.

"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 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 don't strictly belong to those classes. It's 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, Lisi 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.
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 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"[40] 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"[41] 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."[41] Mathematician Bertram Kostant discussed Lisi's work in a colloquium presentation at UC Riverside.[42] In a joint paper with Lee Smolin and Simone Speziale,[43] 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.[44][45]

In September 2010 Scientific American reported on a conference inspired by Lisi's work.[46]

In October 2010, Lisi, Lee 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.[43] 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,"[47] 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.[48] 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.
References

^ a b c d A. G. Lisi (2007). "An Exceptionally Simple Theory of Everything". arXiv:0711.0770 [hep-th].
^ a b c A. G. Lisi; J. O. Weatherall (2010). "A Geometric Theory of Everything". Scientific American 303 (6): 54–61. doi:10.1038/scientificamerican1210-54. PMID 21141358.
^ Greg Boustead (2008-11-17). "Garrett Lisi's Exceptional Approach to Everything". SEED Magazine.
^ Amber Dance (2008-04-01). "Outsider Science". Symmetry Magazine. Retrieved 2008-06-15.
^ Collins, Graham P. (March 2008). "Wipeout?". Scientific American: 30–32. Retrieved 2008-06-18.
^ a b c Lee Smolin (2007). "The Plebanski action extended to a unification of gravity and Yang-Mills theory". arXiv:0712.0977 [hep-th].
^ a b Jacques Distler; Skip Garibaldi (2009). "There is no 'Theory of Everything' inside E₈". arXiv:0905.2658 [math.RT].
^ a b c d A. G. Lisi (2010). "An Explicit Embedding of Gravity and the Standard Model in E₈". arXiv:1006.4908 [gr-qc].
^ a b c d e f g A G Lisi (2011-05-11). "Garrett Lisi Responds to Criticism of his Proposed Unified Theory of Physics". Scientific American. Retrieved 2011-07-30.
^ "Mathematicians Map E₈". AIM. Retrieved 2007-12-30.
^ Roberto Percacci (2008). "Mixing internal and spacetime transformations: some examples and counterexamples". arXiv:0803.0303 [hep-th].
^ a b Roger Highfield (2007-11-14). "Surfer dude stuns physicists with theory of everything". The Daily Telegraph. Retrieved 2008-06-15.
^ a b Urs Schreiber (2008-05-10). "E₈ Quillen Superconnection". The n-Category Cafe. Retrieved 2008-06-15.
^ Jacques Distler (2007-12-09). "A Little More Group Theory". Musings. Retrieved 2008-08-30.
^ a b Jacques Distler (2007-11-21). "A Little Group Theory". Musings. Retrieved 2008-06-15.
^ "The Big Bang: what will we find?". The Daily Telegraph. 2008-03-25. Retrieved 2008-06-15.
^ A. G. Lisi (2002). "Clifford Geometrodynamics". arXiv:gr-qc/0212041 [gr-qc].
^ A. G. Lisi (2005). "Clifford bundle formulation of BF gravity generalized to the standard model". arXiv:gr-qc/0511120 [gr-qc].
^ A. G. Lisi (2006). "Quantum mechanics from a universal action reservoir". arXiv:physics/0605068 [physics.pop-ph].
^ A. G. Lisi (2007-06-09). "Pieces of E₈". FQXi forum. Retrieved 2008-06-15.
^ A. G. Lisi (2007-07-21). "Standard model and gravity". inaugural FQXi conference. Retrieved 2008-06-15.
^ Scott Dodd (2007-10-26). "Surfing the Folds of Spacetime" (PDF). FQXi article. Retrieved 2008-06-15.
^ A. G. Lisi (2007-06-25). "Deferential Geometry". Loops '07 conference. Retrieved 2008-06-15.
^ A. G. Lisi (2007-10-04). "An Exceptionally Simple Theory of Everything". Perimeter Institute talk. Retrieved 2008-06-15.
^ John Baez (2007-06-27). "This Week's Finds in Mathematical Physics (Week 253)". Retrieved 2008-06-15.
^ Sabine Hossenfelder (2007-08-06). "Garrett Lisi's Inspiration". Backreaction. Retrieved 2008-06-15.
^ A. G. Lisi (2007-11-13). "A Connection With Everything". International Loop Quantum Gravity Seminar. Retrieved 2008-06-15.
^ A. G. Lisi (2007-11-20). "An Exceptionally Simple FAQ". FQXi forum. Retrieved 2008-06-15.
^ A. G. Lisi (2008-02-28). "Garrett Lisi: A beautiful new theory of everything". TED talks. Retrieved 2008-10-17.
^ Zeeya Merali (2007-11-15). "Is mathematical pattern the theory of everything?". New Scientist. Retrieved 2008-06-15.
^ Sabine Hossenfelder (2007-11-06). "A Theoretically Simple Exception of Everything". Backreaction. Retrieved 2008-06-15.
^ Luboš Motl (2007-11-07). "Garrett Lisi: An exceptionally simple theory of everything". The Reference Frame. Retrieved 2008-06-15.
^ Peter Woit (2008-05-28). "INSPIRE". Not Even Wrong. Retrieved 2008-08-05.
^ Simeon Warner (2008-05-20). "What's new at the arXiv?". HEP Information Resource Summit. Retrieved 2008-07-22. (The slide containing this statement was subsequently removed from the presentation file.)
^ Jacques Distler (2007-12-09). "A Little More Group Theory". Musings. Retrieved 2008-11-15.
^ Jacques Distler (2008-09-14). "My Dinner with Garrett". Musings. Retrieved 2008-11-15.
^ Carol Clark (2010-03-18). "Rock climber takes on surfer's theory". esciencecommons. Retrieved 2011-07-30.
^ "No 'Simple Theory of Everything' Inside the Enigmatic E₈, Researcher Says". ScienceDaily. 2010-03-26. Retrieved 2011-07-30.
^ a b http://golem.ph.utexas.edu/category/2008/05/e8_quillen_superconnection.html#c016877
^ Roberto Percacci (2008). "Mixing internal and spacetime transformations: some examples and counterexamples". arXiv:0803.0303 [hep-th].
^ a b R. Percacci; F. Nesti (2009). "Chirality in unified theories of gravity". arXiv:0909.4537 [hep-th].
^ Bertram Kostant (2008-02-12). "On Some Mathematics in Garrett Lisi's 'E₈ Theory of Everything'". UC Riverside mathematics colloquium. Retrieved 2008-06-15.
^ a b A. G. Lisi; Lee Smolin; Simone Speziale (2010). "Unification of gravity, gauge fields, and Higgs bosons". arXiv:1004.4866 [gr-qc].
^ "E₈ Theory". FQXi. 2008-08-04. Retrieved 2008-08-05.
^ "FQXi Grants". FQXi. Retrieved 2008-08-08.
^ Merali, Zeeya (September 2010). "Rummaging for a Final Theory". Scientific American. Retrieved 2010-08-25.
^ M. J. Duff (2011). "String and M-theory: answering the critics". arXiv:1112.0788v1.
^ Peter Woit (2011-12-07). "String and M-theory: answering the critics". Not Even Wrong. Retrieved 2011-12-21.

External links

Deferential Geometry - Lisi's wiki, containing detailed mathematical background.
Animation of E₈ - a New Scientist video describing the theory using a visual representation.
The Elementary Particle Explorer - an online E₈ investigation tool for rotating and examining the particle assignments, charges, and interactions in the standard model and Lisi's E₈ Theory.
A beautiful new theory of everything - Lisi presents his theory at TED.

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