An Exceptionally Technical Discussion of AESToE

[garrett]

Hello Mitchell,

I call them either ‘unbroken E8 theory’ and ‘broken E8 theory’, or ‘BF E8 theory’ and ‘modified BF E8 theory’. If you could suggest an appropriate terminology, that would be useful.

I have a slight preference for the latter, but either set is fine.

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.

I cannot figure out exactly what the remaining symmetry is, though. Is it just SO(3,1) x SU(3) x SU(2) x U(1), or is it SO(3,1) x something a little bigger?

It is something a little bigger -- it's basically the symmetry group of the Pati-Salam GUT plus Lorentz, so(3,1)+su(2)+su(2)+su(4). This would then have to break down to the so(3,1)+su(2)+su(1)+su(3) of the standard model, and there are many old descriptions of that.

Anyway, I am still studying these things, but it looks like the unbroken theory should fall foul of the CM theorem;

It doesn't, because the unbroken theory doesn't even produce a spacetime metric, much less the Poincare symmetry necessary for CM to apply.

on the other hand, the broken theory is just a slight modification of the Standard Model and so its quantization should be unproblematic. So regardless of the problems with the unbroken theory, in the broken theory you apparently have a well-defined theory, closely resembling the Standard Model, with no free parameters.

It's a slight modification of the standard model AND gravity, and the quantization of gravity is problematic.

It’s therefore the broken theory which interests me most at the moment, and so I’m trying to understand exactly what it is. Basically it seems to be a topological gauge theory with fermions and Higgses. That sounds like something people could understand and solve. But is that an accurate description?

It's a topological gauge theory with two modifying terms that involve the non-topological gauge fields, the frame-Higgs, and other Higgs. By my thinking, the fermions emerge as the ghosts of the topological part of the gauge field -- but this interpretation of the mathematics is controversial. The tricky part to solve, as always, is gravity.

rntsai,
f4 and g2 are subalgebras of e8. But you're correct that the d4 + d4 + 3x(8x8) breakup is more relevant. The best way I know of to understand the subalgebras and their relationships is to work with their roots. In Table 9, the five major blocks are d4, 8x8, 8'x8', 8''x8'', and d4. You may be able to use GAP, but I'm not familiar with it.

Cold Winter,
The tables in my paper just involve the roots for E8 -- it is not necessary to consider representations. (Though considering the representations may be needed for quantization -- which is a somewhat terrifying prospect.) If you do try to reproduce the Atlas calculation, let me know so I can send your computer a sympathy card.

 

[rntsai]

Garrett,
Do you have a reference for g2 or f4 in e8? I have trouble seeing
that since e8's roots are all the same length wheras g2's ad f4's
aren't?

 

[garrett]

rntsai,
Yes, the best reference I know of is J.F.Adams' book, "Lectures on Exceptional Lie Groups" -- if you send me an email, I'll let you borrow my copy. If you can see how g2 can be a subalgebra of so(6), as a projected subalgebra but not an embedded subalgebra, then you can see how the roots can be "shortened" by the projection.

 

[rntsai]

Thanks for the offer Garrett. Let me try to find these subalgebras on my own first;
it's easier to look for something if it actually exists!. I'll let you know if I get stuck.

I still would like to know what are the underlying subalgebras in both decompositions :
is it f4+g2 for the first and d4 + a2? for the second? and are all subspaces invariant
under these subalgebra actions?

 

[garrett]

rntsai,
There's a 26 that's invariant under f4, and a 7 invariant under g2 -- this 26x7 is in e8. For the second decomposition, d4+d4, there are three blocks of invariant 8x8's in e8.

 

[rntsai]

is the 7 invariant under f4 too?

 

[garrett]

Yes. More than that, the 7 is trivial under f4, and the 26 is trivial under g2.

 

[rntsai]

Excellent. It would have to be trivial I guess since
f4 has no 7 dim reps.

