Tuesday, January 10, 2023

Order out of chaos. Fractals out of qubits

 Looking for an empty USB stick, by pure chance, of course, I have stumbled upon my old presentation. I must say I liked it! Not a bad one. So I will share it here today.

It is much like I am reading now my old papers about conformal infinity. I am trying to understand them and I can't stop wondering: 

-Who wrote this?  

-Was it me?

- Wait a moment, if it was me, why did I write that "it can be easily verified that the whole represented body is contained inside a sphere of radius 1"? 

I open Mathematica, write a line of code:

FindMaximum[{x1^2 + x2^2 + x4^2 + x5^2, x1^2 + x2^2 + x5^2 == 1, 

  x4^2 + x5^2 == 1}, x1, x2, x4, x5]

and look at the result:

{2., {x1 -> 0.707107, x2 -> 0.707107, x4 -> 1., x5 -> 5.71313*10^-10}}

and instantly see that it should be square root of 2 and not 1 as in the paper. Well, it does not matter in this case, but the author (me?) was certainly lousy!

Anyway, here is the presentation I have found this morning:

 


And on the same stick (as you see not quite an empty one) I have found also an mp4 file with this video that I have painfully made from thousands of separate frames: 

So the USB was not empty. It also contains high resolution graphics from the presentation. Like these:







And I have completely forgotten.

Now back to the conformal infinity - the nonlinear cap of our linear Minkowski space.

P.S.1. I asked my friend in Novosibirsk if he knows someone who can help me with my problems with the conformal infinity. Today I received his answer: I should contact Vasily Gorbunov. I have checked the webpage with some of his papers and I am amazed! Before asking my questions I should work really hard to prepare them. The better, I prepare my questions, the more I will be able to get from forthcoming replies. So, this is what I will be doing for next few days: making sure that I am asking questions to which I am not able to find answers all by myself!

P.S.2. I have used Povray to render the above graphics. Of course first I had to do all the necessary algebra by hand and using Mathematica. And, of course, I had to learn (though only superficially) how to use the software. It took me a while.

P.S.3. Yesterday (it was Tuesday)  I was afraid that I have made a mistake representing the doubled infinity as on this picture:
Double conformal infinity


I was thinking that the two singular points may be artefacts of the projection. Today it seems to me that they are really singular. But I am not yet 100% sure. Working on it. Need to understand how light travels when trapped in infinity. Then write another paper on the subject, this time doing everything right. 

But first I have to learn Projective Geometry - the subject I know about next to nothing!

P.S.4. Few days ago I have asked Masahito Saito, a topologist from the University of South Florida in Tampa,  for help with my double infinity. This morning I have received his kind reply. For a topologist the answer is simple: these are just two spheres touching at two points: north and South Poles:
"... So, topologically it seems to be two spheres with two pairs of points identified separately...."
 Can you see it? Can you see these two spheres?

P.S.5. Also this morning one of the authors (RI) of 

"A Mathematica Package for Visualizing Objects Inmersed in R4"


kindly provided me with their package. I still have to learn how to use it and see if it will be of some help.

P.S.6. Saturday 14-01-23, 11:36 AM. Patience, please. Started to write a new post, but had to pause, I am still working on a satisfactory understanding the whole situation. I am not satisfied yet. In fact - very far from being satisfied. Many questions without answers. Thus happiness (because there is so much work needed, and it is clear what kind of work), but no satisfaction - thus excitement.

19 comments:

  1. So I wonder... Could this conformal infinity be related to the scale transformation and renormalisation group? If so in what way?

    ReplyDelete
  2. "- Wait a moment, if it was me, why did I write that "it can be easily verified that the whole represented boy is contained inside a sphere of radius 1"? "...

    Oh my God... It merely reminds me of a dream from that night that has repeated itself. AND S^3 -> S^2...

    "It is a fiction based
    on science.
    But perhaps Nature is already using these
    algorithms at some fundamental, so far unexplored
    by us, level?"

    Well, let's hope so...

    Already, at least, you too are seeing such strange things. That's always better.

    Another sensational note. But still no answer to the question "What next?".

    ReplyDelete
  3. "In EEQT there are quantum jumps, and there is (in
    general non-Hamiltonian) continuous evolution
    between jumps.".

    So continuity or jumps? God, it's all mixed up again.

    ReplyDelete
  4. "So the USB was not empty. It also contains high resolution graphics from the presentation. Like these ones:".

    How have these graphics been made?

    ReplyDelete
  5. Anonymous: Mathilde S.

    "The theory provides
    the probabilistic algorithm for jumps when several
    channels responsible for a continuous monitoring of
    several noncommuting observables are
    simultaneously open.".

    What channels do you mean? I don't understand the meaning of this sentence.

    ReplyDelete
  6. Anonymous: Mathilde S.

    "While the result of each
    particular jump is unpredictable, the resulting
    pattern after millions of jumps is highly organized
    and fractal-like.".

    I can feel this sentence. But I don't understand the previous one at all...

    ReplyDelete
  7. Anonymous: Mathilde S.

    "Circles appear quite often in
    quantum fractals. Why it so happens - is not clear.".

    Why circles? This is a very important question in my opinion. A circle in the mist. All the time. This is great! But why circles? I can't justify it.

    ReplyDelete
  8. Anonymous: Mathilde S.

    "Moebius transformation are conformal".

    Aren't they just locally conformal?

    ReplyDelete
  9. Anonymous: Mathilde S.

    Well, conformal transformations and scale transformations! Now I understand what I dreamed! With this transformation, the metric tensor will be invariant up to scale!

    ReplyDelete
  10. "But first I have to learn Projective Geometry - the subject I know about next to nothing!".

    And I deal with projective unitary groups.

    I have a question.

    PU(n) is the quotient group of the unitary group U(n). How could we go from this to the homotopy of PU(H)? Could PU(H) here be the space in which we classify circle bundles?

    ReplyDelete
  11. Do we choose projective representation, because physical states are only defined up to phase? Are there any other relevant reasons?

    ReplyDelete
    Replies
    1. A projective connection with the (projective) special linear group shows up for relativity too.

      https://arxiv.org/pdf/1110.2159.pdf

      It ends up via the equivalence principle identical to a (global) conformal structure but both math structures are still there so who knows what odd possibilities there might be.

      Delete
  12. My interpretation of the note's title:

    Out of this chaos of physics, order must finally be brought out.... This higher order.

    ReplyDelete
  13. If one were to assume that the photon is a composite particle, consisting of a neutrino-antineutrino pair, would this not change the approach to entangled states and to information theory, also in relation to qubits?

    ReplyDelete
  14. Now I'm reading H. Bacry's book on groups and representations. On this basis, I count the commutators and check if they match.

    Can you recommend a better book on Lie and Poincare algebras?

    ReplyDelete
  15. To see spheres I think you would have to sort of merge all the time-like coordinates (W) together and the two spheres would touch at Z=1 and Z=-1 with W=0. You would get circle slices of one sphere for W going from -1 to 0 and circle slices of the other sphere for W going from 0 to 1.

    ReplyDelete
    Replies
    1. Thanks John!
      Tomorrow I will describe in detail how the topologist described the solution. I will also present how it has been dealt with using Mathematica.

      Delete

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