Constructing the Conformal Infinity

 

This post is thought to be an answer to a comment by Bjab from the previous post.

I do not know who has discovered the stereographic projection and when? I don't know, But it was certainly a great discovery. If we are to believe Wikipedia:

“The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection.[2] Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts.[2] The term planisphere is still used to refer to such charts.”

Here is the picture

Here we embed the real axis with the coordinate x into the unit circle. The point P' on the x-axis is projected, here from the north pole N,  onto the point P on the circle. If we denote by X the coordinate of P' and by x,y the two coordinates of P, we have the following transformation formulae (taken from Wikipedia):

We are going to make one change here: we do not like the north pole! It is on the ocean. We prefer to stay on the land, therefore we will choose the south pole as the projection origin. The formulae will be almost the same, only the sign of the z-axis with change into the opposite. Thus we will use

(X,Y) = (x/(1+z), y/(1+z))

(x,y,z) = ( 2X/(1+X²+Y²), 2Y/(1+X²+Y²), (1-X²-Y²)/(1+X²+Y²) )

Notice that th image of the whole plane is the sphere minus the south pole. The south pole, z=-1, escapes to “infinity” on the plane. Thus the sphere is the plane plus one-point – the infinity. It is called a one-point compactification of the Cartesian plane.

Now nothing prevents us from considering a one point compactification of the three-space with coordinates X,Y,Z. We simply need to add on dimension. Thus we consider for-dimensional space with coordinates x,y,z,v and use a straightforward extension of the above formulae:

(X,Y,Z) = (x/(1+v), y/(1+v), z/(1+v))

(x,y,z,v) = ( 2X/(1+X²+Y²+Z²), 2Y/(1+X²+Y²+Z²), 2Z/(1+X²+Y²+Z²), (1-X²-Y²-Z²)/(1+X²+Y²+Z²⁾)

We may be having problems with imagining the three-sphere in four dimensions, but the algebra is as simple as before. Algebra rules!

We are able now to construct a one-point compactification of Rⁿ for any n. It will be a unit sphere in Rⁿ⁺¹. We simply add one coordinate and use a straightforward generalization of the formulae above.

But: it is not yet the end of the story. For our story to have a happy end, as any good story should, we have to go through yet another adventure.

Let us go back to only one coordinate x. The line becomes a circle. But there, spontaenously, comes the observation that a circle comes from an intersection of the cone with the plane, as on the graphics below:

Do not pay attention to any detail on the picture (I have borrowed it from a random paper on Researchgate), except that the circle is created by the intersection of the light cone with a constant t=1 plane.

Guided by this new brave idea we add an extra variable playing the role of "time". But since it is just an extra variable, not a real "time", we call it w, and we will set w=-1. Why "-1" instead of 1? Well both are good, but to be in agreement with a certain number of conventions used in the literature, lest us agree for -1. Ok?

We also simplify our notation to easily cover a general case. So we will write X for the vector with coordinates X¹,..., Xⁿ, and we write X.X or X² for the sum of squares of the coordinates X¹,...,  Xⁿ . Similarly we write x for the vector with coordinates x¹,..., xⁿ . Our formula for the embedding reads now

(x,v,w) = ( 2X/(1+X²), (1-X²)/(1+X²), -1 )

The sum of squares of the first n+1 coordinates, x²+v² is now automatically equal 1 (it can be verified independently though), so the point (x,v,w) is on the surface of the cone

x²+v²-w² = 0.

The coordinate w=1 intersects this cone, the intersection is the sphere Sⁿ.

We are not quite happy yet. In the formula above we have denominators (1+X²), and we do not like them, even if here they are doing no harm. Our cone is made of generator lines. These are straight lines from the origin, along the cone. It is these lines that are important, not the particular intersection. Thus we replace the formula above by multiplying all coordinates on the right by the 1/2 of the common denominator for n+1 first coordinates. We end up with a new embedding formula:

(x,v,w) = ( X, ½ (1-X²), -½ (1+X²) )

We still have a point on the cone x²+v²-w² = 0. The mapping above is one to-one. We will look at it as a mapping from Rⁿ to generator lines of the cone in Rⁿ⁺².

