Tag Archives: mathematics

Blackboards

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As a college student, I already knew that theoretical physicists weren’t like how they were portrayed in movies. They didn’t wear lab coats, or have universally frizzy, unkempt white hair. I knew they didn’t have labs, or plot to take over the world. And I was pretty sure that they didn’t constantly use blackboards.

After all, blackboards are a teaching tool. They’re nice for getting equations up so that the guy way in the back can see them. But if you were actually doing a real calculation, surely you’d prefer paper, or a computer, or some other method that doesn’t involve an unkempt scrawl and a heap of loose white dust all over your clothing.

Right?

Right?

Over the last few years I’ve come to appreciate the value of blackboards. Blackboards actually can be used for calculations. You don’t want to use them all the time, but there are times when it’s useful to have a lot of room on a page, to be able to make notes and structure the board around concepts. More importantly, though, there is a third function that I didn’t even consider back in college. Between calculation and teaching, there is collaboration.

Go to a physics or math department, and you’ll find blackboards on the walls. You’ll find them not just in classrooms, but in offices, and occasionally in corridors. Go to a high-class physics location like the Perimeter Institute or the Simons Center, and they’ll brag to you about how many blackboards they have strewn around their common areas.

The purpose of these blackboards is to facilitate conversation. If you want to explain your work to someone else and you aren’t using a blog post, you need space to write in a way that you can both see what you’re doing. Blackboards are ideal for that sort of conversation, and as such are essential for collaboration and communication among scientists.

What about whiteboards? Well, whiteboards are just evil, obviously.

Achieving Transcendence: The Physicist Way

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I wanted to shed some light on something I’ve been working on recently, but I realized that a little background was needed to explain some of the ideas. As such, this post is going to be a bit more math-y than usual, but I hope it’s educational!

Pi is special. Familiar to all through the area of a circle \pi r^2, pi is particularly interesting in that you cannot write an algebra equation made up of whole numbers whose solution is pi. While you can easily get fractions (3x=4 gives x=\frac{4}{3}) and even many irrational numbers (x^2=2 gives x=\sqrt{2}), pi is one of a set of numbers that it is impossible to get. These special numbers transcend other numbers, in that you cannot use more everyday numbers to get to them, and as such mathematicians call them transcendental numbers.

In addition to transcendental numbers, you can have transcendental functions. Transcendental functions are functions that can take in a normal number and produce a transcendental number. For example, you may be aware of the delightful equation below:

e^{i \pi}=-1

We can manipulate both sides of this equation by taking the natural logarithm, \ln, to find

i\pi=\ln(-1)

This tells us that the natural logarithm function can take a (negative) whole number (-1) and give us a transcendental number (pi). This means that the natural logarithm is a transcendental function.

There are many other transcendental functions. In addition to logarithms, there are a whole host of related functions called the polylogarithms, and even more generally the harmonic polylogarithms. All of these functions can take in whole numbers like -1 or 1 and give transcendental numbers.

Here physicists introduce a concept called degree of transcendentality, or transcendental weight, which we use to measure how transcendental a number or a function is. Pi (and functions that can give pi, like the natural logarithm) have transcendental weight one. Pi squared has transcendental weight two. Pi cubed (and another number called \zeta(3)) have transcendental weight three. And so on.

Note here that, according to mathematicians, there is no rigorous way that a number can be “more transcendental” than another number. In the case of some of these numbers, like \zeta(5), it hasn’t even been proven that the number is actually transcendental at all! However, physicists still use the concept of transcendental weight because it allows us to classify and manipulate a common and useful set of functions. This is an example of the differences in methods and standards between physicists and mathematicians, even when they are working on similar things.

In what way are these functions common and useful? Well it turns out that in N=4 super Yang-Mills many calculated results are not only made up of these polylogarithms, they have a particular (fixed) transcendental weight. In situations when we expect this to be true, we can use our knowledge to guess most, or even all, of the result without doing direct calculations. That’s immensely useful, and it’s a big part of what I’ve been doing recently.

Physics and its (Ridiculously One-Sided) Search for a Nemesis

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Maybe it’s arrogance, or insecurity. Maybe it’s due to viewing themselves as the arbiters of good and bad science. Perhaps it’s just because, secretly, every physicist dreams of being a supervillain.

Physicists have a rivalry, you see. Whether you want to call it an archenemy, a nemesis, or even a kismesis, there is another field of study that physicists find so antithetical to everything they believe in that it crops up in their darkest and most shameful dreams.

What field of study? Well, pretty much all of them, actually.

Won’t you be my Kismesis?

Chemistry

A professor of mine once expressed the following sentiment:

“I have such respect for chemists. They accomplish so many things, while having no idea what they are doing!”

Disturbingly enough, he actually meant this as a compliment. Physicists’ relationship with chemists is a bit like a sibling rivalry. “Oh, isn’t that cute! He’s just playing with chemicals. Little guy doesn’t know anything about atoms, and yet he’s just sluggin’ away…wait, why is it working? What? How did you…I mean, I could have done that. Sure.”

Biology

They study all that weird, squishy stuff. They get to do better mad science. And somehow they get way more funding than us, probably because the government puts “improving lives” over “more particles”. Luckily, we have a solution to the problem.

Mathematics

Saturday Morning Breakfast Cereal has a pretty good take on this. Mathematicians are rigorous…too rigorous. They never let us have any fun, even when it’s totally fine, and everyone thinks they’re better than us. Well they’re not! Neener neener.

Computer Science

I already covered math, didn’t I?

