Garrett Lisi bid to join Std. Model w/ gravity

[garrett]

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.

 

[selfAdjoint]

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)!

 

[marcus]

...
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]

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.

 

[marcus]

...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

 

[selfAdjoint]

...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]

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?

 

[selfAdjoint]

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?

 

[pavsic]

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.