An Exceptionally Simple Theory of Everything

Elementary particle states assigned to E₈ roots corresponding to their spin, electroweak, and strong charges according to E₈ Theory, with particles related by triality. This eight-dimensional root diagram is shown projected onto a Coxeter plane.

"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. 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.

Overview

Electrons and quarks, with electric (Q) and color (g) charges, make up color-neutral protons (with total electric charge Q=+1) and neutrons (with electric charge Q=0), which make up atoms.

The pattern of weak isospin, T₃, and weak hypercharge, Y_W, and color charge of all known elementary particles, rotated by the weak mixing angle to show electric charge, Q, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass.

The pattern of weak isospin, W, weaker isospin, W', strong g₃ and g₈, and baryon minus lepton, B, charges for particles in the SO(10) model, rotated to show the embedding of the Georgi-Glashow model and Standard Model, with electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes thirty colored X bosons, responsible for proton decay, and three W' and Z' bosons.

The pattern of weak isospin, W, weaker isospin, W', strong g₃ and g₈, and baryon minus lepton, B, charges for particles in the SO(10) Grand Unified Theory, rotated to show the embedding in E6.

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, Y_W (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=${\tfrac {1}{2}}$, g⁸=${\tfrac {1}{2{\sqrt {3}}}}$, W=${\tfrac {1}{2}}$, Y=${\tfrac {1}{3}}$) can interact with a weak boson, W^-, having charges (g³=0, g³=0, W=-1, Y=0), to produce a red down quark, having charges (g³=${\tfrac {1}{2}}$, g⁸=${\tfrac {1}{2{\sqrt {3}}}}$, W=${\tfrac {-1}{2}}$, Y=${\tfrac {1}{3}}$). 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, S_z, and boost, S_t. 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) + 8_V ⊗ 8_V + 8₊ ⊗ 8₊ + 8_- ⊗ 8_-

This decomposition, attributed to Bertram Kostant, relies on the triality isomorphism between eight-dimensional vectors, 8_v, 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 8_V ⊗ 8_V + 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.

Technical Overview

The fundamental geometric idea of E₈ Theory is that our universe and its contents exists as quantum excitations of the largest simple real quaternionic exceptional Lie group, E_8(-24). This is described via an extension of Cartan geometry employing a superconnection. The relevant Cartan geometry is modeled on Klein geometry, beginning with a homogeneous space, G/H, in which the initial Lie group is G = E_8(-24) and the subgroup is H = SL(2,C)xS(U(3)×U(2))xZ₃, in which Z₃ = {1,T,T²} is the cyclic group of order three corresponding to a triality automorphism, T, of E_8(-24).

Usually, in Cartan geometry, the deformation of a Lie group, G, preserving the structure of a subgroup, H, is described by allowing the Lie group’s Maurer-Cartan form, θ, to vary, becoming the Cartan connection,

C = W + Ɛ

The resulting geometry, G̃, is that of a principal bundle, with W the principal H-connection, a 1-form valued in Lie(H) over a base manifold, B, modeled on G/H, with the frame, Ɛ, a 1-form valued in Lie(G/H). If H is a reductive subgroup of G, the curvature of the Cartan connection is

F_C = dC + CC = (dW+WW+ƐƐ) + (dƐ+WƐ+ƐW)

In Lisi’s extension of Cartan geometry, the Cartan connection over B is interpreted as a superconnection,

G = W + E + Ψ

over spacetime, M (a subspace of B), in which, Ψ, a Lie(G/H) valued 1-form over B assumed to be orthogonal to M, is interpreted as a set of three fermionic (Grassmann number) fields over M, valued in Lie(G/H), related by triality, T. According to Lisi, the description of Grassmann number fermions as 1-forms orthogonal to spacetime, valued in a spinor representation, provides a clear geometric understanding of what fermions are. In E₈ Theory, the H-connection is physically the gravitational spin connection, ½ω, plus the Standard Model and X boson gauge fields, H = g + W + B + X, while E is the 1-form frame over M, assumed to equal the gravitational frame 1-form, e, times (in the Clifford algebra sense) the Higgs, E = eΦ. The fermionic part of the superconnection, Ψ, is interpreted as the three generation multiplets of Standard Model fermions. The curvature of the resulting superconnection,

G = ½ω + H + eΦ + Ψ

is

F = dG + GG = (½R - e e Φ²) + F_H + (TΦ + eDΦ) + DΨ + ΨΨ

in which R = dω + ½ωω is the gravitational Riemann curvature 2-form, F_H = d H + H H is the gauge field curvature, T = de + ½ωe + ½eω is the gravitational torsion, DΦ = dΦ + [H,Φ] is the covariant derivative of the Higgs, and DΨ = dΨ + [½ω + H + eΦ,Ψ] is the covariant Dirac derivative of the fermions in curved spacetime.

