An Exceptionally Simple Theory of Everything

[jal]

Hi garrett

I've been following what has been discussed at Bee's blog.
As I expected, you and tony left me behind.

I'm not exactly sure what you mean. But the pretty plots, which you can show to grandma, are projections of all the E8 Lie algebra elements (corresponding to roots in eight dimensions) that show the pattern of interactions for the various particles of the standard model. These patterns are there in E8, all I've done is label them with particle names.

The dance floor is too full.
I hope that eventually you will be able to reduce the patterns (3D) to make it possible to explain to grandma.
jal

 

[garrett]

The dance floor is too full.
I hope that eventually you will be able to reduce the patterns (3D) to make it possible to explain to grandma.

I suggest starting with the dance of the quarks and gluons in G2.

 

[marcus]

as someone who navigates partly by hunches---touch and smell even--- I got a good feeling from this comment by Garrett at Bee's
http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html#c8778500176714763161

Please don't take my word for it. The first person I know of to point out this loophole in Coleman-Mandula was Thomas Love in his 1987 dissertation. There is also a discussion of this loophole in this recent paper by F. Nesti and R. Percacci: Graviweak Unification. Or you can go to the source and look at Coleman and Mandula's paper, in which their condition (1) for the theorem is "G contains a subgroup locally isomorphic to the Poincare group." The G = E8 I am using does not contain a subgroup locally isomorphic to the Poincare group, it contains the subgroup SO(4,1) -- the symmetry group of deSitter spacetime.

always a good sign when the deSitter group shows up instead of the Poincaré, after all the universe is expanding.

Derek Wise thesis was about understanding Mac-Mans gravity. I wonder if he is reading Garrett's paper at this point.

What kind of coalition will form around Garrett's gambit? Research is a relay race and maybe this paper defines a starting line. Sorry about incoherence. Maybe lunch would be a good idea.

 

[CarlB]

What kind of coalition will form around Garrett's gambit?

Well, the empire has struck back, and the paper has been reassigned from physics/hep-th to physics/general. I don't think that will make much of an impression as far as the importance of the paper, but when you mess around with Poincare invariance it does tend to p:ss off the elders.

 

[marcus]

Well, the empire has struck back, and the paper has been reassigned from physics/hep-th to physics/general. I don't think that will make much of an impression as far as the importance of the paper, but when you mess around with Poincare invariance it does tend to p:ss off the elders.

That was beautifully expressed, from beginning "Well..." to the final word.
I had read the news at Bee's blog, where it was brought by just the right messenger with just the right air of satisfied outrage and hysteria.
I expect a statement by Jacques Distler on "musings" which will justify the arxiv action without explicitly taking responsibility, before long.

He may have been already commenting at Bee's, as that particularly stubborn anonymous

 

[Jim Kata]

Thankfully, I'm not afraid of looking ignorant

My knowledge of representation theory and octonions is very limited and definitely not up to snuff to understand all of this paper like taking a real non compact representation of e8 but the thing I find the most interesting is triality.

Another thing is like, Hans and Carl pointed out, the possible connection between tribimaximal matrix and the matrix Garrett uses to embed the su(3) root system into so(6).

I have two questions one is off the wall and one is??

1.) The matrix in Garrett's paper, as he says, can be viewed as the twelve midpoints of the edges of a cube. Now I was not really thinking, but there are 12 leptons, and flavours of quarks, four for each generation. Do these midpoints on the edges of this cube represent each of the generations of quarks and leptons? I wish I could draw what I mean.

2.) Looking at figure 5 this theory seems to predict right handed leptons. Is this true? If so, is there a mechanism to explain why these are not observed?

 

[marcus]

Peter Woit blogged Garrett's paper
http://www.math.columbia.edu/~woit/wordpress/?p=617
as well he might
as for the arxiv gnomes reclassifying the paper from hep-th to gen, fill in the consonants
_acques _ucques.
It's infuriating and depressing that someone at arxiv would stoop to something that dishonorable.

For this reason, and for its merits, I hope we choose this paper as our PF "Beyond" forum Paper of the Year.
It is a small thing but we should have some way, even a small symbolic one, of fighting back.

