# An Exceptionally Simple Theory of Everything

"An Exceptionally Simple Theory of Everything"[1] is a physics preprint proposing a basis for a unified field theory, very often referred to as "E8 Theory,"[2] 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.[3] 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.[2] 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.

The theory received accolades from a few physicists[citation needed] amid a flurry of media coverage, but also met with widespread skepticism.[4] 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.[5]

In a follow-up paper, Lee Smolin proposes a spontaneous symmetry breaking mechanism for obtaining the classical action in Lisi's model, and speculates on the path to its quantization.[6]

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,"[7] 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 argues, in a new paper "An Explicit Embedding of Gravity and the Standard Model in E8,"[8] peer reviewed and published in a conference proceedings, that some assumptions about fermion embeddings are unnecessary and that the antigeneration is not per se a problem sufficient to rule out the one-generation Standard Model.[8][9] In December 2010 and May 2011 Lisi wrote in the popular magazine Scientific American a feature article on the E8 Theory of Everything and an entry in the blog section of the magazine addressing some of the criticism of his theory and how it has progressed, noting that the theory is still incomplete and makes only tenuous predictions, with the three generation issue remaining as a significant problem.[9]

## Overview

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 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.

Lisi states that 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 kind 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, precludes 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. About it, Lee Smolin proposed 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.[6]

### Non-technical overview

Levels of magnification:
1. Macroscopic level - Matter
2. Molecular level
3. Atomic level - Protons, neutrons, and electrons
4. Subatomic level - Electron
5. Subatomic level - Quarks
6. Lie group geometrical representation level

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.[10]

In general, in each gauge theory based on a Yang–Mills action, the symmetries of the Lie group are associated to a specific kind of particles known as gauge bosons (like photons, W and Z bosons, and gluons in the Standard Model; in models involving supersymmetry this is a little more complicated). These gauge bosons can interact with each other and with fermions according to the geometry of the group and its fundamental representations. One of Lisi's challenges is that his theory identifies fermions with the symmetries that are usually associated only with gauge bosons. Generally this is considered not possible and this aspect of the theory still needs to be completed. Another aspect different from the common approaches is that Lisi's theory includes also gravity in the Lie Group E8. While this aspect has been proven to be possible in supersymmetric theories and impossible in a large class of theories (Coleman-Mandula theorem), it is sometimes attempted in other theories and models that do not strictly belong to those classes. It is still not clear if this feature is achievable in Lisi's theory.[11]

In general, a unified theory has a Lie group large enough to contain the Standard Model symmetries. There are many such theories, some of which have used E8 for a long time (like string theory). In Lisi's specific model, as introduced above, each of the 248 symmetries of E8 corresponds to a different elementary particle: Standard Model gauge bosons, gravitons and Standard Model fermions, which can all interact (as usual in this kind of theories) according to the geometry of the group, in this case E8. Lisi states that: "E8 reproduc[es] all known ﬁelds and dynamics through pure geometry."[1]

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.

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 the model at the moment cannot reproduce all the known particles.

Lisi himself acknowledges it as a work-in-progress: "The theory is very young, and still in development,"[12] and the three generation "issue remains the most significant problem, and until it is solved the theory is not complete and cannot be considered much more than a speculative proposal. Without fully describing how the three generations of fermions work, the theory and all predictions from it remain tenuous."[9]

### Algebraic breakdown

Lisi proposes a decomposition of e8, the 248 dimensional Lie algebra of E8, into parts accommodating the gravitational and standard model fields according to the following schema:[1][13][14][15]

 ToE $\mathrm{e}8$ graviweak $\mathrm{so}(7,1) \,$ strong B-L $\mathrm{so}(1,7) \,$ fermions $3 \times (8 \times 8)$ gravity $\mathrm{so}(3,1) \,$ frame-Higgs $4 \times (2 + \overline{2}) \,$ electroweak $\mathrm{su}(2)_L+\mathrm{su}(2)_R \,$ strong $\mathrm{su}(3) \,$ B-L $\mathrm{u}(1)_{B-L} \,$ new bosons $\mathrm{u}(1)+3 \times 6 \,$ gen 1 $8_{S+} \times 8_{S+} \,$ gen 2* $8_{S-} \times 8_{S-} \,$ gen 3* $8_{V} \times 8_{V} \,$

∗ These two generations are only formally identified as second and third generations, this being a problematic aspect of the theory,[9] as explained above.

