Exploring the E8 Theory: A Layman's Explanation

[jal]

From the thread, "An Exceptionally Simple Theory of Everything!"
“I am seeing this reference in several places that E8 is the lie group of an icosahedron (and for that matter E6 is same for a Tetrahedron and E7 is the same for an octahedron). This seems like a very interesting way to approach E8,”

Garrett
All fields of the standard model and gravity are unified as an E8 principal bundle
connection.

Unless I’m mistaken, All that Garrett has done is shown that, “…the leg bone is connected to the hip bone…”

The G2 root system may also be described in three dimensions as the 12 midpoints of
the edges of a cube | the vertices of a cuboctahedron. These roots are labeled g and qIII in Table 2, with their (x; y; z) coordinates shown. These points may be rotated and scaled,
Since the cuboctahedron is the root system of so(6), we have obtained g2 by projecting along a u(1) in the Cartan subalgebra of so(6),

What we have is a 4 legged baby elephant. The fourth leg (gravity) is the same as all the other legs. It doesn’t need to be longer (to Planck scale).
This makes it easier to work with LQG. There are link to LQG which I have gathered at https://www.physicsforums.com/blogs/jal-58039/dynamics-797/
For instance
http://arxiv.org/abs/hep-th/0608210

Loop Quantum Gravity: An Inside View
T. Thiemann
29 Aug 2006

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Wiki is getting better at explaining all of these symmetries. Here are some links.
http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups
List of spherical symmetry groups
http://en.wikipedia.org/wiki/Icosahedron
The icosahedron can also be called a snub tetrahedron
http://en.wikipedia.org/wiki/Tetrahedron
tetrahedron
http://en.wikipedia.org/wiki/Tetrahedral_symmetry
Tetrahedral symmetry
http://en.wikipedia.org/wiki/Polyhedral_compound
polyhedral compound
http://en.wikipedia.org/wiki/Stella_octangula
stella octangula
http://en.wikipedia.org/wiki/Snub_(geometry)
A snub is a related operation. It is an alternation applied to an omnitruncated regular polyhedron.
http://en.wikipedia.org/wiki/Uniform_polyhedron#Definition_of_operations
A uniform polyhedron
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jal

 

[belliott4488]

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

Have a great long weekend.

I think this is a really good concise explanation, and I think it's very close to being exactly correct.

I do have one objection, although I could well be wrong about this. When you talk about projecting the higher-dimensional space in which E8 lives into 3 dimensions, I'm not sure it's right to identify those as the three space dimensions of our regular 4-D world (Minkowski Space). When you project a higher dimensional space to a lower dimensional one, you implicitly assume that the lower dimensional space is embedded in the higher one, i.e. it's a subspace. For example, when I want to visualize a 4-cube (something I do for fun), I first project it into a 3-D subspace, and then to draw it on paper or a computer screen, I have to project that 3-D object to a 2-D subspace. The geometry I use to do this, however, always assumes that I can connect points on the higher-dimensional object to points in the lower dimensional space with straight lines in the higher-D space, i.e. the lower-dimensional space is a subspace of the higher-dimensional one.

Now, in the case of Dr. Lisi's E8 representation, we can certainly project the root space down to three dimensions in order to picture it at all, and then we can project the resulting object down to a 2-D space to draw it on a computer screen or a paper page. These subspaces are just 3- and 2-dimensional geometric spaces, however - in fact, we're really imagining them to be Euclidean spaces, not the curved space in which we live.

More to the point, (and this really is my point, at long last), I don't believe it is correct to say that our 4-D spacetime is a subspace of the higher-dimensional root space of E8. That space is a space of quantum states - a Hilbert Space if you like - and "motion" in that space corresponds to changing states, not to movement in spacetime.

Any thoughts on that?

 

[jal]

belliott4488
More to the point, (and this really is my point, at long last), I don't believe it is correct to say that our 4-D spacetime is a subspace of the higher-dimensional root space of E8. That space is a space of quantum states - a Hilbert Space if you like - and "motion" in that space corresponds to changing states, not to movement in spacetime.
Any thoughts on that?

Sounds good!
When we are "moving" the icosahedron all that we are doing is looking at the different connections. It is still the icosahedron. There is no dynamics, YET, in the connections. The icosahedron is still at t=0. What it will look like when we look at t=1 might be different. Hopefully, the dynamics will be in a "circular" pattern that can last 10^ 35 (a proton).

