An Exceptionally Technical Discussion of AESToE

**MTd2** · Dec16-07, 08:50 PM

Garrett,

it seem Lee Smolin admited he is wrong, and admited that your theory do not include Pati -Salam model:

# Lee Smolin on Dec 15th, 2007 at 8:36 pm

Dear HIGGS

I see, if it is then just a terminological mixup that is of course fine for this issue. I don’t mind making mistakes in public-the time spent studying the Pati-Salam papers was my own and in any case worthwhile-but this shows to me the difficulty of arguing technical issues in the blog environment. Perhaps the experts could find a better way, probably off line, to go through the issues with Lisi point by point and reach a conclusion over the main issues. If so I’d be happy to be involved, so long as everyone involved was patient and professional and no one pretended that the representation theory of non-compact forms of E8 is child’s play.

Thanks,

Lee
# H-I-G-G-S on Dec 15th, 2007 at 10:15 pm

Dear Lee,

I’m glad that we cleared this up, and I appreciate that you admitted error,
in line with your earlier posting on the spirit of science requiring such acknowledgement. I don’t quite agree however that it was a “terminological mixup.” This makes it sounds like there was no real content to the debate, whereas in fact there was. The issue at hand was whether or not Lisi’s embedding contains the Pati-Salam model or not. Jacques showed that it does not. All I did was to provide some helpful clarification. In an earlier post you went on about how “Distler was largely wrong” and so forth, while as far as I can tell, everything he has said has either been correct, or when it was in error, the error was admitted and then clarified. Thus it would be much more appropriate for you to address your admission of error to him than to me. Perhaps if you did so his responses to you would in the future be more temperate.

It is true that blogs are far from the best place to argue technical issues. This discussion was one of the happy exceptions where a point was argued and resolved with all parties in agreement. As for Lisi’s proposal, I believe a conclusion has been reached by the experts.

H
http://cosmicvariance.com/2007/11/16...of-everything/

**moveon** · Dec17-07, 07:24 AM

Originally Posted by garrett

Quote:
Now it seems clear that the Coleman-Mandula question pertains only to the unbroken theory, or to put it another way, that in the broken theory, although the connection is still E8-valued, the action is no longer E8-symmetric.

That's right.

If your connection would be valued in the algebra, its would be expandable in the generators of E8. But some components of your connection are fermionic and thus anticommute, or? How can these possibly satisfy any E8 commutation relations? And if not, what on earth has your construction then to do with E8?

As I was writing over at CV, this is completely different to symmetry breaking (where the proper commutation relations are still satisfied, though non-linearly realized).

**garrett** · Dec17-07, 07:56 PM

Tony,
There are many gradings of E8, most of them interesting. I haven't thought about 7-gradings much. My favorite grading of E8 is a 13 grading corresponding to weak hypercharge -- which currently only works correctly for the first generation.

rntsai,
The Z and the photon fields are (rotated) combinations of W, B_1, and B_2. Specifically, as 1-form coefficients,
$LaTeX Code: \\underline{Z} = \\sqrt{\\frac{5}{8}}\\underline{W}^3 - \\sqrt{\\frac{3}{8}} (\\sqrt{\\frac{3}{5}} \\underline{B}_1^3 + \\sqrt{\\frac{2}{5}} \\underline{B}_2)$
and
$LaTeX Code: \\underline{\\gamma} = \\sqrt{\\frac{3}{8}}\\underline{W}^3 + \\sqrt{\\frac{3}{8}}\\underline{B}_1^3 + \\sqrt{\\frac{2}{8}}\\underline{B}_2$
The $LaTeX Code: B_1^\\pm$ , and a leftover
$LaTeX Code: \\underline{X} = \\sqrt{\\frac{2}{5}} \\underline{B}_1^3 - \\sqrt{\\frac{3}{5}} \\underline{B}_2$
are "new" gauge fields, as in Pati-Salam. (I'm pretty sure I have those right, but I haven't confirmed them.)

The circles are all gauge fields: green for gravitational $LaTeX Code: \\omega_{L/R}^{\\wedge/\\vee}$ , yellow for weak $LaTeX Code: W^{\\pm}$ , blue for gluons, and white for $LaTeX Code: B_1^\\pm$ . The Z, photon, and X are in the Cartan subalgebra at the origin, and are conventionally not plotted.

