Tag Archives: mathematics

Thought Experiments, Minus the Thought

My second-favorite Newton fact is that, despite inventing calculus, he refused to use it for his most famous work of physics, the Principia. Instead, he used geometrical proofs, tweaked to smuggle in calculus without admitting it.

Essentially, these proofs were thought experiments. Newton would start with a standard geometry argument, one that would have been acceptable to mathematicians centuries earlier. Then, he’d imagine taking it further, pushing a line or angle to some infinite point. He’d argue that, if the proof worked for every finite choice, then it should work in the infinite limit as well.

These thought experiments let Newton argue on the basis of something that looked more rigorous than calculus. However, they also held science back. At the time, only a few people in the world could understand what Newton was doing. It was only later, when Newton’s laws were reformulated in calculus terms, that a wider group of researchers could start doing serious physics.

What changed? If Newton could describe his physics with geometrical thought experiments, why couldn’t everyone else?

The trouble with thought experiments is that they require careful setup, setup that has to be thought through for each new thought experiment. Calculus took Newton’s geometrical thought experiments, and took out the need for thought: the setup was automatically a part of calculus, and each new researcher could build on their predecessors without having to set everything up again.

This sort of thing happens a lot in science. An example from my field is the scattering matrix, or S-matrix.

The S-matrix, deep down, is a thought experiment. Take some particles, and put them infinitely far away from each other, off in the infinite past. Then, let them approach, close enough to collide. If they do, new particles can form, and these new particles will travel out again, infinite far away in the infinite future. The S-matrix then is a metaphorical matrix that tells you, for each possible set of incoming particles, what the probability is to get each possible set of outgoing particles.

In a real collider, the particles don’t come from infinitely far away, and they don’t travel infinitely far before they’re stopped. But the distances are long enough, compared to the sizes relevant for particle physics, that the S-matrix is the right idea for the job.

Like calculus, the S-matrix is a thought experiment minus the thought. When we want to calculate the probability of particles scattering, we don’t need to set up the whole thought experiment all over again. Instead, we can start by calculating, and over time we’ve gotten very good at it.

In general, sub-fields in physics can be divided into those that have found their S-matrices, their thought experiments minus thought, and those that have not. When a topic has to rely on thought experiments, progress is much slower: people argue over the details of each setup, and it’s difficult to build something that can last. It’s only when a field turns the corner, removing the thought from its thought experiments, that people can start making real collaborative progress.

What Does It Mean to Know the Answer?

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My sub-field isn’t big on philosophical debates. We don’t tend to get hung up on how to measure an infinite universe, or in arguing about how to interpret quantum mechanics. Instead, we develop new calculation techniques, which tends to nicely sidestep all of that.

If there’s anything we do get philosophical about, though, any question with a little bit of ambiguity, it’s this: What counts as an analytic result?

“Analytic” here is in contrast to “numerical”. If all we need is a number and we don’t care if it’s slightly off, we can use numerical methods. We have a computer use some estimation trick, repeating steps over and over again until we have approximately the right answer.

“Analytic”, then, refers to everything else. When you want an analytic result, you want something exact. Most of the time, you don’t just want a single number: you want a function, one that can give you numbers for whichever situation you’re interested in.

It might sound like there’s no ambiguity there. If it’s a function, with sines and cosines and the like, then it’s clearly analytic. If you can only get numbers out through some approximation, it’s numerical. But as the following example shows, things can get a bit more complicated.

Suppose you’re trying to calculate something, and you find the answer is some messy integral. Still, you’ve simplified the integral enough that you can do numerical integration and get some approximate numbers out. What’s more, you can express the integral as an infinite series, so that any finite number of terms will get close to the correct result. Maybe you even know a few special cases, situations where you plug specific numbers in and you do get an exact answer.

It might sound like you only know the answer numerically. As it turns out, though, this is roughly how your computer handles sines and cosines.

When your computer tries to calculate a sine or a cosine, it doesn’t have access to the exact solution all of the time. It does have some special cases, but the rest of the time it’s using an infinite series, or some other numerical trick. Type in a random sine into your calculator and it will be just as approximate as if you did a numerical integration.

So what’s the real difference?

Rather than how we get numbers out, think about what else we know. We know how to take derivatives of sines, and how to integrate them. We know how to take limits, and series expansions. And we know their relations to other functions, including how to express them in terms of other things.

If you can do that with your integral, then you’ve probably got an analytic result. If you can’t, then you don’t.

What if you have only some of the requirements, but not the others? What if you can take derivatives, but don’t know all of the identities between your functions? What if you can do series expansions, but only in some limits? What if you can do all the above, but can’t get numbers out without a supercomputer?

That’s where the ambiguity sets in.

