Wednesday, November 14, 2007

An exceptionally simple theory of everything

Surfer dude Garrett Lisi has produced a fabulous theory of everything, which, at the classical level at least, unifies the structure of the standard model of particle physics with the structure of general relativity.

The basic idea is that the gauge group of the entire universe is E8, a group which is classified as an exceptional simple Lie group. The gauge field of the entire universe would be represented, at a classical level, by a superconnection upon the total space of an E8-principal fibre bundle over a 4-dimensional space-time. This gauge field subsumes not only the gravitational field, the electroweak field, and the strong field, but all the matter fields as well, including the quark and lepton fields, and the Higgs field.

The diagram here represents the roots of the Lie algebra of E8, each of which purportedly define a possible type of elementary particle. Every Lie algebra has a maximal commuting subalgebra, called the Cartan subalgebra. In each representation of a Lie algebra, the simultaneous eigenvectors of the elements from the Cartan subalgebra are called the weight vectors of the representation, and their simultaneous eigenvalues are called the weights of the representation. In the special case of the adjoint representation, (a representation of a Lie algebra upon itself), the weight vectors are called the root vectors, and the weights are called the roots.

In the case of E8 the Cartan subalgebra is 8-dimensional, hence E8 has 240 roots, and each of these roots is defined by 8 numbers, the eigenvalues of the 8 linearly-independent vectors which are chosen as a basis for the Cartan subalgebra. These 8 numbers are the 'quantum numbers' which define each type of elementary particle.

It's a remarkable paper, which I shall retire to consider at some length.

2 comments:

Garrett said...

Hello Gordon, I'm glad you like the paper. Let me know if you have any questions as you're reading it.

And thanks for writing your excellent book on the standard model. (It's a little pricey though.)

Gordon McCabe said...

Cheers Garrett!

And the book is indeed too expensive; that's Elsevier for you!