An Exceptionally Technical Discussion of AESToE

[Kea]

I think the universe is defined by a set of unitarily inequivalent vacua and the conformal infinity for the AdS. The first is high temperature and end is zero temperature. For spacetime physics, where spacetime has an effective negative heat capacity, this is low entropy to high.

OK, attempting to translate: the series of vacua would be Pitkanen's Planck scale hierarchy, or Riofrio's cosmology scaling hierarchy, or the n-cat complexity hierarchy. Conformal infinity for AdS presumably imposes stringy type duality conditions, but I don't see why we need a classical AdS point of view on this.

 

[Lawrence B. Crowell]

OK, attempting to translate: the series of vacua would be Pitkanen's Planck scale hierarchy, or Riofrio's cosmology scaling hierarchy, or the n-cat complexity hierarchy. Conformal infinity for AdS presumably imposes stringy type duality conditions, but I don't see why we need a classical AdS point of view on this.

The unitarily inequivalent vacua related to each other by Bogoliubov transformations. On my information theory physics forum area

https://www.physicsforums.com/showthread.php?t=115826&page=3

and the page prior to this I indicate this a bit further. The conformal group emerges from the breaking of $E_6$, which of course emerges from the higher energy $E_8$. This in turn is a part of an larger error correction code system. I am not sure how this connects with Matti's idea

The evolution of the universe is a process which maps a set of inequivalent vacua, a purely quantum system of excitons, into a completely classical spacetime configuration with $\rho~=~0$. The universe is then a map between these voids. The complete symmetry of the universe is then some form of quantum error correction code which preserves the total quantum information through the process. The quantum error correction code is then a Golay or Goppa code, such as that defined by the Leech lattice $\Lambda_{24}$, which includes three $E_8$ heterotic groups in a modular system. One of these $E_8$s should be similar to what Lisi's.

A part of what I am doing is trying to build from the ground up, or to use Dennett's idea of a crane he invoked in "Darwin's Dangerous Idea," where things are built up from a lower energy domain up. This is a bit different from Tony Smith's approach where he is working from what Dennett might call a skyhook by building from high up and then down. Maybe the two approaches will result in something in common.

Lawrence B. Crowell

 

[MTd2]

Today, Urs, from N-Cafe Category, made a very interesting point about Lisi's theory:

http://golem.ph.utexas.edu/category/2008/05/e8_quillen_superconnection.html#comments

"Main point, summarized.Whatever the physical viability of the proposal of arXiv:0711.0770, the expression in equation (3.1) on p. 23 is to be interpreted as a Quillen superconnection A on a ℤ 2-graded e 8 associated vector bundle and (3.2) is the corresponding Quillen curvature
F A=A 2.So if one wants to examine the possibility of describing particle physics with this approach, the mathematical structure to determine would seem to be something like “Quillen Yang-Mills theory”.

" Quillen superconnections are different from other notions of superconnections. In particular, Quillen superconnections do not come from a path-lifting property and are not related to an ordinary notion of parallel transport. For a discussion of Quillen superconnections and also of super parallel transport I can recommendFlorin Dumitrscu,

Superconnections and Parallel Transport

(pdf)."

http://etd.nd.edu/ETD-db/theses/available/etd-07212006-131339/unrestricted/DumitrescuF062006.pdf

 

[Cold Winter]

And now folks...

The trick question.

Has all this work produced any "testable" propositions, outside of what Garret started with in the first place?

 

[John G]

A part of what I am doing is trying to build from the ground up, or to use Dennett's idea of a crane he invoked in "Darwin's Dangerous Idea," where things are built up from a lower energy domain up. This is a bit different from Tony Smith's approach where he is working from what Dennett might call a skyhook by building from high up and then down. Maybe the two approaches will result in something in common.

Lawrence B. Crowell

Historically anyways, Tony did much like what Garrett did and started with gravity down at the D2 level. He originally had F4 at the highest level before going to E6 so his spaces could be complex instead of just real. He then made it up to E7 and E8.

 

[John G]

And now folks...

The trick question.

Has all this work produced any "testable" propositions, outside of what Garret started with in the first place?

It would be nice to get the Pioneer anomaly well studied to check it against predictions. I personally think photon decay experiments are messed up so it would be nice to see that looked at to check against predictions. Standard model (force strengths and tree level mass) and neutrino (one step up from tree level mass) calculations already look OK to me (for Tony Smith's variation).

 

[Lawrence B. Crowell]

Today, Urs, from N-Cafe Category, made a very interesting point about Lisi's theory:

" Quillen superconnections are different from other notions of superconnections. In particular, Quillen superconnections do not come from a path-lifting property and are not related to an ordinary notion of parallel transport. For a discussion of Quillen superconnections and also of super parallel transport I can recommend


Florin Dumitrscu,

Superconnections and Parallel Transport

(pdf)."

http://etd.nd.edu/ETD-db/theses/available/etd-07212006-131339/unrestricted/DumitrescuF062006.pdf

I can't find the Quillen paper. I read the start of the Dumitruscu paper. It looks to be fairly canonical differential geometry stuff. I am not sure what the "big idea" is here yet.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Historically anyways, Tony did much like what Garrett did and started with gravity down at the D2 level. He originally had F4 at the highest level before going to E6 so his spaces could be complex instead of just real. He then made it up to E7 and E8.