Soon we will specify n that is of interest for us to be n=4, thus the total space Rⁿ⁺² will be (4+2=6) six-dimensional.

To feel more "at home" with our formulas let us do a little exercise. Namely, let us take a generator line on the cone x²+v²-w² = 0 and find the point X to which it corresponds. In the future instead of "generator line" we will simply use the term "line". Thus we should have

(x,v,w) = a ( X, ½ (1-X²), -½ (1+X²) ),

where a is a proportionality factor, telling us that (x,v,w) and a ( X, ½ (1-X²), -½ (1+X²) ) are on the same line. So, we should have

x = aX,

v = a(1-X²)/2,

w = -a(1+X²)/2.

Here x,v,w are given, and we want to calculate a,X. Subtracting the two last equations we find

v-w = a.

Thus, as long as v≠ w we have a≠ 0 and from the first equation

X=x/(v-w).

When v=w, and we are on the cone x²+v²-w² = 0, x must be 0. This is one generator line (0,v,v), v∈R

.So far being brave did not lead us to any trouble. So now we dare to be even more brave. First of all we specify n=4. By X we mean a vector with coordinates X₁,X₂,X₃,X₄. But now we will think of X₄ as "time coordinate". Therefore by X² we will now mean

X² = (X₁)² + (X₂)² + (X₃)² - (X₄)².

Notice that we have changed the coordinate indices from upper to lower to avoid the confusion about what X² means.

So, we will keep the formula:

(x,v,w) = ( X, ½ (1-X²), -½ (1+X²) )

but with the above understanding what X and X² mean.

Altogether we thus have the signature (+++-) for spacetime, + for v, and – for w. The total signature of the six-dimensional space is (4,2), more specifically (+++-+-).

We will check now carefully if we got into some trouble this way?

Let us analyze the map (*) X ↦ (x,v,w) from M to V. We notice that it is one-to-one, we have an injection. Indeed if X≠X' then the images are also different, simply because, as it is evident form (*), x=X, and it is sufficient for just one coordinate of two points to be different for these two points to be different. Moreover, (x,v,w) and (x',v',w') are certainly not on the same line.

But it is not surjective. Let us find the set on which the inverse map is not defined. In the purely Euclidean case it was just one point, and now what are we going to get? To answer this question we follow the previous method and try to express X in terms of (x,v,w). Thus we write again

x = aX,

v = a(1-X²)/2,

w = -a(1+X²)/2.

And try to solve for a and X in terms of x,v,w, assuming x²+v²-w² =0. As before v-w=a, therefore as long as v≠ w, we have a≠ 0 and

X=x/(v-w).

The "infinity" now is defined as before by v=w. Before it was just one point (one generator line). But now? If v=w, from x²+v²-w² =0 we deduce x² = 0. We now need to find algebraic equations describing our (projective) manifold.

But now x² = (x₁)² + (x₂)² + (x₃)² - (x₄)², and we cannot deduce that x=0 in the Euclidean case. We have a 3-dimensional surface. Let us analyze this surface remembering that we are dealing with lines. We are on the quadric in described in coordinates x,v,w by the formula

x²+v²-w² = 0

In projective geometry these are called homogeneous coordinates. Writing explicitly:

(x₁)² + (x₂)² + (x₃)² - (x₄)² + v² - w² = 0

or

(x₁)² + (x₂)² + (x₃)² + v² = (x₄)² + w² = a

The common value a above cannot vanish, since if it vanishes, all x,v,w would be zero, and the origin is excluded, as it does not define any line. Thus a>0. We can therefore replace x,v,w by (x,v,w)/√ a , and get a pair of equations

(x₁)² + (x₂)² + (x₃)² + v² = 1

(x₄)² + w² = 1

Now we set w=v (our "infinity"):

(x₁)² + (x₂)² + (x₃)² + v² = 1

(x₄)² + v² = 1

We have two equations for five variables. Thus our "conformal infinity" now is not a point, it is a three-dimensional surface in a five-dimensional space of coordinates (x,v). We will explain why this term "conformal" in the coming posts. The point is that the conformal transformations which are singular in space time, are nicely realized by linear transformations from the group SO(4,2) acting in our 6D space of variables x,u,v.

