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CATEGORY: High Energy Physics
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TOPIC: An Exceptionally Simple FAQ
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A new paper, "An Exceptionally Simple Theory of Everything", has attracted a great deal of interest, from both the scientific community and the general public. This forum comment thread will collect my short descriptions of this new theory, in response to questions posed from reporters serving different readerships. It's in chronological order  sorry for the chaos.
Please don't add comments here (they won't be replied to)  if you wish to discuss the theory, an appropriate thread is the FQXi forum post where I first speculated on E8 Theory. If you'd like to read more about me personally, or leave an opinion or encouragement, there is an Exceptionally simple personal FAQ.
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Can you sum up your current work in a way that a layman could understand? I understand that you're applying differential geometry to the problem of uniting general relativity and quantum field theory, but at the risk of sounding too simplistic, how does that work? And exactly what is differential geometry, anyway?
Differential geometry is the study of smooth manifolds, usually in many dimensions  it's calculus on steroids. There are ways of classifying symmetric manifolds, and this links up with all other branches of mathematics; so differential geometry is sort of a hub where a lot of mathematics comes together. Now, there is one manifold in particular  the largest simple exceptional Lie group manifold, E8  that is the most beautiful. The system of roots in the picture I sent you describes the 248 symmetries of E8. What I'm working on is identifying each of the elementary particle fields of the standard model and gravity as one of these symmetries. It turns out that this match is... perfect, as far as I've been able to tell. This model is very new, and there are still things I don't understand about it, but it looks perfect so far. You have to be very careful with these things though, as they can encounter a fatal difficulty at any turn  and when theory contradicts experiment, or requires unreasonable revision, you have to toss it and move on. But this theory of fitting all the standard model and gravitational fields into E8 is working very well so far.
When we have a nice symmetric manifold, like E8, we can mathematically describe how this shape twists and turns over the four dimensional spacetime we live in. This description is called a principal bundle, and the field describing the twists and turns is called a connection, which determines the curvature. What I'm doing is identifying all the standard model and gravitational fields (everything) as parts of an E8 principal bundle connection, and it's working amazingly well  it appears to have all the correct fields and their interactions. Each symmetry of E8 is a different part of this connection, and each symmetry manifests itself as a different type of elementary particle that we have in our universe. When someone unifies gravity with the other fields like this, it's called a Theory of Everything  that's what I'm after.
In the notes for your Perimeter talk, you call this an Exceptionally Simple Theory of Everything, but it looks pretty complicated to me. Is this the kind of theory that you're going to have to know a lot of advanced math and physics theory to understand, or is it something that can eventually be explained in a way that the average person might be able to comprehend?
Heh, I said it was exceptionally simple, I never said it wasn't complicated. Someone is going to need quite a bit of advanced math and physics in order to fully appreciate it, but I think even the average person has a shot at understanding the basics. The symmetric structure of E8 is described by how these points are arranged around the center of the picture I gave you. We can find the interaction between any pair of elementary particles by adding their two corresponding points (as vectors) to get a third. The rest of the theory consists of equations describing the dynamics of these particles. Even if someone can't fully understand the physics and math, there are many beautiful patterns in this E8 root system, and I find it very satisfying that something this mathematically and aesthetically beautiful could be at the foundation of our universe.





Can you talk a little bit about the background of relating symmetries in mathematical structures to physical theories? Can you give our readers an example of when this has been done in the past, and how successful it has been?
Well, there are two different kinds of mathematical symmetry that have been extremely useful for physics.
The first is structural symmetry of the physical laws themselves. The shining example of this was in Maxwell's work on electromagnetism. Basically, Maxwell was playing with the equations as they were known at the time and noticed they were not structurally symmetric  there seemed to be a term missing. When he added this term, the equations came together in a much more elegant whole. With this new, unified theory, he was able to describe the propagation of light as an electromagnetic wave  when previously light had been considered an independent phenomenon. It was a very dramatic theoretical achievement. The biggest advances in physics have often come from unifications: Euler, Lagrange, and Hamilton's reformulation of mechanics; Einstein's special relativity; Feynman's path integrals; etc. The guiding principal is that the equations themselves  the mathematics describing the universe  should be symmetric and beautiful.
The second type of mathematical symmetry is more specific and familiar: the symmetry of patterns and shapes. In mathematics this goes by the name of group theory  and the success of twentieth century high energy physics can largely be attributed to this kind of symmetry. A notable application was the construction of the quark model in the sixties, by Murray GellMann and others. At the time, there was a large zoo of independent particles and no one knew why they had the properties they did. He managed to explain these properties in terms of symmetric patterns  later understood as the group theory of SU(3). Currently, our best model of how the universe works at small scales, the standard model of particle physics, is based on group symmetries.
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Question 1: What exactly is a gauge group?
A gauge group is a high dimensional surface with many symmetries. Basically, it's a pretty shape. The complete set of these shapes was discovered and organized by mathematicians at the beginning of the twentieth century, and they're the centerpiece of modern mathematics and theoretical physics. Among the groups there are five "exceptional" groups that have special structures distinguishing them from the others  of these, E8 is the largest, and the most beautiful, with 248 fundamental symmetries. The symmetries of a gauge group can be described by a pattern of points in eight dimensions, one for each symmetry.
Essentially, what I've done is associate each E8 symmetry with an elementary particle field of physics, including the entire zoo of standard model particles and gravity. Other physicists have previously matched the standard model particles to gauge groups, including E8  these are called Grand Unified Theories  but I'm including gravity as well, which is something new, and technically this makes it a Theory of Everything.
Question 2: What is a "principal bundle"? If the explanation is very technical, perhaps you could explain in which field principal bundles appear? What are they usually used to describe? How do they help you understand the symmetries that you are looking at?
A principal bundle is a playground for group symmetries.
If you have a group, it just sits there, looking pretty  but when you let a group dance over another space, like the one we live in, you get all sorts of fields with interactions and dynamics. Mathematicians call this a principal bundle. These are the main object of study of differential geometry  which is where calculus ends up if you stay in college for ten years. The basic idea is to allow a gauge group, such as E8, to twist around the spacetime we live in  the twists are described by the "connection field." This way, each group symmetry looks to us like an elementary particle field  parts of the connection. The dynamics of the fields, and their interactions, corresponds to the curvature of the twists.
Question 3: You talked (in a separate message) about the symmetries of the particles in the standard model. How do you relate particle symmetries with symmetries in general relativity? That is, what features of general relativity have symmetry and then how do you compare those features with the particles of the standard model, in order to unify them? It seems like trying to compare apples and oranges.
You're right, it's strange. I didn't think it was possible to describe gravity purely as gauge symmetries until I saw Lee Smolin, Laurent Friedel, and Artem Starodubtsev do it in a 2004 paper. They referred to a somewhat obscure discovery made by MacDowell and Mansouri in 1977. They revealed that gravity can in fact be described using a particular Lie group. This was the seed idea that led me to combine gravity with the other fields, and led to the unification of... well, everything, using the E8 gauge group.
I'm not sure why someone didn't come up with this idea long before I did  my guess is the physics community was distracted by string theory. Also, there's a theorem by Coleman and Madula that discourages this sort of unification. But the MacDowellMansouri trick gets around the ColemanMadula theorem by using a different Lie group for spacetime.
Question 4: What do you mean by a "really weird pattern"? How do you analyse the pattern? If it something purely mathematical, how do you interpret it and what features stand out as weird?
I had described the fields as elements of a big matrix, and this matrix was... lopsided. I had never seen a matrix that looked like that related to a group  it looked weird.
Question 5: Once you began to see this connection, how did you test it? What convinced you that it was correct? Did the theory predict features that we see in general relativity and the standard model (e.g. masses of the standard model particles, strength of forces)?
It didn't dawn on me slowly  the realization struck all at once, and my brain buzzed with the implications. So it wasn't a "hmm, maybe this will work" kind of thing, more of a "holy crap, that's it!"
The interactions between the two hundred or so fundamental particle fields we know of are all known. This exact structure is there in the E8 root system. As far as I've been able to tell it's a perfect match of tens of thousands of interactions, all corresponding to the most beautiful mathematical structure there is. How cool is that!
There are still features of this correspondence that I don't completely understand. I'm learning more about it every day. And, you know, as good as these things start out looking, they can always turn out to just not be true about nature. So, this theory I've cooked up may be wrong. But it's looking very good so far. I haven't gotten to the point with the theory yet where I can confidently predict new relationships between the physical constants such as mass, but that's where it's heading. This theory is only a few months old. What I'm going to do is write up what I've found so far and publish a paper, so others can see the details and play with it.
Question 6: Is there a way that we can test your theory? Are there predictions that it makes that will help it stand out from rival theories? Does your theory do away with the need for supersymmetric particles, or predict other particles that we could perhaps look for at the LHC?
That's correct, there are no superparticles in this theory. I made a bet with a pair of string theorists that superpartners won't be seen at the LHC  we'll get to see how that goes. I think I'll be able to use this theory to predict a handful of new particles that might be seen at the LHC, but I'm not yet far enough along to say more. It's very exciting. I have a fun year of work ahead of me.
Question 7: You mentioned that you weren't interested in working on string theory? What advantages does your approach have over rival theories, such as string theory, both in terms of what it can explain, and also aesthetic appeal?
There are lots of good things about string theory. It appears to be quantizeable, and can accommodate gravity in a fairly natural way. It also has restrictions that come out, due to anomaly cancelation, that say what can and cannot be a "good" quantum string theory. Originally people thought this would be enough, when coupled with the right background manifolds, to get all the standard model particle fields to correspond to oscillations of a string. But it's never worked quite right. In order to get it to work at all, string theorists have to bend over backwards and put in all sorts of things by hand. This is the main warning sign that a theory doesn't correspond to nature. What happens is, a theory looks promising, so people invest time in developing it. If it looks like it's matching nature, that's great. But if it doesn't quite fit nature, people have already invested a lot of time in the theory, so instead of abandoning it, they try to revise it  they add stuff and try to patch it up. But the more you have to add by hand, without any experimental guidance, the worse the theory looks and the less likely it is to be true about how nature works.
I suspected at the end of the 90's that string theory had left nature behind, and was going off in its own direction without any connection with reality. And it wasn't spreading out in a proper search either, it was barreling along in one direction like a freight train, guided by a handful of theorists with many followers who were devoting their lives to the theory. I've never been much of a follower, so I walked off to search on my own.
For the record, I do think it's good that very talented people, like Brian Greene, are working on string theory. Everyone should be able to work on what they want, and strings may yet turn out to be the true theory of nature. I just have other ideas.
(If you write about my dislike of string theory, please include that I do think it's worthwhile for some people to work on it.)
This theory I'm working on has the main advantage of being testable  it will very clearly be right or spectacularly wrong about nature. If it's wrong, I'll try to figure out if it's fixable, or I'll abandon it and move on. Another advantage with this theory is its relative simplicity  there is just one geometric field, the connection over spacetime, and the interactions and dynamics come from its curvature. And, at the heart of this theory is the most beautiful geometric structure in mathematics  it's very pretty.
Question 8: What next? If you have unified general relativity and the standard model, does that mean that all the four fundamental forces are now accounted for? What is left for you to do? What do you think you will need to do or demonstrate to gain support for your theory?
Yes, all fields are present and accounted for. Nevertheless, there is a ton of work to do. First, I need to confirm that the particle assignments and interactions are all in agreement with known physics. Any deviations could be fatal. Second, I need to see what new predictions come out: existing particle masses, coupling strengths, new particles, etc. Third, the proper quantum description needs to be worked out. Right now the theory matches up well with methods of loop quantum gravity, which is why I've been hanging out so much with the LQG community. To tell you the truth, it's much too much for one person to work on  but if it keeps going as well as it has been, it won't be just me working on it.
The talks I've been giving have generated a lot of interest  one week ago I delivered a one hour talk to a packed seminar room at the Perimeter Institute. That went VERY well. I'm going to have a paper out on the arXiv soon, with full details, so people will be able to consider it and play with it for themselves.
I feel funny calling it "my" theory, since I've learned so much from the work of others. And if it works as well as it looks, other physicists will play with these ideas and develop the theory far more than I could. It's a little early to say, but what I think, at this point, is that this is nature's theory.
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Question 1: Have theoretical physicists been able to associate the other four exceptional groups with physically observed phenomena? If so, what?
No, as far as I know this hasn't been done before. In the paper, I describe how each of these exceptional groups are related to a known subset of particles and interactions, and how they all combine into E8.
I should have a first draft of the paper ready tomorrow. If you'd like to look at it and you can keep it under your hat until Nov 7 or so, I'd be happy to send you a copy.
Question 2: Are these 248 symmetries just symmetries in rotation, and mirror symmetries? Are there other symmetries?
There are other symmetries. They are not easy to describe, but together they make up E8. The pretty picture of the E8 root system describes how the 248 symmetries are related.
It is possible though to describe the symmetries of E8 in terms of the rotational symmetries of the root system polytope in eight dimensions. If you do this... there are 696,729,600 rotational symmetries! It's a very beautiful and complicated shape.
Question 3: Are there 248 of these particles + forces? Or do you run out of particles and forces to associate with the symmetries and have to postulate the existence of extra particles or forces?
That's right, after the standard model and gravitational fields are fit to the 248 there are a handful of extra particles and forces  new particles. I currently only have some clues as to what these are, so I can't make definite predictions about them yet. It's still a year or so until the LHC turns on...
Question 4: For the cases where they matched the standard model but not gravity to E8, were there "spare" symmetries left over, that hadn't been associated with anything? And if so, why didn't anyone think to stick gravity in?
Two reasons. First, most people were distracted by working on string theory, which handles gravity differently. Second, the ColemanMandula theorem says you can't combine the Poincare group with other groups in a larger group that includes gravity. So people weren't looking for it, and if they were they didn't think it was possible. But there's an obscure way of describing gravity, compatible with Einstein's general relativity, called MacDowellMansouri gravity, invented in 1977. When I learned about this from people in the loop quantum gravity community, I saw how it could all fit together. Using this formulation, gravity can be fit together with the other symmetries, and it dodges the ColemanMandula theorem by not using the Poincare group. The Poincare group is the symmetry of a flat spacetime background  the MacDowellMansouri formulation implies we should use a curved spacetime background instead, called deSitter spacetime. Conveniently enough, this also means there should be a positive cosmological constant, which is what we see.
Question 5: At first, I thought that you were saying that you *begin* by identifying the particles and the forces with symmetries. But here it sounds more like the elementary particles and fields fall out only after you have set up E8 and started manipulating it? Which is correct?
Both.
We know there are a bunch of particles with charges, and we know how they move and interact. And the two are related. I've used these known charges to identify the particles as symmetries of E8, and using the curvature we can write down the action that describes how they move in a way thats compatible with what we know.
In order to know what charges particles have, you have to work backwards from how they move and interact.
 Question 6: How do we see or recognise these fields in the maths? Don't they just manifest themselves as other symmetries? What I mean is, intuitively I could imagine associating particles with symmetries and then letting that group "dance over a space" and seeing gravity and other forces appear as interactions. But you have associated the forces _with_ symmetries too, so what sort of interactions and fields are now being created? What do they correspond with physically?
The interactions only happen through the exchange of particles. As you probably know, the electric force corresponds to the exchange of photons  the field quanta of electromagnetism. A particle, physically, is what you have when you quantize the symmetry fields, mathematically. This can get very tricky, especially when you bring gravity into the picture. No one knows yet how to properly quantize gravity this way.
Each of these fields, corresponding to a group symmetry, dances over spacetime and interacts with the others  producing all the known particles and forces between them.
Question 9: What do you mean by the "root system"?
A "root" is what the points are called in the pretty pattern that describes the Lie algebra. They're also called eigenvalues. Or, if you're talking about this pattern in eight dimensions, these roots are the vertices of the E8 polytope.
A polytope is a fancy word for a polyhedron in a higher dimensional space.
Question 10: What sorts of things do these interactions correspond to? Specific strengths of forces between any two particles?
They correspond to which particles interact with which, to make other particles. For example, an electron can interact with an electroweak boson and become a neutrino, a quark can interact with a gluon and become another quark, etc. These interactions correspond to Feynman diagrams, which is what quantum field theorists use to predict things amazingly well.
Don't worry, Garrett, I'm not interested in printing a story about an imaginary fight between you and the string theorists. It's just that most of our readers have heard of string theory, so they would be interested to know why this is better.
Heh. The fight isn't imaginary.
I'm just one guy, and I'm the first to admit this E8 theory is a longshot, but it looks pretty good. If it works it will be a beautiful unified theory that agrees with the standard model and gravity, as well as predicting a few new particles. And it's better because it works without strings, branes, extra spacetime dimensions, superparticles, CalabiYau manifolds, or other weird string theory inventions that there's no evidence for.
If I thought string theory was better, I'd work on that instead.
Question 12: How do you model the curvature of spacetime using E8?
The Riemann curvature of spacetime is part of the curvature of the E8 connection.
Question 13: So this is an important point for me to clarify. I had thought that if your theory is correct it would do away with the need for people to work on either string theory or loop quantum gravity. But it sounds like maybe your work is showing that LQG is productive. So would the best way to think about it be to say that E8 provides the foundation for LQG?
Close. This E8 theory is a competitor to string theory, but string theorists could easily work on it since it's much simpler. It will be interesting to see how that develops  see if anyone defects. I don't expect string theorists to work on it though, because I'm not in their club.
LQG... A better way to think about it is that LQG provides a foundation for E8 theory. The whole thing came together when I was trying to figure out how to combine the standard model with recent work in LQG. In LQG the field describing gravity is a connection. I was able to use a larger connection to include... everything. I was amazed that it worked! So what E8 theory will be, when it's quantized using the methods of LQG, is Loop Quantum Everything. Heh, that's going to be a good title for someone's paper in the future.





