I reread your paper on neutrino braids last night and have a little better understandings and your post here is almost like coincidence in the numbers (but muon and tauons I do not directly connect from here).
Anyway, I posted it today as Friomnium inspired by your clear paper and Rowlands speculations beyond Dirac.
Again I hope I have not said or misunderstood your worldview, the worldview.
Holy cow. I'm just looking at this N=8 Gell-Mann thing from 1983. In the N=8 theory there are 56 spin 1/2 fermions. In Gell-Mann's proposal, 48 of them are the usual leptons and quarks. The other 8 are "goldstinos" that give mass to the eight "gravitinos".
Gell-Mann classifies them according to how they transform under an SU(3) which is not the SU(3) of color, but rather a diagonal SU(3) in the product SU(3)_color x SU(3)_generation. At least, that's how it's rationalized, but the interpretation is a little confused. Anyway, the up-type quarks are 8+1, the down-type quarks are 6+3bar (or 6bar+3), and the leptons are all just 3 or 3bar. But among the goldstinos, there is a 3 and a 3bar (which account for 6 of the gravitinos). Sounds like your mirror neutrinos!
There are also two "1"s - that is, two more fermionic states which are singlets under this mystery SU(3) which is not color or family symmetry, but some combination of the two; I don't know what they would correspond to. Perhaps the "Gell-Mann mirror neutrinos" should be thought of as transforming under 3+1 and 3bar+1; that would account for all the extra N=8 spin 1/2 states, and all eight gravitinos.
Also of course it is reminiscent of the relationship that there is three times as much "dark energy" as there is "dark matter". The dark matter comes from the singlet mirror neutrinos, the dark energy from the triplet and antitriplet mirror neutrinos.
Sounds good, Mitchell. There is no way I am downloading that Gell-Mann file myself, though, so you will hsve to write something up. I am glad there are no more stau neutrinos! Yes, Riofrio's $3 \Omega_{dm}$ business should be there somehow.
How's this for efficiency: at Theoretical Physics Stack Exchange, I put forward a highly sanitized, totally mainstreamed version of these ideas, focusing entirely on Gell-Mann; and within an hour it has four downvotes and it's off the front page, as if it had never been asked. At least it wasn't deleted entirely!
Anyway, hopefully now you can see that so long as one insists on a commutative geometry $N=8$ theory, one gets Higgs (fairy) stuff floating around in the representations. From the braid point of view, the $1+1+1$ sits on the diagonal of $3 \times 3$ (noncommutative and nonassociative) algebras. There is no reason to associate it to a particle, since particles are specified by the braid set.
Certainly the difference between a geometric and a nongeometric idea of the "extra dimensions" is one of those crucial differences that needs to be expanded upon. I aspire to be comfortable on either side of that divide, but the geometric approach is still the most familiar to me.
I can say that in N=8 supergravity you have 163 fields, 1+28+70 = 99 bosons, and 8+56 = 64 fermions. (It's odd, but I can't find any deep explanation of the 163, or even anyone remarking upon it. But I'm sure it's out there in the literature.) Your extra 1, as you have described it, augments the 56; but it seems like it could go on the other side too, and turn that 99 into 100.
I know. Lubos has blogged about 163 in the past. The N=8 connection seems to be new to him, however (I just asked; normally you say N=8 sugra has 256 states, not 163 fields).
168 is just about my own favorite number, and I really would expect that there is a canonical uplift from 163 to 168 in some branch of mathematics.
163 becomes 256 once you notice that the 1 is a massless spin 2 and the 28 are massless spin 1, so the counting is 1 + 28 + 70 + 64= 163 2*2 + 2*28 + 1*70 + 2*64 = 256 A lot of people prefers to state the number of particles instead of the number of helicity states because when they get mass the number chages.
Hi Kea, mi current opinion is that all of them are explained away with the superBootstrap, except the 1/2 partners of W and Z. Amusingly, the small signal in LHC could come from these partners. So lets wait.
The bad news is that the sBoostrap looks more and more as if string theory was right in the target in 1971. See Acts 9:3-4 then for my current feelings.
And the ribbon graph for the Klein quartic gives a $168$ dimensional representation of the modular group, which is the thing we like to cover with braids, using the bioctonions. That is, we could say
$168 = 56 + 56 + 56$
where $56$ sits over $0$ on the surface and is now associated to the FTS for the $54$ dimensional bioctonion Jordan algebra.
Cool, Alejandro. In the braid setting, Fourier susy maps the $W^{\pm}$ to the left handed $e^{\pm}$ and the $Z$ (color triplet) to the right handed $e^{\pm}, \nu$. So, no sparticle zombies.
And looking at Baez's Klein quartic page, we see that the $3×56=168$ comes from pairs (line,point) on the Fano plane, where the point is incident on the line. In the quartic, the point becomes a cube (with $8=56/7$ vertices) and the line an anticube, which are disjoint. Since there are $21$ ways to choose a (line,point) in the Fano plane, there are $21×8=168$ special (cube,anticube,vertex) triplets in the quartic.
