Photon's eversion

 Our daughter Amy suggested to me a book to read, "Eversion" by Alstair Reynolds

From the master of the space opera comes a dark, mind-bending adventure spread across time and space, where Doctor Silas Coade is tasked with keeping his crew safe as they adventure across the galaxy in search of a mysterious artifact.

‘In mathematical framing, the definition is rather more rigorous,’ Dupin said. ‘But you have the crude sense of it. As a matter of fact, the problem is of interest to me. Consider the question of sphere eversion in a three-dimensional space—’ (...)

 ‘You can’t see it, but it’s possible. It’s a sphere, you see?’ He pinched the paper between two extremities, making it balloon out like a soggy attempt at a paper lantern. ‘It’s a sphere. I think it was always a sphere, and then something went wrong. That’s the topology.’

I looked to the land speeding by us on either side of Demeter. ‘Topography?’

‘No. Topology. Surfaces and volumes. Transformations. Homotopic transformations. Can you see it, doctor?’"

Inversion, Inversion, Eversion Inside out. Conformal transformation. Maxwell equations are conformally invariant. Previous post was about Krishna and photons. I had to redo almost everything, calculate everything from scratch. Here is the result

P.S.1. 28-10-23 17:57 Replaced with a new version. Added Remark 1.4 and the beginning of Sec. 2.

P.S.2. 29-10-23 8:32 This is what my friends and my girls got me for my birthday two days ago: Nobel Prize, socks  and a handy hammer, also a set of (neo)-Platonic (crystal) solids (plus many other goodies not shown ...)

P.S.3. 29-1027 10:33

One of my favorite quotes.

We must regard the present state of the universe as the effect of its past and the cause of its future. Consider an intelligence which, at any instant, could have a knowledge of all forces controlling nature together with the momentary conditions of all the entities of which nature consists. If this intelligence were powerful enough to submit all this data to analysis it would be able to embrace in a single formula the movements of the largest bodies in the universe and those of the lightest atoms; for it, nothing would be uncertain; the future and the past would be equally present to its eyes. ~ Pierre Laplace

Now we need only to make it compatible with chaotic behavior of some  nonlinear systems, and adjust our understanding of synchronicity and quantum phenomena accordingly.

P.S.4/ 30-10-23 12:39 Good news. Essential progress with the paper I am working on. Yesterday night I was still thinking that the only way to get the result that I have envisaged was by using a brute force. But today I realized that, with the right insight, it will cost almost no work at all, and that it can be done in a full generality that I didn't even dream about before!😁😁😁

P.S.5. 30-10-23 13:01 And here is another B-Day gift that I received from my dear friend Pierre Lescaudron.

The Periodic Table of Elements. Pierre passed away on October 19, so the fact that I have received this beautiful gift a week after is an  additional reason to ponder about the mysteries of love, densities and of time.  

P.S.6. 31-10-23 11:18 (Suggested to me by Laura)

P.S.7. 31-10-23 11:44 Signs conventions (from Misner, Wheeler, Thorne "Gravitation")

P.S.8. 31-10-23 13:39 Updated the file. Added beginning of Sec. 2.2 "Action of pure boosts"..

P.S.9. 01-11-23 9:39 The prototype of  the Time Machine based on the Properties of Light and Time Eversion.

P.S.10. 02-11-23 13:39 Completed the proof of Lemma 2.1. Added Acknowledgements at the end. Now it is time for calculating the limit (70). With the handy Lemma it should be now a piece of cake.

P.S.11. 02-11-23 16:22 There is still a possibility (subjectively 50% probability) that the Lemma is false, its proof has errors, and that there is no limit as in Eq. (70). If this reasoning can be fixed - I will find a way to fix it. So more work is needed. I have no idea now how long will it take. That is what I will be working on now.

P.S.12. 02-11-23 18:40 Removed the part containing Lemma 2.1 and its proof. It leads nowhere, even if it is correct. Totally useless work. Discovered one more way how NOT to make the light bulb. Leaning from errors.

P.S.13 05-11-23 12:37 The Dirac Sea

I wrote in a reply top a comment by Natus Videre:  "And what is space?" Perhaps it is the Dirac sea of some kind?

Quoting from the paper by a mathematician  J. Dimock "The Dirac Sea":

" Dirac’s resolution of the problem was to to assume the particles were fermions, invoke the Pauli exclusion principle, and hypothesize that the negative energy states were present but they were all filled. The resulting sea of particles (the Dirac sea) would be stable and homogeneous and its presence would not ordinarily be detected. However it would be possible to have some holes in the sea which would behave as if they had positive energy and opposite charge. These would be identified with anti-particles. If a positive energy particle fell into the sea and filled the hole (with an accompanying the emission of photons), it would look as though the particle and anti-particle annihilated. The resulting picture is known as hole theory. 