 Skipping one space dimension, say x₃, we get a two-dimensional surface in a four-dimensional space

(x₁)² + (x₂)² + v² = 1

(x₄)² + v² = 1

Removing also the variable x₂ we have the intersection of two cylindrical surfaces in 3D:

(x₁)² + v² = 1

(x₄)² + v² = 1

And these were our equations form the previous two posts, though the names of the coordinates were different.

P.S.1. If we were to take care about the physical dimensions, we would have to replace "1" in all the above equations by some constant R (or R squared, as the case may be) - where R is the "radius of the universe". I. Segal et co  used this finite number R to avoid the need for renormalization of quantum field theory at low energies.

P.S.2. Why couldn't we take v=w and x^2=0 as the solution? The problem is that if v=w is different from 0, then x=0 belongs to the solution. While if v=w=0, we have to exclude x=0. This is not a simple algebraic equation. We have to proceed differently.

P.S.3. There is still one problem with our solution. If (x,v,w) satisfies our equations, then (-x,-v,-w) also satisfies them. But these two points are on the same line, so they should be identified. In other words: we have to take the quotient of our solution set by an equivalence relation. We will discuss it later when we will be talking about topology and differential structure on the compactified Minkowski space.

Imagine infinity

The problem from the previous note "Imagine Infinity - A Challenge":

We are in 4D, coordinates X,Y,Z,W. The infinity is given by the intersection of these two cylinders:

1) X² + Y² + Z² = 1

2) Z² + W² = 1

Let us simplify. Skip Y. This corresponds to spacetime being only 1+1 dimensional. So we have

1) X² + Z² = 1

2) Z² + W² = 1

This is a curve in 3D. Let us plot it. This is an intersection of two cylindrical surfaces, perpendicular to each other. We ask Mathematica to plot it:

h = x^2 + z^2 - 1;  

g = z^2 + w^2 - 1;

ContourPlot3D[{h == 0, g == 0}, {x, -1, 1}, {z, -1, 1}, {w, -1, 1}, 

 MeshFunctions -> {Function[{x, z, w}, h - g]}, 

 MeshStyle -> {Thickness[0.05], Red}, Mesh -> {{0}}, 

 ContourStyle -> 

  Directive[Blue, Opacity[0.1], Specularity[White, 30]], 

 PlotPoints -> 60, SphericalRegion -> True]

Here is the result:

 

Instead of two spheres touching each other at two points, north and south poles, we have now two circles. Seems to be correct, but I am not happy with skipping one (and, in fact, two) space dimension(s).

P.S.1. The reader (Bjab) commented that the red circles are not real "circles", they are ellipses instead. To verify if it is indeed the case I have changed the view. And indeed, here is what we get:

Ellipse! Topologically it is (homeomorphic to) a circle. Metrically it is not. But here the metric is unnatural. I will explain it in the next post.

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: 

Parabolic quantum fractal

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.

Imagine Infinity - A Challenge

 At the end of the Universe there is Infinity. What shape does it have? No abstract mathematics is needed in order to answer this question. You need just Light. No algebraic topology, no Hopf algebras, no category theory. It is as simple as it can be - and yet imagining it can be challenging. And so here I am presenting you with this challenge. And I am accepting this challenge myself, since at present I do not have a clear answer.

So here it is.

We are in 4D, coordinates X,Y,Z,W. The infinity is given by the intersection of these two cylinders:

1) X² + Y² + Z² = 1

2) Z² + W² = 1

We notice that necessarily X,Y,Z,W must be in the interval [0,1]. So our infinity is bounded, it is contained within the unit cube of the four-dimensional space.

Next thing we notice is that there are two equations for four variables. Therefore infinity is a two-dimensional surface (4-2=2). So it should be possible to visualize it in a 3D space in which we live. Of course there will be infinitely many ways of visualizing it, depending on our creativity. Some will be more appealing than others.

I will be working on this problem myself with my poor imagination. You can use yours.

