# Garrett Lisi bid to join Std. Model w/ gravity

Mike2 said:
Isn't there a requirement that the number of particles/singularites must effect the metric on the 4-MF so that the more mass there is, the more the spacetime metric curves? Isn't this what you have? Or have you only connected the QFT algebra with a 4-manifold?
Mike2 it is better to discuss this https://www.physicsforums.com/showpost.php?p=839078&postcount=29"

Last edited by a moderator:
CarlB
Homework Helper
marcus said:
Lisi's handle on the Standard Model comes from some 1999 work by Greg Trayling
http://arxiv.org/hep-th/9912231 [Broken]
A geometric approach to the standard model
I just found the above paper. It's very similar to what I'm doing. There are a lot of differences. Baylis and Trayling are assuming 4 hidden spatial dimensions while I'm assuming just one, and that one related to proper time (i.e. the Euclidean relativity). I put the electroweak symmetry breaking into the relationship between the tangent vectors and the Clifford algebra. And I assume a level or two of preons.

Carl

Last edited by a moderator:
garrett
Gold Member
Hi Carl,
It's not so hard to get su(2) out of most any model. The trick is naturally getting su(3). And then it's even trickier to get a chiral su(2).

CarlB
Homework Helper
garrett said:
Hi Carl, It's not so hard to get su(2) out of most any model.
Yes. Given X and Y any two distinct nontrivial canonical basis elements, if one has a complexified Clifford algebra one can assume that they square to unity. From there, if it is either the case that XY = YX or XY = -YX. The latter case produces an SU(2) with {X,Y,XY}. The former case produces a U(1) x U(1). Choosing canonical basis elements at random, you basically have a 50% chance of getting an SU(2).

garrett said:
The trick is naturally getting su(3).
That's where preons come in. If you assume that the elementary fermions are each composed of three subparticles, the SU(3) comes natural.

By the way, I just realized you're Lisa Garrett who's written papers in this area, so you'll understand what I mean when I explain how it is that I came to find out that SU(3) is hard to get (cleanly) in this method.

I wanted to get the charges of the quarks out of Hestenes' geometric algebra. So I looked for solutions to the eigenvector equation:

$$Q|\chi> = 1/3|\chi>$$

where Q and \chi are from the Clifford algebra. Eventually I spent about two weeks looking for idempotents $$\iota$$ with $$<\iota>_0 = 1/3$$. I used all kinds of methods to solve that damned equation but I kept coming up with idempotents that had a scalar element of the form n/2^m and never anything but.

So I basically discovered the spectral decomposition theorem for Clifford algebra idempotents the hard way. (And found it in Lounesto's book, if I recall.) It was only later that I realized that if I assumed preons, the 1/3 factors would come naturally, and that the way that the weak isospin and weak hypercharge quantum numbers work out the preon model falls right into your lap.[/edit]

garrett said:
And then it's even trickier to get a chiral su(2).
I don't see this at all, please explain.

Carl

Last edited:
garrett
Gold Member
CarlB said:
I don't see this at all, please explain.
Chirality was the problem that more or less killed Kaluza-Klein theory in the 80's. Nature couples left-chiral fermions to the electro-weak gauge fields differently than it couples right-chiral fermions. It is hard to come up with a good geometric reason for why.

CarlB
Homework Helper
garrett said:
Nature couples left-chiral fermions to the electro-weak gauge fields differently than it couples right-chiral fermions. It is hard to come up with a good geometric reason for why.
Yes, I did have trouble with this. The solution takes a bit of explaining.

First, you need to assume that the elementary fermions are the left-chiral and right-chiral ones. There is the problem that these are massless and you will later have to figure out a method of putting the masses in, but work is ongoing on that.

That done, the problem then becomes one of symmetry breaking. That is, the primitive idempotents of a Clifford algebra tend to be too danged symmetric.

I think the best way to break the symmetry is to modify the way that the tangent vectors (i.e. $$\partial_x, \partial_t$$, etc.) are connected up with the canonical basis vectors (i.e. $$\gamma_x, \gamma_y$$, etc.)

