
#55
Nov2007, 03:54 PM

P: 182

Great! This is the most fun I've ever had. If group members think it usefull, I'll try to do more of the kind of translation stuff I did yesterday.
I think a study of the cubeoctahedron, especially as it relates to the packing pattern of similar spheres, is one key to understanding this E8 structure. Packing spheres is a real SO(3) problem, and I started studying how it works years ago using marbles and clay. Styrofoam balls and toothpicks work well, too. One thing I found out about packing spheres is that a perfect stack has a density of about .74, where a solid would have a density of one. %74 has recently come up in cosmology as one of the key measures of density of the universe, also. Could they be related? I'll have to get the exact number for the density of stacked spheres, and bookmark the %74 in cosmology next time I see it. Dr. Jack Ng, U of North Carolina, has been working on a model for spacetime exploration involving densely packed spherical Planck sized clocks, but I don't think he has seen or thought about the cubeoctahedral geometry of such a thing. Might be worth looking at, see if it sparks any interest. I'll return with a link. Garrett, for whatever it's worth, I think you did it. Thanks, S. 



#56
Nov2007, 04:00 PM

P: 182





#57
Nov2007, 04:12 PM

P: 182

Still, I think there is something about our percieved three dimensional space changing in one dimension of time that is somehow more real, at least on our scale. Some work was done on triangulations last year that seemed to show that spacetime may be lower dimensional at very small lengths, very high energies. Anyway I think we are all more comfortable with good old SO(3), and a U(1) time, if for no other reason than that is the space where our favorite arts are played (you can include your favorite sport under my catagory art, if you wish.) Anyway I think you are right that the dimensional relationships we are less comfortable with do obey rigourous mathematical rules, and so we can use the different groups just as we would use more familiar things like momentum and energy. 



#58
Nov2007, 06:33 PM

P: 14

Previous posts, in particular #33 by Carl and #41 and #44 by Starkind do a SUPERB job of describing things in regular language. For me, the descriptions become somewhat cloudy at the junction of quantum mechanics and group theory. If I am confused, maybe others are too, so I will keep pushing, (with this post at least) Maybe you guys could elaborate just bit more? Maybe I am seeing a barrier that doesn't exist, or I am so offtrack that I am not even wrong. All I know is something bad happened in the last part of Carl's post on QM, and it happened about the time letters, followed by parenthesis with numbers inside, began to appear. I follow starkind's description of rotations and how this is in the realm of symmetry and group theory, and how an abstract state space is needed to describe the various quantities that appear in the world, but I can't quite get from here back to Carl's description of the mathematics. Grrrr, I don't think I am explaining myself well. If only there were some kind of formal language that we could all speak, maybe with symbols so that it would be concise, so that there wouldn't be any confusion. 



#59
Nov2007, 06:43 PM

P: 182

ok I'll have another go at the next section of CarlB's post. I'll do it offline and post here as or if sections get done.
The letters followed by parenthesis with numbers inside are Lie groups, which wiki does a good job of explaining. Anyway wandering in circles in Wiki got me what I think I know about it. More in a half hour, perhaps. S 



#60
Nov2007, 07:30 PM

P: 14

Sorry, my comment about "letters, followed by parenthesis with numbers inside" was a weak attempt at humor. I do actually know they are groups. I was trying to magnify my ignorance of the issue in a selfdeprecating way. Isn't that funny!!
Also, my comment about the "formal language" was supposed to be funny. It was a statement about how there is a fabulous language of mathematics available to those who take the time to learn it, but I am trying to have someone translate it into my language, because I am dumb and lazy. (some exaggeration here for the sake of emphasis. more self deprecating humor) Please don't put any effort into it beyond what amuses you. I greatly appreciate your efforts already!! 



#61
Nov2007, 07:47 PM

P: 662

RealColbert:
No need for apologies  the whole point of this thread, I believe, is to attempt an explanation that can be understood by nonexperts (or amateurs, or laymen, or jal's grandma, depending on who's doing the explaining). I think confusion over jal's posts in particular is completely excusable, since it's his grandma who seems to define the lower limit of required knowledge for this thread. ;) If his posts are confusing, then he should be asked to explain them, you shouldn't feel deficient. As for your humor  hey, I picked up the ironic tone! But remember, annoying as they may be, that's what smilies were invented for. :P Now ... back to our regularly scheduled discussion (which I'm thoroughly enjoying too, by the way  I just haven't recently had any keen insights into how to explain any of this better than what's been posted.) 



