An Exceptionally Simple Theory of Everything

[marcus]

Okay, I've got my E8 root rotating Java applet up, enjoy:
http://www.measurementalgebra.com/E8.html

I would guess the trouble I am having is special to my system. I tried twice and loading the applet failed both times. So all I get's a blank grey square field with a red X in upperleft corner.

Of course I'm too busy nailing this down as a preon theory to deal with fixing the rough edges in the above.

A short description of the boson / fermion situation, from the point of view of the roots drawn by the above program (and the ones listed in Table 9 of Lisi's arXiv paper).

The fermions carry quantum numbers of +-1/2, but there are an even number of +s (and therefore an even number of -s too). The bosons are then defined by the minimal changes between fermions.

That is, define a distance function on the fermions given by the sum of squares of the differences between the roots. The fermions have even numbers of +s and -s, so this means that two roots have to change. The change is from +1/2 to -1/2 or back. Thus the bosons are all the ways of choosing two quantum numbers out of 8, with those two quantum numbers being +1 or -1 independently.

This has preon model written all over it.

Thanks for the help on this. I am looking forward to the Java illustration when I get it working.

 

[arivero]

An acid comment at Motl's blog asks to recover the units, ie to put all the hbar and c and G in its place. Actually, it could be a good idea in order to see what is preserved in the classical limit, what is lost, what can become a classic field, and what goes to null as field and appears only as particle. As I said before, a lot of the work of a TOE should be to worry about the limits to recover the previous theories.

EDITED: Motl boasts of a triplication of the traffic of its blog. Looking to my own stats, my guess is that all the physics blogosphere have got this *3 factor. (Incidentally, Motl implies about 2500-3500 regular visits to his blog.)

 

[CarlB]

I would guess the trouble I am having is special to my system. I tried twice and loading the applet failed both times.

Marcus, I've got a lot of other simulations that have been out there for a long time but were developed with an older version of Borland's Java. Could you tell me if my gravitation simulation works here:
http://www.gravitysimulation.com/

Actually, now that I look at his scheme again, I see that my version of bosons and fermions is not what he's doing. I had got into doing the roots and ignored the details of the assignments on page 16 of the paper. I like mine more, too bad it doesn't give the standard model!

 

[marcus]

Could you tell me if my gravitation simulation works here:
http://www.gravitysimulation.com/

your orbit applet works fine. It's hard to stop watching it. I was superimposing a spray of Schwarz schild orbits on a spray of Newton.
Everybody's system is different. don't worry about the other which for some reason i can't get.

 

[staf9]

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

Works great for me, may need to update java to get it working right, very nice job Carl

 

[strangerep]

Steven Weinberg showed at pages 12-22 of his book The Quantum Theory of Fields, Vol. III (Cambridge 2000) that Coleman-Mandula is not restricted to the Poincare Group, but extends to the Conformal Group as well.

Now, I honestly don't understand Coleman-Mandula, and I certainly don't know anything about Weinberg's claimed extension to the Conformal group cited here! (Although on the face of it I'm not quite sure it applies, it sounds like Weinberg would have proved that CM applies to anything which has SO(4,2) as a subgroup, but E8 doesn't have SO(4,2) as a subgroup, it only shares the subgroup SO(4,1) in common with SO(4,2)? Are the conditions met or not here?) But it seems to me that if Tony Smith is right then this is an important point. If Weinberg already did, as Garrett puts it, "prove the results of the Coleman-Mandula theorem while weakening condition (1)", then it seems to me there needs to be some response to that. Is there one?

Pages 12-21 of Weinberg vol-III deal with standard C-M. Then on pp21-22 he discusses
the conformal group extension Tony Smith mentioned. To be more thorough, it should be
clarified that Weinberg gives this extension "in theories with only massless particles".

But it seems to me there are other reasons to back off on Coleman-Mandula vs E8
at this time... Some of the other inputs to the C-M theorem are that the "symmetry
generators take 1-particle states into 1-particle states" and "...act on multiparticle
states as the direct sum of their action on 1-particle states". The "1-particle" states
here are the usual momentum/spin unirreps of Poincare.

Now, one doesn't need to read very far into the C-M proof to see that if these
preconditions involving "1-particle states" don't apply, then the C-M proof stops
dead in its tracks very early. In particular, it does not apply to (say) transformations
between inequivalent Fock representations.

