Tag Archives: mathematics

What Can Pi Do for You?

Tomorrow is Pi Day!

And what a Pi Day! 3/14/15 (if you’re in the US, Belize, Micronesia, some parts of Canada, the Philippines, or Swahili-speaking Kenya), best celebrated at 9:26:53, if you’re up by then. Grab a slice of pie, or cake if you really must, and enjoy!

If you don’t have some of your own, download this one!

Pi is great not just because it’s fun to recite digits and eat pastries, but because it serves a very important role in physics. That’s because, often, pi is one of the most “natural” ways to get larger numbers.

Suppose you’re starting with some sort of “natural” theory. Here I don’t mean natural in the technical sense. Instead, I want you to imagine a theory that has very few free parameters, a theory that is almost entirely fixed by mathematics.

Many physicists hope that the world is ultimately described by this sort of theory, but it’s hard to see in the world we live in. There are so many different numbers, from the tiny mass of the electron to the much larger mass of the top quark, that would all have to come from a simple, overarching theory. Often, it’s easier to get these numbers when they’re made out of factors of pi.

Why is pi easy to get?

In general, pi shows up a lot in physics and mathematics, and its appearance can be mysterious the uninitiated, as this joke related by Eugene Wigner in an essay I mentioned a few weeks ago demonstrates:

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

While it may sound silly, in a sense the population really is connected to the circumference of the circle. That’s because pi isn’t just about circles, pi is about volumes.

Take a bit to check out that link. Not just the area of a circle, but the volume of a sphere, and that of all sorts of higher-dimensional ball-shaped things, is calculated with the value of pi. It’s not just spheres, either: pi appears in the volume of many higher-dimensional shapes.

Why does this matter for physics? Because you don’t need a literal shape to get a volume. Most of the time, there aren’t literal circles and spheres giving you factors of pi…but there are abstract spaces, and they contain circles and spheres. A electric and magnetic fields might not be shaped like circles, but the mathematics that describes them can still make good use of a circular space.

That’s why, when I describe the mathematical formulas I work with, formulas that often produce factors of pi, mathematicians will often ask if they’re the volume of some particular mathematical space. It’s why Nima Arkani-Hamed is trying to understand related formulas by thinking of them as the volume of some new sort of geometrical object.

All this is not to say you should go and plug factors of pi together until you get the physical constants you want. Throw in enough factors of pi and enough other numbers and you can match current observations, sure…but you could also match anything else in the same way. Instead, it’s better to think of pi as an assistant: waiting in the wings, ready to translate a pure mathematical theory into the complicated mess of the real world.

So have a Happy Pi Day, everyone, and be grateful to our favorite transcendental number. The universe would be a much more boring place without it.

Explanations of Phenomena Are All Alike; Every Unexplained Phenomenon Is Unexplained in Its Own Way

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Vladimir Kazakov began his talk at ICTP-SAIFR this week with a variant of Tolstoy’s famous opening to the novel Anna Karenina: “Happy families are all alike; every unhappy family is unhappy in its own way.” Kazakov flipped the order of the quote, stating that while “Un-solvable models are each un-solvable in their own way, solvable models are all alike.”

In talking about solvable and un-solvable models, Kazakov was referring to a concept called integrability, the idea that in certain quantum field theories it’s possible to avoid the messy approximations of perturbation theory and instead jump straight to the answer. Kazakov was observing that these integrable systems seem to have a deep kinship: the same basic methods appear to work to understand all of them.

I’d like to generalize Kazakov’s point, and talk about a broader trend in physics.

Much has been made over the years of the “unreasonable effectiveness of mathematics in the natural sciences”, most notably in physicist Eugene Wigner’s famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. There’s a feeling among some people that mathematics is much better at explaining physical phenomena than one would expect, that the world appears to be “made of math” and that it didn’t have to be.

On the surface, this is a reasonable claim. Certain mathematical ideas, group theory for example, seem to pop up again and again in physics, sometimes in wildly different contexts. The history of fundamental physics has tended to see steady progress over the years, from clunkier mathematical concepts to more and more elegant ones.

Some physicists tend to be dismissive of this. Lee Smolin in particular seems to be under the impression that mathematics is just particularly good at providing useful approximations. This perspective links to his definition of mathematics as “the study of systems of evoked relationships inspired by observations of nature,” a definition to which Peter Woit vehemently objects. Woit argues what I think any mathematician would when presented by a statement like Smolin’s: that mathematics is much more than just a useful tool for approximating observations, and that contrary to physicists’ vanity most of mathematics goes on without any explicit interest in observing the natural world.

