TRANSCRIBED interview with Alain Connes - 2017 on the MUSIC of SPACE - video link
If you have ...in particular young students from developing countries, what advice?
That's difficult...The standard ...challenges...the key step in standard mathematics is to understand that the, you know, you don't learn...you make, you do it - until you are really able to take a problem and solve it by yourself, or try to solve it by yourself, you are not doing mathematics.
There are topics that you can learn that - there are some scientific topics that you can learn. But this is not the case for mathematics. For mathematics you have to do it yourself...So it's a little bit like, for instance, if you type... you have to practice. The practice is far more important than whatever...
It's a very democratic subject. There is a key step, a manifestation of this key step is when the student...finds a mistake of the teacher. Because he's able to think by himself. And find out that the result...so this is something which is very important, which is different from other topics.
Your perspective on the symbiotic evolution of math and physics?
I think it's a very complicated issue in the sense that there's one problem that people are trying to solve which is quantum gravity. We know it's quite difficult...In a way, for a mathematician, at least as far as I'm concerned, the issue is even more important from mathematics in the following sense. For instance when Riemann gave his undergrad talk, he was very on the fact that in Riemannian geometry there is no longer point at the very very small scale. And so had already foreseen developments that came much later, in particular in noncommutative geometry. For example the motion of light no longer makes sense... whereas in this notion the function... there's also deviations....
Some people just want to change the rules of physics...so I think must be very careful. At the same time what I will say there - the impact on the evolution of geometry is that experimental physics has provided for us, the inward bound - ...
our perception of the small structure of spacetime by a factor of 10 to the .8 - and that has implications on the geometric model and that implication is fully understood noncommutatively. Spacetime is no longer purely continuum. It's a mixture of the Continuum and the Discrete. So this was a lesson which very very strangely forced the change of the Riemann paradigm... of course Riemann couldn't force it because it involves quantum mechanics.
So the new paradigm of geometry is very close to the Riemannian part but there are nuances from the quantum, from the formalism of quantum mechanics which was discovered by von Neumann. And it tells us that the notion of geometric space becomes more natural and more easy to understand in the quantum formalism.
It's not the immensity only of the University but also the very small scale. How would you define a point?
That's a very interesting question because you can ask...how do you communicate with the extraterrestrial - the place where we are? Because if I tell you that we are extraterrestrial well that won't help because they won't know what we are talking about - they won't understand our language. And then people will say based on general relativity we just have to give our coordinates. that's also foolish - which coordinate system do we take and which knowledge do we have to communicate our coordinate system? And it turns out....it is exactly provided ... how to you communicate the spatial global - not by using a picture. How do you define the coordinates?
Turns out that the best way is to give the MUSIC of the space.
So if you give a shape... each of these shapes have a special musical scale. Which frequencies are the proper frequencies of this shape. Turns out if you want to give invariant space, you have to give the least quantities which are assigned to this space - now the SCALE of the space is noncommutative.
So this is an invariant of the space. And it turns out when animals form the animals equation - animals Laplacian and the inverse of the Dirac Operator... and the one solution then you can actually reconstruct the space. So you need to know a little more than the scale of the space. You need to look at the other points. Each point is defined by a CHORD on the scale. So the point in the space, technically speaking, how do you specify a point? Technically speaking you take the Eigenvectors of the Dirac Operator. They are sections of the variables of space. You evaluate them at the point.
You can't just take a number. You take the scalar products of these variable sections at the point. This gives you the matrix. And it turns out this matrix is exactly ...space is understood by a unique scale and possible chords. And the possible chords are the points.
So in a way what happens is that you reconstruct the space by a kind of Fourier Transform. And I believe this is exactly what the brain does when we see because when we see we have the photons which are coming into a noncommutative eigenspace. And the brain reconstructs the space like we are used to see.
But what is more important, that this is exactly the way we perceive the structure of the Universe. Because we can see the distance universe by looking at the spectrum - or galaxies or spectrals of stars or spectrals of the Universe.
And it's this spectrum that we the information of the Universe. So...we find out that not only it's useful for microscopic differences but it also challenges the point of view on the large distances, but in a way that's perfectly correlated with our perception of the Universe. For instance what happens we know that things are very very distant. You have to remember there was some time that people didn't know that there were things outside our galaxy. It took very bright astronomers to find that. But now we know that things are very very very distant, just because of the Red Shift. So this is again the spectral.
And here there is a concept of distance of unit of length in terms of wavelength?
That also is a very very important step which is so much fun to explain because it relates to very concrete stuff. So the story starts in France more or less during the French Revolution. You see there was a unit of length - there were at least one thousand Units of Length in France! Which means when people were telling tissue and traveling from one place to another they had to measure with respect to the Unit at the entrance of the... hahaha.
Of course the Revolution was the idea to unify things...they decided to, and they are very good scientists, so they decided to try and unify the system by Defining a Unit of Length. So what did they do? They took the LARGEST available object which is the Earth, and they defined a unit of length so that when you multiply this unit of length by million, you obtain the circumference of the Earth...
