Spacetime and anti-space time theory

 As promised in the last post, here is the beginning of the theory behind the images from the previous post:

The three sets correspond respectively to the yellow, blue and the separating areas on the figures in the last post.

Exercises:

1) Prove that [2u]=[u].

2) For any p  define -p, so that it is well defined.

3) To which of the three sets -p belongs? 

4) Show that 

-M_p⁺ = M_p^-, 

-M_p^- = M_p⁺ , 

-∞_p = ∞_p.

5) Prove that M_p⁺ and M_p^-  are non-empty.

P.S.1. Friday January 27, 9:30 AM: replaced yesterday's version of the file, changing a little bit the notation.

P.S.2. Saturday, January 28, 15:44. Added the last section - dealing with the construction work. It is being constructed right now. Pieces added and removed, until the foundation is firm so that it can survive even earthquakes.

Actually it ends with a "Hypothesis". I will try to check now if it is true or not? If it will prove to be correct, it will be a plus for my intuitions. If it will prove to be wrong - it will mean that my intuitions are not yet good, and that there is is an extra work needed in this particular area. And that is important: to state specific, clear and bold hypotheses, and to work hard to prove or disprove them!

P.S.3. Time "flows" backward in anti-space M_p^-. Whatever that means. That's evident to me. It must be so. Otherwise the Universe would be unbalanced - a dreadful thought.

P.S.4. 9:45 AM. If "our" universe is S^1xS^3, then there should be also the next level universe, which is S^1xS3xS^7. 

P.S.5. I will post the new version of the pdf file on Monday afternoon. The "Hypothesis" will then become a "Proposition".

P.S.6. Today I had the following conversation on my old Polish blog:

30.01.2023 X: I do not fit in your blog, so let me ask here.

"Logically, the sentence: if 1=2 then 3=33 is true. "Did the three squeeze into you, or am I missing something ?
30.01.2023 Arkadiusz Jadczyk
don't see the question....

X:30.01.2023
From the falsehood is false - "3=33", and the blog is that we get a true sentence.

30.01.2023 Arkadiusz Jadczyk

The whole sentence "If 1=2 then 3=33 " is logically true.
Whereas without this "if", the " 3=33" itself is false.
Context is important.

X: 30.01.2023

Mother of one, such a compromise of my favorite logic. I hope this remains a secret. ;)

The intuitive absurdity of the implication triumphed over memory, or maybe it was the stupefaction of discussing the integral from dx.

Thank you for your reply.

30.01.2023 Arkadiusz Jadczyk

Success with dx integrals. Thank you for your question. He who asks does not wander.

Source: https://www.ruspeach.com/en/handbook/proverbs/1878/
© www.ruspeach.com - Russian for foreigners

This causes me to think: such situations are not just restricted to abstract logic. They happen in real life. Suppose you tell your friend: If you do this, then I will do that. Time goes on, and, after a while, your friend goes around complaining about you: "He had promised me that he will do that, and he has failed to keep his promise!" The "If" part, that he has failed to fulfil,  is conveniently forgotten. His friends will never learn about the "if" part, and even if some of them do learn, it will be neglected as unimportant. Such is our tendency to look blameless, smart, and innocent in our own eyes. While such a tendency is probably hardwired in our brains - it only shows that our actions do obey the good old Aristotelian logic that we supposedly know so well.  

(Warning: no one should take the above as a "hidden message". Nothing is hidden there, and I did not have in mind any particular person. I am writing about what I consider a general useful wisdom.)

P.S.7. Monday 15:50: I am still not happy with the actual version of the pdf. It still needs verification and corrections. I am currently working on it.

P.S.8 Monday 18:50:  I am not finished yet, as I did not yet describe how the images has been created. This will come tomorrow.

P.S.9. Tuesday 7:50: Today is this "tomorrow". I decided to leave the creation of the images as an exercise for the Reader. All the theory that is needed is already described in details. More important is now to connect the SO(4,2) description with the SU(2,2) description. In their ingenious paper (see the reference in the pdf file) Wojciech Kopczyński and Lech Stanisław Woronowicz provided such a connection. 

W. Kopczyński and L.S. Woronowicz, A geometrical approach to the twistor formalism, Rep. Math. Phys. Vol 2, pp. 35-51 (1971). It is a paper of an indescribable beauty!!!  A true Spring Sonata. Therefore in the next posts I will write about my own little variations on this subject. My main addition, my little twist,  will be: adding the orientation. In the SU(2,2) description, till now, there is no place for "anti-space". Anti-space, anti-time, anti-matter, anti-gravity - that's what I love! There would be no life, no stability without these "antics".  

P.S.10. Tuesday 16:15: Change of mind. I still have to define the conformal structure (i.e. metric up to a variable scale factor). And then to study null geodesics (light rays). This can be done without using SU(2,2). Though SU(2,2) will add an extra perspective. But that's for later. Practical question: can light travel between space and anti-space? We need to know!

P.S.11. Wednesday 9:11: By "chance", searching the net for "graph of a unitary operator" I have found this amazing video demonstrating how linear algebra can enhance and lighten our lives:

 It also shows that in fact there exists a left-handed anti-universe.