 

[starkind]

Hmm. Yes, objects in the space we inhabit seem to obey strict physical laws. Objects in conceptual space do not seem to have to obey any such laws.

I suppose I would have to agree that E8 is more a conceptual space, at this point anyway, than an habitual space. And there do seem to be differences between habitual spaces and conceptual spaces. I guess I am still not certain if the space of particles on E8 is an habitual space or not. Could the fact that habitual objects (particles) seem to fit onto a conceptual structure (E8) imply that there does exist some habitual structure dual to the conceptual structure?

I suppose I can avoid the horns by just saying that I am thinking of the Euclidian dual to habitual space, not habitual space itself. However I have to reserve my doubts. In agreement with the holographic model, our habitual space, which we think of as specially separate from conceptual mathematical space, may not be any more than an artificial construct either.

The difference here seems to be that "real" space time, which we seem to inhabit, is apparently resistant to change. We have to work hard to lift a real stone to the top of a tower, but we can imagine it there instantly without any work at all.

I noticed that Garrett says in the first line of his introduction to his paper "We exist in a universe described by mathematics." I was tempted to quibble with this. After all, we have not yet described it all by math. I thought of suggesting that we exist in a universe describable by math, but that isn't defendable either. Can we be sure everything is describable by some kind of math? Is quantum foam, for example, mathematically describable? We don't know that it does exist, but we do not know that it does not exist either. Part of the universe is chaotic, and it seems to me that chaos is indescribable by definition. We can refer to it and describe it as the opposite of what it is not, but there is no precision possible. And math is nothing if not precise.

I guess I still have to be convinced that there is something special about the three space one time that we seem to inhabit. Plato thought it a shadow of a better reality. We have no idea, really, how consciousness functions. Our habitual world could after all be nothing more than habitual thinking.

But the point is really not arguable in physical terms. Therefore I yield. If you prefer that conceptual space cannot be mapped to ordinary space on the grounds that ordinary space is somehow special, I will not object.

 

[belliott4488]

starkind: I'm afraid I don't agree with your distinction between "conceptual" and "habitual" space. We make physical observations, and those are usually what we loosely think of as "real", although they need not correspond to either distance or time (e.g. charge, field strength, temperature, spin, etc.). We can also predict the values of such observations by use of mathematical calculations, which are abstract by nature. All mathematical spaces are abstract, whether they represent spacetime measurements, phase space measurements, Hilbert space measurements, or what-have-you. I might bump into a physical table in the dark, but I'll never bump into a vector, even if it is a spacetime vector.

My point was only that the parts of our theory that represent spacetime measurements belong to an abstract space that doesn't happen to be a subspace of the abstract space that represents quantum states - they're just separate mathematical beasts. Neither is more or less physical, inasmuch as each one is a bit of mathematics that corresponds to something physical, i.e. some set of observables.

My impression is that there has been some confusion that the assertion we're discussing (that the higher-dimensional E8 representation space might describe the universe we live in) means that our spacetime dimensions are part of that space. I don't believe that's so. There is a subgroup of Dr. Lisi's E8 representation that reflects the symmetries of our 4-D spacetime, and it lives in its own subspace of the E8 root space, but that space is not the same thing as spacetime.

[By the way, I'm not sure why you say that "Part of the universe is chaotic," but if you're referring to chaos theory, then that certainly is describable. That's exactly what chaos theory does - it provides a mathematical framework for describing phenomena that exhibit what appears to be random behavior, typically involving nonlinear differential equations.]

 

[jal]

Hi starkind, … belliott4488!
I understand what you are saying.
Let’s keep in mind that E8 is the MINIMUM LENGTH pattern of the positions of all the SM “particles” that Garrett found that should exist at > 10^-15 and CERN will have to find those other 20 BIG GUYS so that the pattern can be completed.
Think of one of those wonderfull pyramids that are made by those extraordinary chinese atheletes. The curtain is not open all the way and we cannot see all of the atheletes. (20) We do know where the athletes must be positioned so that we can see the (quark) athlete.
Eventually, Garrett and others will make the “pyramid drawings” and we will be able to see the pattern that is required to support the (quark) athletes.
Eventually, the right cuboctahedron will be demonstrated.
Reading the different blogs, it is apparent that this approach is making a lot of people go into denial. Garrett has proposed a model that does not include gravity going past 10^-18. It all depends on what CERN will find. Something tells me that there will be a shortage of paper bags due to hyperventilation.