I'll try to summarize my understanding of some of the
algebra in the paper so far. I'll use algebras instead
of groups; going to groups will complicate things
somewhat and can be done at a later stage.

(1) e8 has a g2+f4 subalgebra; under this :

e8 = (14,1) + (1,52) + (7,26)

(2) e8 has a d4+d4 subalgebra, under this :

e8 = (28,1) + (1,28) + (8,8) + (8,8) + (8,8)

I know others have raised issues with embedding the
group, but at the lie algebra level, is the above beyond
reproach?

 

[Cold Winter]

Cold Winter,
The tables in my paper just involve the roots for E8 -- it is not necessary to consider representations. (Though considering the representations may be needed for quantization -- which is a somewhat terrifying prospect.) If you do try to reproduce the Atlas calculation, let me know so I can send your computer a sympathy card.

LOL, I have no sympathy for iron. I just pound it hard. While the prospect of running my little monster here for 8 weeks straight doesn't bother me ( I have nice cool place for it here in my home ), I want to build a new little monster. The Atlas redo kinda justifies it... sort of gives me an excuse... sort of... . Problem is this is a wait for prices and technology to converge issue. I'm reasonably certain a 16way AMD64 system will be fairly cheap to build in about 18 months.

As I'm sure your aware, once we get to the "table top", quantization ( terrifying or not ) becomes a serious issue. Wait until your asked to see if your theory answers the great cosmological questions I've mentioned above. And when your asked for some decent algebraic reductions to run experiments against...

This entire theory of yours if correct is going to have some really really big implications and the real fun is going to be on that table top. To be honest, I suspect the real terrifying events will come out if we don't do the quantization right. This could be fooling with the core of the universe afterall.

 

[garrett]

rntsai,
Yes.

Cold Winter,
Yep.

 

[rntsai]

do you have a snapshot of the issues with the group embeddings?

 

[garrett]

rntsai,
Sure, the d4+d4 includes gravity, Higgs, and gauge fields via:
$$d4 + d4 = (so(3,1) + su(2)+su(2) + 4 \times (2+\bar{2}))+(su(3)+u(1)+u(1)+3 \times (3 + \bar{3}))$$
This acts on the positive-chiral spinor block, $$8 \times 8$$, in e8 as the first generation of fermions. If we use this assignment, the first generation has exactly the right quantum numbers with respect to the gravitational and standard model fields in d4+d4. Now, there are two other $$8 \times 8$$ blocks in e8, the vector and negative-chiral spinor blocks, related to the first by triality. It seems natural to speculate that these are the second and third generation fermions. However, even though they are equivalent to the first block under triality, these fields do not have the correct standard model quantum numbers unless the d4+d4 is also triality rotated. Without handwaving, this first guess doesn't give the same standard model quantum numbers for the second and third generations as for the first. This means either that the second and third generation particles have different assignments, there's something fancier going on with the relationship to d4+d4, gravity needs to be described differently, or the theory just won't work. This is the main problem with the theory, I think.

 

[mitchell porter]

Garrett,
OK, so I can think of this as "a modified BFE8 theory in which the effective symmetry is Pati-Salam" (and in which further, dynamical symmetry-breaking occurs).

By my thinking, the fermions emerge as the ghosts of the topological part of the gauge field

I find that mysterious since BRST involves additional degrees of freedom, whereas your construction stipulates from the beginning that certain elements of E8 shall be fermions. Are you suggesting that E8 theory itself is the BRST extension of something smaller?

It's a slight modification of the standard model AND gravity, and the quantization of gravity is problematic.

Maybe so. But consider the basic electroweak model (with one generation, one Higgs, and no color) coupled to gravity. Strictly speaking it's unrenormalizable, but that wouldn't stop you from calculating the boson masses, because for that purpose you can just neglect gravity. Can't you do the same for your effective theory?

