Greetings Frank,

I enjoyed reading your essay, and even more once I'd read some of your previous papers, which are cited therein. I would have liked it better if you explained your idea more fully in the essay itself, rather than making me look at your sources to make sense of things. It seems like an interesting idea though. Witten had a paper on BTZ Black Holes (arXiv: 0706.3359) that used the Monster group as a generating object.

I have long been a fan of the "It from Bit" and "Digital Physics" line of reasoning, and a chronicler of what Wigner called the unreasonable effectiveness of Math, in connection with my theory of the Cosmos based on the Mandelbrot Set. It seems perfectly reasonable for me to believe that the interesting objects of Math do act as generators of order in the Physical sciences, although I champion a more pragmatic view in my contest essay.

If I were to pursue a similar road to that suggested by the opening comments of your essay, I would probably note that Gravity (and the motion of material objects) is connected with the Reals, EM waves seem to be well modeled by the Complex numbers, which would suggest we look to Quaternions to understand spin and the Weak force and to Octonions for the Strong force interactions. There seems to be some connection in this reasoning with your essay, as E8 comes out of the Octonions mapped onto themselves. Am I right?

And please a mention on Icosions, as I hadn't heard that term until now. I thought there were only four normed division algebras, and this seems to imply a fifth set in that family.

All the Best,

Jonathan J. Dickau

Thanks for your reply. The monster group is way too much for physics. In my essay

http://www.fqxi.org/community/forum/topic/494

I do some work with the Jordan exceptional algebra. The triality on the three octonions leads to an embedding into the Mathieu or Leech structure. This is I think the penultimate (because the monster might be the ultimate) quantum error correction code for quantum cosmology.

QCD is a breakdown from the G_2 group, where SU(3) is the maximal subgroup of G_2 plus 3 and bar-3. These might be interpreted as the families, where you consider the prospect there are four. G_2 though is the automorphis of the Jordan group, as F_4 and G_2 are centralizer of E_8. The extension to the E_6 valued Jordan matrix leads to twistor space structures and that the norm of states obeys

(φ|ψ) = (2ħ)^{-1}{Ω(φ,ψ) iΩ(φ,gψ)],

where g is a group operation and Ω the symplectic operation. Parentheses are used for bra-ket operations because this editor does not like carrot symbols. The group operation is g_2, which constructs the E_6 Jordan algebra from the Hermitian and anti-Hermitian Jordan matrices.

Cheers LC

Hello again Frank,

As luck would have it; one of the display articles on the FQXi homepage is germane to your essay's topic and to my comments above. It talks about a pair of researchers using Octonions to model quarks and telescope 10 dimensions to 8-d and then to 4-d space-time.

Taking on String Theory's 10-D Universe with 8-D Math

I thought this might be of interest.

All the Best,

Jonathan

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  • Post #24

The icosians are 120 quaterions on H_4. There exist the 120 and 600 cells which are dual to each other. The exceptional group decomposes into two copise of H_4 plus permutations in the Weyl group. The Leech lattice embeds three copies of the E_8, and so this decomposition suggest that the heterotic string is more fundamentally a lattice system similar to solid state physics with a gauge E_8xE_8 field theory on it. It is similar to Bloch waves in solid state mechanics. It is also interesting that the 120-cell will tessellate the AdS_4 spacetime. The AdS_5 spacetime dual to S^5 is then AdS with an additional dimension which has a G_2 holonomy symmetry in the AdS_4xR^1 structure. This G_2 holonomy is the automorphism of the Jordan exceptional algebra.

This is a discrete system, yet the lattice is determined to within a gauge choice. This is the basic departure from solid state physics. String physics emerges in a Skymrion type of model, similar to bundles of relativistic electrons in solid state systems with large ion charges or with graphene.

My article at

http://www.fqxi.org/community/forum/topic/494

discusses the Jordan algebra and its role with black hole complementarity and the cosmological constant.

