The Exceptional Symmetry of E8

This movie attempts to show the beautiful symmetry of the exceptional Lie group E8.

Out of all the known Lie groups, E8 stands out as the largest and most complex exceptional group. It has 248 generating elements, which by themselves have an astounding degree of symmetry. This symmetry can only be fully grasped in 8-dimensional space. But luckily it is also possible to project E8 onto a two-dimensional plane, chosen such that the resulting image preserves a small fraction of its total symmetry. There are different choices for these two-dimensional planes, some preserving more symmetry than others. The movie rotates through a selection of these planes in succession.

© Teake Nutma, 2014

An Exceptionally Simple Theory of Everything

“An Exceptionally Simple Theory of Everything” is a physics preprint proposing a basis for a unified field theory, very often referred to as “E8 Theory,” which attempts to describe all known fundamental interactions in physics and to stand as a possible theory of everything. The paper was posted to the physics arXiv by Antony Garrett Lisi on November 6, 2007, and was not submitted to a peer-reviewed scientific journal. The title is a pun on the algebra used, the Lie algebra of the largest “simple,” “exceptionalLie group, E8. The paper’s goal is to describe how the combined structure and dynamics of all gravitational and Standard Model particle fields, including fermions, are part of the E8 Lie algebra. In the paper, Lisi states that all three generations of fermions do not directly embed in E8 with correct quantum numbers and spins, but that they might be described via a triality transformation, noting that the theory is incomplete and that a correct description of the relationship between triality and generations, if it exists, awaits a better understanding.

Lisi’s model is a variant and extension of a Grand Unification Theory (a “GUT,” describing electromagnetism, the weak interaction and the strong interaction) to include gravitation, a Higgs boson and fermions in an attempt to describe all fields of the Standard Model and gravity as different parts of one field over four dimensional spacetime. 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 Eprincipal 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.

In modern particle physics, the most common approach to describe elementary particles and their interactions is usually through a gauge theory based on a Lie group. A Lie group is a mathematical structure with many complex symmetries, which can be described as an object with a complex geometry. In the corresponding quantum field theory, there is a particle associated with each of these symmetries, and these particles can interact with each other according to the geometry of the group and how the particles are related to the group representation. In Lisi’s model, the Lie group used is E8, a group with 248 parameters.

The complicated geometry of Lie groups, E8 amongst them, is described graphically using group representation theory. Using this mathematical description, each symmetry of a group—and so each kind of elementary particle—can be associated with a point in a weight diagram. The coordinates of these points are the quantum numbers—the charges—of elementary particles, which are conserved in interactions.

Garrett Lisi: A theory of everything (video)