Mathematically he did this in the "voudou" physics. Smith has a lot of work on some representation theory along these lines, but his work IMO comes up a bit short of the dynamics end of things. I have worked out some aspects of what the breaking of an E_6 might look like with conformal gravity and spin-nets. What I have done can be found on

https://www.physicsforums.com/showthread.php?t=115826&page=3

I think if there are several E_8, say E_8xE_8 in string theory or three E_8's in a modular system for the Leech lattice, then there is "enough space" to describe this sort of intermediate energy quantum gravity. This appears to connect up with quantum foam and spin networks that Smolin and others have worked up with LQG. I hope to extend this to arrive at braids and higher systems in the future. In this sense I am trying to not just do pure group irrep work, but trying to tie it to physics (dynamics) as closely as possible.

It is my interest to see if there is some way to embed LQG and string/M-theory into a single system. I think LQG will provide some of the constraints on the stringy stuff that has gone in some cases into lala land.

Lawrence B. Crowell

 

[Kea]

In this sense I am trying to not just do pure group irrep work...

Exactly the point! Classical geometry is simply not rich enough to describe QG observables correctly. If E8 comes into it at all, it is more as an exceptional mathematical structure at the heart of a classification of groups, than as a mere group.

 

[John G]

I think if there are several E_8, say E_8xE_8 in string theory or three E_8's in a modular system for the Leech lattice, then there is "enough space" to describe this sort of intermediate energy quantum gravity. This appears to connect up with quantum foam and spin networks that Smolin and others have worked up with LQG. I hope to extend this to arrive at braids and higher systems in the future. In this sense I am trying to not just do pure group irrep work, but trying to tie it to physics (dynamics) as closely as possible.

It is my interest to see if there is some way to embed LQG and string/M-theory into a single system. I think LQG will provide some of the constraints on the stringy stuff that has gone in some cases into lala land.

Tony does this kind of thing with a 4-dim hyperdiamond Feynman Checkerboard. The "foam" of the Checkerboard is the Clifford Algebra 8-fold periodicity where you can represent any sized Clifford Algebra as Cl(8)xCl(8)xCl(8)... Tony and John Baez had once agreed that the Lorentzian Leech lattice-like E6/F4 would make a great spin foam but there was no known way to make it foamy. You kind of have to drop down to Clifford Algebra (from which Lie Algebras are derived) to make it work.

http://www.valdostamuseum.org/hamsmith/FynCkb.html
http://www.valdostamuseum.org/hamsmith/USGRFckb.html

At high energies before the 8-dim spacetime to 4-dim spacetime dimensional reduction Tony uses E8 as his hyperdiamond lattice and there is a D-brane/string/M/F theory use for this. Interestingly Urs was involved in Tony's work with these 4 and 8-dim lattices.

http://www.valdostamuseum.org/hamsmith/E8.html
http://www.valdostamuseum.org/hamsmith/stringbraneStdModel.html

 

[Kea]

...or three copies of $E_8$ in a modular system for the Leech lattice...

One way that we count generations is via the stringy Euler characteristic of the 6 point genus zero moduli space, which is one of a twistor triple (modelled on $\mathbb{CP}^{3}$). Tony Smith and Matti Pitkanen (together somehow) have considered how this 18d triple descends from a 24d one based on something like 3 copies of $E_8$. But of course, from the category orbifold Euler point of view, the group structure is just a side issue which might end up being useful in recovering Heterotic strings, if they turn out to be useful at all. (Aside: the recovery of LQG structures is much easier to understand through arbitrary restrictions of the categorical structure).

 

[Lawrence B. Crowell]

The three $E_8$ are a modular system of the $\Lambda_{24}$. The $CL(8)$ has the 256 elements of the 248 of $E_8$, or that

$$E_8~=~Spin(16)~+~128,~120~+~128~=~248~dimensions$$

is contained in $Cl(8)~=~1~+~8~+~28~+~56~+~70~+~56~+~28~+~8~+~1$, which is the 240 of the root space of $E_8$, plus the 8 of the Cartan matrix, for the 248, plus 1+3+3+1 in $Cl(8)$ not in $E_8$. For $CL(16)~=~CL(8)\times CL(8)$, a triality copy has some potentially interesting properties, in particular two of these Cartan centers are involved with the interesection or Kahler form in the definition of exotic $M^4$'s and gravitational instantons. There are a number of possible ways to decompose $E_8$, and it might be possible to bury supersymmetric pairs of known particles with their mirror terms, or in the exotic four manifolds this type of theory would predict, which are K^3 type manifolds similar to Calabi-Yau spaces.

The modularity of the Leech lattice is given by a weight 12 modular form (function) defined by the theta function for the E_8 lattice

$$\theta_8~=~1~+~240\sum_{n=1}^\infty div(n)q^{2n}$$

where this is also the Eisenstein E_4. The Leech lattice being composed of three E_8s has a theta function cubic on $\theta_8(q)$ as

$$\Theta_{24}(q)~=~\theta_8(q)^3~-~720 q^2\prod_{n=1}^\infty(1~-~q^{2n})^{24}$$

where the numbers 240 and 720 appear prominantly.

