arXiv, consider if you can do the challenge with another appropriate site.

I’ll start. I’m looking at papers in the “High Energy Physics – Theory” area, announced 6 Jan, 2022. I’ll warn you in advance that I haven’t read these papers, just their abstracts, so apologies if I get your paper wrong!

arXiv:2201.01303 : Holographic State Complexity from Group Cohomology

This paper says it is a contribution to a Proceedings. That means it is based on a talk given at a conference. In my field, a talk like this usually won’t be presenting new results, but instead summarizes results in a previous paper. So keep that in mind.

There is an idea in physics called holography, where two theories are secretly the same even though they describe the world with different numbers of dimensions. Usually this involves a gravitational theory in a “box”, and a theory without gravity that describes the sides of the box. The sides turn out to fully describe the inside of the box, much like a hologram looks 3D but can be printed on a flat sheet of paper. Using this idea, physicists have connected some properties of gravity to properties of the theory on the sides of the box. One of those properties is complexity: the complexity of the theory on the sides of the box says something about gravity inside the box, in particular about the size of wormholes. The trouble is, “complexity” is a bit subjective: it’s not clear how to give a good definition for it for this type of theory. In this paper, the author studies a theory with a precise mathematical definition, called a topological theory. This theory turns out to have mathematical properties that suggest a well-defined notion of complexity for it.

arXiv:2201.01393 : Nonrelativistic effective field theories with enhanced symmetries and soft behavior

We sometimes describe quantum field theory as quantum mechanics plus relativity. That’s not quite true though, because it is possible to define a quantum field theory that doesn’t obey special relativity, a non-relativistic theory. Physicists do this if they want to describe a system moving much slower than the speed of light: it gets used sometimes for nuclear physics, and sometimes for modeling colliding black holes.

In particle physics, a “soft” particle is one with almost no momentum. We can classify theories based on how they behave when a particle becomes more and more soft. In normal quantum field theories, if they have special behavior when a particle becomes soft it’s often due to a symmetry of the theory, where the theory looks the same even if something changes. This paper shows that this is not true for non-relativistic theories: they have more requirements to have special soft behavior, not just symmetry. They “bootstrap” a few theories, using some general restrictions to find them without first knowing how they work (“pulling them up by their own bootstraps”), and show that the theories they find are in a certain sense unique, the only theories of that kind.

arXiv:2201.01552 : Transmutation operators and expansions for 1-loop Feynman integrands

In recent years, physicists in my sub-field have found new ways to calculate the probability that particles collide. One of these methods describes ordinary particles in a way resembling string theory, and from this discovered a whole “web” of theories that were linked together by small modifications of the method. This method originally worked only for the simplest Feynman diagrams, the “tree” diagrams that correspond to classical physics, but was extended to the next-simplest diagrams, diagrams with one “loop” that start incorporating quantum effects.

This paper concerns a particular spinoff of this method, that can find relationships between certain one-loop calculations in a particularly efficient way. It lets you express calculations of particle collisions in a variety of theories in terms of collisions in a very simple theory. Unlike the original method, it doesn’t rely on any particular picture of how these collisions work, either Feynman diagrams or strings.

arXiv:2201.01624 : Moduli and Hidden Matter in Heterotic M-Theory with an Anomalous U(1) Hidden Sector

In string theory (and its more sophisticated cousin M theory), our four-dimensional world is described as a world with more dimensions, where the extra dimensions are twisted up so that they cannot be detected. The shape of the extra dimensions influences the kinds of particles we can observe in our world. That shape is described by variables called “moduli”. If those moduli are stable, then the properties of particles we observe would be fixed, otherwise they would not be. In general it is a challenge in string theory to stabilize these moduli and get a world like what we observe.

This paper discusses shapes that give rise to a “hidden sector”, a set of particles that are disconnected from the particles we know so that they are hard to observe. Such particles are often proposed as a possible explanation for dark matter. This paper calculates, for a particular kind of shape, what the masses of different particles are, as well as how different kinds of particles can decay into each other. For example, a particle that causes inflation (the accelerating expansion of the universe) can decay into effects on the moduli and dark matter. The paper also shows how some of the moduli are made stable in this picture.

arXiv:2201.01630 : Chaos in Celestial CFT

One variant of the holography idea I mentioned earlier is called “celestial” holography. In this picture, the sides of the box are an infinite distance away: a “celestial sphere” depicting the angles particles go after they collide, in the same way a star chart depicts the angles between stars. Recent work has shown that there is something like a sensible theory that describes physics on this celestial sphere, that contains all the information about what happens inside.

