E8 LIE

An Exceptionally Simple Theory of Everything is a preprint proposing a basis for a unified field theory, named E8 Theory, which attempts to describe all known fundamental interactions in physics, and to stand as a possible theory of everything. The preprint was posted to the physics arXiv by Antony Garrett Lisi in November 2007, and has not been published in a formally peer-reviewed scientific journal. The title is a pun on the algebra used, the Lie algebra of the largest “simple,” “exceptional” Lie group, E₈.

The theory “received accolades from a few physicists amid a flurry of media coverage,” but also “widespread skepticism.” Scientific American reported in March 2008 that the theory was being “largely but not entirely ignored” by the mainstream physics community, with a few physicists picking up the work to develop it further. However, as of July 2008, the paper had nine citations from other arXiv preprints, and was the most downloaded preprint on the arXiv.

Overview

Lisi’s model^[2] 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 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.

The fermions enter, via an unconventional use of the BRST technique, as Grassmann number fields valued in part of the E₈ Lie algebra. The bosons are combined with these fermions as one-form and Grassmann number parts of a sort of superconnection, each valued in separate parts of the E₈ Lie algebra. The curvature of this superconnection is calculated, producing the Riemann curvature, gauge field curvature, gravitational torsion, covariant derivative of the Higgs, and the covariant Dirac derivative of the fermions. This curvature is used to build the modified BF action by hand, in an attempt to match the dynamics of the Standard Model and gravity.

In the paper, Lisi describes several deficiencies in this model. The most important deficiency is noted as an incorrect, or “poorly understood,” inclusion of the second and third generations of fermions in E₈, relying on triality. This deficiency, and the incomplete nature of the model, prevents the prediction of masses for new or existing particles. Also, Lisi notes the use of explicit symmetry breaking in building his action, rather than offering a more desirable spontaneous symmetry breaking mechanism. And, no attempt is made to provide a quantum description of the theory—this being left for future work.

In a follow-up paper,^[7] Lee Smolin proposes a spontaneous symmetry breaking mechanism for obtaining the action in Lisi’s model, and speculates on the path to its quantization as a spin foam.

Non-technical overview

Consider a wavy, two-dimensional surface, with many different spheres glued to the surface—one sphere at each surface point, and each sphere attached by one point. This geometric construction is a fiber bundle, with the spheres as the “fibers,” and the wavy surface as the “base.” A sphere can be rotated in three different ways: around the x-axis, the y-axis, or around the z-axis. Each of these rotations corresponds to a symmetry of the sphere. The fiber bundle connection is a field describing how spheres at nearby surface points are related, in terms of these three different rotations. The geometry of the fiber bundle is described by the curvature of this connection. In the corresponding quantum field theory, there is a particle associated with each of these three symmetries, and these particles can interact according to the geometry of a sphere.

In Lisi’s model, the base is a four-dimensional surface—our spacetime—and the fiber is the E₈ Lie group, a complicated 248 dimensional shape, which some mathematicians consider to be the most beautiful shape in mathematics.^[8] In this theory, each of the 248 symmetries of E₈ corresponds to a different elementary particle, which can interact according to the geometry of E₈. As Lisi describes it: “The principal bundle connection and its curvature describe how the E8 manifold twists and turns over spacetime, reproducing all known fields and dynamics through pure geometry.”^[2]

The complicated geometry of the E₈ Lie group 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 diagram. The coordinates of these points are the quantum numbers of elementary particles, which are conserved in interactions. Such a diagram sits in a flat, Euclidean space of some dimension, forming a polytope, such as the E8 polytope in eight-dimensional space.

In order to form a theory of everything, Lisi’s model must eventually predict the exact number of fundamental particles, all of their properties, masses, forces between them, the nature of spacetime, and the cosmological constant. Much of this work is still in the conceptual stage—in particular, quantization and predictions of particle masses have not been done. And Lisi himself acknowledges it as a work-in-progress: “The theory is very young, and still in development.”^[9]

Source: Wikipedia

Tags: E8, Real Forms

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