As for spin-nets or foam, the possible system Tony considered might work. The $F_4/B_4$ defines the additional roots added to spin(8) to define F_4 and these roots define the map

$$spin(8)~\rightarrow~F_{4\setminus 36}~\rightarrow~OP^2$$

which is a property shared by E_6 and E_7. $E_6\times su(3)/(Z/3Z)$ and $E_7\times su(2)/(Z/2Z)$ are maximal subgroups of $E_8$. where both $E_7$ and $E_6$ under signature changes contains the conformal and Desitter groups. The deSitter groups under further decomposition give $E_6~\rightarrow SU(4)\times SU(2)\times U(1)$, where this spin gauge group is a possible model for spin connections on conformal gravity.

Lawrence B. Crowell

 

[Kea]

The modularity of the Leech lattice is given by a weight 12 modular form...

Yes, and for the sake of other readers, let me point out again that the j invariant is also associated to a theta function triplet, related to the E8 function. However, unlike with Witten's current use of the j invariant in the context of a non-physical cosmological constant in 2+1d (AdS/CFT), it arises here in a spatial 3d setting because (a) spatial directions are associated with the triplet and (b) $\Lambda$ is completely replaced by a cosmic time parameter. This makes his physical estimate of the black hole entropy yet another intriguing indication that group triplets, rather then groups, play an important role in the logic of mass generation.

The twistor dimension (Riemann surface) moduli triple is also interesting as a genus (0,1,2) triple, because genus plays the role of time steps, instead of the usual classical directions. These are the three time directions of, for instance, Sparling's twistor theory. In fact, all 12 dimensions of F theory are accounted for this way: 6 compactified directions from the sphere, 3 space and 1 time direction from the torus, and 2 auxiliary dimensions from the genus 2 case. I really can't understand why the string theorists keep insisting that classical geometry is more interesting than this quantum information point of view!

 

[Lawrence B. Crowell]

Yes, and for the sake of other readers, let me point out again that the j invariant is also associated to a theta function triplet, related to the E8 function. However, unlike with Witten's current use of the j invariant in the context of a non-physical cosmological constant in 2+1d (AdS/CFT), it arises here in a spatial 3d setting because (a) spatial directions are associated with the triplet and (b) $\Lambda$ is completely replaced by a cosmic time parameter.

The twistor dimension (Riemann surface) moduli triple is also interesting as a genus (0,1,2) triple, because genus plays the role of time steps, instead of the usual classical directions.

There has been of late some interest in the idea that constants of physics change, or are variable. João Magueijo has suggested various schemes in which the speed of light could vary. The variability of constants is a bit odd. For instance if $\alpha~=~e^2/\hbar c$ were determined by different values of $e,~\hbar,~c$, but where $\alpha~\simeq~1/137$ physics would be absolutely indistinguishable from what we know. So the most proper constants of nature are dimensionless ones, and if we were to track any variation in constants, dimensionless ones would be the choice of what should be detected.

The speed of light is just a conversion factor between space and time: Light travels such a distance in a certain amount of time in a fixed proportion we call c. Much the same is the case with the Planck scale which is that

$$\frac{G\hbar}{c^3}~=~L_p^2,$$

where the Planck length is a conversion factor with units of cm that converts from the Dirac unit of action hbar, to $c^3/G$ that has units of $erg-s/cm^2$. This is "action per unit area," which is the amount of action associated with an area associated with a black hole horizon area. This leads into Beckenstein bounds and in part what I related yesterday about a homomorphism between gravity and QM and how quantum mechanics and its unit of action hbar involves what might be called a "quantum horizon," which is a limit to the detectability of physics, HUP and so forth.

With the Planck unit if we were to half the speed of light we would find that the Planck length is increased by $\sqrt{8}$. Our clocks by the Planck unit of time $T_p~=~\sqrt{G\hbar/c^5}$ would tick away at a rate $\sqrt{32}$ times slower, which by $c~=~L_p/T_p~=~1/2$, would mean that we would observe nothing at all observationally changed by any rescaling of the speed of light! Interestingly if you consider electromagnetism according to Planck units (Planck units of charge, impedance and so forth) you again would observe absolutely nothing at all if you vary the speed of light. The observational consequences for changing the speed of light are absolutely unobservable. One might think that because hbar has not been changed that the fine structure constant should change. But one must realize that the h or hbar was first deduced from the Bohr radius

$$a_0~=~\frac{4\pi\hbar}{me^2}~=~\frac{m_p}{m_e\alpha}L_p,$$

which would appear to change by $\sqrt{8}$ It is assumed the masses of the proton and electron shift with the rescaled Planck mass equally. But there is a hitch here, for we and our experimental set up also rescale by $\sqrt{8}$, which negates any observable scale change in an atom due to the change in c. In effect the experimenters in a $c~\rightarrow~c/2$ world would find an $\hbar'$ and the Bohr radius so that nothing at all changes! In effect their physics published results would be indistinguishable from our own.

The speed of light is a spatial measure associated with projective rays, and these can be rescaled arbitrarily. The Planck unit of action or Dirac's unit $\hbar~=~h/2\pi$ also has what might be called projective properties as well, though physics has not explored this terribly much, where hbar rescales (or the quantum horizon as a projective system) according to how one might change c.

