JOHN BAEZ WRITES ON E8
(Week 95 way back in 1996)http://math.ucr.edu/home/baez/week95.html
(credit from Wikimedia: E8 graph from Peter McMullen work. Many Thanks to aimath.org site! The edge's color is related to relative edge angle. see full image here: http://commons.wikimedia.org/wiki/Image:E8_graph.svg)
In dimension 8 there is only one even unimodular lattice (up to isometry), namely the wonderful lattice E8! The easiest way to think about this lattice is as follows:
Say you are packing spheres in n dimensions in a checkerboard lattice - in other words, you color the cubes of an n-dimensional checkerboard alternately red and black, and you put spheres centered at the center of every red cube, using the biggest spheres that will fit. There are some little hole left over where you could put smaller spheres if you wanted.
And as you go up to higher dimensions, these little holes gets bigger! By the time you get up to dimension 8, there's enough room to put another sphere OF THE SAME SIZE AS THE REST in each hole!
If you do that, you get the lattice E8. (I explained this and a bunch of other lattices in "WEEK65", so more info take a look at that.) In dimension 16 there are only two even unimodular lattices. One is E8 + E8. A vector in this is just a pair of vectors in E8.
The other is called D16+, which we get the same way as we got E8: we take a checkerboard lattice in 16 dimensions and stick in extra spheres in all the holes. More mathematically, to get E8 or D16+, we take all vectors in R8 or R16, respectively, whose coordinates are either all integers or all half-integers, for which the coordinates add up to an even integer. (A "half-integer" is an integer plus 1/2.)
So E8 + E8 and D16+ give us the two kinds of heterotic string theory! They are often called the E8 + E8 and SO(32) heterotic theories.
In "WEEK63" and "WEEK64" I explained a bit about lattices and Lie groups, and if you know about that stuff, I can explain why the second sort of string theory is called "SO(32)". Any compact Lie group has a maximal torus, which we can think of as some Euclidean space modulo a lattice.
There's a group called E8, described in "WEEK90", which gives us the E8 lattice this way, and the product of two copies of this group gives us E8 + E8. On the other hand, we can also get a lattice this way from the group SO(32) of rotations in 32 dimensions, and after a little finagling this gives us the D16+ lattice (technically, we need to use the lattice generated by the weights of the adjoint representation and one of the spinor representations, according to Gross). In any event, it turns out that these two versions of heterotic string theory act, at low energies, like gauge field theories with gauge group E8 x E8 and SO(32), respectively! People seem especially optimistic that the E8 x E8 theory is relevant to real-world particle physics...
Links referred to
WEEK 63 (Lattices & Lie Groups)http://math.ucr.edu/home/baez/week63.html
WEEK 64 (Lattices & Lie Groups)http://math.ucr.edu/home/baez/week64.html
WEEK 65 (A "bunch of lattices")http://math.ucr.edu/home/baez/week65.html
WEEK 90 (About E8)http://math.ucr.edu/home/baez/week90.html
ON THE SAME PAGE, BAEZ WRITES WITH REGARD TO LUCAS NUMBERS IN RELATION TO LATTICES...
Also see related post on this thread regarding possible Lucas/Perfect Number Correspondence based on revised definition of Perect Number...http://forum.physorg.com/index.php?showtop...ndpost&p=285199
It's a bit funny how one of the most curious features of bosonic string theory in 26 dimensions was anticipated by the number theorist Edouard Lucas in 1875. I assume this is the same Lucas who is famous for the Lucas numbers: 1,3,4,7,11,18,..., each one being the sum of the previous two, after starting off with 1 and 3. They are not quite as wonderful as the Fibonacci numbers, but in a study of pine cones it was found that while most cones have consecutive Fibonacci numbers of spirals going around clockwise and counterclockwise, a small minority of deviant cones use Lucas numbers instead.
Anyway, Lucas must have liked playing around with numbers, because in one publication he challenged his readers to prove that: "A square pyramid of cannon balls contains a square number of cannon balls only when it has 24 cannon balls along its base". In other words, the only integer solution of
1^2 + 2^2 + ... + n2 = m2,
is the solution n = 24, not counting silly solutions like n = 0 and n = 1.
he laters goes on to mention...
Anyway, what does all this have to do with Lucas and his stack of cannon balls?
Well, in dimension 24, there are 24 even unimodular lattices, which were classified by Niemeier. A few of these are obvious, like E8 + E8 + E8 and E8 + D16+, but the coolest one is the "Leech lattice", which is the only one having no vectors of length 2. This is related to a whole WORLD of bizarre and perversely fascinating mathematics, like the "Monster group", the largest sporadic finite simple group - and also to string theory. I said a bit about this stuff in "week66", and I will say more in the future, but for now let me just describe how to get the Leech lattice.
First of all, let's think about Lorentzian lattices, that is, lattices in Minkowski spacetime instead of Euclidean space. The difference is just that now the dot product is defined by
(x1,...,xn) . (y1,...,yn) = - x1 y1 + x2 y2 + ... + xn yn
with the first coordinate representing time. It turns out that the only even unimodular Lorentzian lattices occur in dimensions of the form 8k + 2. There is only one in each of those dimensions, and it is very easy to describe: it consists of all vectors whose coordinates are either all integers or all half-integers, and whose coordinates add up to an even number.
INCIDENTALLY, BAEZ MENTIONS THE FOLLOWING BOOK ON RENORMALIZATION
definitely seems worth a read...
1) Laurie M. Brown, ed., "Renormalization: From Lorentz to Landau (and Beyond)", Springer-Verlag, New York, 1993.
It's a nice survey of how attitudes to renormalization have changed over the years. It's probably the most fun to read if you know some quantum field theory, but it's not terribly technical, and it includes a "Tutorial on infinities in QED", by Robert Mills, that might serve as an introduction to renormalization for folks who've never studied it.