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

[SublimeGD]

I don't have much of a math/physics background, undergraduate physics and calculus is where I stopped... yet I find theoretical physics extremely interesting. The recent online hype about the "E8 theory" only really discusses the fact that Lisi likes to surf and doesn't wear a pocket protector. On these forums, your discussions of the theory are way over my head. So, can someone please provide a layman's explanation of E8 theory?

Also... Are the E8 theory and the string-type theories mutually exclusive?

Thanks

 

[jal]

Hi! I don't know how close to the beginning that you need to start.
A good place to start would be to read the definitions in "wiki" of symmetry.
http://en.wikipedia.org/wiki/Symmetry
jal

 

[Mephisto]

i too would like to hear some easier explanation... from what i understand, E8 is this shape in 8D and they project it to 3D, but I'm not sure why... I think they derive something from that projection... And that every vertex basically is an elementary particle, and the interactions between particles are governed by curvatures joining the particles...? Whatever that means...
so really i got nothing :) Plus the above could all be wrong.
It's kinda frustrating considering that I am now taking my second course in Quantum Mechanics, and I am third year in Physics, but yet I can't gather much more than I would even if i went to Sociology or something.

oh and from what i gather it does not go hand in hand with String theory, it is more like competition to it. Plus it doesn't require 11 dimensions etc.

 

[jal]

Yah!
You got it!
I have been trying to get CarlB's E8 to work for me but I cannot get the changes in color.
CarlB could help explain the E8 pattern better than me. ...
There are two things that need to be understood ... the E8 pattern and the Standard model.
Garrett put the two patterns together.
I'm at the stage of still trying to understand the finer points of symmetry.
Most of the discussions were on trying to understand how/why the patterns fit together or if the "rules" were being violated.
jal

 

[arivero]

Also... Are the E8 theory and the string-type theories mutually exclusive?

Hmm, it is an interesting question. String theory has E8xE8, so you could look for an inclusion E8 ---> E8xE8 at the level of representations. It could exist.

 

[starkind]

I am trying to study Lisi's paper using Wiki as a guide. Right now I am getting an idea of what Lie (pronounced 'Lee) groups is about.

My current understanding of E8 is that it is a mathematical object in 234 (IIRC) dimensions. The idea of projecting the object onto a lower dimensional surface is kind of like taking a wire model of a cube, holding it above a sheet of paper in the sun, and tracing the shadows. There are several ways you can turn the wire frame cube to get different patterns on the paper. These are symmetries, all of which are required to get an idea of what the higher dimensional object (in this shadow case, the wire frame cube) actually "looks" like.

Physics isn't about visualization any more. I doubt if humans will ever be able to visualize in much higher than three or four dimensions. Instead, we have to learn the maths. This is hard to do if you are not in a university where such things are taught, but if you are persistant, you can learn a bit on your own. At first, wandering around Wiki is like a maze, but after a while you start to recognise a few things.

My question is about E8 as an object already explored in string theory. Isn't E8 one of the 5 mountains in the landscape? One of the ones supposed to be unified by M theory? I think I recall reading long ago about E8 X E8 as a string object, but I knew less about Lie Groups then than I do now, and Wiki has come a long way in adding onto the maze.

I suppose it is not right to discuss these low level topics here. Just thought some of the other amatures would be interested. Maybe we should find a room of our own to discuss these interesting things further?

S

 

[jal]

There are different levels of Layman.
SublimeGD, you could get more background info by looking up 2d packing, 3d packing, kissing numbers etc. All of these things relate to symmetry. You will eventually end up seeing E8 and Gerrett on wiki. ( Yah, somebody addded Garrett's name to the E8 entry.
The E8 java by CarlB has incorporated a few rules that are not obvious to the layman.
Look at the 3d of the E8 model and you will see that there is a rule being applied to the length and the connections. It is assumed that you know them or that it is obvious. Those who want to understand symmetry have the web to help.
My understanding is that Garrett has taken the "ideal" E8 root pattern and the "ideal" Standard Model pattern and found a match.
The E8 has the "curvature" due to using same "length rule" for the connections.
When the four forces of the SM will be put in then the shape will be affected.

Jump in ... and give your "simple explanations" and guiding hints so that a layman can try to learn about what is being done by Garrett.

 

[CarlB]

I have been trying to get CarlB's E8 to work for me but I cannot get the changes in color.

I'm going to change it so that you pick a color, and then pick a place to put it, where "place" means one of the G4xF2 root subgroups, or the background. That should be more intuitive. And it will show the color scheme while you're doing it.