The theory behind this challenging simple problem I will present in one of the next posts.

The fact is: This is the double cover of the conformal infinity of the Minkowski space as discussed by Roger Penrose - the mathematician who in 2020, was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity.  In the past I have already discussed it here.  But today I doubt  if my proposed solution is a good one. I think it may be either wrong or incomplete.

This photo is not presenting a solution of our challenge 

 but perhaps the true solution will be in some sense similar? I am not sure...

P.S.1. The challenge problem  addresses a "toy infinity", good for (2+1)-dimensional spacetime, thus stripped of one space dimensions. 

P.S.2. Is it a "twisted torus"? Whatever it means.....

P.S.3. Seeking for the solution

The Seekers - Walk With Me

The Carnival Is Over

by The Seekers

P.S.4. We need to know ALL about this surface at infinity. It is infinitely important. Light can travel there in no time, and come back bringing us information. It is our past and our future. It is a homogeneous space for the Poincare group. Therefore a specie of elementary particles is living there. What are these particles? What do they do? What kind of a geometry this infinity carries? All these questions MUST be answered.

I have found a book "Surfaces in 4D space" by three authors Scott Carter,  Seiichi Kamada and Masahico Saito. But it is too difficult for me. I do not know the techniques. And my surface is probably extremely simple: it has no knots etc. But I want to know all the topological invariants - if there are any non-trivial.

P.S.5. Laura is telling me that I should write a book about this Conformal Infinity. Perhaps. But first I need the results. A lot of them. And good quality results.

One Law I want to know

John Archibald Wheeler proposed The One law for the Universe; Boundary of the boundary is zero - BB Principle. 

In symbols:

∂∂ = 0.

From Warner Allen Miller ,"The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle", (pp. 201-227 in Between Quantum and Cosmos: Studies and Essays in Honor of John Archibald Wheeler, Wojciech Hubert Zurek, Alwyn van der Merwe, Warner Allen Miller, Princeton University Press 1988):

All our laws of nature would be then consequences of this one. Looks nice and simple indeed. In topology and in algebraic topology it holds indeed, either as a theorem or as a postulated fundamental property.

In my vision there is also one law: it is the "Law of Free Fall". Everything moves along a geodesic of some kind of "geometry". Can these two visions, Wheeler's and mine, be merged into one vision? Is free fall another expression of BB? Is BB just another expression of Free Fall? And what about Free Will?

And then: why THIS law and not some other? And why one law and not two or three? Or even an infinite number of laws? Why?

While looking for answers, enlarging my knowledge (and my ignorance as well, but only linearly), I am asking myself: "Am I of this world?"

"And so we wander on our way asking ourselves—if truth be known, muttering to  ourselves really—“Am  I  of  this  world  or  the  other?”  and  answering  “I  am  of  both.”  And  we remind ourselves of this as we go along."

 

"Through this circular river symbol, the moat, the tale warns us that this water is not just any water but a certain kind. It is a boundary water, much like the circle the maiden drew around herself to keep the Devil away. When one crosses into or through a circle, one  is entering  into  or  passing  through  to  another  state  of  being,  another  state  of awareness, or lack of one. "

Clarissa Pinkola Estes, "Women Who Run With The Wolves", 

I want to know

 

P.S.1

Found by Laura for me to watch. A good one indeed. Recommend for everybody!

 P.S.2. Just chceked my researchGate mail. Discovered a message from September:

"Hallo Arkadiusz,

I looked to your your Preprint Time of arrival operator in the momentum space, it is interesting and please see my paper (blow) , where one can see the relation to the Fujiwara–Kobe time operator, sec 4 of your preprint.

https://www.mdpi.com/2624-960X/2/2/15
or
https://www.researchgate.net/publication/340491512_Time_Operator_Real_Tunneling_Time_in_Strong_Field_Interaction_and_the_Attoclock_Open_access_journal_Quantum_Report_httpswwwmdpicomjournalquantumrep

P.S.3 While Igor Bayak wrote me 

"Уважаемый Аркадиуш, здравствуйте!

