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Guest post: Tony Smith, “Visualizing E8 Physics” December 13, 2007

Posted by dorigo in internet, mathematics, physics, science.
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Tony Smith needs no presentation to the readers of this blog, since he often contributes to the discussion of physics posts here.  His web site can be found at tony5m17h.net . I received yesterday, and am glad to publish, the following interesting discussion of the properties of the E8 group, which has attracted a lot of attention since the recent paper by Lisi. Enjoy!

Garrett Lisi at hep-th/0711.0770 describes a physics model based
on the 248-dimensional rank 8 exceptional Lie algebra E8 in which
each of the 240 root vectors of E8 are given a physical
interpretation.

The Lie algebra root vectors of E8 form a polytope (called the
Witting polytope) in 8-dimensional Euclidean space. 8 of the 248
generators of E8 are used to form the 8-dimensional root vector
space, and the remaining 248 – 8 = 240 generators of E8 correspond to
the 240 vertices of the E8 root vector Witting polytope.

Garrett Lisi shows a projection of the 240 vertices down into
2-dimensional space

A youtube movie based on a New Scientist article describes some of Garrett Lisi’s physical interpretations of the vertices, and shows how the patterns of vertices transform under rotations.

In this guest post, I want to describe an alternative set of physical interpretations of the 240 E8 root vector vertices and present a movie of how they transform under rotations, so that E8 physics might be more intuitively visualized. In this post, I will abuse notations by using E8 , Spin(16), etc., for both Lie algebra and Lie group, and I will not be careful about group / algebra distinctions, factors of Z2, and other technical matters that might get in the way of exposition.

Like Garrett Lisi’s E8 physics model, this E8 physics model is based on seeing E8 in terms of

248-dimensional E8(8) = EVIII = 120-dimensional adjoint Spin(16) + 128-dimensional half-spinor Spin(16)

and on seeing 120-dimensional Spin(16) as

120-dimensional Spin(16) = 28-dimensional D4 + 28-dimensional D4* + 64-dimensional 8v x 8g

and on seeing 128-dimensional half-spinor Spin(16) as

128-dimensional half-spinor Spin(16) = 64-dimensional 8s’ x 8g + 64-dimensional 8s” x 8g

and on seeing the 240 root vectors of E8 and the 120 – 8 = 112
root vectors of rank 8 Spin(16) as

240 E8 root vectors = 112 adjoint Spin(16) root vectors + 128 half-spinor Spin(16) root vectors =

= 24 D4 root vectors + 24 D4* root vectors + 64-dimensional 8v x 8g + 64-dimensional 8s’ x 8g + 64-dimensional 8s” x 8g

However, in this E8 physics model the physical interpretations of the 240 root vectors are not exactly the same as in Garrett Lisi’s model. Here is how they look in this model.

In this image of this model there are two sets of 24 vertices each:

24 yellow points correspond to the 24 root vectors of D4 which is used to construct Gravity by a generalized MacDowell-Mansouri mechanism based on the 15-dimensional D3 = A3 Conformal Group Spin(2,4) = SU(2,2). To help get started with visualization, here are the 24 yellow points

in the image. Note that the 24 yellow points form three sets:

6 near the top, in a 1 4 1 pattern corrresponding to the 6 vertices of an octahedron;

12 in the middle, in a 4 4 4 pattern corresponding to the 12 vertices of a cuboctahedron;

6 near the bottom, in a 1 4 1 pattern corresponding to the 6 vertices of a second octahedron.

Note also that a 24-cell can be seen as being made up of a cuboctahedron and two octahedra as in this stereo image:

in which the cuboctahdron is green and the two octahedra are red and blue. So, it is clear that the 24 yellow points form a 24-cell, which is the root vector polytope of the D4 Lie algebra.

24 purple points correspond to the 24 root vectors of D4* which is used to construct the U(3) x SU(2) x U(1) Standard Model based on the 15-dimensional D3 = A3 group SU(4) and its 9-dimensional subgroup U(3) and the 6-dimensional SU(4) / U(3) = CP3 Twistor space, with the U(3) giving the SU(3) x U(1) of the Standard Model and the CP3 Twistor space giving (via relation to quaternionic structure) the SU(2) of the Standard Model. Note that the 24 purple points form a pattern similar to that of the 24 yellow points shown above.

Each of the remaining three sets of 64 vertices is of the form 8 x 8g, where 8g denotes the 8 Dirac gamma basis elements of the Dirac gammas of an 8-dimensional Kaluza-Klein spacetime.

64 blue points correspond to 8v x 8g, where 8v corresponds to the 8 basis elements of an 8-dimensional Kaluza-Klein spacetime, so that the 64 blue points correspond to an 8×8 matrix of the 8 spacetime basis elements with respect to 8 Dirac gammas.