This equivalency with respect to projective varieties leads us in some ways to twistor constructions. Twistor geometry is motivated by the fact lightcones are projective geometries, or the projective Lorentz group $PSL(n,~C)$. Twistor space, or twistor theory, applies to four dimensional Minkowksi spacetime, but where the projective structure pertains to the conformal group $spin(4,~2)~\simeq~SU(2,~2)$. The conformal space is six dimensional $R^{4,2}$, and the blow up of a point in this space is $PR^5$ (signature information suppressed). The twistor space is constructed from this projective null space, which is the holomorphic twistor space. The projective twistor space contains the null space, with there being $PT^\pm$ massless $\pm$ helicity states of massless particles. The null projective space is a subspace of the projective twistor space, which has 5 real dimensions, where four of these are complex or components of two complex dimensions. The $PT^\pm$ have the same dimension, but have a twist or helicity state.

This leads to some interesting prospects with spin systems. I argue on

https://www.physicsforums.com/showthread.php?t=115826&page=3

that a spin-net for gravity exihibts a quantum phase transition. This is related to how spin fields, such as those associated with twistors, exist on a Fermi surface. This surface also has some topological features. A space of evolution, which can be a spin-net in the LQG sense, or a D-brane in the string theory (I am not partisan to either theory camp) For the space of evolution, which defines a world volume $V~=~\Sigma\times M^1$, where $V$ is the evolute of the surface $\Sigma$, which has a target map to the spacetime or super-spacetime $M^n$. The compactified winding of a D-brane on this world volume is given by a unitary group $U(n)$, where n is the winding number or coincidence number of these branes. These winding numbers define the brane charges on the voume, which define charges in K-theory groups on the manifold $M^n$, which are closely related to the cohomology $H^p(M^n,~R)$. Within twistor theory for $n~=~4$ this is the sheaf cohomology, where these charges are the $\pm$ helicity states or frequences for the $PT^\pm$ subspaces of twistor geometry.

The topology of this spin space is physically similar to a Fermi surface, which is the standard system in condensed matter physics. Volvik (gr-qc/0005091) has written on how the vacuum state of quantum gravity, and what determines the cosmological constant $\Lambda$. What I want to show is that the K-theory index is identified with the homotopy of the group structure of the Fermi surface $K(X~\subset~M^n)~\sim~\pi_k({\cal G})$, for the space X of k + 1 dimensions.

 

[MTd2]

I can't find the Quillen paper. I read the start of the Dumitruscu paper. It looks to be fairly canonical differential geometry stuff. I am not sure what the "big idea" is here yet.

Lawrence B. Crowell

Urs had the same doubt on the thread, the answer is here:
http://golem.ph.utexas.edu/category/2008/05/e8_quillen_superconnection.html#c016725

It is very enlightening to read all the thread, but you could start at this point.

It would be nice if you and others could also share your opinions on that thread.

 

[MTd2]

The three $E_8$ are a modular system of the $\Lambda_{24}$. The $CL(8)$ has the 256 elements of the 248 of $E_8$, or that

$$E_8~=~Spin(16)~+~128,~120~+~128~=~248~dimensions$$

is contained in $Cl(8)~=~1~+~8~+~28~+~56~+~70~+~56~+~28~+~8~+~1$, which is the 240 of the root space of $E_8$, plus the 8 of the Cartan matrix, for the 248, plus 1+3+3+1 in $Cl(8)$ not in $E_8$. For $CL(16)~=~CL(8)\times CL(8)$, a triality copy has some potentially interesting properties, in particular two of these Cartan centers are involved with the interesection or Kahler form in the definition of exotic $M^4$'s and gravitational instantons. There are a number of possible ways to decompose $E_8$, and it might be possible to bury supersymmetric pairs of known particles with their mirror terms, or in the exotic four manifolds this type of theory would predict, which are K^3 type manifolds similar to Calabi-Yau spaces.

I'd like to point this out: http://golem.ph.utexas.edu/category/2008/05/e8_quillen_superconnection.html#c016745

"We are interested in noncompact real forms (precisely which ones are listed here) of D 4×D 4. While the compact real form of D 4 has a triality symmetry, the noncompact real forms do not. In particular, d 8=h⊕k 3. In the cases of interest, k 1 and k 2 are complex, and complex conjugates of each other. There is no triality symmetry relating them to k 3. And there’s no ℤ 2 grading of the sort you claim."

 

[Kea]

MTd2, Distler et al are discussing a completely different use of E8 to the kind that we have in mind.

 

[MTd2]

MTd2, Distler et al are discussing a completely different use of E8 to the kind that we have in mind.

So, this thread is totally off topic. According to the title, it should be about Lisi's theory, somehow. So, I tried something to get back on topi. But I see that I am also lost here.

 

[Lawrence B. Crowell]

MTd2, Distler et al are discussing a completely different use of E8 to the kind that we have in mind.

Give me a day or two to respond more fully. I have yet to look into the superconductive E_8 much. As for "different uses" for E_8, that is easy to arrive at. There are multiple ways in which it can be decomposed. A part of my thinking with the modular Leech system with three E_8s is that it suggests that at high energy all possible systems may exist, and then at lower energy there is only one E_8 due to the inflaton breaking, landscape-Higgsian configuration or which ever perspective you might prefer.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

So, this thread is totally off topic. According to the title, it should be about Lisi's theory, somehow. So, I tried something to get back on topi. But I see that I am also lost here.

Not off topic per se. Lisi's theory might be wrong after all. It has a certain economy to it, and I suspect somehow it is not completely flawed. We do have a question to ponder as to why physics would "choose" one decomposition or irrep according to subgroups and not another.