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 redantigreen 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 allowed interactions between the particles. :)
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If there was no universe  to my na?Øve thinking  we wouldn't have (or need) string theory or LQG. But the logic of E8, as a pure mathematical object would still exist and the mathematical properties it has would still be true without there having to be an actual universe in which E8 is realized. So, intuitively, it seems to me that E8 should be more fundamental than LQG (and string theory). Does that make sense? And if not, what is wrong in my reasoning?
That's not so na?Øve  that's the Platonist world view, and is shared by many physicists and mathematicians. The E8 Lie group is a more central structure in mathematics than spin networks or strings. But spin nets and and strings are mathematical structures in this same way you're describing  so you can't necessarily exclude them just for being more messy.
What I think is that the universe is pure geometry  basically, a beautiful shape twisting around  and this shape is described by mathematics. This is a slightly different view than believing the universe IS mathematics, but it's close. Since E8 is perhaps the most beautiful structure in mathematics, is very satisfying that nature appears to have chosen this geometry. And quantum mechanics seems necessary for everything to happen.
2) Also, I was reading something Lee Smolin is quoted as saying about LQG. He mentioned that LQG may need to invoke a multiverse, just like string theory. Does E8 do away with the need for a multiverse? (At last, we can explain why everything is the way that it is  and it's down to the shape of E8.)
I'm afraid E8 doesn't say anything directly about quantum mechanics, so it doesn't help that way. Although there are hints of how it might connect to quantum mechanics, algebraically.
The multiverse is what you get when you think about quantum mechanics operating on the scale of cosmology. Personally, I think "many worlds" and "multiverse" are just fanciful ways of saying there are many possibilities for what can happen.
3) Does your analysis of E8 explain the relative strengths and ranges of the forces, and the relative masses of particles? (Sorry, if you have answered this one already.)
This is the goal. Right now it looks promising, but it's not there yet. All the pieces are in place to calculate these things, but some parts aren't perfectly clear, and it's going to take more work to find out one way or the other whether the correct coupling constants (force strengths) and masses come out. This is an "all or nothing" kind of theory  it's either going to be exactly right, or spectacularly wrong. I think it has a good chance of making successful new predictions, which is why I work on it, but it could still turn out to be wrong. It aint over 'til the LHC sings.





You probably answered this in another email that I buried, but I'm not sure whether to refer to E8 as an 8dimensional object or a 248dimenisonal object (or something else.)
The E8 Lie group is a 248 dimensional object. It is a very complicated, and very beautiful, smooth, curved manifold with many different symmetries. ("Manifold" is a fancy name for "surface.") This Lie group is the pretty shape at the heart of E8 theory.
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.
I'm sorry this is so complicated, but I hope that's clear.
Thinking in terms of a hypercube when we say, think of a cube in 3D and then you can think of another axis, orthogonal to the first three and that would give you 4D, etc...and in that way you can keep going and describe an ndimensional cube. In THAT sense is E8 eightdimensional (existing in the same space an 8D cube)? Or is it 248dimensional, existing in the same space as a 248dimensional cube.
The E8 root system is eightdimensional in this sense. And it is a polytope, like the cube and 8D cube are polytopes.
The E8 Lie group is not a polytope, it is a smooth, 248 dimensional surface.
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I also wondered if the NS version is correct, see below.
It has some slight errors (physics has never been just a hobby for me  it's my life), but for the most part it is accurate. I don't see any significant embarrassing errors or misstatements in the article.
Does this mean we do not need any more than four dimensions, for example?
That's correct  this E8 Theory only works in four dimensions.
And how likely is it that the LHC could find one of those 20 new particles?
The theory is very young, and still in development. Right now, I'd assign a low (but not tiny) likelyhood to this prediction. For comparison, I think the chances are higher that LHC will see some of these particles than it is that the LHC will see superparticles, extra dimensions, or micro black holes as predicted by string theory. I hope to get more (and different) predictions, with more confidence, out of this E8 Theory over the next year, before the LHC comes online.
If you could let me know today, that would be great.
Certainly. Things are developing very quickly with this new theory.
I posted the paper to the physics arxiv on Wednesday, Nov 7:
http://arxiv.org/abs/0711.0770
It received an unusual (perhaps unprecedented) amount of attention from physics bloggers (in chronological order):
http://backreaction.blogspot.com/2007/11/theoreticallysimpleexceptionof.html
(This is the best and most informative review of the paper.)
http://motls.blogspot.com/2007/11/exceptionallysimpletheoryof.html
(This is string theorist who, not surprisingly, hates it. Fortunately, his only real arguments against it are vacuous.)
http://www.math.columbia.edu/~woit/wordpress/?p=617
(Another level headed review)
Those are the main reviews and discussions of the paper so far, though there are several others (Physics Forums, etc).
Two hours ago, I presented a one hour talk on the theory to the International Loop Quantum Gravity Seminar:
http://relativity.phys.lsu.edu/ilqgs/
That is a weekly teleconference with a consortium of researchers at fourteen universities around the world. I'm happy to say it went very well.
I'm a little overwhelmed by all the attention, as I'm a bit of a hermit. But I'm quite pleased the physics community is as excited as I am about this new E8 Theory.
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1. First of all, in laymans terms: What the heck are you talking about?! Seriously, is there any way to impart to the average Joe what a unifying theory is, and why it's important?
Most people may be familiar with some of the various kinds of elementary particles, like electrons, quarks (that make up protons and neutrons) and photons (particles of light). These particles have charges. The electron, for example, has an electric charge of negative one. But there are also several other forces we know of, besides the electromagnetic force corresponding to the exchange of photons. The four forces are electromagnetic, weak, strong, and gravity  and each force has a set of particles attached to it, and a different kind of charge. Each elementary particle can be uniquely identified by the charges it has. And there are a lot of elementary particles! About 200 or so. What I found was that the charges of all of these particles are not just random numbers, but that they match a pattern. The pattern represents the most beautiful geometric object in mathematics, E8. This is the "largest exceptional Lie group"  essentially, it's a very large, very beautiful shape. And this is why it's so exciting. If this E8 theory is correct, it means everything in the universe corresponds to how this beautiful shape twists and dances over our four dimensions of spacetime. It would be very satisfying if our universe corresponded to the most beautiful geometry known to mathematics.
2. Your work is showing up among the most popular discussions on the web; what do you think about all the attention?
I enjoy a quiet, contemplative life, interspersed with playing outside. The media attention has been amusing, but stressful for me. I think it's wonderful if I've motivated people to think a bit more about how the universe works. And it's great that people are so interested in this theory I've put forward. It is a beautiful theory about nature, but it might be wrong  that's how science works. It's not enough that it's beautiful, it needs to agree with experiments. Also, this theory is young, and still in development  it may change significantly, or it might turn out to not work. If it doesn't work, I go back to the drawing board. But in any case, it's a good adventure, and there's nothing I'd rather spend my time doing.
3. The criticism I've read seems to span the spectrum, from you're brilliant, the next Eistein, to... Well, you know. What do you think about the critiques?
Einstein's construction of the equations of general relativity was the most amazing and beautiful accomplishment in theoretical physics. And it provided true insight into nature, with testable predictions that have held up for a century. Even if this E8 Theory is successful, it won't equal what he did to revolutionize physics. Also, this E8 Theory did not come from my effort alone. The program to unify the laws of physics has a long history, and many people have built it up over decades. I just hope I've advanced physics a little further with my ideas. And I expect that others will use this paper I've written, and continue developing E8 Theory in ways I haven't imagined. Or, the theory may fail. One needs to maintain a healthy skepticism, especially for ideas without experimental support, however beautiful they are. Criticism of a new theory is healthy. The scientific process will work it out.
4. From a physics stand point, which is better, surfing or snow boarding?
Ha! There's more going on while surfing, with water and wind rushing all around you  it's about chaos and taking chances. Snowboarding lets you draw a cleaner line  it's more geometric and deterministic. The difference between these two is a lot like the difference between particle physics, with it's many quantum particles careening off one another, and Einstein's theory of gravity, the smooth geometry of curved spacetime. I think, for completeness, you need both.
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You added the charges of each quantum particle and came up with the E8 group. Or did you plug the particles into the E8 and found that they corresponded correctly? (What I mean is, would it be closer to likening your process as plotting out different particles as a different symmetry, and overlaying them accurately coincides with E8; or did you calculate the charges of each particle and, on a graph, what emerges is E8?)
The two different descriptions you're giving are related (thus the confusion). The E8 Lie group is a large (248 dimensional) symmetric manifold. It has 248 symmetries, each corresponding to a Lie algebra element. The relationship between these symmetries can be described by a pattern of points in an abstract eight dimensional space, with each point corresponding to a symmetry of E8. This is where the pretty patterns come from. Each elementary particle corresponds to a symmetry of E8, and hence to one of these points in the pattern. And each of these has eight coordinates, corresponding to the eight quantum numbers (charges) for the particle.
The main thing I've done is to figure out all the quantum numbers of the particles in the standard model, and found that these match the points corresponding to E8.
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Can the triality matrix (whcih 'rotates' the different fermion generations), be considered as an 'operator' working on the E8 'eigenstates'? If so, there will probably be some root vectors for which T does not 'commute'. Otherwise the fermion generations would be degenerate states with equal mass. Could the simple search for these roots (without second quantisation) result in the mass gaps for the generations and maybe even the kiode relation? Something like: m(e)=^2==..=factor.^2