But the key reason to back off is that E8 theory is not yet a quantum theory (right, Garrett?).
Hence there is no notion of states of definite particle number, nor even a complete
understanding of how the unirreps are characterized. So let's wait until if/when it
gets developed into a proper quantum theory.

 

[jal]

CarlB
It's fine for me.
However, you would need to include the effects of spin.
ie. Use a 2d surface.
edge,front,edge,back,edge,front (cannot see edges)

 

[Coin]

strangerep, thanks for the clarification.

 

[Nereid]

From Bee's blog:

At 2:03 PM, November 16, 2007, Lumo said...

Dear "Almida",

I assure you that Perelman's precious results have been peer-reviewed several times, for example in http://www.arxiv.org/abs/math.DG/0612069" ' on Garrett?

 

[CarlB]

Given that I wrote the E8 applet last night, I should be sleeping now, but instead I fixed a few bugs and added a group theory color drawing feature.
http://www.measurementalgebra.com/E8.html

I added the ability to change colors according to the F4 and G2 representations. To do this, click the "Colors" button. This will bring up another window where you can change colors of things. To change F4's "8V" component, click the F4 button until it shows "F4 8V". Leave the G2 button as "G2 All". Choose a color, and click OK.

If you want to select colors at the F4 x G2 level, click the F4 and G2 buttons until they show the two components you want to look at, select a color, and click OK. The roots that are in that intersection will change color (if there are any). You can change the color of the background by selecting a color, and clicking "Bkg". When you're done, click "Exit".

This all gets back to Table 9, page 16 of the Lisi paper. There is a bit of a confusion in the bottom 3 lines of the table. These last three lines are labeled as $$q_{II}, q_{III}$$ for the G2 component, but actually the roots described include also $$\bar{q}_{II}, \bar{q}_{III}$$. I'm sure these were abbreviated to keep the table's size under control, but it briefly confused me.

Next, I'll probably add the ability to change the shapes drawn. That should give you most of the capability of making your own

 

[Hans de Vries]

Given that I wrote the E8 applet last night, I should be sleeping now, but instead I fixed a few bugs and added a group theory color drawing feature.
http://www.measurementalgebra.com/E8.html

I added the ability to change colors according to the F4 and G2 representations. To do this, click the "Colors" button. This will bring up another window where you can change colors of things. To change F4's "8V" component, click the F4 button until it shows "F4 8V". Leave the G2 button as "G2 All". Choose a color, and click OK.

If you want to select colors at the F4 x G2 level, click the F4 and G2 buttons until they show the two components you want to look at, select a color, and click OK. The roots that are in that intersection will change color (if there are any). You can change the color of the background by selecting a color, and clicking "Bkg". When you're done, click "Exit".

This all gets back to Table 9, page 16 of the Lisi paper. There is a bit of a confusion in the bottom 3 lines of the table. These last three lines are labeled as $$q_{II}, q_{III}$$ for the G2 component, but actually the roots described include also $$\bar{q}_{II}, \bar{q}_{III}$$. I'm sure these were abbreviated to keep the table's size under control, but it briefly confused me.

Next, I'll probably add the ability to change the shapes drawn. That should give you most of the capability of making your own

Nice applet Carl, has real promises to become a good tool to study the E8 rootsystem.

Regards, Hans

 

[shoehorn]

An acid comment at Motl's blog asks to recover the units, ie to put all the hbar and c and G in its place. Actually, it could be a good idea in order to see what is preserved in the classical limit, what is lost, what can become a classic field, and what goes to null as field and appears only as particle. As I said before, a lot of the work of a TOE should be to worry about the limits to recover the previous theories.

EDITED: Motl boasts of a triplication of the traffic of its blog. Looking to my own stats, my guess is that all the physics blogosphere have got this *3 factor. (Incidentally, Motl implies about 2500-3500 regular visits to his blog.)

Lubos' "claims" are, unsurprisingly, disingenuous junk. He's used google trends to search for the terms "motl", "woit", and "smolin". He fails to point out that the results returned by Google trends are of no relevance to the traffic to any blog; they are simple indicators of the frequency with which each term was searched. As an example, people looking for Lubos' blog by doing a google on "motl" contribute just as much to the google trends results as do those who are looking for other people called "Motl".