MTd2,
H-I-G-G-S was twisting Lee's words, as is clear from his reply (which was visible when you posted).

moveon,
The connection starts out as an E8 valued 1-form. The action (with E8 symmetry broken by hand in my paper, but not in Lee's) introduces dynamical terms for the D4+D4 part of E8, but leaves only the BF term for the rest of E8. These pure gauge degrees of freedom may be replaced by Grassmann fields valued in the non D4+D4 part of E8 -- these are fermions. The resulting generalized connection consists of Lie(E8) valued 1-forms and Grassmann numbers, with the same E8 brackets between Lie algebra elements. This is a fairly standard mathematical construction, used in an unusual way. I'd be happy to expand on it for you or provide references.

**Tony Smith** · Dec17-07, 09:25 PM

Garrett, exactly what is the 13-grading of e8 that you like to use?

Tony Smith

**CarlB** · Dec17-07, 09:42 PM

Let me make a guess for garrett. The 13 is a weak grading, so it's going to correspond to the weak hypercharge quantum numbers of the standard model, that is, it will use the 13 values: (-1, -5/6, -2/3, -1/2, -1/3, -1/6, 0, +1/6, +1/3, +1/2, +2/3, +5/6, +1). To see the assignment, I would start by looking for the weak hypercharge quantum numbers assignment in his paper. Then you assign a particular root to a blade according to its weak hypercharge quantum number.

My recollection of the standard model is that the +- 5/6 quantum numbers are missing. These blades would be particles that don't appear in the standard model. But my concentration has always been on the fermions -- are there some bosons with weak hypercharge +- 5/6?

The peculiar pattern of the weak hypercharge quantum numbers that are actually used in the standard model, that is, leaving off the +- 5/6, has 11 values. Since I'm a density matrix proponent, (which are bilinear rather than the usual state vector formalism which is linear) I'm going to link in a paper which gives those 11 values, rather than all 13, as a solution to a bilinear equation. See chapter 5: http://www.brannenworks.com/dmfound.pdf

**garrett** · Dec17-07, 10:29 PM

Tony,
If we rotate the E8 root system until the vertical axis is weak hypercharge, and rotate out the other axes horizontally to separate the roots a bit, it looks like this:
http://deferentialgeometry.org/blog/hyper.jpg
This makes it visible exactly what is meant by "the charge assignments only work correctly for the first generation," with the other two (smaller triangles) related by triality.

**Tony Smith** · Dec17-07, 11:18 PM

When I count the 13-grading from that image I get:

5 + 6 + 15 + 20 + 30 + 30 + 26 + 30 + 30 + 20 + 15 + 6 + 5

which only add to 238, so I must be miscounting two of them somewhere ???

Anyhow, modulo my error of two, the even graded structure would be

5 + 15 + 30 + 26 + 30 + 15 + 5 = 126-dimensional
(if the missing 2 are even, then 128-dimensional)

and the odd graded structure would be

6 + 20 + 30 + 30 + 20 + 6 = 112-dimensional

so

it seems to me that the odd gradings correspond to the 112 root vectors of the adjoint Spin(16) (120 generators - 8 Cartan subalgebra generators = 112)

and

that the even grading probably have the two I miscounted and are the 128 root vectors corresponding to the half-Spinor of Spin(16).

What bothers me about that is that the fermionic spinor-type things are in the even grading and the bosonic vector/bivector adjoint-type things are in the odd grading,

whereas in Thomas Larsson's 7-grading

8 + 28* + 56 + (sl(8) + 1) + 56* + 28 + 8* = 8 + 28 + 56 + 64 + 56 + 28 + 8

the even grade part is
28 + 64 + 28 = 112 dimensional corresponding to the root vectors of adjoint Spin(16) which seems to represent bosonic vector/bivector stuff
while
the odd grade part is
8 + 56 + 56 + 8 = 128-dimensional corresponding to half-spinor of Spin(16) which seems to represent fermionic spinor-type stuff.

Do you have any thoughts about that ?

Tony Smith

**garrett** · Dec17-07, 11:31 PM

Tony,
Two gluons overlap. There are many such gradings of E8 -- there may be the same kind of 13 grading along a different direction that gives the even/odd 120/128 split you're after.

**sambacisse** · Dec18-07, 10:29 AM

Hi Garrett,

I would like to have your comments on Distler's second post, where he shows that the embedding you propose in E8 is not chiral, contrary to what you and others have claimed, and his proposal (in an appendix to his first post) that perhaps all embeddings in E8 are not chiral, hence rendering your idea of embedding the Standard Model and gravity in E8 rather difficult to realize, to say the least. There has been quite a lot of discussions about this on Cosmic Variance (and it's still ongoing: see comments 90 up to now on this post).

I personally feel Distler's argument is fundamental, relatively easy to follow, and seems to be correct, at least up to the level of my knowledge (perhaps I'm making a mistake). Lee has been trying to address it on Cosmic Variance, but hasn't succeeded in finding a mistake or a loophole in it yet. Do you have anything to say about it?