Посмотрите, пож., на уравнение 3.16 из статьи Хаотическая динамика электрона. Вы ее легко найдете в моем профиле. Там речь о динамическом решении для нулей дзета функции Римана."

Riemann Zeta! Beatiful piece of art!

P.S.4 Another worth considering quotation from "Women Who Run With The Wolves" mentioned in the main text of this note:

"The young and the injured are uninitiated. Neither knows much about the dark predator and are, therefore, credulous. But, fortunately, when the predator is on the move, it leaves behind unmistakable tracks in dreams. These tracks eventually lead to its discovery, capture and containment."

"Wild Ways teaches people when not to act 'nice' about protecting their souls. The instinctive nature knows that being 'sweet' in these instances only makes the predator smile. When the soul is being threatened, it is not only acceptable to draw the line and mean it, it is required."

P.S.5. I am back to conformal infinity. It is a part of U(2). Perhaps I will write a note about it. 

P.S.6. From the battlefield Planet Earth;

P.S.7. I have on my table these two books. Oh God, how much I would like to know their content and master it!. Why it is not possible for us to bring these books close to our heads and download their content (even if it may take a while), and push the button "Understand" and another button "Now"? Why?

But I clearly realize that different people may have extremely different standards of "understanding"....

Different level of "depth". The next button would be "Integrate". Of course the first book would have nothing to ingrate with. Would be instantaneous. But with each new piece of added knowledge integration would take exponentially longer time.

By integration I mean making the whole more than just the sum of parts. For instance: when we learn about complex numbers, we understand much better real numbers about which we have learned before. We create a huge number of connections and interactions between the old and the new knowledge. Thus, approximately, when we have m pieces of old knowledge and add m pieces of new knowledge we can optimally-ideally gain mxn rather than just m+n. But there is also useless knowledge: a knowledge that cannot be integrated with our other knowledge, because it belongs to a different "species" of knowledge. For instance knowledge of blacksmithing or  knitting will hardly add anything useful  to your knowledge of algebraic topology.  We all have finite resources and we have to choose whether we want to know superficially possibly many subjects, or to know really deeply at least one. People with extremely high IQ seem to be naturally gravitating towards the first option.

And, as John Wheeler has noticed: the more we know about a given subject, the more we are painfully aware of how little we, in fact,  know.

P.S.8. You should not miss or neglect this one:

What is impossible takes a little longer

 To each mxn matrix Z such that

Z*Z < I

wehave associated a "symmetry" operator J defined in block matrix form as

The pseudo-unitary group U(m,n) acts on the set of symmetries by

U: J ↦ UJU*

Which translates into linear fractional action on Z, written in a block matrix form

U: Z ↦ Z'

But what if Z*Z = I? It looks as then we encounter a catastrophe. The expression for J has then 0 in the denominator! J explodes. J is in such a case impossible! What to do?

What to do in general when we are facing a catastrophe? Fasten sit belts and keep cold blood. Get smart! That is what we will do now. We adjust and continue as if nothing has happened, except that we avoid explosions. Explosions never do any good. Their effects are, unfortunately long lasting. We have to learn how to control explosions. And that is what we will do now.

We take the bull by the horns. We skip the intermediate steps and dive into the center of the cyclone. Do we really need J? Yes, it is useful, but do we need it NOW? How did we get the formula for transformations of Z? We were considering vectors of the form

When Z*Z < I, we have (z,z) ≥ 0, and = 0 if and only if v=0. Thus, if Z is fixed and v runs over Cn, vectors z generate an n-dimensional positive subspace of X.

What if Z*Z=I? Then ||Zv||2 = ||v||2 and vectors z generate a two-dimensional subspace V of X consisting of isotropic vectors: (z,z)=0. If n≤ m – it is a maximal isotropic subspace.

From now on we we will assume that n≤ m, otherwise we would have to make some adjustements. Anyway, for applications in physics we usually take m=n.

Now, if V is maximal isotropic, and if U is in U(n,m), then V'=UV is also maximal isotropic, since U is an isometry

Warning: We use bold letter U to denote isometries of the indefinite metric space X. We will use normal U to denote isometries (i.e. satisfying U*U=I) from Cn to Cm. U is represented by an (m+n)x(m+n) A,B,C,D block matrix, while U is nxm matrix. Context IS important! 