Cheers LC

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  • Post #25

Hello,

Thanks for the concise description of the Icosians, Lawrence. This is exceptionally cool stuff (pun intended). I shall give your essay a read soon, though I daresay a complete understanding may elude me. However; with each foray into unfamiliar mathematical territory, the better become my chances to come out with something meaningful. This is especially so with folks like you and Frank who delight in the grandeur and beauty of it.

I'd like to see your comments on the above too, Franklin Potter.

All the Best,

Jonathan

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  • Post #26

The icosians are a four dimensional version of the dodecahedron and icosahedron. These are called the dodecahedrachoron and icosahedrachoron. They are dual to each other, and tessellate hyperbolic spaces, as does the dodecahedron in three dimensions, as seen in the attachment.

LC

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  • Post #29

My Oct. 01 post awaits response from the author.

Dear Narendra,

Thank you for more questions in your October 1 post. I have been away for about a week with no Internet access, so I will answer them briefly now.

(1) Why a 4th quark family? The quark states represent the 4 binary rotational symmetry groups that span the 4-D internal symmetry space, each group having two orthogonal states, i.e., the family pair. There are no more such groups available. The groups are [3,3,3], [4,3,3], [3,4,3], and [5,3,3], the last one for the 4th quark family (to correct my error in the Oct 1 response to Lawrence Crowell where I accidentally listed [3,3,3] for the t'-quark and b'-quark family.]

(2) The charges are the same for all quark families, one state with +1/3 and one state with -2/3 unit charges. Of course, there is in 4-D a conjugate internal symmetry space which has the antiparticle states with opposite charge signs and the same mass signs, the latter because these antiparticle states are gauge equivalent and not simply equivalent - that's what the U(1)Y does.

(3) Regarding a look at cosmological effects of this 4th quark family, it is potentially the source of the baryon asymmetry of the universe, as pointed out by W.-S. Hou, because the Jarlskog constant would have the correct value, about a factor of 10^13 larger than with 3 quark families.

Dear Anton,

I have been away for about a week, so I will now respond to one of your many excellent comments in your October 2 post.

(1) Regarding "the universe has to create itself out of nothing..." Mathematically, there is a good possibility for nothing to become something. The fundamental particles known as leptons and quarks are spinors, i.e., spin 1/2 particles whose states must be rotated twice for the wave function to return to its initial value, not just one rotation for vectors.

(2) From a zero-length vector arises two orthogonal spinors, as originally derived by E. Cartan in the 1920's. Here, then, is the mathematical source of something out of nothing. There is a great book by S.L. Altmann called Rotations, Quaternions, and the Double Groups (1986) that provides the details with great insight.

(3) From the zero-length vector in spacetime can arise a quark-antiquark pair. What happens to energy conservation? There is a balance between the mass-energy, charge energy, etc. with the negative potential energy so that the sum remains zero.

Dear Jonathan,

I am back from about a week absence, so I can responds to some of your questions and comments in your October 4 post.

(1) I think that your normed division algebra sequence suggestion may be important, but I might think in a slightly different manner: (i) Reals for the EM vector potential with A because its phase factor involves i eA/c; (ii) Complex for the weak interaction, thereby having a 2x2 matrix in the phase factor; (iii) Quaternions for the color interaction for quarks involving 4x4 matrix rotations spanning the complete 4-D internal symmetry space; and (iv) Octonions in the phase factor for the gravitational interaction.

(2) Note that I am just speculating in (1), for I have not seriously engaged my thoughts with regard to the consequences.

(3) The icosians are the key, at least to connecting our discrete 4-D physical space to discrete 8-D, telescoping upward, yet remaining both a quaternion (because there are only 4 non-zero coefficients) and also being an octonion. No, icosians are not a fifth normed division algebra.

Dear Lawrence,

I have been away, so I can now respond briefly to your interesting comments in your October 5 posting.

(1) Triality of the 3 octonions (telescoped up from discrete 4-D via icosians) for a fundamental interaction of spinor-vector-spinor definitely leads to the Leech lattice and its symmetry, as you state. Also the Golay-24 coding related to the Leech lattice is important for errorless progression, probably through the discrete lattice.