 

[jal]

I see that garrett is carrying on simultaneous conversations in at least 3 places.
I have an observation that might help the beginners.
----------
Is there anyone out there capable of reading a knitting pattern?
You need to know the symbols for each action of the needles and when you apply that action a pattern will emerge.
The needles are flying off into the 3rd dimension and picking up the threads and leaving the threads in a 2d pattern.
There are moves that cannot be done.
Talk to a grandma.
E8 is a template for a knitting pattern. The pattern exists in our perceived 3d.
CERN will be able to look for the patterns that exist from 10^-15 to 10^-18. It will not be able to see the movement of the “needle” doing its dance that created the pattern. LQG will be needed to find out what the “needles” are doing.
I cannot knit, I cannot do LQG and much less E8.
However, if you succeed in writing the knitting pattern, grandma will be able to knit it.
----------
Ref.:
Proton
Mass m = 1.00727646688 ± 0.00000000013 u
Mass m = 938.27203 ± 0.00008 MeV [
Charge radius = 0.875 ± 0.007 fm
(diameter of about 1.6 to 1.7×10−15 m [1], and a mass of 938.27231(28) MeV/c2 (1.6726 × 10−27 kg), 1.007 276 466 88(13) u)
Mean life τ >10^31 to 10^33 years

 

[marcus]

Garrett, want to clarify about the dimension? (for the not too smart spectator)
Baez when he talks about E8 says dimension 248
but I add 222 and 18 and get 240. So I am off by eight.

Just on a general level here is a paragraph on page 27 that I like a whole lot, and any comment or amplification from you would be welcome:

It should be emphasized that the connection (3.1) comprises all fields over the four dimensional base manifold. There are no other fields required to match the fieelds of the standard model and gravity. The gravitational metric and connection have been supplanted by the frame and spin connection parts of A.

The Riemannian geometry of general relativity has been subsumed by principal bundle geometry---a significant mathematical unification.

Devotees of geometry should not despair at this development, as principal bundle geometry is even more natural than Riemannian geometry. A principal bundle with connection can be described purely in terms of a mapping between tangent vector fields (diffeomorphisms) on a manifold, without the ab initio introduction of a metric.

these 8 missing dimensions, in my bad arithmetic, are they the "frame and spin connection parts of A" referred to above---or what, in simple terms, happened to them?

Hey people! Just by the way without the ab initio introduction of a metric means background independent. We are talking about background independent QFT which was, I guess, the whole idea in the first place. It's not surprising if some people are looking pretty happy at this point.

 

[garrett]

Hi marcus,

The other eight Lie algebra elements are the basis elements of the Cartan subalgebra, so they're technically not included with the other roots. They are physical fields though -- six of them are standard and two are new.

With that paragraph you quote, I was saying something I think is important, but might not be widely known by physicists. Conventional GR requires a metric to exist over the manifold -- this is kind of a strange object from the point of view of differential geometry. Nevertheless, physicists are used to thinking of GR as geometric and Yang-Mills as involving algebra. However, Lie algebra elements correspond to vector fields over the Lie group manifold. And a principal bundle can be described purely in terms of maps between vector fields, without a metric, using a tangent vector valued 1-form field over the entire space. In this way, the geometry of principal bundles is more natural than Riemmannian geometry. But this is a very subtle point, and I don't expect it to mean much to most readers.

 

[strangerep]

[Lubos's] only rational attack is based on the Coleman-Mandula theorem, the abstract of which he kindly provides a link to, but evidently didn't read, since the first assumption of the C-M theorem is stated there in the abstract, and doesn't apply in the case at hand, as stated in the paper.

Hi Garrett,

I don't understand the details of your dismissal of the C-M theorem. IIUC, you're basically
saying that it doesn't apply because in your setup we have deSitter instead of Poincare
(right?).