It's terribly sad to see what Lubos has become. He showed such promise at one stage.

 

[Molon Labe]

In this paper http://arxiv.org/PS_cache/hep-ph/pdf/0207/0207124v1.pdf Witten states:

"Of the five exceptional Lie groups, four (G2, F4, E7, and E8) only have real or pseu-
doreal representations. A four-dimensional GUT model based on such a group will not
give the observed chiral structure of weak interactions."

Can someone, in a general sense, explain how Lisi purports to work around this? That is, how does Lisi produce the "observed chiral structure of weak interactions"?

This is absolutely not a challenge to Lisi. I have little understanding of this and I am simply trying to make connections.

 

[datakleptomanic]

i am agog! last time i checked on my IQ it was up there somewhere... but holy moly! batman

 

[check]

Just wanted to jump in and say congrats to garrett! I admit the details of this paper are beyond my comprehension but I understand the basics and its potential.

 

[arivero]

In this paper http://arxiv.org/PS_cache/hep-ph/pdf/0207/0207124v1.pdf Witten states:

"Of the five exceptional Lie groups, four (G2, F4, E7, and E8) only have real or pseu-
doreal representations. A four-dimensional GUT model based on such a group will not
give the observed chiral structure of weak interactions."

Can someone, in a general sense, explain how Lisi purports to work around this? That is, how does Lisi produce the "observed chiral structure of weak interactions"?

This is absolutely not a challenge to Lisi. I have little understanding of this and I am simply trying to make connections.

Thanks for suggesting this reading! I'd add that people interested on GUT should also look for a collection of reprints done by Zee time ago and published with a black soft cover in two volumes (but GUT is mainly the first one).

The chain in formula 7 of Witten paper is the traditional argument up to E8. Also some SO(2n) chains, not only in extra dimensions -as Witten says in the next page- but also in four dimensions. It was not clear if the need extra dimensions were, after all, a question of interpretation. What happens with SO(12) and SO(16) is that they are able to reproduce V-A, but they also have V+A interactions and you must break them. But yes, it seems that the restoration of hopes for E8xE8 come exclusively for extra dimensional theories, because Witten found a way to break the symmetry not into E8 but into a subgroup of E8 getting at the same time only V-A unbroken, and this was using very peculiar choosings in the compactification of the extra dimensions. Not peculiar enough, regretly, and then string theory went to its current stalemate situation. This resumes the rest of the paper.

------------------------------

Now, the point is about a subgroup inside E8xE8, and at the current level of development, even the E8 of Lisi could have some role; do not forget stringers have been very good about taking developments from other fields. A non trivial inclusion $$E8 \subset E8 \otimes E8$$ (not the obvious trivial one) could in turn be a clue to the right choosing of the V-A compliant subgroup.

Why do I still expect something in E8xE8? Because if you look ot representation level, this is build as pair of particles. And then the known peculiarity of quark pairs in the standard model is that all of these pairs, for the five lightest quarks, reproduce again, with the exact counting of degrees of freedom, the particle content of the standard model (only that if taking literally, all these pairs are bosons). This is a very particular (can not tell if "exceptional") point that only happens with three generations and five light quarks.

 

[DataPacRat]

I'm yet another interested reader who understands most science magazines and reporting, but doesn't have a clear understanding of what a 'manifold' is.

I understand that Lisi is proposing that the 240 roots of the E8 group correspond to the various particles and forces, and that each root has 8 coordinates. However, as an amateur, I have trouble deciphering which particle is represented by which combination of letters, superscripts, subscripts, arrows or lines over the top, and so forth, let alone the coloured triangles in Lisi's diagrams (however pretty they are). Would it be possible for somebody to write out a basic text file with, say, 240 lines, each of which has the 8 coordinates, and an English description of Lisi's assigned particle such as "photon" or "bottom anti-quark"? (If there is a specific significance to what any of the coordinate numbers mean, an introductory mention of which one is 'spin' etc would also be nice.)

Thank you for your time.

 

[ericchou]

Congratulations to Garrett!

I don't know about physics. And I can't read english much.

I know your news from Taiwan media. I think you do a great thing.

In my office, I download your paper, and print it. My workmate laugh me " you don't know english, and you don't know physics. why you download it?" Although I don't know physics, But I see figure 2,3,4 in your paper, rotate & rotate. I know some thing happened. a beautiful structure.