Garrett, thanks for offering a venue for the closure to 20th century theoretical physics. Perhaps you shall have liberated some talented theorists who will pick themselves up and move on to new phenomena. A slightly larger "everything." In addition to various "mysteries" that the experimentalists have meekly assembled over the last 20 years while the theorists have been engaged in synthesis, I am hoping for insight into some traditional issues like the existence of conciousness. Is the 'substance' of mind a topic for physics or not? Does some principal bundle describe my sensations of taste? Is my sensation part of everything or not?
Here is another/same issue. Any application of manifolds assumes some degree of smoothness in overlap maps. Often there is an implicit degree of differentialbilty of any vector fields on the manifold. All the more so for principal bundles. What are the consequences of relaxing the differentiability requirements? How should this be done? I am thinking, in Newtonian mechanics, if all the force fields are assumed to be at least once differentiable then there are always unique solutions determined by initial conditions. On the other hand, if one assumes the the force fields are simply continuous, then there are always solutions for the motion of a particle, say, but the solutions may not be unique for any set of initial conditions. (theorem by Cauchy) I am wondering under what criteria the dynamics implied by a connection imply there will exist nonunique solutions.





Garrett,
I actually like your title "An Exceptionally Simple Theory of Everything" and this post is to suggest you that your work might might concern "absolutely everything".
I hate to talk about me  but I need to etll firts that I just completed the report of an experiential learning in socioeceonomy  that ended with a geometrical method to handle problems in those fields with an unexpected  and seemingly universal  manner.
Our experiential developments did drive us at constructing maps, next within the difference group  to which belongs "mathematically speaking" the well known doubleentry accounting system  and from there we constructed a spheroid manifold that may embed all or any reality.
Everything we did is not out of theory but out of experience  and it looks that you only proposed the global theory that may encapsulate  say explain  all of our operational methods.
Of course we only know an experimental side of our methods and still we have to check how much a generalization of your work would fit with them  but in case the answer would be positive, your proposition may lead applications in many other domains than only physics.
In any case, congratulations for your work and your proposition.
(note the report of our experiential work is at Introduction to Holotomial Analysis
Regards
Paul Defourny





Garrett,
This question is surely out of subject, and probably naive, but let me ask this : if particles are travelling in the time dimension in the "opposite" direction, could they be detected ? Are there exemples of particles travelling to "the past". Is the causality principle a scientific concept, and why ?





You mention in the xrv article that symmetry breaking should have a mathematical explanation. Why is that, philosophically? To my mind, symmetrybreakings are where we find all the creativity in the Universe, from the matrix of symmetries (however beautiful they are!)  call it Deontic if you will. It's where all the 'meaning' of interaction, 'chaos  aka complexity', and serendipity (eg. fluctuationdissipation, or even just information transported on momenta transfer) comes out, informing us rather than we informing it. Kind of like those rogue breakers that really get your heart going.
Cheers,
Gary Knight, PhD (Condensed matter and plasmonics)





Garrett,
I have been following the cosmology world for 40 years. For the last 20 I have used this as a thought problem for countless hours on many long drives. By the time Brian Greens book came out, I was comfortable through roughly the first half, and then it went out of control. As you say, working to hard to force a fit.
At times on those drives, I would almost sense a ghost of what was needed, but it was far to complex to imagine. But I did predict to all of my peers that the person who solved it would not one of those sought the answer from bashing particles together, but rather the solution would come from someone who rubbed thoughts together. You proved me right on that.
For what it's worth, although I could only imagine a ghost of the E8, the instant that I saw the shape in the Aim E8 website, I felt the same as you describe above, Holy crap, that could be it. I could never imagine it but when I see it is just what is needed. If it isn't it is is so damn close.
Credentials.....Just a guy whose friends say thinks too much.
Good job and good luck to you in solving the details. Thank you for opening the door closed for 100 years.





Garrett,
First, congratulations for developing a beautiful testable physical theory, for making it reasonably accessible at this early stage to interested nonmathematical laypersons such as myself and for expressing your balanced personal philosophy of life.
My question (and concern)is this. Does the development of your theory have the potential to lead to the development of weapons which will exploit this more unified understanding of the physical structure of the universe and be even more destructive than present weapons?





Garrett:"The Riemann curvature of spacetime is part of the curvature of the E8 connection"
It would of course be wondeful to see this in it's natural setting. Would the Riemann Hypothesis be such a place?
"On the occasion described by Mr. Derbyshire, Hugh Montgomery, a graduate student was chatting with the great physicist Freeman Dyson, when Montgomery happened to mention his findings on the distribution of prime numbers based on Riemann's Zeta function.
Dyson, one of the most highlyregarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higherordered quantum state."
I asked Tommaso Dorigo such a question happening at the heart of the colliders. About gravity.
"It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous primegenerating formula discovered by Euler in 1774.). See more info on Mersenne Prime."
Do you see any relation?





See picture
 attachments:
3.gif 




Garrett,
If I Have understood this correctly the whole idea can be understood by picturing the concept in the same way Mr Carl Sagan did in his televison series from the 80th. In this science program an idea was put forward to give understanding of more than 3 or 4 dimensions to reality as we can persive them. The consept eas explained by relating to a world that had only 2 dimensions  and this world was named Flatland. If a 3dimensional body were to manifest itself in this 2dimensional world it would be like if you cut an apple in two  and "stamped" it on this flat surface. To the inhabitants of Flatland a sudden shape would emerge from out of noware  and leave a mark  and then disapear  without a trace.
So if I have grasped the idea here  the quarks and other manifistations in our space time continium is mearly a manifistation in our 3 or possibly 4 dimension wourld from this 8 or 256 dimensional reality that this E8theory predicts
My question now is  have I got the idea or not?
Regards and keep up the good work





You describe your theory as a "principle bundle"  a mathematical structure (G) of one sort connected to a manifold (M) a "connection": C(G,M) 
if we "connected" different Gs do we get other (but presumably nonactual) selfconsistent physics ? Or does M's structure constrain the choice of G?





I made up a quick blog to try to capture and list the most interesting links and related tidbits regarding this exciting theory.
Link= My blog of links





Oops. Previous post is mine.





Garret  I abandoned physics in the mid 1980s because the GUTs at the time were so ugly. I am delighted that you are proposing such a beautiful theory, and inspired by the way you enthusiastically embrace the importance of experimentation.





I HAVE A SECRET DESIRE THAT MAY BE ENABLED BY THE INFORMATION YOU HAVE ASSOCIATED WITH THE E8 MEGA MATRIX.
I HAVE LONG DESIRED A SIMPLE BREAKFAST CEREAL OF COLOR CODED BITS THAT WHEN COMBINED WOULD GRAPHICALLY FORM ASSOCIATIONS SYMBOLIZING PARTICLE PAIRING/ACTIVITY AT SMALL SCALES AND ASCENDING TO LARGER SCALES. I CAN'T THINK OF A BETTER TEACHING TOOL FOR YOUNG MINDS OR OLDER ONES, FOR THAT MATTER.
STRING THEORY WOULD NOT HAVE ENABLED SUCH A WHIM, NOR WOULD IT HAVE PROVIDED A PAIRING FOR THE TYPE OF GRAVITATIONAL REPRESENTATION NECESSARY THAT YOU SEEM TO HAVE ACCOMPLISHED WITH VECTOR ASSOCIATIONS.
DO YOU THINK YOUR MEGAMATRIX IS A SINGULAR MACRO OR A PAIRING?
ALSO, HOW WOULD YOU MATHMATICALLY REPRESENT A TYPE OF GRAVITY THAT IS NEITHER 'TO OR TOWARD' A FUNCTION WHILE INTEGRALLY RELATED TO THE EXISTENCE OF THAT FUNCTION?
(I HOPE MY NEXT SET OF GRAND CHILDREN AND GREAT GRANDS GET TO PONDER THEIR TABLE TOPS WITH THE CEREAL OF THE FUTURE. AND THANK YOU FOR INDULGING MY WHIM OF THINKING 'INSIDE THE BOX')





I requested that there not be comments here. But there are some that I like enough that I'll respond to them anyway, but not necessarily seriously.
berlin,
Something like this may happen, yes. If the three generations didn't have different E8 quantum numbers, there would be no way for them to be different in this theory, and no way for them to have different masses.
Daniel,
Damn, what the heck's the point of building a secret doomsday device if you're just going to blab about it?
If someone does use E8 to blow up the world, it will be long after we're already gone.
Plato,
Just because I don't see it doesn't mean it doesn't exist. Hmm, but that image does happen to look exactly like my right thumbprint... very odd.
Bo,
The E8 principal bundle over spacetime is a 252 dimensional space, called the "entire space." But, unlike in KaluzaKlein theory, the 248 dimensional E8 Lie group is a "frozen" shape over spacetime points.
Lee,
I never described this theory as a principle bundle, it has no moral fiber.
RD,
Thanks.
Deborah,
I recommend teaming up with Daniel and designing sugar frosted E8, with doomsday machine surprise included in every box. "THEY'RE GRE8!" Please don't shout though, I'm sensitive.





Do you think the day will come when you understand:
1) Why E8 theory describes the universe, and
2) Why E8 theory requires that "something" rather than
nothing exists?
In other words, will you have a eureka moment where you
think, "Aha, that's why it has to be like this"?





The idea of E8 is awesome! We need a great unifying theory, and this seems to fit very well. It does suggest that the universe, etc, must have a designer, much in the same way that a clock's operation suggests the existence of a clock designer and maker. The study of objects seems to have mankind looking for an ultimate of some sort.





If existance requires that the universe be configured as an E8 manifold, and if this perfect geometrical structure allows life, then I am going to start attending church again.
I can understand the idea of trillions of universes with randomly selected laws of nature, a few of which just happen to be conducive to life.
However it seems unlikely that the "life friendly" universes will have perfect and beautiful geometry, or that
such geometry should result in "life".





I'm a complete nonscientist, but in trying to understand your theory, I was reminded of Carl Jung, who believed that mankind often created symbols in an attempt to reconnect with the unconscious " archetypes" that he believed formed the essence of the universe.
Jung was convinced that these archetypes were psychoid, that is, "they shape matter (nature) as well as mind (psyche)" That archetypes are elemental forces which play a vital role in the creation of the world and of the human mind itself.
so when I saw your youtube video of E8 it looks so much like so many primal manmade symbols that Jung would flip over suddenly your theory had a real power for me, even if I couldn't understand the details of the science. so for that  thanks, dude. you're my official new hero.
I don't know if the theory is "factually" correct, but I'm sure it's instinctually correct
as a soul surfer my bet is you've had some of these same thoughts. care to comment?





Andrew wrote: "If existance requires that the universe be configured as an E8 manifold, and if this perfect geometrical structure allows life, then I am going to start attending church again. I can understand the idea of trillions of universes with randomly selected laws of nature, a few of which just happen to be conducive to life."
What makes you believe the unrealizable trillions of universes with randomly selected laws have any reality outside your imagination? The existence of the universe is the greatest mystery, but arguments like this seem a copout shortcut to its solution. I prefer to live with the mystery. Why do you accept three space and one time dimension as mundane, but consider E8 exotic?