Java is a nice programming language, very tightly cast so that it doesn't let me do [as many] stupid errors. It should take 2~3 hours to make the color selection more intuitive. Give me about 2 hours, first I want to surf the net and see what is going on. I should have new code running this evening.

[edit] It's now 4.5 hours later, and I've got the new user interface acceptable. The next step is to do the easy and fun part (the math). But first, a couple hour break.[/edit]

 

[kneemo]

Hmm, it is an interesting question. String theory has E8xE8, so you could look for an inclusion E8 ---> E8xE8 at the level of representations. It could exist.

Heterotic string theory makes use of the compact form of E8 to avoid negative norm states, as Lubos http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html" ).

 

[SublimeGD]

There are different levels of Layman.

from http://chronicle.com/jobs/v45/i47/4547ctlyst.htm

Richard Feynman, the late Nobel Laureate in physics, was once asked by a Caltech faculty member to explain why spin one-half particles obey Fermi Dirac statistics. Rising to the challenge, he said, "I'll prepare a freshman lecture on it." But a few days later he told the faculty member, "You know, I couldn't do it. I couldn't reduce it to the freshman level. That means we really don't understand it."

What I'm seeking is a freshman level understanding of what is going on. I think I understand the concept of symmetry as applied to physics. I will look a little into the mathematics of it, thanks for the link. But again my goal is to just get the gist of it.

 

[starkind]

The basic question has to do with the nature of matter, which is of course made up of atoms and molecules, as every freshman probably knows. Your first course in chemistry will probably teach you about how atoms are held together into molecules by electromagnetic forces, chiefly between the outermost electrons of the atom. You will also learn that the center of the atom is the nucleus, which in turn is made up of protons, neutrons, and some other smaller particles which appear only when the nucleus is broken into smaller bits by a fission reaction.

These smaller particles come in hundreds of varieties, but the standard model of particles has shown that all of them can be explained by adding together the properties of only a few. The key particles are the neutrino, the electron, the muon, the tau, and six quarks, which are called top, bottom, up, down, strange and charm. Almost all ordinary matter is made up of the electron and the up and down quarks.

Each of these particles occurs as a triplet. For example, the electron, muon, and tau share many properties, differing mainly in mass. The neutrino comes in three kinds also, called the electron neutrino, the muon neutrino, and the tau neutrino. The quark triplets involve three colors. Each kind of quark can come in red, blue, or green. These are not colors of light, of course, but just a pretty way of doing some accounting.

In addition to the above, each member of every triplet has an antimatter dual. These particles together make up the fermions, particles which ordinarily tend to get as far away from each other as possible. The bosons are a kind of particle which obey another kind of behavior, in which they tend to cluster together.

There are also “particles” which are thought of as carrying forces. The gluon, the photon, and the Higgs are three of these. However, in my opinion, none of these particles is really a particle in the sense we usually think of matter. They do not have mass in and of themselves, but carry the four forces; electromagnetics for the photons, the weak force for the gluons, the strong force and gravity for the Higgs. All of the particles can be thought of as waveforms in some kind of background.

The particles of the standard model have been observed in colliders, and we know of photons directly from light, but the graviton and the Higgs have not been observed, presumably because they require higher energy collisions to become observable. Recall that higher energy collisions happen in smaller spaces. You can think of the Higgs and the graviton as being very small, therefore very high energy particles. Some scientists are hoping that the Higgs and/or the graviton, or maybe even a black hole, will appear in the new generations of colliders, which should be coming on line in the next few years, and which are able to reach energies in the range of one TeV, a tevatron. I think that means a billion electron volts.

All of this is background for the next stage in physics, which is now called, euphemistically, new physics. This forum, Beyond the Standard Model, interests people who want to know why the standard model particles have the mass, charge, and spin, or quantum numbers, that they do, according to measurable physics. String theory can explain the quantum numbers, but it has five different explanations, and it is thought by many that there must be some more basic theory, with only one explanation. M theory has supposedly connected the five stringy theories into one explanation, but no one seems to know what that explanation is. In any case, it leaves unanswered the fundamental question, what is the zero state, the absolute vacuum, the space-time continuum. If there are waves, what is the stuff that is waving? What is it, when it isn’t waving?

Loop quantum gravity, Dynamic Triangulation, and other ideas have been put forward as a means to explore the fundamental question. It largely comes down to a question of geometry. What is the right geometry, the right mathematics, to describe the most fundamental level that underlies all of matter? The fact that the standard model pieces can be hung on the E8 framework is another proposal for a means of investigation of this question. Essentially, it postulates that the particles of the standard model, along with the graviton and the Higgs, must be an emergent effect of the shape of the universe, which exists in higher dimensions than we poor limited humans can perceive.