(2) You remain fixated on Lie groups and may be correct, but if the b' quark appears at around 80 GeV, then you will definitely need to consider their finite subgroups and the discrete symmetries which I consider to be the underlying basis for the Standard Model.

(3) The Monster has more fingers into many areas of mathematics than I can appreciate. But the Monster has THE intimate connection to elliptic modular functions which definitely play a key role in leptons and quarks, the coefficients of the expansion of the j-invariant in a Fourier series are related to dimensions of its irreducible representations.

(4) And as I have shown, each family has an invariant syzygy with proportionality to the j-function, producing ratios of family masses. There is no need for a Higgs.

(5) The first indication that I might be on the right path was the ability to predict the top quark mass of about 160 GeV in 1992, a few years before its discovery at Fermilab in 1995 at 170 GeV. You may recall that everyone else was predicting values under 95 GeV before 1994.

Hello Frank,

Welcome back. Thanks for the answers to my questions. Some very interesting stuff. No need for a Higgs, with your approach? Way cool.

It seems I will have to do some further study into the Icosians, and may understand your work better after that. This is fun Math to explore! But since it's supposed to be predictive, it will be interesting to see if a fourth quark family appears.

All the Best,

Jonathan

I agree about the importance of modular forms. The Mathieu M_{24} or Leech lattice system has modular structure. The monster F-G group appears to be the ultimate goal. I have thought this was physics and cosmology for the mid 21st century or maybe 100 years from now. Of course given some thing I see coming in the world I question whether we will be around long enough. Of course the M_{24} or Leech is the automorphism sporadic group for the F-G, which makes it terribly important in its own right. In part with the Jordan exceptional algebra I have been focused somewhat on G_2 at the automorphism there.

I think it all comse down to scale. Here is how I see it. String theory operates on a scale of about 10-20 Planck lengths, with the Hagedorn temperatures at sqrt{5pi} times the Planck length. My thinking is that strings are emergent structures, similar to Skymrion field theory, where there are emergent knots of fields. In fact if you read my article

http://www.fqxi.org/community/forum/topic/494

you will see the role of Chern-Simons Lagrangian in the exceptional algebra, which Skymrion theory is based on. This theory extends to a scale maybe down to 5-10 Planck units of scale. This theory is set up to fold it all into the Leech structure which I think holds down to a scale of nearly 2 Planck lengths. Now this is the automorphism of the monster group, so a quantum cosmology based on this might take us to 1 epsilon times the Planck scale.

The monster group contains lots of stuff! An automorphism of it is the 26-dimensional Lorentz group --- the 26 dimensional bosonic string?! It is the ultimate form of group theory, and maybe what ever physical principle or structure can ever possibly exist is coded in it. I will say for now I question whether we can do much with it. You might check out Tony Smith's paper below and his website. he has all sorts of ideas about this, but I question whether this is physics at this time.

If you have ideas about how to actually do physics with the F-G group let me know.

As for discrete and continuum, I see things as an interplay between the two.

Cheers LC

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  • Post #36

i am happy to see your response to m posting. i can understand that you may have something in nothing that existed before the creation of the universe but that is awaited. Though i do not comprehend the 4 th family of quarks indicated by you, i see that these may well be the primordial aprticles generated at the Big bang which subsequently disappeared through ultra fast decay that initiated the ultrafast evolution of the created point during Big bang. he third family and others are then resoonsible for the generation of baryons in the visible matter and the dark matter simply consists of frozen quark matter that no longer has the strong field strength to glue into baryons. Such a picture may help explain initial extra quick evolution of the universe as welll as production of huge amount of dark amtter and associated dark energy. in my esssay i have attempted to understand it through the changing initial strengths of the gravitation and strong nuclear fields the firstb to arrive on the scene in a sequential manner. The Physics of that early universe can not be based on the Phydics we have worked out for the later universe in the past few hundred years! The ehavier quarks decayed ultra fast into lower mass families under very strong field strengths present in those extremely high temperatures cum mass densities, no accelerator in earth can possibly simulate in the years to come!