If E8 theory is correct then:
1) There is only one universe or type of reality,
2) That universe has a structure that is beautiful and symmetrical,
3) And because of that structure intelligent beings must evolve.
Either you believe in a Creator or that life came about through random selection. If you believe the latter then
there must have been a huge number of different types of random universes to be selected from in order to find one with something as improbable as life.
There seems to be no reason that the "lucky winner" would, by chance, have life, and also by chance, be based on a simple, orderly, and beautiful geometry. This would only happen if one required the otherand that idea has hints of religion about it.
As someone once said, "Maybe the universe is complicated, chaotic, and ugly."





What is the main differences between your theory and this (D4D5E7E8) one?
http://www.valdostamuseum.org/hamsmith/d4d5e6hist.html





Andrew:
1) I hope so  maybe.
2) This is a much harder question  probably not.
Phil:
Why is it sensible to believe in a designer that's more complicated than the design?
Clocks can come from evolution.
Andrew:
Could I keep you out of mass if allow you to play with the Higgs VEV dials that determine all the particle masses?
Jonas:
Yes, this is where my own search for meaning has brought me. It will be very satisfying if these symbols work to give correct predictions for our physical world.
Gerry:
I mostly agree, in this universe anyway.
Andrew:
There are plenty of dynamic parameters that could take different values. However, the anthropic principal is based on an assumption that life isn't very flexible, whereas I think the converse is true. I think life could exist in universes we can't even imagine.
Rafael:
There are many differences. For example, Tony likes to remove parts of the Lie algebra structure when it suits his purposes  that's probably the main difference.





"Andrew:
Could I keep you out of mass if allow you to play with the Higgs VEV dials that determine all the particle masses?"
Yesbecause you said so I will just take it on faith that variable particle masses allow me to skip mass w/o fear of eternal discomfort.
OTOH being an "infidel" in regard to your theory, which is not punished, might be the safer course.





GARRETT,
WHEN YOU RESPONDED TO MY CEREAL APPLICATION AS A TEACHING METHODOLOGY, AND DID NOT RESPOND TO THE OTHER QUESTIONS, I CONCLUDED THAT YOU HAD NO NEED FOR MULTIVERSES OR VIRTUAL DIMENSIONS FOR VIRTUAL OR ANY TYPE OF PARTICLE BEHAVIOR; THAT THE E8LIE WITHOUT DISTORTION OR ACTIVITY FROM TRANS UNIVERSES WAS INDEPENDENT.
I DO NOT KNOW ENOUGH ABOUT YOUR PARTICLE ENVIRONMENT DICTATES; HOWEVER, ABOVE YOU REFERENCE MULTIVERSES IN YOUR 12/29/07 RESPONSE ENTRY. IN YOUR BEAUTIFUL PORTRAITURE OF THE E8LIE, I DIDN'T SEE ANY TELL TALE SIGNS SO I HAD TO ASK (I AM A MATHEMATICAL SLUG).
FOR YEARS I HAVE READ THE DOGMA OF STRING THEORY THROUGH M THEORY (AD PUBLISHING DICTATE$ INFINITUM) AND ALWAYS HAD MY OWN TERM, 'PARENTHETICAL', TO USE FOR WHAT THEY SO MISERABLY LACKEDA VECTOR TYPE SHAPE OF PARTICLE ASSOCIATION THAT I WAS REPEATEDLY TOLD COULD NOT RIVAL THE GODDESS OF STRING THOERY AND ITS MANIPULATED/MUTILATED CLONE DISTORTIONS (IT DID NOT FIT CLASSICAL WAVE MECHANICS BUT IT CAN BE PERCEIVED IN THE INTERLACING OF YOUR E8LIE
SO:
1 IS THE E8LIE AN INDEPENDENT ENVIRONMENT?
2 IF NOT, HOW WOULD YOU REPRESENT THE INTERACTION OF
MULTIVERSES(IT WOULD SEEM TO REQUIRE NON CHAOTIC
AND 'NONSTATIC' AND NO POINTS IN BETWEEN (NONPOINCARE)
BEHAVIOR; A NEW AND ENTIRE FIELD ALL TOGETHER.
(THINK ABOUT ITI FIRST THOUGHT IT WAS WHAT ALLOWED YOU TO CONVERGE THE HIERARCHY OF DIMENSIONS INTO THE THREE PLUS ONE (YOU AND EINSTEIN ET AL); NOW I THINK IT IS THE PRIME/ORIGINAL/FIRST SOURCE OF THE 'CASCADE' OF WHAT WE HAVE PERCEIVED/MATHMATICALY DICTATED 'DIMENSIONAL HIERARCHY' TO BE.
MY FAVORITE PERSON WITH WHOM TO INQUIRE OF SUCH THINGS, R.L. RYDER PASSED AWAY A HALF DOZEN YEARS AGO AND I MISS THE SHARPNESS OF THE 'MORE THAN CHEMISRTY' CREATIVITY OF SUCH SHARP, KEEN MINDS. I MUST NOT MISS THIS OPPORTUNITY TO THANK YOU, GARRETT, FOR THE WONDERFUL GIFT YOU HAVE GIVEN TO EVERYONE BY ALLOWING US TO CONTEMPLATE YOUR STUNNING PROTRAITURE OF THE UNDERLYING BEAUTY OF WHERE WE ARE BLESSED TO LIVE.





hihou !!
Me, I find the theory of nothing :





Yahou !!
Me, I found the exeptionnaly simple theory of nothing :
"
"
Garett, What do you think about my theory ?





Dear Dr. Lisi,
just a question regarding TOEs:
Kurt G??dels Incompleteness Theorem ...
so *any* TOE will end up being a "TO almost E".
Guess this doesn't touch your present work at all, but in the long run you must have some sort of opinion about it ... ? Hope it's not too indiscreet to ask about it.
One more thing  I liked your statement about trying to find a beautiful theory. The search for truth and the search for beauty are in a way related  the ancient Indian seers called the Ultimate Reality Satyam, Shivam, Sunderam: the True, the Good and the Beautiful.





Garrett, are you familiar with the work of MS El Naschie? It appears that your TOF and his work (which for the most part appeared in the Elsevier journal Chaos,Solitons & Fractals) have a lot in common? I would appreciate hearing your opinion.





I was thinking about the methods to test the theory: are the current and underconstruction particle accelerators able to discover these new particles? If not, how much time should we wait?





Andrew:
You have successfully proved that making sense is not a barrier to conveying sarcasm  nice.
Deborah:
The consideration of multiverses is largely independent from what I've been doing. Glad you like these new ideas, but I wish you wouldn't type in all caps.
bibou:
Nothing shouldn't be said twice.
N. Tantilov:
(addressed in other thread)
David:
What compelled you to post this same question multiple times?
Raven:
Don't wait at all, go out and play. More seriously, without mass predictions it's impossible to say when new particles associated with this E8 theory might be seen.
(Please don't post in this thread  use one of the others for questions or comments. This thread is mostly for posting interview responses. If you post here anyway, I will make fun of you.)
 this post has been edited by the author since its original submission 




Dear Lisi, please do not make the same mistake of the members of the string community by ignoring competing ideas. Elnaschie has been working in the same direction as for yourself for almost two decades. Before him there were many others such as Ord, L. Nottale, M. Green and J. Schwarz. Could we have your views on this for the benefit of the scientifc community at large.





I have been following the work of M. S. El Naschie for decades. This man has never badmouthed, ignored or downplayed anyone or any contribution. He also acknowledged every single person who contributed to his work unless he genuinely did not know and then he will immediately apologize of the unintended omission. I accept nothing less from Garrett Lisi and look forward to reading his explanation. With genuine good luck and best for both Garrett and Mohamed.





I have been following the work of Mohamed El Naschie for decades. This man has never bad mouthed, ignored or downplayed any one or any contribution. He also acknowlesges every single person who contributed to his work unless he genuinely did not know and then he will immediately apologize for the unintended omission. I accept nothing less from Garrett Lisi and look forward to read his explanation. With genuine good luck wishes for both Garrett and Mohamed





A. Pisko and N. Elsfeld:
El Naschie's work looks rather complicated, and I'd just rather not spend a lot of time with it right now. Also, I seem to need to repeat this again:
Please don't post in this thread  use one of the others for questions or comments. This thread is mostly for posting interview responses. If you post here anyway, I will make fun of you.
It is impressive that your fanaticism overrides respect for my wishes. You aren't perchance also involved with the Ron Paul campaign?





From a recent interview:
Here at SciAm we never jumped on the bandwagon regarding your E8 theory,
I had kind of wondered about that, since the representation theory involved has a visually appealing aspect that would probably make for a good SciAm article. But I certainly appreciate and commend conservatism, especially since some of the media response went overboard.
but I _am_ doing a short "second day" story (more like "second month" at this stage) briefly assessing where things stand now that the shouting has died down.
OK, great, I'll help out however I can.
Mainly I just need to confirm that I have the right impression (from following various threads online, such as the PhysicsForums thread) about what you are doing with regard to the theory. Namely, you continue working on it to try to overcome the various problems that were pointed out (some of which you had already mentioned in your preprint as being areas needing further research).
Yes, that's correct. The largest problem is the description of the second and third generation fermions, and this was discussed extensively in the paper.
One thing I could have done a better job of, but will try to remedy as things develop, is describing the fact that this general technique of describing all fields of the standard model (bosons and fermions) using one connection does work to describe what we currently know. It is only the fit to E8 that is rather tenuous at this point  but this fit to E8 was such an exciting possibility that I ran with it in this paper.
Also, do you have any real collaborators at this point?
I prefer to work on my own, although I've certainly been communicating with a great number more people. It's been so long since I've worked with a collaborator, I'm not sure I remember how. But I have been talking with people as they've worked through the paper and used parts of it to pursue their own ideas, which has been good.
Have you been invited to give talks anywhere recently?
Yes. My earliest upcoming talks will be at the TED conference on Feb 28, then I'll be talking with physicists at UC Davis on March 7.
About the problems... I know from the beginning you've been saying that the issue of getting all three generations of fermions "needs further research."
Yep, right now there are just many interesting clues.
About Distler's claims that even one copy of the proper chiral representation does not sit inside E8 as it needs to, I understand your position is: you're not sure if Distler is right in his claims
Yes, I think he may be making a mistake in his choice of complex structure when he uses conjugation to replace rightchiral fields with antileftchiral fields in describing the standard model. It's a subtle calculation, and it's possible that either of us could be confused on this point. I prefer to work through these things slowly and carefully before enjoining arguments though.
even if he is, using the complex form of E8 instead of the real will, without any doubt, fix the problem.
Yes. If you read his argument, he's unhappy with the way the real spinors in E8 are being twisted in this theory to build complex spinors. If you start with complex spinors in complex E8, there's no problem  at least not for the first generation. But at this point I'd prefer to stick with a real form of E8 if possible.
I have to say the issue that bothers me the most is the fundamental one of mixing fermions and bosons so closely together without having some kind of superalgebra.
Ah. There is actually a large but unfortunately somewhat byzantine literature on this construction  it's part of the BRST formulation.
But the mathematics of this construction are rock solid. The place I first learned about it was in this review paper:
http://arxiv.org/abs/hepth/0201124
where, on page 70, it's defined and described as a "generalized connection":
w = A + C
in which A is a 1form and C is a Grassmann valued field, both valued in parts of a Lie algebra.
Using the algebraic structure of the exceptional groups, the C can be a spinor field algebraically. Unlike in supersymmetry, these A and C never mix  so they can have different units without difficulty. The main advantage of formally adding them is for the distributive property in the Lie bracket  giving an expression referred to as "The Russian Formula." (Have a quick look at page 70 of the above reference).
For a more mathematical description of this kind of BRST extended connection, this recent math paper is typical:
http://arxiv.org/abs/0710.5698
Unfortunately, this sort of BRST symmetry is less familiar to people than supersymmetry, even though it plays an essential role in QFT.
You might have noticed that Peter Woit also likes the idea of possibly interpreting these "ghost" fields, C, as fermions.
I can go into more detail if you're still not happy with this formal addition of A and C  it's bothered other people as well, but I'm quite confident in this aspect of the theory.
In any case, I look forward to your reply.
Sure, let me know if you have other questions. Good luck with the writing.





While I don't understand the incredibly difficult math and other operations you have used to assemble this theory dude I think you may have gotten it.
I have been trying to get a feeling that string theory was the way to go for many years now. To no avail. Calebi Yau? Vibrating strings? While it was getting close, it IMO missed some things. Things that I could not get. Brian Greens book was the best book out there that I read.
Intuitively, it seems that you have something here.
Keep at it and good luck.