----------
For those who do not like wiki, here is reference from mathworld.
http://mathworld.wolfram.com/Cuboctahedron.html
In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron.
--------
If you have not noticed, I have a bias for the hex. packing.
jal

 

[starkind]

Hi belliott4488

I think we may be closer that you suppose. Anyway, I am not locked into an opinion about the difference between mathematical dimensions and “real” dimensions, or even that there is a difference. I think it is an interesting thing to think about. Maybe this is not the place to discuss it. It is pretty slippery and not a matter for measurement.

Really I think the interest here is how the E8 representation does ‘connect’ to the real world. Garrett talks about this in section two, with a nice picture and graph of the part of E8 which includes quarks and gluons and their interactions, table 1 on page 5. You can select one of the quarks, act on it with one of the gluons, and by simple vector addition, out pops the resultant quark. That is pretty neat.

I guess you have a strong point in arguing that the cubeoctahedron on which this occurs is not a space-time figure. There is an element of time in it, as you have to follow the lines, or vectors, to make the connections. But the space is not demonstrable as something into which you might poke a finger. We don’t have any idea what shapes the quarks take inside a particle. I would like to point out, if we cannot say our real space is a subset of E8, we cannot absolutely rule it out as a possibility either.

One might, I suppose, think of it as a sort of table of numbers, like the multiplication table, only in more dimensions. It is true that a farmer might use a multiplication table to figure out how many beans he can expect from an acre, but that does not mean that the multiplication table or its calculations are in any way connected to the soil in which the beans are going to grow. Maybe E8 is entirely like that.

On the other hand, it seems to me provocative that densely packed spheres (oranges or cannonballs) do stack up in a real space exactly dual to the G2 root system. It seems to me to be too good a clue to dismiss without at least some consideration. The promising connection, I think, would have something to do with Planck space-time spheres in a ‘frozen’ four dimensions. A Planck sphere would be the space an event would fill in a Planck time. It would be a sphere two Planck lengths in diameter. An event at this scale would necessarily exclude any other event from cohabiting the same time-space. If two such events did occur in one time-space sphere, they would result in an instantaneous implosion and so be un-measurable even in theory.

I am happy with the status quo that you will take the view that the space we inhabit is not a subspace of E8, while I continue to amuse myself by looking for connections.

As for chaos, my understanding of chaos theory is that if you add three or more periodic motions, the result is a motion which has unpredictable behaviors. It is not just difficult to predict. It is mathematically impossible. Perhaps there are exceptions, but in general three randomly selected periodic waves, when combined, will have periods of relative stability, followed inevitably by a period of increasing amplitude variation, with more frequent extreme events, followed by a phase shift in which the old stability pattern ceases, and a new stability pattern is started. The thing that is impossible to predict is where on the amplitude scale the new pattern will show up. Now that I think of it, this process resets the zero scale of the periodic stability, and may have a bearing on zero point energy problems.

It has been a long time since I studied chaos theory, and my study was informal, not part of a rigorous course in mathematics, so as usual I welcome corrections from people who are better informed.

The cuboctahedron in table 1 is presented in one possible view, that using g3 and g8 as orthogonal axis. I am thinking now about what the relationship of the points might tell me if arranged on a 3d cube, one which I can hold in my hand and rotate at will. My first insight doing this has been that the gluons form a plane, and that the quarks are another plane on one side of the gluon plane, and the antiquarks are a third plane on the other side of the gluon plane from the quarks. If each quark and antiquark and gluon was represented by a sphere, and the spheres were densely packed on the structure, there would be one more sphere in the interior of the 3d structure. Nothing is shown in that position in Table 1. Is this position one of the missing particles in the E8 theory?

Also, where do the up and down quarks that make up most of our universe sit on the cube? Can we infer something about reality from the fact that we see those quarks commonly and the others only rarely?

I’ll be glad to see any comments.

Thanks,

S

ps I am not ignoring you jal but I have to take a break. More later.

 

[jal]

Staying on the subject…. A new field of study … neucleonagraphy.
http://en.wikipedia.org/wiki/Crystallography
jal

 

[cinacchi]

StarKind: I much appreciated your explanation about symmetry of SO(3). Great job! I am going to use it for a small lesson to my niece (she's in 4.th class of secondary school. Thanks!

