Finally, Garrett's model with 3 generations

[MTd2]

Marcus pointed out this yesterday:

http://arxiv.org/abs/1506.08073

Lie Group Cosmology
A. Garrett Lisi
(Submitted on 24 Jun 2015)
Our universe is a deforming Lie group.

*****************************************

This is a non pretentious title, so his solution probably passed unnoticed. Also, he hints on exotic smoothness in 4d, though he hasn't said that directly, but cited Scorpan's book, the Wild World of 4 Manifold.

So, in a way, there is some relationship with Torsten's work.

"One philosophical justification could be that the geometry and topology of four-dimensional manifolds is maximally rich [24], and that the representations of the E8 Lie group are the most numerous, but this is not completely satisfying. Also, there is no good reason, other than the spin-statistics requirement, why the bosonic and fermionic parts of the superconnection must be valued in complementary parts of the Lie algebra. It is possible that this restriction could be relaxed, allowing the existence of BRST ghosts or other particles, or that there may be a natural reason for the restriction that is not yet clear. A better understanding is needed."

The 3 generations appear on p. 32, section 16.

 

[marcus]

Thanks for pointing out the achievement of 3-generations and the connections with other work! There's a chance Garrett will look in here so you might want to post some questions : ^)

 

[garrett]

Thanks MTd2, Marcus. Yes, I'd be delighted to answer questions here, for anyone interested in this paper. (Although I'm saddened you consider the title non-pretentious.)

 

[Demystifier]

(Although I'm saddened you consider the title non-pretentious.)

The title sounds very pretentious to me. In fact, very short titles always sound so.

 

[garrett]

Yes, the only chance to beat "An Exceptionally Simple Theory of Everything" for pretension was to go short.

 

[Demystifier]

Yes, the only chance to beat "An Exceptionally Simple Theory of Everything" for pretension was to go short.

That's fine for the title. But a too concise abstract, in my opinion, is not such a good idea. At the very least, the abstract should explain what kind of argument is used to reach the conclusion.

 

[MTd2]

Garrett,

1.Do you know the work of Torsten? http://arxiv.org/find/gr-qc/1/au:+Asselmeyer_Maluga_T/0/1/0/all/0/1 You hinted exotic smoothness in 4D, that would be worth a look.2. You mention "Following our philosophical desire for geometric unity [16]," [16] E. Weinstein, private communication.
Weinstein is a mystery: http://www.theguardian.com/science/blog/2013/may/23/roll-over-einstein-meet-weinstein
http://www.theguardian.com/science/2013/may/23/eric-weinstein-answer-physics-problems

People talked about it, but no one told what it is.
So, what you presented in your paper is Weinstein's theory? Or something very similar? When you write on the following page (P.24):

"But what if we generalize Cartan geometry and consider deformations of a large Lie group, such as Spin(12, 4), with four-dimensional submanifolds corresponding to spacetime? There is no reason we can’t choose a Spin(1, 4) subroup of Spin(12, 4) and model four-dimensional spacetime on a representative subspace of Spin(12, 4) corresponding to Spin(1, 4)/Spin(1, 3) de Sitter spacetime within the Spin(1, 4) subgroup."

It seems that either his model or your will yield similar stuff. Maybe redundant?

 

[garrett]

I do know of Torsten's work, and that 4-manifolds have the richest structure, but at this point I only use this as vague philosophical motivation for why 4-manifolds might be special.

E. Weinstein is a riddle, wrapped in a mystery, inside an enigma -- but a bit less so now that he's been out to the Pacific Science Institute several times, and we've discussed our ideas extensively. In my opinion, some of his ideas are quite good, and there's been enough cross-pollination that I thought it necessary to cite him, even if he hasn't released his paper yet. He is, in a way, my arch-nemesis, since he is also working on what is, essentially, a unified gauge theory of everything. The details of the structures we use are different, but he is at least playing in the same ballpark, with Ehresmannian geometry in high-dimensional spaces playing the main role. And it's not a ballpark with many players, since most high energy theorists are off playing with strings and branes. But, as well as friends, we're also competitors, and our theories are quite different.

