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

[TheRealColbert]

A starting point

For my listening enjoyment during holiday travels, I spun "Particle Physics for Non-physicists" from the teaching company. In chapter 14, the section on symmetry breaking, there was a quote that seemed to fit this topic.

"There's not always a starting point when you are learning about some complicated thing. It's just a big ol' mess and you just got to jump in somewhere and begin to learn more about it and then the big picture begins to appear."

 

[jal]

Hi coin!
I was hoping for more input from other people too.
I have one more observation (at this time).
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The present status of Garrett’s E8 could be compared to a blank template. It needs to be completed. At present it is ideal for applying the tools of project management. http://en.wikipedia.org/wiki/Project_management
Here is where the amateur can help. E8 can be mapped on a program such as PERT. You could then ask it to show an up quark or a down quark etc..
http://en.wikipedia.org/wiki/PERT
Program Evaluation and Review Technique
The Program (or Project) Evaluation and Review Technique, commonly abbreviated PERT, is a model for project management designed to analyze and represent the tasks involved in completing a given project.
-------
jal

 

[starkind]

I have been looking at the E8 simulations on Youtube



and have noticed a thing or two that may be of interest here. One thing is that the undesignated branches of the E8 particle model are central to the simplest projections. At about 14 seconds into the vid, you can see the eighteen red-blue-green squares congregate at the center of the figure. Of course it is the projection onto two dimensions that seems to congregate, and their proximity to one point only indicates that they are on a common axis in the E8 structure, not that they are "close" to each other. But they do share a common axis, on which all other dimensions have value zero.

Then again at about 32 seconds, a simple configuration occurs in which the colors separate into six outer circles, each a cluster of similarly colored quarks, around one inner circle. Each of the outer circles is centered on a square, indicating an undesignated branch. There is however no square, or anything else, in the center of the central circle.

The other thing I want to mention is the physical meaning of the geometric relationships, as illustrated by table 1 on page 5 of the paper, where the gluons are shown to be related to the quarks in such a way that simple vector addition predicts the result of any quark-gluon interaction. Vector addition, for grandma’s sake, is simply a matter of placing the tail of one vector at the head of the other. The head of the first vector then shows the position on the figure occupied by the result. In this way the red-green gluon is shown to interact with the green quark to produce a red quark.

Presumably the other geometric relationships (where the connecting lines are vectors) are also related to possible interactions and their results.

I would like to be able to select certain groups of particles and suppress the others to get an idea of how, say, electrons rotate through E8. Or perhaps just the electrons and the up and down quarks that commonly make up almost all visible matter. Do the visible matter particles cluster in certain lines and planes also, at a different point in the rotation? What underlying physical reality could be responsible for the preponderance of only two of the six quarks? Can we see something in the structure that might suggest a relationship between the commonly visible bits of matter and the underlying structure of time-space?

S

 

[jal]

I would like to be able to select certain groups of particles and suppress the others to get an idea of how, say, electrons rotate through E8. Or perhaps just the electrons and the up and down quarks that commonly make up almost all visible matter. Do the visible matter particles cluster in certain lines and planes also, at a different point in the rotation? What underlying physical reality could be responsible for the preponderance of only two of the six quarks? Can we see something in the structure that might suggest a relationship between the commonly visible bits of matter and the underlying structure of time-space?

I don't think that Garrett put up a chart of E8 and the flicked paint at the template and called it the standard model representation.
I guess we will just have to wait for the next papers.
---------
inserted: Bee's 2d (6 sphere packing)
http://backreaction.blogspot.com/2006/08/quark-gluon-plasma.html
By Bee on Wednesday, August 23, 2006
Quark Gluon Plasma
The pictures come from this presentation.
http://th.physik.uni-frankfurt.de/~scherer/qmd/cscus2004_stefan_scherer.pdf
jal

 

[jal]

Garrett
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.

Since this thread is for a layman’s explanation of trying to understand Garrett’s E8 Standard Model let’s look at how the particles could be distributed in a simple symmetric pattern? Let’s look at some possibilities by assuming that the proton is a sphere containing those “particles”.
1. You could divide the sphere into 12 inner spheres and divide the 240 “particles” into those 12 spheres. That makes 240/12 = 20 particles per sphere.
2. If you wanted to combine the 12 spheres and the vertex of LQG then you only need one double tetra in the center and assign a group of “particles” to the 8 vertex and to the 4 mid-point of the vertex, for a total of 12 groups of “particles” around the center. Each of those 12 vertex would then contain 20 “particles”. Adding tetras between the spheres could be done but there is the problem of “double counting” of “particles”.
3. There are other combinations that could be made. Garrett will eventually work out the ones that he feel works the best with E8 and tetras.
Here is my image with a hex. packing configuration.
http://www.geocities.com/j_jall/3dspace.gif
If you don’tget the image it is because the site has crashed from too much traffic. You could try doing a search for "12 sphere packing" to get an idea of possible arrangements.
Or start at your search at http://www.grunch.net/synergetics/readings.html
jal

 

[jal]

Did you continue your searching/learning? Did you find the following explanations? Can you make the link with what the “math kids” are doing?
-----------
http://www.scienceu.com/library/articles/isometries/index.html
Introduction to Isometries
------------
http://www.math.uchicago.edu/~farb/papers/isoms.pdf
Isometries, rigidity and universal covers
Benson Farb and Shmuel Weinberger
December 31, 2006
-----------
http://www4.ncsu.edu/~loek/research/res.html
work on symmetric spaces
------------
http://www.verbchu.com/crystals/patterns.htm
Mapping the Hidden Patterns in Sphere Packing
--------------
http://www.mdstud.chalmers.se/~md7sharo/coding/main/node38.html
Applying Coding Theory to Sphere Packing
-------------
http://math.berkeley.edu/~reb/papers/bcqs/bcqs.pdf
A Monster Lie Algebra?