 

[jal]

I'm satisfied that there are enough different explanations for the layperson and the amateur.
I did a search of the web.
Youtube presentation/explanation of E8

Views: 101,702
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An Exceptionally Simple Theory of Everything
Hits 11,200
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E8
Hits 2,330,000
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I think that our explanation was not covered by anyone else.
jal

 

[Coin]

Hi starkind, … belliott4488!
Let’s keep in mind that E8 is the MINIMUM LENGTH pattern of the positions of all the SM “particles” that Garrett found that should exist at > 10^-15 and CERN will have to find those other 20 BIG GUYS so that the pattern can be completed.

So... I am basically in the "layman" camp here and don't understand a lot of this, but I think it's worth observing that as far as I understand things, although there are 18 particles predicted, these 18 particles are all kind of "the same" in a certain sense. They're different particles, but all 18 are colored scalars-- excitations of scalar fields which carry a color charge. The reason why there are 18 of them is there is one such field for each combination of (3 colors X 3 quark generations X (particle+antiparticle)). A colored scalar would be as I understand it quite a strange thing to find, so for E8's purposes I think it might be sufficient to find just one of the 18 fields. No?

This said, I have seen several people in this thread say things along the lines that we "should be able to tell if E8 theory is true" at the LHC at and I don't think this is at all a sure thing. I've not seen any argument that we should specifically expect Lisi's colored scalars to be within the range that the LHC can detect them-- Lisi's theory has the same problem of supersymmetry in this regard, probably worse because we know how to get predictions out of SUSY without actually observing a superpartner and E8 doesn't seem to there yet.

(Of course, Lisi's E8 theory does have an unusually high degree of falsifiability at the LHC, since it's packed EVERYTHING in current physics into the E8 structure and left room for nothing else but those colored scalars. If the LHC discovers ANYTHING, it falsifies Lisi's current formulation of E8, because there is no room for anything new! The only possible thing the LHC can find without breaking Lisi's E8 model is the single higgs-- finding supersymmetry, or a technicolor higgs (unless the colored scalars turn out to be part of the higgs mechanism? I don't know if that's possible) would simply not fit into the E8 root system as this month's paper formulated it.)

 

[starkind]

jal

I am very interested in the way the cubeoctahedron comes out naturally in large enough regions of densely packed spheres. Of course they have to be densely packed, that is the distances between spheres have to be MINIMUM LENGTH, for the pattern to show itself. Is this what you mean?

Could you say more about the “model that does not include gravity going past 10^-18?” I am afraid I have to admit I don’t know what that means. Still I am sure you have better understanding of this model than I do. So far I’ve only got to the quarks and gluons in G2, and I’m still working on getting the implications of that part.

An interesting feature of the close packed stack of similar spheres is that the hex pattern comes out naturally, and is easy to see. There are also orthogonal planes, some rectangular, some square. Then there are parallelograms. It all depends how you split the lattice. I was actually surprised when first thinking of this structure that it also includes a cubic pattern. The most basic 3d shape in there is the tetrahedron.

Finally, I started thinking about how the dense pack structure might develop in an infall regime, where spheres are added to the outside of a growing crystal. It turns out that some very complex surfaces can be generated after the third layer goes on. There is a right hand and a left hand pattern, and if they come into competition for spaces in the lattice, flaws develop where there is still a gap, but the gap is too small for another sphere to be fitted into it. These flaws develop in three dimensions in an infinite variety as the sphere grows.

For a while I played with the idea that the flaws are responsible for the broken symmetries which generate laws of conservation, but I never got an understanding of the Noether principles and decided about then that I couldn’t visualize enough to make any more progress without getting the math.

and more later
S

 

[jal]

coin
The points that you make are not at the layman level.
You said,"If the LHC discovers ANYTHING, it falsifies Lisi's current formulation of E8, because there is no room for anything new! The only possible thing the LHC can find without breaking Lisi's E8 model is the single higgs-- finding supersymmetry, or a technicolor higgs (unless the colored scalars turn out to be part of the higgs mechanism? I don't know if that's possible) would simply not fit into the E8 root system as this month's paper formulated it.)"

Barrett said,
"... The weights of these 222 elements|corresponding to the quantum numbers of all gravitational and standard model fields | exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8.
... After all algebraic elements of the standard model have been fit to the E8 Lie algebra there are a few e8 elements remaining, representing new, non-standard particles.
... Because the 18 algebraic degrees of freedom inhabited by x appear amenable to the same sort of factorization as e (see Table 9), it is natural to factor it into three x fields and three colored and three anti-colored Higgs fields, . It could be possible that this new x gives difierent masses to the different generations of quarks and leptons, producing the CKM and PMNS matrices.