 

[marcus]

Garrett in the conclusions, I think it was, you mentioned the central role played by connections and the fact that this leans in the direction of some type of Loop quantization: suggesting that Loop quantization of some sort might be suitable for LGC. I'd be glad of any more ideas or educated guesses about that.

At the moment I'm interested in the different version of LQG that Bianca Dittrich and others are developing. They just posted a paper with a title like "a new realization of LQG." I started a thread on it. Also Thomas Thiemann, I think, has been working on different treatments of LQG. I can't explain rationally but feel it would be neat if one of these new versions turned out especially compatible with LGC.

I think even though I don't see the quantum gravity in it, as yet, your paper should be on the 2nd quarter 2015 MIP poll. If it is don't be modest (how could you be?) --- be sure to vote for it. : ^)

 

[garrett]

Good question, Marcus. The connection, if you will, between Dittrich's "new realization of LQG," Thiemann's approaches to LQG, and LGC is that these theories are all background free, all use a connection as the fundamental field variable, and are all formulated using a modified BF action (equ (14.1) in LGC, equivalent to a generalized Y-M action). Whichever LQG approach is successful as a description of quantum gravity should extend to LGC, using the same methods, but with a connection valued in a much larger Lie algebra, incorporating matter. Also, LGC, incorporating a description of matter which has been successfully quantized via QFT, may give some insights into how best to develop LQG, in a compatible way. And thanks for including LGC in the MIP poll!

 

[arivero]

Hi Garrett, I see you have a q=4/3 coloured object. My more sincere condolences, and welcome to the club.

How sure are you that it is a gauge boson? Could it be instead a set of scalars? I assume that definitely you can not put it as a fermion (sorry if it sounds a bit mad, but in my case that object happened first as a set of scalars and became a serious headache)

 

[garrett]

Hi Arivero, Yes, I wasn't particularly excited to see that X^4/3 either. There is some flexibility between what is a fermion and what is a boson in LGC, but it's more naturally a boson. Also, it doesn't couple to the frame, so it's a gauge boson and not a scalar. I should emphasize, though, that LGC is a framework for model building using Lie groups, and is not locked to anyone particle assignment, including E8. I won't be convinced of any particle assignment or new particle predictions until I see CKM-PMNS mixing coming out nicely.

 

[arivero]

Hi Arivero, Yes, I wasn't particularly excited to see that X^4/3 either. There is some flexibility between what is a fermion and what is a boson in LGC, but it's more naturally a boson. Also, it doesn't couple to the frame, so it's a gauge boson and not a scalar. I should emphasize, though, that LGC is a framework for model building using Lie groups, and is not locked to anyone particle assignment, including E8. I won't be convinced of any particle assignment or new particle predictions until I see CKM-PMNS mixing coming out nicely.

Well, the good news -to me- are that having the 4/3 thing, the model looks susceptible of superBootstrap (for newcomers, this is my idea of building all the squarks and sleptons from pairs of the five light quarks, using SU(5) flavour)... and then the "diquark" combinations associated to each degree of freedom of the leptons could give a hint of the CKM mixing. This program failed in my naive approach, where the only structure was Pati-Salam, but perhaps here there is more uniqueness.

 

[arivero]

Thanks for pointing out the achievement of 3-generations and the connections with other work! There's a chance Garrett will look in here so you might want to post some questions : ^)

I am not sure if it is an achievement or we are "back to week 91". For history, let me remember that the online community discussed E8 from Baez readings

http://math.ucr.edu/home/baez/week90.html
http://math.ucr.edu/home/baez/week91.html
in 1996. Some of it still permeatez Baez's node19 http://math.ucr.edu/home/baez/octonions/node19.html on octonions.