 

[garrett]

OK, so I can think of this as "a modified BFE8 theory in which the effective symmetry is Pati-Salam" (and in which further, dynamical symmetry-breaking occurs).

Yes, so the d4+d4 part gets modifying terms, and the rest of E8 is pure BF.

I find that mysterious since BRST involves additional degrees of freedom, whereas your construction stipulates from the beginning that certain elements of E8 shall be fermions. Are you suggesting that E8 theory itself is the BRST extension of something smaller?

BRST replaces the gauge degrees of freedom with ghost fields. The pure BF part of the theory, i.e. the non d4+d4 part of the E8 Lie algebra, are pure gauge. These are replaced by Grassmann valued "ghost" fields, so the "extended connection" is a kind of superconnection consisting of d4+d4 valued 1-forms and non d4+d4 valued Grassmann number fields.

Maybe so. But consider the basic electroweak model (with one generation, one Higgs, and no color) coupled to gravity. Strictly speaking it's unrenormalizable, but that wouldn't stop you from calculating the boson masses, because for that purpose you can just neglect gravity. Can't you do the same for your effective theory?

The electroweak breaking and mass assignments are the same as in the standard model. But I won't be able to say anything about fermion masses until the second and third generation are figured out in a way that makes sense.

 

[mitchell porter]

I won't be able to say anything about fermion masses until the second and third generation are figured out in a way that makes sense.

Phenomenology aside, are there simpler theories which exhibit some of the same properties? For example, is there a simpler parameter-free BF-theory-with-constraints in which a Higgs mechanism ends up giving mass to a field?

I actually think this is more important for reader comprehension than the group theory. If people could see how a similar but much simpler theory would actually give rise to some numbers, then it would be easy to believe that E8 theory ultimately makes sense, albeit being more complicated. But at the moment, someone trying to understand the paper has to deal with both the complications of E8 and an unfamiliar formalism (BF theory). In my case, I'm comfortable with perturbative quantum field theory, and can at least make sense of straightforward nonperturbative ideas like lattice calculations. But if I open a paper on BF theory, mostly I see a lot of formal-looking derivations. Where do I have to go if I want to see some calculations - all the way to spin foams?!

Alternatively, the paper by Rovelli and Speziale says that Yang-Mills theory constructed as a perturbation of BF theory behaves exactly the same as Yang-Mills constructed in the normal way. Does that mean, therefore, that for purposes of analysis and computation we can forget about the exotic origins in BFE8 and treat this theory as simply an extended Pati-Salam model? If yes, can the parameter-free-ness and consequent remarkable predictive capability be made explicit at that level of description?

 

[kneemo]

(1) e8 has a g2+f4 subalgebra; under this :

e8 = (14,1) + (1,52) + (7,26)

To see this in terms of derivations, take a look at Tits' construction on pg. 49 of Baez's http://xxx.lanl.gov/abs/math/0105155v4". Also, see pg. 41 where another g2 shows up in the f4 decomposition.

 

[garrett]

Mitchell,
It would be straightforward to formulate a similar theory -- including gravity, electroweakish gauge fields, and real fermion fields -- using only F4 instead of E8.

A modified BF formulation of Yang-Mills is equivalent to Yang-Mills, just vary the action with respect to B and plug that expression back in. The advantage of a modified BF formulation is that it naturally gives the Dirac action for fermions, via the pure BF term for that part of the algebra. It should be very interesting to calculate how these parameters run, and how the formulation connects with approaches to quantum gravity.

kneemo,
Thanks for pointing out the reference.