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.
-----------
http://mathworld.wolfram.com/Cuboctahedron.html

In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron.
--------
Although I spent a bit of time in bed with “Lie”, I’m not familiar enough to answer using Garrett’s approach.
I will try to answer from the point of view of… neucleonagraphy.
http://en.wikipedia.org/wiki/Crystallography
I have a bias for the hex. packing and I think it would give the same answers.
------------
The E8 ---> cuboctahedron ---> 12 spheres in close packing. If you put in the “particle” positions of the SM then there will be 240/12 = 20 “particles” in each sphere. If all of Standard Model can only supply 222 (as presented by Garrett), then there would be a shortage of 18 or 18/12 = 1.5 “particles” per sphere. The pattern would not be completed.
In order to maintain the the symmetry, it is necessary to have the 240 “particle” positions. With a litle bit of spinning the 20 “particles” that are in a “sphere” are replicated into the other spheres. There is still no dynamics involved in drawing the positions of the “particles.”
Garrett does include the Higgs, so if there is no Higgs then using the neucleonagraphy approach, it would be necessary to use a smaller and different pattern than E8.
I expect to see more proposed patterns and eventually there will be one that will prove to be the right one. (There are more than one SUSY proposals)
All of this E8 is for only ONE “nucleon/proton” within the drip line.
Nobody has yet got to the stage of combining multiple E8, “nucleon/proton” together.
If multiple E8 are put together then I would expect to see the emergence of Crystallography.
---------
starkind
“Could you say more about the “model that does not include gravity going past 10^-18?”
That is my prediction not Garrett’s. If there is a “particle” representing gravity, that fits into the E8 pattern or any other pattern then I cannot see how it could be smaller than 10^-18. There are proposals (Randall) that would put gravity “particles” in this size range.
-------
carlB
How is the E8 java coming?
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Maybe, Garrett will have time to give a few comments.

jal

 

[starkind]

Hi Coin

Actually it sounds like you are getting a lot of stuff that I have not yet had time to evaluate. Could you provide the reasoning for your statements, or a link to where the information comes from? For example, about those 18 colored scalars being hard to find…I am not sure what you mean.

On page 15 Lisi says “The weights of these 222 elements---corresponding to the quantum numbers of all gravitational and standard model fields---exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8.” Hence the number, 18, of E8 roots which are not matched to the standard model.

I hope to post next on a study of how the roots of E8 are used to show the quark-gluon relationships in G2. Eventually I would like to understand the fundamental physical-geometric reasons for this amazing correspondence to E8.

I don’t know if we should find confirmation (or denial) for E8 at LHC, but it seems the best current hope that we may find it.

Could you say more about your assertion that “The only possible thing the LHC can find without breaking Lisi's E8 model is the single higgs-- finding supersymmetry, or a technicolor higgs (unless the colored scalars turn out to be part of the higgs mechanism? I don't know if that's possible) would simply not fit into the E8 root system as this month's paper formulated it.)” I thought Lisi was saying there were a handful of particles that could fit in E8 which are not yet discovered. His section 2.4.1 beginning on page 21 is titled “New Particles.”

In his discussion and conclusion section, on page 29, Lisi says “Future work will either strengthen the correlation to known physics and produce successful predictions for the LHC , or the theory will encounter a fatal contradiction with nature.”

In my child-like imagination, it seems to me that Lisi is hanging the known particles and interactions on E8 like bulbs on the branches of a Winter Solstice Conifer. What we know about the standard model of particles fits perfectly, but leaves 18 of the 240 branches bare. If the corespondence to the E8 model is really more than a coincidence, reality will contain some new physics to fill in the remaining 18 branches. Not?

 

[starkind]

Hi jal

The cubocahedron comes naturally out of a stacking of densely packed spheres of equal size (I am going into simpler descriptions here because I want to keep Grandma involved.) For example, the stack of oranges on a grocer’s display table, or the stack of cannon balls in the public square. If anyone is hesitant about this shape, just get some marbles and a not-too-small box, and start by laying in a single thickness of marbles on the bottom of the box. Lay the first row along one wall, and then add the next row in the dimples, so each of row 2 is supported by 2 from row 1. Continue stacking on third and fourth rows, and you will see a triangle shape develop, and if you look closely after there are three rows, you will notice that the triangles fit together in the plane of the bottom of the box into hexagons.