We define a remarkable Lie algebra of infinite dimension, and conjecture that it may be related to the Fischer-Griess Monster group.
The Lie algebra of this paper is indeed closely related to the monster simple group. In order to get a well behaved Lie algebra it turns out to be necessary to add some imaginary simple roots to the “Leech roots”. This gives the fake
monster Lie algebra, which contains the Lie algebra of this paper as a large subalgebra.
See “The monster Lie algebra”, Adv. Math. Vol. 83, No. 1, Sept. 1990, for details.
----------

Chapter 30 of “Sphere packing, lattices and groups” by Conway and Sloane, and Adv. in Math. 53 (1984), no. 1, 75–79. R. E. Borcherds, J. H. Conway, L. Queen and N. J. A. Sloane
----------
http://www.research.att.com/~njas/doc/splag3.pdf
Sphere packing, lattices and groups
Material for third edition, Sept 16 1998
-------------
http://www.research.att.com/~njas/index.html
Neil J. A. Sloane: Home Page
=========
Finally! …. I have reached the end of this simple presentation. ( I think)
If you want to learn …. You got to continue searching.
I found that by doing a search for sphere packing and Isometries that I got the essentials and a simple way to begin to understand the math (Lie) which is used to do physics. It will not make you “a math kid’, but it will make one more person who can have some appreciation of what they are doing.
I hope that all the people who know more than me have not found mistakes in this presentation which would lead the layperson astray.
Good hunting in your quest for understanding!
Jal

 

[starkind]

Thanks for these links, Jal. We had the first real snow of the season and I have been busy all weekend, but today is a study day. I'll take a look at the links and return here later.

S

 

[starkind]

I've been speculating again. I do hope this is not against the forum rules. The speculations began when I looked at

http://th.physik.uni-frankfurt.de/~s...an_scherer.pdf

Beautiful! Shows hex relationships in a quark-gluon liquid model.

Relativistic Pb-Pb collisions produce extremely high energies in extremely small spaces, resulting in a primordial fireball in which quarks and gluons are freed from confinement. Exotic hadrons are found (I didn't quite get this part...who found them? Is this actual data or only part of a model?) involving five quarks in a single particle.

The implication seems to me to be that we can now expect to work with the idea that quarks can be extracted from existing hadrons and recombined into exotic matter. The fireball is so dense with data that models are needed to interpret the results. So it seems to me now that it may be appropriate to speculate on what kinds of models could be tested against the data in hopes of finding a mathematical fit.

Here is my current speculation: the quark-gluon liquid may contain all of the quarks and gluons, not just the few we see in common hadrons. As hadrons freeze out of the q-g liquid phase, they form into lattices in which the up and down quarks are on the visible “surface” of the lattice, while the rest of the quarks are “hidden” “inside”.

Of course, “hidden” only means that our measuring apparatus does not detect them, and “inside” is higher dimensional, so that the inside of the object can very well be bigger than the outside. Think Calabi-Yau.

Don’t panic, the hidden inside dimensions are still measurable, because the hidden structure determines the behaviors we see on the surface. We detect the surface behaviors, and use them to infer the interior structures. We will know when we have the right model if the hidden inner relationships can be shown to determine peculiar behaviors seen on the surface.

Grandma, dear, this is like in the old days before electron microscopes when scientists looked at nearly pure mineral samples and found that they often occur in nearly perfect crystals. Some form cubes, some form octahedrons, some form complicated rhomboid structures. Back then, no one had actually seen an atom, but by using a model in which extremely small spheres of one or two sizes were densely packed, the various crystal lattices could be explained in compelling detail. It was a couple hundred years before x-ray crystallography and electron microscopy actually showed that the tiny spheres really exist, just as the model predicts.

In the old days, scientists used this model to take apart solids and reassemble the parts into other kinds of solids. We call this chemistry, but they thought of it as alchemy. The alchemists were hoping that they could discover how to make gold out of lead, but of course we now know that this cannot be done by rearranging atoms. It can be done, and is done today, at huge expense, by means of various fusion and fission reactions. In these reactions, we rearrange the neutrons and protons that are found in the nucleus of the atom. Unfortunately, the leftovers from the fusion and fission reactions are often deadly poisons. It isn’t profitable or practical to make gold this way.

Now just as we went from banging rocks together (mechanics) to banging atoms together (chemistry) to banging protons and neutrons together (nuclear physics), we now have progressed to the stage of evolution where we can bang together the quarks and gluons that exist within and make up the protons and neutrons. I am not aware that anyone has thought of a suitable name for this new kind of hammering on matter, but the possibilities are interesting.

For example, it may be possible to take apart the quarks in neutrons and protons in a lump of lead and recombine them into the protons and neutrons that make up an equal mass of gold. There would be no leftover poisonous stuff to worry about.

That will be a trivial result. The really exciting goal would be to take apart the protons and neutrons and not recombine them into matter at all. Instead, we may be able to transform them into pure energy in the form of electrons or photons. Again, there would be no leftover poisons. This could even be a way to make energy densities deep enough to warp space and time, deep enough to create gravity fields at will. Warp drive, anyone? Tractor beams? Anti-gravity?

One of the subsets of E8 describes how quarks and gluons inside hadrons (like protons and neutrons) dance around each other. The model shows six quarks and their anti-quarks, each of which has three colors. The colors have to add together to make the hadrons white. The charges on the quarks have to add together to make the hadrons positive, neutral, or negative. These peculiar facts are some of the results we should try to explain using the model.