 

[Coin]

Garrett, I am kind of curious about how Lee Smolin's new paper differs from your approach. I don't want to drag this thread offtopic so I just want to limit this to two technical questions:

Smolin's paper as I understand it covers two subjects: he discusses a general method for integrating LQG with a gauge group unification theory (such as but not limited to E8), then he proposes a different way of incorporating fermions into an E8 symmetry. Much of the paper is taken up by discussion of proposed actions, although I can't tell if this action discussion is part of the LQG/gauge proposal or the fermion proposal or neither (I think it's only part of the LQG/gauge proposal). My questions are:

1. Does Smolin's proposal concerning linking LQG and E8 necessarily require Smolin's proposal considering fermions in E8 to be adopted? Or are they two separate things? (As far as I can tell the answer is that they are separate, but I am not sure...)

2. Does Smolin's suggested alternate method of incorporating the fermions into E8 require actually changing the group decomposition used in your construction, or does it only modify the action?

Thanks!

 

[garrett]

Hi Coin,
The most interesting thing I see in Lee's paper is how he obtains the action for gravity and gauge fields from an initially E8 invariant action. This addresses a dissatisfaction with the "by hand" symmetry breaking in my paper.
1.They're separate. Having the fermions described by non-local links is an interesting and rather speculative idea.
2.Since the full details of this idea aren't worked out, it's hard to say.

 

[Coin]

Garrett, thanks! I may have some more questions later :)

The tables in my paper just involve the roots for E8 -- it is not necessary to consider representations. (Though considering the representations may be needed for quantization -- which is a somewhat terrifying prospect.) If you do try to reproduce the Atlas calculation, let me know so I can send your computer a sympathy card.

LOL, I have no sympathy for iron. I just pound it hard. While the prospect of running my little monster here for 8 weeks straight doesn't bother me ( I have nice cool place for it here in my home ), I want to build a new little monster. The Atlas redo kinda justifies it... sort of gives me an excuse... sort of... . Problem is this is a wait for prices and technology to converge issue. I'm reasonably certain a 16way AMD64 system will be fairly cheap to build in about 18 months.

Hi Cold Winter, you may want to read the ATLAS group's somewhat tortured narrative of exactly how they went about their calculation. The limiting factor turns out to actually be not CPU power, but RAM. Performing the calculation turns out to involve constructing some tables that grow to hundreds of gigs in size, and they found that they essentially had to store the entire table in RAM, as the calculation requires so many more-or-less-random accesses to this table that access times would have been prohibitive had they allowed swapping to disk. It's an interesting read, anyway.

Incidentally, as regards your comment about doing calculations in C, I think actually this is not so necessary as it at first seems. For heavy computation of the exact kind that dominates the E8 stuff-- working with vectors and matrices and such-- C is I think not actually a very good choice. Many higher-level languages can do vector and matrix operations with a highly optimized backend library, removing many of the advantages C would get in this area from being "closer to the metal". On the other hand with C the "close to the metal" nature can actually be a major drawback, since the degree of power given to the programmer in C takes power away from the compiler, thus preventing many useful compiler optimizations from being possible-- and with this kind of stuff a compiler really is usually much better at optimizing than a human is. I can't speak to the efficiency of Mathematica in specific but it would not surprise me if there are language platforms, for example among some of the functional languages, which resemble Mathematica more than C and yet get better performance than C on E8-related calculations. Of course, these languages bring their own problems! And if you are going to be doing something on the scale of the ATLAS calculation I would tend to suspect you have no choice but to use C. Interestingly ATLAS has a package of downloadable software (although I am not sure whether the E8 map program is included) and it is all written in C++.

 

[John G]

Garrett, hi, that Baez link mentioned earlier is actually a link in your paper (apparently four brilliant minds thinking alike (you, Baez, Tits & kneemo and actually a 5th since I think Baez originally got Tits' idea via Tony Smith). You mentioned your E8 idea in simpler form is kind of an F4 one (with real vectors/spinors), and as you mention in your paper you make complex vectors/spinors via E6 and it seems what you get by going up to E8 is a big Jordan Algebra. That big Jordan algebra along with your MacDowell-Mansouri gravity and your D4xD4 bosons are three really interesting things that justify the hype for me and should hopefully stay no matter what you have to change as far as fermions are concerned. Smolin wrote about a big Jordan Algebra for string theory:

