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.

Tuesday, October 11, 2016

Message in a Bottle

I recently received an email from what sounded like a very bright young person, talking about machine consciousness, qualia, and essentia.

Then that email rather mysteriously disappeared. 

So this is a 'message in a bottle,' whereby I hope to reach that individual and ask that they try again.

Saturday, October 01, 2016

The Forces of Nature, Color

Watched this on PBS the other night.

There's nothing here you likely haven't known for a long time.

Except for the fact that they state, quite explicitly, that color is a feature of light itself.

Did our civilization just do a 180 on the subject?

And did no one bother to tell me?

Oooh, that makes me so mad.

Thursday, May 19, 2016

New Support for Alternative Quantum View

May 16, 2016

Of the many counterintuitive features of quantum mechanics, perhaps the most challenging to our notions of common sense is that particles do not have locations until they are observed. This is exactly what the standard view of quantum mechanics, often called the Copenhagen interpretation, asks us to believe. Instead of the clear-cut positions and movements of Newtonian physics, we have a cloud of probabilities described by a mathematical structure known as a wave function. The wave function, meanwhile, evolves over time, its evolution governed by precise rules codified in something called the Schrödinger equation. The mathematics are clear enough; the actual whereabouts of particles, less so. Until a particle is observed, an act that causes the wave function to “collapse,” we can say nothing about its location. Albert Einstein, among others, objected to this idea. As his biographer Abraham Pais wrote: “We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it.”

But there’s another view — one that’s been around for almost a century — in which particles really do have precise positions at all times. This alternative view, known as pilot-wave theory or Bohmian mechanics, never became as popular as the Copenhagen view, in part because Bohmian mechanics implies that the world must be strange in other ways. In particular, a 1992 study claimed to crystalize certain bizarre consequences of Bohmian mechanics and in doing so deal it a fatal conceptual blow. The authors of that paper concluded that a particle following the laws of Bohmian mechanics would end up taking a trajectory that was so unphysical — even by the warped standards of quantum theory — that they described it as “surreal.”

Nearly a quarter-century later, a group of scientists has carried out an experiment in a Toronto laboratory that aims to test this idea. And if their results, first reported earlier this year, hold up to scrutiny, the Bohmian view of quantum mechanics — less fuzzy but in some ways more strange than the traditional view — may be poised for a comeback.



Disclosure: I've long argued for a similar POV.

Sunday, January 10, 2016

Notes on the Revolution, 5

Here's a stunner: Nobelist Frank Wilczek, talking about the same stuff I've been on about for 40 years, regarding color v. action, symmetry, and higher dimensions.

(The first 20 minutes or so are a recap of contemporary physics.)

More on LinkedIn

Sunday, January 03, 2016

Notes on the Revolution, 4

Quantum Computers

Emerging Technology That Will Revolutionize The World

By David Deuchar

  • Google’s Quantum AI team, which works with NASA and D-Wave Systems, recently announced concrete evidence of huge runtime gains for proof-of-principle optimization problems with D-Wave’s latest quantum computer.
  • The 1000+ qubit D-Wave 2X quantum annealer outperforms classical processors by a factor of over 10 to the 8, or 100 million times.
  • Central banks, governments and major aerospace firms already employ the technology on a exploratory basis, but financial researchers/directors see massive potential for portfolio optimization in the vein of HFT trading.
  • This technology not only has massive implications for Google but for the entire financial industry as a whole.

The ability to solve complex, dynamic problems that would take classical computers tens of thousands of years in mere seconds — this has been the promise of quantum computing. The theories and ideas behind quantum computing have been around for decades, but now, Google is starting to demonstrate concrete progress in the quest for a practical quantum annealer that could revolutionize the entire world.

I was happy to see color and consciousness in the scrolling topics. Here’s a little learning to flesh that out a bit.

So long as we adhere to the conventional notions of mind and matter, we are condemned to a view of perception which is miraculous. We suppose that a physical process starts from a visible object, travels to the eye, there changes into another physical process, causes yet another physical process in the optic nerve, and finally produces some effect in the brain, simultaneously with which we see the object from which the process started, the seeing being something “mental”, totally different from the physical processes which precede and accompany it. This view is so queer that metaphysicians have invented all sorts of theories designed to substitute something less incredible.

~Bertrand Russell

If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electromagnetic waves of wavelength in the neighborhood of 590 millimicrons. If you ask him: But where does yellow come in? he will say: In my picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person whose eye it is the sensation of yellow.