I think that the idea of relating triality to generations already surfaced at that time, not sure if Garrett or someone else.

Also while looking for the old weeks google has launched me into some comments in woit's blog, including Weinstein's. Not sure if related: http://www.math.columbia.edu/~woit/wordpress/?p=3665

 

[garrett]

Yes, I'm not yet sure what to make of the X^{4/3}. And until CKM-PMNS mixing comes out naturally, I don't have high confidence in any specific model.

Thanks for reminding me of John Baez's post. Back in 2005 I was working on describing the Standard Model and gravity using one large algebra,
http://arxiv.org/abs/gr-qc/0511120
Then, in 2007, I was wondering if this might be part of some large Lie algebra, and saw John's post,
http://math.ucr.edu/home/baez/week90.html
The match to E_8, with three generations of fermions related by triality, made a big impact on me. But there was a huge problem. With everything described as a E_8 principal bundle over a four-dimensional spacetime base, there is ONE Spin(1,3) subgroup of E_8 that gets picked out as the gravitational Lie group. When this happens, the other two blocks of fermion "generations" in E_8 cannot be fermion generations with respect to that Spin(1,3), as pointed out in http://arxiv.org/abs/0711.0770. Instead, you get an "anti-generation" of mirror fermions, and a "generation" of vector fields -- the fact that Distler and Garibaldi used to "prove" that E8 theory can't work,
http://arxiv.org/abs/0905.2658

Distler's criticism was deceptive though. He didn't claim there were mirror fermions; instead, he said there was not even a single generation in E_8. That misleading claim was so upsetting to me that I spent years finding an explicit embedding of the familiar gravitational and Standard Model generators, including a generation of fermions, and their mirrors, in E_8:
http://arxiv.org/abs/1006.4908
It was good that I did that work, but it took me away from the main hope of describing three generations. Then, a few years ago, while talking with Derek Wise, I realized something interesting about Cartan geometry.

I had heard of Cartan geometry, here on PF and from John Baez and Sharpe's book, as a way of describing gravity and spacetime as "the Spin(1,4) Lie group gone wobbly." That seemed like a really cool description, but when the same framework is applied in the most obvious way to larger Lie groups, it results in very high dimensional spacetimes, which is problematic. But then I realized that one might be able to generalize Cartan geometry in a non-obvious way, by embedding Spin(1,4), containing spacetime, in a larger Lie group, and then letting that embedded four-dimensional spacetime go wobbly. That was the seed idea of Lie Group Cosmology. Then I found that triality could relate three spacetimes within E_8, just like gauge transformations relate different sections of a fiber bundle, and that there could be a single generation of fermions with respect to each one of those three spacetimes, and no mirror fermions. There was also a very natural description of what fermions are within this generalized Cartan geometry. So, by having three triality-related Spin(1,3)'s, I now think I've mostly solved the problem of what the three generations are. Yes, I should have been able to solve that problem years ago, had I just continued with the original E8 theory; but I was dealing with a lot of stressful criticism, and the ideas behind LGC weren't obvious and took awhile to germinate. Fortunately, physicists ignored E8 theory because of the criticism, giving me time to work on it in peace. The next question will be exactly how the embedding and mixing works, and whether CKM-PMNS mixing comes out, and if there are any predictions from that.

 

[arivero]

Yes, I'm not yet sure what to make of the X^{4/3}. And until CKM-PMNS mixing comes out naturally, I don't have high confidence in any specific model.

