Guest post: Tony Smith, “Visualizing E8 Physics” December 13, 2007Posted by dorigo in internet, mathematics, physics, science.
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
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
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,
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.
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.