While it’s generally rude for physicists to propose definitions for mathematics, I’m going to do so anyway. I think the following definition is one mathematicians would be more comfortable with, though it may be overly broad: Mathematics is the study of simple rules with complex consequences.

We live in a complex world. The breadth of the periodic table, the vast diversity of life, the tangled webs of galaxies across the sky, these are things that display both vast variety and a sense of order. They are, in a rather direct way, the complex consequences of rules that are at heart very very simple.

Part of the wonder of modern mathematics is how interconnected it has become. Many sub-fields, once distinct, have discovered over the years that they are really studying different aspects of the same phenomena. That’s why when you see a proof of a three-hundred-year-old mathematical conjecture, it uses terms that seem to have nothing to do with the original problem. It’s why Woit, in an essay on this topic, quotes Edward Frenkel’s description of a particular recent program as a blueprint for a “Grand Unified Theory of Mathematics”. Increasingly, complex patterns are being shown to be not only consequences of simple rules, but consequences of the same simple rules.

Mathematics itself is “unreasonably effective”. That’s why, when faced with a complex world, we shouldn’t be surprised when the same simple rules pop up again and again to explain it. That’s what explaining something is: breaking down something complex into the simple rules that give rise to it. And as mathematics progresses, it becomes more and more clear that a few closely related types of simple rules lie behind any complex phenomena. While each unexplained fact about the universe may seem unexplained in its own way, as things are explained bit by bit they show just how alike they really are.

Where do you get all those mathematical toys?

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I’m at a conference at Caltech this week, so it’s going to be a shorter post than usual.

The conference is on something call the Positive Grassmannian, a precursor to Nima Arkani-Hamed’s much-hyped Amplituhedron. Both are variants of a central idea: take complicated calculations in physics and express them in terms of clean, well-defined mathematical objects.

Because of this, this conference is attended not just by physicists, but by mathematicians as well, and it’s been interesting watching how the two groups interact.

From a physics perspective, mathematicians are great because they give us so many useful tools! Many significant advances in my field happened because a physicist talked to a mathematician and learned that a problem that had stymied the physics world had already been solved in the math community.

This tends to lead to certain expectations among physicists. If a mathematician gives a talk at a physics conference, we expect them to present something we can use. Our ideal math talk is like when Q presents the gadgets at the beginning of a Bond movie: a ton of new toys with just enough explanation for us to use them to save the day in the second act.

Pictured: Mathematicians, through Physicist eyes

You may see the beginning of a problem here, once you realize that physicists are the James Bond in this analogy.

Physicists like to see themselves as the protagonists of their own stories. That’s true of every field, though, to some degree or another. And it’s certainly true of mathematicians.

Mathematicians don’t go to physics conferences just to be someone’s supporting cast. They do it because physics problems are interesting to them: by hearing what physicists are working on they hope to get inspiration for new mathematical structures, concepts jury-rigged together by physicists that represent corners that mathematics hasn’t yet explored. Their goal is to take home an idea that they can turn into something productive, gaining glory among their fellow mathematicians. And if that sounds familiar…

Pictured: Physicists, through Mathematician eyes

While it’s amusing to watch the different expectations go head-to-head, the best collaborations between physicists and mathematicians are those where both sides respect that the other is the protagonist of their own story. Allow for give-and-take, paying attention not just to what you find interesting but to what the other person does, without assuming a tired old movie script, and it’s possible to make great progress.

Of course, that’s true of life in general as well.

Feeling Perturbed?

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You might think of physics as the science of certainties and exact statements: action and reaction, F=ma, and all that. However, most calculations in physics aren’t exact, they’re approximations. This is especially true today, but it’s been true almost since the dawn of physics. In particular, approximations are performed via a method known as perturbation theory.

Perturbation theory is a trick used to solve problems that, for one reason or another, are too difficult to solve all in one go. It works by solving a simpler problem, then perturbing that solution, adjusting it closer to the target.

To give an analogy: let’s say you want to find the area of a circle, but you only know how to draw straight lines. You could start by drawing a square: it’s easy to find the area, and you get close to the area of the circle. But you’re still a long ways away from the total you’re aiming for. So you add more straight lines, getting an octagon. Now it’s harder to find the area, but you’re closer to the full circle. You can keep adding lines, each step getting closer and closer.