If so, then here's the thing that seems strange to me: deSitter contracts to Poincare
(for \lambda\to 0). Contraction is process of continuously changing the
structure constants. Pretty much everything else about the Coleman-Mandula theorem
seems to respect some form of continuity (analytic S-matrix, use of infinitesimal generators, etc).
So if you're relying on deSitter, then shouldn't we get negligible scattering unless
\lambda is significant?

Or am I naively expecting too much from continuity?

 

[garrett]

Strangerep, you have hit the nail on the head. This E8 Theory, which includes MacDowell-Mansouri gravity as an integral part, is not defined for \Lambda=0.

 

[Haelfix]

I was the anonymous person who brought up CM in Bee's thread.

And I am still a little disturbed by it. Desitter space has no Smatrix, and if the action perse (MM) explicitly forbids contraction to regimes where CM applies, I can't see how you will recover the effective standard model field theory.

Read there is no apparent Smatrix in the theory!

Moreover even if there exists such a thing in the theory, I don't see how you will suppress unitarity violating interactions absent imposing a hard cutoff that generates gauge anomalies b/c it breaks the Desitter group. You need to run a general operator analysis to sort the mess out.

The topological sector of your theory is highly nontrivial as well, and I can't see how you will suppress all sorts of very bad instanton processes.

 

[garrett]

I was the anonymous person who brought up CM in Bee's thread.

That's fine, it's certainly not a stupid question.

And I am still a little disturbed by it. Desitter space has no Smatrix, and if the action perse (MM) explicitly forbids contraction to regimes where CM applies, I can't see how you will recover the effective standard model field theory.

It's going to have to be an approximation.

Moreover even if there exists such a thing in the theory, I don't see how you will suppress unitarity violating interactions absent imposing a hard cutoff that generates gauge anomalies b/c it breaks the Desitter group. You need to run a general operator analysis to sort the mess out.

The topological sector of your theory is highly nontrivial as well, and I can't see how you will suppress all sorts of very bad instanton processes.

Why do you think I published the paper? I can't do everything myself.

And I'm first to admit this is just the beginning of a theory that might be wrong.

 

[Haelfix]

Don't get me wrong, that's fine. I happen to love E8 anyway, its a beautiful group

 

[marcus]

I was the anonymous person who brought up CM in Bee's thread. .

It would be great to have more discussion about Coleman-Mandula. On Bee's thread the first mention I could find was Moshe around noon pacific on the 7th
http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html#c978188498139402984

At 12:38 PM, November 07, 2007, Moshe said...

Two quick questions:

1. What is the loophole in the Coleman-Mandula theorem used in this construction? note that the theorem allows constructing theories where internal and spacetime symmetries are unified, as long as those theories are free.

2. When packaging bosons and fermions together, at least one set of fields will have the wrong spin statistics relation. In addition to violating unitarity etc., this definitely is not what is going on in the standard model.

It led to quite a bit of discussion, some of which was echoed a couple of days later over at Peter's
==quote Woit's blog==
# more questions Says:
November 9th, 2007 at 6:57 pm

As long as Coin is asking questions, I didn’t understand (1) why this doesn’t violate the Coleman-Mandula theorem, and, (2) what about the nonrenormalizability of GR?

----

# Coin Says:
November 9th, 2007 at 7:24 pm

MQ, Garrett does seem to offer an argument concerning your (1) in a reply to Moshe in the comments section of the Backreaction post:

1. Yes, the Coleman-Mandula theorem assumes a background spacetime with Poincare symmetry, but this theory doesn’t have this background spacetime — with a cosmological constant, the vacuum spacetime is deSitter. So this theory avoids one of the necessary assumptions of the theorem, and is able to unify gravity with the other gauge fields. On small scales though, Poincare symmetry is a good approximation, and on those scales gravity and the other gauge feels are separate, in accordance with the theorem. (I’m not the first person to dodge C-M this way.)