However, even if we find the answer to the geometry of space-time, there remains the question of what is more fundamental than that? What lies beneath space-time? If the particles can be thought of as being different views of a higher dimensional object, as in the Lisi theory, what is the stuff that causes that higher dimensional object to take the shape it does, and not some other shape?

The stakes are very high. Human culture has entered a cul-de-sac, and we must have some better source of energy than oil if we are to survive as a technological civilization. Atomic energy has given us a clue, but it has some problems, mainly involving the deadly poisonous leftovers of fission. A workable theory of everything may be the key to finding ways to harness energies like the strong force that holds quarks together inside particles, or even the pure energies of mass, and hence gravity, the actual curvature of space-time.

Dr. Lisi’s model may be the best approximation yet to the structure of space-time. Or not. It has the advantage that it can be verified by tests that may be within reach of current technology. Or it can be falsified by those same tests. String, Loop, and Triangulation have suggested no such tests, or at best only a few tests that are not very clear. The Lisi model predicts a few new particles which may soon be within the reach of our technical tools. If these new particles are found, and have the predicted quantum numbers, then the theory will be useful in finding the unification of general relativity with the standard model of particle physics. Lots to look forward to.

Hope this helps. Comments welcome. I am only an independent student, and my understanding is not complete. If anyone here finds I have made a misstatement, I would be very kindly disposed to hear of it.

S

 

[SublimeGD]

The fact that the standard model pieces can be hung on the E8 framework is another proposal for a means of investigation of this question. Essentially, it postulates that the particles of the standard model, along with the graviton and the Higgs, must be an emergent effect of the shape of the universe, which exists in higher dimensions than we poor limited humans can perceive...

...If the particles can be thought of as being different views of a higher dimensional object, as in the Lisi theory...

...The Lisi model predicts a few new particles which may soon be within the reach of our technical tools. If these new particles are found, and have the predicted quantum numbers, then the theory will be useful in finding the unification of general relativity with the standard model of particle physics...

S

Thanks! I was hoping that Feynman quote would motivate someone to give the time of day to a lowly undergraduate like me. I didn't mean to give the impression that I am entirely unversed in physics... although that is probably true from the mathematical sense (undergraduate introductory physics aside). I am aware of the standard model, M-theory, and all that jazz, from reading books by Michio Kaku and Brian Greene (biased string theorists?...) But nonetheless I am at least a little bit aware of what is going on in the physics community.

Can someone please expand, if possible, on the above explanations of "E8 theory." I gather that the standard model "particles" are arranged on the E8 framework, which lies in higher dimensions. By rotating the E8, different arrangements of the particles become "visible" in our 3 dimensional space. Depending on the rotation, the different fundamental forces can be seen. Is this a grossly misled interpretation of the Lisi paper? I've read that the "E8 theory" does not predict higher dimensions... yet doesn't the E8 shape require higher dimensions? Or is the arrangement of the standard model particles not on the higher dimensional E8 shape itself, but on a 3D projection of the E8 shape?? Could rotation through higher dimensions account for quantum uncertainty and weirdness? (particles seemingly popping in and out of existence?)

Thanks for all and any replies

Edit: Okay, I see now that the elementary particles aren't really "placed" around the E8 shape, but are as one of the earlier posters put it.. "every vertex basically is an elementary particle, and the interactions between particles are governed by curvatures joining the particles"

 

[starkind]

The standard model uses the observable 3 dimensions of space and one of time. The Lisi model does not require extra dimensions of space and time. The dimensions of the E8 system are mathematical, not physical dimensions.

The idea that particles pop in and out of physical space from other dimensions is not really it. One version of QG says we live on a higher dimensional brane (a topological surface) and that some particles, specifically gravitons, move through our brane world, becoming briefly part of it, and then they return to a region called the bulk. The bulk is not space as we know it. This idea is invoked to explain why gravity is so much weaker than the other three forces. Some of the force of the graviton leaks into our world, but most of it goes into the bulk.

String theory uses extremely small Calabi-Yau dimensions, curled up approximately at the Planck scale, to explain this same weakness of gravitation, suggesting that the extra gravitational force goes into the small curled up dimensions. Some other theorists have suggested that there are large extra dimensions, hoping to show an effect on gravitation at small distances, for example less than a millimeter. No such effect has been demonstrated as far as I know.