And so on.

This, broadly speaking, is what’s going on when particle physicists talk about loops. The calculation with no loops (or “tree-level” result) is the easier problem to solve, omitting quantum effects. Each loop then is the next stage, more complicated but closer to the real total.

There are, as usual, holes in this analogy. One is that it leaves out an important aspect of perturbation theory, namely that it involves perturbing with a parameter. When that parameter is small, perturbation theory works, but as it gets larger the approximation gets worse and worse. In the case of particle physics, the parameter is the strength of the forces involves, with weaker forces (like the weak nuclear force, or electromagnetism) having better approximations than stronger forces (like the strong nuclear force). If you squint, this can still fit the analogy: different shapes might be harder to approximate than the circle, taking more sets of lines to get acceptably close.

Where the analogy fails completely, though, is when you start approaching infinity. Keep adding more lines, and you should be getting closer and closer to the circle each time. In quantum field theory, though, this frequently is not the case. As I’ve mentioned before, while lower loops keep getting closer to the true (and experimentally verified) results, going all the way out to infinite loops results not in the full circle, but in an infinite result instead. There’s an understanding of why this happens, but it does mean that perturbation theory can’t be thought of in the most intuitive way.

Almost every calculation in particle physics uses perturbation theory, which means almost always we are just approximating the real result, trying to draw a circle using straight lines. There are only a few theories where we can bypass this process and look at the full circle. These are known as integrable theories. N=4 super Yang-Mills may be among them, one of many reasons why studying it offers hope for a deeper understanding of particle physics.

Numerics, or, Why can’t you just tell the computer to do it?

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When most people think of math, they think of the math they did in school: repeated arithmetic until your brain goes numb, followed by basic algebra and trig. You weren’t allowed to use calculators on most tests for the simple reason that almost everything you did could be done by a calculator in a fraction of the time.

Real math isn’t like that. Mathematicians handle proofs and abstract concepts, definitions and constructions and functions and generally not a single actual number in sight. That much, at least, shouldn’t be surprising.

What might be surprising is that even tasks which seem very much like things computers could do easily take a fair bit of human ingenuity.

In physics, I do a lot of integrals. For those of you unfamiliar with calculus, integrals can be thought of as the area between a curve and the x-axis.

Areas seem like the sort of thing it would be easy for a computer to find. Chop the space into little rectangles, add up all the rectangles under the curve, and if your rectangles are small enough you should get the right answer. Broadly, this is the method of numerical integration. Since computers can do billions of calculations per second, you can chop things up into billions of rectangles and get as close as you’d like, right?

Heck, ten is a lot. Can we just do ten?

For some curves, this works fine. For others, though…

Ten might not be enough for this one.

See how the left side of that plot goes off the chart? That curve goes to infinity. No matter how many rectangles you put on that side, you still won’t have any that are infinitely tall, so you’ll still miss that part of the curve.

Surprisingly enough, the area under this curve isn’t infinite. Do the integral correctly, and you get a result of 2. Set a computer to calculate this integral via the sort of naïve numerical integration discussed above though, and you’ll never find that answer. You need smarter methods: smart humans doing the math, or smart humans programming the computer.

Another way this can come up is if you’re adding up two parts of something that go to infinity in opposite directions. Try to integrate each part by itself and you’ll be stuck.

firstplot

But add them together, and you get something quite a bit more tractable.

Yeah, definitely a ten-rectangle job.

Numerical integration, and computers in general, are a very important tool in a scientist’s arsenal. But in order to use them, we have to be smart, and know what we’re doing. Knowing how to use our tools right can take almost as much expertise and care as working without tools.

So no, I can’t just tell the computer to do it.

How (Not) to Sum the Natural Numbers: Zeta Function Regularization

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$1+2+3+4+5+6+\ldots=-\frac{1}{12}$

If you follow Numberphile on YouTube or Bad Astronomy on Slate you’ve already seen this counter-intuitive sum written out. Similarly, if you follow those people or Sciencetopia’s Good Math, Bad Math, you’re aware that the way that sum was presented by Numberphile in that video was seriously flawed.

There is a real sense in which adding up all of the natural numbers (numbers 1, 2, 3…) really does give you minus twelve, despite all the reasons this should be impossible. However, there is also a real sense in which it does not, and cannot, do any such thing. To explain this, I’m going to introduce two concepts: complex analysis and regularization.