http://xxx.lanl.gov/abs/hep-th/0104050

and I know Tony Smith like it for string theory/spin foam too (and Smolin certainly likes spin foam-type models). Smith and Ark Jadczyk are the two physicists I've read about the most. Ark isn't into Jordan Algebras but he is into Clifford Algebra and Dirac Gammas and Tony I know can talk about an E8 model using Gammas instead of Jordan Algebra so in my mind string theory, spin foams, Jordan Algebra, and Dirac Gammas are all kind of related and found in E8 above E6. You seem to be using Jordan Algebra in a spin foam sense too, is that true?

That D4xD4 for bosons is something I've never seen before. Cause of your use of D4xD4, Tony Smith actually added a way of looking at his model in a D4XD4 way so now I've got not only your model but a new version of Tony's to try and learn the best I can (thanks I think). I think I really like the use of D4xD4 though perhaps not for the reason you use it. It seems like even though you only have a 4-dim spacetime that extra D4 kind of creates an extra 4-dim spacetime. Tony actually has an 8-dim spacetime but I don't fully understand yet his D4xD4 (or yours) as well as I understand Tony's version using only one D4. Anyways thank you very very much for making this kind of stuff more mainstream, mainstream physics no longer seems so depressing to me!

 

[garrett]

John,
Yep, many connections...

 

[mitchell porter]

It would be straightforward to formulate a similar theory -- including gravity, electroweakish gauge fields, and real fermion fields -- using only F4 instead of E8.

Let's give this a shot then! Just to be clear, my objective is to work this through until I can see to my own satisfaction that it is a well-defined quantum theory. So let me sketch in advance how it looks like things are supposed to go. Your equations 3.7 and 3.8 will still hold, except that things are now f4-valued. Gravitational so(3,1) will drop out, and there will be fermions and gauge bosons left over.

First question: which parts of f4 will play the role of $$\underset{.}{\Psi}$$? It looks like I can break it down as f4 = so(8)+(8+8+8) or as f4 = so(9)+16 - would these lead to distinct "modified BF F4" theories?

 

[garrett]

mitchell,
Things aren't going to be much easier with F4. One 8 will be the first generation of leptons, but... Majorana I think. And the other two 8's will be related by triality, but that leaves the same generation issue as with E8.

 

[Chronos]

Tony Smith used to hang here. He is brilliant, and unorthodox. I'm a big fan [despite getting booted from 'Arxiv' for no reason]. I think garrett is on the same track with his approach. The E8 concept looks bullet proof to this point.

 

[mitchell porter]

How about SO(9)$$\oplus$$Spin(9)? Again, this is just for didactic purposes - for someone who wants to be shown how they can get, say, Feynman rules for a theory like this. So far as I can see, there should be a modified BF F4 theory with SO(9) bosons and Spin(9) fermions in which several of your constructions can be carried out.

One thing that had been troubling me was where the uniqueness (no free parameters) comes from. I couldn't follow it down to the phenomenological level. But I guess it's just that the field couplings are determined by the structure constants, and then the masses are determined by the couplings and the Higgs VEVs.

 

[Tony Smith]

f4 and e8

Mitchell Porter asks about f4 = so(8)+(8+8+8)

Here is how I see that:

f4
=
so(8) 28 gauge bosons of adjoint of so(8)
+
8 vectors of vector of so(8)
+
8 +half-spinors of so(8)
+
8 -half-spinors of so(8) (mirror image of +half-spinors)


Therefore, you can build a natural Lagrangian from f4 as

8 vector = base manifold = 8-dim Kaluza-Klien 4+4 dim spacetime

fermion term using 8 +half-spinors as left-handed first-generation particles
and the 8 -half-spinors as right-handed first-generation antiparticles.

a normal (for 8-dim spacetime) bivector gauge boson curvature term using
the 28 gauge bosons of so(8).