64 red points correspond to 8s’ x 8g, where 8s’ corresponds to D4 +half-spinors and to the 8 first-generation fermion particles (electron, neutrino, red up quark, green up quark, blue up quark, red down quark, green down quark, blue down quark), so that the 64 red points correspond to an 8×8 matrix of the 8 first-generation fermion particles with respect to 8 Dirac gammas.

64 green points correspond to 8s” x 8g, where 8s” corresponds to D4 -half-spinors (mirror image to +half-spinors) and to the 8 first-generation fermion antiparticles,, so that the 64 green points correspond to an 8×8 matrix of the 8 first-generation fermion antiparticles with respect to 8 Dirac gammas.

Note that the 24 yellow D4 + 24 purple D4* + 64 blue = 112 adjoint Spin(16) vertices are in some sense fundamentally bosonic, physically corresponding to gauge bosons or spacetime vectors,

while

the 64 red and 64 green = 128 half-spinor Spin(16) vertices are in some sense fundamentally fermionic, physically corresponding to fermion particles and antiparticles.

which is characteristic of exceptional Lie algebras being constructed by combining adjoint-type and spinor-type repesentations.

To see how the 240 root vectors of E8 transform under rotation, I used a root vector rotation web applet by Carl Brannen and took a bunch of screen shots and used them to make an image-sequence movie. There may be a little glitch about half-way through the 34 second movie (I may have messed up by hitting a reset button, or by taking screen shots a little off center, or etc), but to me it seems that, even so, the movie gives interesting visualization insights into how the 240 root vectors of E8 fit together to describe physics.

Click here to see the .mov movie.

Using the basic components described above, it is natural to construct a Lagrangian

with the 64 blue points (8-dimensional Kaluza-Klein spacetime) as base manifold

with the 24 D4 and 24 D4* yellow and purple points (Gravity and the Standard Model gauge groups) forming curvature terms

with the 64 red fermion particle and 64 green fermion antiparticle points forming fermion terms.

The blue 64 and red 64 and green 64 are related by Triality inherited from the Spin(8) triality among vectors, +half-spinors, and -half-spinors. Instead of using the triality for fermion generations, this model uses Triality to show a subtle supersymmetry between fermions and gauge bosons, seeing the gauge bosons as related to bivectors constructed from the blue 64 vectors, and using the Triality to relate them to the red 64 fermion particles and the green 64 fermion antiparticles.

In the interest of keeping this expository guest post somewhat simple, I will only mention in passing such things as that the 8-dimensional Kaluza-Klein is motivated by the work of Batakis, the second and third generations of fermions are composites of the first generation fermions, the Higgs mechanism comes from a geometric construction due to Meinhard Mayer, the force strengths as particle masses are calculated using structures related to bounded complex domains in the spirit of Armand Wyler, etc. For such details and more, as well as references, see my web page entitled E8, Cl(16) = Cl(8) (x) Cl(8), and Physics Calculation or the corresponding 82-page pdf version.

I will try to reply to comments here not only about the visualization movie, but also about any questions that might arise from the 82-page detailed paper.

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Comments

1. Kea - December 13, 2007

Thanks, Tony! This is a great summary of your paper for dummies like me. A 3x(twistor space) triality would be highly reminiscent of G. Sparling’s work on a three time physics, associated with the variable c ideas of Louise Riofrio, but I recall that you prefer the convention of a /\ for the 1-time (D3) sector, as in conventional approaches. Since we would like to break E8 (which is just a classical symmetry after all) with the addition of the extra degrees of freedom, is there a sense in which you see this picture as an emergent description?

2. Anonymous - December 13, 2007

Umm, I’m going to try to be as non-insulting as I can, but… why does anyone take Tony Smith seriously? Sure, he can play around with algebras in comment sections a lot. But his website is RIDICULOUS. Crying about how his treatise on “Sufi Islam, IFA, the Rig Veda, and Physics and the multicultural backgrounds of Jesus and Mary Magdalene” was “banned by Cornell” (actually the arXiv)… as if it were some kind of controversial social work, instead of just a bad pseudoscience paper. Quantum consciousness with Clifford algebras and closed timelike loops which correspond directly to abstract thoughts… and claims that “ETs” might be taking over “our” sector of the Milky Way because we’re not communicating to them in proper Clifford algebra form. I could go on, but I just can’t bear to continue clicking; it’s too painful.

So, I ask, why do people actually take this guy seriously?

3. Kea - December 13, 2007

Why do people actually take this guy seriously?

They do?

4. carlbrannen - December 14, 2007

Okay, that does it, I need to improve the applet. Thanks for the reference, Tony, I’m going to add a feature to that E8 program so that it can run canned scripts. It will basicallly consist of a copy of the code I wrote for the gravity simulator applet.