This discussion is not going to be mathematically rigorous, but it should give an authentic and accurate view of where these results come from. If you’re interested in the full mathematical details, a later discussion by Numberphile should help, and the mathematically confident should read Terence Tao’s treatment from back in 2010.

With that said, let’s talk about sums! Well, one sum in particular:

$\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+\frac{1}{6^s}+\ldots = \zeta(s)$

If s is greater than one, then each term in this infinite sum gets smaller and smaller fast enough that you can add them all up and get a number. That number is referred to as $\zeta(s)$ , the Riemann Zeta Function.

So what if s is smaller than one?

The infinite sum that I described doesn’t converge for s less than one. Add it up in any reasonable way, and it just approaches infinity. Put another way, the sum is not properly defined. But despite this, $\zeta(s)$ is not infinite for s less than one!

Now as you might object, we only defined the Riemann Zeta Function for s greater than one. How do we know anything at all about it for s less than one?

That is where complex analysis comes in. Complex analysis sounds like a made-up term for something unreasonably complicated, but it’s quite a bit more approachable when you know what it means. Analysis is the type of mathematics that deals with functions, infinite series, and the basis of calculus. It’s often contrasted with Algebra, which usually considers mathematical concepts that are discrete rather than smooth (this definition is a huge simplification, but it’s not very relevant to this post). Complex means that complex analysis deals with functions, not of everyday real numbers, but of complex numbers, or numbers with an imaginary part.

So what does complex analysis say about the Riemann Zeta Function?

One of the most impressive results of complex analysis is the discovery that if a function of a complex number is sufficiently smooth (the technical term is analytic) then it is very highly constrained. In particular, if you know how the function behaves over an area (technical term: open set), then you know how it behaves everywhere else!

If you’re expecting me to explain why this is true, you’ll be disappointed. This is serious mathematics, and serious mathematics isn’t the sort of thing you can give the derivation for in a few lines. It takes as much effort and knowledge to replicate a mathematical result as it does to replicate many lab results in science.

What I can tell you is that this sort of approach crops up in many places, and is part of a general theme. There is a lot you can tell about a mathematical function just by looking at its behavior in some limited area, because mathematics is often much more constrained than it appears. It’s the same sort of principle behind the work I’ve been doing recently.

In the case of the Riemann Zeta Function, we have a definition for s greater than one. As it turns out, this definition still works if s is a complex number, as long as the real part of s is greater than one. Using this information, the value of the Riemann Zeta Function for a large area (half of the complex numbers), complex analysis tells us its value for every other number. In particular, it tells us this:

$\zeta(-1)= -\frac{1}{12}$

If the Riemann Zeta Function is consistently defined for every complex number, then it must have this value when s is minus one.

If we still trusted the sum definition for this value of s, we could plug in -1 and get

$1+2+3+4+5+6+\ldots=-\frac{1}{12}$

Does that make this statement true? Sort of. It all boils down to a concept from physics called regularization.

In physics, we know that in general there is no such thing as infinity. With a few exceptions, nothing in nature should be infinite, and finite evidence (without mathematical trickery) should never lead us to an infinite conclusion.

Despite this, occasionally calculations in physics will give infinite results. Almost always, this is evidence that we are doing something wrong: we are not thinking hard enough about what’s really going on, or there is something we don’t know or aren’t taking into account.

Doing physics research isn’t like taking a physics class: sometimes, nobody knows how to do the problem correctly! In many cases where we find infinities, we don’t know enough about “what’s really going on” to correct them. That’s where regularization comes in handy.

Regularization is the process by which an infinite result is replaced with a finite result (made “regular”), in a way so that it keeps the same properties. These finite results can then be used to do calculations and make predictions, and so long as the final predictions are regularization independent (that is, the same if you had done a different regularization trick instead) then they are legitimate.

In string theory, one way to compute the required dimensions of space and time ends up giving you an infinite sum, a sum that goes 1+2+3+4+5+…. In context, this result is obviously wrong, so we regularize it. In particular, we say that what we’re really calculating is the Riemann Zeta Function, which we happen to be evaluating at -1. Then we replace 1+2+3+4+5+… with -1/12.

Now remember when I said that getting infinities is a sign that you’re doing something wrong? These days, we have a more rigorous way to do this same calculation in string theory, one that never forces us to take an infinite sum. As expected, it gives the same result as the old method, showing that the old calculation was indeed regularization independent.