If you let the second and third fermion generations be composites of the first,
i.e., if the 8 first-gen particles/antiparticles are identified with octonion
basis elements denoted by O,
and
you let the second generation be pairs OxO
and the third generation be triples OxOxO
and
if you let the opposite-handed states of fermions not be fundamental,
but come in dynamically when they get mass,
then
f4 looks pretty good IF you can get gravity and the standard model
from the 28 so(8) gauge bosons.

Recall that n=8 supergravity etc had problems because
the 12-dim Standard Model SU(3)xSU(2)xU(1)
does NOT fit inside 28-dim Spin(8) in a nice subgroup way.

If you want to make gravity from 15-dim Conformal group so(2,4) by McD-M
then
you have 28-15 = 13 so(8) generators left over,
which are enough to make the 12-dim SM,
BUT
the 15-dim CG and 12-dim SM are not both-at-the-same-time
either Group-type subroups of Spin(8)
or Algebra-type Lie algebra subalgebras of so(8).

If you try to get both the 15 CG and 12 SM to fit inside the 28 so(8),
you see that they do not fit as Lie Group subgroups
and
you see that they do not fit as Lie algebra subalgebras
so
what I have done is to look at them as root vectors,
where the so(8) root vector polytope has 24 vertices of a 24-cell
and
the CG root vector polytope has 12 vertices of a cuboctahedron
and
the remaining 24-12 = 12 vertices can be projected in a way that
gives the 12-dim SM.

My root vector decomposition (using only one so(8) or D4) is one of
the things that causes Garrett to say that I [Tony]
have "... a lot of really weird ideas which I[Garrett] can't endorse ...".

So,
from a conservative point of view, that you must use group or Lie algebra
decompositions,
f4 will not work because one copy of D4 so(8) is not big enough for
gravity and the SM.

Also,
f4 has another problem for my approach:
f4 has basically real structures,
while
I use complex-bounded-domain geometry ideas of Armand Wyler to calculate
force strengths and particle masses.

So,
although f4 gives you a nice natural idea of how to build a Lagrangian
as integral over vector base manifold
of
curvature gauge boson term from adjoint so(8)
and
spinor fermion terms from half-spinors of so(8)

f4 has two problems:
1 - no complex bounded domain structure for Wyler stuff (a problem for me)
2 - only one D4 (no problem for me, but a problem for more conventional folks).

So,
look at bigger groups:

e6 is nice, and has complex structure for me,
so I can and have constructed an e6 model,
but
it still has only one D4 (which is still a problem from the conventional view),

so

do what Garrett did, and go to e8
and notice that
if you look at EVIII = Spin(16) + half-spinor of Spin(16)
you see two copies of D4 inside the Spin(16)
(Jacques Distler mentioned that)
which are enough to describe gravity and the SM.

I think that Garrett's use of e8 is brilliant,
and have written up a paper about e8 (and a lot of other stuff) at

http://www.valdostamuseum.org/hamsmith/E8GLTSCl8xtnd.html

which has a link to a pdf version
(there is a misprint on page 2 where I said EVII instead of EVIII,
and probably more misprints, but as I said in the paper
"... Any errors in this paper are not Garrett Lisi's fault. ...".

I use a different assignment of root vectors to particles etc
I don't use triality for fermion generations,
since my second and third generations are composites of the first,
as described above in talking about f4.

For an animated rotation using Carl Brannen's root vector java applet from

http://www.measurementalgebra.com/E8.html

see my .mov file at dotMac at

http://web.mac.com/t0ny5m17h/Site/CB4E8snp.mov

In it:

24 yellow points are one D4
24 purple points are the other D4

64 blue points are the 8 vectors times 8 Dirac gammas (of 8-dim spacetime)

They are the 24+24+64 = 112 root vectors of Spin(16)

64 red points are the 8 fermion particles times 8 Dirac gammas

64 green points are the 8 fermion antiparticles time 8 Dirac gammas.