#62
Nov2007, 07:48 PM

P: 182

The Dimensionality of the Quantum Numbers
CarlB said: “You get to have one quantum number for every motion you can do with your symmetry that is "independent" sort of.” Maybe we should go back a little and talk about what a quantum number is. If you have had chemistry you will recall that when two electrons are in an orbital around a nucleus of an atom, one has to have “spin up” and the other has to have “spin down.” There are no spin states for electrons in an orbital that are in between values, like spin half way between up and down. It is always and has to be one or the other, up or down. The quantum numbers for particles are always exclusive like that. No two particles can have the exact same quantum numbers. This is kind of like the way two objects cannot occupy the same space. However the space now has more dimensions than our familiar SO(3). Lisi has found that the space of E8, which we are trying to imagine, can contain eight quantum numbers, and these eight quantum numbers describe all the known particles, and a few more, with each particle having a unique address in the E8 structure. The eight dimensions Lisi uses can be seen along the top row of table nine on page16 of his paper. You see all the particles listed there under the E8. They are listed showing both their standard model symbol, and the icon Lisi uses in his diagrams for each particle. So you see the various circles, triangles, squares, and combinations of triangles or squares in the left hand side of the table under E8, and these are the icons. On the right side under E8 are the formula notations that physicists use for the names of the corresponding particles. The quantum numbers along the top row on the right of E8 start with two quantum numbers labeled with some numbers and letters. You will recognize the ½, onehalf, and the three index above the S or T respectively. The squiggle in the center of all that is an omega, and the one on the left has a subscript “i” next to the two. That means, I think, that the denominator in that quantum number is imaginary, which is a mathematical topic we don’t need to go into right here. I suspect that the T and S have to do with spinners and tensors, another math topic which we may as well ignore for now. Just to note that they are ways the particle can react in certain fields. The omega and the S and T make me think the fields are gravitational. Then there is a “U” number and a “V” number, each with a superscript 3. These are followed by a w, then an x, y, and z, and then an F4 and a G2. Finally in the top row is a #, which is a count of how many different particles are defined in that row. I hope someone who knows more about the different quantum numbers will come in now and tell us what these quantum numbers mean, physically. I am not up to that yet. What I do think I know is that each of these quantum numbers represents a dimension, or degree of freedom, or direction in which the particles can move. Some move in one direction, some in another direction, and how they move under that quantum number tells us some of the information we need to know to tell what particle we are seeing. CarlB goes on to tell us about some of the features of the geometry which makes all this happen: Two small motions are independent if it doesn't matter what order you do them in (i.e. they "commute" as in obey the commutation law of multiplication AB = BA so order doesn't matter). Independent motions are great. They're easier to analyze because you can fiddle with one without screwing up the other. For the example of SO(3), the three small rotations do not commute. It might be obvious that rotation around Albuquerque doesn't commute with moving Albuquerque North/South. To see that moving Albuquerque North/South doesn't commute with moving Albuquerque East/West we can discuss the puzzle: Suppose two people have good GPS systems and start hiking from the same point. Person X goes 1 mile East, and then 1 mile North. Person Y goes 1 mile North and then 1 mile East. Do they end up at the exact same point? The answer is that, in general, they do not. To see why, get a globe, and see what happens if you increase the 1 mile to 1000 miles. Assuming that the starting point is Albuquerque (which is in the Northern hemisphere), you will find that the person who starts going North first, will end up farther to the east. The reason is that when you travel East at a higher latitude (i.e. more northerly) you cross more lines of longitude. The same effect occurs for very small rotations. And the result of careful calculations is that none of the small rotations in SO(3) commute and so you can't break things up. By contrast, with E8 you can pick out 8 small rotations that commute. Therefore the quantum numbers of an E8 state requires 8 quantum numbers to specify. This idea of commutation works as in the GPS example in our good old SO(3), but it also works between some others of the dimensions, and how we do calculations depends on if the two dimensions we are multiplying are going to commute or not. 