So if we choose $2$ lines in the Fano plane ($21$ ways) and then on each line choose a point away from the line intersection, we get $84 = 21 \times 4$ geometric objects. The $168 = 2 \times 84$ puts an ordering on the two lines. On the quartic there are $84$ edges. Now there are also $84$ configurations coming from the set of cubes and anticubes for the Fano subdiagram, namely an anticube pair with two associated cubes.
Mitchell has also noticed some exploits of octonions and the Fano plane to justify the infamous 84 geometric objects as all the quarks and leptons excepting the top quark. I suspect of a dual formulation where instead of excepting the top, it is the neutrino sector which is left out. Still, I agree that we have too many small numbers, some will go, some will fail.
The sBootstrap map for the W+ supermultiplet is a lot trickier. No clue still about the fermion, it looks as a baryion but it is not. For the scalar, the point is that it is axial and then it fails to couple to color nor to B-L electric charge. So a diquark uu is really a colourless particle of charge +1, when it is incorporated in the supermultiplet. Ugly.
Kea, I never did, until my own way directed me towards them. Probably it is wise, when your way cross the standard lore, to bounce away from it, to avoid to repeat the same mistakes. On the other hand, one can read the lore and think what is wrong in the standard way (so our thread in PF is called "the wrong turn").
On the other hand, one can read the lore and think what is wrong in the standard way ...
There is no standard way. That would imply that the establishment would never make a total mess of everything. The opposite is obviously true, and the ghost of Kuhn will taunt the bones of all the stringers.
Hi Kea,
ReplyDeleteI reread your paper on neutrino braids last night and have a little better understandings and your post here is almost like coincidence in the numbers (but muon and tauons I do not directly connect from here).
Anyway, I posted it today as Friomnium inspired by your clear paper and Rowlands speculations beyond Dirac.
Again I hope I have not said or misunderstood your worldview, the worldview.
The PeSla
Hi, The Pe Sla. You have a great blog, and misrepresentation is hardly an issue when we make so many mistakes along the way!
ReplyDeleteHoly cow. I'm just looking at this N=8 Gell-Mann thing from 1983. In the N=8 theory there are 56 spin 1/2 fermions. In Gell-Mann's proposal, 48 of them are the usual leptons and quarks. The other 8 are "goldstinos" that give mass to the eight "gravitinos".
ReplyDeleteGell-Mann classifies them according to how they transform under an SU(3) which is not the SU(3) of color, but rather a diagonal SU(3) in the product SU(3)_color x SU(3)_generation. At least, that's how it's rationalized, but the interpretation is a little confused. Anyway, the up-type quarks are 8+1, the down-type quarks are 6+3bar (or 6bar+3), and the leptons are all just 3 or 3bar. But among the goldstinos, there is a 3 and a 3bar (which account for 6 of the gravitinos). Sounds like your mirror neutrinos!
There are also two "1"s - that is, two more fermionic states which are singlets under this mystery SU(3) which is not color or family symmetry, but some combination of the two; I don't know what they would correspond to. Perhaps the "Gell-Mann mirror neutrinos" should be thought of as transforming under 3+1 and 3bar+1; that would account for all the extra N=8 spin 1/2 states, and all eight gravitinos.
Also of course it is reminiscent of the relationship that there is three times as much "dark energy" as there is "dark matter". The dark matter comes from the singlet mirror neutrinos, the dark energy from the triplet and antitriplet mirror neutrinos.
Sounds good, Mitchell. There is no way I am downloading that Gell-Mann file myself, though, so you will hsve to write something up. I am glad there are no more stau neutrinos! Yes, Riofrio's $3 \Omega_{dm}$ business should be there somehow.
ReplyDeleteUnder the $57$ of triality we would have $1+1+1$, not just two singlets.
ReplyDeleteHow's this for efficiency: at Theoretical Physics Stack Exchange, I put forward a highly sanitized, totally mainstreamed version of these ideas, focusing entirely on Gell-Mann; and within an hour it has four downvotes and it's off the front page, as if it had never been asked. At least it wasn't deleted entirely!
ReplyDeleteFuckem, Mitchell, let 'em be.
ReplyDeleteAnyway, hopefully now you can see that so long as one insists on a commutative geometry $N=8$ theory, one gets Higgs (fairy) stuff floating around in the representations. From the braid point of view, the $1+1+1$ sits on the diagonal of $3 \times 3$ (noncommutative and nonassociative) algebras. There is no reason to associate it to a particle, since particles are specified by the braid set.
Certainly the difference between a geometric and a nongeometric idea of the "extra dimensions" is one of those crucial differences that needs to be expanded upon. I aspire to be comfortable on either side of that divide, but the geometric approach is still the most familiar to me.