Garrett,
Can BANG CRUNCH BANG be right if it is a balanced equation.
The energy equation for exnihilo does not balance it is 2E=2M*2C^2.
But the energy equation for the big crunch balances it is.
1/3 APPLE+ 1/3 ORANGE+ 1/3 ORANGE= 1 APPLE/ORANGE.
Iam amazed by the intolerance of this community.
The equation for the universe must balance for evolution since you cannot get energy from outside the universe from another dimension like in exnihilo.
The four states of matter become one.
1/3 FERMI DIRAC EQUATION+ 1/3 BOSE EINSTEIN+ 1/3 BOSE EINSTEIN= 1 BOSE EINSTEIN/FERMI DIRAC.
And one unstable state of matter explodes to become four.
Of course this can be a better bomb but it can also be a better nuclear reactor.
Particles X an Y cannot be in the SAME STATE at the SAME TIME in the SAME PLACE except before the big bang.
The previous universe may have been a Godel one and a contradiction in terms of time so that it gives rise to a non contradictory explosion and the laws to our universe.
What is so silly about that that you can't print it.
Are people not allowed to talk anymore about religion.
Steve





Jim,
Thanks for the encouragement.
Richard/Steve,
I will put this as politely as I can: As far as I can tell, you are spouting complete nonsense. I don't wish to insult or anger you, but you should consider whether you want to waste your time communicating with someone (me) who doesn't appreciate your effort.





The following question is directed to Dr. Garrett Lisi: Your theory is that of unification and not only grand unification but quantum gravity unification of all fundamental forces. In such a case the unification coupling constant is one of, if not the most important result. It is the illusive point where all the four fundamental forces meet. What is the value of this coupling? El Naschie claims that he found the exact value of this coupling to be 1 divided by 26 assuming super symmetry. So what is the value coming out of your own calculations?





Hello Garrett,
I'm only a physics amateur, I write from Pesaro, Italy; I would ask you this question (or more probably to correct my question if badly formulated):
Which kinds of "charge" are beyond the 8 dimensions of the E8 root space illustrated in the famous beautiful animation ?
Perhaps I did't understand enough your phrase ("And each of these has eight coordinates, corresponding to the eight quantum numbers (charges) for the particle"), but I would expect somewhat like:
Axis 1: electric charge
Axis 2: strong interaction "colour"
...and so on.
Of course, I'm an enthusiastic fan of your E8 theory, I'm 45 and it is what I hoped to see when I was 20 and I read about electroweak unification... Please continue to keep us prophanes informed! Thank you and good work!





Dear Dr. Lisi
I am 85 years old and thought the following may encourage you to continue your excellent work. What the young W. Heisenberg had to do to get his Nobel prize and invent quantum mechanics was to make sense of thousands of numerical results found
experimentally. Without really knowing what he was doing, he told me he went on making patterns out of these numbers and invented rules to make them follow any system at all. Later on his Prof. Max Born discovered that his student Heisenberg had rediscovered an obscure part of mathematics called matrix multiplication which does not commute. You and your colleagues are not doing numerology when you are fitting the standard model into E8. You are doing basic and fundamental physics modeling. Those who say this is numerology do not understand either physics or mathematics, or in fact numerology which was an important part of the work of Backmeister Fuller.
With my best wishes to you, El Naschie, Lee Smolin and all free thinking and adventurous young men like you.





Hi Garrett,
Have you had an opportunity to read my book on “New Approaches Towards A Grand Unified Theory”?
Upon reading your theory again (and I still don’t understand everything!), I think I can better translate the bosonic component of your theory into my theory. As a Particle Physicist, I preferred to call my theory a GUT, because TOE is String Theory nomenclature, and I was slow to embrace the importance of multiple (10, 12, 26?) dimensions. In Sections 7.3 and 7.4 of my book, I came to the conclusions that the first ten dimensions are composed of spacetime and two hierarchal (in the sense that one is more physically relevant to our spacetime than the other) 3branes; that these dimensions unite as a 10dimensional SU(11) “Bosonic GUT”; and that the forces relate to dimensions as follows: Strong (1st and 2nd D), Electromagnetic (3rd D), Weak (4th D), Hyperflavor (5th and 6th D), Gravity (7th D), and WIMPGravity (8th through 10th D). In Section 7.5, it looks as if the three dimensions of WIMPGravity might collapse into one dimension that interacts with tardons, and two dimensions that interact with tachyons; which implies that my 10D SU(11) “Boson GUT” may collapse into an effective 8D SU(9) “Boson GUT” of order 8 x 10 that could coexist with fermions in your 8D E8 TOE of order 8 x 31. I agree with the possibility of an effective 8D GUT/TOE, but String Theorists will not be happy with such a small number of dimensions.
I thoroughly enjoyed your G2 approach to the Strong force. It is very similar to my U(1) x SU(2) x SU(3) approach to HyperflavorElectroweak. The difference is that your 2D approach uses nested 2simplices of quark and antiquark colors, whereas my (2+1+1)D approach uses nested 3simplices of quark and lepton hyperflavorweak isospins.
The biggest difference between our approaches is that I introduced a (2+1+1)D HyperflavorElectroweak and a (3+1)D Gravity/WIMPGravity (that might collapse into a (1+1)D Gravity/WIMPGravity model), whereas you used a (2+1)D PatiSalam Electroweak model and a (1+1)D Cl(3,1) Gravity model. It looks like you have projected the four dimensions of HyperflavorElectroweak into the three dimensions of a PatiSalam Electroweak, so you have potentially ignored details associated with my GUT’s fifth and/or sixth dimensions.
In Section 2.4.2, you implied that the E8 triality may break down. Figure 2 implies that the 240 roots of E8 may decompose as 240 = 8 x 30, and 30 is divisible by three. Are there any significant 5fold symmetries (30 is also divisible by 5) within this geometry? My GUT theory implies five generations of fermions.
Occam ’s razor is a good philosophical approach (I also like KISS – Keep it simple, Stupid! and PhD – Push here, Dummy!), and we shouldn’t introduce more complexity into the theory than is required to explain our observations (String Theory with 10500 parameters is definitely guilty of this, and my GUT might be as well). However, I think that our approaches towards the Weak and Gravitational interactions imply that there is more to both forces, and we shouldn’t call this a TOE if these 8 dimensions imply extra dimensions that aren’t described. For that matter, can a True Theory of Everything exist in less than 26 dimensions?
Good Luck!
Ray Munroe





Dear Garrett
I am sure you are inundated by too many irrelevant comments and confused remarks all apart of unfounded and poisonous socalled criticism on a site ironically called “The Reference frame”. By contrast I hope I could bring to you some constructive suggestions which may be of help. It seems to me it is important to embed E8 in some kind of spacetime and create a substitute to ordinary classical dynamics. This point has been made by Mohamed Elnaschie in frequent lectures which I have attended and also various publications. Elnaschie started following the usual counting of the degrees of freedom of spinors in order to fix the gauge. In his case this is merely fixing the scaling. Starting by 10 dimensions, we have 32 + 32 = 64 complex component equal to 128 degrees of freedom for a Dirac spinor. One goes on halving this number to 64 majorana, 32 majoranaWeyl, 16 light cones and 8 onshells. Adding all together we find exactly your 248 dimensions of E8. Subsequently Elnaschie proceeded to embed E8 in 4D spacetime and found 252. Remembering that his average scaling exponent is 2 we see that doubling this number leads to 504. This is the dimension of the simple linear Lie group SL (2, 8) = 8(641) = (8)(63) = 504 and corresponds to a standard model with 126 particles or when spin up and spin down are not counted as two different particles, we have 126 divided by 2 equal 63 particles. This result as I will reason is not accurate. In fact it is wrong as Elnaschie pointed out because you need at least 8 dimensions to embed E8 and not 4. Nevertheless 63 and 126 are consistent with Heterotic string theory. In this theory the number of first level massless states is 8064. Thus dividing by the corresponding spinor degrees of freedom namely 32 + 32 = 64, one finds 126 which are 63 multiplied by 2. By contrast if we give E8 the minimum embedding one would find 248 + 8 = 356 which leads to 64 particles. When not embedding E8 at all, one finds 248 divided by 4 to be 62 particles or (62)(2) = 124 particles counting up and down as different particles. The main point which Elnaschie is demonstrating is that embedding E8 in the 26 bosonic spacetime dimensions of string theory leads to 248 + 26 = 274. This is exactly twice the value of the inverse fine structure constant of electromagnetism (2)(137) = 274. The doubling corresponding to E8E8 which have (2)(248) = 496 must be (2)(274) = 548 which is Elnaschie wellknown total dimensions of the exceptional Lie symmetry group hierarchy involving the sum of the dimension of E1 to E8 as shown in various of his published work. The total number of particles in this case is 137 or 68.5 if spin up and spin down are not counted as two different particles. Thus there are still 137 – 120 = 17 elementary particles or equivalently 8.5 elementary particles to be discovered in addition to the already experimentally discovered 60 or (60)(2) = 120.
Much of what I have written here and more is on Elsevier science direct in several papers by Elnaschie. I recommend: “String theory, exceptional Lie group hierarchy and the structural constant of the universe. Chaos, Solitons & Fractals,” 35(2008) 712 and “Light cone quantization, heterotic strings and EInfinity derivation of the number of Higgs bosons”, Chaos, Solitons and Fractals, 23(2005) pp. 19311933.
Summing up, I think embedding E8 in D = 26 would solve a great deal of problems for the Garrett Lisi model. However and in all events, I would like to congratulate Lisi on his achievement.
R. Marek





Garrett
Your theory as I understand it has a sort of fields democracy. You can derive any field from E8 by letting it, as you like to say, “dance” on our 4 D spacetime. Well this sounds pretty similar to Elnaschie. He uses the golden mean transformation, in fact simple scaling to deform E8 into a Penroselike fractal tiling. This Penrose universe is in fact an example of a non commutative space as explained in detail in the classical books of A. Connes. This space is homomorphic to the compactified Klein modular curve and possesses 336 + 3 = 339 hierarchical degrees of freedom. Using the 496 of E8E8 and the 20 of Einstein’s gravity tensor, Elnaschie found the electromagnetic inverse constant to be 496 – (339 + 20)= 137.
I think these results and the connections to noncommutative geometry may be quite important to your work. The particular paper in question is “On Penrose’s view of transfinite sets and computability and the fractal character of Einfinity spacetime” published in Chaos, Solitons & Fractals, 25(2005) pp. 531 – 533.
With my best wishes,
Dietmar Kohlhass





Garrett – it seems you have set a trend toward maximal simplicity. I read about a new theory in the Telegraph using what is called Adellic function for prime numbers. The PAdic theory was given for P=2 in one quarter of a line:
2Adic of 137 = 1
Or in more intelligible terminology, looking at 137 from a P=2 reversed magnifying glass it is exactly equal to 1. The physical interpretation of this mathematical scaling of a number field is the tantalizing bit. We know 137 is the inverse electromagnetic constant which is the weakest coupling. But 1 is the largest coupling possible and is believed to be that of the Planck mass to the Planck spacetime or Planck Aether. Second,137 is the exact number of elementary particles in the standard model while there is only one type of particles in the Planck Aether. This is remarkable confirmation of the Planck Aether theory which was developed by one of Heisenberg’s students who is a retired Professor at the University of Nevada in the USA.
The Telegraph is referring to a paper published by El Naschie in Chaos, Solitons & Fractals. It would be great to know your views on the ramification of this remarkable unification which must be deeply related to the theory of PAdic quantum mechanics.
J. Schiffer





Garrett – it seems you have set a trend toward maximal simplicity. I read about a new theory in the Telegraph using what is called Adellic function for prime numbers. The PAdic theory was given for P=2 in one quarter of a line:
2Adic of 137 = 1
Or in more intelligible terminology, looking at 137 from a P=2 reversed magnifying glass it is exactly equal to 1. The physical interpretation of this mathematical scaling of a number field is the tantalizing bit. We know 137 is the inverse electromagnetic constant which is the weakest coupling. But 1 is the largest coupling possible and is believed to be that of the Planck mass to the Planck spacetime or Planck Aether. Second,137 is the exact number of elementary particles in the standard model while there is only one type of particles in the Planck Aether. This is remarkable confirmation of the Planck Aether theory which was developed by one of Heisenberg’s students who is a retired Professor at the University of Nevada in the USA.
The Telegraph is referring to a paper published by El Naschie in Chaos, Solitons & Fractals. It would be great to know your views on the ramification of this remarkable unification which must be deeply related to the theory of PAdic quantum mechanics.
J. Schiffer