I don't claim to understand the details of Lisi's idea. But I think I have gotten a pretty good grasp of the dimensions thing. There is commonly a confusion about "higher dimensions" being some kind of 'space' that we just don't see. Physicists don't help clarify this by talking about things like phase space, representational space, momentum space, and so on. These are clearly (I think) not spaces in the sense of a room which you can occupy or not occupy as you so choose. Really a mathematical dimension is any measurement. You can have any number of measurements, or dimensions, but usually physicists like to have as few as is necessary to describe an event. You can describe the shape of any static physical object with three dimensions, the shape of any dynamic object with four dimensions, or maybe five. The one or two additional dimensions are those of time.

But what if you need to describe a system that has charge, mass, spin, and so on, which are not really shapes at all? You need more mathematical dimensions to do that. Lisi's model does describe the quantum numbers, so it needs more dimensions, mathematic dimensions, to do that.

This coffee house and internet hot spot is closing, but I will look for corrections tomorrow.

S

 

[belliott4488]

starkind: Yes, you did misspeak on a few points. For example, gluons are the carriers of the strong interaction, not the weak, which is carried by the W+,W-, and Z particles, which are massive. The Higgs is not a gauge boson, and as such is not a carrier of a gauge interaction. It does couple to other particles and gives rise to their masses by virtue of this interaction, but that's not the same as gravity (and has nothing to do with the strong interaction). Gravitons are indeed the carriers of gravity in quantum field theories of gravity, but they're not part of the Standard Model. There are a few other places where I'd question your description of the SM and related matters, but I'll leave it at that.

SublimeGD: I think a key point to understand here is why people talk about symmetry groups, and Lie groups in particular, at all in particle theory. E8 is obviously getting a lot of attention, but I think it would be far easier for you to start with the SU(2) X U(1) representation of the Electroweak interaction. If you can get to the point where you're comfortable with how that group represents the gauge symmetries of that interaction, then move on to SU(3) and the strong interaction. It's more complicated, but is even more satisfying (to my eye), once you get how the charge carrying particles (quarks) and the gauge particles (gluons) are described by the gauge symmetry group. There are many descriptions of all this out there (Google is your friend), so just dive in. Once you're comfortable with those ideas, then the extension to E8 as the ultimate symmetry group might be a bit easier.

 

[Chronos]

In mathematics, a dimension is merely the freedom to occupy different states. Spacetime is a construct of human perception. Only 4 quantities are necessary to fix the position of an entity in spacetime [with respect to any given observer]. This does not, as starkind noted, limit the number of additional states an entity may occupy: such as color, charge, mass, spin, etc. These cannot be reduced to units of position, hence are just as fundamental [dimensional] as spacetime coordinates in mathematical terms. In other words, what starkind said.

 

[Mephisto]

starkind even though you possibly made a few small errors i greatly appreciate your long post; and thanks to belliott for clarifying. I think however that the point of "E8 Layman's explanation" is that you don't need to dwell into details of symmetry groups to understand it. But from what i gather from the above posts there may not even be such an explanation

 

[SublimeGD]

I think however that the point of "E8 Layman's explanation" is that you don't need to dwell into details of symmetry groups to understand it. But from what i gather from the above posts there may not even be such an explanation

Exactly. Its like if I asked in an optometry forum for a layman's explanation of how the eye worked, then got redirected to wiki articles on cell biology and biochemistry. Maybe your right, and there just is no possible layman's explanation for this type of topic.

In regards to my question about "quantum weirdness," I retract the question entirely. It was unwise of me to extrapolate about the reasoning behind something I don't understand, from something that I don't understand.

To make something clear, I'm using "E8 theory" to mean what is proposed in the Lisi paper, not the E8 mathematical structure itself. I think in one of the other threads I saw Lisi express that he didn't want it called the "Lisi theory" or anything like that.

Also in regards to the dimensions question... I understand that dimensions in mathematics are not the same thing as spatial dimensions. But starkind, you say...

It largely comes down to a question of geometry. What is the right geometry, the right mathematics, to describe the most fundamental level that underlies all of matter? The fact that the standard model pieces can be hung on the E8 framework is another proposal for a means of investigation of this question. Essentially, it postulates that the particles of the standard model, along with the graviton and the Higgs, must be an emergent effect of the shape of the universe, which exists in higher dimensions than we poor limited humans can perceive.

So you state that elementary particles are an emergent effect of the shape of the universe, which exists in higher dimensions. Then you state that the "E8 theory" does not require more dimensions? I'm confused as to what is going on here...This quote is directly from Lisi in the thread about his paper...

"E8 lives in 8D. Sure, it can be projected down to 3D. It then needs to be projected to 2D to be shown on a screen or paper. If we used holograms instead of screens, I'd be making tons of 3D plots. But, as it is, I just project from 8D to 2D, because if I go from 8D to 3D to 2D, the perspective would make a mess of things."