Use the standard Dirac matrices as an example. What we are going to do is to loosen the definition of the Dirac operator:

$$\nabla \psi = (\partial_x\gamma_x + \partial_y\gamma_y + ...) \psi$$

Instead of using the canonical basis VECTORS, $$\gamma_\mu$$, we will instead use arbitrary canonical basis ELEMENTS $$\Gamma_x, \Gamma_y, \Gamma_z, \Gamma_t$$ provided only that the usual Dirac relations are satisfied. That is,

$$\Gamma_x^2 = \Gamma_y^2 = \Gamma_z^2 = -\Gamma_t^2 = 1$$

and the $$\Gamma_\mu$$ anticommute.

Now in the usual spinor model of quantum mechanics, this modification leaves the Dirac equation unchanged. That is, $$\Gamma_\mu$$ makes just as good a set of generators of a Clifford algebra as $$\gamma_\mu$$ does. So the change doesn't alter any of the usual physics.

But in Trayling's model of the elementary particles, the change from $$\gamma_\mu$$ to $$\Gamma_\mu$$ amounts to a remapping of the elementary particles.

The remapping by the above naturally preserves all addition and multiplication in the Clifford algebra so all the remapped particles preserve their quantum numbers with respect to the remapped operators. But the remapping does not preserve the squared magnitude of the Clifford algebra. Since the squared magnitudes are what we associate with probabilities, this means that while the particle propagation is unchanged (same Dirac equation), the particle interactions are altered. This is exactly the kind of symmetry breaking you need.

I've been assuming just one small cyclic hidden dimension and a complexified Clifford algebra. This gives 8 primitive idempotents. Before breaking symmetry, these 8 idempotents all have squared magnitude 1/8. The symmetry breaking can only raise the squared magnitude of a primitive idempotent. But on the other hand, the primitive idempotents still sum to one so the squared magnitude of the sum of all the idempotents is still going to be 1.

Another way of saying this is to say that the primitive idempotents become unstable. Typically, four of them become slightly unstable (tending to form up in pairs), while the other four become extremely unstable (also tending to form up in pairs). A typical set of squared magnitudes is:

$$8|\iota_{---}|^2 = 1.31$$
$$8|\iota_{--+}|^2 = 1.31$$
$$8|\iota_{-+-}|^2 = 1.31$$
$$8|\iota_{-++}|^2 = 1.31$$
$$8|\iota_{---} + \iota_{--+}|^2 = 2.000$$
$$8|\iota_{-+-} + \iota_{-++}|^2 = 2.000$$
$$8|\iota_{+--} + \iota_{+-+}|^2 = 2.000$$
$$8|\iota_{++-} + \iota_{+++}|^2 = 2.000$$
$$8|\iota_{+--}|^2 = 231.7$$
$$8|\iota_{+-+}|^2 = 231.7$$
$$8|\iota_{++-}|^2 = 231.7$$
$$8|\iota_{+++}|^2 = 231.7$$

Without the symmetry breaking, all 8 of the primitive idempotents in the above would have squared magnitude 1/8, and the four pairs would have squared magnitude 1/4.

For example, you've now reduced your set of 8 primitive idempotents as potential elementary particles to 4 idempotents (the ones that have squared magnitude of 2.000). These idempotents can interact by making small changes, but the probabilities are very different between the two pairs. Hence, the symmetry is broken in probability / amplitude.

To match up with the weak hypercharge, weak isospin structure of the standard model, so far I've had to assume that the elementary particles correspond to certain combinations of four primitive idempotents each.

I presented a poster at the PANIC05 conference in Santa Fe last month on this subject that goes into more detail on this:
http://brannenworks.com/PPANIC05.pdf

The detailed calculations for the symmetry breaking are on pages 14-16 of this unfinished paper:
http://brannenworks.com/long_PANIC_Not_Complete_.pdf

An older paper with a less complete derivation of the particle symmetry breaking is here:
http://brannenworks.com/PHENO2005.pdf

Carl

Last edited:
garrett
Gold Member
Hey Carl,

I looked over the stuff you linked to, and you do have a lot of good ideas there. We seem to share a lot of thinking on the importance of Clifford algebras in constructing pretty particle multiplets. (I also first got into the subject by reading Hestenes.) And I agree the quantum mechanical density matrix gets short shrift in most QFT treatments. Its importance only becomes clear when tackling quantum optics. It's a good structure to play with when trying to understand QM geometrically. Also, it makes sense to try a preon model if you're determined to use a low dimensional Clifford algebra, since you need a bigger one if you're going to fit in the known fermions as fundamental fields. On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)

But, as you point out, your model is incomplete. What I've managed to pull off is to put together a complete model using many of these same elements. And I've kept as close as possible to the conventional formalism, using only the most elementary differential geometry, and only things that already have a fundamental place in the standard formalism of QFT and GR. The key was to see a Clifford algebra as a big Lie algebra. I had that idea in my head when I saw Freidel's papers on restricted BF gravity -- and bang, it all came together at once. When I combined the Clifford version of BF gravity with Trayling's model, it was as if they were made for each other. When the Higgs multiplet popped out, multiplying the vierbein, I nearly fell out of my chair! So I had to write it all up.