Let us find a general form of a maximal isotropic subspace, say V. V is necessarily nxn dimensional (recall that we assume n≤ m). Let z be a non-zero vector of V. Written as a column vector {w,v}, it is evident that v is a non-zero vector, otherwise we would have (z,z)<0. If z,z' are two vectors in V, then if v=v' we must have w=w', otherwise z-z' would be negative and it should be isotropic.  Thus, for z in V w is uniquely determined by v. By linearity is is easy to see that w must depend linearly on v, moreover we must have ||w||=||v||. Therefore z is of the form z={Uv,v}, where U is an isometry, i.e U*U=I. Conversely any such U detrmines a maximal isotropic subspace consisting of vectors {Uv,v}.

Now, let U be an element of U(n,m). Then Uv consists of vectors {U'v',v'}, for some other isometry U'. Writing U in a block matrix form, we get

U'v'=(AU+B)v

v'=(CU+D)v

exactly the same way as we did it before for Z.

Exactly the same way as before, for Z, we deduce that CU+D must be invertible, since if there would exist a nonzero v' such that (CU+D)v'=0, then z' would have to be of negative (z',z'), which is impossible, since it is isotropic, or zero, which is impossible since U is invertible.  Therefore the same linear fractional transformation formula holds

The maximal isotropic subspaces of X form what it is called the Shilov boundary of the domain of maximal positive subspaces considered before.

In the coming posts we will discuss its relation to our four-dimensional spacetime for m=n=2.

P.S.1. In physics we start with m=n=2. The elements of X are called twistors. They are bi-spinors for six-dimensional extended spacetime with metric of signature (4,2) - the space of conformal relativity. Our division into blocks is nothing but a representation of a bi-spinor as a pair of spinors. Our U, elements of U(2) group,  will correspond to the events of our four-dimensional space time. What is perceived as a simple point in our space time is, as we will see, a Plato projection of a null geodesic (light ray) in six dimensions. Transition from 6 to 4 (or from 4 to 6)  is like a phase transition. Then we can go from 6 to 8, from 8 to 10, and from 10 to 12 (Burkhard Heim's world  and seventh density) by adding each time two dimensions, always of signature (1,1). BTW: Distance between 4 and 12 is 8, so there is a place for octonions ....

P.S.2. And another by the way:

"Contraryto the majority of studies that have focused on health effect of redmeat, this study argues that total meat consumption, in general,benefits people health, which leads to greater longevity. This hypothesis is supported by a study conducted by Campbell advocatingthat total meat consumption may offset the detrimental effect of redmeat on people’s health. "

P.S.3. From an email I have sent to my Friend this morning:

Here are my thoughts on the subject. The graph taken from Irina's "Riemann" paper:

Graphs taken from my Kairons:

Elementary solution of the Kairon wave equations have support on hyperplanes tangent to the light cone. Therefore, in particular, the source can produce wave packets that allow for propagation of information so that  we can learn about the "true" («Истинный») state of the source.

By the way, I think that my Hilbert space carries also a natural unitary representation of the conformal group, and that the whole construction can be extended to U(2) or its double cover. But this I left for the future. For convenience I am attaching the Kairons paper again.

Still working on the explicit form of the action of SU(2,2) (or rather SO(4,2)) on the double cover. Step by step.

Best,

ark

P.S.4 

The double cover of U(2) - A+L

P.S.5 Pretty soon we will  have to ask a help from the PC, as there will be a need to do some tiresome calculations. Much of these calculations can be done with Mathematica as it can crunch both symbols and numbers, and also produce graphics. Unfortunately it is quite expensive - unless you are some kind of a student -  then it gets cheaper. For a review of Mathematica see here. Symbolic noncommutative calculations are often simpler to deal wisth using FREE Reduce computer algebra software.