This paper shows that the celestial theory has a property called quantum chaos. In physics, a theory is said to be chaotic if it depends very precisely on its initial conditions, so that even a small change will result in a large change later (the usual metaphor is a butterfly flapping its wings and causing a hurricane). This kind of behavior appears to be present in this theory.

arXiv:2201.01657 : Calculations of Delbrück scattering to all orders in αZ

Delbrück scattering is an effect where the nuclei of heavy elements like lead can deflect high-energy photons, as a consequence of quantum field theory. This effect is apparently tricky to calculate, and previous calculations have involved approximations. This paper finds a way to calculate the effect without those approximations, which should let it match better with experiments.

(As an aside, I’m a little confused by the claim that they’re going to all orders in αZ when it looks like they just consider one-loop diagrams…but this is probably just my ignorance, this is a corner of the field quite distant from my own.)

arXiv:2201.01674 : On Unfolded Approach To Off-Shell Supersymmetric Models

Supersymmetry is a relationship between two types of particles: fermions, which typically make up matter, and bosons, which are usually associated with forces. In realistic theories this relationship is “broken” and the two types of particles have different properties, but theoretical physicists often study models where supersymmetry is “unbroken” and the two types of particles have the same mass and charge. This paper finds a new way of describing some theories of this kind that reorganizes them in an interesting way, using an “unfolded” approach in which aspects of the particles that would normally be combined are given their own separate variables.

(This is another one I don’t know much about, this is the first time I’d heard of the unfolded approach.)

arXiv:2201.01679 : Geometric Flow of Bubbles

String theorists have conjectured that only some types of theories can be consistently combined with a full theory of quantum gravity, others live in a “swampland” of non-viable theories. One set of conjectures characterizes this swampland in terms of “flows” in which theories with different geometry can flow in to each other. The properties of these flows are supposed to be related to which theories are or are not in the swampland.

This paper writes down equations describing these flows, and applies them to some toy model “bubble” universes.

arXiv:2201.01697 : Graviton scattering amplitudes in first quantisation

This paper is a pedagogical one, introducing graduate students to a topic rather than presenting new research.

Usually in quantum field theory we do something called “second quantization”, thinking about the world not in terms of particles but in terms of fields that fill all of space and time. However, sometimes one can instead use “first quantization”, which is much more similar to ordinary quantum mechanics. There you think of a single particle traveling along a “world-line”, and calculate the probability it interacts with other particles in particular ways. This approach has recently been used to calculate interactions of gravitons, particles related to the gravitational field in the same way photons are related to the electromagnetic field. The approach has some advantages in terms of simplifying the results, which are described in this paper.

Classicality Has Consequences

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Last week, I mentioned some interesting new results in my corner of physics. I’ve now finally read the two papers and watched the recorded talk, so I can satisfy my frustrated commenters.

Quantum mechanics is a very cool topic and I am much less qualified than you would expect to talk about it. I use quantum field theory, which is based on quantum mechanics, so in some sense I use quantum mechanics every day. However, most of the “cool” implications of quantum mechanics don’t come up in my work. All the debates about whether measurement “collapses the wavefunction” are irrelevant when the particles you measure get absorbed in a particle detector, never to be seen again. And while there are deep questions about how a classical world emerges from quantum probabilities, they don’t matter so much when all you do is calculate those probabilities.

They’ve started to matter, though. That’s because quantum field theorists like me have recently started working on a very different kind of problem: trying to predict the output of gravitational wave telescopes like LIGO. It turns out you can do almost the same kind of calculation we’re used to: pretend two black holes or neutron stars are sub-atomic particles, and see what happens when they collide. This trick has grown into a sub-field in its own right, one I’ve dabbled in a bit myself. And it’s gotten my kind of physicists to pay more attention to the boundary between classical and quantum physics.

The thing is, the waves that LIGO sees really are classical. Any quantum gravity effects there are tiny, undetectably tiny. And while this doesn’t have the implications an expert might expect (we still need loop diagrams), it does mean that we need to take our calculations to a classical limit.