It's a nice unification. The fields of the standard model and gravity are in a single Clifford valued connection,
A = phi e + omega + W + B + G + nu + e + u + d
and the dynamics are in its curvature.

But it doesn't work perfectly. I don't see where generations come from, and the masses don't come out. I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's. Back when people were saying "Wow, look how well this little model works -- I'll bet if we play with this we'll be getting particle masses in no time!" Of course, they were wrong then... and I could well be wrong now. But I've only written up the most conservative approach possible that fits the standard model. In the paper, I've described in detail a minimal set of building blocks, and I do wish others like you would go play with them, as I'm only one guy and I'm very slow. There are hundreds of different ideas to try: preon models in smaller Clifford algebras, su(5) GUT's in bigger ones, Kaluza-Klein versions, Cartan geometry, sigma models, modified actions, etc. And those paths don't even yet include quantization. The motivation for my model was recent work in Loop Quantum Gravity -- in which the pre-quantized field variables are the chiral (self-dual) spin connection and vierbein. And since my model only involves the same connection variables in a bigger algebra, the same path to a spin-network formulation should be accessible, as well as the recent work in topological perturbation based on BF theory. A good quantum gravity theory is going to reproduce standard quantum field theory when this bigger connection is plugged in.

There are just so many things to do it boggles my mind. But please do take what I've done and use the pieces you like. They really are all the most conventional bits of differential geometry. And there's a decent chance you or me or someone will put a scheme together that connects a whole bunch of these 31 dots.

CarlB
Homework Helper
garrett said:
On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)
Well, as a non academic who presents papers at APS meetings, it is traditional to start with an "Einstein Was Wrong!!!" foil. The multivariable speed of light is a natural generalization for the variable speed of light that has been used recently to explain the inflation problem in cosmology. (See "VSL" in arxiv.org). But what got me into it was that I really wanted to see the elementary particles fall out as the natural small oscillations in a media.

If I gave you a glass jar filled with a liquid that treated right and left polarized light differently you would likely conclude that it was filled with a liquid that was not right-left symmetric. The same thing should apply to spacetime. Of course a jar is easy because we are outside the jar.

Hestenes has an interesting argument in favor of a flat metric (which therefore requires a variable speed of light). If I recall, it amounts to requiring, from a sort of mathematical point of view, that the tangent vectors be embedded in the manifold, and therefore that the manifold be flat. He didn't say it quite like that.

garrett said:
What I've managed to pull off is to put together a complete model using many of these same elements.
I look forward to reading your papers. I just copied them off the web and will read them at home probably tomorrow night. But I'm really not a gravitation guy and I bet that a lot of that is just going to go over my head.

garrett said:
I don't see where generations come from, and the masses don't come out.
I'm going to assume that mass is essentially an interaction where a left-handed particle turns itself into a right-handed one and at the same time reverses its direction. There's a way of writing this out in Feynman diagrams that can be resummed to get the massive propagator from the massless propagators. The bad news is that it is highly Lorentz asymmetric. But my guess is that this is a clue on how nature does it. By the way, Feynman hints at this technique in a footnote in his popular book "QED: The strange theory of matter and light".

Anyway, this method of giving masses to the particles implies that you have to allow the left and right handed particles to change back and forth to one another. Since you have three preons, this implies that you have some (hidden from the standard model) freedom in how to choose their relative phases. By the rules of Feynman diagrams, the preon phase is preserved, or multiplied by a constant, in this change, and the amplitude gives the branching ratio. This implies that you can write a matrix that gives the branching ratios and phase changes between the three preons.

