An SO(2,2) Iterated Function System Part 3 - Weyl spinors and null vectors

 Introduction.

In the early 1960s, Penrose was deeply preoccupied with a fundamental problem in physics: how to describe the geometry of spacetime in a way that naturally incorporated quantum mechanics and the behavior of light. He was frustrated with the standard mathematical tools and felt there must be a more profound, elemental description of reality.

The pivotal moment came in 1963. Penrose was a visiting professor at the University of Texas at Austin. He was not in his office, but was driving with a colleague (some accounts say it was the physicist Ivor Robinson) outside of the city.

As he was gazing out the car window at the flat, featureless Texas landscape, his mind began to wander.

The long, straight highway and the vast, open horizon triggered a shift in his perspective. He started thinking about the paths of light rays—how they could be seen as fundamental, and how points in spacetime might be a derived concept from the way these light rays intersect.

The key insight was this: Instead of thinking of space as the primary concept and light rays moving through it, what if he reversed the roles? What if the light rays (the "null lines" or paths of photons) were the primary objects, and a "point" in spacetime was defined as the set of all light rays passing through it?

This was the genesis of twistor theory. In that moment, he realized he needed a new mathematical space—what he would later call twistor space—where each point represents a light ray in our physical spacetime. The geometry of our universe could then be encoded in the complex geometry of this twistor space.

The Humorous Aftermath
The anecdote often includes a charmingly human detail. The flash of inspiration was so intense and all-consuming that Penrose, excitedly trying to explain his radical new idea to his colleague, began scribbling equations and diagrams on the car's dashboard.

One can imagine the driver's mixed feelings about having their car used as a blackboard for groundbreaking theoretical physics!

We continue from An SO(2,2) Iterated Function System Part 2. The two-dimensional real vector space R² of Weyl spinors for Spin(2,2) carries an SL(2,R)-invariant bilinear form ε. In order to distinguish spinors from vectors, from now on, we will use Greek letters ψ etc. to denote the elements of R² endowed with this form. Thus

ε(φ,ψ) = φ^Tωψ.                (1)

It is then natural to introduce what physicists call the Dirac conjugated spinor

φ^bar = φ^Tω,                 (2)

so that the invariant bilinear form ε can be written as

ε(φ,ψ) = φ^bar ψ.                (3)

Exercise 1. Verify that φ^bar φ = 0 for all φ.

Null Vectors from Weyl spinors

Now we we are ready redefine the construction of determinant zero matrices from spinors. We define now

X(φ,ψ) = φ ψ^bar = φ ψ^Tω.                (4)

Now, for (S,T) in SL(2,R)⨉SL(2,R), we have

X(Sφ,Tψ) = S X(φ,ψ) T^-1,                 (5)

the standard transformation law of vectors x represented by 2⨉2 matrices x^.

Exercise 2. Verify Eq. (5).

Note. Our "spinors" are spinors of the group SO(2,2), which is the conformal group of R^1,1. Therefore we should, in fact, call them "twistors". They are of course our toy baby  twistors. The "adult" twistors of Roger Penrose are spinors of SO(4,2).

Spinors from null vectors.

In the construction below we will first take a purely pragmatic approach, without discussing its geometrical meaning. So, let

A={{a,b},                    (6)
      {c,d}}

be any nonzero matrix of determinant zero. We will show that A is necessarily of the form

A =  φ ψ^{bar                (7)}

for some φ, ψ.

Since it is at non-zero matrix, at least one of its elements must be non-zero. Suppose it is the first row, first column, element A₁₁ = a. Let us define  φ' to be the column vector equal to the first column of A:

         a
φ' = (    ),                (8)
        c
and let ψ'^bar be the first row

ψ'^bar = (a,b).                (9)

Construct A' =φ' ψ'^bar . This the matrix

A' ={{aa,ab},{ca,cb}}.                (10

The zero determinant condition implies ad = bc. We can thus substitute cb in A' by ad. Then A' becomes

A' = {{aa,ab},{ca,ad}} = a {{a,b},{c,d}} = a A.                (11)

Therefore setting

         1
φ = (    ),                (12)
       c/a

ψ'^bar = (a,b),                (13)

solves our problem.