Spacetime and anti-spacetime

 Here is the graphics representing the Minkowski space (yellow) and Minkowski anti-space (blue) embedded in the double cover of the conformal compactification. They are separated by (double cover of) the conformal infinity. Can light travel between spacetime and anti-spacetime? I am going to find an answer to this fascinating question.

And from the yellow side:

To be continued.....

P.S.1 Currently I am recalculating everything and finding really annoying misprints in my old papers. They drive me crazy!  Yesterday I have started an exchange with a well known mathematician in Novosibirsk, one of the authors of a Springer monograph about Riemannian geometry and geodesics,  also interested in conformal infinity. He tried to read my old papers and wasn't very happy, so I am preparing a CLEARLY WRITTEN DOCUMENT FOR HIM. Being very careful this time.

P.S.2.My new friend, mentioned in P.S.1, send me today his new paper - and introduction to Segal's chronometric theory (in Russian). There are some points when we agree (topology) and some when he disagrees with me (differentiable structure). We have to agree on everything before the publication of the paper. So I am now busy so that I do not have time to eat!

P.S.3. In his paper I have found new to me realization of the universal covering of the group U(2) in terms of quaternions. How could I not see it all by myself?! So finally we have some good use of quaternions in physics!!! Progress!

P.S.4. We now have snow in the sothern France. Yesterday is also snowing. Looks like we have a climate change? A while ago one of our intertnet cables got disconnected from the pole. Last time it happened on February 24 last year. It was the time of the start of the Russian "denazification" of Ukraine. Is it going to be a sign of a new decisive Russian offensive?

P.S.5. Valera (first name of my Russian friend, mathematician) sends me a fourth version of his 21 pages long paper - after his midnight!. It becomes more and more clear (the paper). His approach is through the unitary group  U(2), while I prefer to play with isotropic subspaces in R^6. Now even one result from my Quantum Fractals book is being quoted as useful, though it is concentrated on somewhat different issues.

P.S.6. I have to learn about "manifolds with corners". Perhaps the conformal infinity has what is called a "corner"? A corner of the universe? (Ark Fleet Ship B is being used  there). I don't know yet for sure.

P.S.7. Probably the best book on smooth manifolds: John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer 2012. Table of content can be perused here. Includes concise appendices on linear algebra, on topology, sections on categories and functors, manifolds with corners, lot of examples.

.P.S.8. While working on the final version of one of the papers (in order to make the reviewers and the editor happy), and it will be sent, updated,  to the publisher today or tomorrow, so I will be able to return to the problem of light travelling along the conformal infinity on the boundary between space and anti-space) I have found (by chance of course) a very interesting paper that opens for me some new doors I was looking for a long time:

"Explicit construction of a time superoperator for quantum unstable systems"

November 2001 Chaos Solitons & Fractals 12:2591-2601

DOI: 10.1016/S0960-0779(01)00074-1

Gonzalo E Ordonez, Tomio Petrosky, Evgueni Karpov, I. Prigogine

Abstract: 

A time superoperator T conjugate to the Liouville superoperator LH=[H,] is constructed for a quantum system with one excited state or unstable particle. While there is no time operator conjugate to the Hamiltonian in the wave function space due to the positivity of energy, T may exist in the density matrix space as the spectrum of LH covers all the real axis. This is the first example of an observable that can only be formulated in the Liouville–von Neumann space of density matrices. In our example the expectation value of T gives the lifetime of the unstable particle. Once the time superoperator is obtained it is easy to define an entropy superoperator.

P.S.9. A mathematically more precise paper on the same fascinating subject:

Adolfo R. Ordonez, "Rigged Hilbert Spaces associated with Misra-Prigogine-Courbage Theory of Irreversibility", published in ; Physsica A 252 (1998) 362-376

P.S.10. In fact Prigogine's idea of considering "superoperators" fits perfectly the subject of this note. In finite-dimensional case the space in which superoperators act is H⊗H*, where H* is the dual Hilbert space (anti-space), and the spectrum of super-Hamiltonians made of ordinary Hamiltonians consists of the differences of the eigenvalues, thus is always symmetric: if a1-a2 is positive then a2-a1 is negative.

P.S.11. In fact I was playing with "superoperators" probably before the term was made famous by Ilya Prigogine. Here is a Remark I have made in my second published paper (that became a part of  my PhD Thesis) :

:

P.S.12 And today (Thursday January 26) we will have a close encounter: "In fact, this is one of the closest approaches by a known near-Earth object ever recorded"

 "The asteroid will be closest to Earth at 4:17 p.m. EST (2117 GMT) that day. At that point, it will be about 2,200 miles (3,600 kilometers) above the planet's surface, according to NASA."

P.S.13. I am still trying to find an aesthetically satisfactory way to justify mathematically the two images from this post. I am getting closer and closer.