P.S.6. Good news! Physicists are now rediscovering EEQT!. Except that they rename it into a commercial name: "Hybrid Quantum-Classical Master Equations" (2014), and pretend they have discovered it all by themselves, like Lajos Diosi, who knows all my papers but will avoid quoting me.
Some of them, surprisingly, still remember that before "Hybrid Systems" there was EEQT. Example: "The constraints of post-quantum classical gravity" , by Jonathan Oppenheim and Zachary Weller-Davies 2022.
But their understanding of the subject is still rather superficial. Hopefully, with time, they will take all the goodies from my papers, re-own it, and sell to the wide public.
I fully understand why physicists in my Alma Mater town of Wroclaw will never quote me - according to them I have made "wrong choices", I am a "black sheep" - they are all "white wolves", following main stream and a "proper" religion and a "proper" politics. They think that one day I will feel sorry for my choices, perhaps after my death, so they hope.

P.S.7. It so happened that I am forced to visit the forest of homotopy groups. I have zero knowledge of this forest. Beasts are hiding behind every tree. I am completely dumb. Right now I will be looking at Trautman's "Double covers of pseudo-orthogonal groups". It's all new for me! Except, perhaps, of Clifford algebras, where my knowledge is slightly more than just simple zero. Adventures, adventures. Life is certainly not boring.

P.S.7. By "chance" (of course, as always) I have just received a message from Igor Bayak on Linkedin.com, announcing his paper "Chaotic dynamics of an electron". And what do we see in the Abstract?

Abstract

First, we construct the image of the torus on the two-layer shell of the sphere and note that the isometries of the image of the torus on the sphere generate the unitary group U(2), and then we establish that, as a result of the action of the modular group on the sphere, it is factorized in such a way that the minimal (one-element) equivalence classes are given by the set of primes. 

The group U(2)!  When I am just looking for the action of SU(2,2) (in fact O(4,2)) on its double covering space! 

Paying attention left and right

 I continue reading Langan's 52-pages long "book":  "Introduction to Quantum Metamechanics (QMM)".

In September 2022 on substack.com forum Langan wrote:

"Some readers may have heard of the television show “The Secret of Skinwalker Ranch”. Skinwalker Ranch, located in Utah, is one of the world’s premier hotspots for UFO sightings, paranormal mysteries, and “High Strangeness”. "

and then he continued with

"I'm usually quite sympathetic to reports of paranormal phenomena. In fact, my own paranormal experiences were part of what drove me to construct a sophisticated "big picture" theory able to accommodate explanations. The world is swimming in such phenomena, but we've been conditioned not to see or hear them...or rather, to see and hear only the ones intended by their sources, whatever those may be. Unfortunately, some people have muddied the waters by counterfeiting them."

But in his QMM, he is concentrating on quantum mechanics claiming that QM quite often called for explaining consciousness and its problems,  has problems with Reality itself:

"QM merely yields statistical predictions or their outcomes. QM does not include definitions or attributions of being, existence, or reality."

Yet there is a problem with this statement, as Langan is not precise about which QM he is talking about? There are many versions. Some of these versions, the most popular ones, indeed have the status described by Langan above. But if Langan would be paying attention left and right and study the subject deeply, not just reading mainstream papers, he would certainly notice that there are versions of quantum mechanics that were created exactly to overcome these problems. One example is EEQT. In EEQT we do not have problems with reality, at least no more than we have with classical mechanics that Langan is leaving untouched with his criticism.

True, in EEQT as well we predict only statistical averages of observables. We can also simulate individual processes of events, but such simulations are of no practical use, since each particular sequence of events is not reproducible - as in real life.

A similar objection concerns those who state that it is impossible to reconcile quantum theory and general relativity, because "these belong to different categories". It depends on which quantum mechanics one has in mind. In EEQT version there is one common category, namely the category of coupled classico-quantal systems (a precise definition of which  is yet to be done).

While it is true that EEQT is incomplete, and while it is true that it is almost unknown by mainstream physicists, it shows that the above mentioned objections to QM miss the point, and miss it badly.

