An Exceptionally Simple Theory of Everything

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"An Exceptionally Simple Theory of Everything" is a proposed basis for a unified field theory, which attempts to describe all known fundamental interactions in physics, and to stand as a possible theory of everything. Posted to the physics arXiv by Dr. Antony Garrett Lisi in November 2007,[1] the theory quickly stirred public interest and drew a wide range of reactions from other physicists. The title is a pun on the algebra used for Wilhelm Killing's model (E8), which is both a "simple" and an "exceptional" Lie group.

Contents

[edit] Overview

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A visual representation of E8 polytope, a manifestation of the structure of E8.
A visual representation of E8 polytope, a manifestation of the structure of E8.

Lisi's model is a classical gauge theory which has some similarities to grand unified theories (GUTs) such as left-right, Pati-Salam, Georgi-Glashow, SO(10), or E6 based approaches. The choice of E8 structure by Dr. Lisi, representing the largest (and arguably, most beautiful[2]) exceptional Lie group, was in part driven by the proposition: "The mathematics of the universe should be beautiful. A successful description of nature should be a concise, elegant, unified mathematical structure consistent with experience."[1]

In order to form a theory of everything, the model must eventually predict the exact number of fundamental particles, all of their properties, masses, forces between them, the nature of space-time, and the cosmological constant. Much of this work is still on the conceptual stage, in particular quantization and predictions of particle masses, and Lisi himself acknowledges it as a work-in-progress: "The theory is very young, and still in development. Right now, I'd assign a low (but not tiny) likelihood to this prediction."[3] He further remarks: "This is an all-or-nothing kind of theory - it's either going to be exactly right, or spectacularly wrong."[4]

[edit] Description

Lisi proposes a decomposition of the 248 dimensional Lie algebra of E8 according to the following schema:


 
 
 
 
 \mathrm{E}_8 = \mathrm{F}_4 + \mathrm{G}_2 + (26 \times 7)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \mathrm{F}_4~ (graviweak)
 
 
 
 \mathrm{G}_2~ (strong)
 
 ( 26 \times 7 )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \mathrm{SO}(7,1)~+~(8+8+8)
 
 
 
 \mathrm{SU}(3) + 3 + \bar{3}
 
 (8+8+8~+~1+1)
~~\times (3 + \bar{3} + 1)
 

Note that the '8's that appear in the above schematic can refer to different representations. Also, the above decomposition is only given for the complexified Lie algebra. Such a decomposition does not appear to be possible for the needed real form.[5][6]

The strong force is embedded in the SU(3) subalgebra, and gravity in a SO(3,1) subalgebra of SO(7,1). The electroweak force is mapped to a SU(2)L + SU(2)R subalgebra of SO(7,1), as in the Pati-Salam model. Notably, Lisi includes fermions along with bosons in the same representation, and also proposes that the three generations of fermions of the standard model may be described by a triality rotation (relating to the three-fold symmetry of SO(8)).

In order to specify dynamics, Lisi postulates a modified BF theory action,

S = \int <\overset{.}\underset{=}{B}\underset{=.}{F}+\frac{\pi G}{4}\underset{=}{B}^G\underset{=}B^G\gamma-\underset{=}{B}'*\underset{=}{B}'>, and decomposes it to MacDowell-Mansouri action for gravity, electroweak and graviweak parts, and an SO(8) part that contains both "...gluons and a first guess at the action for the new fields." He also remarks that the actions for the second and third fermion generations are "... related by triality in a way that is not presently understood well enough to write down."[1] It should be noted that this Lagrangian is not invariant under the E8 symmetry. Lee Smolin has proposed a way to obtain the bosonic part of Lisi's action (plus higher-order terms) from a fully E8-symmetric theory.[7]

With 'Graviweak F4' and 'Strong G2' as the two primary elements of E8, the following model is created:


 
 
 
 
 Theory of Everything
E8 = F4 + G2 + (26X7)
 
 
 
 
 
 
 
 
 
 
 
 Strong force G2: su(3)
 
 
 
 
 Graviweak F4:
so(7,1)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Gravity
D2: SO(3,1)
 
 
 
 
 
 Electroweak force
D2: su(2)L + su(2)R
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Weak force
 
 
 
 
 Electromagnetism
u(1)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Electric force
 
 
 
 
 Magnetic force
 
 
 


Note: the description provided above is just a brief summary. The full description is entailed in the paper in the physics arXiv.

[edit] Consequences

Lisi's model aims at reproducing all known fundamental fields and dynamics in nature through pure geometry. Lisi's model has 20 elements out of the 248 basis elements of E8 that do not correspond to known particles or forces. These may include new quantum numbers, a new set of colored Higgs scalars, as well as fields that mix leptons and quarks and have forces that vary depending on fermion family. Lisi remarked that: "The lack of extraneous structures and free parameters ensures testable predictions, so it will either succeed or fail spectacularly."[1]

It is generally agreed in public reviews that in its current form, the model by itself is not yet a complete theory of everything. Most importantly, it has not been quantized. Quantization is the most difficult part in producing a theory of quantum gravity. Lisi acknowledges certain issues, but states: "I consider this to be a developing theory that is worth my time to work on, as a long shot."[8]

[edit] Publicity and controversy

Lisi's paper was quickly publicized after its release on 6 November 2007, and spun-off a variety of mostly controversial debates across various blogs and online discussion groups. Numerous news sites from all over the world reported this new theory, noting the personal background of Dr. Lisi.

[edit] Initial reception by the scientific community

After addressing an international meeting on loop quantum gravity at the Perimeter Institute for Theoretical Physics, Lee Smolin, who sits on the advisory panel at FQXI (the institute that funded Lisi's research), remarked on Lisi's work as:

One of the most compelling unification models I've seen in many, many years.

