https://www.youtube.com/watch?v=Z52ZAPrRbqE&ab_channel=ColloqueWright
Alain Connes:
There are what we call "emission spectra" - if you take Natrum, for examnple, and you look at the light that goes through a prism, you obtain spectral lines, that create a kind of bar code which is totally specific to that specific chemical product.
These chemists understood that most spectral lines were connected - these bar codes were connected to chemical products. And they invented a NEW chemical body called Helium which was supposed to have a specific bar code. And miraculously in the 20th Century during the eruption of Vesivium - analyses were made and indeed Helium was confirmed as indeed having that kind of spectral line.
Chemists and physicists studied these spectral lines and observed that there was Ritz-Rydberg's Law. When expressed in form of FREQUENCIES NOT IN TERMS OF WAVELENGTH - certain spectral lines add up to give new spectral lines.
And they understood that if you want to understand that kind of law - you had to use not ONE index (alpha and beta) but TWO indices. If you study spectral lines under that point of view you realize that certain lines are the addition of two different spectral lines.
This was a miraculously, wonderful discover that was made, thanks to Heisenberg. Heisenberg understood that this law of composition which is called Ritz-Rydberg's Law led immediately - if you are a physicist you concentrate on observable values - led to Matrice Mechanics. Of course mathematicians know about that but no physicists. If you make a product of two matrices you use precisely this Ritz-Rydberg law. You obtain the "ik" form the sum of "ijk" and "jk." The discover of Heisenberg was that these matrices were NOT COMMUTING.
The order of the terms have a Vital part to play... E=mc2 but you can't inverse the terms of this equation in this specific case. Commutativity does no longer hold in the phases of a microscopic system. This might be a difficult challenge but we tend to know that kind of phenomenon, because when we write things down using lanuage, we know that we have to take into account the ORDER in which we have to write the letters: if we don't we have sometimes the case of anagrams. In other words if you invert the letters, sometimes you can have a different sentence.
If you go from the quantum world to the normal world you lose meaning sometimes.
john milnor, eigenvalues of the laplace operation on certain manifolds, 1964
where he shows that the natural invariant of a shape given by its spectrum, I gave, explained to you why, is NOT enough to characterize the shape. You have to know a slightly more about it, you have to know the CHORDS and not only the [frequency] scale, in musical terms.
If you take a disc, for example, this disc is going to have all sorts of frequencies and certain frequencies, if you can listen to them. The frequencies of this disc will depend on two data: the number of the change when you go around the disc. Here for example you have five and then there is another number, which is the number of vibrations the more you far away from the center. The note you can hear is higher. A specific disc has what we call a spectrum which is a scale, canonically associated which can be calculated in mathematical terms and also has a spectrum that you can see going from infrared to ultraviolet.
So mathematicians discovered that it's just not possible to characterize a space through the data of its scale. I showed you the disc and the specific frequencies of discs, which characterize the shape, when there is a shape dimension to you obtain a parabola. What's important is that you can find examples of shapes having the SAME spectrum [isospectral frequency but NOT isometric] which ARE different.
After the example of Milner, in 16 dimensions....Mark Kac ["Can we find the shape of a drum?"] Gorden Web and Wolpert found a very simple example in two dimensions. To show you the invariant to use - the first shape is made of a triangle and square.
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Connes:
The obvious question that comes to mind is to find if there's a form that matches the spectrum [of the Western music equal-tempered scale] and this question is going to come up. And when you look - you look at various areas and it doesn't work. Why? Because the spectra explains that the form that we're looking for is zero. And the reason we understand this is because we see this via the spectrum that is going exponentially because it's the same number at various powers [the pitch equivalence of octaves]. So the dimension obviously has to be zeros. So there's a fantastic answer: The quantum sphere which is a Spectral Triplet. And it has a fantastic advantage compared to other possible solutions.
There's a minus sign. So it's devilish. It's a noncommutative object which is infinity. But it's really a Triplet spectral. It's an interval with a new geometry. And when we calculated the spectrum and compared to the other spectrum - we were quite spell bound by the matching between the two. What's actually devilish, diabolic, if you look further in the frequencies, you have the feeling that it's not going to work.
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