Experimentally, we need to look for starting conditions which may have an effect on the types of hadrons which freeze out of the fireball. Are there initial conditions which produce more protons than neutrons? Are there initial conditions which produce more anti-matter than matter? Are there initial conditions which produce only energy and no matter at all? Which initial conditions can we control and modify for experiments? Electromagnetic fields? Angle of collision? Presence of strong acelleration fields, such as those which may occur near a black hole?

We have a lot of studying to do.

 

[jal]

Hi!
Simple description!
Could you fix your link.

 

[jal]

The following link has some very interesting images.
It should make you wonder if the mechanisms and the pattern for these structures are also at a smaller scale (inside the proton).
http://wwwphy.princeton.edu/~steinh/quasiphoton/
Experimental Measurement of the Photonic Properties
of Icosahedral Quasicrystals
Weining Man, Mischa Megens, Paul M. Chaikin and Paul J. Steinhardt

 

[staf9]

Is it just me being crazy, does the first-ever imaging of the 3d quasicrystal Brillouin zone look familiar?

I'm going to go read their paper.

 

[starkind]

jal
Here is the correct link. I found it on your link to Bee's 2d (6 sphere packing) as "stefan's talks". For some reason I don't understand, the link gets mutilated every time I try to copy it here. It shows up complete on my edit screen, but only partial on the board. So I am going to try to work around... the below has replaced each back-slash with an asterisk. I guess if you want to follow the link from here, copy it to your address bar with backslashes insterted in place of asterisks.



http:**th.physik.uni-frankfurt.de*~scherer*qmd*cscus2004_stefan_scherer.pdf

Otherwise, go to the link for Bee's backreaction in jal's post, number 94. I see that the link to stefan's paper is also mutilated in post 94. But if you click the backreaction link in 94, you can find the paper by clicking Bee's link in her blog to "stefan's paper".

 

[starkind]

Is it just me being crazy, does the first-ever imaging of the 3d quasicrystal Brillouin zone look familiar?

I'm going to go read their paper.

The crystal lattice model used to be just that...a model, without direct physical evidence, until the kind of crystallography as shown in the images linked above. Now we can say with a very high degree of certainty that macroscopic physical matter is made of what amounts to tiny spheres in dense packed structures.

This was enough to provoke the speculation that within the atom there are parts which also form a dense packed spherical structure, namely the nucleus, made primarily of protons and neutrons. As far as I know, there are still no images made directly from the protons and neutrons in a nucleus, but it is usual in college level courses to depict the protons and neutrons as small, hard, mutually exclusive spheres.

We are now pushing this crystal lattice model another level down the spectrum, thinking that the quarks and gluons in a proton or neutron are also composed of some smaller scale densely packed spheres. As far as I know there is no physical evidence of any kind to suggest that the crystal lattice model still holds at the quark scale. However, the work Lisi has done is highly provocative.

It is still problematic that the SO(3) geometry calls for a lattice structure connecting all the kinds of quarks and gluons, and in extension to E8, all the kinds of particles. The behaviors of protons and neutrons can be entirely accounted using only two kinds of quarks, the up and down. If we suggest that E8 and SO(3) are physical spaces inside the proton and neutron, then we are saying that all the other quarks and gluons are somehow present physically inside the proton or neutron.

We are a long way from having direct evidence to support this idea, and furthermore, it calls for a much more complicated picture than the current idea that protons and neutrons are composed only of three quarks of two kinds, along with their related gluons. By Occam’s razor, such apparently needless complications should be cut away. Worse, the idea that all those other quarks and gluons are present inside the hadrons requires us to wave into existence some kind shielding to explain how they can be present and yet not affect the known behaviors.

Still, all is not lost. Geometry is one of the oldest applications of mathematics, and geometric rules have been shown to apply in a physical way to chemistry. It also works very well in conceptualizing structures in nuclei. It certainly has applications in explaining the composition of nucleons. And, some respectable academic researchers sitting on piles of credentials have seen fit to explore even more remote regions of physical knowledge using geometry to explain the behaviors of space and time at the Planck scale.

So we are not entirely out of order in thinking about how nucleons may be composed of spherical quarks in a dense packed lattice structure. But any idea we may put forward will have to be compelling if it is going to stand. We will have to have a simple easy model that explains known behaviors on the basis of a lattice geometry in which most of the components of the lattice are invisible. The model will have to explain the known behaviors, and also have a mechanism to explain the invisibility.

I am going to suggest a phase structure in which the three generations of the standard model come from our measurement “in the present instant” being bracketed by instants immediately past and instants immediately next to come. Physical objects in the immediate future may be in a state analogous to a gas, physical objects in the present instant of measurement may be in a state analogous to a liquid, and physical objects in the immediate past may be in a state analogous to a solid. All of our measuring apparatus is in the present or liquid phase. Only at the extreme limits of measurement do we get a means to infer the physical nature of the generation just past and the generation just to come.

This phase shift becomes more obvious as we measure smaller and smaller spatial separations. As the spatial component of the measuring process becomes small, the time component gets closer and closer to unity with the spatial component. At the Planck scale, time and space is one thing, while at the Fermi scale, space predominates to the extent that time units become infinitely small. The present instant becomes a two dimensional space-time surface with no measurable time-like thickness.

Then we may think of the up and down quark, along with related gluons, as embedded in the present instant, while the next and past instants contain the other two generations of the standard model. In this way, the unification of space-time joins smoothly with the macroscopic realm at the Fermi limit. Below the Fermi limit, the “objects” are seen as embedded in a space-time geometric lattice, while above the Fermi limit, the “objects” are seen as having three extended spatial dimensions and a single instantaneous two dimensional layer in a foliated time-like sequence.