Figuring out how to do this has been surprisingly delicate, and full of unexpected insight. A recent example involves two papers, one by Andrea Cristofoli, Riccardo Gonzo, Nathan Moynihan, Donal O’Connell, Alasdair Ross, Matteo Sergola, and Chris White, and one by Ruth Britto, Riccardo Gonzo, and Guy Jehu. At first I thought these were two groups happening on the same idea, but then I noticed Riccardo Gonzo on both lists, and realized the papers were covering different aspects of a shared story. There is another group who happened upon the same story: Paolo Di Vecchia, Carlo Heissenberg, Rodolfo Russo and Gabriele Veneziano. They haven’t published yet, so I’m basing this on the Gonzo et al papers.

The key question each group asked was, what does it take for gravitational waves to be classical? One way to ask the question is to pick something you can observe, like the strength of the field, and calculate its uncertainty. Classical physics is deterministic: if you know the initial conditions exactly, you know the final conditions exactly. Quantum physics is not. What should happen is that if you calculate a quantum uncertainty and then take the classical limit, that uncertainty should vanish: the observation should become certain.

Another way to ask is to think about the wave as made up of gravitons, particles of gravity. Then you can ask how many gravitons are in the wave, and how they are distributed. It turns out that you expect them to be in a coherent state, like a laser, one with a very specific distribution called a Poisson distribution: a distribution in some sense right at the border between classical and quantum physics.

The results of both types of questions were as expected: the gravitational waves are indeed classical. To make this work, though, the quantum field theory calculation needs to have some surprising properties.

If two black holes collide and emit a gravitational wave, you could depict it like this:

All pictures from arXiv:2112.07556

where the straight lines are black holes, and the squiggly line is a graviton. But since gravitational waves are made up of multiple gravitons, you might ask, why not depict it with two gravitons, like this?

It turns out that diagrams like that are a problem: they mean your two gravitons are correlated, which is not allowed in a Poisson distribution. In the uncertainty picture, they also would give you non-zero uncertainty. Somehow, in the classical limit, diagrams like that need to go away.

And at first, it didn’t look like they do. You can try to count how many powers of Planck’s constant show up in each diagram. The authors do that, and it certainly doesn’t look like it goes away:

An example from the paper with Planck’s constants sprinkled around

Luckily, these quantum field theory calculations have a knack for surprising us. Calculate each individual diagram, and things look hopeless. But add them all together, and they miraculously cancel. In the classical limit, everything combines to give a classical result.

You can do this same trick for diagrams with more graviton particles, as many as you like, and each time it ought to keep working. You get an infinite set of relationships between different diagrams, relationships that have to hold to get sensible classical physics. From thinking about how the quantum and classical are related, you’ve learned something about calculations in quantum field theory.

That’s why these papers caught my eye. A chunk of my sub-field is needing to learn more and more about the relationship between quantum and classical physics, and it may have implications for the rest of us too. In the future, I might get a bit more qualified to talk about some of the very cool implications of quantum mechanics.

Science, Gifts Enough for Lifetimes

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Merry Newtonmas, Everyone!

In past years, I’ve compared science to a gift: the ideal gift for the puzzle-fan, one that keeps giving new puzzles. I think people might not appreciate the scale of that gift, though.

Bigger than all the creative commons Wikipedia images

Maybe you’ve heard the old joke that studying for a PhD means learning more and more about less and less until you know absolutely everything about nothing at all. This joke is overstating things: even when you’ve specialized down to nothing at all, you still won’t know everything.

If you read the history of science, it might feel like there are only a few important things going on at a time. You notice the simultaneous discoveries, like calculus from Newton and Liebniz and natural selection from Darwin and Wallace. You can get the impression that everyone was working on a few things, the things that would make it into the textbooks. In fact, though, there was always a lot to research, always many interesting things going on at once. As a scientist, you can’t escape this. Even if you focus on your own little area, on a few topics you care about, even in a small field, there will always be more going on than you can keep up with.

This is especially clear around the holiday season. As everyone tries to get results out before leaving on vacation, there is a tidal wave of new content. I have five papers open on my laptop right now (after closing four or so), and some recorded talks I keep meaning to watch. Two of the papers are the kind of simultaneous discovery I mentioned: two different groups noticing that what might seem like an obvious fact – that in classical physics, unlike in quantum, one can have zero uncertainty – has unexpected implications for our kind of calculations. (A third group got there too, but hasn’t published yet.) It’s a link I would never have expected, and with three groups coming at it independently you’d think it would be the only thing to pay attention to: but even in the same sub-sub-sub-field, there are other things going on that are just as cool! It’s wild, and it’s not some special quirk of my area: that’s science, for all us scientists. No matter how much you expect it to give you, you’ll get more, lifetimes and lifetimes worth. That’s a Newtonmas gift to satisfy anyone.