In the end, whether or not we have the full analytic answer is a matter of degree. The closer we can get to functions that mathematicians have studied and understood, the better grasp we have of our answer and the more “analytic” it is. In practice, we end up with a very pragmatic approach to knowledge: whether we know the answer depends entirely on what we can do with it.

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?

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.

Trust Your Notation as Far as You Can Prove It

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Calculus contains one of the most famous examples of physicists doing something silly that irritates mathematicians. See, there are two different ways to write down a derivative, both dating back to the invention of calculus: Newton’s method, and Leibniz’s method.

Newton cared a lot about rigor (enough that he actually published his major physics results without calculus because he didn’t think calculus was rigorous enough, despite inventing it himself). His notation is direct and to the point: if you want to take the derivative of a function f of x, you write,

$f'(x)$

Leibniz cared a lot less about rigor, and a lot more about the scientific community. He wanted his notation to be useful and intuitive, to be the sort of thing that people would pick up and run with. To write a derivative in Leibniz notation, you write,

$\frac{df}{dx}$

This looks like a fraction. It’s really, really tempting to treat it like a fraction. And that’s the point: it’s to tell you that treating it like a fraction is often the right thing to do. In particular, you can do something like this,

$y=\frac{df}{dx}$

$y dx=df$

$\int y dx=\int df$

and what you did actually makes a certain amount of sense.

The tricky thing here is that it doesn’t always make sense. You can do these sorts of tricks up to a point, but you need to be aware that they really are just tricks. Take the notation too seriously, and you end up doing things you aren’t really allowed to do. It’s always important to stay aware of what you’re really doing.

There are a lot of examples of this kind of thing in physics. In quantum field theory, we use path integrals. These aren’t really integrals…but a lot of the time, we can treat them as such. Operators in quantum mechanics can be treated like numbers and multiplied…up to a point. A friend of mine was recently getting confused by operator product expansions, where similar issues crop up.

I’ve found two ways to clear up this kind of confusion. One is to unpack your notation: go back to the definitions, and make sure that what you’re doing really makes sense. This can be tedious, but you can be confident that you’re getting the right answer.

The other option is to stop treating your notation like the familiar thing it resembles, and start treating it like uncharted territory. You’re using this sort of notation to remind you of certain operations you can do, certain rules you need to follow. If you take those rules as basic, you can think about what you’re doing in terms of axioms rather than in terms of the suggestions made by your notation. Follow the right axioms, and you’ll stay within the bounds of what you’re actually allowed to do.

Either way, familiar-looking notation can help your intuition, making calculations more fluid. Just don’t trust it farther than you can prove it.

Pi in the Sky Science Journalism

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You’ve probably seen it somewhere on your facebook feed, likely shared by a particularly wide-eyed friend: pi found hidden in the hydrogen atom!

From the headlines, this sounds like some sort of kabbalistic nonsense, like finding the golden ratio in random pictures.

Read the actual articles, and the story is a bit more reasonable. The last two I linked above seem to be decent takes on it, they’re just saddled with ridiculous headlines. As usual, I blame the editors. This time, they’ve obscured an interesting point about the link between physics and mathematics.

So what does “pi found hidden in the hydrogen atom” actually mean?

It doesn’t mean that there’s some deep importance to the number pi in nature, beyond its relevance in mathematics in general. The reason that pi is showing up here isn’t especially deep.

It isn’t trivial either, though. I’ve seen a few people whose first response to this article was “of course they found pi in the hydrogen atom, hydrogen atoms are spherical!” That’s not what’s going on here. The connection isn’t about the shape of the hydrogen atom, it’s about one particular technique for estimating its energy.

Carl Hagen is a physicist at the University of Rochester who was teaching a quantum mechanics class in which he taught a well-known approximation technique called the variational principle. Specifically, he had his students apply this technique to the hydrogen atom. The nice thing about the hydrogen atom is that it’s one of the few atoms simple enough that it’s possible to find its energy levels exactly. The exact calculation can then be compared to the approximation.

What Hagen noticed was that this approximation was surprisingly good, especially for high energy states for which it wasn’t expected to be. In the end, working with Rochester math professor Tamar Friedmann, he figured out that the variational principle was making use of a particular identity between a type of mathematical functions, called Gamma functions, that are quite common in physics. Using those Gamma functions, the two researchers were able to re-derive what turned out to be a 17th century formula for pi, giving rise to a much cleaner proof for that formula than had been known previously.

So pi isn’t appearing here because “the hydrogen atom is a sphere”. It’s appearing because pi appears all over the place in physics, and because in general, the same sorts of structures appear again and again in mathematics.

Pi’s appearance in the hydrogen atom is thus not very special, regardless. What is a little bit special is the fact that, using the hydrogen atom, these folks were able to find a cleaner proof of an old approximation for pi, one that mathematicians hadn’t found before.

