Garett Lisi's 248-dimensional theory?

[Matthewkind]

I'm sorry to be so bothersome; however, I've recently been hearing a lot of good things about this Garett Lisi person. Apparently, he can explain the fundamental forces by using Lie Groups, which is a branch of mathematics that I am currently not familiar with. I only know that it works through some sort of crazy circle distortion thing. Can anyone explain to me how Lie groups work? Also, can you explain to me exactly what Mr. Lisi is positing when he claims to have reconciled quantum mechanics and general relativity?

 

[Kevin_Axion]

He essentially used the Largest Exceptional Lie Group $$E_8$$ and put gravity and quantum mechanics into the $$E_8$$ structure. The transformations of $$E_8$$ structure produce the particle interactions et cetera. Particle physics already uses Lie Groups such as the Electroweak Unification uses $$SU(2) \otimes U(1)$$ which is a sub-group in the larger $$E_8$$ group. It's more of a GUT then a full theory of everything as many other problems aren't addressed and the theory is still in its infancy.

"More specifically, Lisi combines the left-right symmetric Pati-Salam GUT with a MacDowell-Mansouri description of gravity, using the spin connection and gravitational frame combined with a Higgs boson, necessitating a cosmological constant. The model is formulated as a gauge theory, using a modified BF action, with E8 as the Lie group. Mathematically, this is an E8 principal bundle, with connection, over a four dimensional base manifold. Lisi's embedding of the Standard Model gauge group in E8 leads him to predict the existence of 22 new bosonic particles at an undetermined mass scale." - Wikipedia

An example of a Lie group is a rotation in the complex plane about a circle with radius 1. This is called the $$U(1)$$ Lie group. I believe the transformations follow the equation $$e^{i\theta} = cos \theta+ isin\theta$$

Wiki: http://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything
TED Talk:
Elementary Particle Explorer: http://deferentialgeometry.org/epe/EPE3.html

 

[suprised]

He essentially used the Largest Exceptional Lie Group $$E_8$$ and put gravity and quantum mechanics into the $$E_8$$ structure.

Nope, there is nothing about a unification of gravity and quantum mechanics in this model. On the contrary, it is in violation of basic quantum mechanical principles like the spin-statistics theorem.

 

[martinbn]

Which $$E_8$$ group does he use?

 

[lpetrich]

More properly, which E8 representation does he use?

From what I gather, it's the 248 representation, which is both the fundamental one and the adjoint one.

Fundamental -- every other irreducible representation can be derived from product representations. If X_i is the rep's vector space, the product representations are X_i*X_j*X_k...

Adjoint -- the rep's vector space is the generators of the algebra. Gauge fields must be in the adjoint rep of their gauge algebra.

So gauge fields must be in the E8 248 rep.

The next larger E8 rep has size 3875, which is even worse.

For the 3D angular-momentum algebra, a.k.a. A1, B1, SU(2), SO(3), etc.:
Fundamental: spinor (spin 1/2, dimension 2)
Adjoint: vector (spin 1, dimension 3)

-

It's rather obvious that no experimentally-accessible particles have unbroken E8 gauge symmetry, so it must be broken in some way. But there are lots of ways to break E8:

D8 = SO(16)
A7*A1 = SU(8)*SU(2)
A5*A2*A1 = SU(6)*SU(3)*SU(2)
A4*A4 = SU(5)*SU(5)
D4*A3 = SO(10)*SU(4)
E6*A2 = E6*SU(3)
E7*A1 = E7*SU(2)
A8 = SU(9)

D7*U(1) = SO(14)*U(1)
A6*A1*U(1) = SU(7)*SU(2)*U(1)
A4*A2*A1*U(1) = SU(5)*SU(3)*SU(2)*U(1)
A4*A3*U(1) = SU(5)*SU(4)*U(1)
D5*A2*U(1) = SO(10)*SU(3)*U(1)
E6*A1*U(1) = E6*SU(2)*U(1)
A7*U(1) = SU(8)*U(1)

F4*G2

Some of these subgroups contain not only the Standard-Model symmetry, SU(3)*SU(2)*U(1), but also such much-discussed supersets of it as SU(5), SO(10), and E6.

The Pati-Salam model, mentioned in the OP, has symmetry SU(4)*SU(2)*SU(2) or SO(6)*SO(4)

It's a subset of SO(10), and thus, E6 and E8.

 

[martinbn]

More properly, which E8 representation does he use?

Well, first I want to know which group.

 

[lpetrich]

What possibilities are there?

 

[martinbn]

What possibilities are there?

There is the complex group. There are three real forms and two non trivial double covers. I am guessing he uses a real group and it is either the compact form or the split one.

 

[lpetrich]

Such global properties usually don't matter much in particle physics. What's usually important are the local properties -- the Lie algebra. That gentleman is apparently using a 248-dimensional representation, since it's the adjoint one.

 

[martinbn]

Such global properties usually don't matter much in particle physics. What's usually important are the local properties -- the Lie algebra.

If it is the algebra then why call it group! Then my question is which algebra of type E8 is he using?

 

[lpetrich]

The exceptional Lie algebra, as far as I can tell.

 

[garrett]

$$E_{8(-24)}$$
http://arxiv.org/abs/1006.4908"

 

[martinbn]

$$E_{8(-24)}$$
http://arxiv.org/abs/1006.4908"

Thanks.