### Predictions

By matching 226 known standard model particles to some of the 248 symmetries of E8, the theory is able to predict the existence and quantum numbers of 22 new particles.[1] Three of these are the same new $\mathrm{su}(2)_R \,$ and $\mathrm{u}(1)_{B-L} \,$ gauge bosons as predicted in the Pati–Salam model, the W' and Z' bosons. Another is a new $\mathrm{u}(1) \,$ gauge boson, with a corresponding new quantum number. And the remaining 18 new bosons predicted are new colored fields, interacting with the strong force. Lisi states that some of these 22 particles might be seen at the Large Hadron Collider.[16]

Since Lisi does not specify masses for these particles their prediction is not falsifiable by non-discovery in any given experiment, because the masses could exceed the experiment's reach. However, the discovery of new particles that do not fit in Lisi's classification, such as superpartners, would fall outside the model, and falsify Lisi's match to E8. Also, because the theory at the moment fails to predict all the known particles and the matching of the three fermion generations is tentative and problematic in the model, Lisi places a low confidence in these predictions.

### Q and A

Q: How does time arise from symmetry breaking in E8?

Lisi: It's not precisely clear yet, but there are four time-like directions in E8 that mix to give the direction of time in spacetime.

## Chronology and reaction

Three previous arXiv preprints by Lisi deal with mathematical physics related to the theory. "Clifford Geometrodynamics,"[17] in 2002, endeavors to describe fermions geometrically as BRST ghosts. "Clifford bundle formulation of BF gravity generalized to the standard model,"[18] 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,"[19] 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,[20] at an FQXi conference,[21] and for an FQXi article.[22] Lisi gave his first talk on E8 Theory at the Loops '07 conference in Morelia, Mexico,[23] soon followed by a talk at the Perimeter Institute.[24] John Baez commented on Lisi's work in "This Week's Finds in Mathematical Physics (Week 253)",[25] and Lisi was interviewed on Sabine Hossenfelder's "Backreaction" blog.[26] 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,[27] and responded to press inquiries on an FQXi forum.[28] He presented his work at the TED Conference on February 28, 2008.[29]

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[12] and New Scientist,[30] 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.[31] 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.[32] 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.[33][34] 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.[15][35][36] These arguments were distilled in a paper written jointly with Skip Garibaldi, "There is no 'Theory of Everything' inside E8,"[7] 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,"[37][38] 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.[9] 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,"[8] 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. Addressing the three generation case, Lisi agrees that three generations of fermions cannot be directly embedded in E8, but suggests that a gauge transformation related to triality might be used to relate the 64 mirror fermions and 64 other E8 generators to two other generations of 64 fermions.[9]

The group blog, The n-Category Cafe, provides some of the more technical discussions, with posts by Lisi, Urs Schreiber,[13] Kea,[39] and Jacques Distler.[39]

Sixteen 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.[6] Roberto Percacci's "Mixing internal and spacetime transformations: some examples and counterexamples"[11] addresses a general loophole in the Coleman-Mandula theorem also thought to work in E8 Theory.[9] Percacci and Fabrizio Nesti's "Chirality in unified theories of gravity"[40] confirms the embedding of the algebra of gravitational and Standard Model forces acting on a generation of fermions in $so(3,11) \oplus 64$, mentioning that Lisi's "ambitious attempt to unify all known ﬁelds into a single representation of E8 stumbled into chirality issues."[40] Mathematician Bertram Kostant discussed Lisi's work in a colloquium presentation at UC Riverside.[41] In a joint paper with Lee Smolin and Simone Speziale,[42] 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,"[8] Lisi describes E8 Theory using explicit matrix representations.

On August 4, 2008, FQXi awarded Lisi a grant for further development of E8 Theory.[43][44]

In September 2010 Scientific American reported on a conference inspired by Lisi's work.[45]

In October 2010, Lisi, Smolin and Simone Speziale published a partially related paper on unification, in a peer-reviewed journal, proposing an action and symmetry breaking mechanism, and using an alternative treatment of fermions.[42] In December 2010 Scientific American published a feature article on E8 Theory, "A Geometric Theory of Everything",[2] written by Lisi and James Owen Weatherall.

In December 2011, in his paper, "String and M-theory: answering the critics,"[46] 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.[47] 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.'

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