Monday, February 21, 2022

Why we feel [space]time is passing: The noncommutativity of Alain Connes from the noncommutative music chords - as a pdf

 http://quantumostinato.blogspot.com/2015/06/disentangling-how-can-time-emerge-from.html

I knew I had seen this before but I couldn't track it down....

https://indico.math.cnrs.fr/event/782/material/slides/3 

Sure enough - that's it!!

https://indico.math.cnrs.fr/event/782/attachments/1851/1997/Connes.pdf 

So this is before he is EXPLICITLY relating it back to music theory - yet it's still based on the same "shape of a drum" paper - can we HEAR the SHAPE of a drum? No because it is in 3D then it is noncommutative....

So he is saying you can have the SAME Spectrum or Scale even though the two different shapes have different "chords" due to the change in phase. So Isospectral is frequency of area but NOT isomorphic as geometry. Type One and Type II are isospectral.  But they don't have the same SECOND invariant.

So it's the same "scale" or spectrum but they are an "octave' apart!

So then the von Neumann "Type II" classification is the conversion to a symmetric 0 to infinity space that is based on the geometric continuum.

So this is like the piano, if you want, in which you can play, in both cases, because there are three different kinds of notes. But somehow I will call something a chord if there exists a point at which the corresponding eigenfunctions both don't vanish. OK? So and either it could be three [Perfect Fifth on the piano]. It turns out that the chord which is blue-red is not possible for shape two but it is for shape one.

If you want a new invariant it's intimately related to a point. And the point in that view which is spectral is given by correlations between the eigenvariables. So that the points, in fact, because after all if we want to say where we are, we not only have to give the space, the geometry, but we also have to specific the point where we are. ...an emission matrix of scalar products of spinors of the point where we are. You have eigenspinors and you look at the corresponding matrix and that would be exactly what you need to look at the point.

The point is the emission matrix scalar product of spinors of the eigenvariables. Because the blue-red is an octave difference as a chord with the same frequency of "3" that is noncommutative therefore it is not allowed for the shape II as the continuous conversion that is not an octave difference. In other words 3/2 is C to G but 2/3 is C to F as the octave of 3:4.

So the octave is the "zero" equivalence as squared symmetry or "even" while the 1/2 and 1/4 is the "odd" equivalence with 3/2 or 3/4 based on the "odd" or 3 transposition that is noncommutative.

youtube lecture is in English!

 The description is in French - that's why I had not tried watching it....

So he's saying von Neumann said you use "discrete variables" that are noncommutative as the "inner space" or "inner fluctuations" - to then square them back into a symmetric continuum.

I believe that the true variability is quantum and that the true variability is exactly the fact that when you take a quantum observable it doesn't have a single value. That is has many possible values which are given by the spectrum of the operator. And that belief, once you have found this, plus the fact that the discrete variables can not coexist with the continuous variables without the quantum formalism... How can the times that we know a match from this considerations, and in a very strange manner, it will be related to the end of the talk.. I know it's difficult but in my mind it's backed up by an intuition that's come from many years of work and this is very difficult to transmit. So I will try...

 We are wrong to try to write things in time. Because of our minds, which are logical, we always trying to reconstitute a logical past... Just because we want to feel happy about it. What I'm saying is that things might be different and that there could be a fundamental quantum variability...

Alain CONNES
Mardi 19 mai 2015 16:30 - 17:00

Alain Connes - Temps et aléa du quantique

So the "system" as the octave is commutative. It's a problem of factorization of the inner space, of dividing a system into two subsystems.

[observable space] Time emerges from the factorization.

Type II is a space (scale) classified by a real dimension.

That real dimension can either be bounded dimension like (0, 1) [Type One] or an infinity, (0, infinity). [Type Two that is noncommutative]

So you take the inverse of the exponential as the factor.

So Connes Ph.D. 1972 work was proving that time evoluiton was not based on the symmetric space... but on the noncommutativity itself as time evolution from the negentropic thermodynamics itself. The time evolution only changes locally from the inner automorphisms.

 Alain Connes, French mathematician who won the Fields Medal in 1982 for his work in operator theory.

 My feeling is that the passing of the time could very well come from the fact that we UNable to know all observables in the quantum mechanics.... we are unable to control all the observables of the universe. We are only able to control a small part of them which is a factorization of this kind. And because of our lack of knowledge of the full observables, we have the feeling that time is passing.

[Space]Time is emergent and a corollary of our lack of knowledge.... There is a relation between thermodynamics and the Big Bang...

 – Qigong Grandmaster Effie P. Chow

I feel that the past, and the present and the future is all in one state. They’re operating at different frequency levels. It isn’t the past here, the present here and the future here. We are all at one state. The past, present and future are all here. That’s why some people have deja vu….

– Qigong Grandmaster Effie P. Chow

Connes:

Out of this problem, which is a purely mathematical problem, ...von Neumann classified factors, Type I to III, then came the time evolution...and the uniqueness of the time evolution which completely, you weren't allowed to classified factors... So the idea then is the following: parameter T, there is one parameter group of things which ... On the other hand if you take very seriously the idea that the origin of variability doesn't come from the passing of time, but comes from simply the formalizing of Quantum Mechanics, from the Hilbert space, that's it. Then it's absolutely viable that you can relate to ordinary time and time evolution as we know. Von Neumann provides a solution....it depends that you have a subsystem, it depends on a factorization.

Type One does not involve infinitely degrees of freedom. Type Two - evolution is unavoidable, it occurs infinitely.... You can not suppress it. It's not an inner automorphism. It has the amazing property that, it is in the center of the group of automorphisms....it doesn't depend on any choice. ....It is repetition that will allow you to SEE this time evolution... factorizations as infinite repetitions - otherwise you would not see it.
It's the factorization...The motivation has nothing to do with the ...topology...It's extremely strong, that's why it's the FIRST time I've given this talk. The passing of time is due to our partial knowledge... Then how do we single out this subsystem? I am unable to know.

.....

 The direction of time is specified. You can not change H to -H because you need a state. And this is why you can't have reverse time...The time evolution would not be the same if you changed it to -T [minus T]. If it would be the same then it would be trivial. ..If you give a factorization there is only one notion of time....You can be locally out of equilibrium.

 Factor for instance 4 x 4 matrices as 2 x 2 times 2 x 2. Now take a vector as pure in four dimensional Hilbert Space....Then you do the infinite transfer product of the Four dimensional Hilbert Space according to this vector. In that Hilbert Space you will have all the left algebras will act together and all the right algebras will act together. And that will be a factorization in the analysis of Type III....unless a vector is a trace vector of which there is only one....You take a 4 x 4 matrix which is 2 x 2 times 2 x 2 and you just repeat it. You repeat the same. That's enough to get a Type III to get the time evolution. The factorization is in the subpart.

 So I think the Type One is the 4 x 4 matrix like the "double octave" in music! Fascinating.


 


 




























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