Essay Abstract

In 2007, A. Garrett Lisi published "An Exceptionally Simple Theory of Everything" (TOE) in which he presented a geometrical approach towards TOE based on E8 and the Gosset lattice. Although Lisi's approach has been very well received by FQXi members and pop culture, it has received some serious physics critique - most notably from Prof. Jacques Distler of the University of Texas. Distler's fundamental complaint is that E8 is not large enough to properly contain three chiral generations. Still, it seems appropriate to consider Lisi's geometrical approach a reasonable way to model an approach towards a TOE - a "toy model" TOE as such. The author recently posted "A Case Study of the Geometrical Nature of Exceptional Theories of Everything" and published a book on "New Approaches Towards A Grand Unified Theory". These two papers present the possibility of a geometrical approach towards a TOE. Geometry enters into this approach to TOE in two different ways: 1) Yang-Mills Boson GUT's are derived by recognizing similarities between certain crystal symmetries and certain SU(N) Lie Algebra symmetries, and 2) Particle multiplets are constructed from Simplices, and the product of these Simplices builds representative multi-dimensional lattices. It is anticipated that this geometrical approach may be an axiomic breakthrough that allows us to bypass the apparent complications of Gödel's Incompleteness Theorem and ask the question "What is Ultimately Possible in Physics?" - A Geometrical Approach Towards a TOE.

Author Bio

Dr. Ray B. Munroe, Jr. received all of his degrees in Physics from Florida State University with a Ph.D. in High-Energy Physics Phenomenology in 1996. He is currently CEO-in-waiting of his family's retail business.

Download Essay PDF File

Dear Readers,

This "What is Ultimately Possible in Physics?" essay refers to two papers with limited availability. Both files are rather large, and may load slowly. They are posted here as [link:www.lulu.com/preview/paperback-book/new-approaches-towards-a-grand-unified-theory/1296633]Ref.[4][/link] and Ref. [3] (attached below).

Sincerely,

Ray MunroeAttachment #1: A_Case_Study_3.3.pdf

This looks similar to the paper you sent to me in August, though more condensed. I will give this a read by tomorrow.

Cheers LC

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Hi Ray,

Glad to see that you are participating in this contest. I will read you paper carefully in the following weak. Good luck in this contest!

  • [deleted]

Hi Dr Cosmic Ray ,

Happy to see your paper on the contest .

It's well condensed your work ,and relevant .

Good luck .

Steve

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Dear Ray Munroe,

A theory of everything would endow us with God-like powers, masters of the laws of the Universe. The idea of getting a theory of everything is the idea of the end of fundamental physics [Giovanni Amelino-Camelia]. Thus, if somebody creates TOE, we can close all physics laboratories because we are masters of the laws of the Universe.

Consequently a TOE cannot be created for following reasons:

1. We don't know the nature of space-time, matter, gravitation, inertia, ets. There are a lot of 'holes' in all physics theories. Can we create a TOE using our poor knowledge about space-time and matter?

2. TOE cannot be created because it will be the end of fundamental physics.

Thus I have a question. Why theorists look for TOE if this theory cannot be created by definition? Maybe they must look for less important theories like unification of three interactions. Do you think TOE must be a mathematical model only?

Sincerely,

Leshan

Dear Lawrence,

You have the old version of Ref. [3]. This is a condensed version with a different theme and presentation. I enjoyed your contest paper. Now I need to read your larger "Jordan" paper. Good luck in the contest!

Dear Florin,

I enjoyed your contest paper. Could a geometrical approach to a TOE bypass Gödel's Incompleteness Theorem? Good luck in the contest!

Dear Sphere-keeper Steve,

Thank you for the good wishes.

Have Fun!

Ray Munroe

Dear Leshan,

Your perspective is an interesting one. Truly, I despise the implications of "TOE", but follow industry-wide nomenclature regarding a unification of matter and interactions.

Even if we understood everything/ nearly everything, we may still have limitations such as the speed of light and Planck's constant. So the gap between human-like vs. God-like may be the gap between knowing how something works in theory vs. being able to apply that knowledge in a practical manner. I don't consider this knowledge God-like. If I did, I probably wouldn't share it with anyone else.