Then, when you have enough in the first layer, add a second layer, placing the new layer marbles in the dimples of the first layer. Notice that now each marble of the second layer is supported by three marbles in the layer under it. Continue to add layers, and eventually you will see a pyramid develop.

Now the cubeoctahedron is inside this pyramid. It is kind of hard to visualize, but anyone should be able to do it with some work. It consists of one marble in the center, with twelve marbles surrounding it. Or, it may be easier to visualize by thinking of the single layer first, where each marble (except the ones on the edges) is surrounded by six others. If this is still too hard to visualize, start with pennies on a table, and then you can easily verify that one penny laid on the surface can be surrounded by exactly six pennies laid touching each other all around it.

Now imagine the seven marble hexagonal unit suspended in mid-air, and you may be able to see that you can add exactly three marbles on each side of the hexagonal plane. When you have done this, you will see that you have one marble in the center of twelve marbles. Styrofoam balls and toothpicks work well to build this part of the structure, also. You can find a nice picture of this structure in the link jal provided to Stephan Wolfram’s Mathworld page on the cubeoctahedron earlier in this thread.

There are actually two ways to add the three marbles to the top of the hexagonal plane. If you experiment a little, you will see that placing the first marble determines where you can place the next two marbles. There are actually six dimples available, but as soon as you place one marble, it becomes impossible to fit a marble into the other three dimples. You have to use the two remaining dimples where there is enough room.

Because of this fact, it is possible to make two different dense packings of the thirteen marbles. In one form, the cubeoctahedron, the marbles on each side of the hex layer are opposite each other, and in the other, they are opposite to the alternate dimples, the ones that are blocked by the placement of the first marble. This is hard to see in words. If you really want to see this, get some Styrofoam balls and toothpicks. Or, marbles and modeling clay works, too.

Jal, I wonder if your scheme of dividing by 12 is affected by this explication of the geometry. Should you divide by thirteen instead of twelve? If not, why not? Perhaps justification could be made on the basis of the desired effects being confined to a surface?

Thanks,

More later

S

 

[jal]

If we assume minimum length, then the smallest sphere (living in 3D) that can be made is with 6 “particles” on it’s surface. The sphere would have a surface area of 24 units.
If it was a cube then there would be one particle on each face.
If we assume 20 “particles” on the surface of a sphere then the size of the sphere would be 80 units. (X12=960 units) Obviously, by assuming minimum length it should be possible to distribute the 20 “particles” inside the sphere without violating minimum length.
I have not done the calculations to arrive at the smallest sphere that could contain the 20 particles or what kinds of symmetrical arrangement would be possible.
I expect that carlB will need to do it in order to make his E8 java model.
These 20 “particles” will be in a symmetrical arrangements. Repeating the pattern 12 times around a central point and you got a 3d E8.
As you can now imagine it is possible to build different spheres containing more than 6 “particles” up to the 20 “particles” for E8.
Those patterns in those 12 spheres must fit into the size of a proton (10^-15).
One of those options will end up matching with the “particles’ of the Standard Model with the complement of the data from CERN.
Apparently, SU(5) as a unifying theory does not work.
We are at the beginning of neucleonagraphy.
----------
jal

 

[Coin]

The points that you make are not at the layman level.
You said,"If the LHC discovers ANYTHING, it falsifies Lisi's current formulation of E8, because there is no room for anything new! The only possible thing the LHC can find without breaking Lisi's E8 model is the single higgs-- finding supersymmetry, or a technicolor higgs (unless the colored scalars turn out to be part of the higgs mechanism? I don't know if that's possible) would simply not fit into the E8 root system as this month's paper formulated it.)"

Barrett said,
"... The weights of these 222 elements|corresponding to the quantum numbers of all gravitational and standard model fields | exactly match 222 roots out of the 240 of the largest simple exceptional Lie group, E8.
... After all algebraic elements of the standard model have been fit to the E8 Lie algebra there are a few e8 elements remaining, representing new, non-standard particles.

Right, but what I was trying to say is that although there are 18 "new" mystery particles in Lisi's E8, corresponding to the "leftover" roots 223 through 240, I do not think it is the case that these particles could be just anything-- it is not known what EXACTLY those particles are, but Lisi's formulation does predict certain properties for those particles.