In my preon theory, these three preons are what is giving you SU(3) in your quarks, so they must be treated cyclically. If you assume geometric 3-vectors that distinguish those three preons, you get that the branching ratios have to be in the ratio of 4 to 1 to 1. (You can show this by computing the usual (1+cos)/2 factors for three vectors equally distributed on a cone.) This implies a 3x3 matrix with entries as shown in equation (6) of this paper:
http://arxiv.org/PS_cache/hep-ph/pdf/0505/0505220.pdf [Broken]

As it turns out, the above matrix, which uses the Cabibbo angle for some of the phase changes, postdicts the electron, muon and tau masses accurately to experimental error. You get the three generations and you get their masses.

garrett said:
I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's.
Yes, I bet there's a lot of string theory people who wish it were back in the late '80s again.

I will certainly try to use what you've written.

By the way, you can't possibly understand how much of a relief it is to find someone who can actually understand what I'm working on.

Carl

Last edited by a moderator:
garrett said:
Hey Carl,
I looked over the stuff you linked to, and you do have a lot of good ideas there. We seem to share a lot of thinking on the importance of Clifford algebras in constructing pretty particle multiplets. (I also first got into the subject by reading Hestenes.) And I agree the quantum mechanical density matrix gets short shrift in most QFT treatments. Its importance only becomes clear when tackling quantum optics. It's a good structure to play with when trying to understand QM geometrically. Also, it makes sense to try a preon model if you're determined to use a low dimensional Clifford algebra, since you need a bigger one if you're going to fit in the known fermions as fundamental fields. On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)
But, as you point out, your model is incomplete. What I've managed to pull off is to put together a complete model using many of these same elements. And I've kept as close as possible to the conventional formalism, using only the most elementary differential geometry, and only things that already have a fundamental place in the standard formalism of QFT and GR. The key was to see a Clifford algebra as a big Lie algebra. I had that idea in my head when I saw Freidel's papers on restricted BF gravity -- and bang, it all came together at once. When I combined the Clifford version of BF gravity with Trayling's model, it was as if they were made for each other. When the Higgs multiplet popped out, multiplying the vierbein, I nearly fell out of my chair! So I had to write it all up.
It's a nice unification. The fields of the standard model and gravity are in a single Clifford valued connection,
A = phi e + omega + W + B + G + nu + e + u + d
and the dynamics are in its curvature.
But it doesn't work perfectly. I don't see where generations come from, and the masses don't come out. I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's. Back when people were saying "Wow, look how well this little model works -- I'll bet if we play with this we'll be getting particle masses in no time!" Of course, they were wrong then... and I could well be wrong now. But I've only written up the most conservative approach possible that fits the standard model. In the paper, I've described in detail a minimal set of building blocks, and I do wish others like you would go play with them, as I'm only one guy and I'm very slow. There are hundreds of different ideas to try: preon models in smaller Clifford algebras, su(5) GUT's in bigger ones, Kaluza-Klein versions, Cartan geometry, sigma models, modified actions, etc. And those paths don't even yet include quantization. The motivation for my model was recent work in Loop Quantum Gravity -- in which the pre-quantized field variables are the chiral (self-dual) spin connection and vierbein. And since my model only involves the same connection variables in a bigger algebra, the same path to a spin-network formulation should be accessible, as well as the recent work in topological perturbation based on BF theory. A good quantum gravity theory is going to reproduce standard quantum field theory when this bigger connection is plugged in.
There are just so many things to do it boggles my mind. But please do take what I've done and use the pieces you like. They really are all the most conventional bits of differential geometry. And there's a decent chance you or me or someone will put a scheme together that connects a whole bunch of these 31 dots.
Hi Garrett,
did you know the work of Tolksdorf (math-ph/0503059). He is doing similar things: to derive the standard model including Gravity by using the Clifford bundles and its Dirac operator.

Torsten

Gold Member
Dearly Missed
marcus said:
Garrett, occasional visitor here, posted this today:
http://arxiv.org/abs/gr-qc/0511120
Clifford bundle formulation of BF gravity generalized to the standard model
A. Garrett Lisi
24 pages
"The structure and dynamics of the standard model and gravity are described by a Clifford valued connection and its curvature."
congratulations.
this was the original post on this thread
which started to be about Garrett's paper but got into an intense discussion of Torsten-Helge

selfAdjoint just noticed gr-qc/0511120 in a recent comment at Woit's blog and reminded us of it. I'd be happy if we could get more of this paper explained.