So is the shape of the universe "8D"? Or does the shape of the universe have nothing to do with the Lisi paper?

I also have a question about the various animations, the one on youtube, and the java applet from CarlB... How is the E8 shape being rotated in these animations? Does the rotation require "higher dimensions"? It doesn't seem like its just a bunch of points of a 3D structure being rotated about a center axis... or is that precisely what the animation is showing?

Chronos, what do you mean space-time is a human construct? I thought general relativity pretty much showed that space-time is actually a "something" that can be bent and whatnot?

Thanks again for everyones contributions.

 

[belliott4488]

Exactly. Its like if I asked in an optometry forum for a layman's explanation of how the eye worked, then got redirected to wiki articles on cell biology and biochemistry. Maybe your right, and there just is no possible layman's explanation for this type of topic.

Sorry, but I don't think this is really right. What I believe you're looking for is a layman's explanation of how E8 - and specifically its dimensionality - works to describe fundamental particles. I think any such explanation would be a special case of an answer to the more general question, "How does any Lie Group work to describe fundamental particles?" There are many, many such layman's explanations of the latter, so I was suggesting that you start with those. Most of them are much clearer than anything I could come up with.

Also in regards to the dimensions question... I understand that dimensions in mathematics are not the same thing as spatial dimensions. But starkind, you say ... that elementary particles are an emergent effect of the shape of the universe, which exists in higher dimensions. Then you state that the "E8 theory" does not require more dimensions? I'm confused as to what is going on here...This quote is directly from Lisi in the thread about his paper...

"E8 lives in 8D. Sure, it can be projected down to 3D. It then needs to be projected to 2D to be shown on a screen or paper. If we used holograms instead of screens, I'd be making tons of 3D plots. But, as it is, I just project from 8D to 2D, because if I go from 8D to 3D to 2D, the perspective would make a mess of things."

So is the shape of the universe "8D"? Or does the shape of the universe have nothing to do with the Lisi paper?

I also have a question about the various animations, the one on youtube, and the java applet from CarlB... How is the E8 shape being rotated in these animations? Does the rotation require "higher dimensions"? It doesn't seem like its just a bunch of points of a 3D structure being rotated about a center axis... or is that precisely what the animation is showing?

I think the confusing thing here is the difference between the dimensionality of the group itself and that of spacetime. E8 (as well as other Lie groups) describes the symmetries of a geometrical space, so we can try to picture those symmetries by thinking about the higher-dimensional spaces in which those symmetries exist, which requires the projections Lisi describes in your quotation. No one is suggesting that any of those dimensions correspond to spacetime dimensions, as you've noted yourself. Well, actually, I think starkind is suggesting that the dimensions of the space in which E8 operates are the physical dimensions of the universe, and I think I'd object to that. Perhaps I've misunderstood that point, however - I didn't really follow his explanation, I'm afraid.

 

[TheRealColbert]

How does any Lie Group work to describe fundamental particles?

YES! That is my question. I am having trouble making the connection between my undergrad physics and group theory. I am a global learner to the extreme, so I am trying to get an overall sense of the big picture, but it seems to be presented in a sequential format in the places I have looked, with a lot of unfamiliar vocabulary and symbolism. Maybe that is how it has to be, though. I bought a modern algebra textbook, and I have checked quite a few internet sites, but I can't make the connection.

If it is just a classification system for particles, I think I can understand that. Maybe a simple analogy would be the multiplication tables a grade school kid would use? The rules are based on algebra, but geometry can be useful for gaining insight. (like illustrating the symmetry)

If group theory is really a periodic table for subatomic particles, how is this a theory of everything? how does F=MA or Maxwell's equations come out of that?

Am I even on the right track?

 

[jal]

I’m going to be the devil’s advocate.
A layman can be a dud waiting for the next wave, the 15 year old down the street, the single mom, the doctor who took out your gall bladder or grandma doing her knitting.

There are only a handful of “math kids” who understand what Garrett has proposed.

If our simple explanation is really “wrong” I’m sure that Garrett, Tom Smith, or even John Baez would be more that willing to step in with a clarification.

I think that what has been said so far is a big help to the layman who wants to seek further understanding.

I’m going to ask two simple question based on the following info., to try and focus on E8 and see if the answers end up being simple enough to give greater clarity. ( SM can come later)

---------

starkind
But what if you need to describe a system that has charge, mass, spin, and so on, which are not really shapes at all? You need more mathematical dimensions to do that. Lisi's model does describe the quantum numbers, so it needs more dimensions, mathematic dimensions, to do that.

from https://www.physicsforums.com/showthread.php?t=196498&page=7
http://www.measurementalgebra.com/E8.html
post #105 11-16-2007, 03:28 PM

I’ll use CarlB’s java animation for a base of reference for my questions.