P.S. Mathematically EEQT formalism is within the category of Semigroups of Positive Maps in Banach *-algebras (cf. Banach algebra with involution - somewhat more general than C*-algebra) and induced Piecewise Deterministic Stochastic Processes. All classical mechanics and field theory (in particular General Relativity) fits into this category if we restrict ourselves to the subcategory of Abelian algebras. Without this restriction, for general algebras, we are dealing with classico-quantal systems, the classical part being represented by the center of the algebra. In the future this category will need to be extended so that the very *-algebra structure becomes an object (dynamical variable).

Quoting from "Towards the theory of matter, geometry and information":

"... Continuing the analogy : in the same way as a gravitational field curves space-time [f6], the information field may curve the state space. May change the geometry of the space of quantum states. May enable the flow of information and of energy through new channels. Now quantum matter gets a worthy partner, just as the gravitational field was a worthy partner to classical matter. The same way as gravitational field is local[f7] in space-time, the information field is local in Hilbert space where " near" means " similar". The geometry of the information field must be, as we have said, a nonlinear geometry. Only in this way can we explain the stability of structures, such as the structure of life. With the phenomenon of life we can in this way, associate a topological invariant (a kind of a vortex) in the nonlinear field of information."

P.S.2.General Relativity will need to replaced by something better. Indication about how that should be done  (pre-metric formulation by Hehl and co-authors) can be found in the paper by Wei-Tuo Ni  "Spacetime structure and asymmetric metric from the premetric formulation of electromagnetism", where, by the way, one of my own papers is being quoted.

P.S.3 It is also interesting how much Langan's statements are close to those contained in some of Laura's writings! Did he secretly read The Wave and High Strangeness?  Or it is  just an accidental coincidence?

P.S. 4. From another discussion group:

On Fri, Dec 30, 2022 at 2:48 PM Kineman-SBOC wrote:

    This is an  update on data from JWST recently released showing candidate galaxies in large number as early as 250my after the supposed big bang. This counts as a revolutionary discovery because theory precluded finding fully formed galaxies that early. Here’s a blurb and a picture:

Paul Werbos

16:19 (1h ago)

do scientific-basis-of-consciousness, Biological, nature

On Fri, Dec 30, 2022 at 2:48 PM Kineman-SBOC wrote:

    This is an  update on data from JWST recently released showing candidate galaxies in large number as early as 250my after the supposed big bang. This counts as a revolutionary discovery because theory precluded finding fully formed galaxies that early. Here’s a blurb and a picture:

    An image of deep space showing hundreds of galaxies against the black void. Galaxies are red, yellow, white and blue

> In fact, astrophysicists are already finding the early universe might be a lot busier than they expected. Stars may have started forming at a much faster rate than some models have predicted. How did matter coalesce and start to form these galaxies early on? We don't know yet. But Webb is, seemingly, already rewriting what we thought we knew about the beginning of, well, everything.

>

> It's an astronomical revolution. So strap in. It's going to be one hell of a ride Implications?  All the current theories of the early cosmos are wrong in some major way, including mine. While the kinematic model predicted we would find fully formed galaxies as far back as we look, it also predicted a general lensing effect and magnification of the view that is not showing up where it should have been. In fact, these fully formed galaxies are at a distance where some standard models predicted quantum soup and no model of galaxy formation allowed for such early development. Everyone is back to the drawing board. The universe is looking flatter and more infinite than anyone expected.

General Relativity and its cosmological models will have to be redone. Together with Quantum Theory.

P.S.5 A thought: consciousness/information is related to "indefinite metric" and to "the universe of negative probability events" (physics speaks about "ghost states", and for a good reason). That is why I insist on *-algebras rather than the subcategory of C*-algebras.  I do not know yet how in details to fill in the blanks. Notice though that Clifford algebras may be considered as particular cases of C*-algebras.

P.S.6 And an interesting recent update:

Paul Werbos

17:27 (37 minut temu)

do Chantal, Mark', Scientific

MANY years ago, I saw articles by Arp in Ap.J., THE astrophysics journal, giving real data which hints at some of the alternate mechanisms. If amount of red shift varies a lot by TYPE of galaxy, it suggests an interaction of light with SOMETHING. That was long before we had photos of dark matter."

I love it that he mentions Arp!!! You may like to check "ORIGINE DE LA VIE Synthèses des théories existantes" Didier Salvignol 2010.