4th density geometry

[woneill1701]

Can anyone recommend a good book (or reading list) that would allow me to transition from my simplistic Undergrad Physics-based view of Geometry to being able to understand the Maths-based world of Geometry, and the Geometric relationship with Clifford Algebras?

Never mind!!!

Got to love it! After years of trying and bouncing off, then finally asking the question, I finally figured it out: The missing link I was looking for was the subject of Abstract Algebra. It explains, Groups, Rings, and the Geometric side of it expands into Vector Spaces. Exactly the correspondance that I was looking for...

I can now complete my roadmap that will allow me to progress my physics-based understanding of 3D geometry to a level where I am able to understand (and maybe one day contribute to) the discussions in this wonderful thread!

I studied the regular 3D Geometric Algebra of Grassman/Hestenes about 10 - 20 years ago and it immediately answered all my questions about the link between Matrices, Vectors, and simple Geometry that nobody at University could actually explain. (The professors seemed to get quite testy when I asked them to explain why the relationship between Vectors and Matrices was what it was... That said, they always got testy whenever I asked them the whys of anything!)

Anyway I realized that the properties of Geometric Algebra could explain a lot more than the link between Vectors and Matrices, but for it to be able to explain Wave Particle Duality, Time, and a link between Mass, Charge, and the Lorentz transform, a lot more dimensions were needed than just our regular 3. (I never accepted Minkowski Spacetime as real, and have yet to see anything that can convince me that Time is an actual dimension any more than Frequency/Wavelength, the population of rabbits in a field, or any other independent degree of freedom is...)

Sorry about the distraction!

 

[Guest 15258]

Got to love it! After years of trying and bouncing off, then finally asking the question, I finally figured it out: The missing link I was looking for was the subject of Abstract Algebra. It explains, Groups, Rings, and the Geometric side of it expands into Vector Spaces. Exactly the correspondance that I was looking for...

I know a bit of abstract algebra and it's good stuff to know, so well worth your time I think. If you're into physics then perhaps you'd want to focus more on groups and representation theory, as well as Lie groups. Having said that rings are nice as well, I don't know how much they come up in physics though.

If you're anything like me though it's not easy, and is much harder to do by yourself. I'd be happy to help you as much as I can here. Maybe you could create an abstract algebra thread if you have questions. (although, of couse, stack exchange is pretty much dedicated to this.)

Sorry about the distraction!

Don't be sorry.

 

[woneill1701]

I know a bit of abstract algebra and it's good stuff to know, so well worth your time I think. If you're into physics then perhaps you'd want to focus more on groups and representation theory, as well as Lie groups. Having said that rings are nice as well, I don't know how much they come up in physics though.

If you're anything like me though it's not easy, and is much harder to do by yourself. I'd be happy to help you as much as I can here. Maybe you could create an abstract algebra thread if you have questions. (although, of couse, stack exchange is pretty much dedicated to this.)



Don't be sorry.

Appreciated!

The reason I (think I) want to understand Rings is that when I got deeper into the Geometric Algebra, and wanted to go beyond our 3\4 D, it expanded to a 16 D Ring algebra that was non commutative - with a bunch of constraints regarding which of the 16 dimensions could interact. (The interaction diagram between dimensions looked remarkably like the icon John G uses - or the Kaballah Tree of Life...)

When I hit this I was incapable of moving forward because my maths was so primitive.

The details have so-far been beyond me, and as John G knows, I refer to this 16D algebra as the “Ring 16” version of Geometric Algebra.

As my other posts would indicate, I fundamentally believe that science has been suppressed/hi-jacked by all major governments in an attempt to prevent someone developing the next super weapon. Minkowski spacetime being one of the hi-jacks, Vectors being another...

I don’t blame them, but, as is usual for me, I have an OCD need to find the truth...

So I want to go back to basic (fundamental) principles and research the properties that a 16D algebra-based universe would posess. (After 30 years, I am so detached from the current orthodoxy that, in my arrogance, I believe I might stand a chance of seeing through any facade that has been put in place...)

For me, the key is that in our 3D “universe”, one one hand there are Matter, time, mass, gravity, 3 dimensions, and the interference artifacts of a quantum wave packet (with “spooky action at a distance”, but nothing measurable). On the other hand we can see/measure photons/light waves that, based on special relativity, have no mass, do have momentum, and do not experience either our universe or time as we know it.

At the same time, magnetism is a phase-transformed version of electricity, but gravitation isn’t - even though it has the same basic structure...