The three dimensional space-time lattice is then fundamental at least down to the Planck scale. At macroscopic scales we are measuring such large spaces that the time dimension seems to become continuous.

This model may be tested by examining data on standard model particles from current and near-future collision experiments. What signature might we see to support the idea that the uncommon generations of particles are in advanced and retarded time frames?

The universe is expanding. Future generations would seem much larger than present generations. Past generations would seem much smaller. Energy is a function of size. Mass is a function of energy. I am going to go look for the mass relations among standard model generations.



Richard

 

[belliott4488]

The crystal lattice model used to be just that...a model, without direct physical evidence, until the kind of crystallography as shown in the images linked above. Now we can say with a very high degree of certainty that macroscopic physical matter is made of what amounts to tiny spheres in dense packed structures.

Well, you want to be careful with that metaphor; even at the molecular level, there are no "hard spheres."

This was enough to provoke the speculation that within the atom there are parts which also form a dense packed spherical structure, namely the nucleus, made primarily of protons and neutrons. As far as I know, there are still no images made directly from the protons and neutrons in a nucleus, but it is usual in college level courses to depict the protons and neutrons as small, hard, mutually exclusive spheres.

Well, that's unfortunate, and it should certainly be taken as a cartoon of reality at best.

We are now pushing this crystal lattice model another level down the spectrum, thinking that the quarks and gluons in a proton or neutron are also composed of some smaller scale densely packed spheres. As far as I know there is no physical evidence of any kind to suggest that the crystal lattice model still holds at the quark scale. However, the work Lisi has done is highly provocative.

Not that his work actually suggests anything about a physical crystalline structure, right? I wouldn't want people to get the wrong impression about what he has proposed.

... If we suggest that E8 and SO(3) are physical spaces inside the proton and neutron, ...

But WHY would we ever say such a thing? Are you thinking that the geometric properties of the Lie Groups that describe fundamental symmetries have a physical realization? (see my final comments, below.)

then we are saying that all the other quarks and gluons are somehow present physically inside the proton or neutron.

Well, it's part of the standard theory of quantum fields that you have "soup" of virtual particles all popping in and out of existence for minute periods of time, but I don't think that should be confused with the existence of real observable particles.
*****************
Once again, I have to stress that it is not correct to think that the space of quantum states (Hilbert Space) is not the same thing as space-time. More to the point, there is nothing about the use of Lie Groups in Quantum Field Theory in general, or in Lisi's model in particular, that suggests that the representation space of the group in question (i.e. the space where the multi-dimensional geometric structure lives) contains or is equivalent to the space-time in which we live.

You've seen the description of the group O(3) as the group of rotations in 3 spatial dimensions. Maybe you also recall where it was stated that it is mere coincidence that in this case the representation space and the space in which the rotations take place are both 3 dimensional. That's because they're entirely different spaces. (As a counter-example you could consider the group of rotations in 2-D space, which is a 1-dimensional group, requiring only one parameter to define a rotation, or group element.)

I really think it's important to keep these spaces straight and not to confuse them. And of course, I hope I haven't just confused things all the more ...

 

[starkind]

Thanks belliott4488

I am glad you are here. You are right of course about there being no “hard spheres” in the absolute sense, at least macroscopically. I add the “at least” because we don’t know anything about what really goes on below the particle scale. And even a spherical chunk of the hardest stuff we can obtain is not really absolutely hard, in the sense of inelastic. Every surface we can touch is made up of tiny bits of matter held in place by electrostatic forces. The tiny bits are truly tiny; there is much more space than there is bits in any available object. So I need to be clear that “hard spheres” are only hard in the sense that elasticity is extremely limited, and then only until the energy of a probe exceeds the binding energy of the particles involved.

I hit on the term “hard sphere” when trying to find a commonly descriptive way to separate the bits from the spatial fields in which they exist. Grandma doesn’t get the idea that matter isn’t really solid. She isn’t going to understand that in field theory there are no absolutely hard bits of any kind at all. I know you already get this, but I assume there may be some reader who does not have field theory to work with. Really, I am writing the kind of explanations I might have understood myself at the age of 15 or so when I first encountered and was fascinated by the concept of special relativity. That was a year before I first got to study chemistry.

I wrote: “it is usual in college level courses to depict the protons and neutrons as small, hard, mutually exclusive spheres.”

You replied: “Well, that's unfortunate, and it should certainly be taken as a cartoon of reality at best.”

Indeed. I guess anyway I should have said it was usual in college level courses when I took them, thirty years ago. Even then, the professor was careful to caution us not to take the drawings too seriously.

Nowadays I have come to the understanding that most if not all of what we think we know is little more than caricature. Table surfaces aren’t really hard, atoms aren’t really hard, protons aren’t really hard. But the dense pack spheres model does work for crystallography, doesn’t it? Each atom in a crystal is pretty much trapped in a spherical shell from which other atoms are excluded. The shell mostly stays where it is in relation to its neighbors. No other atom of the same size or larger can squeeze between them without the application of enough force to break the lattice.

The atoms are not hard little balls, and the teacher will be sure to tell you so, but in a shorthand sort of way, when doing chemistry, we can think of them like that. Do introductory organic chemistry classes still use those little colored wooden balls with holes in them in which you place sticks or springs to attach them to each other? Did Watson and Crick use the little wooden balls and springs and sticks when building their model of DNA? It works well enough when dealing with objects all on a similar scale.