They looked at the stars and they measured angles. So they just needed to measure an angular portion to get a gradient. And they chose a gradient portion between France and Spain (Barcelona). And in 1782 - the full Revolution - they sent two people - were sent out - to do the following: The idea was that they would have a base - so there they would lie down a significantly long distance - some bar if you want. And take that as a base. Now they were only measuring Angles which was a very smart idea. So they were putting telescopes on top of... and by doing triangulation they were comparing the base between distance. And out of that was defined the unit of length which was actually a little more..
It was an interesting story, there was all sorts of developments in the story - one of the guys - I don't know - had to make measurements in Spain. And of course - he was mentioning angles by his telescope - he had lots of trouble because there was a war between France and Spain at the time. And he had to explain to the Spanish Army that by putting this telescope on top of the eave and looking - he was not a spy. haha. But he was trying to define the unit of length. haha.
So there were all sorts of very interesting developments. I love to tell this story. I don't know why.
And then so this unit of length was deposited near Paris - the Meter - deposited near Paris. So - this is not particular because if you want to measure a bed... of course... So that was the situation of the time...
But then some very interesting events happen. So there were - around 1870s - they noticed the Platinum Bar that was defining the Meter was changing! How did they notice that? By actually measuring its length very precisely by comparing it with the Krypton wavelength.
And gradually they decided to take the right step and the right step of course was to take this Wavelength as the new definition of the unit of length. So that took some time. That took some time.
So what is very interesting to know is that there are instruments that are sold in the shop - you can buy them - and these instruments are based again on the wavelength. It's no longer Krypton. It's Cesium because it turns out Cesium is a ... the wavelength of Cesium is microwave. So it's like when you heat something in the microwave - it's on the order of ...meters.. it allows you to measure lengths of up 12 decimals. So it's absolutely incredible.
And this is now what is taken as the unit of lengths - of course people will tell you it's a Unit of Time, not a unit of length. But because of the constancy of the speed of light, the speed of light has been fixed to a very specific number.... So things have evolved.
And now what you see from that - is that there was a complete change ...because the unit of length is no longer a LOCALIZED object which is somewhere. But it's a Spectral Data from quantum mechanics from Noncommutative Geometry. It's exactly parallel to this change of parity in physics.
So it's very very concrete and the advantage - the enormous advantage is that if we add to for instance unified geometry system NOT on EArth - but in the Galaxy for instance. If you tell the people - come to Paris - and compare ... haha. with the Unit of Length... they would laugh at you.... there would be wars actually because people would say, "We HAVE our unit of length."
If you tell to people, take a chemical element, of course Cesium is a little bit complicated...like helium or hydrogen. Hydrogen is essentially anywhere. Helium comes from a very exceptional supernova. So there are elements in the Universe it's not so clear. There a specific patterns in the spectral rays of Hydrogen. Then one would have to find the hyper-fine splitting. The advantage of hyper-fine splitting is a difference of energy which is very very small. That would, in the Universe, it would generate microwave wavelengths.
Which is much more practical. Whereas if you take a huge difference of frequency you would get a very very tiny unit of length. What I'm saying is that if you communicate with people - by sending a Probe - and if you are able to tell them what is your unit of length - this is marvelous. And you just send a spectral ray of hydrogen and you explain which one you want to find out - this is very simple. If they are smart they will understand.
And this description of the fine structure of spacetime - a spectrum of an operator...?
It's a little more complicated. The spectrum of the operator gives the unit of lengths. It allows to combine the discrete and the continuum. Essentially it's a mixture of the discrete and the continuum. What experimental physics unveiled over a century is exactly what's the structure of discrete space. So at first the discrete space with quantum Eigen Operators - what we found - at first we are positing from the bottom up approach. We were taking from experiment and trying to figure out what was going on. And gradually we found what finite space should be. But in the recent work about two or three years ago... with ...we were very amazed. Because we were asking a purely geometry problem which was motivated by noncommutative geometry which was totally disjoined from physics and astronomy and so one. By developing this problem in Dimension Four - we found exactly the same space and geometry which was put in before.
Why is discrete important?
What is it important to have a Discrete Space? The most obvious problem that you have - if you don't have this discrete space - is that the Higgs Boson - the - died just one year before the particle was discovered - it doesn't fit with standard geometry. Why? Because with standard geometry if you take a function of space you will differentiate it and you get a gauge potential... The differentiation depends on the DIRECTION in which you differentiate. This is why you get something which is Spin One which depends on the direction. But the Higgs Particle which is Spin Zero - so it doesn't depend on the direction.
So you wonder how can you obtain geometrically a particle of Spin Zero. Now image that instead of just manifold there is a discrete element - the discrete element is telling me whether I'm on the top or the bottom. So now - I have more information. I know whether if I am on the top or on the bottom and I think of a function. Now this function will have a value - will have a development here and under. They don't have to be the same.
So I can differentiate the function UP or DOWN but I can also take the finite differential course - it doesn't depend on which direction I take. That's the Boson of Spin Zero and that corresponds to the Higgs Boson. So the Higgs Boson was the discrete sign of which direction.
So all the masses of the particles comes from this mechanism. ...what you find out is that the main thing which is the Matrix of Masses and the angles of particles - is in fact the Line Element for the Finite Structure. So the line element of the finite structure contains this formation. With this model the Discrete contains the information about the masses and the mixing.Thank you very much.
http://denisevellachemla.eu/gnc-ac.html
More Alain Connes transcribed at that link.
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