Regarding "closing all physics laboratories", I worry that our great experiments such as the LHC are getting so expensive and expansive that we may not be able to (easily) trump the last great experiment. For the sake of all future Supercolliders (my thesis modeled phenomena for the International Linear Collider ILC that hasn't been built yet), I hope that the LHC discovers something - the Higgs would be nice...

I think that the form of an SU(11) boson GUT is large enough to contain any Theory of Quantum Gravity - even though we do not currently have that Theory. I referred to the need to enumerate Feynman Diagrams and Lagrangian components in my "What is Ultimately Possible" Conclusion. Furthermore, I think that the form of an SU(13) boson GUT is large enough to contain the effects of Fermion Generations, and the CKM Quark and PMNS Neutrino Mixing Matrices (more details in the attached Ref. [3]) - even though we may not yet fully understand all of these effects either.

In other words, I don't have all of the answers, but maybe I have the right framework and maybe WE (humanity) can work together to flesh out the details. As they say "The Genius is in the Generalities and the Devil is in the Details".

As a Particle Physicist, I love symmetries. If these symmetries lead to THE TOE, then that is wonderful. If we can't find THE TOE because of philosophical or mathematical (Gödel's Incompleteness Theorem) limitations, then hopefully it will bring us closer to a full understanding of particles and interactions. Why do physicists chase grand theories? Why do dogs chase cars? It is part of our nature that we must always ask the difficult questions and keep pushing at the horizon.

The unification of the three strongest interactions has been done: Standard Model and Georgi-Glashow SU(5). We need to unify Gravity with those three forces, and we need to unify boson and fermion particle content. I think my Geometrical model is large enough to accomplish these goals.

Is the TOE just a mathematical model? Mathematics doesn't even like the idea of a TOE!

Good luck on your essay!

Sincerely, Ray Munroe

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I read your paper. I post in the following from what you wrote:

An important decay route for K12' is K12SU13SO24E8 , where the interpretation is that the SU13SO24of rank 12 and order 444 is a Super Yang-Mills Boson GUT with tensor, vector and scalar boson content - many of which are hypothetical and as yet undiscovered (Ref. [3] has an expansion of the prior Icosahedral example - Equation 3), and the E8 of rank 8 and order 240 is a Fermion particle multiplet. From its Dynkin diagram, E8 has symmetries of 240 82 35, and thus exhibits two-fold "duality", three-fold "triality" and five-fold "pentality" symmetries in an eight-dimensional "octality" space. To the author's knowledge, Lisi never identified the pentality symmetry. Curiously, H4 has the same symmetries of 120 4 ´ (2 ´ 3´ 5) in a four-dimensional "tetrality" space.

This appears to take off from what we communicated about last August. One point which is worth pointing out is that SU(13) ⊂ SO(26), which is a bosonic realization of the 26-dimensional string. Hence the open string with α'M^2 = n - 1 is replaced by α'M^2 = 4(n - 1). The SO(24) multiplet of physical states for the open string are doubled, into SO(24) x SO(24) in the closed string. The gravitational massless state is the symmetric part of |Ω^{ab}> = α^a_{-1}α^b_{-1}|0> in the closed string. The state ½ |Ω^{(ab)}> is a spin 2 field, which is the graviton. The antisymmetric portion is a second rank tensor. The antisymmetric portion will then define the gauge fields.

I will have more comments to follow.

Cheers LC

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I have some further comments as I look at your essay again. One problem with Lisi's program was that he framed the weak interactions with gravity. In order to do this and keep within the strictures of the Coleman-Mandula theorem is to use supergenerators or Grassmannians. Supersymmetry was advanced in part for this purpose.

Cheers LC

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Dear Leshan ,

You say

"TOE cannot be created because it will be the end of fundamental physics."

I think what a theory evolves ....and the fundamentals show us the road ...

It's the same with the general relativity ,this theory evolves like all .

A real theory permits to evolve and never shows an end road of fundamenatls of physics ,because a real theory inserts the fundamenatls ,simply .