That, if anything, is the interesting part of this news story, but it’s also part of a broader trend, one in which physicists provide “physics proofs” for mathematical results. One of the more famous accomplishments of string theory is a class of “physics proofs” of this sort, using a principle called mirror symmetry.

The existence of “physics proofs” doesn’t mean that mathematics is secretly constrained by the physical world. Rather, they’re a result of the fact that physicists are interested in different aspects of mathematics, and in general are a bit more reckless in using approximations that haven’t been mathematically vetted. A physicist can sometimes prove something in just a few lines that mathematicians would take many pages to prove, but usually they do this by invoking a structure that would take much longer for a mathematician to define. As physicists, we’re building on the shoulders of other physicists, using concepts that mathematicians usually don’t have much reason to bother with. That’s why it’s always interesting when we find something like the Amplituhedron, a clean mathematical concept hidden inside what would naively seem like a very messy construction. It’s also why “physics proofs” like this can happen: we’re dealing with things that mathematicians don’t naturally consider.

So please, ignore the pi-in-the-sky headlines. Some physicists found a trick, some mathematicians found it interesting, the hydrogen atom was (quite tangentially) involved…and no nonsense needs to be present.

Using Effective Language

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Physicists like to use silly names for things, but sometimes it’s best to just use an everyday word. It can trigger useful intuitions, and it makes remembering concepts easier. What gets confusing, though, is when the everyday word you use has a meaning that’s not quite the same as the colloquial one.

“Realism” is a pretty classic example, where Bell’s elegant use of the term in quantum mechanics doesn’t quite match its common usage, leading to inevitable confusion whenever it’s brought up. “Theory” is such a useful word that multiple branches of science use it…with different meanings! In both cases, the naive meaning of the word is the basis of how it gets used scientifically…just not the full story.

There are two things to be wary of here. First, those of us who communicate science must be sure to point out when a word we use doesn’t match its everyday meaning, to guide readers’ intuitions away from first impressions to understand how the term is used in our field. Second, as a reader, you need to be on the look-out for hidden technical terms, especially when you’re reading technical work.

I remember making a particularly silly mistake along these lines. It was early on in grad school, back when I knew almost nothing about quantum field theory. One of our classes was a seminar, structured so that each student would give a talk on some topic that could be understood by the whole group. Unfortunately, some grad students with deeper backgrounds in theoretical physics hadn’t quite gotten the memo.

It was a particular phrase that set me off: “This theory isn’t an effective theory”.

My immediate response was to raise my hand. “What’s wrong with it? What about this theory makes it ineffective?”

The presenter boggled for a moment before responding. “Well, it’s complete up to high energies…it has no ultraviolet divergences…”

“Then shouldn’t that make it even more effective?”

After a bit more of this back-and-forth, we finally cleared things up. As it turns out, “effective field theory” is a technical term! An “effective field theory” is only “effectively” true, describing physics at low energies but not at high energies. As you can see, the word “effective” here is definitely pulling its weight, helping to make the concept understandable…but if you don’t recognize it as a technical term and interpret it literally, you’re going to leave everyone confused!

Over time, I’ve gotten better at identifying when something is a technical term. It really is a skill you can learn: there are different tones people use when speaking, different cadences when writing, a sense of uneasiness that can clue you in to a word being used in something other than its literal sense. Without that skill, you end up worried about mathematicians’ motives for their evil schemes. With it, you’re one step closer to what may be the most important skill in science: the ability to recognize something you don’t know yet.

Bras and Kets, Trading off Instincts

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Some physics notation is a joke, but that doesn’t mean it shouldn’t be taken seriously.

Take bras and kets. On the surface, as silly a physics name as any. If you want to find the probability that a state in quantum mechanics turns into another state, you write down a “bracket” between the two states:

$\langle a | b\rangle$

This leads, with typical physics logic, to the notation for the individual states: separate out the two parts, into a “bra” and a “ket”:

$\langle a|$ , $|b\rangle$

It’s kind of a dumb joke, and it annoys the heck out of mathematicians. Not for the joke, of course, mathematicians probably have worse.

Mathematicians are annoyed when we use complicated, weird notation for something that looks like a simple, universal concept. Here, we’re essentially just taking inner products of vectors, something mathematicians have been doing in one form or another for centuries. Yet rather than use their time-tested notation we use our own silly setup.

There’s a method to the madness, though. Bras and kets are handy for our purposes because they allow us to leverage one of the most powerful instincts of programmers: the need to close parentheses.

In programming, various forms of parentheses and brackets allow you to isolate parts of code for different purposes. One set of lines might only activate under certain circumstances, another set of brackets might make text bold. But in essentially every language, you never want to leave an open parenthesis. Doing so is almost always a mistake, one that leaves the rest of your code open to whatever isolated region you were trying to create.

