lie group: E8

aequinoctium

it seems that something has been solved regarding this 248 dimensional problem.
i am unfamiliar with it but it seems that it's applications are very important in string theory etc.
maybe someone would like to say more...

Son Goku

Well it's the largest of the Exceptional Lie Groups, being 248 dimensional, as you said.

Lie Groups have been very useful since the 1950s when it was realised that they describe symmetries of the different forces. SU(3) describes the strong force and U(1) X SU(2) describes the electroweak force.
Lie groups are basically the reason we thought up quarks.

I'd be able to rant about Lie groups and their use in particle physics until the cows come home,(It's one of my favourite topics in all of theoretical physics) but what do you want to know?

Do you want to know what a group is? What a Lie group is?
Or how we use Lie Groups in physics to figure stuff out?
Or just E8? Or all four?

The explanations are surprisingly simple and it makes particle physics make much more sense.

r3nu4l

I'd like to know a bit about the practical applications of this E8 research.

Will it actually solve any problems or prove any existing theories? Or will it just lend more support to existing theorems.



Oh...and will it make the physics engine in GTA games any better*




*this bit may be a joke

Son Goku

r3nu4l wrote:

Oh...and will it make the physics engine in GTA games any better*

Funnily GTA uses the SU(2) lie group for texture rendering.


Anyway Lie Groups in general are used in physics when you discuss symmetries.
An example is football. A football is symmetric under rotations about the x,y,z axis, in the sense that it would look the same no matter what way you rotated it.
(Drawing a picture on one face of the ball would spoil the symmetry because you'd be able to see the picture rotating. This is called symmetry breaking.*)

When you describe rotations using mathematics you need to use 3 X 3 matrices to describe the rotation. These 3 X 3 matrices are called SO(3), a lie group.
When these 3 X 3 matrices act on a ball, it makes no difference, so we say the ball is invariant (doesn't change) under SO(3). Or, to be a bit more jargony, SO(3) is a symmetry group of a football.

Similarly quarks come in three colors (American spelling is always used) called red, green and blue. They can also be a mix of colours like red and green, i.e. a brown quark. However it makes absolutely no difference to the physics if a red quark became a green quark or a brown quark became a purple(red and blue) quark.
So quarks can have their color changed and act the exact same way, therefore quarks have color symmetry.

Again, just like there was a set of 3 X 3 matrices to handle rotating a football, there is a set of 3 X 3 matrices that handle changing quark color. These matrices are called SU(3). So SU(3) is a symmetry group of quarks.

Basically it was the fact that quarks are SU(3) invariant that allowed us to figure out how the strong force works. Most mathematical models of the strong force would not allow quarks to be SU(3) invariant, so it really narrowed the search.

Now, E8 is another group. It's known that the Strings of String Theory are invariant under this group, so it helps us to figure out how strings work and narrows the work involved in finding out String Theory's predictions.

However Lie Group's main application in physics has always been in helping us find particle symmetries and the forces between particles. Strings and E8 are just a recent continuation of this trend.

Anyway that's a very quick run down.

*Symmetry breaking is the reason electromagnetism and the weak force look like two separate forces.