Dear Narendra,

Thank you for your further comments. I can respond thusly:

(1) Once I discovered that the 3 lepton families when assigned to the 3 finite binary rotational subgroups of SU(2)L x U(1)Y known as [3,3,2], [4,3,2] and [5,3,2], i.e., the 3-D rotational symmetries, provided the mass ratios 1:108:1728 via connections with the j-invariant function, I had no choice on assigning the quark families to the 4-D rotational symmetries of the 4 finite binary rotational subgroups. They are the only ones remaining that fit the same scheme, therefore 4 quark families. Notice that the leptons are 3-D entities and the quarks are 4-D entities in this scheme, not point particles.

(2) Then I discovered that the SU(3)-color symmetries can also fit in 4-D by working out the 4-D rotations which occur in two planes simultaneously, thereby reproducing the color interaction and the three color charges. So the Standard Model fits into a 4-D internal symmetry space and does not require a larger space.

(3) If I now ask whether other fundamental particles are possible, I would say NO: no supersymmetry particles, no unparticles, no dark matter, no Higgs, etc. They would not fit this same scheme.

(4) Consequently, I would doubt that your suggestion "dark matter simply consists of frozen quark matter that no longer has the strong field strength..." will survive. The color interaction is built into the origin of the quark particle states, i.e., their 4-D quality itself. Putting 3 4-D quark states together mathematically makes one 3-D baryon, etc.

(5) I would also doubt that any of the fundamental interactions, including gravitation, have changed their strengths over time, but I cannot provide an argument yet.

(6) For any alternative approach to cosmology, one needs to explain baryogenesis. One way is the one provided by the standard model of cosmology - big bang, inflation, etc., which is a complicated model with lots of patchwork over the years. What needs to be explained by alternatives is essentially the creation of about ONE hydrogen atom per cubic meter per 10 million years to account for the matter/energy density of the universe as we know it.

I have an idea about how to work the monster into things, or at least indicate how it is in the background of things. The J^3(O) eceptional matrix can be extended to the C* valued J^3(CxO). This may be further extended to the quaterionic J^3(HxO) with the inclusion of anti-Hermitian cubic matrix terms and a G_2 fibration between the C* ant anti-C* elements. This leads to 26-dimensional lightcone realizations, which are Lorentzian group realizations of the Leech lattice. This is a 26-dimensional automorphism group over the Fischer-Greiss group (monster).

More later,

Lawrence B. Crowell

17 days later

Dear Franklin Potter,

Most of the outcomes from collider experimentations are probabilistic rather than deterministic, may be due to Gödel's incompleteness and it's contradictory to Einstein's statement, 'God does not play dice'. This may be of mathematical constrains in observability or physical inconsistency in reality and I think this impediment may be of physical inconsistency in reality. I agree with you that nature is numerical, but the numeric order what nature exhibit is rather different from the mathematical systems we adapt to observe nature by theoretical formulations.

If we think of a wave propagation through spin matrix in that quaternions number system and norms of geometric origin are applicable, then we may have to identify 18 hadrons in pairs with opposite spins, to observe a segment of surface of a section in the numeric hierarchy the nature have, in that the finite simple groups may be applicable.

With best wishes,

Jayakar

Dear Jayakar,

Thank you for your comments. I'll try to respond to some of them.

(1) Quantum mechanics has two parts (i) a deterministic part which Penrose calls Schrödinger evolution U and (ii) a non-deterministic part called state reduction R. I.e., one first has a linear combination of possible state outcomes according to the rules of QM, and then in the apparatus appears one of them with its probability of occurence.

(2) In my approach, all particles that can be isolated such as an electron, proton, etc. require a 3-dimensional space for their state definition. But we live in a 4-D spacetime which can be shown to behave like a 4-D space when both are discrete - i.e., can both be seen as rotations of a tesselated Riemann sphere. What I'm getting to is that our fundamental particles may really have that 4th spatial dimension for their use even though somehow it is shared with out time dimension. What we interpret as probabilistic collapse to a single outcome may be more determined than we think. I'm only guessing here. Then perhaps 'God does not play dice.'