Several more posts over the course of that thread drill down on this point further…

---

# Garrett Says:
November 9th, 2007 at 7:26 pm

more questions:

(1) The first person I know of to point out this loophole in Coleman-Mandula was Thomas Love (a visitor here) in his 1987 dissertation. There is also a discussion of this loophole in this recent paper by F. Nesti and R. Percacci: Graviweak Unification. Or you can go to the source and look at Coleman and Mandula’s paper, in which their first condition for the theorem is “G contains a subgroup locally isomorphic to the Poincare group.” The G = E8 I am using does not contain a subgroup locally isomorphic to the Poincare group, it contains the subgroup SO(4,1) — the symmetry group of de Sitter spacetime.

(2) I’m banking on the LQG community to crack this one. So multiply the odds of this E8 Theory being right times the odds of LQG finding the right answers for quantizing the theory… and I’m first to admit it’s a long shot. But I think it’s got a chance, which is why I work on it.
==endquote==

In his reply Garrett gives links for the Percacci-Nesti paper and for the Coleman-Mandula one.
I for one could use any further discussion of Coleman-Mandula that comes along and what especially interests me is that John Baez spent several weeks here at Beyond forum convincing us that the right local symmetry group for quantum field theory was SO(4,1) and not the Poincaré.
And I was convinced. So when people sound as if they think Garrett is fudging by using the deSitter group as an inferior substitute---to dodge the C-M no-go obstacle---then I wonder because it seems to me the reverse. It is the older work which uses the inferior substitute and this new work uses the RIGHT local symmetry.
And Colemandula be damned. If this is dumb, please help me see why.

 

[jal]

I want to bring your attention to a drawing.
https://www.physicsforums.com/attachment.php?attachmentid=11509&d=1194846037
posted by Hans de Vries at

https://www.physicsforums.com/showthread.php?p=1501369#post1501369

If you do the same thing for E8, the shadow of the 3d construction produces the familiar E8 pictures.
A search, using “image”, will find it.
-------------
The 3d construction, if orientated properly, produces the tetras.
The hard question ….
Since a 3D construction of E8 can produce the 2D construction, What is the obstacle to assuming that the E8 is a construction existing in 3D?

 

[garrett]

E8 lives in 8D. Sure, it can be projected down to 3D. It then needs to be projected to 2D to be shown on a screen or paper. If we used holograms instead of screens, I'd be making tons of 3D plots. But, as it is, I just project from 8D to 2D, because if I go from 8D to 3D to 2D, the perspective would make a mess of things.

 

[Hans de Vries]

I want to bring your attention to a drawing.
https://www.physicsforums.com/attachment.php?attachmentid=11509&d=1194846037
posted by Hans de Vries at

https://www.physicsforums.com/showthread.php?p=1501369#post1501369

That image was generated with the (freeware) ray-tracer Povray.
All the projections come for free It's just setting up three light
sources on the principle orthogonal axis.

So the projections are actually shadows. You can give them colors as
I did by using transparent colored objects and/or colored light sources.Regards, Hans

 

[Hans de Vries]

It would be great to have more discussion about Coleman-Mandula.

Indeed, Theorems like that of Coleman and Mandula could be in
the same league as Lorentz invariance and Gauge invariance, with
respect to the guidance they can bring.

Weinberg handles the proof in appendix B of the first chapter (ch:24)
in Volume III handling the possible internal symmetries in combination
with the Poincaré group. Haven't studied the proof though ...

I don't think you can really "circumvent" Coleman Mandela by going
to another space-time group. You'll have to deal with the appropiate
version of the theorem for that specific group instead. I don't know
if there exist something along that line for SO(4,1). Weinberg refers
to chapter 32, the last chapter in Vol.III, for the application of
Coleman Mandela to higher dimensional spaces and he mentions
p-brane theories where the Theorem "does not apply"


Regards, Hans

 

[CarlB]

With that paragraph you quote, I was saying something I think is important, but might not be widely known by physicists. Conventional GR requires a metric to exist over the manifold -- this is kind of a strange object from the point of view of differential geometry. Nevertheless, physicists are used to thinking of GR as geometric and Yang-Mills as involving algebra. However, Lie algebra elements correspond to vector fields over the Lie group manifold. And a principal bundle can be described purely in terms of maps between vector fields, without a metric, using a tangent vector valued 1-form field over the entire space. In this way, the geometry of principal bundles is more natural than Riemmannian geometry. But this is a very subtle point, and I don't expect it to mean much to most readers.