As I understand the way Lisi's paper was constructed, he does predict the quantum numbers of these particles-- Lisi's current construction assigns specific spin values and specific charges (?) of various types to the 18 new particles, and says "look for particles that look like this". This means that although there's lots of different functions those 18 extra roots could perform (like assigning masses, as you quote) there's only a limited number of things they could be. They couldn't be, say, fermionic superpartners of the bosons, as far as I know, not without changing the E8 construction majorly, because that wouldn't fit the predicted quantum numbers. So if we found a fermionic superpartner at the LHC, Lisi's E8 wouldn't be able to explain that even though there are still those 18 unassigned roots.

This is to be viewed as a positive feature of Lisi's E8 formulation-- it is good to be specific and it's good to be falsifiable.

Actually it sounds like you are getting a lot of stuff that I have not yet had time to evaluate. Could you provide the reasoning for your statements, or a link to where the information comes from? For example, about those 18 colored scalars being hard to find…I am not sure what you mean.

I'm sorry, I don't mean that they will be hard to find-- I just mean that we don't know whether they will be hard to find or not. Maybe they are easy to find and they will show up at the LHC. Maybe they will be hard to find and they will not become visible until some far-future accelerator. I am just saying, I don't think we know enough about this theory yet to say whether one should expect to see those particles at the LHC. One can hope, of course!

Could you say more about your assertion that “The only possible thing the LHC can find without breaking Lisi's E8 model is the single higgs-- finding supersymmetry, or a technicolor higgs (unless the colored scalars turn out to be part of the higgs mechanism? I don't know if that's possible) would simply not fit into the E8 root system as this month's paper formulated it.)” I thought Lisi was saying there were a handful of particles that could fit in E8 which are not yet discovered. His section 2.4.1 beginning on page 21 is titled “New Particles.”

So to clarify, my statements about the 18 extra particles and "colored scalars" are based on the section 2.4.1 you cite:

After all algebraic elements of the standard model have been fit to the E8 Lie algebra there are a few e8 elements remaining, representing new, non-standard particles. There are two new quantum numbers, X and w... This field factors into three generations, x1=2=3, corresponding to different w quantum numbers, and a new Higgs scalar, PHI, for each color and anti-color. The new field, xPHI, is a joining of x and PHI in the same way ePHI is a joining of the gravitational frame, e, and the Higgs, PHI.

I'm a little confused about some of what that means. But you'll note here that besides just positing these extra fields exist, Garret has specified a number of fairly specific things about what they should "look like". Anyway you'll notice here that beyond just positing the new X and W, Garrett says specifically what to do with them-- you use them to create new Higgs-like scalar fields, exactly 18 of them, in fact, each identified in a specific way. The rest of the section consists of speculations by Garrett as to what these Higgs-like scalar fields might be doing.

I think that this is the correct way to read all this, for one thing because Sabine at Backreaction seems to have read it that way:

He finds a few additional particles that are new, which are colored scalar fields

And also because if you look on page 16 of Garrett's paper, in the big table where he identifies the mappings of roots to particles-- where it is explained what the "symbols" in the diagram pictures mean, that is, and what the quantum numbers for each one are-- the last three rows consist of the xPHI symbols, and exactly 18 of them are listed.

Finally, I actually asked Garret about it, or I tried to anyway. This is from the comments section of Not Even Wrong (emphasis mine):

Also a little confused: so counting up all the fields we expect to see in nature we find they fit with 222 of the roots in your E_8 root system, leaving 18 “extra” roots whose properties as fields are described on page 22 of the paper. The paper seems to be saying that these 18 new fields each act kinda like the Higgs, and each one is identified with a specific one of three generations and a specific color or anti-color. If this reading is correct, what do these generation/color identifications refer to? Does this have to do with the color or anti-color of quark that the field is able to interact with, or is the idea that the field carries color charge, or…?

Yes, exactly so. These new scalar fields have color quantum numbers, and so interact with the quarks and gluons.
In my dreams at night, these new Higgs fields give the CKM matrix, but I don’t know how that works when the sun comes up.
They’re also a potential dark matter candidate, but I don’t say that in my paper because I think that’s a cliche.