1. How many points did you use? Why?
2. What is the distance between the points? Why?

---------------
Don’t answer by saying CarlB has already explained it.
Challenger your communication skills and see if you can use the right words so that the concept will get through.

 

[starkind]

Thanks, belliott4488, for the corrections. I am trying to understand this as a layperson myself, having started with only college physics and calculus. I have been studying these ideas (QG, string, loop, triangulations, cat theory, Lie algebra, etc.) for five years now, on my own and on the internet (mostly Wiki and PF) using books from Dover press, among others, available in national bookstore chains. I have never taken a course that mentioned any of this stuff. So it is no surprise to me to hear that I have made some misstatements. Part of my reason for writing here is to learn to use the language of physics clearly and correctly, so I appreciate any hints you or others may give me to identify areas where I have misunderstood or confused the concepts. At university, you are required to write papers and tests to give you a clue where you may be lacking, but here we have to rely on others for corrections.

I would appreciate if you could say more about your assertion that the Higgs “does couple to other particles and gives rise to their masses by virtue of this interaction, but that’s not the same as gravity.” How is mass different from gravity, aside from the English usage of mass as something a particle can possess, and gravity as a force (or a curved space-time) that affects mass. I am afraid that mass and gravity seem inextricable in my mind. In fact, I think we could do without either idea, (that of curved space-time and that of particles throwing gravitons back and forth at each other) and simply ask what is inertia? Why does it cause some objects to resist acceleration, including gravitational acceleration?

SublimeGD, I agree with belliott4488 that we should first start with trying to get used to how SU(2) and U(1) are used to describe the electroweak interaction. I am trying to study this myself now, and find that Lisi’s paper makes a good motivator. Lisi says the electroweak gauge W is an element of su(2) and the electroweak gauge B is an element of u(1). I think I have learned that SU(2) and U(1) are Lie groups, and that su(2) and u(1) are the algebras associated with those groups.

In the table on page 2, I wonder what the electroweak gauge B is. I guess the electroweak gauge W has to do with the W bosons of the weak force. I don’t know what happened to the third weak force boson, the Z. And I can make a guess that the electroweak gauge B must then be the electro- part of the electroweak, since I have seen B used as a symbol for the magnetic part of the electromagnetic field.

I wonder if ‘spin’ as used in a Lie group algebra so(3,1) has anything at all to do with spin as a quantum number?

The e in the table is a frame, but what is a frame? I know relativistic frame, is it the same?

 

[starkind]

jal

I'm not sure what you mean by question one. I have a microsoft browser and so can't view the applet. (Ba-a-a-ahhh).

E8 has 248 dimensions, so I guess 248 points. Lisi projects this down to three dimensions as a cubeoctahedron, one of my favorite objects, as on page 5, table one. The cubeoctahedron has only eight vertices, but the projection is onto the midpoints of the lines of the cube, which makes twelve points. The other points are not lost, they are just obscured. Many points of the E8 must project down onto one point in 3 dimensions.

The distance between the points, so far as I know, is undefined. Some lines on a projected image look shorter than others, but in the higher dimensional object, they may be the same length, or even longer. If E8 obeys topological rules, as I suspect it must, the distances can be any length or any relationship.

Is this what you meant?

 

[jal]

Hi starkind!
I'm trying to get those that say that they want to understand to do a little bit of thinking and to do some observations.
Those that have an understanding, I want them to give a layman's explanation.
Here is some more help.
--------
Pick an outside “dot”… you will see that it rotated from zero to 90 degrees and then back.
“Stop” the rotation and you will see that there is a mirror image in each quadrant (4). What is happening in the first quadrant is also happening in the other quadrants.

Now, … disregard the other 3 quadrants. Find the other symmetries in the first quadrant.

 

[SublimeGD]

Sorry, but I don't think this is really right. What I believe you're looking for is a layman's explanation of how E8 - and specifically its dimensionality - works to describe fundamental particles. I think any such explanation would be a special case of an answer to the more general question, "How does any Lie Group work to describe fundamental particles?" There are many, many such layman's explanations of the latter, so I was suggesting that you start with those. Most of them are much clearer than anything I could come up with.