Sometimes we don’t have a better way of doing the calculation, and that’s when regularization techniques come in most handy. A particular family of tricks called renormalization is quite important, and I’ll almost certainly discuss it in a future post.

So can you really add up all the natural numbers and get -1/12? No. But if a calculation tells you to add up all the natural numbers, and it’s obvious that the result can’t be infinite, then it may secretly be asking you to calculate the Riemann Zeta Function at -1. And that, as we know from complex analysis, is indeed -1/12.

Amplitudes on Paperscape

Elegance, Not So Mysterious

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You’ll often hear theoretical physicists in the media referring to one theory or another as “elegant”. String theory in particular seems to get this moniker fairly frequently.

It may often seem like mathematical elegance is some sort of mysterious sixth sense theorists possess, as inexplicable to the average person as color to a blind person. What’s “elegant” about string theory, after all?

Before explaining elegance, I should take a bit of time to say what it’s not. Elegance isn’t Occam’s razor. It isn’t naturalness, either. Both of those concepts have their own technical definitions.

Elegance, by contrast, is a much hazier, and yet much simpler, notion. It’s hazy enough that any definition could provoke arguments, but I can at least give you an approximate idea by telling you that an elegant theory is simple to describe, if you know the right terms. Often, it is simpler than the phenomenon that it explains.

How does this apply to something like string theory? String theory seems to be incredibly complicated: ten dimensions, curled up in a truly vast number of different ways, giving rise to whole spectrums of particles.

That said, the basic idea is quite simple. String theory asks the question: what if, in addition to fundamental point-particles (zero dimensional objects), there were fundamental objects of other dimensions? That idea leads to complicated consequences: if your theory is going to produce all the particles of the real world then you need the ten dimensions and the supersymmetry and yadda yadda. But the basic idea is simple to describe. An elegant theory can have very complicated consequences, but still be simple to describe.

This, broadly, is the sort of explanation theoretical physicists look for. Math is the kind of field where the same basic systems can describe very complex phenomena. Since theoretical physics is about describing the world in terms of math, the right explanation is usually the most elegant.

This can occasionally trip physicists up when they migrate to other careers. In biology, for example, the elegant solution is often not the right one, because evolution doesn’t care about elegance: evolution just grabs whatever is within reach. Financial systems and economics occasionally have similar problems. All this is to say that while elegance is an important thing for a physicist to strive for, sometimes we have to be careful about it.

What’s up with arXiv?

Blackboards, Again

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Recently I had the opportunity to give a blackboard talk. I’ve talked before about the value of blackboards, how they facilitate collaboration and can even be used to get work done. What I didn’t feel the need to explain was their advantages when giving a talk.

No, the blackboard behind me isn’t my talk.

When I mentioned I was giving a blackboard talk, some of my friends in other fields were incredulous.

“Why aren’t you using PowerPoint? Do you people hate technology?”

So why do theorists (and mathematicians) do blackboard talks, when many other fields don’t?

Typically, a chemist can’t bring chemicals to a talk. A biologist can’t bring a tank of fruit flies or zebrafish, and a psychologist probably shouldn’t bring in a passel of college student test subjects. As a theorist though, our test subjects are equations, and we can absolutely bring them into the room.

In the most ideal case, a talk by a theorist walks you through their calculation, reproducing it on the blackboard in enough detail that you can not only follow along, but potentially do the calculation yourself. While it’s possible to set up a calculation step by step in PowerPoint, you don’t have the same flexibility to erase and add to your equations, which becomes especially important if you need to clarify a point in response to a question.

Blackboards also often give you more space than a single slide. While your audience still only pays attention to a slide-sized area of the board at one time, you can put equations up in one area, move away, and then come back to them later. If you leave important equations up, people can remind themselves of them on their own time, without having to hold everybody up while you scroll back through the slides to the one they want to see.

Using a blackboard well is a fine art, and one I’m only beginning to learn. You have to know what to erase and what to leave up, when to pause to allow time to write or ask questions, and what to say while you’re erasing the board. You need to use all the quirks of the medium to your advantage, to show people not just what you did, but how and why you did it.

That’s why we use blackboards. And if you ask why we can’t do the same things with whiteboards, it’s because whiteboards are terrible. Everybody knows that.

4 gravitons

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