Thanks for reminding me of John Baez's post. Back in 2005 I was working on describing the Standard Model and gravity using one large algebra,
http://arxiv.org/abs/gr-qc/0511120
Then, in 2007, I was wondering if this might be part of some large Lie algebra, and saw John's post,
http://math.ucr.edu/home/baez/week90.html
The match to E_8, with three generations of fermions related by triality, made a big impact on me. But there was a huge problem. With everything described as a E_8 principal bundle over a four-dimensional spacetime base, there is ONE Spin(1,3) subgroup of E_8 that gets picked out as the gravitational Lie group. When this happens, the other two blocks of fermion "generations" in E_8 cannot be fermion generations with respect to that Spin(1,3), as pointed out in http://arxiv.org/abs/0711.0770. Instead, you get an "anti-generation" of mirror fermions, and a "generation" of vector fields -- the fact that Distler and Garibaldi used to "prove" that E8 theory can't work,
http://arxiv.org/abs/0905.2658

Distler's criticism was deceptive though. He didn't claim there were mirror fermions; instead, he said there was not even a single generation in E_8. That misleading claim was so upsetting to me that I spent years finding an explicit embedding of the familiar gravitational and Standard Model generators, including a generation of fermions, and their mirrors, in E_8:
http://arxiv.org/abs/1006.4908
It was good that I did that work, but it took me away from the main hope of describing three generations. Then, a few years ago, while talking with Derek Wise, I realized something interesting about Cartan geometry.

I had heard of Cartan geometry, here on PF and from John Baez and Sharpe's book, as a way of describing gravity and spacetime as "the Spin(1,4) Lie group gone wobbly." That seemed like a really cool description, but when the same framework is applied in the most obvious way to larger Lie groups, it results in very high dimensional spacetimes, which is problematic. But then I realized that one might be able to generalize Cartan geometry in a non-obvious way, by embedding Spin(1,4), containing spacetime, in a larger Lie group, and then letting that embedded four-dimensional spacetime go wobbly. That was the seed idea of Lie Group Cosmology. Then I found that triality could relate three spacetimes within E_8, just like gauge transformations relate different sections of a fiber bundle, and that there could be a single generation of fermions with respect to each one of those three spacetimes, and no mirror fermions. There was also a very natural description of what fermions are within this generalized Cartan geometry. So, by having three triality-related Spin(1,3)'s, I now think I've mostly solved the problem of what the three generations are. Yes, I should have been able to solve that problem years ago, had I just continued with the original E8 theory; but I was dealing with a lot of stressful criticism, and the ideas behind LGC weren't obvious and took awhile to germinate. Fortunately, physicists ignored E8 theory because of the criticism, giving me time to work on it in peace. The next question will be exactly how the embedding and mixing works, and whether CKM-PMNS mixing comes out, and if there are any predictions from that.

Well, a lot of readers appreaciated the idea of triality => generations already in the first work. Let me recall that it spawned a revision of trialities here in PF https://www.physicsforums.com/threads/triality-and-its-uses.175959/

and a direct reference in a note of Boya:

Fourth, there is a hint for family triplication, as the group F4, with 3-Torsion, is related to octonion triality; for example, in the ”Mercedes” Dynkin diagram for O(8) triality is manifest; adding to Spin(8) the three equivalent representations, we reach F4

After the first draft of the paper was sent off, we became aware of the (now famous) preprint of G. Lisi [15]. In fact, we subscribe unconsciously to the Pati-Salam “ leptons as the fourth color” philosophy, as Lisi does; he also uses polytopes and the F4 group, and hints to a relation between triality and generations. He is more ambitious, though, as he considers gravitation as well, but does not adhere to supersymmetry.

EDITS:
- kneemo tracks the idea generations=triality to a paper of Silagadze http://arxiv.org/abs/hep-ph/9411381 In fact S. himself told us in the discussion following week 253 http://mathforum.org/kb/message.jspa?messageID=5799515

- two mathoverflow questions related to the topic: http://mathoverflow.net/questions/116666/triality-of-spin8 http://mathoverflow.net/questions/75875/why-su3-is-not-equal-to-so5[

 

[garrett]

Yes, the idea of using triality for generations has been talked about for quite some time. The main problem though, in my opinion, was that when you include gravitational spin(1,3) acting on three triality-related generations, in the usual way, you don't get three generations, but instead get mirror fermions and vectors, which make the idea intractable. What I've tried to do in LGC, among other things, is describe how you can have three triality-related spin(1,3)'s acting on three generations of fermions, in a way that's natural and makes sense. What I'm excited about here is that all of this, including how fermions appear as Grassmann valued spinors, can be understood as deformations of a Lie group, via generalized Cartan geometry.