I think the confusing thing here is the difference between the dimensionality of the group itself and that of spacetime. E8 (as well as other Lie groups) describes the symmetries of a geometrical space, so we can try to picture those symmetries by thinking about the higher-dimensional spaces in which those symmetries exist, which requires the projections Lisi describes in your quotation. No one is suggesting that any of those dimensions correspond to spacetime dimensions, as you've noted yourself. Well, actually, I think starkind is suggesting that the dimensions of the space in which E8 operates are the physical dimensions of the universe, and I think I'd object to that. Perhaps I've misunderstood that point, however - I didn't really follow his explanation, I'm afraid.

I suggested that "analogy" to illustrate my frustration with trying to get a dumbed down explanation of the contents and implications of the Lisi paper. I am aware that theoretical physics is a different beast all together, and requires a different approach to gain even a grain of understanding. Thanks for the suggestion of looking into how Lie groups describe fundamental particles. But I still feel that a clear explanation of the contents of the Lisi paper could be made from a conceptional point of view, rather than strictly mathematical. Then again, I'm in no position to make that claim, since I don't understand the Lisi paper

Is anyone "suggesting that the dimensions of the space in which E8 operates are the physical dimensions of the universe"?? This is really one of the key points that must be driven home in a "layman's explanation." Is the E8 symmetry or whatever it is simply a mathematical tool that happens to describe our universe? Or does the "fact"(just assume for sake of argument) that the E8 symmetry describes our universe imply something about our universe, perhaps the "shape" of our universe? This is really what I set out to find out by asking for a layman's explanation. I do want to learn more so I can really understand physics, but I want a simple dumbed down explanation right now!

YES! That is my question. I am having trouble making the connection between my undergrad physics and group theory. I am a global learner to the extreme, so I am trying to get an overall sense of the big picture...

Stephen? Colbert? Really? Did you secretly cause the writer's guild strike so you could have some time to learn about particle physics?

I think that what has been said so far is a big help to the layman who wants to seek further understanding.

Very true, thank you guys very much.

SublimeGD, I agree with belliott4488 that we should first start with trying to get used to how SU(2) and U(1) are used to describe the electroweak interaction. I am trying to study this myself now, and find that Lisi’s paper makes a good motivator. Lisi says the electroweak gauge W is an element of su(2) and the electroweak gauge B is an element of u(1). I think I have learned that SU(2) and U(1) are Lie groups, and that su(2) and u(1) are the algebras associated with those groups.

Yea, I have been looking into this too. Somewhere on this forum I saw a thread that showed what type of background one needed to understand the Lisi paper. Graduate level particle physics, group theory... I don't expect to read a couple of wiki articles and suddenly understand the Lisi paper (but I am reading them). That is why I'm trying to hard to get a dumbed down explanation now.

 

[belliott4488]

Thanks, belliott4488, for the corrections. I am trying to understand this as a layperson myself, having started with only college physics and calculus. I have been studying these ideas (QG, string, loop, triangulations, cat theory, Lie algebra, etc.) for five years now, on my own and on the internet (mostly Wiki and PF) using books from Dover press, among others, available in national bookstore chains. I have never taken a course that mentioned any of this stuff. So it is no surprise to me to hear that I have made some misstatements. Part of my reason for writing here is to learn to use the language of physics clearly and correctly, so I appreciate any hints you or others may give me to identify areas where I have misunderstood or confused the concepts. At university, you are required to write papers and tests to give you a clue where you may be lacking, but here we have to rely on others for corrections.

Hey, I'm glad to help in any way I can - I'm just a layperson myself. As jal points out, though, there are different levels of laypeople. I actually managed to complete my graduate studies in Theoretical Physics (long ago), but that's where my career as a professional Physicist ended, hence my status as a layman.
I studied the Standard Model in grad school (note that I say "studied", and not "learned"), and I'd still like to pursue a deeper understanding of it. Maybe if I can help to lead you through some of the underbrush we can each get closer to what we're looking for.

I would appreciate if you could say more about your assertion that the Higgs “does couple to other particles and gives rise to their masses by virtue of this interaction, but that’s not the same as gravity.” How is mass different from gravity, aside from the English usage of mass as something a particle can possess, and gravity as a force (or a curved space-time) that affects mass. I am afraid that mass and gravity seem inextricable in my mind. In fact, I think we could do without either idea, (that of curved space-time and that of particles throwing gravitons back and forth at each other) and simply ask what is inertia? Why does it cause some objects to resist acceleration, including gravitational acceleration?