Quickly publicized,[3] Smolin had to clarify this as a spontaneous comment, and noted subsequent press coverage as premature.[9]

Sabine Hossenfelder, who helped arrange for Lisi's participation, also clarifies that while she believes "... Garrett's paper has the potential to become a very important contribution, and his approach is worth further examination", she also notes:[10]

Given today's status, Garrett's model does not naturally lead to a unification of the Standard Model interactions with gravity (he has to choose the action by hand that contains both), it does not allow us to understand quantum gravity (since there's nothing said about quantization), it does not explain the parameters in the standard model (since there isn't yet a mechanism for symmetry breaking), it does not explain the cosmological constant or its value (as said above, to claim there has to be one, it would be necessary to show there's no way to do it without one), it does not explain the hierarchy problem (and I see no way to do so), it does not explain why we live in a spacetime with 3 spatial and 1 timelike dimensions, it does not in my very humble opinion yet qualify for being called a Theory of Everything.

Other preliminary feedback was mixed; Carlo Rovelli, a leading quantum gravity physicist, commented:[11]

When I started to read the article I was a sceptic. When I'd finished it, I asked myself why I hadn't had the idea before.

John Baez described the theory as speculative. While stating that he felt it was more mathematically natural to combine bosons and fermions within a Z2 grading of an E6 Lie group rather than within a Z2 grading of an E8 Lie group, he added that only time will tell. He has recently said,[12]

[Lisi] certainly makes no serious attempt to get the Standard Model Lagrangian in all its detail. Nor does it seem possible (without further feats of genius). For example, his Lagrangian has no place for the ∼25 adjustable constants contained in the Standard Model: particle masses, coupling constants, etc. Nor does he attempt to derive these constants.

David Finkelstein, emeritus professor of the Georgia Institute of Technology said:[4]

Some incredibly beautiful stuff falls out of Lisi's theory, I think that this must be more than coincidence and he really is touching on something profound.

[edit] Criticism

Jacques Distler from the University of Texas at Austin demonstrated that it is not possible to embed the fermions of the Standard Model within E8 and concludes:[5][6]

The overlap between the set of people who know some group theory and those who are (still) interested in giving Lisi’s 'Theory of Everything' a passing thought is empty.

Luboš Motl, former assistant professor at Harvard University (2004–2007) commented:[13]

Every high school senior excited about physics should be able to see that the paper is just a long sequence of childish misunderstandings.

Distler, Motl and others also argue[13] that it is impossible to have a theory with internal and external symmetries unified in any non-trivial way, as this violates the Coleman-Mandula theorem. Lisi and Smolin have claimed that the Coleman-Mandula theorem is not applicable in this theory.

Dr. Lisi responded in Distler's blog on embedding the second and third generation fermions:[14]

I have discussed this inadequacy clearly in the paper, going so far as to explicitly state it is currently the main problem with the theory.

It has also been noted[15][16] that there are problems with the embedding of particles of different spin (spin 1/2 fermions, spin 1 gauge bosons, and spin 2 gravitons) and statistics in the same group representation, as proposed by Lisi. This is related to the Spin-statistics theorem that holds for all Lorentzian quantum theories.

Lee Smolin has further written:[17]

Does Lisi’s proposal survive Distler’s second post? Even if there is some truth to Distler’s argument, is the result the end of Lisi-like proposals or are there alternatives which evade it? For example, by going to the complexification? Or might it be that Lisi’s proposal works for the Euclidean spacetimes but not for Lorentzian? Might it be that it only works if only part of the Lorentz algebra is gauged, as in the Ashtekar or Thiemann formulations? All these are interesting possibilities...

[edit] Putting the theory to the test

Lisi has stated that his theory is testable, and might be put to the test by the Large Hadron Collider particle accelerator, which is currently under construction near Geneva, Switzerland. He states that the theory predicts the existence of new, colored Higgs particles at an undetermined mass scale; such predictions are not falsifiable because the mass can always be placed beyond the reach of a given experiment. However, the discovery of new particles, such as superpartners, that do not fit in Lisi's classification would falsify the theory.

[edit] References

  1. ^ a b c d A. Garrett Lisi, An Exceptionally Simple Theory of Everything, Cornell University Library, Submitted on 6 November 2007
  2. ^ Hermann Nicolai, in (AIM) Mathematicians Map E8 (retrieved 30 December 2007)
  3. ^ a b Highfield, Roger. "Surfer dude stuns physicists with theory of everything", The Daily Telegraph, 2007-11-14. Retrieved on 2007-11-23. 
  4. ^ a b Is mathematical pattern the theory of everything? - fundamentals - 15 November 2007 - New Scientist. Retrieved on 2007-11-18.
  5. ^ a b Musings (Distler, 24 December; retrieved 30 December 2007)
  6. ^ a b Musings (Distler, 9 December 2007; retrieved 30 December 2007)
  7. ^ Lee Smolin: The Plebanski action extended to a unification of gravity and Yang-Mills theory on arXiv
  8. ^ Lisi on Jacques Distler's blog "Musings" (22 November 2007)
  9. ^ Lee Smolin on Cosmic Variance
  10. ^ Sabine Hossenfelder's blog (retrieved 1 January 2008)
  11. ^ Baez, John. This Week's Finds in Mathematical Physics (Week 253).
  12. ^ Baez, John. Comment on "The n-Category Cafe".
  13. ^ a b Motl, Lubos (2007-11-07). Garrett Lisi: An exceptionally simple theory of everything. Retrieved on 2007-11-23.
  14. ^ Backreaction (Hossenfelder; retrieved 30 December 2007)
  15. ^ Comments at Backreaction
  16. ^ Dynamics of Cats
  17. ^ Lee Smolin on Cosmic Variance blog (17 December 2007; retrieved 5 January 2008)

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