 

[Alberto Garcia]

Hi, does exist a graviton in the theory? If it does exist, is it an elementary particle with spin 2?

 

[garrett]

Yes. Although it is not the most natural choice of variable, one can expand the metric for small perturbations away from a background, and quantize that, describing spin 2 gravitons. If one does that, the natural choice of background in LGC is de Sitter spacetime. But a more natural choice for variables to quantize are the spin connection and frame.

 

[MTd2]

Garrett, has Jacques Distler already checked your paper?

 

[garrett]

Right, because he's the most unbiased proofreader...

The true test of this LGC framework won't be what potshots critics take at it, but whether I, or someone else, can use it to correctly match the masses and mixings of the Standard Model, and make some predictions.

 

[E8(8)]

This is not a serious paper, that it is easy to see in a first lecture: it is inconsistent an incorrect. About this other paper of Garret Lisi:

http://inspirehep.net/record/859554?ln=en

It has been shown to be incorrect here:

http://inspirehep.net/record/1236254

So end of discussion. I am very surprised that Garrett Lisi's papers are considered here as if they were serious research papers. This might be misleading for studients without the experience to distinguish solid research from nonsense decorated with fancy words.

 

[MTd2]

E8(8), you are so boring. Why don't you go discuss this with Distler?

 

[garrett]

http://inspirehep.net/record/1236254 proves that the "extended GraviGUT algebra," which contains gravity and the Standard Model with one generation of fermions, is contained in E_8. But I guess I don't understand E8(8)'s point, since that's nothing but good news.

 

[E8(8)]

http://inspirehep.net/record/1236254 shows that this paper:

http://inspirehep.net/record/859554?ln=en

is incorrect. That is clear even from the abstract of http://inspirehep.net/record/1236254. So first of all, Lisi has sent a completely incorrect paper to arxiv. Then, they fix Lisi's construction and they embed and extension of the complexified so(14), namelu so(14)_{C}, into the complex E_8. So OK, they embed an extension of the complexified so(14)_{C} into the complex E_8, that does not support any of your claims.

 

[arivero]

I wonder if you have contemplated neutrinos out of this setup

a hint could be already in the decomposition

248=28+28+64+64+64

if each of the 64 means a full generation, a neutrinoless generation should be a subset of 56. So perhaps is a subset that interacts with the previous 28+28 "spin(4,4)+spin(8)" .

 

[garrett]

E8(8): In http://inspirehep.net/record/859554?ln=en, what is claimed is that the algebra of spin(11,3) acting on a generation of 64 fermion degrees of freedom is contained in E8. It was never necessary in that paper that spin(11,3) + 64 should, itself, be considered a Lie subalgebra of E8. In http://inspirehep.net/record/1236254, spin(11,3) + 64 is extended by a 14 dimensional ideal, and complexified, and, now as a Lie algebra, shown to embed in complex E8. None of this shows that http://inspirehep.net/record/859554?ln=en is incorrect. Also, it's potentially of use to LGC.

 

[garrett]

arivero: Now that is an interesting question! Since spin(1,3) + su(2)_L + u(1) + su(3) won't fit in spin(4,4) + spin(8), some of the boson generators would have to live in the 64's. Interestingly, the embedding and triality automorphism can be chosen such that W^{\pm} lives in the 64's, taking the place of some right handed neutrino dof. Since the action of the 64's on each other mixes them, this is possibly the origin of CKM-PMNS mixing. If neutrinoless double beta decay is observed, this model will remain sound. If it isn't, then the only hope will be embedding, with right handed neutrinos, in complex E8. In either case, one must use and relate different Cartan subalgebras for weak vs mass eigenstates.