Engineering

Think about how mathematicians think about physicists, and you’ll know how physicists think about engineers. They mangle our formulas, ignoring our pristine general cases for silly criteria like “ease of use” and “describing the everyday world”. Just lazy!

Philosophy

What do these guys even study? I mean, what’s the point of metaphysics? We’ve covered that, it’s called physics! And why do they keep asking what quantum mechanics means?

These guys have an annoying habit of pointing out moral issues with things like nuclear power plants and worry entirely too much about world-destroying black holes. They’re also our top competition for GRE scores.

Economics

So, what do you guys use real analysis for again? Pretending to be math-based science doesn’t make you rigorous, guys.

Psychology

We point out that surveys probably don’t measure anything, and that you can’t take the average of “agree” and “strongly agree”. Plus, if you’re a science, where is your F=ma?

They point out that we don’t actually know anything about how psychology research actually works, and that we seem to think that all psychologists are Freud. Then they ask us to look at just how fuzzy the plots we get from colliders actually are.

The argument escalates from there, often ending with frenzied makeout sessions.

Geology?  Astronomy?

Hey, we want a nemesis, but we’re not that desperate.eyH

Ansatz: Progress by Guesswork

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I’ve talked before about how hard traditional Quantum Field Theory is. Building things up step by step is slow and inefficient. And like any slow and inefficient process, there is a quicker way. An easier way. A…riskier way.

You guess.

Guess is such an ugly word, though…so let’s call it an ansatz.

Ansatz is a word of German origin. In German, it is part of various idiomatic expressions, where it can refer to an approach, an attempt, or a starting point. When physicists and mathematicians use the term ansatz, they mean a combination of all of these.

An ansatz is an approach in that it is a way of finding a solution to a problem without using more general, inefficient methods. Rather than approaching problems starting from the question, an ansatz approaches problems by starting with an answer, or rather, an attempt at an answer.

An ansatz is an attempt in that it serves as researcher’s best first guess at what the answer is, based on what they know about it. This knowledge can come from several sources. Sometimes, the question constrains the answer, ruling out some possibilities or restricting the output to a particular form. Usually, though, the attempt of an ansatz goes beyond this, incorporating the scientist’s experience as to what sorts of answers similar questions have had in the past, even if it isn’t understood yet why those sorts of answers are common. With information from both of these sources, a scientist comes up with a preliminary guess, or ansatz, as to answer to the problem at hand.

What if the answer is wrong, though? The key here is that an ansatz is only a starting point. Rather than being a full answer with all the details filled in, an ansatz generally leaves some parameters free. These free parameters represent unknowns, and it is up to further tests to fix their values and complete the answer. These tests can be experimental, but they can also be mathematical: often there are restrictions on possible answers that are difficult to apply when creating a first guess, but easier to apply when one has only a few parameters to fix. In order to avoid the risk of finding an ansatz that only works by coincidence, many more tests are done than there are parameters. That way, if the guess behind the ansatz is wrong, then some of the tests will give contradictory rules for the values of the parameters, and you’ll know that it’s time to go back and find a better guess.

In the end, this approach, using your first attempt as a starting point, should end up with only a few parameters free, ideally none at all. One way or another, you have figured out a lot about your question just by guessing the answer!

The use of ansatzes is quite common in theoretical physics. Some of the most interesting problem either can’t be solved or are tedious to solve through traditional means. The only way to make progress, to go beyond what everyone else can already do, is to notice a pattern, make a guess, and hope you get lucky. Well, not just a guess: an ansatz.

Why I Am Not A Mathematician

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(No relation to Russel’s Why I Am Not A Christian. Well, not much.)

I am a theorist. I study theories. Not the well-supported theories of the AAAS definition, but simply potential lists of particles, and lists that, further, are almost certainly not “true”.

Most people find that disconcerting. Used to thinking of scientists as people who investigate the real world, people whose ideas are always tested in the fire of experiment, the idea of a scientist whose work has no direct connection to the real world is a major source of cognitive dissonance…for at least a few minutes. After that, a light dawns in most people’s heads, as they turn to me with a sigh of relief and say,

“Oh. So you’re a Mathematician.”

No.

No, I am not a Mathematician. There is a difference, subtle but vast, between what I do and a mathematician does.

An illustrative example: Quantum Electro-Dynamics, or QED, is the most successful theory in the entirety of science. Yes, I do mean the entirety of science. Quantum Electro-Dynamics, the theory of how electrons and light behave, agrees with experiments to ten decimal places. Ten digits of detail, predicted then observed. That’s more confirmed accuracy than anything else in physics, in science at all, has ever achieved.

And if you ask a mathematician who specializes in this sort of thing, they’ll tell you that QED probably doesn’t exist.

Now, by this they don’t mean that electrons don’t exist, or that light doesn’t exist. What they mean is that, if you follow the theory’s implications all the way, you get a contradiction. You can calculate each step of the way, getting reasonable results each time, results that keep agreeing perfectly with experiments…but if you were to go all the way, off to infinity, you get results that make your whole theory stop making any sort of reasonable sense.

But as physicists, we keep using it. Because before reaching infinity, for any real calculation, it works. Perfectly.

That’s the difference between a theoretical physicist and a mathematician: for a mathematician, everything must be completely rigorous, and every implication, out to infinity, has to be vetted. For a physicist, if a theory gives reasonable results, we don’t really care whether it is completely clear how it works mathematically. We use physical reasoning, using concepts that work in the physical world, even if we’re studying a theory that doesn’t actually exist in the physical world. And while that sounds like a poor way to study abstract ideas, it allows us to take risks mathematicians can’t, which sometimes means we can make discoveries that even the mathematicians find interesting.