(The CKM matrix is this gadget, which according to wikipedia "The CKM matrix describes the probability of a transition from one quark q to another quark q'". Garrett also suggests this possibility that the "new higgses" could have something to do with the CKM matrix in the paper, and in the paper he also suggests there could be relevance to "the PMNS matrix"; according to wikipedia PMNS is to neutrinos as CKM is to quarks, and PMNS appears to explain why http://www.ps.uci.edu/~superk/nuosc.html happens. Incidentally I don't think the dark matter comment should be taken too seriously, a later commenter made some points giving reasons why a colored scalar would not do a good job as a dark matter candidate, which I don't think Lisi responded to.)

--- --- --- ---

...okay, have I lost you all yet?? I might have gone too far away from "layman" terminology here. Let me try to phrase this simpler:

To sum up: Lisi predicts 18 new particles. But he doesn't just predict any old particle: He is able to predict some specific things about the new particles, enough so that if we detect a new particle in a particle accelerator we should be able to say whether that is a particle which Lisi's E8 predicted or not. The particles Lisi predicts are extremely special and distinctive. They are "colored scalars". What does this mean? Well, the "scalar" part means the particles are spin 0. Particle "spin" has to do with how many degrees of freedom that a particle field has. Spin 0 is the least freedom that a field can have-- a spin 0, "scalar" field is the simplest kind of field you can have. When you have a field of this type, you basically just have a number assigned to every single point in space, and those numbers are the "field". Sometimes there is a ripple that passes through the numbers in the field, and we call this ripple a "particle".

Even though they are as simple as a field can get, scalars do a lot of stuff. The most famous (actually I think it might be the only) scalar field that we know about right now in nature is the Higgs field, and the Higgs is just all kinds of useful. For example it is the reason why particles have mass. One of the main things the LHC is trying to do is prove the existence of the Higgs. The LHC is hoping that it will observe a particle called a "Higgs Boson", which is a ripple in the Higgs field. If we see this ripple in the Higgs field, then we will know this field exists. However, maybe we will see something different! There are nonstandard theories that say there is more than one Higgs, and that the different Higgses do different things. (This theory is called for example "technicolor" theory-- although technically supersymmetry predicts more than one Higgs as well.)

So Garrett predicts we'll just see one normal Higgs, like the standard model predicts. However in addition to this he predicts 18 fields that are like the higgs, but special. These higgs-like fields interact with the "color force", also called the "strong nuclear force" which is the thing that holds things like protons and neutrons together. Every quark has a "color" (not actually a color, they just call it that), and the way the colors attract each other binds the quarks together into things like protons very tightly. Garrett's 18 scalars also have colors-- each of the scalar field interacts with one particular kind of quark. So for example one of the scalars interacts with the red generation-I quark and another one interacts with the blue generation-II anti-quark. This is very special! Looking on google I find there are other theories which have tried to incorporate colored scalars before, but this is very very rare. A colored scalar would probably do very interesting things, and it would be very easily identifiable if you built a big enough accelerator to see it (in other words it would be "easy to find", but maybe/maybe not "easy to find at the LHC").

Of course, this is just how I understand things so far, based on the things I quote above. (There are some things I am worried I could be wrong about: First off, maybe it is possible that a future version of E8 theory could take those "x" and "w" quantum numbers that result in the 18 colored scalars, and break them down in some different way that produce some different kind of particle; I just don't know. Second off, in the quote from 2.4.1 of Garret's paper I put above, I ellipsised past something about "a non-standard pair of fields B... interacting with right-chiral fermions". I clipped this part because I don't understand it, and also I can't find any further discussion on it which implies it's not such a big deal. But I don't know what this "B" refers to and maybe it is a bigger deal than I thought. If anyone who understands this more than I do could correct any errors I've made here I'd appreciate it.) So again a reminder, take everything I say with a grain of salt!

 

[jal]

Some layperson that you are!
I think that your post came in at just the right time in the development of this simple explanation.
I read Bee's thread and could not follow all (heheh most) of the discussion.
Now, I feel, it is where the other amateurs on this forum will feel more comfortable with what Garrett is doing. (that should be just about everyone heheh)
I hope others will repond ... I'm going to listen and learn.
Thanks!
jal

 

[SublimeGD]

Wow, this thread has far exceeded my expectations. Everyone keep up the good work.

 

[marcus]

Wow, this thread has far exceeded my expectations. Everyone keep up the good work.

Congratulations on an excellent thread Sublime! the last couple of posts by Coin are really helpful. Quick somebody ask a question, in hopes that Coin will continue

 

[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).
--------
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.