https://www.youtube.com/watch?v=m20KxUB5lMU
It was discovered by Ritz and Rydberg a long long time ago, that in fact the spectral rays that you see, when you view them not in wavelengths but in frequencies - So what you do you plug in the frequencies and you plug in the corresponding spectral rays. There is a natural way to label them by two indices, not by one index. And when you label them by two indices, what you find, for things that are labeled by i and j and j and k and j is common to the two, then the frequencies, it's not a group, it's not a group, but it's something that has a "partial law of composition" -
Now this fact is strange when you see it at first but the guy who completely deciphered it is Heisenberg. What he found is completely amazing because he found it for physical reasons. His reasoning, in fact it amazed, it was with Dirac, Dirac was amazed by the discovery of Heisenberg. So Heisenberg was looking at spectral rays...what Heisenberg found is that in fact the key was in the Ritz-Rydberg combination transform. What he did, he said, "I don't have a group" - if I had a group, he was talking about action, angle as Fourier transform. Normally if I was doing classical transform I would have a group of frequencies. But I can do the same transform you would do with the action of angular variables. Use the Ritz-Rydberg combination to define a product, you'd say, in modern language, he's defining the Groupiod algebra... what he found are matrices. And this was absolutely amazing because he had no idea that matrices were known to mathematicians. And this was found later by Born and Jordan.
And this opened up a complete world - and in the end, of course, of Schroedinger, the spectral were completely deciphered - in Hilbert space - ...
Alain Connes: On the Notion of Space
Sep 4, 2021
The discovery of Heisenberg is really the first place of noncommutative geometry....This phase space you can not talk it to the commutative....
and what we shall see very shortly is that the spectral paradigm which I'm explaining now, in fact has exactly this power that it embodies not only the gravitational force but it embodies ALL the other forces that we know so far...
except if you use Dirac, if you use Clifford matrices, so they anti-commute so when you raise the sum of them as the square, the cross terms disappear and you get the sum of squares. And it turns out in the spectral paradigm, which is, for the moment, noncommutative geometry, what happens is that it is the Dirac propagator, the inverse of the Dirac Operator, which becomes a line element.
The way you measure distances is no longer by making or taking an arc and taking its minimal length between two points. No - it's by sending a wave and this wave has a kind of bound to its frequency because it's a commutator of the function of the Dirac Operator that's bounded by 1. And that way you measure distance but you're formula for distance is much more adaptable because it doesn't require space to be connected. It works in many many situations in which the other paradigm would not work.
And it gives you the freedom - fine tuning, fine details of the space....
The geometry was are talking about is already quantum.
Bose-Einstein condensation theory for any integer spin: approach based in noncommutative quantum mechanics
A Bose-Einstein condensation theory for any integer spin using noncommutative quantum mechanics methods is considered.
BOSE-EINSTEIN CONDENSATION
. We show that nonlocality caused by the spin noncommutativity depends on the spin of the particle; for spin zero, nonlocality does not appear, for spin half, ΔxΔy≥θ2/2, etc. In the relativistic case the noncommutative Dirac equation was derived. For that we introduce a new star product. The advantage of our model is that in spite of the presence of noncommutativity and nonlocality, it is Lorentz invariant.
So that is the semi-classical approach when in fact the "Planck Cell" needs to be defined by noncommutative time-frequency phase space.
So instead of a spatial symmetry you get a time minimum of the Dirac Operator as 1 with the inverse as the noncommutative time-frequency Dirac Propagator using Clifford 4 x 4 matrices.
that gets into the noncommutative Pythagorean Theorem....
A right triangle with sides , , has an interior angle of. Thus,
https://link.springer.com/article/10.1140/epjc/s10052-019-6794-4
One of the basic principles of quantum mechanics, Heisenberg’s uncertainty principle, requires that a localization in spacetime can be reached by a momentum transfer of the order of , and an energy of the order of [2,3,4]. On the other hand, the energy must contain a mass , which, according to Einstein’s general theory of relativity, generates a gravitational field. If this gravitational field is so strong that it can completely screen out from observations some regions of spacetime, then its size must be of the order of its Schwarzschild radius . Hence we easily find , giving . Thus the Planck length appears to give the lower quantum mechanically limit of the accuracy of position measurements [5]. Therefore the combination of the Heisenberg uncertainty principle with Einstein’s theory of general relativity leads to the conclusion that at short distances the standard concept of space and time may lose any operational meaning.
.it follows that the nonlocality induced by the noncommutativity of the coordinates will be a function of the energy.
Back to Alain Connes:
Moreover, a certain number of things are already there. One would have to prove a fantastic generalization of the reconstruction theorem...What is very satisfactory is the following. You see when people show the Heisenberg Commutation Relations - they extended them out by regroup normalizations...Amazing work, Wigner, ... But by doing that you can not of course to expect to get geometries by doing that. You will get a few parameters, you won't get the proper notations that are themselves geometry and that's a big input of the Heisenberg representations. Among the representations you will those that come from spin manifolds....But the main idea is that by passing from the ordinary Heisenberg relation to this higher Heisenberg relation than you find that geometry is born in "inner space" so it is born in quantum mechanics. And that mere statement should be a very strong indication for people who are doing physics because gravity is the story of geometry. But then you have to find it compatible with the quantum and that's already a sign, that the compatibility is amazingly difficult because it will be of course very very difficult to prove a theorem of this kind. Already we know that at least all the manifolds give you such a representation.... It was really a geometric problem - we had no intention of finding the "star" model at all...We were describing the sphere using a 2 x 2 matrix... There is a way to encode the algebra in a way which can be - which is almost coming out of Riemann - you're giving it a free projection of a 2 x 2 matrix with a projection. It's not the algebra of functions on a sphere but it's the algebra of 2 x 2 matrices.
If you want the algebra that is used by the Universe it is of the same type - it is something that is very simple ...but it can not be commutative because the commutative creates a collapse.
If you apply the commutative to the language all the anagrams are the same - so you collapse the meaning, so you collapse the information which is apriori available into a dessert by passing to the commutative. In fact noncommutative is not strange - no because we speak, because we write - we are used to keeping the information from the order of the letters and the order of the words... So and this is the essence of we are trying to do.
I told Noam Chomsky about noncommutative phase as the secret meaning of language. He said he wished he had time to research it more. Thanks Alain Connes for figuring out the secret of the Universe.
Connes:
It's a very far fetched goal - the goal that we are addressing is a much simpler goal. It's to understand why are on Earth aren't we just living in gravity?
Dynamics of a Dirac fermion in the presence of spin noncommutativity
In this work, we gain further insight in the physical aspects of the spin noncommutativity, which mix spacetime and spin degrees of freedom in a noncommutative scenario, in a Lorentz invariant way. Due to the combination of noncommutativity and Lorentz invariance, time also becomes nonlocal so the modified Dirac equation in general contains an infinite tower of time derivatives.
where instead of the angular momentum, the commutator of coordinates is proportional to the spin.
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