In this geometric description, physical four-dimensional spacetime, M, is considered as a sheaf of gauge-related subspaces of G̃. For the case in which the curvature vanishes, F = 0, there is no excitation of the Lie group, G, and the Higgs field has a vacuum expectation value, Φ=Φ₀, corresponding to a positive cosmological constant, Λ = - 12 Φ₀², with the vacuum spacetime, as a subspace of G, identified as de Sitter spacetime, satisfying R = -6Λee.

Within a Lie group, the Maurer-Cartan form, θ, is the natural frame and determines the Haar measure for integration over the group manifold. With the Killing form of the Lie algebra, this also determines a natural metric and Hodge duality operator on the group manifold. For a deforming Lie group, the Maurer-Cartan form is replaced by the superconnection, G, defined over the entire deforming Lie group manifold via gauge transformation. This superconnection, G, determines the Hodge duality operator, ★, and the curvature, F, of the deforming Lie group. The action for E₈ Theory is the Yang-Mills action, integrated over the entire deforming Lie group.

S = ½ $\displaystyle \int$ (F,★F)

Since the structure of the H subgroup and fermionic directions of B are preserved, this action reduces to an integral over spacetime,

S = ½ V $\displaystyle \int$ {(ee,★ee) Φ⁴ - (R,★ee) Φ² + ¼ (R,★R) + (DΦ,★DΦ) - (T,★T) Φ² + (F_H,★F_H) + (DΨ,★DΨ)}

in which V is a constant volume factor, and the Hodge star, ★, is now the Hodge star for M determined by the gravitational frame, e. As well as the usual Einstein-Hilbert action for gravity, Yang-Mills action, and Higgs action, this action includes a quadratic torsion term, a quadratic curvature term, and a quadratic spinor Lagrangian.

Chronology and reaction

Three previous arXiv preprints by Lisi deal with mathematical physics related to the theory. "Clifford Geometrodynamics", in 2002, endeavors to describe fermions geometrically as BRST ghosts. "Clifford bundle formulation of BF gravity generalized to the standard model", in 2005, describes the algebra of gravitational and Standard Model fields acting on a generation of fermions, but does not mention E₈. "Quantum mechanics from a universal action reservoir", in 2006, attempts to derive quantum mechanics using information theory.

Before writing his 2007 paper, Lisi discussed his work on an Foundational Questions Institute (FQXi) forum, at an FQXi conference, and for an FQXi article. Lisi gave his first talk on E₈ Theory at the Loops '07 conference in Morelia, Mexico, soon followed by a talk at the Perimeter Institute. John Baez commented on Lisi's work in "This Week's Finds in Mathematical Physics (Week 253)", and Lisi was interviewed on Sabine Hossenfelder's "Backreaction" blog. Lisi's arXiv preprint, "An Exceptionally Simple Theory of Everything", appeared on November 6, 2007, and immediately attracted a great deal of attention. Lisi made a further presentation for the International Loop Quantum Gravity Seminar on November 13, 2007, and responded to press inquiries on an FQXi forum. He presented his work at the TED Conference on February 28, 2008.

Numerous news sites from all over the world reported on the new theory in 2007 and 2008, noting Lisi's personal history and the controversy in the physics community. The first mainstream and scientific press coverage began with articles in The Daily Telegraph and New Scientist, with articles soon following in many other newspapers and magazines.