Is this referring to the work by the Cambridge geometry group?

http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/gravity.html

 

[garrett]

No Carl, you're not that lucky. I was referencing this:

http://deferentialgeometry.org/#[[Ehresmann principal bundle connection]]
http://en.wikipedia.org/wiki/Ehresmann_connection

 

[garrett]

Also, I'll be presenting a talk tomorrow, bright and early:

http://relativity.phys.lsu.edu/ilqgs/

The pdf for the talk just went up five minutes ago, but there seems to be some problem displaying them on windows machines. If anyone has any suggestions on how to fix that, it would be appreciated. (I have a mac)

 

[CarlB]

This windows machine displays the Acrobat file beautifully. It makes a good addition to the arXiv article. It used to be my experience that very large Acrobat files (the above is 1.8MB) should be downloaded rather than opened directly.

 

[Hans de Vries]

Also, I'll be presenting a talk tomorrow, bright and early:

http://relativity.phys.lsu.edu/ilqgs/

The pdf for the talk just went up five minutes ago, but there seems to be some problem displaying them on windows machines. If anyone has any suggestions on how to fix that, it would be appreciated. (I have a mac)

Success with your talk Garrett. You might say that you "have set the stage" now,
more then you probably would have imagined.


Regards, Hans

 

[garrett]

Thanks Hans, you're not kidding. I'm looking forward to going back to being a hermit again after tomorrow, playing with equations instead of with people.

I think we worked out the pdf problem. Thanks for the tip though Carl.

 

[jal]

E8 lives in 8D. Sure, it can be projected down to 3D.

Well ... it's better than dealing with 11D.
Believe me ... I'm trying with what I got ...?
From your statement I get a visual of a proton iceberg.
I looked at your slides ... it should give an interesting presentation.

 

[marcus]

...I'm looking forward to going back to being a hermit again after tomorrow, ...

It's been wonderful being able to watch the baby debut. thanks for being here at PF during.
Really smart of Jorge Pullin to invite you immediately to do ILQGS. It is already starting to be runup to next July QG².

On your sllde #45 in the concluding "discussion of E8 theory" section you say

Quantization:
* Coupling constants run.
* Large Lambda compatible with UV fixed point.
* Just a connection — amenable to LQG, spin foams, etc.

In Reuter's papers Lambda indeed gets large as k -> infty, so he has Lambda run in the right direction for E8 theory's needs.

But the other constant he has running is G_Newton which he has go to zero (!) as k -> infty. It seems like an unexpected thing that might cause a stumble, so I mention in case you get into discussing that bullet of slide 45.

Have fun with the ILQGS talk! it will be good for both theories and I hope they can converge some.

 

[strangerep]

Desitter space has no Smatrix, [...]
Read there is no apparent Smatrix in the theory!

Hi Haelfix,

I have some followup questions (for yourself, or anyone else who
knows the answers)...

Where can I read more about how DeSitter space has no S-Matrix?
I'm interested in the precise assumptions that lead to that conclusion.
Is the answer trivial in that, if one looks backwards in an expanding space,
there's no such thing as infinitely-separated effectively-free particles
for t \to -\infty? Or is there more to it than that?

If the former, I'm wondering how Garrett could get any cross-sections
out of such a theory which could be meaningfully compared with particle
physics experiments.

For that matter, what are the Casimirs and unitary irreps for the DeSitter
group? I tried some googling, but couldn't find a good exposition of this.

 

[marcus]

...Where can I read more about how DeSitter space has no S-Matrix?
...
Is the answer trivial in that, if one looks backwards in an expanding space,
there's no such thing as infinitely-separated ...

It'll be interesting to see what Haelfix says. Thanks for asking. About your guess of a trivial explanation, note that deSitter space contracts and then expands, so if you look back in time you can have infinitely separated paths.

but even without that, after two particles are a lightyear apart who cares? S-matrix could still be a good effective approximation, even if local reality is deSitter shaped. Just my two cents. We'll see what Haelfix says.