As I recall astrophysics, massive ordinary stars sometimes collapse to neutron stars….made all of neutrons. The neutrons are not really hard little balls, but to some degree they act that way. When the density of the star gets to such a point that the hard little balls break down, the matter in the star continues to collapse and become even more dense as it forms a black hole. The fact that the neutron star resists further collapse until a certain energy density is reached suggests that the hard little ball model of neutrons works in there, as well.

So I suggest we may continue to use the hard little ball caricature at least into the nucleus, and since it has done us such service, we may as well allow ourselves to use it provisionally when thinking about quarks and gluons. Maybe it is only coincidence that E8 has an SO(3) cubeoctahedral subset which can be modeled with densely packed hard spheres and which encompasses quarks and gluon behaviors. Probably it is only coincidence. Surely it is only coincidence. Almost certainly. But we cannot rule out the application of the model on a provisional basis until we find contradictory evidence.

Garrett Lisi presumably would accept a career in academic physics if offered by the right school. Of course he has to be very careful not to say anything that may compromise his future. What a disaster for a career in academics to assert something that later proves erroneous. I am lucky that I don’t have to worry about that sort of thing. I can wander down dark alleys looking for an open door others might have missed. If there is no open door and the alley is a cul-de-sac, I am free to back out of it and continue my search elsewhere.

I don’t speak for Garrett Lisi, but only from my own understanding, and to other amateurs who may be looking for non-mathematical conceptual tools to grapple with this interesting topic. I think I can safely say that Garrett Lisi’s work is provocative. It provoked me anyway, and evidently quite a few other people. I will take any responsibility required of the assertion that E8 may represent some physical relationship that enforces its mathematical behavior.

You wrote: “But WHY would we ever say such a thing? Are you thinking that the geometric properties of the Lie Groups that describe fundamental symmetries have a physical realization?”

Well now I have to turn this around and ask if WHY is ever a question that can be answered by physics. But nevermind. I’ll entertain it anyway. Why not?

Not to evade the question. The geometric properties of the Lie Groups that describe fundamental symmetries do have a physical realization. It is realized in the physical behavior of quarks and gluons, which are accepted as part of the standard model. If it is not the geometric properties of the Lie Groups, then WHY do they behave that way?

Consider this. The multiplication table has nothing, physically, to do with the surface area of a bean field. And yet it is useful as a model of the field when calculating harvest yields and fertilizer applications. It could even be physically laid out on the field, just as it is on a piece of paper, to prove a point. It fits exactly. The fact that E8 model is a mathematical description does not rule out the possibility that it might be laid out exactly in a physical space to describe and predict physical behaviors.

I think the problem may be that you believe somehow physical space is qualitatively different from mathematical space. It is a common assertion. But I should be able to ask you what evidence you have to support your view. What is special about physical space that makes it unique and separate from mathematical space?

You wrote: “Well, it's part of the standard theory of quantum fields that you have "soup" of virtual particles all popping in and out of existence for minute periods of time, but I don't think that should be confused with the existence of real observable particles.”

Are you saying if a particle doesn’t last long enough to be observed that it is not real? How is it not real? Is Hawking radiation real? IIRC Hawking radiation is formed when a virtual particle pair is formed in a place where one member of the pair is trapped inside the horizon, while the other one is left outside, in our ‘real’ world. Unruh radiation, again IIRC, is virtual particles made ‘real’ by a horizon-like separation caused by the accelerated field of the observer. Again, if you think there are two conditions, real and virtual, that should not be confused, I should ask you to explain in what critical way they are different.

You wrote “it is not correct to think that the space of quantum states (Hilbert Space) is not the same thing as space-time. More to the point, there is nothing about the use of Lie Groups in Quantum Field Theory in general, or in Lisi's model in particular, that suggests that the representation space of the group in question (i.e. the space where the multi-dimensional geometric structure lives) contains or is equivalent to the space-time in which we live.”

I agree that there is nothing about Lisi’s model suggesting representation space is equivalent to the space-time in which we live. However you seem to be asserting that they are in fact different, in which case again you should be specific about how they are different. (By the way, I am assuming the double negative in the quote above was unintentional. Please correct me if I am mistaken about this point.)

In summary, how is mathematical space uniquely different from physical space in such a way that it is important not to confuse them?

Thanks for an interesting couple of hours. I hope we get more of them.

S.

 

[belliott4488]

starkind:

If I tried to respond to all that you've written here, it would probably take an entire forum page. More importantly, it would almost certainly take this thread even farther from its original topic than it already is. For that reason, I'd like to respond in detail directly to you and to post only a general high-level response here. If the details of our conversation are interesting to others, then I suggest that we start another thread to continue it there.

I don't seem to have made my point very clearly, so let me try again. First, the distinction between "physical space" and "mathematical space" is yours; I don't believe I made it, but if I did, then I was possibly being a little sloppy. I meant to distinguish between different mathematical spaces, all of which are abstract by definition, and all of which correspond to various physically measurable quantities.

Physics is the business of creating mathematical models of physical observables with a well-defined correspondence between the observables and the entities in the mathematical theory. One possible set of observables is that of spatial measurements, such as relative position. We can model this set with a 1-d mathematical space, perhaps to describe the position of a bead on a wire; a 2-d space, as we regularly do with street maps; a 3-d space, as we do for all sorts of problems in classical mechanics; or a 4-d space if we're doing relativistic mechanics. These spaces are all abstract, but they correspond to physical observables, specifically to relative positions.

There are also other physical observables that can be modeled by other mathematical spaces. Those abstract spaces also have various properties, which might well correspond to relationships between the physical observables being modeled. An example is momentum space, which is often used to model classical mechanics. It's no more or less real than position space, but you shouldn't confuse the two because points, or coordinates, in each of these spaces correspond to different (and incompatible) physical observables.

