As they say "Knowledge protects, ignorance endangers."
And so recently, thanks to AI, I have learned that what we are doing here is being done by other people, though with a somewhat different goal. The domain of research that concerns us is called "Conformal Field Theory". Here are two pages from the PhD Thesis "Relations between 2D and 4D Conformal Quantum Field Theory", by Daniel Meise, University Göttingen, 2011.
We will derive here formulas similar to (2.58), but not by the method described in there. It is in order to have simple formulas like (2.58) that I have changed my formulas for Clifford algebra generators eμ. Then I doubted if I have it all correct, but today I realized that all seems to be ok.
So, I will be adding new stuff to my "Notes".
Afternotes
17-08-25 19:21 I know. I promised a new post. But I was busy. Finally, using our triangular matrices, I managed to create this "fractal".
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| Scottish torus |
In the next post, hopefully on Monday, I will describe how I got it. The torus is, of course, our conformally compactified 1+1 dimensional universe.
18-08-25 19:06 I will not finish writing my new post today. Will finish tomorrow.
While waiting for Anna to get better, I have a question not directly related to recent stuff here, but concerning the interpretation of experimental data.
ReplyDeleteIt is said that muon rate at the surface of the Earth is direct evidence of time dilatation and validity of Lorentz transformations. Muon lifetime at rest is cca 0.2 microseconds which translates to cca 500-600 m path at the speed of light before decaying. Assuming muon flux at the surface comes as secondary cosmic rays produced in the atmosphere, meaning they traveled ~ 10 km from the point of their production, what would be interpretation of this experimental observation that would be in line with Galileo transformations, sort of bypassing time dilatation and Lorentz transformations?
Here is what Grok has to say on the subject:
DeleteYes, alternative interpretations of the cosmic ray muon experiment have been proposed by professional theoretical physicists. The most prominent and empirically equivalent alternative is the neo-Lorentzian (or Lorentz ether theory) interpretation of special relativity. This approach, while yielding the same predictions as the standard Einsteinian formulation, differs philosophically by positing a preferred reference frame (often associated with the cosmic microwave background rest frame or an undetectable "ether") where time dilation and length contraction are absolute physical effects caused by motion relative to that frame, rather than symmetric and purely relative phenomena between observers.In the context of the muon experiment—where high-energy muons produced in the upper atmosphere reach Earth's surface in greater numbers than expected from their short laboratory decay time due to relativistic effects—the neo-Lorentzian view attributes the extended muon lifetime to an absolute slowing of the muons' internal decay processes as they move at high speeds relative to the preferred frame. This contrasts with the standard relativistic explanation, where time dilation is observed from Earth's frame (muons' clocks run slower) or length contraction from the muons' frame (the atmosphere's thickness appears shorter). The neo-Lorentzian framework maintains empirical agreement with the experiment but rejects the principle of relativity's full symmetry, treating the effects as asymmetric and tied to absolute motion.This interpretation has been advocated by several notable theoretical physicists:John Stewart Bell (known for Bell's theorem in quantum mechanics): In his 1976 paper "How to Teach Special Relativity," Bell argued that relativity should be taught starting from the Lorentzian perspective for pedagogical clarity, emphasizing that effects like length contraction are real physical distortions due to absolute motion through an ether-like medium, rather than mere coordinate artifacts. He viewed Einstein's approach as potentially misleading despite its elegance.
Franco Selleri (Italian theoretical physicist specializing in quantum and relativity foundations): Selleri critiqued special relativity for paradoxes related to simultaneity and causality, favoring a neo-Lorentzian reformulation with a preferred frame to resolve them while preserving Lorentz invariance empirically. He saw this as a more realist interpretation aligned with classical intuitions.
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Antony Valentini (theoretical physicist focused on quantum foundations and pilot-wave theory): Valentini supports neo-Lorentzian views as compatible with non-local quantum effects and a preferred frame, arguing it better integrates with foundational issues in physics like hidden variables and non-equilibrium quantum states.
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These physicists emphasize that the neo-Lorentzian interpretation is mathematically equivalent to special relativity in predictions (including for the muon experiment) but philosophically preferable for reasons such as compatibility with absolute simultaneity, realism about physical effects, and potential alignment with quantum non-locality (e.g., Bell's inequalities). However, it is not widely adopted in mainstream physics due to its reliance on an undetectable preferred frame (seen as unnecessary by Occam's razor), increased mathematical complexity, and inconsistency with the symmetric elegance of Einstein's formulation, which integrates seamlessly with general relativity and quantum field theory.
