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, 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]

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.

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 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 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",[6] 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.

Lisi continued to promote variations on his original proposal in the years after the Distler and Garibaldi paper.

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, YW, 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, 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${\displaystyle {}={\tfrac {1}{2}}}$ , g8${\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}}$ , W${\displaystyle ={\tfrac {1}{2}}}$ , Y${\displaystyle {}={\tfrac {1}{3}}}$ ) 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 ${\displaystyle {}={\tfrac {1}{2}}}$ , g8${\displaystyle {}={\tfrac {1}{2{\sqrt {3}}}}}$ , W${\displaystyle {}={\tfrac {-1}{2}}}$ , Y${\displaystyle {}={\tfrac {1}{3}}}$ ). 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.

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

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.[7] 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.[citation needed]

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. As of 2020, none of the particles predicted by any version of E8 Theory have been detected.

History

Before writing his 2007 paper, Lisi discussed his work on a Foundational Questions Institute (FQXi) forum,[8] at an FQXi conference,[9] and for an FQXi article.[10] Lisi gave his first talk on E8 Theory at the Loops '07 conference in Morelia, Mexico,[11] soon followed by a talk at the Perimeter Institute.[12] John Baez commented on Lisi's work in "This Week's Finds in Mathematical Physics (Week 253)",[13] Lisi's arXiv preprint, "An Exceptionally Simple Theory of Everything", appeared on November 6, 2007, and immediately attracted attention. Lisi made a further presentation for the International Loop Quantum Gravity Seminar on November 13, 2007,[14] and responded to press inquiries on an FQXi forum.[15] He presented his work at the TED Conference on February 28, 2008.[16]

Numerous news sites 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,[17] 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.[18] Peter Woit commented, "I'm glad to see someone pursuing these ideas, even if they haven't come up with solutions to the underlying problems".[19] The group blog The n-Category Café hosted some of the more technical discussions.[20][21] Mathematician Bertram Kostant discussed the background of Lisi's work in a colloquium presentation at UC Riverside.[22]

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.[23][24][25] These arguments were distilled in a paper written jointly with Skip Garibaldi, "There is no 'Theory of Everything' inside E8",[6] 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 response, Lisi argued that Distler and Garibaldi made unnecessary assumptions about how the embedding needs to happen.[26] 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",[27] eventually 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.[26] "The explanation for the existence of three generations of fermions, all with the same apparent algebraic structure, remains largely a mystery," Lisi wrote.[27]

Some follow-ups to Lisi's original preprint have been published in peer-reviewed journals. Lee Smolin's "The Plebanski action extended to a unification of gravity and Yang–Mills theory" proposes a symmetry-breaking mechanism to go from an E8 symmetric action to Lisi's action for the Standard Model and gravity.[28] Roberto Percacci's "Mixing internal and spacetime transformations: some examples and counterexamples"[29] addresses a general loophole in the Coleman–Mandula theorem also thought to work in E8 Theory.[26] 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".[30] In a joint paper with Lee Smolin and Simone Speziale,[31] published in Journal of Physics A, Lisi proposed a new action and symmetry-breaking mechanism.

On August 4, 2008, FQXi awarded Lisi a grant for further development of E8 Theory.[32][33]

In September 2010, Scientific American reported on a conference inspired by Lisi's work.[34] Shortly thereafter, they published a feature article on E8 Theory, "A Geometric Theory of Everything",[2] written by Lisi and James Owen Weatherall.

In December 2011, in a paper for a special issue of the journal Foundations of Physics, Michael Duff argued against Lisi's theory and the attention it has received in the popular press.[35][36] 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