 

[arivero]

Interestingly, the embedding and triality automorphism can be chosen such that W^{\pm} lives in the 64's, taking the place of some right handed neutrino dof.

Had you some SUSY in, perhaps it could be some difference between spin 1 particles which are partners of s=3/2 and spin 1 which are partners only of s=1/2.

Another question: do you have some hint of the top quark?

I mean, there are two "mass protections" in the standard model fermions. On one side, the electric charge protects all the particles except neutrinos against a majorana mass. On other side, some unknown mechanism protects all the particles except top quark of getting a dirac mass of "electroweak size". Sometimes we have speculated here of the numerology of both mass protections, where it happens that we have the same number of unprotected dof: either three neutrinos on one side or three colours of a quark on the other. Moreover, the number of protected complex dof in each case is 84, so a long shot is that the same math that in the M2-M5 brane duality is happening here.

 

[mitchell porter]

So far I am failing to understand the three "regional" space-times. But I have a tentative analogy, perhaps you can comment on its appropriateness.

Suppose we have a two-dimensional physical system containing three fields, φ(x,y), χ(x,y), ψ(x,y).

And then someone proposes as a model of this, a three-dimensional theory containing three fields, Φ(x1,x2,x3), X(x1,x2,x3), Ψ(x1,x2,x3).

And then they say, this is the mapping: Φ(x1,x2,-) -> φ(x,y), X(x1,-,x3) -> χ(x,y), Ψ(-,x2,x3) -> ψ(x,y)

where e.g. Φ(x1,x2,-) means the restriction of the field Φ to some (x1,x2)-submanifold of the (x1,x2,x3) space.

So the final claim is that the dynamics of Φ on that submanifold, happens to reproduce the dynamics of φ in the physical system of interest. And similarly that X and Ψ reproduce the dynamics of χ and ψ, when one focuses on the relevant submanifolds.

It looks to me as if you are doing something similar, only with the three generations; since you refer, on page 27, to "partially overlapping Spin(1,4)’s in Spin(4,4)".

So it sounds like there is a set of degrees of freedom ("Spin(4,4)"), and there are three ways of picking out a subset of them that can play the role of geometric degrees of freedom in space-time ("Spin(1,4)"), and you want the space-time base for each SM generation of fermion fields to be a different subset each time. But the subsets aren't disjoint ("partially overlapping").

That's what my analogy has tried to illustrate in vastly simplified form. But have I got it right?

 

[garrett]

arivero: There is a Higgs and interaction with the different fermion generations, but I don't yet have a good model of their different masses and mixings.

 

[garrett]

Mitchell: Yes, this is an excellent analogy. Some additional information is that the embedding space is not three dimensional, but has some high dimension, such as 248. The "-" you write can be considered as fibers when this high-dimensional space is considered as the total space of a fiber bundle. And there's a special automorphism of the space, called triality, that maps the three fields to each other, and can be considered a large gauge transformation. Spin(4,4), which is 28-dimensional, is the smallest space I know of for which this makes sense for a 4-dimensional spacetime.

 

[arivero]

Garrett, any hope of connecting this E8 to some symmetry of a 11D M5-brane?

(Baez http://math.ucr.edu/home/baez/week237.html speaks of M2-branes related to E8. Well, both are related via duality)

 

[garrett]

There appears to be a connection, but it's a bit of a stretch, and there would be a lot of theoretical convolutions needed. I'm finding it much more interesting to simply consider deformations of the E_8 Lie group.

 

[E8(8)]

What do you mean by a "deformation of E_8"?

 

[garrett]

A "deformation" of a Lie group is meant in the same way that our lumpy spacetime is a deformation of de Sitter spacetime.