Lawrence B. Crowell

 

[jal]

I just discovered that "kagome lattice" is being used as an approach by some people.
http://arxiv.org/abs/0711.3471
Thermodynamics of Ising spins on the Triangular Kagome Lattice: Exact analytical method and Monte Carlo simulations
Authors: Yen Lee Loh, Dao-Xin Yao, Erica W. Carlson
(Submitted on 23 Nov 2007 (v1), last revised 28 Apr 2008 (this version, v2))
-------
I saw the resemblance with the drawing on p. 15.
I'll leave it with you to evaluate the similarities.
jal

 

[CarlB]

Regarding triality, I had this sudden realization that triality could be related to a new kind of quantum statistics.

Given a multiparticle state with identical particles, one considers the swap operator "S" that swaps two particles. The swap operator squares to unity and so has eigenvalues of +1 (bosons) and -1 (fermions). It is a postulate of quantum mechanics that all quantum states are eigenststates of the swap operator. Any permutation may be written as a product of swaps so one finds that quantum states of identical particles are eigenstates of any permutation operator. From this one can derive the two types of quantum statistics. From quantum statistics, thermodynamics follows by counting states.

The above works great for the known particles but it is only an assumption that the postulate extends to all particles. The next least complicated alternative statistics would be defined by assuming that the quantum states are eigenstates of the "P" operator that cyclically permutes three objects. Rather than defining all possible permutations, cyclic permutation operators generate only the even permutations. Acting on a 3=particle state, the cyclic 3-permutation "C" is:

C |a,b,c> = |b,c,a>

The cyclic 3-permutation cubes to unity and so its only possible eigenvalues are cubed roots of unity. One root is real, +1, and there are two complex roots, $$\exp(\pm 2\pi/3)$$.

A possibly related concept is the "tripled Pauli statistics" that Lubos Motl found when examining the thermodynamics of small vibrations of black holes. See page 21 (or page 1155) of:
http://arxiv.org/abs/gr-qc/0212096v3

Motl's paper was on the edge between quantum mechanics and general relativity, so it could be that, uh, "Brannen statistics", rather than Bose statistics or Fermi statistics is needed to unify gravitation and particle mechanics.

 

[arivero]

I have recovered a conference remark of my old boss, LJ Boya, about some "siblings" (my word) of E8: Sp(1) and Oct(8).

the point is that E8 x E8 is well known to have the same dimension, 496, that SO(32).

And also happens that
Sp(1) x Sp(1) ~ O(4), with dim 6
Oct(1) x Oct(1) ~ O(8), with dim 26.

The numbers 6, 26 and 496 are perfect primes. Of course, 13 and 248 are the two extremes of the exceptional groups (Oct(1) is G2, isn't it? Smolin relies in this, in the paper hep-th/0104050, does him?). In any case, no clue about why 52, 77, and 133 have not got any similar role.

Also, note that a perfect prime is also a hexagonal number. And sBootstrap uses hexagonal numbers to fix the number of generations of particles: half of the smallest even hexagonal number. Thus I would be not surprised it 13 and 248 happened to have an interpretation as "generation number" in some models.

 

[Kea]

Brannen statistics...

LOL, Carl! But according to etiquette, you should refer to it as Motl statistics, or as ternary statistics, following kneemo's terminology.

 

[Lawrence B. Crowell]

Regarding triality, I had this sudden realization that triality could be related to a new kind of quantum statistics.

I had thought of some similar ideas. The first is with supersymmetry, where if we think of a SUSY doublet $\Phi~=~\phi~+~\xi\psi$, for $\xi,~\psi$ as a Grassmann variable and a Dirac field and $\phi$ a boson field, then we can think of there being $a,~a^\dagger$ type operators which interchange the fermionic and bosonic field. Now think of there being a field analoguos to the polarization vector in EM such that ${\vec P}~=~e\sigma$, and that there is a coupling of this SU(2) vector with the photon-like state operator. We would then have a covering over the SUSY states with three directions, which might imply some underlying symmetry we think of as associated with Boson-Fermion statistics.

The other idea is that the Galois field $F_4$ is the Dynkin diagram for $D_4$, and the diagram looks like a Mercedes-Benz symbol. This would then have the angular distribution you are thinking of. Oh, and BTW, $F_4$ is a group discription for an elementary spinor field and also is what emerges as the Galois field from the Schild's construction of general relativity. So these two ideas in some ways might connect together.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

The numbers 6, 26 and 496 are perfect primes. Of course, 13 and 248 are the two extremes of the exceptional groups (Oct(1) is G2, isn't it? Smolin relies in this, in the paper hep-th/0104050, does him?). In any case, no clue about why 52, 77, and 133 have not got any similar role.

Also, note that a perfect prime is also a hexagonal number. And sBootstrap uses hexagonal numbers to fix the number of generations of particles: half of the smallest even hexagonal number. Thus I would be not surprised it 13 and 248 happened to have an interpretation as "generation number" in some models.

These numbers are related to Mersenne primes as

$$6~=~2^1(2^2~-~1)$$
$$26~=~2^2(2^3~-~1)$$
$$496~=~2^4(2^5~-~1).$$

It is an interesting pattern. I am not sure whether this amounts to an accident of some sort or whether there is actually something going on here. If there is a connection it involves some connection between number theory and algebra nobody appartently understands.