So I think you are right.

Thank you.

eric

 

[staf9]

In recent news on people "jumping the gun," I've heard of people doing some serious work on E8 Theory even though the paper was posted a whole 11 days ago and we can't really confirm it until the LHC confirms/disproves the existence of the missing particles in E8.

Personally I think there's a good chance of E8 being right, so I'm kind of jumping the gun by saying that but certainly not as much as people who are already starting to research other relationships in particle physics using E8 as a basis.

I'm sure this has been done over and over again when people see some light in a theory though, so nothing new. And who knows, someone may discover something that could prove E8 right or wrong.

 

[marcus]

...I've heard of people doing some serious work on E8 Theory even though the paper was posted a whole 11 days ago ...

that's exciting. do you happen to know where? connected with what institution?
(never mind that's just how my mind works )
it's an awful lot to ask that a dark horse independent approach be taken on at a good research institution but I sure hope that happens

 

[staf9]

that's exciting. do you happen to know where? connected with what institution?
(never mind that's just how my mind works )
it's an awful lot to ask that a dark horse independent approach be taken on at a good research institution but I sure hope that happens

It's more of individuals (string theorists) who are trying to disprove E8 Theory. This was mostly a word-of-mouth thing that I heard and I don't mean to be spreading any rumors, but it isn't too tough to believe since E8 does rule out string theory.

It would be really great if a research institution did take a closer look at the theory though. That would get my hopes up.

When I read Garrett's paper, there were a bunch of unexplained phenomena that ran through my mind when he mentioned missing particles in E8. Plus, it got me thinking about the different possibilities of unification of gravity and the standard model if using E8 is wrong. Hopefully, it did that same thing for someone much smarter than I am.

My ever changing opinion on this topic is at the point where, realistically, I think the theory has a small chance of being right.

But, the light at the end of the tunnel is that E8 theory and the work that results from E8 may clarify certain aspects of particle physics and the unification of gravity and the standard model so that someone in the future may be able to answer these questions.

In other words, I don't believe E8 theory has to be 100% correct in order to answer a few of our questions about the universe.

 

[Hdeasy]

Yes, congrats to Garrett. It is really a wonderful achievement. Elsewhere I made the point that there are certain similarities with Heim Theory. E.g. even though HT doesn’t have explicitly E8 in it, it’s dimension law throws up 57-dim. The symmetries of E8 represent a 57-dimensional solid .

I found out, by playing with Heim's dimension law for n dim embedded in higher dimensional space of N dim::

N = 1 + Sqrt(1+ n.(n-1).(n-2))

Heim's solutions to this were
n = 4, N = 6 (basic HT for mass formula etc.)
n = 6, N = 12 (EHT with 8 dimensions as the main ones, but 4 additional time like ones that are involved in determining the QM probabilities - and I think something of this sort was mentioned for Lisi's theory too... or am I imagining it?)

The next solution after the ones listed by Heim was:

n = 57, N = 420

And indeed there are no others for n < 1000.

Interesting to see gravi-electroweak effects in Lisi’s theory – reminiscent of the gravito-photons of HT, which successfully predicted the Tajmar artificial gravity effect to within an order of magnitude, beating GR by more than 20 orders in that sense. One or 2 groups are lending tentative confirmation to the Tajmar effect – more definite reproduction efforts should report in soon. If they are positive, then it looks good for HT’s 2 extra G-forces, and so a SO (10) unification might be on the cards. It would then be a question of how that tied in with E8.

 

[arivero]

How is it that nobody is intrigued about the fermion doubling in table 9 of the paper? Is it some obscure interplay between complex and real representations?