 

[E8(8)]

Can you give a precise sensible mathematical definition of what you mean by "deformation of E_8"?

 

[Jimster41]

A "deformation" of a Lie group is meant in the same way that our lumpy spacetime is a deformation of de Sitter spacetime.

I'm just a layperson trying to get a sense of what what the view is here. Your paper has sent me into a fugue of wikipedioia.
is there any intuitive way to understand (enough to appreciate the view) the difference between the number of dimensions and the number of roots, of an algebraic structure like the e8?

Your statement above seems relevant to my confusion here, I have been trying to tie the E8 and representation theory to the experience of looking at objects in the world. I get the idea that the algebra describes the constrained behavior of a set of objects - potentially those objects are the basis of the SM, the fundamental structures of matter in space time.

If that is in some sense getting the meaning of it, Do you have any thoughts on what process is causing deformation? What is causing the evolution of species, as it were, such that we don't just have the raw basis, but instead... The world with all its richness, in time. Is there a sense in which the algebra must be "run"?

 

[garrett]

Yes, E8(8), a deforming Lie group is described by a variation of a Lie group manifold's Maurer-Cartan connection away from zero curvature.

 

[E8(8)]

Can you give a a precise definition of what you mean by "deformation of E_8"? What you just said does not seem to have any sense. Can you at least point out to a mathematical reference where such concept is defined and explored?

 

[garrett]

I'm just a layperson trying to get a sense of what what the view is here. Your paper has sent me into a fugue of wikipedioia.
is there any intuitive way to understand (enough to appreciate the view) the difference between the number of dimensions and the number of roots, of an algebraic structure like the e8?

Hello Jimster41,
Sure, for an N-dimensional Lie algebra (corresponding to an N-dimensional Lie group), there can be a maximum of R mutually linearly independent commuting generators, which span what is called a Cartan subalgebra. With respect to these mutually commuting generators, acting via the Lie bracket on the rest of the Lie algebra, there will be N-R eigenvectors, called root vectors, with eigenvalues called roots.

Your statement above seems relevant to my confusion here, I have been trying to tie the E8 and representation theory to the experience of looking at objects in the world. I get the idea that the algebra describes the constrained behavior of a set of objects - potentially those objects are the basis of the SM, the fundamental structures of matter in space time.

Yes, in physics, the roots, and specifically the eigenvalues with respect to the mutually commuting generators, correspond to elementary particle charges.

If that is in some sense getting the meaning of it, Do you have any thoughts on what process is causing deformation? What is causing the evolution of species, as it were, such that we don't just have the raw basis, but instead... The world with all its richness, in time. Is there a sense in which the algebra must be "run"?

This is a harder question. The best answer I know of comes from quantum mechanics, which basically says nature tries everything.

 

[garrett]

Can you give a a precise definition of what you mean by "deformation of E_8"? What you just said does not seem to have any sense. Can you at least point out to a mathematical reference where such concept is defined and explored?

Yes, that is the point of the entirety of the paper. What part is confusing to you? Do you not understand that the Maurer-Cartan form is a Lie algebra valued 1-form, with vanishing curvature, defined on the Lie group manifold? And if you do understand that, is it not sensible to consider variations of this 1-form? If you clarify the problem you're having with this, I may be able to help by providing more information (that's in the paper), but I can't help if your objection is vague.

 

[E8(8)]

My objection is not vague. What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

 

[Jimster41]

Sorry if I this covered in the paper, does the deforming Spin(1,4) Lie group containing the rigid Spin(1,3) sub algebra have to be DeSitter to work, to allow GR to be represented as the Cartan Connection between them (I hope that's not butchering it too much), or can it also/instead be AdS?

 

[garrett]

My objection is not vague. What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

Yes, the main idea and mathematical formulation of Cartan geometry has been explored in many previous works, perhaps most extensively in Sharpe's book on the subject. What is provided in LGC is a generalization and application to physics.