Okay - good questions, with rather far-reaching implications.
The Higgs mechanism is the means by which particles acquire mass in the Standard Model. The Higgs field is a scalar field (i.e. spin zero), and when it interacts with the other particles in the theory, which would otherwise be massless, the effect can be observed as their having mass. A Google search on "Higgs Mechanism" will produce numerous explanations.
As far as the relationship of mass and gravity, of course you're right that they're pretty inextricably related, but that doesn't make them equivalent. Electrons have mass, and are thus affected by gravity, but they are hardly "carriers" of gravity or otherwise responsible for its existence (other than that they do have their own tiny gravitational fields, as does anything with mass or energy). Gravity is the field; mass - or energy, which is equivalent in relativistic theories - is what the field couples to, i.e. it is the quantity that determines the strength of the resulting force (speaking classically).
You are also right to bring up the equivalence of inertial and gravitational mass, although I suspect you might already be aware that this equivalence is the basis for Einstein's General Theory of Relativity.

SublimeGD, I agree with belliott4488 that we should first start with trying to get used to how SU(2) and U(1) are used to describe the electroweak interaction. I am trying to study this myself now, and find that Lisi’s paper makes a good motivator. Lisi says the electroweak gauge W is an element of su(2) and the electroweak gauge B is an element of u(1). I think I have learned that SU(2) and U(1) are Lie groups, and that su(2) and u(1) are the algebras associated with those groups.

In the table on page 2, I wonder what the electroweak gauge B is. I guess the electroweak gauge W has to do with the W bosons of the weak force. I don’t know what happened to the third weak force boson, the Z. And I can make a guess that the electroweak gauge B must then be the electro- part of the electroweak, since I have seen B used as a symbol for the magnetic part of the electromagnetic field.

Okay, this is more that I can explain in detail right here, but let me just try this for now: There are three quantum fields associated with the SU(2) part of the Electroweak symmetry group, SU(2) X U(1) (specifically with the generators of the group), and one for the U(1) group. These don't quite correspond to the physical gauges bosons (read: force carriers) that we see, but instead get "mixed", in the Quantum Mechanical sense.
[This is an example of something that happens a lot in quantum theory, where a field can exist in one "eigenstate", that is, a state where it has a definite value with respect to something, or it can be in a mixture of such states, where it then has a definite value with respect to something else, but it can't be in both.]
In this case, one of the three bosons that correspond to the SU(2) part of the group SU(2) X U(1) (called W0) and the one that corresponds to the U(1) part (called B) mix, and by virtue of the Higgs mechanism, we get two different combinations of these two particles, which are electrically neutral. One becomes the photon and is massless (it doesn't interact with the Higgs field), corresponding to a "mixed" U(1) part that "breaks off" to become the symmetry of the E-M field. The other does interact with the Higgs, and become the massive Z particle. It, along with the two W+/- particles, conveys the Weak interaction. It is their relatively large masses that limit the range of the Weak force.

I recommend the Wikipedia pages on this subject to get a better description than what I've given here.

 

[belliott4488]

By the way - this is directed to the moderator - should this thread be place elsewhere, since it's not strictly "Beyond The SM" any more? The OP was asking specifically about the use of E8 as a symmetry group, but we've kind of backed up to the gauge symmetries of the SM, for now at least.

 

[jal]

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

 

[SublimeGD]

By the way - this is directed to the moderator - should this thread be place elsewhere, since it's not strictly "Beyond The SM" any more? The OP was asking specifically about the use of E8 as a symmetry group, but we've kind of backed up to the gauge symmetries of the SM, for now at least.

No, the question is still there. Can someone please provide a quick dumbed down conceptual summary of the contents of the Lisi paper?

 

[SublimeGD]

I actually managed to complete my graduate studies in Theoretical Physics (long ago), but that's where my career as a professional Physicist ended, hence my status as a layman.

Just a layman with a graduate degree in theoretical physics...

I just want a "conceptual" explanation. If that is not possible, can someone please just say its not possible?

For example, if I asked for a layman's explanation of general relativity I would expect something like "gravitational acceleration can be described by the curvature of space and time." I wouldn't be content with a link to a wikipedia article on Riemannian geometry, and a suggestion to "start here".

 

[staf9]

I am perhaps one of the world's absolute worst teachers, so I'm not even going to try to give a simple explanation of E8 Theory because I know I'll over-complicate it.

However, I can give a little bit of help: Whenever I'm reading a paper on arXiv that I don't understand, I open wikipedia and search every word, then every group of words that I don't understand.

If I find an explanation that I don't understand on wikipedia, I use google or keep digging deeper into wikipedia.

Now, I'm not saying that this is the best way to learn particle physics (some of the stuff on wikipedia is wrong), but searching some key words and reading up can help a little.

Here's what I think would be worthy to look up just to understand what the paper is about: differential manifold, differential structure, E8 (mathematics), principle bundle, and connection (might be listed under principle bundle).


Of course, I may have already overcomplicated it.

 

[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...