What I need from you is a set of numbers that describe the motion you like and a set of colors. You can send that to me at my usual address carl at brannenworks dot com. Probably the easiest way to send me the data is to take a screen shot of your computer screen showing the color scheme (which you get by pressing the [color] button), and the 16 numbers describing the two ends of the root axes.

5. Fred - December 14, 2007

Umm, I’m going to try to be as non-insulting as I can, but…
So, I ask, why do people actually take this guy going under the name of Anonymous seriously? The fear of revealing one’s own identity must be a terrible thing to live with considering we espouse the joys of living in a free and open society. Taking pot shots at someone from behind a mask is the sure sign and mark of a coward.

6. Tony Smith - December 14, 2007

Kea, you ask “… is there a sense in which you see this picture as an emergent description? …”.
Yes. For example:
1 – the whole thing can be seen as emergent in a sense that Cl(8) structure might emerge like Yin-Yang of I Ching etc. See
http://tony5m17h.net/simph.html
with the chains of Cl(8) emerging as described in
http://tony5m17h.net/ClifTensorGeom.html
2 – as to breaking the entire E8 symmetry, the key is dimensional reduction of the full 8-dim spacetime into a 4-dim physical spacetime plus a CP2 internal symmetry space, and that is driven by freezing out of a preferred quaternionic subspace of the original E8 8-dim octonionic space.
That is what makes the second and third fermion generations emerge as composites of the first, and the Higgs to emerge from a Tquark condensate, and a conformal / varying c phase of cosmological gravity emerge.

Carl – Thanks very much for your E8 root vector applet.
Your color program is now OK and easy to use,
as is your way of entering the parameters.
What I had to do by hand was to take snapshots
(attempted every half second or so) to put together to
make a movie, and if you could automate a string of
snapshots that would be nice.

Fred – Thanks for your comment.

Anonymous – whoever you are, why do you read my web site if it causes you pain, and why do you care whether or not anyone takes me seriously ?

Tony Smith

7. Doug - December 15, 2007

Hi Tony,

1 – The AIM has a representation of the ‘crystal graph for E8′ which appears to be a different perspective of the E8 root system.
To me this suggests some type of torus?

http://aimath.org/E8/e8graphinfo.html

2 – Plasma physics appears to recognize the link among electromagnetism, torus and helix.
Is this some type of ‘Monster toroidal symmetry’ once referred to as a ‘folded-doughnut’ with repect to the Borcherd proof?

http://physicsworld.com/cws/article/print/24295/1/PWfus3_03-06

8. Tony Smith - December 15, 2007

Doug, the AIM crystal graph is of all 248 elements of E8,
while the root vector polytope has only 248 – 8 = 240 vertices,
where the 8 elements not represented are 8 Cartan subalgebra
elements that are represented by the 8 basis vectors of the
8-dim Euclidean space in which the root vector polytope lives.

You are indeed correct that the E8 root vector polytope
“suggests some type of torus”.

To see that, note that half of the 240 vertices can be projected
into the 120 vertices of a 600-cell in 4-dim space,
and
that the 600-cell is dual to the 120-cell in which has
120 dodecahedra as 3-dim faces (since it is a 4-dim polytope)
with the centers of the 120 dodecahedra being 120 vertices
that are half of the 240 of the E8 8-dim polytope.

Now, here is the fun part:
Look at Coxeter’s CBMS book/pamphlet “Twisted Honeycombs” (AMS 1970) on pages 19-20 where he says:
“… Imagine a pillar formed by ten solid dodecahdra …
Adjacent dodecahedra … share a horizontal pentagonal face …
The five edges of this pentagon form a neck round which there is ample room to hang a necklace of five new dodecahedra …
When all ten horizontal pentagons … have been decorated with necklaces, we have altogether a pillar consisting of 10 + 50 = 60 dodecahedra ….
The original ten are entirely covered by the fifty,
and the exposed faces of the fifty form … 200 pentagons …
So far, we have regarded the sixty dodecahedra as being in ordinary 3-space … The next step is to put them into a Euclidean 4-space … and to bend the original pillar into a ring … producing a ring-shaped … torus …
A complete polytope …. is obtained by making a second complex of sixty cells … and interlocking the two rings …
the polytope is regular … the total number of cells is 120 …[it].. is called the 120-cell” …”.

So, as Coxeter shows beautifully, the 120-cell is two interlocking tori of dodecaheda, and two 4-dim 120-cells can be put in 8-dim space to form the 240 vertices of the E8 root vector polytope.