Charge can be positive/negative, but mass is always positive (unless it has negative energy)...

I see a very simple symmetry there that I am incapable of describing mathematically. It bugs!!!

Visually, I see interactions happening within dimensional sub-universes that each have their own experience of time, and which are completely and blissfully unaware of each other except for the results of interference patterns of waves that exist in the other sub-universes.

Time and space are coupled within these sub-universes, but the interactions are constrained...

The resulting itch has bugged me for 30 years, and now it needs to be scratched!

So, now I am trying to learn a language that can be used to represent and manipulate the concepts involved.

If, given this, you can put me on a better path than the one I am looking at, or if you still think Representation theory and Lie groups are the answer, then I will gratefully accept your direction.

I abandoned my love of Physics because nobody at Manchester University even cared to ask even the most basic questions about what they were “teaching” us about how the universe worked.

I concluded it was all a lie. As you can tell, I have never recovered.

 

[Bluegazer]

my questions about the link between Matrices, Vectors

Wow! I'm surprised. It's something I had thought of very recently.

Is the Matrix a hyperdimensional arithmetic operation?

Before we start this thread, a couple of clarifications. I know that the forum as a living entity, is an ocean of information that is constantly evolving. Such is the rhythm, that it is very difficult to keep up with it and to be up to date. For that reason, and in spite of the search function...

cassiopaea.org

My theory is as follows:

Number 33 and code 353535 is the result of a hyperdimensional arithmetic operation, which is based on the manipulation of the vectors that define the 3rd density by using the 4th density when traveling in space-time. Since 4th density supersedes 3rd density thanks to the capacity of physical variability, i.e. it can alter the vectors and therefore the unit, then it is possible to create a space-time loop between 3rd and 5th density*. And for this arithmetic operation to work, one of the vectors to be manipulated is intimately associated with how human beings perceive the world, that is to say in a LINEAR way. The concept of linear time that we have been lied to in our heads IS A VECTOR!

In linear algebra, a rotation matrix is the matrix that represents a rotation in Euclidean space. In three dimensions, rotation matrices represent rotations in a concise way and are frequently used in geometry, physics and computer science. Although most applications consider rotations in two or three dimensions, rotation matrices can be defined in spaces of any dimension.

A cylinder, a donut, are objects capable of rotation, on a symmetrical axis. One could hypothetically think that the 4th Density has as its rotating property. And that's when the image of the Tesseract appears.

And of course, a rotation matrix uses the special orthogonal group SO(n).

*NOTE: In reference to the loop that occurs as an effect of a karmic condition generated to re-generate the loop.

 

[Guest 15258]

Hi woneill1701 (star trek reference?)

I don't know much about geometric algebra. I've looked at some stuff and read some of the Wikipedia page and I'd like to learn. The axioms don't seem so bad, a vector space with an inner product which is quadratic or something. Exterior products are interesting also, I think they're like k-forms from differential geometry, but I haven't played around with either.

Lie groups and representation theory are interesting and I think are good to know if your into physics. The rotation matrices are a representation of a Lie group, for example, and they're good to know for physicists. You need to know a bit of differential geometry (tangent spaces) for Lie groups.

I think it best though if you start at the beginning of the abstract algebra stuff and come back to this stuff later. Maths builds upon itself and to learn it you need to start at the bottom.

My friend says that doing mathematics is like standing at the base of an infinite cliff with an infinite plain behind you. Then you climb a little and there's still an infinite cliff in front and an infinite plain behind. There's always more to learn and it's always hard.

I also disagree with the Minkowski stuff. I think it'd be correct if we replaced all the η's with δ's and used O(4) or SO(4) (4-dim rotation group) instead of the Poincare group (Lorentz transformations). I'd like to know why you don't like the Minkowski stuff. I think it's possible to understand the physics to some degree without formalizing it, so perhaps we can discuss this parallel to the maths.

Here's a video from the Electric Universe people on the logical inconsistency of special relativity,

Wow! I'm surprised. It's something I had thought of very recently.

The 353535 stuff is very confusing IMO. I remember (memory could be scrambled) Laura writing somewhere that she figured out that it relates to genetic codes from the book in search of the double helix by John Gibbons (book title and author name may be scrambled to.) The transcripts also seem to imply that there might be some number theoretic content to this stuff.

 

[Bluegazer]

The 353535 stuff is very confusing IMO.

