Achieving Transcendence: The Physicist Way

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.

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5 thoughts on “Achieving Transcendence: The Physicist Way

  1. Konstantinos

    Very interesting! I had no idea of such a concept (then again, I am an applied math person).

    I’m curious though. What can you infer about a function or a number from that degree of transcendentality? What’s the difference between \pi and \pi^2 w.r.t that?

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    1. 4gravitonsandagradstudent Post author

      The trick is not so much learning something about a function from its degree of transcendentality, but rather using a predicted degree of transcendentality to predict what functions you expect. So for example, there is a quantity called the six point remainder function that is guaranteed to have degree of transcendentality of two times the number of loops at which you calculate it.

      For a simpler example, suppose you have a function with one variable, and you know it has degree of transcendentality one. Then its behavior is restricted, it has to be made of just logarithms and pi. If it has degree of transcendentality two, it can contain logarithms squared, pi squared, pi times logarithms, or dilogarithms (mentioned in the above post). And so on. So rather than having to try absolutely every function ever you’ve got a much better handle on what to expect.

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      1. Konstantinos

        I see! It’s nice that you can do such a characterization then. It must be very useful in the cases you mention.

        It also tells me that you guys must be working a lot with logarithms!

        Thanks for the reply!

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