1. ^ A. G. Lisi (2007). "An Exceptionally Simple Theory of Everything". arXiv:0711.0770 [hep-th].
2. ^ a b c A. G. Lisi; J. O. Weatherall (2010). "A Geometric Theory of Everything" (PDF). Scientific American. 303 (6): 54–61. Bibcode:2010SciAm.303f..54L. doi:10.1038/scientificamerican1210-54. PMID 21141358.
3. ^ Greg Boustead (2008-11-17). "Garrett Lisi's Exceptional Approach to Everything". SEED Magazine. Archived from the original on 16 April 2018.
4. ^ Amber Dance (2008-04-01). "Outsider Science". Symmetry Magazine. Archived from the original on 5 July 2008. Retrieved 2008-06-15.
5. ^ Collins, Graham P. (March 2008). "Wipeout?". Scientific American. 298 (4): 30–32. doi:10.1038/scientificamerican0408-30b. PMID 18380135.
6. ^ a b Jacques Distler; Skip Garibaldi (2010). "There is no 'Theory of Everything' inside E8". Communications in Mathematical Physics. 298 (2): 419–436. arXiv:0905.2658. Bibcode:2010CMaPh.298..419D. doi:10.1007/s00220-010-1006-y. S2CID 15074118.
7. ^ Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512. Archived from the original on 2008-10-09. Retrieved 2015-12-15.
8. ^ A. G. Lisi (2007-06-09). "Pieces of E8". FQXi forum. Archived from the original on 2 June 2008. Retrieved 2008-06-15.
9. ^ A. G. Lisi (2007-07-21). "Standard model and gravity". inaugural FQXi conference. Retrieved 2008-06-15.
10. ^ Scott Dodd (2007-10-26). "Surfing the Folds of Spacetime" (PDF). FQXi article. Retrieved 2008-06-15.
11. ^ A. G. Lisi (2007-06-25). "Deferential Geometry". Loops '07 conference. Retrieved 2008-06-15.
12. ^ A. G. Lisi (2007-10-04). "An Exceptionally Simple Theory of Everything". Perimeter Institute talk. Retrieved 2008-06-15.
13. ^ John Baez (2007-06-27). "This Week's Finds in Mathematical Physics (Week 253)". Archived from the original on 30 June 2008. Retrieved 2008-06-15.
14. ^ A. G. Lisi (2007-11-13). "A Connection With Everything". International Loop Quantum Gravity Seminar. Archived from the original on 22 May 2008. Retrieved 2008-06-15.
15. ^ A. G. Lisi (2007-11-20). "An Exceptionally Simple FAQ". FQXi forum. Archived from the original on 2 June 2008. Retrieved 2008-06-15.
16. ^ A. G. Lisi (2008-02-28). "Garrett Lisi: A beautiful new theory of everything". TED talks. Archived from the original on 18 October 2008. Retrieved 2008-10-17.
17. ^ Zeeya Merali (2007-11-15). "Is mathematical pattern the theory of everything?". New Scientist. Archived from the original on 12 May 2008. Retrieved 2008-06-15.
18. ^ Sabine Hossenfelder (2007-11-06). "A Theoretically Simple Exception of Everything". Backreaction. Archived from the original on 26 May 2008. Retrieved 2008-06-15.
19. ^ Woit, Peter (November 9, 2007). "An Exceptionally Simple Theory of Everything? | Not Even Wrong". Not Even Wrong. Retrieved 2020-10-12.
20. ^ Urs Schreiber (2008-05-10). "E8 Quillen Superconnection". The n-Category Cafe. Archived from the original on 2008-06-19. Retrieved 2008-06-15.
21. ^ "The n-Category Café". utexas.edu. Retrieved 20 February 2017.
22. ^ Bertram Kostant (2008-02-12). "On Some Mathematics in Garrett Lisi's 'E8 Theory of Everything'". UC Riverside mathematics colloquium. Archived from the original on 28 June 2008. Retrieved 2008-06-15.
23. ^ Jacques Distler (2007-11-21). "A Little Group Theory". Musings. Archived from the original on 12 May 2008. Retrieved 2008-06-15.
24. ^ Jacques Distler (2007-12-09). "A Little More Group Theory". Musings. Retrieved 2008-11-15.
25. ^ Jacques Distler (2008-09-14). "My Dinner with Garrett". Musings. Archived from the original on 2008-11-19. Retrieved 2008-11-15.
26. ^ a b c A G Lisi (2011-05-11). "Garrett Lisi Responds to Criticism of his Proposed Unified Theory of Physics". Scientific American. Archived from the original on 2011-07-02. Retrieved 2011-07-30.
27. ^ a b A. G. Lisi (2010). "An Explicit Embedding of Gravity and the Standard Model in E8". arXiv:1006.4908 [gr-qc].
28. ^ Lee Smolin (2009). "The Plebanski action extended to a unification of gravity and Yang–Mills theory". Physical Review D. 80 (12): 124017. arXiv:0712.0977. Bibcode:2009PhRvD..80l4017S. doi:10.1103/PhysRevD.80.124017. S2CID 119238392.
29. ^ Roberto Percacci (2008). "Mixing internal and spacetime transformations: some examples and counterexamples". Journal of Physics A: Mathematical and Theoretical. 41 (33): 335403. arXiv:0803.0303. Bibcode:2008JPhA...41G5403P. doi:10.1088/1751-8113/41/33/335403. S2CID 1211477.
30. ^ R. Percacci; F. Nesti (2010). "Chirality in unified theories of gravity". Physical Review D. 81 (2): 025010. arXiv:0909.4537. Bibcode:2010PhRvD..81b5010N. doi:10.1103/PhysRevD.81.025010. S2CID 119225258.
31. ^ A. G. Lisi; Lee Smolin; Simone Speziale (2010). "Unification of gravity, gauge fields, and Higgs bosons". Journal of Physics A: Mathematical and Theoretical. 43 (44): 445401. arXiv:1004.4866. Bibcode:2010JPhA...43R5401L. doi:10.1088/1751-8113/43/44/445401. S2CID 118507772.
32. ^ "E8 Theory". FQXi. 2008-08-04. Archived from the original on 2008-08-09. Retrieved 2008-08-05.
33. ^ "FQXi Grants". FQXi. Archived from the original on 3 July 2008. Retrieved 2008-08-08.
34. ^ Merali, Zeeya (September 2010). "Rummaging for a Final Theory". Scientific American. 303 (3): 14–17. Bibcode:2010SciAm.303c..14M. doi:10.1038/scientificamerican0910-14. PMID 20812465.
35. ^ M. J. Duff (2011). "String and M-theory: answering the critics". Foundations of Physics. 43 (1): 182–200. arXiv:1112.0788. Bibcode:2013FoPh...43..182D. doi:10.1007/s10701-011-9618-4. S2CID 55066230.
36. ^ Peter Woit (2011-12-07). "String and M-theory: answering the critics". Not Even Wrong. Retrieved 2011-12-21.