Yeah, I think so too. But I can't help but notice the periodicity of the number. That's where the time loop seems to be. I recently started reading the part of Laura's books that talks about Game Theory and Economics. There you can also find matrices and vectors, and I assume that by looking into economics you will find the mathematical arrangements that manipulate economics. Imagine for a moment that you extrapolate the operation of economic manipulation to the manipulation of the 3rd Density, by means of the 4th Density. Because after all, money is also ENERGY. And that monetary energy we know is related to surplus value, that is, the exchange of work for money. All work, even physical work, is given by an investment of consciousness in that work.

 

[woneill1701]

Hi woneill1701 (star trek reference?)

Yup!

I don't know much about geometric algebra. I've looked at some stuff and read some of the Wikipedia page and I'd like to learn.

For me, the only place to go to in order to understand the implications of Geometric Algebra is the book by David Hestenes - "New Foundations for Classical Mechanics"

It's hard to buy, but easy to download... I'm sure you would devour it in a couple of days.

He actually created a full series of books applying it to conventional Physics - and in doing so, simplifying everything from Maxwell's equations to SR/GR...

For context, while still at school I had mastered the use of Homogeneous Matrix Operations on 2D and 3D Vectors in the context of Computer Graphics using the book "Mathematical Elements for Computer Graphics" by David Rogers. It was good about explaining what to do, but delivered none of the whys...

I am OCD, and my learning style is based on understanding - not memory, so I hoped that when I went to University they would enlighten me as to the fundamental reasons why of the Universe (this being one of them)...

To my horror, it was just as bad as being at school... We were given an even more primitive toolkit of vector operations to use than I had already got from my computer graphics book, and when I asked them why things were the way they were, they got very testy!

Anyway, it was only when I read the Geometric Algebra book 10 years later that everything became so obvious and clear! The stuff they wouldn't/couldn't explain about vector/matrix operations just fell out of a very simple mathematical construct.

In short, by adopting the Vector format we all know today, they had excluded 80% of the mechanisms behind dimensional transformations/interactions from view. Geometric Algebra, by adding in the complete formulation, makes it SOOOOO much simpler while at the same time actually reducing mathematical complexity.

To me, it was one of the most important "lost" aspects of physics that prevents any aspiring physicist from being able to derive their own understanding from basic principles. You are just given a toolkit and graded on your ability to use it constructively...

I also disagree with the Minkowski stuff. I think it'd be correct if we replaced all the η's with δ's and used O(4) or SO(4) (4-dim rotation group) instead of the Poincare group (Lorentz transformations). I'd like to know why you don't like the Minkowski stuff. I think it's possible to understand the physics to some degree without formalizing it, so perhaps we can discuss this parallel to the maths.

In terms of Spacetime, I think it is a great modeling tool, and simply extends the usual graphical representation of anything against time into 4D. Nothing wrong with that! Just like any other representation that juxtaposes some function against time, it makes visualization possible and easy!

My problem comes when they say that this is reality, and that time is the 4th dimension!

I believe this was a deliberate attempt to prevent any further legitimate research into additional dimensions because the folks in the establishment already understood the implications of anyone being able to manipulate dimensional transforms that could affect external dimensions...

To me, Time cannot be a dimension because it is impossible to move through it. To me it is an artifact - an emergent property - of the Lorentz transformation. (In my original work with vectors for Computer Graphics I did a lot of work with perspective transformations, and the Lorentz Transformation, to me, is nothing more than a perspective transform.

But, as of today, I do not have the maths to either prove or dis-prove it. I feel like a chimp digging into an ant-hill with an old twig compared to the tools you guys have!

Anyway, I will follow your suggestions - Lie groups do seem directly related to GA - and I will get to them once my core Maths is back up to speed.

I was originally taught Maths as a process toolkit. This time I'm going to learn it as a language from basic principles!

Here's a video from the Electric Universe people on the logical inconsistency of special relativity,

The EU folks definitely have a lot of stuff right! Plasma is definitely the driving force behind most of Cosmology! But, I think they have over-reached with their formulation of Electric Gravity (things in a Faraday Cage would all be weightless if they were correct)...

I'm still not 100% sold on what to think about SR vs. the AEther.

SR does seem to have withstood many of its tests - and there have been many!

At the same time, SR could explain why EM waves have physical presence and momentum, but no physical matter or mass in our universe.