(3) One other advantage of the specific finite groups for the leptons and quarks as the basis for all physics, other than being subgroups of the gauge group of the Standard Model, is that their linear fractional transformations in the discrete spacetime can be shown to lead to the conservation laws in the limit of a continuous spacetime. The connection may not be so direct with other groups.

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  • Post #41

Franklin,

I happened to look in on your site and found you responded to a post of mine 3 weeks ago.

You are covering some of the same ground I work in. My work on the exceptional Jordan matrix is meant to take things to the Leech lattice and the 26-dimensional Lorentz group, which is the automorphism of the Monster group. I am not sure exactly why the gauge structure for QCD is discrete due to the large mass ot the T-quark.

The automorphism group for the Jordan exceptional algebra are the G_2 and F_4 groups. These are also the centralizers on the heterotic E_8. As such a transformation of connection terms by the g_2 holonomy generates transformations on f_4. This is in part seen in my diagram I drew up a week ago or so. In this way the curvature of the space is determined by a Skymrion abelian field term. Everything is reduced to something very simple --- indeed to something used in solid state physics.

The F_4 is connected to the symplectic group in 8 dimensions, which is reduced further to SU(4). G_2 is reduced to SU(3), which is a subgroup of SU(4). I wrote this argument up in a bit of a jarring way in the paper. Another way to see this relationship is that G_2 fixes a vector in spin(7) since spin(7)/G_3 = S^7. F_4 defines a triality of SO(8) ~SO(O), and the inclusion of G_2 in spin(8) which again has spin(7) sets the two groups in a duality of 7 vs 8, where in the Hopf fibration

S^7 - -> S^{15} - -> S^8,

There is a homotopy on how 7 spheres are tied as knots in 15 dimensions according to the Chern-Simons link invariant ∫ρ/\dρ evaluated on the sphere of dimension 2n - 1, here n = 8. So there exists a function f:S^{2n-1}  S^n, such that for ω is a volume form on S^n, here n = 8 again, then f*ω is a closed form, and further since the n-th cohomology of S^{2n-1} H^n(S^{2n - },R) = 0 this form must then be exact as well f*ω = dρ, for ρ an n-1 form.

This then set the Skyrmion field theory, which naturally indicates the underlying fermionic structure. The basis elements on S^7 for the G_2 are define as

e^a = ψ^(x)γ^aψ(x)

for γ^a Dirac matrix elements of Cl_{7,1}, and gauge connections are

A_μ = ψ^(x)∂_μψ(x)

The Skyrmion field in the basic Jordan matrix in 27 dimensions is abelian, which is quantizable almost trivially. For extended Jordan matrices in 78, 133, and 248 dimensions are nonabelian, which correspond to the complex, quaternionic and octonionic J^3(O)

The G_2 and F_4 automorphisms are centralizers so gauge transformations by one group is matched by transformations of the other. This then results in an emergent Skyrmion field that is similar to a Fermi quasi-particle field. This quasi-particle field can be thought of as induced by the classical nonlinear field theory, just as knots or topological solitons in condensed matter systems are induced by an underlying system of electrons and phonons. The classical spacetime physics though emerges at low energy from this system. For higher Jordan matrix algebras with E_6 or E_8 algebra will be non-abelian, but higher groups such as the E_6 -Jordan matrix algebra things are quantizable. So curiously there is a kind of double emergence.

I have been playing with this for the last several weeks, and this appears to indicate that gravitation does not have to be quantized directly. I found this structure with the exceptional matrix model recently as a way of understanding how fermionic quasi-particles could renormalize the cosmological constant. The recent Fermi Gamma Ray observational results, with no frequency dependency on the speed of light, also suggest that ideas of violent quantum foam in spacetime are simply wrong. Gravitation does have an action in string theory

S = ∫d^nx sqrt{g}R α'R^{abcd}R_{abcd} ...

expanded around the string length for length >> than the string length. The second term is the trace of the Bel-Robinson tensor is an O(sqrt{Għ}), or order Planck length, term which is a small quantum correction to the classical term.

If you look on my blog area I have some posts on these developments, in particular detail on the g_2 algebra as a holonomy over the 7-sphere.

Cheers LC