Thanks a lot! :-)

**moveon** · Dec19-07, 05:03 AM

Originally Posted by garrett

The resulting generalized connection consists of Lie(E8) valued 1-forms and Grassmann numbers, with the same E8 brackets between Lie algebra elements. This is a fairly standard mathematical construction, used in an unusual way. I'd be happy to expand on it for you or provide references.

Please! To my knowledge the only algebras that contain both bosonic and fermionic generators are superalgebras, and E8 is not one of them. How can the commutation relations close into E8?

**Tony Smith** · Dec19-07, 10:40 AM

moveon said "... the only algebras that contain both bosonic and fermionic generators are superalgebras ...".

No, that is not true.
As Pierre Ramond in hep-th/0112261 said "... exceptional algebras relate tensor and spinor representations of their orthogonal subgroups ...",
and
the exceptional algebra 248-dim E8 contains
120 generators corresponding to the tensor/bosonic part 120-dim adjoint Spin(16)
and
128 generators corresponding to the spinor/fermionic part 128-dim half-spinor Spin(16)
and
their commutation relations do close into E8.

However, Pierre Ramond went on to say in that paper:
"... Spin_Statistics requires them [ the adjoint/bosonic and half-spinor/fermionic ] to be treated differently ...",
so
any model you build with E8 must somehow treat them differently.

For example, you might just construct a Lagrangian into which you put
the 128 half-spinor fermionic generators into a fermion term
and
8 of the 120 bosonic generators into a spacetime base manifold term
and
120-8 = 112 of the 120 bosonic generators into a gauge boson curvature term.

Then you might have disagreement as to how natural (or ad hoc) is such an assignment of parts of E8 to terms in a Lagrangian,
but all should agree that you have "treat[ed] them differently" as required by Spin-Statistics.

However, in Garrett's 13-grading decomposition of the 240 root vectors of E8

5 + 6 + 15 + 20 + 30 + 30 + 28 + 30 + 30 + 20 + 15 + 6 + 5

some of the graded parts contain both bosonic terms and fermionic terms,
for example the central 28 has both circles (bosons) and triangles (leptons and quarks),
which has led Thomas Larsson to complain (on Cosmic Variance):
"... both fermions and bosons belong to the same E8 multiplet. This is surely plain wrong. ...".

I think that the point of Thomas Larsson is that
the model must treat the fermions and bosons differently to satisfy Spin-Statistics
so
the fermionic generators must be put into some part of the model where the bosonic generators are not put
so
if you decompose the generators into multiplets some of which contain both fermionic and bosonic generators (as in Garrett's 13-grading decomposition) then you are not respecting your multiplets when you, from a given multiplet, put some of them into a fermionic part of the model and some of them into a bosonic part of the model.

This is not merely an objection of ad hoc assignments of generators to parts of the model,
it is an objection that the assignments do not respect the chosen decomposition into multiplets.

Tony Smith

PS - It is possible to choose a decomposition that does keep the bosonic and fermionic generators separate, the simplest being 64 + 120 + 64
where the 120 is bosonic and the 64+64 = 128 is fermionic.

**moveon** · Dec19-07, 11:12 AM

Originally Posted by Tony Smith

moveon said "... the only algebras that contain both bosonic and fermionic generators are superalgebras ...".

No, that is not true.
As Pierre Ramond in hep-th/0112261 said "... exceptional algebras relate tensor and spinor representations of their orthogonal subgroups ...",
and
the exceptional algebra 248-dim E8 contains
120 generators corresponding to the tensor/bosonic part 120-dim adjoint Spin(16)
and
128 generators corresponding to the spinor/fermionic part 128-dim half-spinor Spin(16)
and
their commutation relations do close into E8.

Oh yes, this is of course very well known since ages. But those tensor and spinor rep generators are all bosonic, and close into the usual E8 commutator relations. My point is, apparently still not appreciated, that if some of the generators are made fermionic (as it happens for superalgebras), then they cannot produce the E8 commutation relations (and jacobi identities etc) any more. The opposite seems to be claimed here all over, so I'd like to see, how. Please prove this by writing them down!

And if the E8 commutation relations are not there, there is no E8 to talk about. There is "somewhat" more to E8 than a drawing of the projection of its polytope...

**John G** · Dec19-07, 11:51 AM

This from Tony's website might be good to look at (I'm sure Tony can say more if needed):

http://www.valdostamuseum.org/hamsmi...eStdModel.html

**Thomas Larsson** · Dec19-07, 02:45 PM

One needs to distinguish between spin and statistics.

There are two types of statistics: fermions, which anticommute and obey Pauli's exclusion principle, and bosons, which commute.