If it wasn't for Quantum Physics, I would probably still be challenging SR the same way I challenge Minkowski, but given that Quantum Physics has to be right for any of our Solid State technology to work, I see SR as the mechanism why we cannot detect any materialistic waves associated with Quantum Physics.

(I see Quantum waves traveling at the speed of light through a universe that has collapsed in size down to nothing, with no observable time on the part of the wave. I imagine other denizens of that universe having the same conversation that we are today, but with our quantum waves behaving like E/M in that universe...)

 

[Bluegazer]

Here's a video from the Electric Universe people on the logical inconsistency of special relativity,

Q: (A) Okay. UFT. This is one of these things that I don't know what it is good for, because the Wave will erase everything and make everything new. Yet, it is in me, so let me ask. I don't know what it is good for, but I want to do it. Einstein was working on his UFT for like 30 years. Maybe more. He was changing his methods. At some point, did he realize that he found a solution? During all these thirty years, was there a point where he came upon the right solution?

A: Yes, but sadly, his solution for UFT largely erased TOR.

Q: (A) Once he found this solution, did he reject it because it erased TOR?

A: No. His progenitors sealed it, in order to keep intact the status quo.

Q: (L) His progenitors? Isn't that your parents?

A: Other definitions apply. [A source from which something develops.]

Q: (A) Can we have an idea of which year Einstein found the solution that erases TOR?

A: Sure, it was 1938.

 

[woneill1701]

Do you have a theory?

I separate SR from Minkowski and GR.

SR is purely a question of whether the Lorentz transform applies universally, or whether the Michelson-Morley experiment was measuring the AEther...

Minkowski was a separate imposition on SR, and if I remember properly, Einstein was not happy... “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore!”

Passive-aggressive resistance!

Now GR is a different creature compared to SR...

Do we know which one the Cs were referring to?

 

[woneill1701]

Do you have a theory?

I separate SR from Minkowski and GR.

SR is purely a question of whether the Lorentz transform applies universally, or whether the Michelson-Morley experiment was measuring the AEther...

Minkowski was a separate imposition on SR, and if I remember properly, Einstein was not happy... “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore!”

Passive-aggressive resistance!

Now GR is a different creature compared to SR...

Do we know which one the Cs were referring to?

As an FYI, while I believe that Einstein was a self-promoting opportunist, he did not come up with SR in a vacuum. He acted as a moderator when the entire Physics community was in a crisis!

All the big names in Physics at the time were struggling with the question of how to explain the apparent invariant speed of light. They were fighting over different explanations, and mechanisms, and all Einstein contributed was the question, “Why not just accept it as a reality and work from there?”

Because nobody could come up with a better answer, his approach was accepted...

Despite all the huff and puff, Einstein never received a Nobel for either SR or GR.

He DID receive a Nobel for his photoelectric theory, but even that was just him providing an interpretation of what others had already found...

Einstein wasn’t a charlatan or the bad guy. Similarly, SR wasn’t really his. He was really standing on the shoulders of an army of giants!

So the REAL question is whether the Lorentz transform is real, or whether the measurements were wrong and there really is an AEther...

 

[seiw83]

HIS TOR was simple and UFT is one step more.
TOR is simple but you need to look at right example.
UFT is simple but you need a right object.
Any idea for object to destribe UFT in 3d?
object has to be 3d/4d manipulative in two axis.

 

[John G]

The reason I (think I) want to understand Rings is that when I got deeper into the Geometric Algebra, and wanted to go beyond our 3\4 D, it expanded to a 16 D Ring algebra that was non commutative - with a bunch of constraints regarding which of the 16 dimensions could interact. (The interaction diagram between dimensions looked remarkably like the icon John G uses - or the Kaballah Tree of Life...)

When I hit this I was incapable of moving forward because my maths was so primitive.

The details have so-far been beyond me, and as John G knows, I refer to this 16D algebra as the “Ring 16” version of Geometric Algebra.

Yes a matrix ring is just the non-commutative version of a matrix algebra; it's a ring via matrix multiplication/addition forming a ring. The M16(R) matrix ring would be a 16x16 matrix for five of the nine Cl(8) signatures (16x16=2^8=256). The other four are M8(H) with H for Hamilton/quaternion thus we have 8x8x4=256. I don't play with it directly either. I cheat and play with a 16x16 Hodge Star map version of it and even cheat at that by playing with a 16x16 rule space partitioning (a cellular automata thing) version of a Hodge Star map.