They are the 64+64 = 128 root vectors of a half-spinor of Spin(16).

If you watch them rotate,
you can see how they are related in interesting ways.


Tony Smith

 

[Cold Winter]

Hi Cold Winter, you may want to read the ATLAS group's somewhat tortured narrative of exactly how they went about their calculation. The limiting factor turns out to actually be not CPU power, but RAM. Performing the calculation turns out to involve constructing some tables that grow to hundreds of gigs in size, and they found that they essentially had to store the entire table in RAM, as the calculation requires so many more-or-less-random accesses to this table that access times would have been prohibitive had they allowed swapping to disk.

I've looked at the Atlas code. Yes it seems to be all "C", but with a few modules that seem more GUI related. I am perhaps a bit more jaded when it comes to iron than others. For e.g. I have a supplier pawning DDR2 ram for $9.99/GB... that puts 128GB in the $1278 range. Same source has a 16 way SATA2 controller at around $600 and of course, 250GB SATA2 drives are now in the $120 range ( 16X means $1920 ) Motherboards are now in the $1500 range for 4 socket Opterons... that leaves 4 cpus ( AMD has announced 4 core units for 2009 ? ) typically in the $1200/piece range. I figure I can build one H...! of a monster for around $10,698 in todays terms.

In 18 months that could be under $4000... which I think I can swing at that time.

BTW, that's 2Terabytes of disk mirrored and striped, so getting around the other difficulties ( capacity and I/O speed ) noted in that article isn't a biggy. That SATA2 controller at 300MB/sec will be hitting all disks at about 192Mb/sec... which should translate into a run time guesstimate of 11,000 seconds ( <200 minutes? ).

I'm inclined to go with FP math on this so that conversion will double the bandwidth requirement at a bit higher speed. Although I have to check the SSE capability with the 32 bit integers that the Atlas programmers originally used. That could certainly impact the run times in both directions.

In effect, given some time ( and a budget to fit my limited means ) I should be able to hammer E8 quite nicely. Certainly for much less than what the LieGroup are talking about.

 

[garrett]

mitchell,
I don't want to go too far with tangents like so(9)+9 because the thread will get confusing. But the modified BF setup in the paper is a very adaptable way to take algebras like this and get models with bosons and fermions. What you said about the couplings (from the structure constants) and the masses (from the Higgs VEVs) is correct.

Hi Tony,
Welcome back to PF. I think it's great that you and several other people have taken this E8 idea and run with it. It's good to have people searching in all different directions. In the paper, I tried to use a bare minimum of mathematical structure, but it's possible a little more will be needed in order to solve the generation question. Even if I can't solve it minimally, it will be satisfying to me if others take the mathematical ideas and tools in the paper and use them in their own models.

 

[rntsai]

To see this in terms of derivations, take a look at Tits' construction on pg. 49 of Baez's http://xxx.lanl.gov/abs/math/0105155v4". Also, see pg. 41 where another g2 shows up in the f4 decomposition.

Thanks kneemo for the reference. I didn't realize it before but there's
quite a bit on the algebra decompositions in Baez' paper; he just put
towards the end of the paper...long after the octanion setting has worn
me out. The use of quaternions (and octanions, clifford algebras,...)
is probably intersting in its own right and I think it helps if you
have a deep pool of understanding of such things that you can draw
on to clarify things. Unfortunately I don't, so they end up obscurring
rather than clarifying things for me. For the sake of what's in Garrett's
paper, Lie algebras over the complexes (or reals) is enough.

What I was (and am still) looking for is an explicit description of these
decomposition...something I can run explicit calculations with. I have such
a thing for the f4/d4 : f4=28+8+8+8 case (see an earlier post in this thread);
I'm looking for the e8/(d4+d4) and e8/(g2+f4) equivalent. Something like :

(1)a basis for e8; Cartan basis is good, Chevalley even better since that's
what I have already. The structure constants of e8 in either of these
basis are well known and are accessible for my calculations.