Exercise 2. Can a similar reasoning be used assuming, for instance, that it is b ≠ 0, instead of a ≠ 0 as above?

Exercise 3. Show that the decomposition (7) is essentially unique, that is that if  φ', ψ' is another solution of (7), the there is a constant λ≠0, such that

φ' = λφ, ψ'= (1/λ)ψ.                (14)

Hint: the exercise may need the concepts of a kernel and a range of a matrix considered as a linear operator. Using AI for help is allowed, and even encouraged.

In the next post we will solve the same problem but replacing real numbers with integers. Since division is not allowed within integers, we have have to use a more sophisticated approach in that case.

To be continued...

AI at work, Part 2

From my today's chatting with GPT: Response to my Prompts

Created by Kandinsky 3.5

Title: "The Quest for Twistor Truth"

Concept Description:

1. The Labyrinthine Library:

A vast, surreal library with endless twisting hallways and floating bookshelves. Books have keywords and references written on them, such as "Null Lines," "Minkowski Space," and "Twistor Theory."

2. The AI Guide:

A glowing orb with a halo of spinning binary code. The AI is holding a large magnifying glass and appears to be frantically searching through books and scrolls.

3. The Patient Scholar:

A calm, wise scholar wearing a whimsical outfit and a thinking cap adorned with mathematical symbols and equations. The scholar has a bemused smile.

4. Mischievous Imps of Misinformation:

Small, mischievous creatures scattering fake books and incorrect references around the library. They have playful, cheeky expressions.

5. The Twistor Vortex:

A swirling vortex in the center of the library, representing the complex topic of twistor theory. Null lines and mathematical symbols spiral into it.

6. The Illuminated Path:

A bright, illuminated path leading the scholar and the AI guide towards a glowing book labeled "Truth." This book represents accurate and verified information.

Surrealistic Elements:

Floating clocks showing different times, symbolizing the timeless nature of the search for knowledge.

Books morphing into birds and flying around, representing the elusive nature of finding the right information.

Twisted, Escher-like staircases leading to nowhere, symbolizing the complexity and confusion of the topic.

Created by DALL-E 3 v.1

"I apologize for the repeated errors in my responses today. Here are a few hypotheses that could explain why these errors occurred:

Misinformation in Source Data:

The information I provided may have been based on inaccurate or outdated sources. Although I strive to offer accurate data, occasionally, errors from source materials can propagate into responses. 

 

Miscommunication or Misinterpretation:

There might have been a miscommunication or misinterpretation of the specific materials and references you were seeking. Ensuring a precise understanding of your needs is crucial. 

 

Complexity and Specificity of Topic:

The relationship between null lines in Minkowski space and twistors is a specialized and complex topic. This complexity increases the likelihood of referencing inaccuracies or less accessible materials. 

 

System Limitations:

As an AI, I rely on a vast array of data but may lack specific, up-to-date bibliographic details on niche academic papers. This limitation can result in incorrect references.

To address these issues more effectively, I will ensure that I use more reliable databases and references for your specific request. I can assure you that there is no deliberate programming or intention by my developers or handlers to provide incorrect information in response to questions containing specific keywords. The goal of this AI is to provide accurate, helpful, and reliable information to the best of its ability.

The errors in my responses can be attributed to a few common factors:

Complexity of Topic:

Niche and highly specialized topics like the relationship between null lines in Minkowski space and twistors are complex, which increases the chance of unintentional inaccuracies. 

 

Source Data Limitations:

The AI relies on a vast dataset compiled from diverse sources. Occasionally, errors or outdated information from these sources can influence responses.

Context and Interpretation:

Misinterpretation of your requests or the nuances of specific academic references can lead to mistakes. Ensuring precise understanding and context is challenging. 

 

System Limitations:

Despite extensive training, there are inherent limitations in the AI's ability to access real-time data or verify specific details like ISBN numbers or paper availability in real-time.

There is no motive or programming directive to provide incorrect information. The intention is always to be as accurate and helpful as possible. If you have further questions or need assistance with other topics, I'm here to help! "

 

Created by DALL-E 3 v.2

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!