Lawrence B. Crowell

 

[arivero]

Yep, my boss put this pattern between a catalog of pending problems, in a meeting of the Royal Academy of Science of Zaragoza, a couple years ago. I independently met the question of even hexagonal numbers in the sBootstrap theory last year. So actually we have three coincidental patterns, one finite and two infinites (dimensions of exceptional groups are 14, 52, 78, 133, 248 ):

the dimension of GxG when G is an exceptional group
28, 104, 156, 166, 496
the perfect numbers (related, as you say, to Mersenne primes).
6, 28, 496, ... 2^(k-1) * (2^k -1)
the even hexagonal numbers.
6, 28, 66, 120, 190, 276, 378, 496, 630,... 2n(4n-1).

Note that some people speaks of "A1,G2,F4,E6,E7,E8." as "the traditional Cartan exceptional group sequence", thus really the number 6 is also included in the finite sequence.

$$6~=~2^1(2^2~-~1)$$
$$26~=~2^2(2^3~-~1)$$
$$496~=~2^4(2^5~-~1).$$
It is an interesting pattern. I am not sure whether this amounts to an accident of some sort or whether there is actually something going on here. If there is a connection it involves some connection between number theory and algebra nobody appartently understands.

Edit
6, 28, 120, 496... are also Sloane http://www.research.att.com/~njas/sequences/A006516 http://www.research.att.com/~njas/sequences/A007691

 

[Lawrence B. Crowell]

(dimensions of exceptional groups are 14, 52, 78, 133, 248 ):

the dimension of GxG when G is an exceptional group
28, 104, 156, 166, 496
the perfect numbers (related, as you say, to Mersenne primes).
6, 28, 496, ... 2^(k-1) * (2^k -1)
the even hexagonal numbers.
6, 28, 66, 120, 190, 276, 378, 496, 630,... 2n(4n-1).

Note that some people speaks of "A1,G2,F4,E6,E7,E8." as "the traditional Cartan exceptional group sequence", thus really the number 6 is also included in the finite sequence.

I made a type with $28~=~2^2(2^3~-~1)$. I had not thought about whether there are integer patterns or sequences associated with the root dimension of groups. It appears as if this pattern pertains to the complexification of these groups $G(R)\times G(R)~=~G(C)$. I might spend a little bit of time numerically generating these numbers to see whether this gives patters with the dimensions for the Leech lattice and its subgroups. For $dim(\Lambda_{24})~=~196560$ is divisible by 32760, 196560 -:- 32760 = 6, and $32760 ~=~2^7(2^8~-~1)$. It is hard to know if there is really something here of interest to physics, or whether this is numerology.

Lawrence B. Crowell

 

[arivero]

As I told elsewhere, in my case the hexagonal number pattern appears when you ask for the sBootstrap conditions, a coincidence between bosonic and fermionic degrees of freedom that happens in the QCD string. Half of this number (ie 3, 14, 33, 60, ... ) is the number of generations needed for the sBootstrap to exist.

I suspect that some quantisation of flavour will produce at least SO(32), if not E8xE8. This accounts for the 496. But no hint about Leech lattice.

 

[CarlB]

LOL, Carl! But according to etiquette, you should refer to it as Motl statistics, or as ternary statistics, following kneemo's terminology.

The Koide formulas for the mesons are justifiable without any need for fancy statistics because a meson has only one quark and one antiquark (for the simpler mesons which are more appropriate for the formula), and these are distinguishable so there is no need for statistics. But to get the formula to apply to the baryons, I need something else. Otherwise the formulas / color bound state model can't work for things like uuu or uud baryons.

But this is a long shot idea by a complete amateur so I don't have to follow etiquette. And since Lubos has repeatedly called everyone else a complete idiot (including me) I think we can safely assume that the universe isn't going to reward Lubos with a "statistics". More to the point, the statistics he gave in his paper don't work. They're just suggestive that something strange could be going on.

 

[Lawrence B. Crowell]

As I told elsewhere, in my case the hexagonal number pattern appears when you ask for the sBootstrap conditions, a coincidence between bosonic and fermionic degrees of freedom that happens in the QCD string. Half of this number (ie 3, 14, 33, 60, ... ) is the number of generations needed for the sBootstrap to exist.

I suspect that some quantisation of flavour will produce at least SO(32), if not E8xE8. This accounts for the 496. But no hint about Leech lattice.

I am not sure what sBoostrap is, and I suppose that the hexagonal number pattern is related to this. These generations, such as the 60, is half the number of spinor on the $M_{12}$ Mathieu group on the 120-cell or icosahedron. I am not sure if degeneracies are considered here, but 196560 is divisible by 32760 with 6 as the answer. I am not sure if there is any significance to this

I think a small verision of quantum gravity is the trio group $S^3\times SL_2(7)~ \subset~M_{24}$. The Leech lattice contains $M_{24}$ as a quantum error correction code, which embeds three $E_8's$ --- an $E_8\times E_8$ for the graded heterotic supergravity field theory and the third for this configuration of all possible spacetimes. In the restricted $S^3\times SL_2(7)$ this is a thee dimensional Bloch sphere where each point on it is a "vector" in a three space spanned by the Fano planes associated with these three $E_8$'s. $S^3\times SL_2(7)$ has 1440 roots and is itself a formidable challenge, but this represents a best first approach. $M_{24}$ has 196560 roots and clearly an explicit calculation of those is not possible at this time.