(...)

The basic idea is that if an n-fermion state is modeled by an n-fold wedge product of Hilbert spaces, then the Dirac sea should be descibed as an infinite wedge product of Hilbert spaces."

Sailing the Dirac Sea.

P.S.14. 06-11-23 11:49

Paul Dirac and Poetry

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it’s the exact opposite!  

Paul Dirac

P.S.15. 08-11-23 8:29 Dreaming about the Ripples on the Dirac Sea

"Geoff Landis has just completed a postdoctoral fellowship at NASA’s Lewis Research Center in Cleveland. He writes science fiction grounded in the hard sciences, but his first story, “Elemental,” a Hugo nominee for best novella of 1984, dealt with magical matters in a scientific context and appeared in Analog. Later work has been published in Isaac Asimov’s Science Fiction Magazine, Pulphouse, and Amazing Stories.

About his award-winning short story, Geoff writes, ” ‘Ripples in the Dirac Sea’ was an experimental story for me. Quite a number of disparate threads wove into the final narrative. One important thread was my feeling that a story involving time travel should have a nonlinear narrative to reflect the discontinuous way the characters experience time.

“I also wanted to see if it was possible to write a story in which real physics is presented. Very little of modern SF goes beyond the early quantum mechanics of Heisenberg and Schrodinger, work which is admittedly remarkable and beautiful, but by no means the end of the story. Here I tried to invoke some of the strangeness and beauty— I might even say sense of wonder—of the physics of Paul Adrien Maurice Dirac. In ‘Ripples’ I decided to explore the inconsistency between Dirac’s relativistic quantum mechanics and the mathematics of infinity developed by Cantor and others (as far as I can tell, a quite real inconsistency). The Dirac sea is also real, not an invention of mine— despite the very science-fictional feel of an infinitely dense sea of negative energy that surrounds and permeates us."

But I'm telling you: The infinite dense sea of 
Bose-Einstein condensate of light
is even more intriguing!

P.S.16. 09-11-23 8:17 

• R.L. Forward "Negative matter propulsion".
• G.A. Landis "Comment on 'Negative matter propulsion'".
  
  Negative mass the same as negative energy? I don't think so. What about "anti-photons?"

P.S.17. 09-11-23 10:53 My negative mass/energy space propulsion project:

P.S.18 09-11-23 17:28 Changed notation. Replaced omega with E (for energy). Fixed Proposition 2.1. Added its Proof. Now it is time to verify and to find and correct possible errors.

P.S.19. 10-11-23 7:49 Something is still wrong. The matrix is indeed orthogonal, but it does not do what I am expected it to do. Back to the drawing desk. Chase the error. The devil must be hiding in some tiny detail. Errare humanum est. But WHY? (Bjab?) Why some people make more errors than other?????? Errare humanum est, perseverare diabolicum. I know. But one thing is to know, and completely different thing is creatively apply this knowledge to fix one's errors for good!

P.S.20 16-11-23 7:50 Finished Sections 2.1 and 2.2. Added plots of vector fields.

 Animation of the combed sphere  came not the way I intended.... Can cause a headache!

But I do not have time now for playing with animations of rotations. Now calculating the torsion tensor of the induced teleparallel connection. Deadline for finishing the paper: December 30.

The vector fields e1 and e2, are mutually orthogonal and, in x-space satisfy the equations

div e1 =div e2 = 0. But they should be thought of as Hertz potentials, and not as electric and magnetic fields.

P.S.21 19-11-23 10:41 Houston, we have a problem.

P.S.21 200-11-23 10:52

P.S.22 20-11-23 11:54

"... His truth is marching on. ..."

P.S.23 21-11-23 13:13 Laura is telling me that it is not Elvis, who died in 1977, singing this song which was written by Leonard Cohen in the 80s. And no, it is not time travel either. It is Ron Jesse who is singing the song.

P.S.24 21-11-23 18:19 In stereographic projection coordinates the teleparallel affine connection and torsion come out beautifully! Somewhat different than those in the  papers by Staruszkiewicz, but so much simpler, prettier and more elegant! Tomorrow will update the file. Haleluja!

P.S.25 23-11-23 12:12 Reading "The Late Summer Passion of a Woman of Mind" by Rebecca Goldstein:

"... Eva had come to the conclusion that there is something seriously amiss with her book.

(...)

Time, Eva now thought. It was time that contained the ambiguity. She had not come to terms with the concept of temporality in her book. She had not shown, had not even taken up the question, of how the systems of Plato and Spinoza, which presented reality as eternal and changeless, held by tenseless logic in the stasis of necessity, could accommodate even the notion of passing time. How is it that our sense of the flow of time, which so dominates our experience of thi world, can arise? To say that it is ultimately an illusion does not solve the problem. How can there be room for even this deception, in such systems as these?"