Lisi's paper spawned a variety of reactions and debates across various physics blogs and online discussion groups. The first to comment was Sabine Hossenfelder, summarizing the paper and noting the lack of a dynamical symmetry breaking mechanism. Luboš Motl offered a colorful critique, objecting to the addition of bosons and fermions in Lisi's superconnection, and to the violation of the Coleman-Mandula theorem. In the presentation "What's new at the arXiv?" on May 20, 2008, Simeon Warner stated that Lisi's paper is the most downloaded article on the arXiv. Among the physicists early to comment on E₈ Theory, Sabine Hossenfelder, Peter Woit and Lee Smolin were generally supportive, while Luboš Motl and Jacques Distler were critical.

On his blog, Musings, Jacques Distler offered one of the strongest criticisms of Lisi's approach, claiming to demonstrate that, unlike in the Standard Model, Lisi's model is nonchiral — consisting of a generation and an anti-generation — and to prove that any alternative embedding in E₈ must be similarly nonchiral. These arguments were distilled in a paper written jointly with Skip Garibaldi, "There is no 'Theory of Everything' inside E₈", published in Communications in Mathematical Physics. In this paper, Distler and Garibaldi offer a proof that it is impossible to embed all three generations of fermions in E₈, or to obtain even the one-generation Standard Model. In a press release from his university, "Rock climber takes on surfer's theory", Garibaldi states that his article with Distler is a rebuttal of Lisi's theory. In response, Lisi argues that Distler and Garibaldi made unnecessary assumptions about how the embedding needs to happen. Addressing the one generation case, in June 2010 Lisi posted a new paper on E₈ Theory, "An Explicit Embedding of Gravity and the Standard Model in E₈", peer reviewed and published in a conference proceedings, describing how the algebra of gravity and the Standard Model with one generation of fermions embeds in the E₈ Lie algebra explicitly using matrix representations. When this embedding is done, Lisi agrees that there is an antigeneration of fermions (also known as "mirror fermions") remaining in E₈; but while Distler and Garibaldi state that these mirror fermions make the theory nonchiral, Lisi states that these mirror fermions might have high masses, making the theory chiral, or that they might be related to the other generations.

The group blog, The n-Category Cafe, provides some of the more technical discussions, with posts by Lisi, Urs Schreiber, Kea, and Jacques Distler.

Thirty-eight arXiv preprints have cited Lisi's work. Lee Smolin's "The Plebanski action extended to a unification of gravity and Yang–Mills theory", December 6, 2007, proposes a symmetry breaking mechanism to go from an E₈ symmetric action to Lisi's action for the Standard Model and gravity. Roberto Percacci's "Mixing internal and spacetime transformations: some examples and counterexamples" addresses a general loophole in the Coleman-Mandula theorem also thought to work in E₈ Theory. Percacci and Fabrizio Nesti's "Chirality in unified theories of gravity" confirms the embedding of the algebra of gravitational and Standard Model forces acting on a generation of fermions in spin(3,11) + 64₊, mentioning that Lisi's "ambitious attempt to unify all known fields into a single representation of E₈ stumbled into chirality issues". Mathematician Bertram Kostant discussed Lisi's work in a colloquium presentation at UC Riverside. In a joint paper with Lee Smolin and Simone Speziale, published in Journal of Physics A, Lisi proposes a new action and symmetry breaking mechanism. In "An Explicit Embedding of Gravity and the Standard Model in E₈", Lisi describes E₈ Theory using explicit matrix representations. In "Lie Group Cosmology", Lisi describes the extension of Cartan geometry underlying E₈ Theory.

On August 4, 2008, FQXi awarded Lisi a grant for further development of E₈ Theory.

In September 2010, Scientific American reported on a conference inspired by Lisi's work.

In December 2010 Scientific American published a feature article on E₈ Theory, "A Geometric Theory of Everything", written by Lisi and James Owen Weatherall.

In December 2011, in his paper, "String and M-theory: answering the critics", for a Special Issue of Foundations of Physics: "Forty Years Of String Theory: Reflecting On the Foundations", Michael Duff argues against Lisi's theory and the attention it has received in the popular press. Duff states that Lisi's paper was incorrect, citing Distler and Garibaldi's proof, and criticizes the press for giving too much positive attention to an "outsider" scientist and theory.

External links

• Deferential Geometry - Lisi's wiki, containing some mathematical background
• The Elementary Particle Explorer - an online tool for rotating and examining the particle charges and interactions in the Standard Model, GUTs, and E₈ Theory
• An 8-dimensional model of the universe - Lisi presents his theory at TED