Hi Garrett,
Fashion as much as dogmas reign in science as in any other aspects of human endeavors. It should not come as a surprise that the cheapest shot in the trade is asking where is the Lagrangian? A little bit of history of science may help although it will never cure.
The action principle was introduced in science as a mere alternative albeit more formal way of arriving at Newton’s equation of motion. It is connected to the name of Maupertius although he dealt only with the elementary problem of minimizing a work function. The more profound problem of minimizing a function was solved by Euler. It then became fashionable to formulate the laws of mechanics without drawing a single picture or diagram in contrast to Newton. At the end we had two schools of thinking, that of the imaginative H. Poincare and that of the sterile Bourbacki group. An abstract method such as the action and variational principle is without doubt of great help in a field such as particle physics where pictures and diagrams are not as helpful as in classical mechanics. But this and quantum field theory was fiercely resisted in the USA as in the Soviet Union. However a detente took place, then a change of guards in the USA as in Russia brought the opposite situation and without a Lagrangian you are not supposed to make a single move. Habit and mental inertia do the rest. Lisi’s work is free of such artificial constraints. He seems to work in three steps just like in the work of Elnaschie. First a clear model, then an enlightened counting, then algebraic manipulation to find what he expects to find. Other physicists like Lisa Randal marvel at the enormously complex mathematic and algebraic computation she is capable of doing as if this is what it is all about. Others like Lisi seek maximum simplicity to find an answer to a physical question and not to demonstrate a supernatural talent for computation as for instance in the case of the proof of the four color problem. Lisi and his followers are theoretical physicists in the mould of Poincare and Einstein. The majority nowadays are mathematical physicists of the Bourbacki type. Lisi has to live with that until the tide of fashion changes.
Robert Fisher





If I understand Lisi’s work correctly, he does not develop an ad hoc model for the spacetime of particle physics nor put a model of particle physics into a spacetime of this or any other dimensionality. In other words, he does not proceed as in string theory or in loop quantum gravity. He also does not extend quantum field theory to become a quantum gravity field theory.
What Lisi does is take an already existing symmetry group and its corresponding Gosset, i.e. a fourdimensional polytope upon which the E8 manifold is based. Then he works carefully a onetoone correspondence between the different points marked by eightdimensional vectors called octonions and particles with the aim of recovering the standard model and making some prediction for extending it to conform at ultra high energy to the idea of unification, by showing that all the fields of all the different fundamental interactions could be obtained from the same E8 manifold. Professor Lee Smolin an openminded guiding light in quantum gravity concluded that it is a neat way to seek the unification of all fundamental interactions in this basically nonconventional way.
Lisi’s work is therefore combinatoric in the spirit and quite near to that of Sir Roger Penrose’s program. Also not surprisingly it does not collide with the loop quantum gravity philosophy or with the basic ideas of Mohamed Elnaschie EInfinity theory except for not admitting irrational numbers at a fundamental level. Seen in this light, Lisi’s theory is not restricted by mathematics used traditionally in this field which is an advantage.
I think people tend to forget that the founding fathers of quantum mechanics were appalled by Feynman’s path integral and string theory took even much longer to be accepted. All what I am saying is give Lisi a chance. I am not directing this toward the young researcher but I am addressing the establishment, particularly the strings community.
As for Lisi, my only advice is that he extend combinatoric as the factorial function was extended to gamma function and topological dimension was extended to noninteger Hausdorff dimension. This would be a revolution similar to Einstein bending spacetime and making it curved.
Dr. L. M. C.





I am a lay person, and I have read Brian Greene's book about string theory. String theory does not allow for the existance of singularities and quantum chaos. What implications does your "Exceptionally Simple Theory of Everything" have regarding singularities and quantum chaos.





The comment written by anonymous on May 4 2008 is worth considering. He is both right and wrong. First he is somewhat wrong because topological singularity theory is used in string theory. However these are structurally stable singularities called by R. Thom catastrophe theory. You can read about that in the McGraw Book published in 1990 in London and authored by Elnaschie, a Professor of Engineering Mechanics at Sibley School of Aeronautics and Astronautics in Cornell, U.S.A. A brief account maybe found in a book by M. Kaku published by Springer. Also physically whenever you have a mini black hole you have a singularity and string theory uses length scale equal to the radius of a Planck mass which is a mini black hole.
The second point is more involved. Quantum chaos theory is not a classical chaos theory because the quantum suppresses ordinary chaos. But the anonymous comment is quite potent! One could describe the work of Nobel Laureate G. ‘tHooft, as well as the same Engineering Professor mentioned above Elnaschie, as searching for the common roots of classical mechanics and quantum mechanics and finding that in classical chaos. However the exact relation of ‘’tHooft and Elnaschie’s work to the work of G. Casati and Boris Cherecov and the quantum chaos community in general is far of being clear at the moment as far as I am aware. But in general you are right. Neither string theory nor loop quantum gravity have place for the fuzziness of chaos, classical or quantum and it would be a great achievement if Garrett theory could incorporate chaotic symmetries in the sense of field and Glotobiski as discussed in many popular writings by the very talented Ian Stewart.
Dr. Robert Fisher





Maybe I missed something, but I'm confused as to the relationship of this model with time. Maybe I'm being to simple here, but is there any way this model reflects the concept of infinty? The E8 design is a very defined shape that appears to have a definate beginning and end.





Yes of course you have týme ýn Lýsý's E8. It ýs the same sýtuatýon as wýth Eýnsteýn. If Einsteýn's space týme ýs compact and closed, then ýt ýs lýke on a sphere. There ýs no begýnnýng and no end although ýt ýs closed and compact. The E8 you see ýs a projectýon of what ýs close and compact but has ýnfýnýte týme. There ýs no problem here at all. However Lýsý dýd not embed E8 ýn any spacetýme. But he can very easýly. Someone on thýs sýte poýnted out that embeddýng 248 ýnto the 26 bosonýc dýmensýons you get at the end 548 dýmensýons or 4 x 137. It ýs somewhere on thýs sýte. You wýll easýly fýnd Sara.
Dr. Robert Fýsher





Yes of course you have time ýn Lisi’s E8. It is the same situation as with Einstein. Ýf Einsteýn’s space time is compact and closed, then it is like on a sphere. There is no beginning and no end although it is closed and compact. The E8 you see is a projection of what is closed and compact but has infinite time. There is no problem here at all. However Lisi did not embed E8 in any spacetime. But he can very easily. Someone on this site pointed out that embedding 248 into the 26 bosonic dimensions you get at the end 548 dimensions or 4 x 137. Ýt is somewhere on this site. You will easily fýnd it Sara.
Dr. Robert Fisher





Disillusioned by conventional quantum mechanics, Richard Feynman invented path integral. I think we are facing a similar situation today with Lisi’s E8 proposal. In fact it is possible to interpret Elnaschie’s method as moving from ordinary path integral to summing over all exceptional Lie symmetry groups. This intriguing point is however this: there are finite numbers of exceptional Lie and Stein manifolds. This way the problem such as Gribor copies is illuminated in a totally unexpected way.
M. Kavorkian





Sorry dear friend but the correct word is Gibov copies. I hope it is only a misprint. There are many similar maladies in the quantum theory of fields. All these problems can be solved by summing over all compact and noncompact harmonic exceptional two and three stein spaces. For an indepth consideration of this and related problems, see Chaos, Solitons and Fractals in science direct. The paper is titled “The internal dynamics of the exceptional Lie symmetry groups hierarchy and the coupling constants of unification”. See also related papers by L. Crnjac, JiHuan He, Ayman Okaby as well as the various papers by Garrett Lisi and Lee Smolin.
D. Wong





On its own as a single Lie symmetric group E8 cannot do the entire job of unification. On the other hand by summing over all exceptional Lie groups it can be done. This was the program of Prof. Mohamed El Naschie with whom I have had the honour of collaborating on this subject for some time.





To Wong and Kavorkian. You have both spelled the name of the great Russian Theoretical Physicist wrong. He is Professor V. Gribov. He is the first to point out that the usual procedure for fixing the gauge freedom in nonAbelian gauge theories is ambiguous. This puts classical theories of quarks confinement in doubt. This has to do with super conductivity of magnetic monopoles as well as gauge invariance. That is why Elnaschie used summing over exceptional Lie and stein spaces and used a different argument for deriving confinement from phase transition of spacetime to a Planck Aether with a single Planck mass as mini black hole.
L. Cran





A mathematical derivation of the mass spectrum using essentially Lisi’s E8 theory extended to exceptional family has been given recently. The paper is in an Israeli journal published by Freund in Tel Aviv “International Journal of Nonlinear Sciences and Numerical Simulation”. 9(3) pp. 307 – 208 (2008). The paper is called MontonenOlive duality and the mass spectrum of elementary particles via EInfinity.
A. Mehra





At long last Scientific American took notice of E8. On page 16 of the April 2008 issue, Graham P. Collins gives a somewhat mixed up review of the theory and comments on Lisi’s work. Of course he does not mention the work of Green, Schwarz, He, Crnjac, Elnaschie or anyone else, only rejoice that Lisi’s paper was wiped out from the internet archive. The present Stalinistic regime of theoretical physics ayotallahs do not permit that a wonderful theory such as that of Lisi’s becomes respectable. The beautiful small world complexity neural network is not allowed to exist because of the archbishops of superstrings and quantum field theory. People like Lee Smolin and ‘tHooft are rare species nowadays and the Telegraph proved to be more scientifically minded than Scientific American.
B. Kerek





Hamish is unifying the unpublished unified theory imagine with my quantum wierdness equation.
And Garrett needs a carrot if you can experss the geometry of your theory as equations.
Then you can add them 1/3 APPLE+ 1/3 ORANGE+ 1/3 ORANGE= 1 APPLE/ORANGE.
You can add all the equatons of your theory to Einsteins and Quantum mechanics and come up with one equation for all.
Don't believe me.Try my method and see.
Steve





Richard Thomas theory of everything simple as Ockhams rasor qualifies as a theory of everything.
Wheras Garrett Lises doesn't qualify.
A smart scientist.





I really admire Lisi and all who are in a similar position for putting up with utterly trivial comments to say the least. When did this decline in standards start. For Gods sake, these are important scientific issues in which only elevated people should participate by combining any idea of any value. What is the scientific content of the three lines of the gentleman calling himself “A smart scientist.” I think the answer is nothing. And as for Steve why can’t you say clearly what you want to say. Ok on this site the majority like Wong, MarekCrnjac, Cran, Mehra and Kerek make sense. However in a scientific debate there is no place for nonsense and I wish all nonsensical comments be taken out of Lisi’s site.
Bob Meyers





Dear Garrett,
I think I have worked out the similarities and differences between my ideas and your ideas, and the answer is E12 with a rank and effective dimension of 12 and an order of 684. Of course, several mathematicians just said “There’s no such thing”, and maybe someone can prove that my “E12” should be defined differently, but that’s what I used for my TOE.
Your ideas inspired me so much that I had to write a new Section, reference your work, and publish the Second Edition of my book “New Approaches Towards A Grand Unified Theory”. I published this book through Lulu.com, and free partial previews of “Section 7.7 – A Deceptively Simple E12 Theory of Everything” are available through Lulu. Lulu will have the 2nd edition – I’m not sure which edition (if any) the other online book retailers will have this month.
I didn’t need any ghost particles – I was able to fill all available particle states with my SU(11) boson GUT and Hyperflavor fermions. I concluded that a Supersymmetric Exceptional group MUST have fermion singlet representations, and that these fermions become the basis vectors for the adjunct Supersymmetric representation. A simple example is your G2 of color, with 2 basis vectors: g3 and g8; twelve roots: 6 gluons, and 6 up quarks/ antiquarks; and a 2dimensional fermion color singlet: electron/ positron. In the adjunct Supersymmetric representation, the 2 basis vectors are now: selectron/ antiselectron; the twelve roots are: 6 gluinos, 6 up squarks/ antisquarks; and the 2dimensional singlet is gluino3/ gluino8. When we add a 4dimensional fermion singlet to an F4 group, then our 24plet of fermions with a 3fold triality symmetry becomes a 28plet of fermions with the 7fold “septality” symmetry that is characteristic of Hyperflavor.
Table 19 is a symbolic E12 TOE. Unfortunately, Lulu’s preview software didn’t like Table 19, so I excluded it from the preview. Dr. Robert Fisher is correct that we need a Lagrangian, but I have exhausted this month’s inspiration…
Any new ideas from you?
Have Fun!
Ray Munroe





