Finally, Garrett's model with 3 generations

1. Jun 29, 2015

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.

Last edited: Jun 29, 2015
2. Jun 29, 2015

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 : ^)

3. Jun 29, 2015

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.)

4. Jun 30, 2015

Demystifier

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

5. Jun 30, 2015

garrett

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

6. Jun 30, 2015

Demystifier

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.

7. Jun 30, 2015

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

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?

8. Jun 30, 2015

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.

9. Jun 30, 2015

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. : ^)

Last edited: Jun 30, 2015
10. Jul 1, 2015

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!

11. Jul 3, 2015

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)

Last edited: Jul 3, 2015
12. Jul 4, 2015

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 any one 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.

13. Jul 4, 2015

arivero

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.

14. Jul 5, 2015

arivero

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

15. Jul 5, 2015

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.

16. Jul 5, 2015

arivero

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:

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[

Last edited: Jul 5, 2015
17. Jul 6, 2015

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.

18. Jul 7, 2015

Alberto Garcia

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

19. Jul 7, 2015

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.

20. Jul 7, 2015

MTd2

Garrett, has Jacques Distler already checked your paper?