It is even more fun to read, in the book “Shaping Space” (ed. by Senechal and Fleck, Birkhauser 1988) the article by Banchoff (chapter 16, pages 221-230) entitled “Torus Decompositions of Regular Polytopes in 4-Space” where he says:
“… Torus decompositions are especially well suited for describing the Hopf mapping … The familiar central projection of the hypercube suggests a decomposition of the hypersphere into solid tori, and this decomposition carries over to other polytopes as well … Similar treatments of the 120-cell and the 600-cell are implicit in the work of several mathematicians … The coordinates for the 24-cell … are very similar to those which appear in Coxeter’s discussion … in “Twisted Honeycombs” … although he does not explicitly use the Hopf mapping … Professor Coxeter pointed out that these coordinates also appear … in the 1951 dissertation of G. S. Shephard. …”.

Tony Smith

PS – It is sort of interesting to compare the cooperative tone of mathematicians dealing with such geometric things
with the bitter hatred that seems to pervade much of the theoretical physics community nowadays.

9. Doug - December 17, 2007

Hi Tony,

1 – Thanks for explaining the difference between the E8 crystal graph and root vector polytope.

2 – Thanks for the Coxeter 1970 and Senechal / Fleck 1988 references. I should be able to get them via interlibrary loan over the next 2-4 months.

3 – The Banchoff (chapter 16), “Torus Decompositions of Regular Polytopes in 4-Space” is referenced in two more recent books on Google book search. Pages are viewable except for p2-3 of a.
a – RV Moody – Science – 1997 – 555 pages, The Mathematics of Long-Range Aperiodic Order – Page 7 in Colin C Adams ‘Knotted Tilings’ p1-8, [p2-3 not shown] NATO ASI Series.
b- Peter McMullen, Egon Schulte – Mathematics – 2002 – 566 pages, Abstract Regular Polytopes – Page 520 in the bibliography for section 6B ‘Regular Polytopes on Space Forms – Locally Spherical Polytopes’ p152-162.

Have you read these by any chance?

4 – This work appears to be related to Ergodic Theory which Terence Tao will discuss on his blog What’s New in the post ‘254A: Topics in Ergodic Theory’.

10. Tony Smith - December 18, 2007

Doug, thanks for the references, which I had not read.

I looked on Amazon about “The Mathematics of Long-Range Aperiodic Order” and was dismayed to find that the list price was $399.00 and the cheapest copy available was secondary sellers was $386.61, so I might remain ignorant of some nice stuff because of that.

However, I have spent the $138.00 (still very high) price for the book “Abstract Regular Polytopes” to order it from Amazon.

The stuff does seem to be related to ergodic theory.
For instance, the Hopf map is related to the structure of the 3-sphere S3 and flows on S3 may be important in ergodic theory and the work of Perelman on which Terence Tao has been involved lately.

Tony Smith

11. Doug - December 20, 2007

Hi Tony,

With books this expensive, I find it prudent to first look at them through the interlibrary loan program.

At Google book search, there is a feature “Find this book in a library”, that sometimes may require a search in ‘other editions’, for example:

a – The mathematics of long-range aperiodic order lists 51 libraries [domestic and foreign] with their distance from your location.

b – Abstract regular polytopes lists 232 libraries for all 4 editions.

12. dorigo - December 21, 2007

Doug, very good suggestion. I did not know about that search feature!

Cheers,
T.

13. My Domains » Blog Archive » Guest post: Tony Smith, “Visualizing E8 Physics” - January 3, 2008

[...] Original post by dorigo [...]

14. Alejandro Rivero - May 10, 2009

last year (mid 2008) LJ Boya suggested to try speculatively to relate the 4-dim 24-cell polytope to a susy theory of 128 fermions and 128 bosons, as 11D supergravity. The 24 cell has 1-24-96-96-24-1 elements, so a total of 121+121. Were it possible to argue to mute 1 to 8, it would be a 128+128.

LJ stresses that: 1) it is better to separate gauge and higgs: 24 degrees of freedom for SU(3)xSU(2)xU(1), 8 dof for the higgses. 2) the MSSM with masive neutrinos has exactly 128 bosons + 128 fermions.

Now, let me think. Could the SM emerge from supergravity? The 44 components of the 11D graviton reduce to 2 dof in 4D. Of the extant 42, we should take 24 for the gauge group. From the other 18, plus the 84 11D scalars, we must to build the 96 sfermions and the higgs mechanism, which can not be the MSSM one. Or we need to conjure 2 dof from the tower of kaluza klein.

15. Siul Segrob - October 13, 2009

Hi, I am looking for Tony Smith. None of the previous e-mail addresses that I had from him are working now.

If somebody knows his e-mail address, please could you contact him and let him know that I would like to discuss something with him ?
He knows me.

My e-address is : siul.segrob@gmail.com
Cheers


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