If you're asking, "what is this?", some of us have typed up a quick "Layman's explanation" of quantum numbers of Lie groups at Physics Forums.
If you like the applet, you might also enjoy my simulation of orbits around a black hole. It goes through a sequence of lessons in orbital dynamics using Painleve and Schwarazschild coordinates, with comparison with Newton's theory. And I guess I should link to my old clunky and incomplete crystal drawing program. My specialty is Clifford algebra, specifically C(4,1), which is related to what Garrett is doing in certain ways. This algebra is 64 dimensional and calculations by hand are exhausting and prone to error. So I have a Clifford algebra calculator do the work.
This website is called "MeasurementAlgebra" because it is devoted to Julian Schwinger's Measurement Algebra. The measurement algebra is defined in two papers of his, The Algebra of Microscopic Measurements (1959) and The Geometry of Quantum States (1960). These almost entirely ignored papers give a very elegant foundation to quantum mechanics and quantum field theory. It would be well for you to read these papers before exploring my reasoning for why a broken E8 is natural to find in the quantum numbers of the point particles of an effective field theory. In short, the measurement algebra is peculiarly suited to modeling composite point particles where one ignores the gauge bosons and only tracks the valence particles. If you're wondering, "who are these amateurs doing "New Physics" and what are they doing? a place to start might be Marni Sheppeard's blog post on NP2008.
If you're interested in what I do, I'm an amateur working in elementary particles. My regular job involves stuff like driving forklifts and filling out environmental forms for Liquafaction Corporation's ethanol plant. They are looking for outside investors so if you want to make a lot of money and help the environment, give us a ring. If you're not an insider in the ethanol industry, you will probably learn a lot by reading Robert Zubrin's new book, Energy Victory
While I don't publish papers, my work, especially the extension of Koide's equation to the neutrinos, is referenced in the peer reviewed journal papers. I think it's kind of funny when academics have to mention my home page in their papers. I also have a blog. I can use LaTeX in the blog so I've been putting my most recent work there. I suppose they'll eventually have to reference that, LOL.
If by some reason you get the urge to understand density matrices as a replacement for the state vector formalism of quantum mechanics, you might read the first few chapters of my incomplete book on density matrices. The basic idea is to use qubit Feynman diagrams arXiv:0704.2121 to build a version of QFT that works for non perturbative bound state calculations. This is explained in a long series of blog posts starting here.
NOTES.
VIEW: You're looking at the projection into 2 dimensions of a set of
240 root vectors, each of which is 8 dimensional. There are a lot of
ways of picturing a set of 8 dimensional vectors in 2 dimensions.
The image you see is 2 dimensional, so to create it, we need two
projections, a horizontal projection and a vertical projection.
These two projections give the coordinates (h,v) of a point on the
screen.
To define the two projections requires two 8 dimensional vectors.
But you can see that at the bottom of the applet there is room for
you to enter four 8 dimensional vectors. It is these extra vectors
that give the image movement.
There are two horizontal vectors, "Horizontal" and "Alt Horiz",
that you can write to. (At power up, these are set randomly.)
The vector that is used for the horizontal projection, call it HP, moves
between these two vectors and their negatives by an equation
that looks like:
HP = Horizontal * cos^2(t) + Alt_Horiz * sin^2(t).
Vertical is handled similarly. If you write new numbers into the
applet, you have to left click "Update" in order to have the applet
read the new results.
COLORS: To change the colors, click on the "Colors" button. With some browsers, you will
have to click twice. A color choosing panel will open up. You can move this
panel around so you can select colors while seeing the main panel.
To change colors, left click on a color, and then left click on the thing
you want to color. The rows are F2 reps, the colums G4 reps. If you color
a column or row header, it will color all the roots that fall in that rep.
If you click inside the matrix, it will color that particular segment of
the roots. You can also color the background by clicking on the "Bkg"
button.
When I get around to it, I will program the "Lisi" button to give the
colors he used in his paper. The color interface I've programmed only
lets you specify one color for each segment, or about 20 colors. Lisi
was kind to send me his color list. I haven't looked at it carefully,
but it seems to specify a different color for each 240 roots. The
data structures will handle this. All I have to do is to take his
assignments, put them in the order I write the roots, and then convert
his color codes to RGB format.
Carl Brannen