When we talk about the symmetry groups of fundamental particles, those groups have representations, which may well be described by geometry. The points in this space do not correspond to points in position space, however; they correspond to particle states. Just because, for example, an up quark might appear to the left of a down quark in a particular visualization of the group, that has nothing to do with where it is in position space, and therefore has nothing to do with where we might measure its position (assuming we could even do that).

My point boils down to this: the mathematical model has parts that correspond to positions in space and parts that don't. The root space that we've all seen in Lisi's E(8) theory has geometric properties, but they do not correspond to geometric properties of relative positions of particles.

 

[jal]

You guys are doing great!
I want to underline your comment...

The points in this space do not correspond to points in position space, however; they correspond to particle states. Just because, for example, an up quark might appear to the left of a down quark in a particular visualization of the group, that has nothing to do with where it is in position space, and therefore has nothing to do with where we might measure its position (assuming we could even do that).

Do you want to try to answer another question?
Electricity and magnetism inside the proton. What is doing it?
jal

 

[belliott4488]

o you want to try to answer another question?
Electricity and magnetism inside the proton. What is doing it?
jal

Hm ... "doing it"? I'd have to say it's those -1/3e and +2/3e charged quarks ... any other suggestions?

 

[jal]

=========
belliott4488 We’ll leave electricity and magnetism as an open question.
Let’s look at the gravity question.
I want to talk at the level of “amateur”.

Garrett
It should be emphasized that the connection (3.1) comprises all fields over the four dimensional base manifold. There are no other fields required to match the fields of the standard model and gravity. The gravitational metric and connection have been supplanted by the frame and spin connection parts of : A. The Riemannian geometry of general relativity has been subsumed by principal bundle geometry | a significant mathematical unification.
Devotees of geometry should not despair at this development, as principal bundle geometry is even more natural than Riemannian geometry. A principal bundle with connection can be described purely in terms of a mapping between tangent vector fields (difieomorphisms) on a manifold, without the ab initio introduction of a metric.

I don’t know how it’s done, (a mapping between tangent vector fields).
I’m doing some reading to get ready for an explanation that I might understand. I’m finding that there are ways to include gravity into QCD. (Ie. SO(10), SUSY and other ways)
http://arxiv.org/abs/gr-qc/0506063
The roots of scalar-tensor theory: an approximate history
Authors: Carl H. Brans
(Submitted on 10 Jun 2005)
---------
http://en.wikipedia.org/wiki/Vector_field
Vector field
--------------
http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Flow/flow.html
Flows of Vector Fields
----------
http://en.wikipedia.org/wiki/Scalar_field
Scalar field
---------
http://en.wikipedia.org/wiki/Scalar_field_(quantum_field_theory)
Scalar field theory
---------------
I’m also finding some "Old" approaches.
http://en.wikipedia.org/wiki/Brans-Dicke_theory
In theoretical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan-Brans-Dicke theory) is a theoretical framework to explain gravitation. It is a well-known competitor of Einstein's more popular theory of general relativity. It is an example of a scalar-tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general relativity.
-----------
http://en.wikipedia.org/wiki/Self-creation_cosmology
Self-creation cosmology (SCC) theories are gravitational theories in which the mass of the universe is created out of its self-contained gravitational and scalar fields, as opposed to the theory of continuous creation cosmology or the steady state theory which depend on an extra 'creation' field.
As an alternative gravitational theory SCC is a non-standard cosmology in which the Brans-Dicke theory (BD) has been modified to allow for mass creation. It relaxes the requirement of the conservation of energy-momentum (or four-momentum) so the scalar field may interact directly with matter.
------------
At this time, there are similar questions in “An Exceptionally Simple Theory of Everything!”. Could an “OLD” approach work?

 

[Dave0101]

My Apologies

I'm sorry guys, but if I can be that dope who wants to bring the layman's version down to a total dummies version... I'm hoping someone can lay this bag of snakes out straight for me.

As I understood it, there are two separate sets of rules for describing the behavior of things... one for very tiny things, and one for very large things. I was also under the impression that the two sets of rules don't relate to each other well, and that the goal of a "theory of everything" was to create one set of rules that works for both. Is that right?

Now as I'm trying (and I think, failing) to wrap my head around this "exceptionally simple" theory, I'm getting the impression that what it actually does is describe, geometrically, all existing particles and their behaviors. I'm seeing it, as someone else said, more like a periodic table of everything physicists have seen and hope to eventually see? I'm getting the impression that it's reliability is derived from it's ability to make everything we know about fit, somehow, on the vertices of this E8 model in a way that properly describes their known behaviors? If so, what about the model illustrates each particle's properties?

Is this at all accurate? Is it actually a unification theory?

Again, my apologies... I've never had a physics class and my math is so bad I can barely balance a checkbook. I'm just trying to get a dummies picture of what this means to people smarter than me.

 

[belliott4488]

As I understood it, there are two separate sets of rules for describing the behavior of things... one for very tiny things, and one for very large things. I was also under the impression that the two sets of rules don't relate to each other well, and that the goal of a "theory of everything" was to create one set of rules that works for both. Is that right?

Yes, that's about it.

Now as I'm trying (and I think, failing) to wrap my head around this "exceptionally simple" theory, I'm getting the impression that what it actually does is describe, geometrically, all existing particles and their behaviors. I'm seeing it, as someone else said, more like a periodic table of everything physicists have seen and hope to eventually see? I'm getting the impression that it's reliability is derived from it's ability to make everything we know about fit, somehow, on the vertices of this E8 model in a way that properly describes their known behaviors? If so, what about the model illustrates each particle's properties?