 

[garrett]

Sorry if I this covered in the paper, does the deforming Spin(1,4) Lie group containing the rigid Spin(1,3) sub algebra have to be DeSitter to work, to allow GR to be represented as the Cartan Connection between them (I hope that's not butchering it too much), or can it also/instead be AdS?

The symmetric space resulting from modding Spin(1,4) by Spin(1,3) is de Sitter spacetime -- in this case, no deformations are necessary, and de Sitter spacetime can be considered the "rest state," described by the Maurer-Cartan form on the Lie group. One only needs to consider deformations in order to describe our lumpy spacetime, with matter in it, described by a Cartan connection. If you wanted to describe Anti-de Sitter spacetime, you'd have to start with Spin(2,3).

 

[E8(8)]

Yes, the main idea and mathematical formulation of Cartan geometry has been explored in many previous works, perhaps most extensively in Sharpe's book on the subject. What is provided in LGC is a generalization and application to physics.

I am obviously not asking about the mathematical formulation of Cartan geometry. I know that is a solid theory of mathematics, and I know Sharpe's book. You are obviously evading the question. I will repeat it:

What do you mean by "deforming a Lie group"? What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

Can you AT LEAST give one mathematical reference where this is properly defined , because in your paper it is not? And by this I evidently don't mean a general book a bout Cartan geometry, I mean a reference where the deformations that you use are defined and studied. Can you at least do that? It is not that hard if you know what you are talking about. Because right now it looks like you don't know what "deforming a Lie group" is and that your paper is nonsense.

 

[garrett]

I provided a solid reference, and a succinct description. Perhaps you don't know Sharpe's book, or anything for that matter, as well as you think you do. This is from Sharpe:

A Cartan geometry on M consists of a pair (P, \omega), where P is a principal bundle H \to P \to M and \omega, the Cartan connection, is a differential form on P. The bundle generalizes the bundle H \to G \to G/H associated to the Klein setting, and the form \omega generalizes the Maurer-Cartan form \omega_G on the Lie group G. In fact, the curvature of the Cartan geometry, defined as \Omega = d \omega + \frac{1}{2} [\omega, \omega], is the complete local obstruction to P being a Lie group.

The manifold P may be regarded as some sort of “lumpy Lie group” that is homogeneous in the H direction. Moreover, \omega may be regarded as a “lumpy” version of the Maurer-Cartan form. The Cartan connection, \omega, restricts to the Maurer-Cartan form on the fibers and hence satisfies the structural equation in the fiber directions; but when Ω \ne 0 we lose the “rigidity” that would otherwise have been provided by the structural equation in the base directions and that would have as a consequence that, locally, P would be a Lie group with \omega its Maurer-Cartan form. Thus, the curvature measures this loss of rigidity.

I prefer the descriptive "deforming" to "lumpy," but the concept is the same. Perhaps you should compose a note to Sharpe to inform him that his work is nonsense.

 

[garrett]

The LGC idea begins with Cartan geometry, as a deformation (or excitation) of a Lie group G', and then generalizes to what can happen when G' is a subgroup of a larger group, G. And, yes, it's simple, and, yes, that's all one needs to build a ToE.

 

[E8(8)]

The LGC idea begins with Cartan geometry, as a deformation (or excitation) of a Lie group G', and then generalizes to what can happen when G' is a subgroup of a larger group, G. And, yes, it's simple, and, yes, that's all one needs to build a ToE.

You are not generalizing anything. You are just using Cartan geometry and you are not even doing that right. The fact that you are calling "deforming Lie group" a Cartan geometry, which is something very classical and very well known, is simply absurd, ridiculous. The abstract of your paper: "Our universe is a deforming Lie group.", can be thus rewritten as:

"Our universe is a manifold."

Because a Cartan geometry has as underlying space a manifold. So great, Garret, you really have obtained a TOE by modelling our universe using a manifold. I think no one thought of that one before! ;)