 

[arivero]

The old E8
http://www.slac.stanford.edu/spires/find/hep/www?j=PRLTA,45,859

Grand Unification with the Exceptional Group E8

I. Bars and M. Günaydin *
Physics Department, J. W. Gibbs Laboratory, Yale University, New Haven, Connecticut 06520

Received 14 April 1980

A truly unified model of the basic gauge interactions, except for gravity, based on the exceptional group E8 is proposed. The fundamental fields belong to the smallest possible single representations for each spin. In addition to accounting for the three "observed" SU(5) families, this Letter predicts the existence of three more conjugate SU(5) families below 1 TeV.

and its bandwagon (well, the whole train if you click "Next")

http://www.slac.stanford.edu/spires/find/hep?c=PRLTA,45,859 Note also
http://arxiv.org/abs/hep-ph/0401212
http://arxiv.org/abs/hep-ph/0201009
Should $E_8$ SUSY Yang-Mills be Reconsidered as a Family Unification Model?
Authors: Stephen L. Adler

 

[starkind]

arivero

Could you say more about this? I looked at the table and noted that the fermions in lines 4 through 15 are double, but it seems that the doubles here are matter-antimatter pairs, if I am right about the bar notation above the lower set of pairs meaning antimatter. Is that what you meant? I am struggling in deep water here and anything you say might give me a lifeline...

Thanks

 

[Fredrik]

It's more of individuals (string theorists) who are trying to disprove E8 Theory. This was mostly a word-of-mouth thing that I heard and I don't mean to be spreading any rumors, but it isn't too tough to believe since E8 does rule out string theory.

Does it? If you don't mind, can you explain how? Preferably in a way that someone like me can understand? (I know the basics of QFT and GR, but not the fiber bundle stuff. I know what a bundle is, but not much more beyond that (for the moment)).

 

[arivero]

arivero

Could you say more about this? I looked at the table and noted that the fermions in lines 4 through 15 are double, but it seems that the doubles here are matter-antimatter pairs, if I am right about the bar notation above the lower set of pairs meaning antimatter. Is that what you meant? I am struggling in deep water here and anything you say might give me a lifeline...

Thanks

Well, the last column in each row of the table is "#" and it counts the number of components, from the 248, that are accounted for in the row. For 4 fermions in a row, the # counts 8 components for a row of leptons (or a row of antileptons) and, obviously, a factor 3 for a row of quarks (or a row of antiquarks). Now, this does that the leptons for one generation, when you count particles and antiparticles, have a total # of 16, while in reality they have a total 8 degrees of freedom, so they? So it seems that the particles are duplicated; if it is so, well, duplication is an effect already noticed in old 1980 E8 unification and not completely out of the orthodoxy. But I'd expect it to be explained with some detail in this new context. Is it?

 

[Coin]

So I've got various questions about this E8 thing, and I'm not sure whether to ask them here or in the "layman" thread (I am in the somewhat awkward position of being very much a layman with regard to the physics here but having a somewhat more technical perspective with regards to the math). I'm interested in E8 because I have had a lot of trouble understanding yang-mills/gauge theories, and it seems like-- even as speculative as it is-- studying E8 might be a good way to learn about the general principles of these things, because E8 is in a certain sense simple compared to the Standard Model (even if it is only simple because there is so much we don't know about it yet ). Before I can do that though I want to be able feel like I understand the mathematical structure that is E8 itself.

From looking at wikipedia and this page (I think from the people who "mapped" E8 awhile back?), the impression I get is that E8 consists of those vectors of length 8 that can be formed from adding together integral multiples of the members of a basis of "root" vectors. The group operation appears to be vector addition, and the "root" vectors consist of all 8-vectors of the form
<±1, ±1, 0, 0, 0, 0, 0, 0>
or
<±0.5, ±0.5, ±0.5, ±0.5, ±0.5, ±0.5, ±0.5, ±0.5>
Because there are 8-vectors which it is not possible to construct by adding together these roots, E8 forms a proper subset of the set of {all vectors of length 8 consisting of integer or half-integer values}.

Is all this correct? Okay, so: If so, is this the E8 Lie group or the Lie algebra? In either case, what is the corresponding algebra/group? And in the case of the algebra, what is the lie bracket? (The atlas page says only that the lie bracket for E8 is "very hard to write down". Oh.) And finally, is it "weird" that E8 is a lie group/algebra-- yet has only a countably infinite number of members, and is apparently constructed entirely of discrete structures? I've thus far only encountered lie groups which are continuous, where it makes sense to talk about things like "infinitesimal generators". There doesn't seem to be anything infinitesimal about E8 at all. (Mind you, I'm not complaining-- I have a CS background and I am WAY more comfortable with anything discrete than I am with anything continuous! It just seems jarringly different from the way I understood people to use lie groups/algebras previously, and I'm confused how I missed this.)