Calculations of the Past

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Last week was a birthday conference for one of the pioneers of my sub-field, Ettore Remiddi. I wasn’t there, but someone who was pointed me to some of the slides, including a talk by Stefano Laporta. For those of you who didn’t see my post a few weeks back, Laporta was one of Remiddi’s students, who developed one of the most important methods in our field and then vanished, spending ten years on an amazingly detailed calculation. Laporta’s talk covers more of the story, about what it was like to do precision calculations in that era.

“That era”, the 90’s through 2000’s, witnessed an enormous speedup in computers, and a corresponding speedup in what was possible. Laporta worked with Remiddi on the three-loop electron anomalous magnetic moment, something Remiddi had been working on since 1969. When Laporta joined in 1989, twenty-one of the seventy-two diagrams needed had still not been computed. They would polish them off over the next seven years, before Laporta dove in to four loops. Twenty years later, he had that four-loop result to over a thousand digits.

One fascinating part of the talk is seeing how the computational techniques change over time, as language replaces language and computer clusters get involved. As a student, Laporta learns a lesson we all often need: that to avoid mistakes, we need to do as little by hand as possible, even for something as simple as copying a one-line formula. Looking at his review of others’ calculations, it’s remarkable how many theoretical results had to be dramatically corrected a few years down the line, and how much still might depend on theoretical precision.

Another theme was one of Remiddi suggesting something and Laporta doing something entirely different, and often much more productive. Whether it was using the arithmetic-geometric mean for an elliptic integral instead of Gaussian quadrature, or coming up with his namesake method, Laporta spent a lot of time going his own way, and Remiddi quickly learned to trust him.

There’s a lot more in the slides that’s worth reading, including a mention of one of this year’s Physics Nobelists. The whole thing is an interesting look at what it takes to press precision to the utmost, and dedicate years to getting something right.

Of p and sigma

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Ask a doctor or a psychologist if they’re sure about something, and they might say “it has p<0.05”. Ask a physicist, and they’ll say it’s a “5 sigma result”. On the surface, they sound like they’re talking about completely different things. As it turns out, they’re not quite that different.

Whether it’s a p-value or a sigma, what scientists are giving you is shorthand for a probability. The p-value is the probability itself, while sigma tells you how many standard deviations something is away from the mean on a normal distribution. For people not used to statistics this might sound very complicated, but it’s not so tricky in the end. There’s a graph, called a normal distribution, and you can look at how much of it is above a certain point, measured in units called standard deviations, or “sigmas”. That gives you your probability.

Give it a try: how much of this graph is past the 1\sigma line? How about 2\sigma?

What are these numbers a probability of? At first, you might think they’re a probability of the scientist being right: of the medicine working, or the Higgs boson being there.

That would be reasonable, but it’s not how it works. Scientists can’t measure the chance they’re right. All they can do is compare models. When a scientist reports a p-value, what they’re doing is comparing to a kind of default model, called a “null hypothesis”. There are different null hypotheses for different experiments, depending on what the scientists want to test. For the Higgs, scientists looked at pairs of photons detected by the LHC. The null hypothesis was that these photons were created by other parts of the Standard Model, like the strong force, and not by a Higgs boson. For medicine, the null hypothesis might be that people get better on their own after a certain amount of time. That’s hard to estimate, which is why medical experiments use a control group: a similar group without the medicine, to see how much they get better on their own.

Once we have a null hypothesis, we can use it to estimate how likely it is that it produced the result of the experiment. If there was no Higgs, and all those photons just came from other particles, what’s the chance there would still be a giant pile of them at one specific energy? If the medicine didn’t do anything, what’s the chance the control group did that much worse than the treatment group?

Ideally, you want a small probability here. In medicine and psychology, you’re looking for a 5% probability, for p<0.05. In physics, you need 5 sigma to make a discovery, which corresponds to a one in 3.5 million probability. If the probability is low, then you can say that it would be quite unlikely for your result to happen if the null hypothesis was true. If you’ve got a better hypothesis (the Higgs exists, the medicine works), then you should pick that instead.

Note that this probability still uses a model: it’s the probability of the result given that the model is true. It isn’t the probability that the model is true, given the result. That probability is more important to know, but trickier to calculate. To get from one to the other, you need to include more assumptions: about how likely your model was to begin with, given everything else you know about the world. Depending on those assumptions, even the tiniest p-value might not show that your null hypothesis is wrong.