I am not sure if there is any magical numerology here, but with the 8 additional weights for each of the $E_8$'s, which is then doubled to a total of 48 weights due to the double covering on the Bloch sphere, this gives 1488 as the size of the trio group, which when divided by 6 gives 248, or when divided by 3 gives 496. I am not sure, but this might have some relevancy to Carl Brennan's triality approach to things.

Lawrence B. Crowell

 

[CarlB]

This thread is getting old, so I'll type in the latest Koide fits. These are supposed to be practical applications of triality to quarks interacting to form mesons. As such, they might give a clue on how to fit the quarks better in with the leptons and quarks.

The first observation is that the (handed) quarks and anti quarks are 2/3 or 1/3 of the way between leptons and anti leptons in quantum numbers, and they come in 3 colors, so one ends up with a (1,3,3,1) multiplet with the 1's two types of leptons, say left handed positron and right handed electron, while the 3's two types of quarks, say right handed up quark and left handed anti-down quark. The implication is that the quarks and leptons could be built from three preons each of eight types, charged or neutral, left handed or right handed, and preon or antipreon (with positive charge). Then the electron is composed of three preons with charge +1/3 each, the up quark is made from two +1/3 preons and a neutral preon, etc.

Koide's formula for the masses of the charged leptons reads as follows (ignoring an overall mass scale factor of 25.054 sqrt(MeV)):
$$\sqrt{m_{en}} = \sqrt{0.5} + \cos(2/9 + 0\pi/12 + 2n\pi/3)$$
A similar formula fits the neutrino oscillation mass differenes (with a mass scale of 0.1414 sqrt(eV) also not included):
$$\sqrt{m_{\nu n}} = \sqrt{0.5} + \cos(2/9 + 1\pi/12 + 2n\pi/3)$$
The formula for the charged leptons is quite old and famous. I found the neutrino mass formula a couple years ago and it's now in the literature in various places, for example, Mod. Phys. Lett. A, Vol. 22, No. 4 (2007) 283-288:
http://www.worldscinet.com/mpla/22/2204/S0217732307022621.html

If the quarks are to be composed of a mixture of preons, which form should they follow?

Thinking of the above formulas as resonance conditions for the preons, perhaps a meson made from two quarks could resonate either way. The simplest place to test this is on the mesons that are most carefully and exactly studied, the b-bbar (Upsilon) and c-cbar (J/psi) mesons.

In the particle data group information on the b-bbar and c-cbar mesons:
http://pdg.lbl.gov/2007/listings/contents_listings.html
there are six of each type given:

The Upsilon b-bbar mesons are:
Name, quarks, I^G(J^PC) mass(error) koide_type
\Upsilon(1S) b/b 0^-(1^{--}) 9460.30(26) 1
\Upsilon(2S) b/b 0^-(1^{--}) 10023.26(31) 1
\Upsilon(3S) b/b 0^-(1^{--}) 10355.20(50) 0
\Upsilon(4S) b/b 0^-(1^{--}) 10579.40(120) 1
\Upsilon(10860) b/b 0^-(1^{--}) 10865.00(800) 0
\Upsilon(11020) b/b 0^-(1^{--}) 11019.00(800) 0

The Koide_type is 0 or 1 according as that mass is part of a triplet of states that follow the Koide formula with 0 or 1 copies of pi/12 in the angle. The resulting equations for the Upsilon masses are (leaving off the factor of 25.054 again):
$$\sqrt{m_{\Upsilon 1 n}} = 3.994433 - 0.128815\cos(2/9 + 1\pi/12 + 2n\pi/3)$$
$$\sqrt{m_{\Upsilon 0 n}} = 4.137251 - 0.077550\cos(2/9 + 0\pi/12 + 2n\pi/3)$$

The measured and calculated masses are as follows (MeV):
9460.3(3) ~= 9451.8
10023.2(3) ~= 10041.0
10355.2(5) ~= 10355.1
10579.4(12) ~= 10569.1
10865.0(80) ~= 10864.3
11019.0(80) ~= 11019.5

which is considerably more accurate than random chance would suggest. To put this into perspective, the mass difference between the charged and neutral pions is about 5 MeV.

Fitting the Koide formula to six masses like this is similar to how one would fit a spin-1/2 splitting to six masses. In that case one looks for how one can put the six masses into three pairs of masses with the same difference between the masses. If that were the case for the Upsilons, you can be sure that there would be papers showing why a quark interaction causes this kind of splitting. After you divided the six particles up into three pairs, you need only four degrees of freedom to describe the particles, say the three averages of the pairs, and the split amount. Similarly, with the above Koide fit, you end up removing two degrees of freedom from the six masses. The new four degrees of freedom are 3.994433, -0.128815, 4.137251, and -0.077550.

