An Exceptionally Technical Discussion of AESToE

[starkind]

DataPacRat

Look at table 9 on page 16. Add up the numbers in the last column, headed '#'.

Each row contains a group of similar particles. For example, row 4 contains left and right electrons and electron neutrinos. Each occurs in a plus or minus version under column two, so in the final column the row represents eight particles, eight roots.

Corrections?

S

 

[garrett]

DataPacRat,
As starkind said, the complete particle label assignments are in Table 9. These are the standard coordinates for the E8 roots. To figure out which physical quantum numbers these correspond to, you'll have to skim through the paper. The other Tables, 2, 3, 4, 5, 6, and 7, fill in the details necessary to see the exact assignments in Table 9. We can transform from these standard coordinates to physics coordinates using the matrix at the bottom of p18. Also, keep in mind that the second and third generation assignments are speculative at this time. If you can't put it together, send me an email and I'll write back with the explicit particle list attached.

For what all these particles are called, in English, and good explanations of particle physics in general, http://particleadventure.org/" is a great site. As an example to get you started, "\mu_L^{\wedge}" is a left-chiral, spin-up muon.

rntsai,
Yes, you have everything exactly right.

 

[mitchell porter]

Hello Garrett,

I have been thinking about your section on dynamics. To explain my points, I need to be able to distinguish between (i) the theory which would result if your action (3.7) only contained the first term (ii) the theory resulting from the full action. 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.

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. 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?

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

 

[rntsai]

A couple of questions about the e8 decompositions (page 18) :

e8 = f4 + g2 + 26x7

I don't think f4 or g2 occur as subalgebras of e8. If these
are only subspaces, are they invariant under any particular
subalgebra action? It's possible the e8 has a d4 subalgebra
(so(7,1) so the f4 decomposes as invariant subspace of that;
same thing for g2/a2...so is this decomposition into subspaces
invariant under d4+a2 or something else?

Then again maybe the f4/g2 decomposition isn't as relevant as
the one on page 21 :

e8 = so(7,1) + so(8) + (8x8) + (8'x8') + (8''x8'')

but for this case too. What are the subspaces invariant under?
or how else are these subspaces characterized?

 

[Cold Winter]

I'm afraid my question is rather simple, though it is mathematical...
(If you'd like, feel free to think of this as the 'how do we show this theory in popular science magazines?' question. A table with 240 entries, listing the roots, what each number means, and what particle corresponds to each root, would seem to be a fairly easy way to more fully communicate what's involved to people who haven't taken calculus.)

Well, you've been pointed to the LieGroup website which "may" answer your question.

My interest is in duplicating experiments. And in this case, I'm looking to "duplicate" the E8 computation done by the Atlas program which is the source of these tables. If it can't be duplicated or I come up with different results...

Once I'm satisfied that the requirements of the experimental method are satisfied, I can take Garretts theory and start asking some "tabletop experiment" type questions. Sorry, gents, I don't have a "super collider" in my pocket so I do with what I got.

Then I think I'll ask the current "great questions" in cosmology... some of which at least at the scale of leptons can probably be done on a tabletop.

 

[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.