#63
Nov2007, 08:05 PM

P: 182

Hi RealColbert and belliott4488
Sorry I missed the humor. Anyway, I would probably have gone on in the same vein even if I had got the joke the first time. I am taking the tack, as belliott4488 said, of trying to explain this so my grandma would understand it. An additional advantage of doing it this way is that if I am in error about something, which is not unlikely as we all know, it will be an error for which we all may be able to understand the correction. And I do ask for any corrections, even if I get to be the butt of the joke. Sometimes I even make myself laugh, which is not a good trait in a standup comedian. So I'll just continue on as far as I am able, and hope that when you are done laughing at me, you will let me in on the joke, too. Anyway it is a pet peeve that specialists in any field do not define their jargon, at least the first time they use it in a paper, or at least give a reference of some kind so we ordinary mortals have a chance to look it up on Wiki. So I try to start from the grass roots and work my way up. Thanks for keeping me company on this journey. s 



#64
Nov2007, 08:20 PM

P: 182

Ok, well, the rest of CarlB's post assumes some familiarity with QM. I've never had any formal training in this stuff and my familiarity with QM is spotty. So, I am going to go follow Garrett Lisi in his generous offer to let us use his work on the fx site. I'll get back when I think I have something grandma would understand.
S 



#65
Nov2007, 08:54 PM

P: 640

Hi!
I think the groundwork has been done ... proceed. If there are questions ... I'm sure someone will ask and someone will rephrase in their own words. We are approaching the "amateur" level. TheRealColbert ... An error on my part .... Although I used your name it was meant as a general comment "learning is a self generating process" meant for all the readers who want to be spoon feed rather than putting in the effort required to learn. We can only make things easier when we use everyday language. Precission is thereby lost but would be regained by the "seeker" of a deeper understanding. Okay??? jal 



#66
Nov2007, 09:00 PM

P: 182

Ah, well, it is closing time at the coffee hole again. I read Lisi's posts at fqxi, and it all sounded very sensible and promising, and fit for a general public who really don't need to understand anything but they like to look at the pretty pictures. Do we really have to dive into the maths to go any further? Maybe so.
I'm going to go back into working my way through the paper. It could be that elaborations of CarlB's computer images will be the best way to study the E8 without having to do the maths. Best regards... S 



#67
Nov2107, 01:12 PM

P: 640

Going from a layperson to an amateur just means doing a little bit more digging to get a better understanding.
The first step is to dig in the citations. Below are three that could be interesting and informative. citations (# 1) http://arxiv.org/abs/0704.3091 Triacontagonal coordinates for the E(8) root system Authors: David A. Richter (Submitted on 24 Apr 2007) The next step is to find out what some of the words mean. http://mathworld.wolfram.com/CoxeterGraph.html The Coxeter graph is a nonhamiltonian cubic symmetric graph on 28 vertices and 42 edges. Three embeddings are illustrated above … http://mathworld.wolfram.com/CubicSymmetricGraph.html cubic symmetric graph  (# 2) http://arxiv.org/abs/0705.3978 Mapping the geometry of the F4 group Authors: Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, Antonio Scotti (Submitted on 28 May 2007 (v1), last revised 1 Oct 2007 (this version, v2)) The third step is to “click” on the authors and see the other work that they have published that might give you more information. http://arxiv.org/find/mathph,math/1.../0/1/0/all/0/1 Showing results 1 through 5 (of 5 total) for au:Cacciatori_S The fourth step do a search for some of the terms that are used in the papers http://www.gregegan.net/SCHILD/Spin/SpinNotes.html Greg Egan's Spin Networks Mathematical Details Or http://www.gregegan.net/APPLETS/30/30.html In the E8 lattice in 8 dimensions, each hypersphere makes contact with 240 nearest neighbours; 112 of these lie at the centres of the 112 6cubes of an 8cube centred on the representative hypersphere, while the other 128 lie on half the 256 vertices of another 8cube which is half the size of the first. The 240 neighbours lie on 120 rays which split up into 15 octads of orthogonal rays, which again determine the colouring scheme. http://jdc.math.uwo.ca/spinfoams/ Spin networks, spin foams and loop quantum gravity Dan Christensen  (3 #) http://xxx.lanl.gov/abs/grqc/9704009 Is ``the theory of everything'' merely the ultimate ensemble theory? Authors: Max Tegmark (Institute for Advanced Study, Princeton) (Submitted on 3 Apr 1997 (v1), last revised 1 Dec 1998 (this version, v2))  Depending on your level of layperson you will now be an amateur. As you progress in your understanding and learning, you will soon realize that “the more you know the more questions you will have”. ( Not to mention, that you will realize the huge amount of training and learning that is necessary to be able to do the work that you read in those papers.) Cheers for the “math kids”. Although life dealt you a different kind of hand, with the web, you can get a greater appreciation of science than any other previous generation. Grandma can knit and I can’t. (I envy her skills too) jal 



