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.
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.
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ReplyDeleteI have this strange question today: can escape to the spiritual world or the world of structures be both an escape from temporality and mental pain? What is the relationship between mathematical structures and perception? Why do these structures become different in our minds when we look at the world from a higher perspective?
ReplyDeleteAnd another question: is this higher perspective not precisely what physics is looking for? And is it possible to build a meta-level physical theory without using meta-mathematics?
It certainly makes sense that an afterlife between incarnations might ease the regrets of the current incarnation and be less stuck in time. Perceiving a thought seems like lots of spacetime vertices would be connected without time restrictions in order to form that thought.
DeleteAt the highest level I think it all looks like integer number patterns like in the Pascal triangle. How that codes a thought I don't know. I can think of something like personality having a Pascal triangle-like information structure, but for getting more into the details of a thought, I have no idea.
And now the question is what structures to use to best represent the experience at that level at this level?
DeleteHow do we see elements of these experiences in other structures and in the scientific articles we read?
So this is again a question that comes up often: How do we bring physics to life? What is it missing, why does it seem to be suffering so much?
Obviously one problem most physicists have is that they will do all sorts of ad hoc things to maintain a rigid version of causality and time. I once read about some very complicated extra things David Deutsch was doing just to maintain a rigid causality for his many worlds model. If your basic model is telling you to ease up on a rigid causality then maybe you should look at that possibility too.
DeleteSimilarly I once saw Garrett Lisi conversing in the comment section of a blog when someone said "isn't that just the conformal group" and Lisi answered "no I won't go there". People often won't go where there own models are suggesting because it's too outside rigid mainstream views.
Fundamentally maybe you can start just thinking of spacetime basis vectors with a transverse, longitudinal and time structure and then realize there's also an internal spacetime version of this and then realize something like personality has the same structure and then see where all these basis vectors lead and I think that includes fun things like differentiable manifolds. This spacetime basis vector structure would relate to Pascal's triangle via Bott periodicity.
@ John G
DeleteThis is one of the most beautiful descriptions I have ever read!
I see it very much the same way. Thank you for being there. You are someone who confirms to me that my thoughts are flowing in the right direction!!!
@ John G
ReplyDeleteOther question:
Are there errors at the foundations in physics, errors so profound that they cannot be corrected and physics has to be created from scratch, or is there a way out that involves exceptionally clever bypassing patterns? How do you see it?
Is physics wrong? After all, it agrees multifacetedly with our observation. But somewhere there are contradictions that seem impossible to reconcile. Can they be reconciled? Or did the contradictions arise because the definitions were incorrect from the very beginning?
From things I've read, I tend to think of major problems like renormalization or getting to the center of a black hole as eventually being eliminated via off the beaten path kinds of ideas. For renormalization supposedly the dilation of the little used conformal group might help; for black holes, you can think of a Planck mass black hole as a quantum not classical particle (sum over histories of all first generation particles allowed at a single vertex by the Pauli exclusion principle) and maybe this will help. So I think it's more impossible tends to be impossible in a conventional way and almost everybody wants to stick to the conventional way.
Delete@ John G
ReplyDeletePhysics and psychology are in fact very similar. Both sciences are like exploring the world - it is essentially the same world every time. Only the name is different. However, the worlds of physics and psychology are separated by time.
I understand myself better by reading about mathematical concepts and formulating new ones. I formulate them better when I learn about psychology and how to understand aspects of myself that I don't like.
Whatever we study - we study essentially the same thing. But we are learning to use different tools, we are learning many different methodologies. Later we can see something in them and transfer it to our new concepts. Life is essentially a religion...
"November 2001 Chaos Solitons & Fractals 12:2591-2601
ReplyDeleteDOI: 10.1016/S0960-0779(01)00074-1
Gonzalo E Ordonez, Tomio Petrosky, Evgueni Karpov, I. Prigogine".
Here is the article in full, but the PDF cannot be downloaded:
https://www.sciencedirect.com/science/article/abs/pii/S0960077901000741
http://library.lol/scimag/10.1016/s0960-0779%2801%2900074-1
Delete@Ark
ReplyDeleteI have a problem with this time and entropy. I think this entropic arrow of time is a meaningless model. The existence of a correlation between two phenomena does not yet prove their relationship. I think that is what has happened with time and entropy.
'To be able to define an entropy operator, it is thus necessary to overcome this degeneracy.
However, why do we introduce an entropy operator? What is this entropy? Is entropy always increasing? But in this view, its maximum value also increases assuming that inflation occurs.
Unfortunately I got the link wrong. I was too happy too soon, but the whole paper is not there either....
This could also be interesting:
ReplyDeletechrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.fuw.edu.pl/~kostecki/scans/fujiikamei1989.pdf
I found something like this: https://www.scirp.org/journal/paperinformation.aspx?paperid=108792
ReplyDeleteAnd there:
"The article shows that the special theory of relativity (STR) created in the last century was based on postulates due to the lack of the required experimental information and turned out to be incorrect, as its principle of light speed non-exceedance was refuted by studies of special processes in linear electric circuits in the 21st century.".
Sounds weird... What do you think about it?