garrett
Gold Member
marcus said:
this was the original post on this thread
which started to be about Garrett's paper but got into an intense discussion of Torsten-Helge
That's OK, the differential structure tangent was interesting. It's good to look down at what you're standing on every once in a while.

selfAdjoint just noticed gr-qc/0511120 in a recent comment at Woit's blog and reminded us of it. I'd be happy if we could get more of this paper explained.
Great, I'm around and would be happy to help anyone with it. By way of encouragement, I should point out that I'm a pretty conservative guy and the mathematical structures described in the paper are all standard bits of differential geometry. So there's nothing in there that you would spend time on and have it be a waste -- since every piece in there is in standard use. And I've laid out the calculations to be easily reproduced. The interesting result of the paper is that these pieces fit together to describe the whole enchilada succinctly, with just one Clifford bundle connection breaking up to give gravity as well as the gauge fields, fermions, and Higgs of the standard model.

Staff Emeritus
Gold Member
Dearly Missed
What impressed me about the paper just from a brief scan I did this morning is the careful way you introduce the mth used one step at a time as it's used, which is so much better than the usual presentations in arxiv papers. It gave me a little frisson, I can tell you and I'm eager to get at it as soon as I get home from my Chistmas visit.

BTW, not to hijack the thread again, but my new grandaughter Elizabeth (b. 12/21/05) is beautiful and healthy, and I'm in seventh heaven about that (too)!

Gold Member
Dearly Missed
...
BTW, not to hijack the thread again, but my new grandaughter Elizabeth (b. 12/21/05) is beautiful and healthy, and I'm in seventh heaven about that (too)!
future generations are what it's all about
warmest congratulations
the solstice, when our luck turns, is a good birthday
greetings to new Elizabeth from one or more at PF physics

garrett
Gold Member
Hey selfAdjoint, congratulations on the new... operator.

I try to make everything as clear as I can, but I was just thinking the other day how great it would be if there were wiki-esque hyperlinks off of variables and mathematical structures. It seems like whenever I'm reading a math paper I'm always hitting a symbol and going "what's that?" So when I write I try to set things up explicitly as I go, or define them right where I first use them so a reader doesn't have to look around.

But really, I'd love to be able to click a mouse on a symbol and have a window pop up with a contextually relevant definition. Wouldn't that be a great way to structure a "paper"? The wikipedia math entries are good that way, but really I'd like to see something like that on a smaller level -- so you could just click on symbols and get links "back" to how they are defined, "sideways" to alternative definitions, and "ahead" in various directions to how they get used. It's possible this way of presenting "networked" information would mesh nicely with categorical thinking, but I don't know enough about that yet. I've seen similar software tools for textually represented concepts, called "mind mapping" software, etc., but none well adapted to mathematical expressions.

Ahh, the future...

Anyway, I'm very happy to hear you appreciate what I've done so far.

Gold Member
Dearly Missed
garrett said:
...thinking the other day how great it would be if there were wiki-esque hyperlinks off of variables and mathematical structures...
as a first approximation, if you were doing an HTML layout of gr-qc/0511120, what wikipedia articles would you insert links to? maybe the answer is obvious, but that shouldn't deter anyone

Staff Emeritus
Gold Member
Dearly Missed
garrett said:
...you could just click on symbols and get links "back" to how they are defined, "sideways" to alternative definitions, and "ahead" in various directions to how they get used.
Wow! Where do I sign up? Isn't this the fulfillment of Tim Berners-Lee's dream? How about an example for, say Connection? Clifford Algebra?

garrett
Gold Member
marcus said:
as a first approximation, if you were doing an HTML layout of gr-qc/0511120, what wikipedia articles would you insert links to? maybe the answer is obvious, but that shouldn't deter anyone
I was really thinking more about making internal, bidirectional links within a document. Or, bidirectional links between nodes in a more flexible display system. And I hadn't thought about external links, but I guess all these obvious ones come to mind:
http://en.wikipedia.org/wiki/Clifford_algebra
http://en.wikipedia.org/wiki/Representations_of_Clifford_algebras
http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
(1)
http://en.wikipedia.org/wiki/Standard_Model
http://en.wikipedia.org/wiki/Connection_(mathematics)
(2)
http://en.wikipedia.org/wiki/Fiber_bundle
http://en.wikipedia.org/wiki/Gauge_theory
http://en.wikipedia.org/wiki/BRST_formalism
as well as about a hundred others.