Conformal compactification of the Minkowski space and its double cover

In the previous note Constructing the conformal infinity we have started with one-point compactification of the Euclidean space and ended up with what is called the conformally compactified Minkowski space. In this note we will start with the end of the previous note and study the emerging structure in some detail. Therefore, let V denote R⁶ with Cartesian coordinates x,v,w,

x= (x¹,x²,x³,x⁴) ∈ R⁴,

endowed with the quadratic form

q(x,v,w) = x²+v²-w²,

where

x²=(x¹)²+ (x²)²+ (x³)2- (x⁴)².

We define the “cone” C as the set of all (x,v,w) satisfying

q(x,v,w)=0, (x,v,w) ≠ 0.

So we have removed the origin x=v=w=0.

C inherits its topology from R⁶: open sets in C are intersections of open sets in R⁶ with C.

There are two equivalence relations in C that are of interest, we will denote then R and R+. They are defined as follows

(x,v,w) R (x',v',w') if and only if (x',v',w') = a(x,v,w), a∈R, a≠ 0

(x,v,w) R+ (x',v',w') if and only if (x',v',w') = a(x,v,w), a∈R, a>0

We take the quotients of C by these equivalence relations and denote them

M^c and M^2c respectively:

M^c = C/R,

M^2c = C/R+

We call M^c the compactified Minkowski space, and M^2c the double compactified Minkowski space.

We endow both sets with the quotient topology. Thus open sets in M^c are defined as those of which the preimages by the natural map from C to C/R are open in C. Similarly with M^2c .

We have a natural covering map from M^2c to M^c . Let us denote by [(x,v,w)] the equivalence class of (xv,w) with respect to R, and by

[(x,v,w)]₊ its equivalence class with respect to R+. Then [(x,v,w)] is the set

[(x,v,w)] ={a(x,v,w): a ≠ 0}

while

[(x,v,w)]₊ ={a(x,v,w): a > 0}

We can say that [(x,v,w)]₊ is a half-ray through (x,v,w), while [(x,v,w)] is the whole ray (excluding 0) through (x,v,w). Every half-ray is contained in a unique ray, and half-rays [(x,v,w)]₊ and half-rays [(x,v,w)]₊ and [-(x,v,w)]₊ are contained in the same ray. Therefore the natural map M^2c to M^c is 2:1.

The group O(4,2) is the group of linear transformations of R⁶ leaving invariant the quadratic form q. Its subgroup SO(4,2) consist of transformations given by 6x6 matrices of determinant 1. Since the transformations are linear, they transform equivalence classes of R and of R+ into equivalence classes, therefore they define transformations of M^c and M^2c.

We now show that the action of SO(4,2) on M^2c is transitive and that M^2c is (pathways) connected.  

Let [x,v,w]₊ and [x',v',w']₊ be two different points in M^2c. We have q(x,v,w)=0 and q(x',v',w')=0. Therefore

(x¹)²+ (x²)²+ (x³)² + v² = w²+(x⁴)²

and the expression above must be >0 since the origin is excluded. We can now replace (x,v,w) by the unique element in the same equivalence class for which the right hand side =1. We do the same with the second point. So we have

(x¹)²+ (x²)²+ (x³)² + v² = w²+(x⁴)²=1

(x'¹)²+ (x'²)²+ (x'³)² + v'² = w'²+(x'⁴)²=1

We can now continuously rotate (x¹, x², x³, v) to (x'¹, x'², x'³, v') using a rotation from SO(4) and, at the same time, continuously rotate (w,x⁴) to (w',x'⁴) by an SO(2) rotation. The two rotations together, written as a block diagonal (4+2)x(4+2) matrix,  give us a rotation in SO(4,2). The path is continuous in M^2c , because the topology in the quotient space is defined so that the quotient map is continuous. Therefore SO(4,2) acts transitively on M^2c and also M^2c is pathways connected. By projection (which is continuous) we get the same results for  M^c .

Assuming only one space dimension, therefore neglecting x² and  x³, here is what I got for a graphic representation of  M^2c , and the double conformal infinity (v=w):

Note: The numbers on the graphics are irrelevant.

Mathematica code used for the graphics:

torus2[a_, b_][fi_,psi_] := {(a + b Cos[fi]) Cos[psi], (a + b Cos[fi]) Sin[psi], b Sin[fi]}

mc = ParametricPlot3D[torus2[8, 3][fi, psi], {fi, 0, 2 Pi}, {psi, 0, 2 Pi}, Mesh -> None, PlotStyle -> Directive[Opacity[0.6], LightGray, Specularity[White, 10]]]

inf1 = ParametricPlot3D[torus2[8, 3][fi, fi], {fi, 0, 2 Pi}, PlotStyle -> {Black, Thick}]

inf2 = ParametricPlot3D[torus2[8, 3][fi, Pi-fi], {fi, 0, 2 Pi}, PlotStyle -> {Black, Thick}]

Show[{mc, inf1, inf2}]

Notice that M^2c is, topologically, S¹x S³ rather than S¹x S¹ as on the picture, where space is one- instead of 3-dimensional.

In the next post we will draw images of the Minkowski space on the torus, and we will see how time translations are represented there.

P.S.1. Something else to think about, that Laura brought my attention to:

 By the way a very good example of how to give a talk!

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: