E8 Theory

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“An Exceptionally Simple Theory of Everything” is a physics preprint proposing a basis for a unified field theory, 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”, “exceptional” Lie 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.

The theory is presented as an extension of the grand unified theory program, incorporating gravity and fermions. 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 must 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.

The theory received accolades from a few physicists amid a flurry of media coverage, but also met with 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. In a follow-up paper, Lee Smolin proposed a spontaneous symmetry-breaking mechanism for obtaining the classical action in Lisi’s model, and speculated on the path to its quantization. In July 2009, Jacques Distler and Skip Garibaldi published a critical paper in Communications in Mathematical Physics called “There is no ‘Theory of Everything’ inside E8“, arguing that Lisi’s theory, and a large class of related models, cannot work. They offer a direct proof that it is impossible to embed all three generations of fermions in E8, or to obtain even the one-generation Standard Model without the presence of an antigeneration. In response to Distler and Garibaldi’s paper, Lisi argued in a new paper, “An Explicit Embedding of Gravity and the Standard Model in E8“, peer reviewed and published in a conference proceedings, that Distler and Garibaldi’s assumptions about fermion embeddings are incorrect and that the antigeneration is not by itself a problem sufficient to rule out the one-generation Standard Model. In July 2010, a group of mathematicians and physicists, including David Vogan, Garibaldi, and Lisi, met for a week-long conference in Banff to discuss the mathematics and physics related to the exceptional groups. In December 2010, Scientific American published a feature article on “A Geometric Theory of Everything”, authored by Lisi and James Owen Weatherall.[2] In May 2011, Lisi wrote an entry in the blog section of Scientific American addressing some of the criticism of his theory and how it had progressed, noting that the theory was still incomplete and made only tenuous predictions, with a precise description of the three generations of fermions and their masses remaining as the largest outstanding problem. In June 2015, Lisi posted a paper, “Lie Group Cosmology”, describing the geometry of E8 Theory as an extension of Cartan geometry, and providing a description of the three generations of fermions via triality, while not predicting their masses.

The goal of E8 Theory is to describe all elementary particles and their interactions, including gravitation, as quantum excitations of a single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E8. A Lie group, such as a one-dimensional circle, may be understood as a smooth manifold with a fixed, highly symmetric geometry. Larger Lie groups, as higher-dimensional manifolds, may be imagined as smooth surfaces composed of many circles (and hyperbolas) twisting around one another. At each point in a N-dimensional Lie group there can be N different orthogonal circles, tangent to N different orthogonal directions in the Lie group, spanning the N-dimensional Lie algebra of the Lie group. For a Lie group of rank R, one can choose at most R orthogonal circles that do not twist around each other, and so form a maximal torus within the Lie group, corresponding to a collection of R mutually-commuting Lie algebra generators, spanning a Cartan subalgebra. Each elementary particle state can be thought of as a different orthogonal direction, having an integral number of twists around each of the R directions of a chosen maximal torus. These R twist numbers (each multiplied by a scaling factor) are the R different kinds of elementary charge that each particle has. Mathematically, these charges are eigenvalues of the Cartan subalgebra generators, and are called roots or weights of a representation.

In the Standard Model of particle physics, each different kind of elementary particle has four different charges, corresponding to twists along directions of a four-dimensional maximal torus in the twelve-dimensional Standard Model Lie group, SU(3)×SU(2)×U(1). The two strong “color” charges, g3 and g8, correspond to twists along directions in the two-dimensional maximal torus of the eight-dimensional SU(3) Lie group of the strong interaction. The weak isospin, T3 (or W), and weak hypercharge, YW (or Y), correspond to twists along directions in the two-dimensional maximal torus of the four-dimensional SU(2)×U(1) Lie group of the electroweak interaction, with W and Y combining as electric charge, Q. Whenever an interaction occurs between elementary particles, with two coming together and becoming a third, or one particle becoming two, each type of charge must be conserved. For example, a red up quark, having charges (g3 = 1 2 {\displaystyle {}={\tfrac {1}{2}}} , g8 = 1 2 3 {\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}} , W = 1 2 {\displaystyle ={\tfrac {1}{2}}} , Y = 1 3 {\displaystyle {}={\tfrac {1}{3}}} ) can interact with a weak boson, W, having charges (g3 = 0, g8 = 0, W = −1, Y = 0), to produce a red down quark, having charges (g3 = 1 2 {\displaystyle {}={\tfrac {1}{2}}} , g8 = 1 2 3 {\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}} , W = − 1 2 {\displaystyle {}={\tfrac {-1}{2}}} , Y = 1 3 {\displaystyle {}={\tfrac {1}{3}}} ). The complete pattern of all Standard Model particle charges in four dimensions may be projected down to two dimensions and plotted in a charge diagram.

In grand unified theories (GUTs), the 12-dimensional Standard Model Lie group, SU(3)×SU(2)×U(1) (modded by Z6), is considered as a subgroup of a higher-dimensional Lie group, such as of 24-dimensional SU(5) in the Georgi–Glashow model or of 45-dimensional Spin(10) in the SO(10) model (Spin(10) being the double cover of SO(10), and having the same Lie algebra). Since there is a different elementary particle for each dimension of the Lie group, in addition to the 12 Standard Model gauge bosons there are 12 X and Y bosons in the SU(5) Model and 18 more X bosons and 3 W’ and Z’ bosons in Spin(10). In Spin(10) there is a five-dimensional maximal torus, and the Standard Model hypercharge, Y, is a combination of two new Spin(10) charges: “weaker charge”, W’, and baryon minus lepton number, B. In the Spin(10) model, one generation of 16 fermions (including left-handed electrons, neutrinos, three colors of up quarks, three colors of down quarks, and their anti-particles) lives neatly in the 16-complex-dimensional spinor representation space of Spin(10). The combination of these 32 real fermions and 45 bosons, along with another U(1) Lie group (corresponding to Peccei–Quinn symmetry), constitute the 78-dimensional real compact exceptional Lie group, E6. (This unusual algebraic structure, reminiscent of supersymmetry, of gauge fields and spinors combined in a simple Lie group, is characteristic of the exceptional groups.)

As well as being in some representation space of the Standard Model or Grand Unified Theory Lie group, each physical fermion is a spinor under the gravitational noncompact Spin(1,3) Lie group of rotations and boosts. This six-dimensional Lie group has a two-dimensional maximal torus (technically a hyperboloid) and thus two kinds of charge, spin, Sz, and boost, St. A Dirac fermion (consisting of fermion and anti-fermion) has eight real degrees of freedom corresponding to its real vs. imaginary parts, left or right chirality, and being spin up or down. Using the Lie group equivalence of Spin(1,3) and SL(2,C), and the chirality of Standard Model weak force fermion interactions, each fermion (and each anti-fermion) can be described as a two-complex-dimensional left-chiral Weyl spinor under gravitational SL(2,C). Accounting for the up or down spin for each of the 16 left-chiral fermions of one generation (or 15 fermions if neutrinos are Majorana), each fermion generation corresponds to 64 (or 60) real degrees of freedom.

In GraviGUT unification, the gravitational Spin(1,3) and Spin(10) GUT Lie groups are combined (modded by Z2) as parts of a Spin(11,3) Lie group, acting on each generation of fermions in a real 64-dimensional spinor representation. The remaining parts of Spin(11,3) include the 4-dimensional spacetime frame and a Higgs field transforming as a 10 under Spin(10). The resulting gauge theory of gravity, Higgs, and gauge bosons is an extension of the MacDowell-Mansouri formalism to higher dimensions. Several physicists objected to the apparent violation of the Coleman-Mandula theorem, which states the impossibility of mixing gravity and gauge fields in a unified Lie group over spacetime, given reasonable assumptions. Proponents of GraviGUT unification and E8 Theory claim that the Coleman-Mandula theorem is not violated because the assumptions are not met.

In E8 Theory, it is observed that the GraviGUT algebra of spin(11,3) acting on one generation of fermions in a real positive-chiral 64-spinor, 64+, can be part of the 248-dimensional real quaternionic e8 Lie algebra,

e8 = spin(12,4) + 128+

The strongest criticism of E8 Theory, stated by Distler, Garibaldi, and others, including Lisi in the original paper, is that given an embedding of gravitational spin(1,3) in the spin(12,4) subalgebra of e8, the 128+ includes not only the 64+ of a generation of fermions, but a 64 “anti-generation” of mirror fermions with non-physical chirality. Since we do not see mirror fermions in nature, Distler and Garibaldi consider this to be a disproof of E8 Theory. Lisi has voiced two responses to this criticism. The first response is that these mirror fermions might exist and have very large masses. The second response, stated in the original paper and in his latest work, is that there is not a single embedding of gravitational spin(1,3) in e8, but three embeddings related by triality, with respect to which the 64 contains a second generation of physical fermions, and the third generation of fermions is contained within spin(12,4).

The algebraic breakdown of the 248-dimensional e8 Lie algebra relevant to E8 Theory is

e8 = spin(4,4) + spin(8) + 8V ⊗ 8V + 8+ ⊗ 8+ + 8 ⊗ 8

This decomposition, attributed to Bertram Kostant, relies on the triality isomorphism between eight-dimensional vectors, 8v, positive-chiral spinors, 8+, and negative-chiral spinors, 8, relating to the division algebra of the octonions. Within this decomposition, the strong force su(3) embeds in spin(8), three triality-related gravitational spin(1,3)’s embed in spin(4,4), the three generations of 60 fermions embed in 8V ⊗ 8V + 8+ ⊗ 8+ + 8 ⊗ 8, and the gravitational frame, Higgs, and electroweak bosons embed throughout, with 18 colored X bosons remaining as new predicted particles.

In E8 Theory’s current state, it is not possible to calculate masses for the existing or predicted particles. Lisi states the theory is young and incomplete, requiring a better understanding of the three fermion generations and their masses, and places a low confidence in its predictions. However, the discovery of new particles that do not fit in Lisi’s classification, such as superpartners or new fermions, would fall outside the model and falsify the theory.


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