P.S.26 23-11-23 19:23 Added Sec 2.3.1. Stereographic projection 

and 2.3.2 The connection coefficients. Added references to Staruszkiewicz's two papers. Updated the file.

P.S.27 25-11-23 17:15 Taking break for studying recent papers on the same or related subjects. Learning and thinking. Thinking and learning. Meditating. Brainstorming.

P.S.28 26-11-23 11:11 Laura's find on X:

Unable (temporarily) to answer another important question: "What is time?", I consider related question: What is (timeless) photon? And how it relates to the leptonic magnetic monopole of Georges Lochak? Clifford algebras are there.

P.S.28 26-11-23 11:36 Human consciousness is a local debugging unit with the ability to read the wave. It has also very limited wave writing capability. Most of these units are faulty. But that is how Nature works - creating abundances of everything (except for magnetic monopoles which seem to be very rare a specie).

P.S.29  26-11-23 12:47 Coincidence? After writing the previous P.S. I went home to get a lamp (I need more photons!) and found Laura in the laundry room reading this: 

She did not read the previous P.S. But she must have read my mind Or I read her mind. Or simply colinearity - reading the same wave at the approximately the same time.

P.S.30 26-11-23 18:14 Added three new highly relevant references to the paper in progress. In the next couple of days will add new stuff with my comments about these references.

P.S.31 27-11-23. From "Single Photon Manipulation". IntechOpen 2020, p. 15

"During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. "

P.S.32 27-11-23 15:08 And so, from today on,  I become a member of the Editorial Board of the MDPI journal "Mathematics", section Mathematical Physics.  Perhaps because I feel as being more a mathematician than a physicist... I respect and adore physics, but my true love is (applied) mathematics.

P.S.33 01-12-23 10:30

While reading Daniel Kastler, "Introduction à l'électrodynamique quantique", Dunod (Orléans, Impr. nouvelle), 1960, In "Avant propos":

"...Indépendamment de son utilite, l'unité des sciences physico-mathématiques nous semble être buts de l'humanisme scientifique."

P.S.34 02-12-23 12:07

The Emotional Journey of Creating Anything Great

 P.S.35 02-12-23 12:27

Oppenheimer

P.S.36 02-12-23 18:26 Reading Christine Li, "5-minute self-discipline Exercises", p. 137

"List five things that you are thankful for:"

P.S.37 03-12-23 10:06 Reading Philip Goff, "Why? The Purpose of the Universe"

"(...) Philosophy for me isn’t just an abstract exercise but something I live out. During my ethical subjectivist teenage years, I kissed my best friend John’s girlfriend, telling him to his face that it was okay because ‘good’ and ‘bad’ depend on how you feel and it didn’t feel bad. For his part, John decided on a creative response. With the help of a few friends, he kidnapped me, tied me up, and bleached my hair bright white. Even in the absence of moral objectivity, there are ways of stopping people doing shitty things. (...)"

Ibid:

"My favourite interpretation of quantum mechanics is the most boring and unsexy of all of them: the pilot wave interpretation." 

I criticized it in my Quantum Fractals (perhaps partly because I was jealous?) . But now, when I read that it is "boring and unsexy" - perhaps will change my views, reconsider. It happens once in a while that what we at some point of time dislike and even hate, later on becomes our favorite!

11:37: I still think that it can be happily merged with EEQT. It will then become more sexy and less boring (discrete events!), but without losing its attractive features. Or, better, add the pilot-wave feature to EEQT! Should be doable! In EEQT between jumps the classical system follows the classical mechanics. But it may as well be governed by the pilot-wave guided evolution. Except that  it will be now dissipative.

 P.S.38 05-11-23 19:39 While reading "Non-Commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts" by 7 authors (two of them I have met in the past)

There are anti-photons!

Bjab should be happy!

P.S.39  06-12-23 18:36 Philip Goff is is overusing the Ockham razor principle. As Laura has noticed, when we were discussing the subject,: if Nature would use the Ockham razor, we would not be here discussing the question "Why?". 

P.S.40 10-12-23 Failure! Defeat! I am depressed. Tried to read the paper

"Action of the conformal group on steady state solutions to Maxwell's equations and background radiation" 

by Bertram Kostant, Nolan Wallach

After five or six pages the mathematics there became completely unknown to me. And I know that it is a very important paper - necessary for understanding consciousness. But in the paper we find category theory, exact sequences,  algebraic topology,  harmonic analysis - and lot of other things that I have rough idea about, but not working knowledge. Disaster. It will take me a month or more to learn, to fill out the holes in my mathematical education. They say:

Perhaps. Should I feel inspired by my failure? I should and I am.

P.S.40 15-12-23 19:42 Finally finished preparing my complete list of publications with links to all published papers. Here it is. It has a title line in Russian, because it was prepared for Math-Net.Ru.