Open parentheses make programmers nervous, and that’s exactly what “bras” and “kets” are for. As it turns out, the states represented by “bras” and “kets” are in a certain sense un-measurable: the only things we can measure are the brackets between them. When people say that in quantum mechanics we can only predict probabilities, that’s a big part of what they mean: the states themselves mean nothing without being assembled into probability-calculating brackets.

This ends up making “bras” and “kets” very useful. If you’re calculating something in the real world and your formula ends up with a free “bra” or a “ket”, you know you’ve done something wrong. Only when all of your bras and kets are assembled into brackets will you have something physically meaningful. Since most physicists have done some programming, the programmer’s instinct to always close parentheses comes to the rescue, nagging until you turn your formula into something that can be measured.

So while our notation may be weird, it does serve a purpose: it makes our instincts fit the counter-intuitive world of quantum mechanics.

Lewis Carroll, Anti-String Theorist?

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You all know the real meaning of Alice in Wonderland, right?

No, I’m not talking about drugs, or darker things. I’m talking about math!

The 19th century was a time of great changes in mathematics, and Charles Dodgson, pen name Lewis Carroll, was opposed to almost all of it. A very traditional mathematician, Dodgson thought of Euclid’s Elements as the pinnacle of mathematical reasoning. Non-Euclidean geometry, symbolic algebra, complex numbers, all of these were viewed by Dodgson as nonsense, perverting students away from the study of Euclidean geometry and arithmetic, subjects that actually described the real world.

Scholars of Dodgson/Carroll’s writing have posited that the craziness of Wonderland was intended to parody the craziness Dodgson saw in mathematics. When Alice encounters the Caterpillar, she grows and shrinks non-uniformly as the Caterpillar advises her to “keep her temper”. “Temper” here refers not to anger, but to ratios between different parts: something preserved in Euclidean geometry but potentially violated by symbolic algebra. Similarly, the frantic rotations around the table by the Mad Hatter and his tea party are thought to represent imaginary numbers and quaternions, concepts used to understand rotation which had to postulate extra dimensions to do so.

Dodgson was on the wrong side of history, and today mathematics deals with even more abstract concepts. What amuses me, though, is how well Dodgson’s parodies match certain criticisms of string theory.

String theorists often study theories with two properties not found in the real world: conformal symmetry and supersymmetry.

In a theory with conformal symmetry, distances aren’t fixed. Different parts of objects can grow and shrink different amounts, and the theory will still predict the same physical behavior. The only restriction is that angles need to be preserved: two lines that meet at a given angle must meet at the same angle after transformation. In other words, keep your temper.

Alice, undergoing a conformal transformation.

I’ve talked about supersymmetry before. A supersymmetric theory can be “turned” in certain ways, related to exchanging different types of particles. If you “turn” the theory twice in the same “direction”, you get back to where you started, sort of like how if you square the imaginary number i you get back to the real number -1. Supersymmetry sees a group of particles and declares that “it’s time to change places!”

I thought the string theory skeptics among my readers might find the parallels here amusing. With parody, if not always with science, the best work was often done long, long ago.

Calculus Is About Pokemon

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Occasionally, people tell me that calculus was when they really gave up on math. It’s a pity, because for me calculus was the first time math really started to become fun. After all, it’s when math introduces the Pokemon.

What Pokemon? Why, the special functions of course.

By special functions I mean things like $\sin x$ , $\cos x$ , $e^x$ , and $\ln x$ . Like Pokemon, these guys come in a bewildering variety. And in calculus, you learn that they, like Pokemon, can evolve.

$x$ integrates into $\frac{1}{2}x^2$ !

$\frac{1}{x}$ integrates into $\ln x$ !

$\sin x$ integrates into $-\cos x$ , and $\cos x$ integrates into… $\sin x$ .

Ok, the analogy isn’t perfect. Pokemon don’t evolve back into themselves. But the same things that make Pokemon so appealing are precisely why calculus was such a breath of fresh air. Suddenly, there was a grand diversity of new things, and those new things were related.

College gave me new Pokemon, in the form of the Bessel functions. Nowadays, I work with a group of functions called Polylogarithms, and they’re even more like Pokemon. Logarithms are like the baby Pokemon of the Polylogarithms, integrating into Dilogarithms. Dilogarithms integrate into Trilogarithms, and so on.

Polylogarithms, in turn, evolve into Poliwrath

To this day, the talks I enjoy the most are those that show me new special functions, or new relations between old ones. If a talk shows me a new use of multiple zeta values, or new types of Polylogarithm, it’s not just teaching me new physics or mathematics: it’s expanding my Pokemon collection.

4 gravitons

Stories about physics from someone who's been there