In the extrapolations of Ray ,Lawrence ,Lisi ,Florin,.....it's mathematical and this technic needs limits and fundamenatls .But these extrapolations are interestings for the geometry and the symmetry .

Of course the physicality is not present but if the fundamenatls are inserted and if some realities are considered thus it's relevant .

I think what when the two senses are harmonized ,phys/math,thus it's very relevant .I dlike see a superimposing of these works with a kind of sorting with fundamenatls ,for me the rotating spheres implying mass .

Steve ,The theory of spherisation ,a UTE or GUT of Rotating Spheres ,the physicality is an effect ,the mass .

Sincerely

Steve

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Dear Lawrence,

Of course we have been talking about these ideas for at least a couple of months, I referenced your blog comments in the Ref. [3] paper posted above. It is interesting to me that the maximum dimensionality of the bosonic operators keeps coming up one short of the space dimensionality. In my 12-D K12' space, my highest-dimensioned bosons are 11-D. Similarly in your 27-D space, we have an SO(26) with a 26-D bosonic string.

It is good that you found a graviton in the mathematics. I felt certain that this model was large enough to include Gravity, a Gravity-brane , and a related short-ranged WIMP-Gravity force.

I obtained much inspiration from Lisi's paper, but still think that the E8 is large enough for a particle multiplet, but not all of the known interactions as well.

I tried to find the form of a Supersymmetric K12' in Ref. [3].

Dear Steve,

Perhaps the GUT is an evolution process. Perhaps we will never truly know everything. As soon as I work out K12', Lawrence will have J27 ("J" for Jordan?) finished. My simplices would probably get along with your spheres.

Have Fun!

Ray Munroe

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The real crux of the matter seems to be with G_2 automorphism over the octonions. This determines a holonomy which for N fixed points or Killing spinors determines the number of supersymmetries in the theory and the conformal structure of holographic principle.

Cheers LC

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At the bottom of page 3 you write:

These geometrical constraints may be related to Clifford bivectors and the first-class constraints RB Munroe Jr, What is Ultimately Possible in Physics - A Geometrical Approach Towards a TOE 4 of BRST formalism (Becchi, Rouet, Stora and Tyutin) [9]. Note that antiparticles could simply be the inversion operator applied to these particle states, thus yielding a nested dual tetrahedron.

I am intrigued by this comment and I was wondering if you had further developments on this.

Cheers LC

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Hello ,

Dear Ray or Lawrence ,

Could you explain me what is this holographic principle ,please ?

Regards

Steve

  • [deleted]

Dear Lawrence,

Is it a G2 or an I2(7)? See the supersymetric part of the discussion section of Ref.[3] - attached above as "A Case Sudy 3.3.pdf".

These two nested tetrahedra collectively comprise a cube (diagram and basis transformation in Ref.[3]). BRST and/ or Faddeev-Popov "ghosts" are relevant to this geometry - they comprise the "fifth point" in the E8 Pentality symmetry.

Dear Steve,

I haven't worked directly on the holographic principle, but a reference is here.

Dear FQXi Friends, OUCH! It is ironic that all of my scores have been 8's, 7's or 1's. I guess people either "love" or "hate" my ideas. Please "hate" them in moderation - I'm not the only person with "crazy" ideas in this contest.

Have Fun!

Ray Munroe

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The isometry of G_2 is on 2-planes tangent to C^5. There are 10 possible such planes and given any choice of basis, and up to 10 possible Killing spinors which can exist accord to the holonomic action of G_2 with dΩ = 0 on a system of Pfaffian forms. The holonomic action of G_2 is then set by dK = 0, or equivalently above dΩ = 0. The condition means Ω is covariantly constant, which is a condition it being a Killing spinor, and the existence of additional covariantly constant field-form restricts the G_2 holonomy so the Killing spinor equation has more than one solution and the 4 dimensional field theory has extended N > 1 supersymmetry. So the G_2 as the automorphism of the octonions in J^3(O) is what determines the supersymmetries.

The I_2(7) is a discrete group identical to the Fano plane. This is a different structure entirely that determines the multiplication tables (168 of them) for the octonions.