As my other posts would indicate, I fundamentally believe that science has been suppressed/hi-jacked by all major governments in an attempt to prevent someone developing the next super weapon. Minkowski spacetime being one of the hi-jacks, Vectors being another...

I don’t blame them, but, as is usual for me, I have an OCD need to find the truth...

So I want to go back to basic (fundamental) principles and research the properties that a 16D algebra-based universe would posess. (After 30 years, I am so detached from the current orthodoxy that, in my arrogance, I believe I might stand a chance of seeing through any facade that has been put in place...)

16x16 is from 8 Clifford algebra basis vectors. You get 16 by first splitting the 8 into two groups of four (spacetime plus Kaluza-Klein dimensions) then using all 2^4=16 multivectors for both spaces.

For me, the key is that in our 3D “universe”, one one hand there are Matter, time, mass, gravity, 3 dimensions, and the interference artifacts of a quantum wave packet (with “spooky action at a distance”, but nothing measurable). On the other hand we can see/measure photons/light waves that, based on special relativity, have no mass, do have momentum, and do not experience either our universe or time as we know it.

At the same time, magnetism is a phase-transformed version of electricity, but gravitation isn’t - even though it has the same basic structure...

Charge can be positive/negative, but mass is always positive (unless it has negative energy)...

I see a very simple symmetry there that I am incapable of describing mathematically. It bugs!!!

Visually, I see interactions happening within dimensional sub-universes that each have their own experience of time, and which are completely and blissfully unaware of each other except for the results of interference patterns of waves that exist in the other sub-universes.

Time and space are coupled within these sub-universes, but the interactions are constrained...

Yeah magnetism is kind of a reference frame dual of electricity but conformal gravity (includes aether math) is for me more a Hodge dual of EM (and the nuclear forces). If you have a many-worlds interpretation and think about Feynman path integrals calculating probabilities in the present using paths in the future, things can get spooky. Plus and minus charge would be two of the Kaluza-Klein dimensions. The other two Kaluza-Klein dimensions would be translations and special conformal transformations. Mass might relate to a volume form/affine connection bimetric that can be built along with the conformal group via having the translation/conformal basis vectors.

The resulting itch has bugged me for 30 years, and now it needs to be scratched!

So, now I am trying to learn a language that can be used to represent and manipulate the concepts involved.

If, given this, you can put me on a better path than the one I am looking at, or if you still think Representation theory and Lie groups are the answer, then I will gratefully accept your direction.

I abandoned my love of Physics because nobody at Manchester University even cared to ask even the most basic questions about what they were “teaching” us about how the universe worked.

I concluded it was all a lie. As you can tell, I have never recovered.

Matrix rings and Clifford/Geometric algebra are the place to be. As the 2^N dimensions suggest it is very information theory related and the Cs mentioned it as being better than group theory. Also though as the Cs suggested, group theory is useful for understanding the parts of the Geometric Algebra.

 

[Guest 15258]

For me, the only place to go to in order to understand the implications of Geometric Algebra is the book by David Hestenes - "New Foundations for Classical Mechanics"

It's hard to buy, but easy to download... I'm sure you would devour it in a couple of days.

Thanks for the recommendation, I'll keep an eye out for it. Is it free to download?

Anyway, it was only when I read the Geometric Algebra book 10 years later that everything became so obvious and clear! The stuff they wouldn't/couldn't explain about vector/matrix operations just fell out of a very simple mathematical construct.

See, you know more about it than I do.

To me, Time cannot be a dimension because it is impossible to move through it. To me it is an artifact - an emergent property - of the Lorentz transformation. (In my original work with vectors for Computer Graphics I did a lot of work with perspective transformations, and the Lorentz Transformation, to me, is nothing more than a perspective transform.

I agree that Lorentz transformations are perspective transforms. I also think that perspective transforms are the way to go because we observe events by looking at it. I just think that Lorentz transformations are the wrong ones to use in the case of SR.

SR does seem to have withstood many of its tests - and there have been many!

Ok, so I'm a bit of a conspiracy theorist. I think the PTB have created an SR marketing campaign to make it look like SR is true. I think this marketing campaign is so successful that anyone who disagrees with the mainstream is marginalized and ostracized. It seems to me that part of the success of this is due to the fact that SR is difficult to test.

Do we know which one the Cs were referring to?

From the context it must be either GR or both SR and GR. Fingers crossed for both.

So the REAL question is whether the Lorentz transform is real, or whether the measurements were wrong and there really is an AEther...