(2)another basis of e8 in terms of (1) that exhibits the decomposition.
This could be just a 248x248 matrix where so for example rows 1 to 52
span f4,...

Table 9 in Lisi's paper in principle has the same information for e8/(d4+d4)
so it should be usable if I can work out the mapping between the 8 columns
(1/(2i))w_T^3,(1/2)w_S^3),U^3,V^3,w,x,y,z and an accessible basis of e8.
Altrenatively I can start with the Chevalley basis that I have and mimic
the rotation/projections Gerrett Lisi describes; but each step is susceptible
to misinterpretting conventions (right vs left matrix action for example), typos,...

 

[Berlin]

combining quarks

Garrett,

Is it possible to assign the real physical u-quark particle to a combination of the u-L and C-L roots of your table 9 such that its a vector with w-quantum number of zero, just like the t, b quarks? (and the same w=0 for all leptons and quarks). Would this lift the degeneracy of the quark masses due to the higgs fields? Could the neutrino's get their mass from the new x-i.phi fields rotating them to a (+/- one) w quantum number?

berlin

 

[garrett]

rntsai,
I think the basis for the roots in Tables 8 and 9 are pretty standard. You may be able to construct or match up a basis of e8 generators from John Baez's paper, but I haven't worked this out explicitly yet.

Hello Berlin,
Yes, these are all ideas worth playing with. There are many ways to take the framework in this paper and develop it in various directions to try and resolve the remaining mysteries.

 

[kneemo]

Unfortunately I don't, so they end up obscurring
rather than clarifying things for me. For the sake of what's in Garrett's
paper, Lie algebras over the complexes (or reals) is enough.

What I was (and am still) looking for is an explicit description of these
decomposition...something I can run explicit calculations with. I have such
a thing for the f4/d4 : f4=28+8+8+8 case (see an earlier post in this thread);
I'm looking for the e8/(d4+d4) and e8/(g2+f4) equivalent.

For explicit calculations, Baez's paper isn't the best reference. If you want to see how the F4 and E6 derivations are constructed see pgs. 29-33 of http://arxiv.org/abs/hep-th/0302079" . The E7 and E8 infinitesimal transformations are merely extensions of these.

 

[rntsai]

For explicit calculations, Baez's paper isn't the best reference. If you want to see how the F4 and E6 derivations are constructed see pgs. 29-33 of http://arxiv.org/abs/hep-th/0302079" . The E7 and E8 infinitesimal transformations are merely extensions of these.

I posted a question about this in the GAP forum and a Scott Murray
was kind enough to send me explicit basis for both d4+d4 and g2+f4.

Looking at the last three columns of Table 9, it seems there's a
relationship between the two decompositions. What's the nature
of this relationship?

We have under d4+d4 : e8=(28,1)+(1,28)+(8,8)+(8',8')+(8'',8'');

The table implies for example that 64 dimensional (8',8') breaks up
as 8(l)+8(l')+24(qI)+24(qI') (under g2? or f4?).

 

[garrett]

I posted a question about this in the GAP forum and a Scott Murray
was kind enough to send me explicit basis for both d4+d4 and g2+f4.

Looking at the last three columns of Table 9, it seems there's a
relationship between the two decompositions. What's the nature
of this relationship?

The first d4 is a subalgebra of f4, and g2 is a subalgebra of the second d4.

We have under d4+d4 : e8=(28,1)+(1,28)+(8,8)+(8',8')+(8'',8'');

The table implies for example that 64 dimensional (8',8') breaks up
as 8(l)+8(l')+24(qI)+24(qI') (under g2? or f4?).

The 8's are acted on by d4 in f4, and the 1's and 3's for l's and q's are acted on by a2 in g2.