Revising Space Time Geometry: A Proposal for a New Romance in Many Dimensions by Renate C.-Z. Quehenberger

Renate,

Congratulations on a nicely written essay. I too agree that quaternions, and the algebra they represent, SU(2), have the potential to reveal certain physical relationships are are essential. It's no accident that very much work in elementary particles is done according to that.

Higher dimensional forms for relativity, such as Kaluza-Klein theory are an interesting alternative to look into. I believe the "capu nili" difficulty you mention is related to the splitting of the Lorentz transformation into parity classes depending on whether the transformation is proper and orthochronous. That is dealt with in the essay Is Kinematics Compatible With Field Symmetries?

Now the especially interesting thing you can see in that essay is that the higher dimension of mass normally associated with Kaluza-Klein type approaches collapses the common space-time relativistic relations back into equations of 3 dimensional space showing the dispersion of energy while at the same time being equivalent to the 4 dimensional expressions!

Best regards,

Steve Sycamore

Dear Renate

Plato's cave = holographic universe?

See my essay http://fqxi.org/community/forum/topic/1413

Renate,

You really should read my essay The Algebra of Everything. I think I do a reasonable job of tying physical reality to the 8 dimensions of Octonion Algebra. The Quaternions you like manifest themselves within their 7 subalgebras of O. Their chiral choices determine the full variability in the definition for O. This variability and the easy assumption it should have no impact on the description of physically observable phenomenon mandates the form the mathematical description of reality must take. Electrodynamics is best and most fundamentally demonstrated in an O framework, not an H or Minkowski space-time. Take a look, it will be worth your time.

Rick

Thanks for the flowers, Vladimir!

What about a toplogy dinner ? : http://quantumcinema.uni-ak.ac.at/site/qc-goes-public/qc_arts-birthday-2012/

Sorry for the late response: I got lost in France: http://membres-liglab.imag.fr/nesme/founqi2/

Very interesting that flatland is still so much around, M. V. Vasilyeva

... if we consider that the idea was fist mentioned by Gustav T. Fechner (1846) too.

You see how much one story, - or two- may evoke !

Good luck with your story !

Dear Steven,

I must well return your compilmets towards your article. Thank you very much for your link!

It provides deeper inights into the relevant mathematical structures.

I see, your conclusion that only"c^2 is the invariant (...)

[and] "not c as it is in Special Relativity and Lorentz theory " meets well the quaternionic approach,- isn't it ?

Good luck and best regards,

Renate

Dear Yuri, may I correct: Plato's world ideas meets the higher dimensional quantum world, David Bohm's holographic universe on level-1

Plato's cave demonstrates our 3D perception

Good luck!

Just in case

My advice to read

http://www.galiulin.narod.ru/ufn022f.pdf

http://www.galiulin.narod.ru/crys03_6.pdf.pdf

Dear Rick,

many thanks for your link to your fascinating essay!

Now I guess, I understand why octonionic algera exceeds he Hilbert space ...

Here you might find the preliminaries for a geometry of a diecrete space using grid Z5 which should support the geometric visualisation octonions (- work in progress).

www.researchcatalogue.net/view/22616/22617

I wish to proceed far enough to visualize one day soon your complex field equations;-)

Best wishes !

Renate

Dear Renate,

I enjoyed reading your essay! The historical context is very interesting. It's nice to have this all in one place. A couple of questions and remarks.

1. I think Maxwell's poem was referring to the fact that all (one-dimensional) knots come undone in four-dimensional space. Is this what you were referring to on page 8?

2. The quaternions and octonions (defined by Cayley) are related to Hopf fibrations, which are geometrically beautiful and also physically relevant (for instance, in quantum information theory).

3. You mention Klein's program in regard to covariance (i.e. "Lorentz invariance.") I think it's interesting to consider the possibility that this is only an approximate concept. This is one of the topics I discuss in my essay On the Foundational Assumptions of Modern Physics.

4. Another interesting thing to consider is non-integer dimension (fractal dimension, emergent dimension, etc.). This is particularly relevant in discrete models.

Thanks again for the interesting read! Take care,

Ben Dribus

Thank you so much for this invaluable links, Yuri!

Dear Quehenberger,

In the conjecture of Tetrahedral-branes in Coherently-cyclic cluster-matter paradigm of universe, reductionism of extra-dimensional expressions is by the non-descriptive complexity of the coordinates of configuration space, in that its generalized coordinates are time, string-length and the central angle of transformed eigen-rotation. Thus in this scenario of dimensionality, three-dimensional structures emerge with fractals.

With best wishes

Jayakar

Hey Renate, love the prose and historical intertwines. Great piece. A nice example of ScienceArt, both with capitals. Although to my personal taste there's a bit to much geometry here and a bit too little topology here. :) We all meanwhile know where babies come from, but where do dimensions come from? Where do lines come from?

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process.

Sergey Fedosin

good text

Cheers Reni.

Like the Nietsche quote.

Thanks Han,

if so, you may find more Nietzsche here:

http://quantumcinema.uni-ak.ac.at/site/wp-content/uploads/A_Digital_Vision.pdf

- a prosecution referring to sociological and cognitive effects concerning the common agreement of the perceptions of higher dimensions touching the foundations of the current WELTBILD.

Many thanks for your personel review, Prof. Bob!

Indeed, one could easily draw a line of thoughts from the abolished Limbo, via the abandoned ether and Plato's forgotten „order of heavens" to the hierarchy problem.

A topologist, could take a bundle (F) of hyperbolic spaces with some handles and loopholes in it, apply the Fourier transform, arrive in 6D and - wait for the babies.

A geameter exists already there,- in Plato's ideal world, filled with (Maxwell's and Faraday's) interfering lines of forces, forming triangles over and over, entangled towards manifolds of all possible higher order.

Didn't Klein tell, that all geometries and topology are merging in higher dimensions?

Good question; - can we realy explain „dimensions" with "degrees of freedom" only ?

I guess they are necessary to avoid Hilberts trap and mix up a beer mug with ...