Past this, the biggest thing that is confusing me here are the "roots". First off, although this is probably not all that important, how on Earth were they chosen? That is to say, was someone just playing around with addition on different sets of basis vectors, and went "oh hey this particular combination of 248 vectors acts kinda weird, everyone else come look at this"? Or was E8 first discovered as some other kind of structure, and it was later realized that the 8-vectors above are a convenient representation of that structure? Second off and more importantly, I am dreadfully confused by these root "diagrams" such as one finds all over Lisi's paper. As far as I can tell, the idea is that we plot each of the roots as a point in eight-dimensional space. (I take it that we plot them by simply treating each 8-vector as a coordinate?) However, then we for some reason draw lines between some of the roots! Why on Earth do we do this? What do the lines mean?

I'm similarly a little bit confused by this "simple root" thing that wikipedia describes. As far as I can tell, the "simple root"s are an alternate integral basis for E8, consisting of the eight vectors found in the rows of this matrix:

Wow, that's convenient! What's confusing me here though is, why on earth do we bother using the 248 roots described above, when we could just use these 8 simple roots and be done with it? Another thing confusing me: Wikipedia offers a "dynkin diagram" (which I take it is different from the "root diagrams" used with the 248-root system) which looks very deep and beautiful:

http://upload.wikimedia.org/wikipedia/en/d/d3/Dynkin_diagram_E8.png

... but I can't for the life of me figure out what it's supposed to mean. Wikipedia says that this is a graph where vertices represent members of the simple root system, and edges are drawn between any two members of the simple root system (I assume this means a 120 degree angle when we treat the simple roots as coordinates in 8-space.) Okay, that's nice, but why? Why do we care which members of the simple root system are at 120 degree angles to one another?

I have a couple more questions related to what Garrett in specific is doing, but these are just my questions about the E8 [group? algebra?] itself. Any help in figuring these things out would be appreciated. In the meanwhile, something vaguely frustrating me is that there does not seem to be any specific information on E8 in the obvious places. It is clearly a well-researched subject but the best I can find is these very vague wikipedia-style summaries, and John Baez's writeups (which are invariably exhaustive and lucid, but everything I've found which Baez has written covering E8 seems to be primarily about other things, like octonions, and only indirectly concerned with E8). Is there some particular thing, perhaps a book, I would be best served by going and reading if I am curious about the mathematics of E8?

 

[staf9]

Does it? If you don't mind, can you explain how? Preferably in a way that someone like me can understand? (I know the basics of QFT and GR, but not the fiber bundle stuff. I know what a bundle is, but not much more beyond that (for the moment)).

At a first glance, it's easy to think that E8 and string theory can coexist on some level.

There were a few things that made them mutually exclusive in my eyes after a while though, some of which were pointed out on forums that Garrett posted on.

String theory (specifically supersymmetry) uses a super Lie algebra to package bosons and fermions together. E8 uses simple (and possibly graded in the future, depending on where the theory goes) lie algebras to replace anti-commuting 1-forms with Grassmans.

String theory was vague to me when it came to determining how many dimensions to use to describe the universe, though I believe 10 spacetime dimensions (or 11 spacetime dimensions for M-Theory) is what is used. E8 Theory uses a 4 dimensional base manifold with 8 mathematical dimensions since the e8 root system can be rotated in that many dimensions. E8 theory is a "do or die" theory, it states its parameters meaning it can be verified or proven wrong about these things unlike many of the other theories today.

I may have missed many other things; just the basis of the two theories seem very different and at some points contradictory to me.

 

[staf9]

To Coin - here's some helpful websites for E8:

http://aimath.org/E8/e8.html

http://math.mit.edu/~dav/E8TALK.pdf (the narrative, http://atlas.math.umd.edu/kle8.narrative.html, is a pain to read)
E8 is an exceptional Lie group, specifically a Lie group for an icosahedron, or 20-faced polyhedron, maybe looking up search terms with regards to that might help. When I read Garrett's paper I had to look up Lie algebra and groups on wikipedia to remember a bunch of the math.