Pleased to read Ray Munroe’s recent comment dated May 22, 2008. This is the sort of comments acceptable on a respectable site. Yes without digging deeper there is no such thing as E12 because E8 is the largest exceptional Lie symmetry group. Any larger group will have an infinite dimensional Lie algebra. However, Prof. H. Nicolai from Max Planck Einstein Institute in BerlinGermany worked with E10 and E11. These are special forms of Exceptional Lie group extended beyond the initial idea. It all started by H. Gorgi, M. Elnaschie, J. Schwarz and many others who noticed that by systematically modifying the Dynkin diagram one will find that SO(10) may be called E5 while SU(5) is E4.
Subsequently, M. S. Elnaschie at FrankfurtGermany proposed to work with a hierarchy of Exceptional Lie Symmetry group leading to a total symmetry group dimension equal to 548. By including all two and three stein spaces, he finds not only 4 alpha bar =548 where alpha bar = 137 but also 5 alpha bar + 1 = 686 as dimensions. From all of that we can easily conclude that Ray Munroe may be well justified in inventing E12. He said it is 684 dimensional which means only 2 less than what Enaschie has calculated and only 1 less than (5)(137) = 685. It maybe worthwhile that Munroe looks at Elnaschie’s work and vice versa and that both should be thankful to the work of Lisi and this site. Most of Elnaschie’s work is published in Nonlinear Dynamics Journals. Here are few samples:
(1) One and two stein space hierarchies in High energy physics, Chaos, Solitons & Fractals. 36(2008) pp. 11891190.
(2) The internal dynamics of the exceptional Lie symmetry groups hierarchy and the coupling constants of unification. Chaos, Solitons & Fractals (2008) doi:10.1016/j. Chaos. 2008.04.028.
(3) MontonenOlive duality and the mass spectrum of elementary particles via EInfinity. Int. Journal of Nonlinear Science and Numerical Simulation. 9(3), 307308.
Bob Meyers





Two brand new papers came to my attention and they may be more than relevant to Garrett Lisi’s research. The papers are by a Saudi scientist at King Abdullah Institute for Nano and Advanced Technologies, KSU, Riyadh, Saudi Arabia. Both papers are on Elsevier science direct.
(a) Eeight exceptional Lie groups, Fibonacci lattices and the standard model.
(b) Towards a quantum field theory without Gribov copies and similar problems.
The two papers are published by Elsevier and the name of the Journal is Chaos, Solitons & Fractals and author’s name is M. S. Elnaschie.
A. Kasim





Dear Bob Meyers,
Thank you for the observation that 5 x 137 = 685 is only one element larger than my 684plet of “E12”. I have been aware of Sir Arthur S. Eddington’s old work regarding the Fine Structure Constant for many years, but I have not followed Prof. Mohamed S. Elnaschie’s work. Personally, I am a proponent of Five Fundamental Forces, Five Generations of Fundamental Fermions (there’s that number FIVE again), and I’m a big fan of Dirac’s Large Numbers Hypothesis regarding the number ~10^40. But I’ve always been suspicious of theories built around the number 137, because the Fine Structure “Constant” of QED varies with the renormalization mass scale (for instance, the Weakscale alpha bar is ~128), and how does that affect the theory?
Nonetheless, your observations have raised my interest. I need to visit Florida State University’s Dirac Science Library, and read Elnaschie’s ideas.
Sincerely,
Ray Munroe
p.s. – A correction to my earlier posting: The G2 of color bosons contains basis: g3, g8; roots: 6 gluons & 6 squarks/ antisquarks; and singlet: selectron/ antiselectron. The adjunct G2 of color fermions contains basis: electron/ positron; roots: 6 gluinos & 6 quarks; and singlet: gluino3/ gluino8. I think it is sloppy to mix bosons and fermions in the same representation group, when we know that there is an adjunct Supersymmetric representation.





Dear Ray,
Thank you for responding to my comments. I can give you more definite things of which I have just become aware. In hyperbolic geometry volume is an invariant. There is a hyperbolic manifold called M4 studied by some Swiss mathematician in Zurich and used by Mohamed El Naschie in high energy physics. Believe it or not, the socalled two volume of M4 is exactly your dimension 684. El Naschie published that in several journals of nonlinear dynamics in various versions of varying sophistication. It is incredible because M4 is based on a Coexter polytope representing a 120 cell gossett for E8. You need two of them to construct E8. Since dimension is invariant just as volume, I am now convinced your E12 is the mother of all exceptional Lie groups. In a sense you have reached summing over all existing stein and exceptional groups by simply calculating the hyperbolic volume. On reflection this is not astonishing at all. Volume is a higher dimensional area and you get an area by integration. Integration is summation. So it is merely tautology. We are just using different languages. But the message is the same.
You don’t need to go to Florida. One of the closest students of Prof. El Naschie is Nasr Ahmed working in Newcastle. He works with a prominent student of Steven Hawking, Prof. Ian Moss. Nasr will send you all of El Naschie’s work free of charge because I think he has it. And if not, he can put you in contact with Prof. El Naschie. I know El Naschie is as elusive as an electron and I wonder sometime if he is real or a collection of scientists with a pseudo name like Bourbake in France because he works in politics, philosophy, literature, engineering and science.
Regarding Eddington’s work and El Naschie’s interpretation, this is nothing to be suspicious of at all. It is straightforward mathematics. Let me give you the simplest form of it. You know that E8E8 which is 469 takes care of all interactions. Now particle physics consumes 336 + 3 of them. Gravity has 20. Subtracting both all what is left is 137 for electromagnetism. But 137 itself is variable in El Naschie’s theory. It is 137 at our low energy scale. It is 128 or 127 at various electroweak scales. It is 42 at grand unification and it is 26 at complete quantum gravity unification. Finally it is exactly equal to 1 at the Planck Aether scale. Being the coupling of the Planck masses to the Planck Aether. So it is of course a variable. It is a variable in disguise and it varies from 137 to 1. El Naschie published a remarkable equation based on PAdic analysis. The equation says the PAdic norm for P = 2 of 137 is exactly 1. There is much more to say for instance in El Naschie’s theory alpha bar is not 137 only. It is 137.082039325. This is exactly equal to 20 multiplied by the inverse of the golden mean to the power of 4. So El Naschie doesn’t build his theory from 137 but from the golden mean which is the basic element of the generalization of the Platonic bodies including the E8 gossett. Without the golden mean, there are no E8 gossetts and no octonions and there is no Lisi’s theory. In fact most of the Platonic bodies will also disappear.
Thank you very much for giving me the opportunity to have this exchange of thoughts and my sincere wishes for the success of your forthcoming book. I really hope you, Lisi and El Naschie and all the people working with you succeed in bringing us one step further rather than staying in this stalemate from which theoretical physics is suffering because of the internal civil war between the different factions of the socalled mainstream. I recall a word I read in an article by El Naschie about Thomas Mann’s book Death in Venice, Prof. Achenbach’s friend said: Do you know what lies at the bottom of the mainstream……..Mediocrity!
Bob Meyers





To Bob Meyers
I was really flabbergasted to see a paper published four years ago using a manifold with exactly (2)(342) = 684 something. This is exactly the order of Ray Munroe’s E12 not one centime less.
To Ray Munroe:
I said 684 something because this was not called dimension but twice the fourdimensional volume invariant of a manifold called M4. This manifold is based on a 120cell coexter polytope and therefore is related to the E8 Gosset. Elnaschie noted that (26 +k)(26 +k) = 685. This volume invariance may be regarded as a substitute to dimension. The paper titled: Supersymmetry, transfinite neural networks, hyperbolic manifold, quantum gravity and the Higgs, is a clear validation of Munroe’s E12 which has F theory spacetime dimension as 12. The paper is published in Chaos, Solitons and Fractals No 22 (2004) pp. 9991006. The author is M. S. Elnaschie from Cobham, Surrey, UK. It is amazing that the hyperbolic volume of M4 is equal to E12 dimensions. However, it is unbelievable that it is almost equal to the 686 of the sum of all exceptional Lie symmetry groups and stein spaces. These are entirely different theories and formulations leading to exactly same results. It is beautiful and must be true.
A. Kasim





Dear Garrett:
To whom it may be of interest, I would like to say that the E12 proposal of Ray Munroe has taken me by surprise for more than one reason. I am a follower of the work of Professor Ruth Kellerhals who is a student of the famous Swiss mathematician Prof. Hof, University of Basel, Switzerland. Her wonderful review article is published in Mathematical Intelligencer. My then Ph.D. coadvisor, Prof. Mohamed El Naschie made several references to her paper. She established the hyperbolic 4manifold M4 found by Dr. M. Davies based on 120cell coxeter polytope that has a hyperbolic volume 104 multiplied by pi square divided by 3 which is 342.146286. Since El Naschie needs two of them to compare to E8E8, one must multiply by two and find 684.232572. He subsequently reasoned that the exact expression is simply alpha bar divided by 2 and multiplied by 10. In other words it is the 137 alpha bar multiplied by 5. Thus Ray Munroe has found an exceptional symmetry group hyperbolic manifold because (137)(5) = 685. El Naschie calls the exact expression, the transfinitely exact expression:
(137.082039325)(5) = 685.410197
If we would have taken only the integer part from the outset we would have found Ray’s value which is:
(2)(342) = 684
The implication is breathtaking because El Naschie obtained the same results using path integral and YangMills theory combined in his paper titled “Topics in the mathematical physics of EInfinity” which unfortunately is published in an Elsevier Journal – Chaos, Solitons & Fractals  rather than freely on the worldwideweb (www). Prof. El Naschie is an enormously kind person who did himself a bad service by boycotting internet publications and relying mainly on periodicals which with the exception of Nature, Science and Physics Review, no one reads any more. However a particular paper on this 685 hyperbolic manifold published in the International Journal of Nonlinear Science and Numerical Simulation, an Israeli Journal, was chosen by Thomson  ESI Essential Science Indicator as a most cited hot paper. This is found on the internet at: www.esitopics.com/nhp/2006/September06MohamedElNaschie.html. The title of the paper is: “On a Fuzzy Kahlerlike manifold which is consistent with the two slit experiment” and in the same journal, vol. 6, issue 2 pp. 9598 (2005). Editor in chief Prof. JiHuan He, Shanghai, People’s Republic of China.
I must end by congratulating Ray Munroe on his E12 discovery. However this would have not happened or at least would not have been appreciated without G. Lisi, Lee Smolin and the courageous Telegraph science writer.
Sonja Kaliski





Bob – I live in Tallahassee, Florida. The Dirac Science Library is on the other side of town, and they subscribe to many of El Naschie’s favorite journals. I agree that modern research has become “big business” and too many capable researchers have sold out to the mainstream. It’s “publish or die” and the mainstream controls most of the journals, which is why I published on Lulu after two years of rejections by journals who wouldn’t say much more than that my paper “wasn’t appropriate” for their journals. Some good friends of mine have had copies of my book since November 2007, but I haven’t heard good or bad critiques or comments from any of them. They might have too much to lose from siding with an outsider like me. It’s OK – I understand. I’m not a tenured Professor with hundreds of publications, but I do have a Doctorate in Particle Physics and a few publications. That should qualify me to discuss these topics, whether other researchers choose to agree with me or not.
Bob, A. Kasim, and Sonja – El Naschie and I both defy the mainstream, but we might be different flavors of nonconformity. However, the ties between E12, 5 x 137, and M4 plus E8 are amazing. El Naschie is working with a Minimal Supersymmetric Standard Model of Particle Physics, whereas I have introduced new force quanta, new Hyperflavor/ KaluzaKlein types of fundamental fermions, and new generations of leptoquark fermions. Although my fundamental representation might be E12, I also have singlet states and Supersymmetric partners. Adding up the degrees of freedom, I have at least 1,416 = 12 x 118 different elements in my theory. If we subtract the four 12plets of singlet states from the 118 sets of 12plets, and add the 12 dimensions back in, then we have 118 – 4 + 12 = 126, which looks similar to alpha bar at the electroweak scale. How’s that for a little bit of El Naschielike numerology?
You might all enjoy reading Chapters 3 and 4 of my book “New Approaches Towards A Grand Unified Theory” on Lulu.com. I have extended the free preview to include these Chapters about my efforts to fit the fundamental coupling constants (including the fine structure constant) with Quantum Statistical Grand Unified Theory (a thermodynamic GUT/ TOE of the lowenergy coupling constants).
Prof. M.S. El Naschie – Your ideas are interesting, but moderately difficult to find. Have you considered organizing your best ideas into one book? Lulu.com makes selfpublishing easy and affordable, and nonconformists are welcome. I would buy your book if it was reasonably priced!
Garrett – Sorry for hijacking your blog site… What are you up to? I haven’t heard from you in a while.
Sincerely, Ray Munroe





such a symmetry...
as it's "everything": there is no more left&right...
how we can more speaking about "symmetry"
is it a "point" symmetrical?
cherio:
e