The C's mention an Aether here and there when talking about this stuff.

I started a thread on it years ago: Newtons laws, relativity & the ether

HIS TOR was simple and UFT is one step more.
TOR is simple but you need to look at right example.
UFT is simple but you need a right object.
Any idea for object to destribe UFT in 3d?
object has to be 3d/4d manipulative in two axis.

Yeah, I think UFT is simple once you know what it is. And once you have it 4D geometry makes itself apparent.

 

[woneill1701]

Thanks for the recommendation, I'll keep an eye out for it. Is it free to download?

There are plenty of sites where it can be downloaded as a free pdf. Just be careful to use a virus scanner...

See, you know more about it than I do.

Yeah right!

Ok, so I'm a bit of a conspiracy theorist. I think the PTB have created an SR marketing campaign to make it look like SR is true. I think this marketing campaign is so successful that anyone who disagrees with the mainstream is marginalized and ostracized. It seems to me that part of the success of this is due to the fact that SR is difficult to test.

Me too!

But they do appear to use relativistic corrections on satellite clocks, Newtonian mechanics needed relativistic corrections to correctly deal with the orbit of Mercury, and I do keep coming back to Quantum Physics: Unless we can find some DeBroglie Pilot Waves, we need to explain wave/particle duality - and SR represents a beautifully elegant solution to both sides of the problem.

It's funny me defending SR, though, because I had the same argument against SR, with my SR lecturer at university, as the guy in that video you linked to. I was hoping to be "schooled" (I learn fastest when challenged), but he actually couldn't refute my analysis, so he simply banned me from any of his subjects...

His only argument was, "But Einstein said..."

He clearly didn't understand what he was trying to teach - like a lot of folks there...

At the time I couldn't understand why he got so upset with me...

I eventually figured out the answer about SR for myself, and passed the course based on what the Lorentz transform implied about light in our universe - but it is one of those circular arguments that only works if you accept that Lorentz is universal.

(Basically the guy in the video is applying classical Galileian Relativity to a Lorentz/Einstein construct whereby light doesn't experience our 3 dimensions at all, and thus cannot have "dimensional components of velocity" like any regular inhabitant of our universe. At light speed, the two different forms can never be reconciled! Either Galileian Relativity is correct, or SR is correct - there is no half-way house at the speed of light...)

 

[woneill1701]

Yes a matrix ring is just the non-commutative version of a matrix algebra; it's a ring via matrix multiplication/addition forming a ring. The M16(R) matrix ring would be a 16x16 matrix for five of the nine Cl(8) signatures (16x16=2^8=256). The other four are M8(H) with H for Hamilton/quaternion thus we have 8x8x4=256. I don't play with it directly either. I cheat and play with a 16x16 Hodge Star map version of it and even cheat at that by playing with a 16x16 rule space partitioning (a cellular automata thing) version of a Hodge Star map.

16x16 is from 8 Clifford algebra basis vectors. You get 16 by first splitting the 8 into two groups of four (spacetime plus Kaluza-Klein dimensions) then using all 2^4=16 multivectors for both spaces.

Yeah magnetism is kind of a reference frame dual of electricity but conformal gravity (includes aether math) is for me more a Hodge dual of EM (and the nuclear forces). If you have a many-worlds interpretation and think about Feynman path integrals calculating probabilities in the present using paths in the future, things can get spooky. Plus and minus charge would be two of the Kaluza-Klein dimensions. The other two Kaluza-Klein dimensions would be translations and special conformal transformations. Mass might relate to a volume form/affine connection bimetric that can be built along with the conformal group via having the translation/conformal basis vectors.

Matrix rings and Clifford/Geometric algebra are the place to be. As the 2^N dimensions suggest it is very information theory related and the Cs mentioned it as being better than group theory. Also though as the Cs suggested, group theory is useful for understanding the parts of the Geometric Algebra.

Wow! As always, I'm back to being a chimp with my pet twig when you speak!

Plus and minus charge would be two of the Kaluza-Klein dimensions.

Wouldn't plus and minus charge be just different values on the same "charge" dimension?

(One of my issues with time as a dimension is that there are no examples of negative time...)

I always figured that this was what was going on with spin too: the spin states were just +/- on a dimension that translated into ours as this thing called "spin" that has an inherent angular momentum, but nobody can physically explain...

Matrix rings and Clifford/Geometric algebra are the place to be.

I will get there! This thread has re-ignited the fire in me that 3 years at university extinguished!