 

[jal]

Coin!
you said, "... the impression I get is that E8 consists of those vectors of length 8 that can be formed from adding together integral multiples of the members of a basis of "root" vectors... "

wiki says, "Each of the root vectors in E8 have equal length. It is convenient for many purposes to normalize them to have length √2."

and http://aimath.org/E8/e8.html says something interesting, "...This lattice, sometimes called the "8-dimensional diamond lattice", has a number of remarkable properties. It gives most efficient sphere-packing in 8 dimensions, and is also the unique even, unimodular lattice in 8 dimensions."
------
The layman thread is not up to this speed ...yet.
--------
staf9
Your links used a different approach to explain. It wasn't good for me. I probably will help the "pros."
jal

 

[Coin]

Jal, I'm sorry, I slipped into CS-ism there. When I said "vector of length 8" I meant something more like "vector of dimension 8". I was trying to express that there were 8 items in the vector. (Actually I guess technically "dimension" is the wrong word there too-- although E8 itself is eight dimensional...)

(And the length √2 thing applies not to E8 in general, but only to the root system people like to use... hmm, come to think of it, is that why we use the 248-element root system rather than the 8-element "simple" root system as the basis? I mean, is the appeal that the elements of the 248-element root system are all the same vector length, √2, whereas the elements of the simple root system are of different lengths?)

 

[Fredrik]

At a first glance, it's easy to think that E8 and string theory can coexist on some level.

There were a few things that made them mutually exclusive in my eyes after a while though, some of which were pointed out on forums that Garrett posted on.

String theory (specifically supersymmetry) uses a super Lie algebra to package bosons and fermions together. E8 uses simple (and possibly graded in the future, depending on where the theory goes) lie algebras to replace anti-commuting 1-forms with Grassmans.

Let's see if I understand what you're saying... String theory implies supersymmetry. Garrett's E8 theory doesn't contain supersymmetry at all. Therefore, the two are incompatible. Is that the gist of it?

String theory was vague to me when it came to determining how many dimensions to use to describe the universe, though I believe 10 spacetime dimensions (or 11 spacetime dimensions for M-Theory) is what is used. E8 Theory uses a 4 dimensional base manifold with 8 mathematical dimensions since the e8 root system can be rotated in that many dimensions.

I don't understand that last sentence. Does E8 theory say that space-time is four-dimensional?

 

[jal]

A quote from Garrett
http://fqxi.org/community/forum.php?action=topic&id=107
TOPIC: An Exceptionally Simple FAQ
For a magazine image, you may want to use the image of G2 and use some arrows and text box overlays to describe how the interactions between particles correspond to visually adding together the points in these diagrams. This will be the best part of this theory for most people -- you can actually determine how all the particles interact by how these points add together in these pretty pictures.

For example, if you look at the picture of the G2 root system in the paper: Take the green up triangle (that's a green quark) and add the blue circle on the far right (a red-anti-green gluon) and you get the red up triangle (a red quark). This is how the quarks interact with the gluons. It's vector addition -- maybe you can overlay some arrows over the G2 picture to describe how this works for your readers.

When we do the same thing with the points in any of the E8 pictures, we get all the interactions between all the particles. :)

The E8 root system, which is what's shown in the pictures, is a pattern of 240 points in 8 dimensions. This pattern of points describes the shape of the E8 Lie group, through its Lie algebra. The pattern in 8 dimensions is projected onto the 2 dimensional page from different angles to make the different pictures. By understanding this pattern, we get a better understanding of the E8 Lie group.
The elementary particles correspond to points in the E8 root system, which correspond to elements of the E8 Lie algebra, and thus to symmetries of the E8 Lie group.
That's correct -- this E8 Theory only works in four dimensions.
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coin ... go and do a search/revue for the concept that are contained in the quote. It might help to do some different visualizing.
http://aimath.org/E8/e8.html says something interesting, "...This lattice, sometimes called the "8-dimensional diamond lattice", has a number of remarkable properties. It gives most efficient sphere-packing in 8 dimensions, and is also the unique even, unimodular lattice in 8 dimensions."
jal

 

[staf9]

Let's see if I understand what you're saying... String theory implies supersymmetry. Garrett's E8 theory doesn't contain supersymmetry at all. Therefore, the two are incompatible. Is that the gist of it?

Somewhat, it's more of the way they treat bosons and fermions that makes them different.

I don't understand that last sentence. Does E8 theory say that space-time is four-dimensional?

Yes, E8 theory only works in 4 spacetime dimensions, and all types of string theory describe many more dimensions than that. That's really the major contradiction in my eyes.