This sort of segues into one possible problem with your program. Again you have this matter of the gravi-weak sector. It appears to be a framing of the SO(4) P-S electroweak theory with SO(3,1) gravitation. On page 4 your write:

"Curiously, H4 has the same symmetries of 120 4 2 35in a four-dimensional "tetrality" space. Table 2 represents these E8 component symmetries as products of Simplices within a 12-dimensional K12' lattice. We already reviewed the 3-Simplex of Electro-Color, and learned that this sub-theory implies that leptons possess the neutral Strong Color charges of white and anti-white. The next new physics is revealed in a study of the 4-Simplex of "Gravi-Weak"."

which might give you a way out. The H_4 is a 5dim octahedral (octahedrachoron) which can tessellate AdS spacetime. This gravi-weak sector might then be a multiplication of the EW fields with quivers of quaternions in this spacetime tessellation. This all of course pertains at a higher energy than standard string theory. The whole point of this with the Leech lattice is that strings are skymrion field effects due to quantum error correction codes or sphere packings. The (H_4xE_8)^2 is an intermediate energy state of affairs between the Leech lattice (penultimate symmetry) and E_8xE_8. At the string level the gravitons are framed according to supersymmetric graded structures with spins (3/2, 2).

I use the term penultimate symmetry, because the Leech lattice or Mathieu M_{24} is the automorphism group over the Fischer-Greiss group. I dare not try to go this far, for this is huge, but it is the ultimate set of symmetries for the 26 dimensional space of Lorentizian symmetries (the good ole fashioned bosonic string). A monster group theory of fields might in 50 to 100 years be seen as the ultimate theory of physics.

Lawrence B. Crowell

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Dear Lawrence,

G2 has the right symmetries for supersymmetry. I expect a five-fold symmetry to yield 5 particle multiplets: 1) scalar bosons, 2) matter fermions, 3) vector bosons, 4) gravitino fermions, and 5) tensor bosons. On the other hand, the 168 of I_2(7) is also interesting - it is half of Klein's Chi(7) (order 336), and a quarter of the K12' roots (672).

These symmetries are all relevant. The 672 roots of K12' appear as shallow holes in the Leech lattice four times (Conway & Sloane, "Sphere Packings, Lattices & Groups" 3rd ed. pp.517-520). If the TOE is the Monster Group, it will take a very long time to understand all of the symmetries and physical implications. The TOE may be an evolution of ideas.

It seems that we started at opposite ends of the problem. I wanted to "fix" Lisi's seemingly simple ideas, and ended up with something that might be related to 11-D M-Theory. You started out with the Leech lattice and 26-D String Theory and tried to work out that end of the problem. I refuse to say that one is right and one is wrong because I think they are related - my view may be a simplification or a transient decaying solution of your view.

I like your "Quiver of Quaternions" and did refer to this in Ref.[3].

Have Fun!

Ray Munroe

  • [deleted]

Hi Ray,

The TOE may be an evolution of ideas.....super .

You are two genius ,a pleasure to read your extrapolations .I learn in math with both of you .

Thanks for that .

Take care

Steve

Ray,

Your scheme is rather complex in some ways. I think the framing you have of gravitation with electro-weak interactions is a bit which meeds to the thought out a bit more. The Leech lattice has in terms of Tacobi theta function s 3 E_6's and so the breakdown to two copies of H_4xE_8 might reflect a framing of gravity with gauge fields ir EW according to a Bloch wave function defined by the 120-cell lattice.

Your connection between K12 and I_2(7) above is interesting. When I get the chance I will try to think about this more.

You might want to consider how this vast zoo part fields or particles you have are an aspect of dark matter. Dark matter is probably in part supersymmetric pairs of matter we understand fairly well.

When it comes to the monster group, we hardly have the techniques to understand this outside of rather superficial aspects. A physical theory based on the Conway groups and the F-G monster is far beyound our abilities right now. The Mathieu and Leech lattice systems are the automorphism groups over the monster and provide a formidable challenge to work with now in the early 21st cnetury.

Cheers LC