Wow! Where do I sign up? Isn't this the fulfillment of Tim Berners-Lee's dream? How about an example for, say Connection? Clifford Algebra?
It is annoying as hell that even after all this time there isn't a good standard way of displaying mathematical expressions on web pages. It should have been in there from the beginning. MathML is struggling, but probably will get there eventually. But I don't know if they intend to allow for hyperlinks off of symbols or collections of symbols within expressions. If they do, that will provide most of the functionality I'm suggesting.

To some degree, academic papers have always incorporated similar internal linking through equation reference numbers, such as (1). But with a web document I'd like to see similar links from the symbols to make everything complete and easy.

Also, I'd like to see bidirectional links. For example, if you go to the wikipedia entry on connections, (2), I'd like to see the math symbols linked back to the pages describing what they are. But I'd also like to see links at the bottom of the entry that point to how connections are used -- i.e. what entries link BACK to this connection entry, with these reverse links sorted Google style by popularity (i.e. how many things link to them.) These backwards links would point to examples for the use of what you're looking at, and it wouldn't take any more human effort to encode them since they could be automatically assembled by crawling the network.

I think all this stuff will happen. Just not soon enough. Is anyone here familiar enough with MathML to know if you can make expressions within expressions into hyperlinks?

Staff Emeritus
Gold Member
Dearly Missed
Maybe instead of MathML you should approach Google with these ideas. They already have the technology for those reverse links, and they are sufficiently nerdish to appreciate the need.

Wiki's internal links are great but shallow. You get back to the bare definition of a group pretty fast, whereas i'm thinking, and I believe you're thinking of a link-supported tutorial function. That reverse link idea is so great! It could become a replacement/adjucnt for citation counting: How many reverse links does YOUR research support, Sir? How many times last year were YOUR links activated, Ma'am?

garrett said:
Hey Marcus,
Yes, I've read Matej Pavsic's papers and corresponded with him. And our work shares many common lines. But I came to the conclusion after playing with it a bit that the standard model just doesn't naturally fit in Cl_{1,3}. This isn't to say I don't think his work is good, and he never says it does fit -- he just says "it might." But, most of the stuff I've laid out in my paper are independent of the dimension and signature of what Clifford algebra you want to work in. It's a fairly compact introduction to model building with Clifford valued forms. I just picked Trayling's model as the one that looks by far the best in the end, especially when joined with gravity.
It is only yesterday, that I first saw this discussion.

The gauge group U(1)xSU(2)xSU(3) does not fit in Cl_{1,3} in the way you played with it. I played in a slightly different way.

You are right that the Clifford group does not contain U(1)xSU(2)xSU(3).
But in my papers
http://arxiv.org/abs/gr-qc/0507053
http://arxiv.org/abs/gr-qc/0511124
http://arxiv.org/abs/hep-th/0412255
I discuss the generalized Dirac equation. I introduce the concept of Clifford valued field. The latter field can be expanded in terms of the basis elements which span fourindependent minimal left ideals. The elements of each ideal represent spinors.

As discussed in the papers, a Clifford valued field can be transformed from the left and/or from the right. There is no problem, if the Clifford group acting on \psi from one side only does not contain U(1)xSU(2)xSU(3). My point is that the combined group, acting from the left and from the right can contain it. I show that the transformations acting on \Psi, rerpresented as a 4x4 matrices, can be written in the form of a matrix U which is the direct product of two matrices R and S. Although the transformations R that act from the left do not form the group U(1)xSU(2)xSU(3), and so do not the transformations S that act from the right (but only "nearly form it,as you showed), we have that the combinded transformations U = R \ocross S indeed form U(1)xSU(2)xSU(3).

In your illuminating discussion (sent privately to me) you have considered only a part of the story. The full story indeed contains the GROUP of the standard model. Whether such theory indeed describes the standard model, together with its particle content and all other properties, remains to be investigated. But I think that a promising step has been done in realizing the possibility that a Clifford valued "wave function" \Psi, that can be represented as a 4x4 matrix (four independeent minimal left ideals of Cl_{1,3}), can form a representation of the standard model group. My point is that such \Psi, if we allow for complex valued components, has enough degrees of freedom (namely 32) to describe the standard model particles of the first generation (e,\nu)_{L,R}, [(u,d)_{r,g,y}]_{L,R} and corresponding antiparticles.

Last edited: