What is E8 Theory and How Does It Relate to String Theory?

[belliott4488]

belliott4488
You have not said one word that grandma could understand.
jal

Hey! You never met my grandma!

Well, dang ... I did try. I think I was shooting for what I thought was starkind's level of interest and understanding, which was possibly inconsiderate of SublimeGD, who was the original poster, after all.

Okay. I have no idea how to explain at a 'grandma' level what E8 means in this context. I might be able to start with the general idea of symmetries in Quantum Theory, but even that would be difficult for poor granny. I think it would require a lot of careful thought, but at the end I'd have no more to offer than is already available on the web, so perhaps I should take this as my cue to bow out as gracefully as I can ...

 

[william donnelly]

Part 1 of a basic introduction to E8 is now posted at http://sigfpe.blogspot.com/

So far it's just an introduction to the concepts of Lie group and Lie algebra, but it is written in a very accessible way.

 

[CarlB]

A Complete Idiot's Guide to E8

No one seems to have stepped up to the plate here, so let's have an amateur take a swing at the ball.

I'm going to talk about E8 as compared with the more familiar symmetry group SO(3) or SO(3,R). Wikipedia entries:
http://en.wikipedia.org/wiki/Rotation_group
http://en.wikipedia.org/wiki/E8_(mathematics)

The Symmetry Manifold and its Size

SO(3) is the group of rotations around the origin in 3 dimensions. When you rotate something, you get to choose an axis of rotation and how much to rotate around that axis. The axis of rotations choice is like picking a direction away from the origin. Let's count them.

You can move two perpendicular directions which gives you 2 dimensions. Or you can rotate around a spot and that gives you 1 more. Thus the SO(3) rotations are a 3 dimensional "manifold". Another way of saying the "3" is that if you begin with no rotations, there are basically three small movements you can make.

Get a globe and find Albuquerque, my home town. Think about what you can do to the globe, symmetry wise, relative to Albuquerque. You can move Albuquerque to the north/south, or to the east/west, or you can spin the globe on an axis around Albuquerque in a clockwise/counterclockwise direction. That is 3 dimensions and so SO(3) is a 3-manifold.

Our purpose is to talk about quantum numbers, but first let's talk about the dimensionality of the quantum numbers. That means "how many" quantum numbers each particle gets.

The Dimensionality of the Quantum Numbers

You get to have one quantum number for every motion you can do with your symmetry that is "independent" sort of. 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.

Operators In Quantum Mechanics

The subject we are applying this theory to is quantum mechanics and so should discuss it a little. In quantum mechanics, the quantum states are "eigenvectors of operators". What that means is that if you write the operator as a matrix A, the quantum states are vectors \psi that satisfy the equation:
A \psi = \lambda_A\psi where \lambda_A is the quantum number. This will be familiar to people who've studied even the most elementary quantum mechanics book.

[note]Author is a proponent of the density operator formalism. In that formalism, the above is much sexier, but to discuss it here would unduly confuse most readers. Accordingly, with effort, he will suppress the urge to preach to you sinners.[/note]

To fully characterize a quantum state, you first choose as many operators A, B, C, ... which commute, and then define the quantum states as being eigenvectors of ALL these commuting operators. The reason for doing this is that it is always possible (due to some math theorems), and it gives you a nice clean way to describe the quantum states, namely their eigenvectors for A, B, C, which we can write as a vector (\lambda_A,\lambda_B,\lambda_C,...).

With SO(3), only one operator can be chosen, so there is only one quantum number. SO(3) isn't used as much in QM as the very similar symmetry group SU(2). In SU(2), that quantum number is called "spin". With E8, you can pick out 8 commuting operators, so to define a quantum state, you have to define 8 quantum numbers. With SU(2), you only have one commuting operator (which is usually chosen to be "spin in the z direction") so an SU(2) state gets only one quantum number.

Now if you've been paying attention in your elementary particles classes, you know that to distinguish an electron from a neutrino requires more than just 1 quantum number. To get all the particles into one group requires a more complicated symmetry group than SU(2). What Garrett did was to fit the known elementary particles into E8 by carefully assigning their quantum numbers. And he did it in a way that somehow respects gravity in a way that I do not understand yet but certainly got the Perimeter Institute to applaud.

Those who have studied beginning quantum mechanics learned that spin comes in various "representations". In spin 1/2, the quantum number (spin) is either -1/2 or +1/2. The difference between these two quantum numbers is 1.

In spin 1, the spin is either -1, 0, or +1. The difference between consecutive spins is 1. In spin 3/2, the spin is either -3/2, -1/2, +1/2, or +3/2. The difference between each is again 1. This difference between quantum numbers is consistent, and this is true in general. Note that with spin 3/2, you could talk about a difference of 2 or 3 between spin values instead of 1, but that would be a waste of time because 2 and 3 are multiples of 1.

For E8, there are 8 quantum numbers, so in any representation of E8, the difference between two different quantum numbers has to be given by an 8-vector. Similar to the differences between the quantum numbers of 3/2, you can choose a set of differences between quantum numbers of E8 (which are therefore 8-vectors), that are sufficient to get you anywhere you want to go, and are minimal in that you couldn't get rid of one. These weights are the origin of those diagrams with the little circles connected by lines.

Getting back to the pretty applet, the 8-vectors each correspond to the quantum numbers of a particle (in a specific representation of E8).

Did that help? By the way, note that the "Complete Idiot" in the title of this post is me.

 

[jal]

Hi CarlB!
I like the changes that you made to you E8 java. I've been playing around with it.
I understand (most) of the wiki E8
http://en.wikipedia.org/wiki/E8_(mathematics)
and what you did to make your E8 java.
http://www.measurementalgebra.com/E8.html
There is a difference between a layman and an amateur. You are the prime example of amateur and grandma is the layman.
The interest in Garrett's work has been demonstrated in layman's blogs.
I do think that it is possible to explain things (without using BIG WORDS) so that the layman can understand.
That's why I wanted to use your java as a supporting visual and walk through it by observing the different symmetries and reducing them. ( You did read my previous post and my two simple questions?)
Everyone can read from wiki E8.
"A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.

The E8 root system is a rank 8 root system containing 240 root vectors spanning R8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. Each of the root vectors in E8 have equal length. It is convenient for many purposes to normalize them to have length √2."

An amateur could understand that but not a layman.
I've done enough babbling ... time for the pros to speak.

 

[Prologue]

Thanks william donnelly for posting that link, it is very helpful.

Actually thanks to everyone discussing this, I am very grateful. I love getting into the conceptual nature of all of these things but lack the math background (I am getting there though). I know we can all relate to knowing something very well and then it becomes hard to remember where the sticking points are that beginners might have, thanks for doing it though!

 

[jal]

william donnelly
Part 1 of a basic introduction to E8 is now posted at http://sigfpe.blogspot.com/
So far it's just an introduction to the concepts of Lie group and Lie algebra, but it is written in a very accessible way.

jal
I've done enough babbling ... time for the pros to speak.

When he gets to it he might use CarlB's java.
jal

 

[starkind]

Jal, I am also trying for a layman’s explanation. In fact, I first started this internet journey at superstringtheory.com, where I met self-Adjoint, both of us under other screen names. I followed him here to physicsforums.com, a little sneakily I’m afraid, as he was hesitant to pose one of my questions here, but he finally gave in and posted a link to PF from SST.

(As I recall, the question had to do with gravity as an acceleration, and if it would be consistent with known physics to consider gravity as an actual outward acceleration of matter from its center of mass. There were no objections to using this interpretation, and I have held it ever since. It has some implications for the idea that the universe is expanding, and the caveat that locally, particles, planets, and galaxies are not expanding. I would have to rephrase this caveat, suggesting that local matter is expanding, but at a slightly different rate, based on relativistic time distortions.)

SelfAdjoint was very kind to me, and to many other people who were trying to learn outside of the university atmosphere. He is fondly and respectfully remembered here at PF. Of course I can’t speak for him, and I don’t pretend to know what he was trying to do, but it seemed to me that he would have been very sympathetic to our trying to find a layman’s explanation of QG. In fact, he seemed a little regretful when I decided I had to learn the math, and not just continue to rely on non-mathematical language. Of course that is my perception, not a fact about sA.

So I, for one, would vote, if I had the status, for continued attempts to find a common-English description of what is happening in theoretical physics. But we have to be very patient, and careful. English language is a slippery thing. Definitions of words change, new words appear, old usages either persist, or disappear. I am not really sure of how to begin such a project. It seems to me it could be that Wiki would be a better format for this kind of discussion, because it allows many people to work together on a single page. It is possible to edit and add to each other’s work. There are side pages for discussion and history.

I will post this now, and go offline to compose some other things on the E8 ToE.

S

 

[starkind]

Hey! You never met my grandma!

Well, dang ... I did try. I think I was shooting for what I thought was starkind's level of interest and understanding, which was possibly inconsiderate of SublimeGD, who was the original poster, after all.

Okay. I have no idea how to explain at a 'grandma' level what E8 means in this context. I might be able to start with the general idea of symmetries in Quantum Theory, but even that would be difficult for poor granny. I think it would require a lot of careful thought, but at the end I'd have no more to offer than is already available on the web, so perhaps I should take this as my cue to bow out as gracefully as I can ...

Please don't go away until we find an appropriate place to continue this discussion. I have noticed in the past that the moderaters here (hi ZZ) are not shy, or even polite, about shutting down what they see as useless threads.

 

[ccdantas]

What about Baez's This Week Finds 253?

http://math.ucr.edu/home/baez/week253.html

 

[starkind]

What about Baez's This Week Finds 253?

http://math.ucr.edu/home/baez/week253.html

Would John Baez like to host a long discussion of these things in simple English on his blog?

I am thinking we should maybe move to a Wiki site.

 

[starkind]

CarlB wrote: “You can move two perpendicular directions which gives you 2 dimensions. Or you can rotate around a spot and that gives you 1 more. Thus the SO(3) rotations are a 3 dimensional "manifold". Another way of saying the "3" is that if you begin with no rotations, there are basically three small movements you can make.

"Get a globe and find Albuquerque, my home town. Think about what you can do to the globe, symmetry wise, relative to Albuquerque. You can move Albuquerque to the north/south, or to the east/west, or you can spin the globe on an axis around Albuquerque in a clockwise/counterclockwise direction. That is 3 dimensions and so SO(3) is a 3-manifold.

I rewrote the above like this:

We want to be able to talk in a general way about the rotational symmetries of any shape or kind of object, so it would be better to think of how to turn a sphere, like a globe of the earth. If we can talk easily about the symmetries of turning a globe, then we can easily talk about the symmetries of any shape of object, by imagining that it is totally encased in a ball of clay that turns it into a sphere. We can talk about turning the sphere easily, and then when we want to relate this to the object, we only have to take away the clay to see what it looks like.

So let's talk about turning a sphere in ordinary three space. Or, as physicists might say, consider the SO(3) group of symmetries.

If you are holding a simple sphere in your hand, like you would hold a small plastic ball, and you hold it by two fingertips on opposite sides of the ball, there are three basic ways you can turn it. (Really there are lots more ways to turn it, but any of the other ways can be described by turning it a little in one of the three basic ways, and then a little in another of the three basic ways, and repeating this over and over until you get the exact same movement.)

There are three basic ways to turn the ball in your fingers. Let's count them. First, you can turn your whole hand so that one of the two fingers that hold the ball is closest to you, and the other is on the farther side. The most you can turn in this fashion is when it looks like the top finger is right in the center of the part of the ball that is visible, and the other finger is on the other side. Take the ball away and hold your fingers like that, and you would see the top finger and the bottom finger are exactly in a line away from your eye, with the top finger closer.

A second way you can turn the ball is to turn your hand so that the two fingertips holding the ball are on opposite edges, from your point of view. Now they are both the same distance from your eye and you can see both of them on the edge of the ball at once. You could possibly do this so that one finger is closer to the top of the ball than the other, but that would make the two fingertips unequal…one closer to the top and one closer to the bottom. Instead, to keep our discussion as simple as possible, it would be better to turn it so that the two fingers are both equally far from the top and from the bottom. That is where the orthogonal part of the SO(3) comes from. In an orthogonal description, you will see that poles (where the fingers are touching the ball) turn from the top-bottom line, or axis, to a position in a line at right angles, ninety degrees from each other. We can say that these two lines of axis, top and bottom, and left and right, are perpendicular to each other. In fact, these are two of the basis lines we use to describe the rotations of objects in our ordinary three dimensions of space.

There is one more possible movement you can make, and that is no movement at all. So let's count: one, fingertips are at the top and bottom, two, fingertips are front and back, three, fingertips are on opposite edges. In mathematics, we call these three different axis lines x, y and z. X is generally up/down, y is generally side to side, and z is generally front to back. Mathematicians call this kind of description a manifold.

Another example could be a globe of the earth. Pick any spot on the earth, say your home town. By turning the globe, you can move your home town east or west, or by changing the N-S axis of the earth, you can move your home town north or south. Finally, you can turn the globe on an axis which runs right through your home town, right through the center of the globe, and out the other side. Using this axis instead of the North and South poles, you can turn your town around and around, with the rest of the globe spinning in circles around it. So there are three ways you can move your town: up or down, left or right, or you can spin it round and round, clockwise or counterclockwise. These are the three rotations of the SO(3) group.

 

[starkind]

CC:

I see John Baez is going to go on vacation! And then to Loops'07 in Mexico. I wish I could go to Loops'07, but I spent my fun money at Waterloo week before last.

CarlB:

I finally got to see the E8 rotation video on Youtube. It is certainly beautiful. I wish I could see your applet but I am one of the microsoft sheep you mentioned in the source document.

As I watched the Youtube vid, I wondered what the thing would look like if you traced all the orbits of all the particles? Or if you looked only at the orbits of the subgroups, like only the gluons, or only the quarks, or only the fermions, or only one generation of the fermions, and so on like that.

 

[william donnelly]

CC:

I see John Baez is going to go on vacation! And then to Loops'07 in Mexico. I wish I could go to Loops'07, but I spent my fun money at Waterloo week before last.

Actually Loops 07 was back in June (TWF 253 is from June). But you can get some of the audio and slides online.

Also John didn't go to Loops, if you read carefully you will see he is talking about Garrett going to Loops.

 

[starkind]

CarlB wrote:
“Our purpose is to talk about quantum numbers, but first let's talk about the dimensionality of the quantum numbers. That means "how many" quantum numbers each particle gets.”

In the above, we have talked about the meaning of the SO(3) group in ordinary English. Our goal here is to describe the E8 group in ordinary English. It turns out that the SO(3) group is one part of the E8 group. The E in E8 stands for Extraordinary, and it certainly is.

Before we look at the rest of the parts of E8, let's talk about quantum numbers. Quantum numbers are simply the ways physicists describe the behaviors of sub-atomic particles, like electrons, protons, neutrons, neutrinos, quarks, and many others. Each kind of particle has its own quantum numbers, and there are rules to tell us how different particles stick together or fall apart.

We have seen that the SO(3) group of symmetries describes objects in ordinary space. We will go on to talk about other symmetry groups, which describe how objects behave in other sets of dimensions. But first, let's talk about the dimensionality of the quantum numbers.

When we talk about the dimensionality of ordinary space, as we have seen, we can describe any object using only three numbers. Each number is a measurement of the object’s shape in three dimensions. This means that every point on the surface and interior of the object can be given a name made up of its position related to x, its position related to y, and its position related to z. Every position on the object has its own set of numbers, and no other position has the same numbers. For example, the center of the sphere can be called the origin, and given the numbers (x,y,z)=(0,0,0) The “north pole” of the sphere can be given the numbers (0,1,0). You see that the north pole is one distance unit from the center along the y axis. The south pole would be called (0,-1,0). I hope you can figure out for yourself what the other points we have discussed would be called. If not, we can go further into it to see how it works.

When physicists talk about quantum numbers, they are talking about measurements of the charge, mass, spin, and color of an object. The objects are too small to be seen, but they have these properties which can be measured. Each kind of particle has its own set of quantum numbers. You will notice that I have not mentioned position in ordinary 3 dimensional space here. Nor have I mentioned time.

It turns out that we can think of the quantum numbers as if they were set up as points on an object in ordinary 3 dimensional space, except there can be more than three numbers involved. How do you imagine a space like our 3 dimensional space, if there are more than three numbers?

It is kind of hard to think about, but we do have some familiarity in the ordinary visible world with a fourth number that describes ordinary objects, and that number is time. Using this fourth number, we can talk not only about the object as a static shape, but also about how it moves and changes. We are all familiar with objects that move and change. In classical physics, we talk about this in dynamics, which is exactly the study of how objects move and change in three dimensions of space and one dimension of time.

In dynamics, numbers like x,y,and z are used, but it quickly becomes clumsy to describe motion and change using just those numbers. So we add the fourth number, t for time, and then we talk about momentum, energy, force, and other things about the object which we can measure in four dimensions. In fact, physicists talk about “momentum space,” which is not really the space we think of when we are playing with a ball or entering or leaving a room. Momentum space has to do with the mass of an object, and its velocity, which is to say how far it moves in how much time. Momentum is just mass multiplied by velocity, and momentum space has numbers which describe a body in terms of mass, change in time, and change in position. Often it is possible to talk about change in position on just one of the three dimensions of ordinary space. We don’t even have to talk about which direction the object moved. We can simply say that it had x grams of mass, it moved y units of distance, and it moved that far in z units of time.

Now you see that we have changed our basis units. Instead of up down, we have distance, forward or back. Instead of right left, we have units of time. Instead of z, we have units of mass. Of course if we are talking about simple motion of a single object, the mass isn’t likely to change much, so we put mass on the z axis where we can’t see changes very well anyway.

Then we can make a picture of how the object moved in space and time, as we do when we want to consider the acceleration of an object, say, as it falls off a table. All of this is to help you see that dimensions do not have to be spatial, or even temporal. We can talk about mass as a dimension, and even draw it as an axis on a graph, just as if it were an ordinary distance measurment. When physicists talk about momentum space, or phase space, or quantum space, they are inventing an imaginary space, in some ways like our three dimensional space, and in some ways different. The important thing is, “objects” in “phase space” are shaped, and move and change, with the same mathematical rules that we use to describe objects in our ordinary three dimensional space.

Clearly phase space is not a room you can walk into or out of. I think it might have been better if people had chosen some other word, because it would have avoided a lot of confusion. There is no ‘creature from dimension x,’ because dimension x, if x is greater than three, is not likely to be a place you can pop in and out of. But it was done that way and I guess we are stuck with it. So we have to teach people who are starting out that the tenth dimension is not “another world” where monsters might live. Actually, the things that live in the tenth dimension are not monsters, but are quite well behaved, and they obey the rules and generally do things that would not be likely to upset grandma’s weekly church tea.

When physicists talk about the dimensionality of quantum numbers, they are just talking about how many quantum numbers are needed to talk about what a quantum object does as it goes about its daily business. To do this, they need to know the rules of behavior in the different dimensional sets, and that is one of the uses of the Lie symmetry groups, like S0(3). The rules of behavior in SO(3) are called the so(3) algebra.

 

[starkind]

Actually Loops 07 was back in June (TWF 253 is from June). But you can get some of the audio and slides online.

Also John didn't go to Loops, if you read carefully you will see he is talking about Garrett going to Loops.

Thanks, william donnally! I guess I knew that but was interested in getting ahead of myself and didn't pay attention to what I was doing. Who in academia would be going on vacation in November? I would like to go to Mexico, where Daniel Sudarski teaches at ICN-UNAM, if my notes from Waterloo are right. He gave an interesting talk called "The Quantum Origin of the Cosmic Structure: an arena for quantum gravity phenomenology." In it he talks about his paper gr-qc/0508100.cqg, 2317 (2006), written in collaberation with A. Perez (of Utrect) and H. Sahlmann (of Penn. State and Utrecht.) His talk was about the shortcomings of standard lore (in physics), the need for an extra element (tied to QG) and then the tests and predicitions which we may expect to see in the comming years. I didn't take many notes because I was trying to listen very closely to what he said. Anyway I imagine the talk is available at Perimeter, online.

 

[jal]

starkind !
Now you are on a roll.
Next thing we will hear is the the wave dudes explaining their SO(3) in the pipe. hehehe
Don't ask me ... ask Garrett ... he's the surfer. heheh
jal

 

[belliott4488]

... The E in E8 stands for Extraordinary, and it certainly is.

Heh ... Actually, the "E" stands for "Exceptional", but you're right - it certainly is extraordinary!

(That's half of Lisi's pun in the title of his paper, BTW.)

 

[garrett]

Hey all, sorry I haven't checked in -- been kind of busy. I appreciate the effort to produce a good description for the interested public. You're welcome to pull whatever you like from my descriptions at FQXi:
http://fqxi.org/community/index.php
Here are my descriptions of the theory:
http://fqxi.org/community/forum.php?action=topic&id=107
And here's some personal background for the curious:
http://fqxi.org/community/forum.php?action=topic&id=108
Please feel free to extend, change, or incorporate these into your descriptions however you like.

Garrett

 

[CarlB]

Hi y'all. Plane rides are the best time to think. I just got off one and realized that the next thing to add to that applet is the ability to choose views from a pulldown menu, and the ability to upload and download stuff you've typed into it.

You mess around with it, eventually it does something you like. You click a button and a "text area" appears. The text area contains all the parameters you used, color, and view. You click ^a ^c ^v and you copy the information to your own data file to save it. (I can't save data files on an internet Java applet without a lot of significant pain. Probably because I'm just an amateur java programmer.)

If you later want to see that again, you click the same button. The text area appears, and you write over the information there with the new data. It then displays this instead.

From there, you can email me the information here on Physics Forums and I can add that view to the pull down menu so that others can use it without having to mess with the textArea. Or you can send the data file to your collaborators directly and they can upload it to the applet the same way.

Also as a result of the plane ride, I've thought more carefully about how E8 arises naturally in a composite model:
http://carlbrannen.wordpress.com/2007/11/20/broken-e8-as-a-result-of-composite-particles-i/

 

[TheRealColbert]

Bridging the gap

Thanks guys! That stuff is gold! A couple more questions to help me bridge the gap, if that is ok... and please correct any misconceptions in the following.

I see that you need to create a multi-dimensional space to describe all the degrees of freedom a system can have, and Garrett used E8 as the substrate for his "world". Now I try to relate this with something I am familiar with, say mechanics in 3d space. There is the substrate, an XYZ coordinate grid, and some scalar values such as mass. With this you can make a differential equation that you solve to describe the state of the system relative to another variable, such as time. In quantum mechanics, you solve the differential equation (Schrodinger's), to get a wave solution, which you turn into a probability function to tell you stuff about the system.

So does Garrett's theory, in an analogous fashion, provide a differential equation in a wacky multi-space which contains all possible variables (quantum numbers) which you can solve (using whatever relevant boundary conditions) to describe any possible configurations of any possible "stuff" in the universe? Thus a theory of everything?

Or is there another layer of mathematics I am oblivious to? Is the group theory stuff used as a "shortcut" to find solutions? I didn't read Garrett's links yet, the answer might be there...

 

[jal]

Imagine that you are looking through a microscope into a crystal ball. (R4)
You are observing the Standard Model particles (strong su(3), electroweak su(2) x u(1), gravitational so(3,1) doing their dance. You are drawing the patterns on piece of paper. When you look up and look at the paper, it hits you … E8 can represent the pattern that you have drawn. More than that, … You think that the dance could be interpreted as geometric patterns. Lot’s of work left to do.
From Garret’s paper
http://arxiv.org/pdf/0711.0770
p. 6
The G2 root system may also be described in three dimensions as the 12 midpoints of
the edges of a cube - the vertices of a cuboctahedron. These roots are labeled g and qIII in Table 2, with their (x; y; z) coordinates shown. These points may be rotated and scaled, ….
----------
Reference for refreshing http://en.wikipedia.org/wiki/Cuboctahedron
“Both triangular bicupolae are important in sphere packing. The distance from the solid's centre to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's centre.”
http://mathworld.wolfram.com/Tetrahedron.html
The vertices of a tetrahedron of side length can also be given by a particularly simple form when the vertices are taken as corners of a cube (Gardner 1984, pp. 192-194). One such tetrahedron for a cube of side length 1 gives the tetrahedron of side length having vertices (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)
------------
Now, …. I also have a crystal ball (R4) that I have been examining with a microscope. It does not have any Standard Model particles in it. Go read my blog and you will see what I saw.
jal

 

[TheRealColbert]

You are drawing the patterns on piece of paper

Abstract patterns like Feynman diagrams or observable characteristics like trails through a bubble chamber?

I am not yet up-to-speed on how to think about "strong su(3), electroweak su(2) x u(1), gravitational so(3,1)", but the previous posts are helpful. My apologies if the level is too "low" for the thread. I know people get tired of the same old questions, but sometimes it is nice to "talk" to someone. Thanks for your patience!

 

[TheRealColbert]

From Garrett's FAQ,

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.

Now I am confused again. Is this an abstract space, describing how quantum numbers (observable properties of stuff) change and interact with each other, or is it a literal description of the "real" world, where things like curvature relate to actual "stuff" out there that could be demonstrated with an appropriate experiment? Or is it a silly question, because regardless of how you model "reality", if you model it completely and correctly your model will give the same result whether in your "math" world or the "real" world, thus they would be indistinguishable? In other words, there would be no experiment you could do in one world that you couldn't do in the other, (hypothetically, assuming magical experimentation abilities)

 

[jal]

TheRealColbert
I would suggest that you use wiki and refresh and re learn all about geometry and symmetry so that you can reorganize the information that you have in your head. You will certainly learn new info which will help you understand how Garrett has put all of his information together.
The only person who can put an image in your head is yourself.
Until you review what you think you know, there is no way for you to know if ?
... garbage in ... garbage out ...
jal

 

[starkind]

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 space-time 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.

 

[starkind]

Heh ... Actually, the "E" stands for "Exceptional", but you're right - it certainly is extraordinary!

(That's half of Lisi's pun in the title of his paper, BTW.)

Thanks for keeping me straight. I'm proud to be accepted in this company.

 

[starkind]

From Garrett's FAQ,



Now I am confused again. Is this an abstract space, describing how quantum numbers (observable properties of stuff) change and interact with each other, or is it a literal description of the "real" world, where things like curvature relate to actual "stuff" out there that could be demonstrated with an appropriate experiment? Or is it a silly question, because regardless of how you model "reality", if you model it completely and correctly your model will give the same result whether in your "math" world or the "real" world, thus they would be indistinguishable? In other words, there would be no experiment you could do in one world that you couldn't do in the other, (hypothetically, assuming magical experimentation abilities)

Some math oriented people seem to take the stand that our "ordinary" 3space1time dimension view of "reality" is no more or less real than the mathematical higher dimensional worlds which are invisible to most of us. Invisible? Unimaginable, maybe.

Still, I think there is something about our perceived 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 category 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.

 

[TheRealColbert]

I would suggest that you use wiki ...

Your post implies that I am highly confused I thought I was approaching a verbal understanding, but in reality I am either highly confused or have done a poor job communicating my "gaps". I don't want to wreck a good thread, so I will happily shut-up if I am taking the thread down the wrong path. (and I will take your advice on wiki).

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 off-track 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.

 

[starkind]

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

 

[TheRealColbert]

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 self-deprecating 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!