Monday, February 20, 2017


Many of the foundations of wave mechanics are based on the analyses and equations that Rayleigh derived for the theory of acoustics in his book The Theory of Sound. Erwin Schrödinger, a pioneer in quantum mechanics, studied this book and was familiar with the perturbation methods it describes. 
Excited again — found a number of articles by Atiyah, du Sautoy et al., which begin to tie together for me various threads regarding harmonics and sound.
For mathematical physicists, this material will be old news — but for the fact that they've missed the connection to what we actually hear, owing to sound's putative status as a "mental" thing. The scare quotes are there because "mental" is one of those words we all understand until we start to think about it.
Riemann discovered that the physics of music was the key to unlocking the secrets of the primes. He discovered a mysterious harmonic structure that would explain how Gauss's prime number dice actually landed when Nature chose the primes.
What Riemann discovered was that Gauss's graph is like the fundamental note played by an instrument, but that there are special harmonic waves that, when added to this graph, gradually change it into the true graph or "sound" of the primes, just as the harmonics of the clarinet change the sine wave into the square wave.
~du Sautoy
"Mathematics in the 20th Century," by Sir Michael Atiyah

Monday, January 09, 2017

Perfect Harmony

'Harmonics,' by Cory Ench
When starting down a new path, I like to find the simplest book I can find on the subject at hand.
The eminent mathematician Edward Frenkel has provided a great service here in regard to the 'Langlands program,' which has been aptly described as the Grand Unified Theory (GUT) of mathematics.
I was quite excited to learn of a good number of intersections between that vast and lively body of work* and my own, including symmetry, projective geometry, spectral theory, Riemannian surfaces, quantum field theory (QFT), and harmonic analysis.
Although symmetry is arguably the most important theme here, gauge symmetries are fairly abstract, whereas harmonic analysis provides a trove of correlations between simple physical theory and what we directly experience in sight and sound.
Let's review.
The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound. A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency f and integer multiples of that frequency.
The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.
OK, now here's Frenkel.
The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other. The idea is that a general sound wave is a superposition of harmonics, the way a symphony is a superposition of the harmonics corresponding to the notes played by various instruments. Mathematically, this means expressing a given function as a superposition of the functions describing harmonics, such as the familiar functions sine and cosine. Automorphic functions are more sophisticated versions of these familiar harmonics. There are powerful analytic methods for doing calculations with these automorphic functions. And Langlands' surprising insight was that we can use these functions to learn about much more difficult questions in number theory.
Well, this is just a taste, but that's enough for today.
* The Langlands program has deep roots in number theory, but I've only scratched the surface of that sprawling topic. For the time being, here's a nice bridge in regard to theory vis-à-vis experience.

It was not until the advent of quantum mechanics in the twentieth century that absorbtion spectra were given a satisfactory theoretical explanation. They were shown to correspond with eigenvalues of appropriate Schrödinger operators. A given atom could absorb or emit light only at certain frequencies, corresponding to the energy levels of bound states represented by different eigenvalues. The mathematical spectra of differential operators thus carried fundamental information about the physical world, which even now seems almost magical.
The analogy with number theory is through spectra of other differential operators. These are Laplace-Beltrami operators (and variants of higher degree) attached to certain Riemannian manifolds. The spectra of these and other operators are expected to carry fundamental information about the arithmetic world, a possibility that also seems quite magical.