There are also two types of spin: spinors, which have half-integer spin, and tensors and vectors, which have integer spin.

The spin-statistics theorem asserts that physical fermions always have half-integer spin and physical bosons have integer spin. But this is non-trivial and surprisingly difficult to prove. In contrast, BRST ghosts are fermions with integer spin, and therefore unphysical. Physical and unphysical fermions are not the same.

What is quite easy to prove is that statistics is conserved, i.e.

[boson, boson] = boson
[boson, fermion] = fermion
{fermion, fermion} = boson.

People like Lee, Peter and Bee know this, of course, and it must be obvious that putting both bosons and fermions into the same E8 multiplet violates this fundamental principle. That they don't emphasize this simple fact but instead complain about manners is something that I find surprising and quite disappointing.

**Tony Smith** · Dec19-07, 03:50 PM

Here is what I hope is a concrete example of what I think that Thomas Larsson is saying (please feel free to correct my errors):

If you were to (not what Garrett did) make a physics model by decomposing E8 according to its e17 5-grading:

g(-2) = 14-dim physically being spacetime transformations
g(-1) = 64-dim physically being fermion antiparticles
g(0) = so(7,7)+R = 92-dim physically being gauge bosons
g(+1) = 64-dim physically being fermion particles
g(+2) = 14-dim physically being spacetime transformations

then that would be consistent with spin-statistics because
the products fermion(-1) times fermion(+1) would be gauge bosons(-1+1=0)
the products of gauge bosons(0) times gauge bosons(0) would be gauge bosons(0+0=0)
the products of gauge bosons(0) times fermions(-1) would be fermions(0-1=-1)
the products of gauge bosons(0) times fermions(+1) would be fermions(0+1=+1)

etc

The point is that if you have fermions and bosons mixed up together in the same part of the graded decomposition, you do not get good spin-statistics,
but
it is possible to decompose in a way that you do get good spin-statistics
and
that is something that should be taken into account in model-building.

Tony Smith

PS - Sorry for burying stuff like fermion(-1) times fermion(-1) giving spacetime(-2) into an "etc" (sort of like spinor x spinor = vector) but in this comment I am just trying to make a point and not build a complete model here.

**garrett** · Dec19-07, 08:21 PM

sambacisse,
The issue is a bit more complicated than it appears because of how the real representations are mixed together in exceptional groups into complex representation spaces, relying on an inherent complex structure. This sort of thing is described halfway through John Baez's TWF253 for the case of E6. When describing so(3,1) reps in terms of sl(2,c) this is further complicated, and when swapping in conjugated anti-fermions it's more complicated still -- because one has to be clear in each step which complex structure one is conjugating with respect to. I thought I had this figured out several years ago, but I don't like to make statements about complicated things without having slowly worked through them in detail. So I've stayed out of the arguments. Of course, I can say that the worst case scenario is that one might have to use a complex E8.

moveon,
Tony addressed this a bit, and I'll try to summarize the specific case in the paper. The E8 Lie algebra may be naturally decomposed into a D4+D4 subalgebra, and everything else. In terms of the number of elements, this decomposition is:
(28+28)+64+64+64
which I don't consider a "grading," but it relates to gradings. The important thing is the Lie brackets. If we label the D4+D4 elements "bosons," and the rest "fermions," the brackets are as Thomas Larsson has helpfully described. Now, if the E8 symmetry is broken such that the "fermion" part of the Lie algebra is pure gauge, then that part of the connection may be replaced by Lie algebra valued Grassmann fields. We end up with a D4+D4 valued connection 1-form field, $LaTeX Code: \\underline{H}_1+\\underline{H}_2$ , and three other fields, the first of which is the first generation fermions, $LaTeX Code: \\Psi$ , which are Grassmann valued E8 Lie algebra elements. Because of the structure of E8, the Lie brackets between these give the fundamental action:
$LaTeX Code: [\\underline{H}_1+\\underline{H}_2,\\Psi] = \\underline{H}_1 \\Psi - \\Psi \\underline{H}_2$
The brackets between two $LaTeX Code: \\Psi$ 's are in D4+D4, but these terms vanish in the action. Notice that there is no symmetry here relating the fermions to bosons. That symmetry was destroyed when we broke the E8 symmetry by adding the terms we did to the action. I did that by hand in my paper, and Lee talks about how that can happen dynamically in his. There is a cute trick in the BRST literature whereby these objects can be formally added in a generalized connection:
$LaTeX Code: \\underline{H}_1+\\underline{H}_2+\\Psi$
Since I like cute math tricks, I used it -- allowing all fields to be written as parts of this "superconnection," with the dynamics coming from its generalized curvature.