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Other purported alternatives, such as claims of experimental inconsistencies or non-relativistic mechanisms (e.g., muons traveling superluminally or having variable lifetimes due to environmental factors), appear in fringe literature but are not proposed or endorsed by established professional theoretical physicists and lack empirical support.
Thank you very much for taking the time and effort to grok the answer!
DeleteAnother partly related issue to the above one that have been mulling over for some time is with Cherenkov radiation, experimental method for detecting very high energy particles, including photons, that are supposedly transdimensional according to the C's, in the atmosphere and also on the surface in so called water detectors.
Unfortunately, don't have a nicely formulated question about it yet...
As soon as I had switches to something else for a minute, you are discussing acute and smart questions here! :) However, I could not imagine that Sasha would blame me for the pause in the Blog publishing...)
DeleteInterpretations of relativity effects are eternal stumbling block for me, I cannot decide on which side I am: I like the generally accepted version for its symmetry and mathematical elegance and also like neo-Lorentzian view for its intuitive clarity. A naive justification for the absolute frame of reference might be as follows: light does not move in time, so it does not move at all. Then, it is the massive matter that moves with respect to light, not vice versa.
This time AI provided a good review of the question, I liked especially the explanation of the symmetry of reference frames (where time goes slower and where distance becomes shorter).
This AI text was enlightening and useful for me. Ark, Sasha, thank you for generating it!
All the credits belong to Ark; I just gathered the courage to ask something that's been in the back of my mind for quite some time.
DeleteAnd you're not to blame for anything, except maybe for providing very useful feedback, as Ark pointed out. ;p
If you check the date and page number of the Notes file in previous post, it was updated at least twice since your "being slightly ill" comment, on Aug 8th and yesterday, reaching 10 pages in total in the latest version, from initial 6 of them, IIRC.
So, no real pause in posting afterall. :)
Oh, yes, Saša, indeed I did not notice that the text has reached "10 pages in total in the latest version". Thank you for noting, I will read the "new" part with pleasure.
DeleteBut why did you say "while we are waiting" in the above comment? I guess it was not only a polite figure of speech, a good introductory phrase, but a wake-up call for me, right? ;)
Well, I understood Ark's comment/reply to you in previous post;
Delete"Arkadiusz Jadczyk, August 5, 2025 at 7:57 AM
Take care of your sore throat. And me know when you are well";
that is its last part that it meant 'waiting for you to get well', so I kinda used that phrase to ease my anxiety about bothering Ark with asking the question that followed the phrase and was not related to what Ark was posting recently.
Nevertheless, I'm glad you are feeling better and that you found the question and the answer Ark provided to be helpful.
Ark, please note a few misprints:
ReplyDeletepage 3:
"It is to verivy that" --> It is to verify that
Just for the text completeness, please, specify what is η_μν.
page 7:
"to O(2, 2)" --> to O(4?, 2)
"Since the actions of L and R commute" -- why do they commute?
As far as I remember, L and R are arbitrary matrices and elements of group SL(2,R), or I am missing something important?
and one more, page 9, formula (48):
DeleteUT(a) --> RT(a)
Thanks. Fixed.
Deleteif L and R are SL(2,R) matrices and xhat is determinant zero matrix, then (L xhat)R=L (xhat)R, so left and right actions commute, even if L and R do not commute.
Correction:(L xhat)R=L (xhat R)
DeleteArk, but why x-hat is a determinant-zero matrix? We did not require that before. I remember that we used that L and R are determinant-one matrices to prove (40). At that time I managed to obtain:
Deleteω R^-1 = R^T ω, where ω = {{0, 1}, {-1, 0}}
and the same for L.
However, these expressions are of no use to show that
(L xhat)R = L (xhat R).
Couldn't we simply appeal to associativity property of SL(2,R) matrices to justify this fact?
In general xhat can be any matrix. But we are mainly interested in the null cone of R^{2,2} - that is Q(xhat)=0, but Q(xhat)= -det(xhat).
DeleteAnd you are right in that the commutativity of left and rifgt action follows from the associativity property of Cl(2).
Many thanks, Ark, you confirmed all my expectations.