Again, I think you've got it about right. The comparison with the Periodic Table of the Elements (PT) that we all saw in high school is apt, because both it and the current understanding - the Standard Model (SM) - show fundamental particles (or "elements" in the case of the PT) that fit nicely into an apparently well-ordered structure but with no explanation at all of why that particular structure exists.

In the case of the PT, the explanation came from Quantum Mechanics, which predicts the observed structure from a basic set of simple assumptions. In the case of Garrett Lisi's paper, what he's doing is trying to start with a basic assumption (the group E(8) as the fundamental symmetry group of elementary particles) and from that to derive not only the nice ordered structure of the SM, but also the required symmetries of gravitational theory.

As for your question about what properties of fundamental particles are described by this model, it has to do with "Quantum numbers" that are used to describe the different states in which particles can exist. In the SM certain different "flavors" of particles are forever different, i.e. there's no way to see an electron and a quark as two different states of the same thing. In Lisi's model, these particles are related by "symmetry transformations" or "gauge transformations", so that they really are just different states of same thing. You can think of rotations of the group's root structure, which move one vertex to another, as a way of identifying physical processes that allow such transformations.

This kind of thing is not new, since the SM allows such transformations, e.g. from an electron to a neutrino by the emission of a W+ particle. Lisi's theory extends this by allowing all fundamental particles to be related by such transformations; presumably there would be a mechanism for an electron to transform into a quark.

Is this at all accurate? Is it actually a unification theory?

To assess its degree of accuracy, physicists will have to use it to come up with experimental predictions and then to perform the experiments to see how well they match the predictions. That's a long way off. Nonetheless, if the theory is mathematically sound (the jury's still out), then to the extent that it at least reproduces the predictions of the SM and gravity theory, then it would be as accurate as they already are. The real test is to look for new and unique predictions from it, however.

Again, my apologies... I've never had a physics class and my math is so bad I can barely balance a checkbook. I'm just trying to get a dummies picture of what this means to people smarter than me.

Well, I'd say you're doing pretty well for a "dummy"!

 

[jal]

Tony Smith has been at this long enough to produce a lot of input for E8
http://www.valdostamuseum.org/hamsmith/E8GLTSCl8xtnd.html
--------
I must say that when I first got on the web, Tony's page was one of the first that I found. His explanations/presentations have improved ... a lot...
My understanding improved only a little bit. Tony is moving too fast for me to catch up.

 

[starkind]

Hi Belliot

You said: “The points in this space do not correspond to points in position space, however; they correspond to particle states. Just because, for example, an up quark might appear to the left of a down quark in a particular visualization of the group, that has nothing to do with where it is in position space, and therefore has nothing to do with where we might measure its position (assuming we could even do that).”

This is fine. I surrender the point, although I suspect it will continue to resurface. For future reference, may we agree that: “there is no evidence in position space, which has observable objects moving around each other in three spatial dimensions and one temporal dimension, that it is special or preferred or primal to (in short, more ‘real’ than) any other abstract mathematical space.”?

Hi Dave0101

You asked “what about the model illustrates each particle's properties?”

I would like to expand on this a little.

On page five of Lisi’s paper, you will find in Table 1 a list of particles in the column labeled G2. These particles are quarks and gluons, each with a superscript indicating its color properties. The quarks (q’s in the bottom part of the table, shown in the picture as triangles) are red, green, and blue, and each one has an anti-particle, shown with a bar over the letter. The anti-particle is anti-matter. The red green and blue are just quantum numbers, not really colors as our eyes see them.

The gluons, g, have two superscript colors each, because they act on quarks to change one color into another color. To see how the diagram works geometrically, you should start by labeling each triangle and each circle according to its coordinates on the diagram. If you are looking at the page in color, this will be easy, because each particle is shown in a different color….real color this time, a color you can see with your eyes. You can just match the symbol under column G2 in the table with the symbol on the diagram. I happen to be working with a black and white copy, so I have to look at the values in the table under g-superscript-3 and g-superscript-8. These ‘g’ are just labels of the axis of the two dimensional plane on which the diagram is shown. You can see these labels in the diagram at the top and at the right of the drawing, in faint gray. (Well, mine are gray.) Just think of the top vertex as having value one on the g8 scale and zero on the g3 scale. The quark on the right of the diagram has value one on the g3 and zero on the g8. If you calculate or measure the other positions as on any two dimensional graph, you will see that the circles and triangle positions are shown in the g3 and g8 columns, identifying which particle goes at which position.

Then you can do vector addition to see the possible reactions. Vector addition just means that you place the head of one vector at the tail of the other, keeping the lengths and angles of the vectors unchanged. In this case, the vectors are shown as the faint lines connecting the particle symbols.

For example, the circle on the far right of the diagram represents the red anti-green gluon. If this gluon reacts with the quark at -1/2, 1/2sqrt3, which you can read off the table as a green quark, you simply take the line that goes from the center to the green quark and slide it to the right until the bottom end, or tail, of the line is located at the head of the line from the center to the red anti-green gluon, at the right of the picture. Don’t change the angle or the length of the line, which is to say the vector, and you will find that the head of the transported vector is now on the red quark. This means that a red anti-green gluon reacts with a green quark to produce a red quark.

The column labeled V_beta is a table of root vectors, aka eigenvectors, spanning the E8 space. Each root vector corresponds to a type of particle. The fact that they span the space means that they are present in every view of E8. There is no part of E8 which they do not reach across. This is not true of every vector in the subspaces, which may appear or not appear in different views of the E8 structure.