#68
Nov2107, 03:45 PM

P: 360

Hey guys, just checking in real quick. IMO, the best references to start with would cover the basics of Lie algebras and representations:
http://deferentialgeometry.org/paper...esentation.pdf http://deferentialgeometry.org/paper...20Algebras.pdf 



#70
Nov2207, 01:15 PM

P: 182

So for the short easy answer, how about something like "E8 is an extraordinarily beautiful geometric figure in higher dimensions, which projects onto our three dimensions of space and one of time in a way that fits the particles of the standard model and their interactions along with gravity into a single object. This model unites the standard model of particle physics with the theory of general relativity, and predicts new particles and interactions which are expected to be confirmed or denied by experiments at LHC Cern in the near future."
Have a great long weekend. 



#71
Nov2307, 10:41 AM

P: 640

From the thread, "An Exceptionally Simple Theory of Everything!"
“I am seeing this reference in several places that E8 is the lie group of an icosahedron (and for that matter E6 is same for a Tetrahedron and E7 is the same for an octahedron). This seems like a very interesting way to approach E8,” This makes it easier to work with LQG. There are link to LQG which I have gathered at http://www.physicsforums.com/blogs/j.../dynamics797/ For instance http://arxiv.org/abs/hepth/0608210 Loop Quantum Gravity: An Inside View T. Thiemann 29 Aug 2006  Wiki is getting better at explaining all of these symmetries. Here are some links. http://en.wikipedia.org/wiki/List_of...ymmetry_groups List of spherical symmetry groups http://en.wikipedia.org/wiki/Icosahedron The icosahedron can also be called a snub tetrahedron http://en.wikipedia.org/wiki/Tetrahedron tetrahedron http://en.wikipedia.org/wiki/Tetrahedral_symmetry Tetrahedral symmetry http://en.wikipedia.org/wiki/Polyhedral_compound polyhedral compound http://en.wikipedia.org/wiki/Stella_octangula stella octangula http://en.wikipedia.org/wiki/Snub_%28geometry%29 A snub is a related operation. It is an alternation applied to an omnitruncated regular polyhedron. http://en.wikipedia.org/wiki/Uniform..._of_operations A uniform polyhedron  jal 



#72
Nov2307, 12:06 PM

P: 662

I do have one objection, although I could well be wrong about this. When you talk about projecting the higherdimensional space in which E8 lives into 3 dimensions, I'm not sure it's right to identify those as the three space dimensions of our regular 4D world (Minkowski Space). When you project a higher dimensional space to a lower dimensional one, you implicitly assume that the lower dimensional space is embedded in the higher one, i.e. it's a subspace. For example, when I want to visualize a 4cube (something I do for fun), I first project it into a 3D subspace, and then to draw it on paper or a computer screen, I have to project that 3D object to a 2D subspace. The geometry I use to do this, however, always assumes that I can connect points on the higherdimensional object to points in the lower dimensional space with straight lines in the higherD space, i.e. the lowerdimensional space is a subspace of the higherdimensional one. Now, in the case of Dr. Lisi's E8 representation, we can certainly project the root space down to three dimensions in order to picture it at all, and then we can project the resulting object down to a 2D space to draw it on a computer screen or a paper page. These subspaces are just 3 and 2dimensional geometric spaces, however  in fact, we're really imagining them to be Euclidean spaces, not the curved space in which we live. More to the point, (and this really is my point, at long last), I don't believe it is correct to say that our 4D spacetime is a subspace of the higherdimensional root space of E8. That space is a space of quantum states  a Hilbert Space if you like  and "motion" in that space corresponds to changing states, not to movement in spacetime. Any thoughts on that? 


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