It would be easy to suppose that this is random chance, but the c-cbar mesons also come in exactly six masses, and these also are very closely fit by four Koide parameters. In this case the mass formulas are:
$$\sqrt{m_{\Psi 1 n}} = 2.442070 - 0.249554\cos(2/9 + 1\pi/12 + 2n\pi/3)$$
$$\sqrt{m_{\Psi 0 n}} = 2.510507 - 0.089433\cos(2/9 + 0\pi/12 + 2n\pi/3)$$
and the mass fits are unnaturally accurate:

J\psi(1S) c/c 0^-(1^{--}) 3096.916(11) 1 ~= 3096.9
\psi(2S) c/c 0^-(1^{--}) 3686.093(34) 0 ~= 3686.1
\psi(3770) c/c 0^-(1^{--}) 3771.1(2.4) 1 ~= 3773.8
\psi(4040) c/c 0^-(1^{--}) 4039(1) 0 ~= 4040.4
\psi(4160) c/c 0^-(1^{--}) 4153(3) 0 ~= 4149.8
\psi(4415) c/c 0^-(1^{--}) 4421(4) 1 ~= 4418.4


I've not yet figured out how to derive this from the assumption that the quarks are composites made from the same things that make up the leptons. I think it has to be doen with perturbation theory. My instinct is that the quarks are acting as a body that has two possible resonances, the type 0 (electron - muon - tau) and the type 1 (neutrinos). Either of these two resonances shows up in threes just like the generations of particles do. But only one resonance can be excited at a time. The result is six resonances that satisfy a Koide relationship.

Among the 8 numbers that give the four Koide fits here, some of them are rather close to rational numbers, or square roots, or what have you. For example, the first of the 8 Koide fit numbers, 3.994433, is very close to 4. But I don't see an overall pattern to the numbers.

My suspicion is that if this can be put into a perturbation expansion, we will see how to derive the 8 Koide fits from simpler assumptions. But I haven't figured out how to do this yet. This may be the reader's opportunity to score a quick paper. Like I mention above, my suspicion is that one should model this as a system that has two available, but mutually exclusive, resonances. And I'm working on this, but I haven't yet got anything worth writing up.

It may or may not help to read the incomplete paper I'm writing that is driving the search for these kinds of coincidences:
http://www.brannenworks.com/qbs.pdf

 

[CarlB]

New paper out on E8:
http://www.iop.org/EJ/abstract/1751-8121/41/33/332001/

You can download the acrobat version at the above link.

Meanwhile, Kea and I are working on the CKM (quark mixing) and MNS (neutrino mixing) matrices. Kea has pointed out the usefulness of 1-circulant and 2-circulant 3x3 matrices for these things.

1-circulant are what I used to call "circulant" while 2-circulants are the reverse order. The even permutations on 3 elements uses the 1-circulants while the odd permutations on 3 elements uses the 2-circulants. The matrix sum given below is of a real 1-circulant and an imaginary 2-circulant. The MNS (neutrino mixing) matrix can be written in a peculiarly simple 3x3 form as the sum of a 1-circulant and a 2-circulant matrix:
$$\sqrt{1/6}\left(\begin{array}{ccc} +\sqrt{2}&1&0\\0&+\sqrt{2}&1\\1&0&+\sqrt{2}\end{array}\right)\;\;\pm i\sqrt{1/6}\left(\begin{array}{ccc} -\sqrt{2}&1&0\\1&0&-\sqrt{2}\\0&-\sqrt{2}&1\end{array}\right)$$

That is, when you take the squared magnitude of the entries of the above 3x3 complex matrix, you get the MNS matrix under the "tribimaximal" form which is a good approximation of current experimental measurement:
$$\left(\begin{array}{c|ccc} &\nu_1&\nu_2&\nu_3\\ \hline e&2/3&1/3&0\\ \mu&1/6&1/3&1/2\\ \tau&1/6&1/3&1/2\end{array}\right)$$

The above matrix has all rows and columns sum to 1. In addition, the complex matrix (i.e. the sum matrix given at the top), has all rows and columns sum to
$$(\sqrt{2} +1 \mp i\sqrt{2}\pm i)/\sqrt{6}$$
which has magnitude 1. So you can multiply by the complex conjugate of this sum and convert the complex matrix given above into a form which is "doubly magic" in that its rows and columns all sum to 1, and the sums of the squared magnitudes of its rows and columns also sum to 1.

Matrices that have this property (double magic) are kind of unusual. It's interesting in that the "double magic" is a linear and bilinear property. It is linear in that the sum of the rows and columns all add to 1; therefore we can take two such matrices and sum them to obtain a new matrix that also has the property that its rows and columns add to a constant. If the two objects being summed are scaled, then we can arrange for the sum matrix to have all rows and columns add to 1. On the other hand, the sum of the squared magnitudes is a bilinear property, a requirement of normalization.

Of course we're looking at how to do the CKM matrix, preferably as a function of the MNS matrix.

 

[Kea]

New paper out on E8:
http://www.iop.org/EJ/abstract/1751-8121/41/33/332001/

Thanks for the link on E8. I see he refers to Bars, Gunaydin, Duff, Witten and other interesting authors, so I look forward to reading it.

Note that another way to write the tribimaximal matrix is as a product of quantum Fourier operators $F_3 F_2$, one associated to three dimensions (mass quantum numbers) and one to two (spin operators). Carl has used a 6x6 matrix version of related operators to derive idempotents for all the standard model fermions. The construction is essentially unique.

 

[Berlin]

Just seen the new version of Smolins paper on the Plebanski action. It says that a new paper is in preparation together with Lisi and Speziale on gravity+EW unification, building on Smolins framework.

http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.0977v2.pdf

I would call this "exceptionally" good news! E8 is still alive.

berlin