For the rest, what Belliot said.

Hope this helps. It’s all in the paper. I recap it here to improve my own grasp of the mechanics, and in hopes that others will point up any flaws in my explanation.

S.

 

[patfla]

Quantum variables

Hi – newbie here. Or if I want to be kinder to myself “educated layperson”.

I’ve read the whole topic and know where that puts me: at the bottom of the totem pole. Which is just fine since then there’s no way to go but up.

Have read several Smolin’s books; Peter Woit’s Not Even Wrong and others. So my ears perked up when I first learned of Garrett and his latest paper.

Well there are things that I knew already; things that I’ve learned over the last mth or whatever reading around; and now I have a whole new set of questions. I’ll limit myself to just one of those here. (although as you can see below, it'll hardly be a single sentence).

The components (observables?) of the 8-vectors which are the objects that inhabit the E8 Lie algebra (its operator being the ‘bracket’ or commutator). The components would be the quantum numbers. I’m trying to figure out just what they are.

This topic pointed me to Tbl. 9 on p. 15 of Garrett’s paper. The 8 components seem to be columns 2-9 and they read something as follows (my first stab at TeX):

\frac{1}{2i}\omega ^{3} _{T} \;\;\; \frac{1}{2}\omega ^{3} _{S}<br /> \;\;\; U^{3} \;\; V^{3} \;\; w \;\; x \;\; y \;\; z \;\;

You should see 8 terms above.

Scroll up just slightly from Tbl. 9 in Garrett’s paper where he explains what these are.

The first four are from F4. 2 are associated with so(3,1) gravity and the other 2 are the 2 fields associated with the electroweak. I’m guessing that the omegas on the left are so(3,1) gravity and U^{3} and V^{3} are the electroweak’s 2 fields?

That’s the first half of my question. The other half consists of the remaining 4.

Here, Garrett explains, one has 3 and 1. 3 are the fields associated with the electrostrong and the remaining 1 is something associated with u(1)_{B-L} (whatever that is).

The division of labor here would seem a little clearer: the 3 are x\;\;y\;\;z. And the final one (u(1)_{B-L} ) is w.

Is that right?

All for now – pat

 

[jal]

Hi!
I'm reviving this old thread to ask a few questions and I don't want to litter the technical discussion.

Is having more than one copy on the standard model a problem?
What would it mean?

What model would you use to try to describe what quarks were doing when they were not confined, which would be in the early universe, before decoupling?
jal

 

[starkind]

hi jal

I've been cutting teeth on the other thread too.

I guess you mean having more than one copy OF the standard model...just a typo, probably, but let me know if you mean ON because I don't get that usage.

I wonder if the three copies could represent a time progression...one instantaneous moment and its immediate future and past instants?

 

[jal]

Yes, it's a typo.
You did get my drift ... physical meaning ...how is it to be interpreted.
jal

ps. I had trouble with the audio of J. B. intro to the E 8 presentation.
The audio went out a one of his explanation ... how he got more spheres of the same size in between the other spheres and still keept them one unit away from the center.
If someone could clarify.

 

[jal]

Let’s leave the questions, in the previous post, aside for now.
Since the boys at An Exceptionally Technical Discussion of AESToE are standing around kicking dust I went and did some searching that might help the amateurs.
If you do a search for, "Pierre Darriulat" ELEMENTARY PARTICLES; you will get a doc file that will give a good explanation of elementary particles and their interaction.
I expect that there are probably new info that might modify some of these explanations.
jal

 

[starkind]

Yes, it's a typo.
You did get my drift ... physical meaning ...how is it to be interpreted.
jal

ps. I had trouble with the audio of J. B. intro to the E 8 presentation.
The audio went out a one of his explanation ... how he got more spheres of the same size in between the other spheres and still keept them one unit away from the center.
If someone could clarify.

Hi jal

I'm also still having problems with the vid player. Very frustrating. MIT has a whole world of courses I can't access because the player won't play. I suppose this is because of my Microsoft platform. I'm not savy enough to try to change over to one of the open software platforms. Probably buy an apple next time.

About the one unit away from center question, I think he is saying that the added spheres are kissing the original sphere...so that they are one unit away (one radius) by definition. Of course fourth dimensional problems do not scale the same way three dimensions do. I think you can add the fourth dimensional spheres in between the three dimensional ones because their volume can be extended in the fourth dimension, which we don't see in 3d representations. This is analogous to the way same-sized 3d objects in a 2d picture (such as a photograph) can appear to be different sizes (far away objects look smaller). A 3d rep of a 4d object can be larger or smaller depending on the viewpoint of the observer. Rotations and translations that preserve an object's volume in 4d may rotate and translate in or out of the 3d space such that the 3d cross section is larger, smaller, or even non-existant in the 3d space. So, fitting more identical volume 4d objects into a 3d rep is not a problem.

If this is still difficult, think about a 2d painting of a large crowd of people. The people are all about the same size in the 3d world, but any number of them can be fit into the 2d picture because the ones that are farther away in 3d look smaller in the 2d pic. So 4d spheres can easily fit into the space left over in a 3d dense packing.

 

[jal]

Hi starkind!
Although your description sounds good, it still leaves a funny feeling.

I hope that the amateurs read and SAVE the doc from "Pierre Darriulat". Things have a habit of dissappearing from the web.

I have another bothering some question.

We have measured gravity down to the size of a hair.
We have gravity when we have matter/particles.
We can justify extrapolating gravity down to the size of quarks.
However, when working with scalars, there are no particles.
What is the justification for assigning one of the scalar to gravity?
jal