DeleteI am also very proud of myself for doing that calculation below demonstrating isomorphism between Lambda transform of coordinates x_mu and the matrix transform x' = LT x TR^-1. It is a really new insight for me!
Thanks, Anna. In the meantime I have had problems with my calculations. Finally ends meet, so I think tomorrow I will add some new stuff. I still have to figure it out how to write about it.
DeleteCan you please help me "see" the geometry behind the matrix representation of our toy Cl(2) as R(2,2)?
DeleteCould it be "understood" as composed/generated by two 'orthogonal' planes, (x1,x2)×(x3,x4), where 'distances' as quadratic forms of each of them sort of 'cancel' each other or 'subtract' one from another?
Or would it be more suitable to 'view' it as two R(1,1) spaces or 'space-times' spanned by coordinates in Eq.50?
Or to just take it as it is, as one sort of 4dim space R(2,2) without bothering the 3dim linear part of the mind to try to understand and 'visualize' it somehow in geometric sense?
I tend to just visualize it as a conformal structure complex domain thing. Thus SO(2,2)/SO(1,1)xU(1)=4 dim=2 complex-dim. This would be the toy for SO(2,4)/SO(1,3)xU(1)=8 dim=4 complex-dim.
DeleteHow do you visualize "conformal structure complex domain thing", what does it look like in your visualization?
DeleteAnd please don't say it looks like conformal structure complex domain thing as I have no idea what that looks like or even what it means exactly. :)
For the 4 complex domains the best description I've seen is to have a time disk and a space hypersphere aka the 2 and 4 of SO(2,4). The real part would be the surface thus 1 dim and 3 dim and the complex would point from both surfaces to both interiors. I think things get messy via the Shilov boundary of the Cartesian product of the time disk and the space hypersphere but it's messy enough already for me just saying that without thinking about it.
DeleteThank you!
DeleteBy the way, let me make a quick remark about visualization. Recently we discussed the "mysterious" Koide formula for the masses of leptonic triad: electron, muon, and tau lepton https://dzen.ru/a/aJ7XKkJYqQ30H2Vv.
DeleteThis Koide formula really calls for a geometric representation! See, e.g., [4] in the post. To my surprise, a very simple reorganization of the Koide formula gave me an equation of 'absolute' curve, which remains invariant under projective transformations in hyperbolic geometry. Then I used WolframAlpha to plot this equation. The resulting picture is also published in the post. Unfortunately, no one knows how to interpret this visualization, although someone finds it cute... :)
Probably, some of the readers here will look at the cone and say "Eureka!"
@Saša
Delete"Can you please help me "see" the geometry behind the matrix representation of our toy Cl(2) as R(2,2)?"
I wish I could!
Well, that 'color inverted' Scottish donut was nice touch, so maybe not intentionally, but you did provide some useful help.
DeleteThanks!
Анна, а не могли бы вы дать ссылку на книгу Кагана, в которой рассматриваются изоморфные лоренцевой группе круговые преобразования.
ReplyDeleteIgor, I have only .djvu files of the 2 books by V.F. Kagan. If you give me your mail address, I will be glad to send it to you. This is mine: asidorova@mail.ru
DeleteAfter my forced lazy time, I'm trying to catch up with the Cl(2,2) machinery.
ReplyDeleteLet me deal with the "simple example" of triangular matrices.
Want to show that Lambda transform of coordinates x_mu is equivalent to the matrix transform x' = LT x TR^-1, where
matrix x = {{x1+x3, x2+x4}, {x2-x4, -x1+x3}}
For LT = {{1, 0}, {a, 1}} and RT = {{1, a}, {0, 1}},
Lambda 4x4 matrix is as follows:
{{1+a^2/2, 0, a^2/2, -a}
{0, 1, 0, 0}
{-a^2/2, 0, 1-a^2/2, a}
{-a, 0, -a, 1}}
Then, we calculate x' as the product LT x RT^-1 =
{{1, 0}, {a, 1}} {{x1+x3, x2+x4}, {x2-x4, -x1+x3}} {{1, -a}, {0, 1}}
and happily discover that
x'1 = (1+a^2/2) x1 + a^2/2 x3 - a x4
x'2 = x2
x'3 = -a^2/2 x1 + (1 - a^2/2) x3 + a x4
x'4 = -a x1 - a x3 + x4
which is exactly the same transformation of x values as by the Lambda 4x4 matrix.