In practice, unfortunately, we usually can’t estimate all of those assumptions in detail. The best we can do is guess their effect, in a very broad way. That usually just means accepting a threshold for p-values, declaring some a discovery and others not. That limitation is part of why medicine and psychology demand p-values of 0.05, while physicists demand 5 sigma results. Medicine and psychology have some assumptions they can rely on: that people function like people, that biology and physics keep working. Physicists don’t have those assumptions, so we have to be extra-strict.

Ultimately, though, we’re all asking the same kind of question. And now you know how to understand it when we do.

Discovering New Elements, Discovering New Particles

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In school, you learn that the world around you is made up of chemical elements. There’s oxygen and nitrogen in the air, hydrogen and oxygen in water, sodium and chlorine in salt, and carbon in all living things. Other elements are more rare. Often, that’s because they’re unstable, due to radioactivity, like the plutonium in a bomb or americium in a smoke detector. The heaviest elements are artificial, produced in tiny amounts by massive experiments. In 2002, the heaviest element yet was found at the Joint Institute for Nuclear Research near Moscow. It was later named Oganesson, after the scientist who figured out how to make these heavy elements, Yuri Oganessian. To keep track of the different elements, we organize them in the periodic table like this:

In that same school, you probably also learn that the elements aren’t quite so elementary. Unlike the atoms imagined by the ancient Greeks, our atoms are made of smaller parts: protons and neutrons, surrounded by a cloud of electrons. They’re what give the periodic table its periodic structure, the way it repeats from row to row, with each different element having a different number of protons.

If your school is a bit more daring, you also learn that protons and neutrons themselves aren’t elementary. Each one is made of smaller particles called quarks: a proton of two “up quarks” and one “down quark”, and a neutron of two “down” and one “up”. Up quarks, down quarks, and electrons are all what physicists call fundamental particles, and they make up everything you see around you. Just like the chemical elements, some fundamental particles are more obscure than others, and the heaviest ones are all very unstable, produced temporarily by particle collider experiments. The most recent particle to be discovered was in 2012, when the Large Hadron Collider near Geneva found the Higgs boson. The Higgs boson is named after Peter Higgs, one of those who predicted it back in the 60’s. All the fundamental particles we know are part of something called the Standard Model, which we sometimes organize in a table like this:

So far, these stories probably sound similar. The experiments might not even sound that different: the Moscow experiment shoots a beam of high-energy calcium atoms at a target of heavy radioactive elements, while the Geneva one shoots a beam of high-energy protons at another beam of high-energy protons. With all those high-energy beams, what’s the difference really?

In fact there is a big different between chemical elements and fundamental particles, and between the periodic table and the Standard Model. The latter are fundamental, the former are not.

When they made new chemical elements, scientists needed to start with a lot of protons and neutrons. That’s why they used calcium atoms in their beam, and even heavier elements as their target. We know that heavy elements are heavy because they contain more protons and neutrons, and we can use the arrangement of those protons and neutrons to try to predict their properties. That’s why, even though only five or six oganesson atoms have been detected, scientists have some idea what kind of material it would make. Oganesson is a noble gas, like helium, neon, and radon. But calculations predict it is actually a solid at room temperature. What’s more, it’s expected to be able to react with other elements, something the other noble gases are very reluctant to do.

The Standard Model has patterns, just like the chemical elements. Each matter particle is one of three “generations”, each heavier and more unstable: for example, electrons have heavier relatives called muons, and still heavier ones called tauons. But unlike with the elements, we don’t know where these patterns come from. We can’t explain them with smaller particles, like we could explain the elements with protons and neutrons. We think the Standard Model particles might actually be fundamental, not made of anything smaller.

That’s why when we make them, we don’t need a lot of other particles: just two protons, each made of three quarks, is enough. With that, we can make not just new arrangements of quarks, but new particles altogether. Some are even heavier than the protons we started with: the Higgs boson is more than a hundred times as heavy as a proton! We can do this because, in particle physics, mass isn’t conserved: mass is just another type of energy, and you can turn one type of energy into another.

Discovering new elements is hard work, but discovering new particles is on another level. It’s hard to calculate which elements are stable or unstable, and what their properties might be. But we know the rules, and with enough skill and time we could figure it out. In particle physics, we don’t know the rules. We have some good guesses, simple models to solve specific problems, and sometimes, like with the Higgs, we’re right. But despite making many more than five or six Higgs bosons, we still aren’t sure it has the properties we expect. We don’t know the rules. Even with skill and time, we can’t just calculate what to expect. We have to discover it.