Dear Ray:
You can contact Prof. El Naschie directly or through his student nasr2000@gawab.com.
As far as I am aware El Naschie abhors internet, doesn’t use it and he doesn’t read it. He is really truly oldfashioned in this respect and guards his privacy jealously. But I can tell you could become friends.
However friendship must be based on true understanding. The expression El Naschielike numerology is a complete misrepresentation of what it is. First the Balmer formula was a constructive piece of numerical simulation. You see nature and you try to simulate it. It was of course Bohr who improved things and then came Schrodinger and showed simply they are Eigen value of an Eigen value problem. This is how we at last understood the atom and found a deeper theoretical justification for the numerical simulation of the Balmer formula. Yet we don’t know why we have to use complex numbers in quantum mechanics and this caused all the development which took place from Lie to Lisi. Your 126 is not numerology although in your text you obtained it in a numerological way. There is a world of difference between numerology and enlightened counting. Without enlightened counting as Nobel Laureate Steven Weinberg calls it, we could have no quantum field theory. El Naschie combined enlightened counting with deep theoretical and mathematical reasoning. Let me show you how your 126 comes about. Forget spin up and spin down, then we have exactly 48 fermions and 12 bosons – all experimentally well documented forming our standard model. This makes 60 physically present types of elementary particles. Now we haven’t added the graviton nor two additional bosons similar to the w, one charged positively and the other charged negatively. We could also say we have 3 bosons, 2 charged and one neutral and forget for the time being gravitons. Either way we end with 63 particles 60 real and 3 still to be confirmed. This is what El Naschie and Nobel Laureate Weinberg call enlightened counting.
Next comes the theoretical part. Hetoretic string theory predicts that the total number of massless states is the multiplication of left and right movers and this comes to 8064 massless states. In this counting every spin degree of freedom is considered a different particle. This is unrealistic and we can show that there are 128 different directions. If we consider up and down to be different particles, then they are 64 directions which should be eliminated by dividing 8000 by 64, we get our 126 particles. This is what you have got and it is ridiculous to call El Naschie’s procedure numerology. Well the correct result is of course 137. The simplest way to demonstrate that is by embedding the 126 in supergravity’s 11 dimensions and get 137. The more sophisticated way is to use the Kahler manifold with fuzzy dimensions which is discussed in the paper of El Naschie  referred to in earlier discussions.
There are many misunderstandings due to superficial readings of most theories and if all famous scientists and Nobel Laureates would have been right all the time, there would have been no history of science.
To Garrett, I don’t think we hijacked your blog site because really everything said here was stimulated by your work and what we are saying is very relevant to it.
Bob Meyers





Dear Bob,
Yesterday was not my best work. This reconciliation between my SUSY E12 and alpha bar has been bothering me for days. My SU(11) boson GUT needs more Goldstone/ Higgs scalars to break the SU(11) symmetry down to an SU(7) and supply longitudinal degrees of freedom to the massive Q, R, U and V Grand Bosons that are sequestered on the gravitybrane. I might also need spin 3/2 leptoquarks. And we haven’t even begun to consider nonminimal Supersymmetric models. I could easily increase my particle content from 118 x 12 up to 137 x 12. It would be interesting if a 12dimensional SUSY TOE had a particle content of 137 x 12, but it doesn’t yet feel natural to me. I haven’t given up. It is a work in progress…
The comment “El Naschielike numerology” came across rudely, and I apologize to Prof. El Naschie, his followers, and you. What I meant is that 137, 248 and 684 are just numbers. Truly, some numbers may contain more “enlightenment” than other numbers. I understand that concept and my work is full of such kinds of numbers.
Anonymous “e” – Left and right still exist. After all, the lowenergy symmetries still prefer left over right (Table 8 and Figure 3 in my book help clarify how that still occurs). It appears that we have a bodycentered cubic lattice of fundamental fermions in hyperspace dimensions. As such, how we define “GUT/ TOE” depends on how many nearestneighbors, nextnearestneighbors, etc. we choose to include. My hyperflavor theory includes nearestneighbor fermions, and increases fermion degrees of freedom by a factor of 7 (consider a simple cubic lattice with the origin, and one unit to the left, right, front, back, up and down). Seven is one of my “enlightened” numbers, and it carries on into SO(8) 28plets and their respective role in E12 = 12 x (2 x 28 + 1).
Garrett – Thank you and the FQXi Community for providing a forum to discuss these ideas. I’m sure you must be busy, but we would really like to hear your ideas as well.





I don’t understand why a wellestablished Elsevier Journal such as Chaos, Solitons & Fractals with the highest impact factor amongst all international Journals of nonlinear dynamics should be considered moderately difficult to find. The solution for the present deadlock in theoretical physics must come from an interdisciplinary direction. Consequently, a particle physicist must read across the artificial limits of specialization if he wants to impact particle physics. Interestingly both Garrett Lisi and Mohamed El Naschie have both a nonlinear dynamics background. Nonlinear dynamics, chaos and fractals are by definition interdisciplinary.
A.Kasim





To Ray Munroe
You are almost right but not completely in stating the difference between you and El Naschie. Your theory is essentially a socalled Technicolor. Elnaschie states clearly that his is transfinitely exact. Both of you are invoking far more particles than could be ever discovered. However, we are all talking about energy under one tesla as far as the standard model is concerned. The rest is theory – to come down to one tesla in a consistent manner. String theory is no different. They work with 8064 coming from 496 and end up with 126 or 63. El Naschie comes from 8872 down to 685 the 548 and ends up with 137 or 68.5. You start with 684 and if you do all correctly you end with something very close to 68.4 or 136.8. There is no fiddling here. It is all consistent with the EInfinity action principle which El Naschie derives from the sphere packing density in higher dimensional space by summing over all exceptional Lie groups in analogy to Feynman’s path integral.
Sonja Kaliski





Dear A. Kasim,
Yes, all of my degrees are in Physics from the same University (Florida State U.), and all of my journal articles are about Particle Physics simulation and prediction. That must appear to be a narrow field of study, and you probably wonder how I ever fell out of the mainstream? I also studied Solid State Physics and Plasma Physics in graduate school at the University of Texas. I guess you could say that crystalline symmetry groups and thermodynamics contaminated my Particle Physics Worldview. I agree that we need more “generalists” to balance out all of the “specialists” in this field of study.
Certainly, the local science library subscribes to Elsevier’s Chaos, Solitons & Fractals Journal, but I don’t personally, and $31.50 US for one article via internet is a steep price.
Dear Sonja Kaliski,
No, my theory is not Technicolor. I first developed a version of Quantum Statistical Grand Unified Theory in 1981, while I was a graduate student at U. Texas. I understood that I needed an extra level of quantization, and I relied on Technicolor for that purpose. My Quantum Statistical Professor didn’t like my usage of Technicolor, Technicolor went out of fashion, and I later realized that String Theory could supply this extra level of quantization. I think the difference is that Technicolor relies on deeper levels of fundamental constituents (i.e. going from composite protons to composite quarks to fundamental preons?) whereas my Hyperflavor electrons are supermassive fundamental particles that probably better correspond to Kaluza – Klein electrons. Their greater masses might make Hyperflavor electrons look like a new generation of leptons beyond the tau, thus the “flavor” part of the name. And we might have lattices of fundamental fermions in hyperspace, thus the “hyper” part of the name.
Yes, we need to first understand the physics under 1 TeV. I hope that the LHC can find the light Higgs boson. If not, the proposed International Linear Collider (ILC) will have a better Signal to Noise ratio for certain types of events (I studied that machine’s performance for my 1996 doctoral thesis – It takes too long to build these machines because too many people believe that the Standard Model or the Minimal Supersymmetric Standard Model is all there is and no one wants to spend $20,000,000,000 US to measure the next two decimal places of a particle mass or an astrophysical constant). Is Supersymmetry at the Weak Scale of 1 TeV, or my Gravity Scale of 20,000 TeV? We need to carefully analyze the cosmic ray data at the 10,000 to 100,000 TeV scale and determine if we can justify a supercollider even more powerful than both the LHC and ILC.
136.8 is close to 137. But I have extra Supersymmetric and singlet states that aren’t part of the 684, and thought I was working in 12 dimensions, not 10?
Sincerely, Ray Munroe





Dear Ray,
In this letter I just want to clarify once and for all times this point about numbers because we all feel very strongly about it. There is a deeply seated misunderstanding in this respect which must be eradicated. No dear friend, 137, 248 and 684 are not just numbers. Of course they are numbers but not in this case. They have been derived with a particular meaning from a definite model attached to what we human beings call physical meaning. So let me stress this point because people do not understand the difference between numbers, number theory, numerical simulation and the number coming out from a theory.
If we are searching and we are searching for the number of Higgs, what do you think the end result would be – a fancy Greek letter with many tensor indices? Of course not, we will find a number, namely the number of particles. Even if somebody decides to denote it with a Cyrillic character, we have to attach to this character a number. In fact the most important thing in superstring theory is a number  namely 496 massless gauge bosons that we start with. It could not have been 500 without changing the theory. Our standard model is happy with only 12. However we have to add other things to it, namely 48 fermions.
Numerology is different. For instance Wolfgang Pauli died in a hospital suffering from cancer. The number of the room where he was hospitalized was 137. Now a superstitious scientist will feel strongly that this is a hint from a being living in higher dimensions that 137 is the secret of everything and Providence made it in such a way that the room number where a great theoretical physicist moved from the here to the hereafter is 137. I am of course exaggerating. But our procedure is by George extremely different. You remember that using WeinbergElnaschie enlightened counting, we reasoned that we have 63 elementary particles. We didn’t count up and down as different. If we do and of course we should, then we just multiply by 2 and get 126. If we want to consider a supersymmetric theory, then we have to have equal number of fermions and bosons and that will require us to multiply by 2 once more and we get 252. This result we reinforced using a sophisticated theory, Heterotic superstring. We start by Fock space  multiplying left and right movers we get 504. This is exactly equal to the dimension of the simple linear Lie group for n= 8. It is simply 63 x 8 = 504.
If you attempt to reduce it to what we have just counted 252, then you divide by 2. Said differently, we know that the holographic boundary of our 496 exceptional Lie manifold is Kleinmodular curve with 336 symmetries. This is a wellknown result and you find it using the simple Lie symmetry group for n = 7 which comes to 7 x 48 = 336. The corresponding instanton density is as well known 24. You find that in any textbook on superstring theory. The total number of instantons is simply the multiplication of the holographic boundary 336 x 24 which comes to 8064. This is exactly the number of the first level of massless states of Heterotic string theory.
To come down to the supersymmetric model you divide by 32 degrees of freedom of the corresponding spinors and the result is our 252. Should you have wanted to find the 63, you should have of course divided by the maximal total number of degrees of freedom of the spinors which is 128.
Now that we understand it all, it is really trivial. But it wasn’t always that trivial. The snag is however that this is all approximation. The correct theory should have given us 137 particles or 274 particles and sparticles as a super symmetric model. So dear Ray, all these numbers didn’t come from empty vacuum or transcendental meditation. These are all stiff analytical results. To get the 24 you must understand the theory of Kahler manifold and you can find then the Betti number and add them together or you use the wedge product of the field strength and integrate over the 4dimensional volume. The exact value is however 26.18033989 and to find it you have to consider transfinitely fuzzy 4dimensional Kahler manifold. This all can be found in the work of Elnaschie.
But now to a big surprise even for me. Please look into figure 15 on page 594 of volume 30 issue 3 November 2006 of Chaos, Solitons & Fractals. The title of the paper where you can find this figure is: “Elementary prerequisites for EInfinity” by M. S. Elnaschie. In this figure he shows us a Penroselike fractal tiling but with a hetoretic string proportionality. I should have said transfinite hetoretic superstring proportionality. It is incredible but the invariant area is exactly equal to 685.410968. You may recall that this is also the exact volume of twice M4 manifold and I would bet my bottom dollar that if you make your E12 transfintely exact then its dimension will be precisely the same number. Now could you put your hand on your heart and swear this is numerology  Of course not. In one stroke, in this ingenious combination which Sir Roger Penrose in England, Alain Connes in France and Medhat Gazzaly also in France suspected, Regge quantum gravity, hetoretic string, hyperbolic manifold, exceptional Lie and steinspaces and your E12 are connected. The only person who ever noticed this and completed the theory is Mohamed Elnaschie. However, he clearly didn’t know that your E12 existed as a single exceptional group. You see after E8 the Lie algebra is infinite dimensions. Elnaschie knew that there is something like E12 existed but only as a super position of many compact and noncompact exceptional Lie group. But he never knew that a single group E12 could exist. We have not seen your analysis and I haven’t communicated with Elnaschie and I don’t know his views. But I suspect he will agree with what I have said here because I studied his work meticulously. He is of course a nonlinear dynamics man in the first place. He is an engineer by training and otherwise a selftaught person. Very frequently he knows the answer before he could find rigorous mathematics to support it. But this particular piece I think is a brilliant combination brought about by luck circumstances for which Garrett Lisi has played a major role.
I sincerely hope we will never come back again to this number business and as I heard Elnaschie say quite often: “You have to use everything at your disposal to understand the phenomena”. All tools are valid  experimental, theoretical, philosophical and number theoretical including numerical simulation. That is why the Americans were so successful at many things which eluded the sophisticated Europeans. Bobacki is great but for my money Poincare is greater and Einstein is supreme.
We have to make everything as simple as possible but not simpler. That is what the great Albert used to say.
Have a nice weekend.
Bob Meyers