ReplyDeleteI can say that in N=8 supergravity you have 163 fields, 1+28+70 = 99 bosons, and 8+56 = 64 fermions. (It's odd, but I can't find any deep explanation of the 163, or even anyone remarking upon it. But I'm sure it's out there in the literature.) Your extra 1, as you have described it, augments the 56; but it seems like it could go on the other side too, and turn that 99 into 100.
Oh, but the number $163$ is a nice moonshine number. Of course, it will never make sense within the stringy geometric paradigm.
ReplyDeleteHere is Ronan's site on the number $163$. Personally, I like that $163 = 54 + 54 + 54 + 1$.
ReplyDeleteThere's probably an "extension" to 168 (symmetries of the Fano plane).
ReplyDeleteNo, Mitchell. In moonshine, $163$ is a very special number. Look at Ronan's page.
ReplyDeleteI know. Lubos has blogged about 163 in the past. The N=8 connection seems to be new to him, however (I just asked; normally you say N=8 sugra has 256 states, not 163 fields).
ReplyDelete168 is just about my own favorite number, and I really would expect that there is a canonical uplift from 163 to 168 in some branch of mathematics.
163 becomes 256 once you notice that the 1 is a massless spin 2 and the 28 are massless spin 1, so the counting is
ReplyDelete1 + 28 + 70 + 64= 163
2*2 + 2*28 + 1*70 + 2*64 = 256
A lot of people prefers to state the number of particles instead of the number of helicity states because when they get mass the number chages.
Ah, good point, Alejandro. So what is your current opinion on fairies?
ReplyDeleteMitchell, then you will know that $168 = 7 \times 24$. These are the heptagons of the Klein quartic tiling.
ReplyDeleteHi Kea, mi current opinion is that all of them are explained away with the superBootstrap, except the 1/2 partners of W and Z. Amusingly, the small signal in LHC could come from these partners. So lets wait.
ReplyDeleteThe bad news is that the sBoostrap looks more and more as if string theory was right in the target in 1971. See Acts 9:3-4 then for my current feelings.
And the ribbon graph for the Klein quartic gives a $168$ dimensional representation of the modular group, which is the thing we like to cover with braids, using the bioctonions. That is, we could say
ReplyDelete$168 = 56 + 56 + 56$
where $56$ sits over $0$ on the surface and is now associated to the FTS for the $54$ dimensional bioctonion Jordan algebra.
Cool, Alejandro. In the braid setting, Fourier susy maps the $W^{\pm}$ to the left handed $e^{\pm}$ and the $Z$ (color triplet) to the right handed $e^{\pm}, \nu$. So, no sparticle zombies.
ReplyDeleteOops, duh. I mean $56$ for the octonion FTS and then $2 \times 56$ for the bioctonion one.
ReplyDeleteAnd looking at Baez's Klein quartic page, we see that the $3×56=168$ comes from pairs (line,point) on the Fano plane, where the point is incident on the line. In the quartic, the point becomes a cube (with $8=56/7$ vertices) and the line an anticube, which are disjoint. Since there are $21$ ways to choose a (line,point) in the Fano plane, there are $21×8=168$ special (cube,anticube,vertex) triplets in the quartic.
ReplyDeleteSo if we choose $2$ lines in the Fano plane ($21$ ways) and then on each line choose a point away from the line intersection, we get $84 = 21 \times 4$ geometric objects. The $168 = 2 \times 84$ puts an ordering on the two lines. On the quartic there are $84$ edges. Now there are also $84$ configurations coming from the set of cubes and anticubes for the Fano subdiagram, namely an anticube pair with two associated cubes.
ReplyDeleteCheck out Sloane's OEIS, particularly A198343 at http://oeis.org/A198343 . You will see many 'magic numbers'.
ReplyDeleteMitchell has also noticed some exploits of octonions and the Fano plane to justify the infamous 84 geometric objects as all the quarks and leptons excepting the top quark. I suspect of a dual formulation where instead of excepting the top, it is the neutrino sector which is left out. Still, I agree that we have too many small numbers, some will go, some will fail.
ReplyDeleteThe sBootstrap map for the W+ supermultiplet is a lot trickier. No clue still about the fermion, it looks as a baryion but it is not. For the scalar, the point is that it is axial and then it fails to couple to color nor to B-L electric charge. So a diquark uu is really a colourless particle of charge +1, when it is incorporated in the supermultiplet. Ugly.
ReplyDeleteAlejandro, I never took classical supermultiplets seriously, and I most certainly do not now.
ReplyDeleteKea, I never did, until my own way directed me towards them. Probably it is wise, when your way cross the standard lore, to bounce away from it, to avoid to repeat the same mistakes. On the other hand, one can read the lore and think what is wrong in the standard way (so our thread in PF is called "the wrong turn").
ReplyDeleteOn the other hand, one can read the lore and think what is wrong in the standard way ...
ReplyDeleteThere is no standard way. That would imply that the establishment would never make a total mess of everything. The opposite is obviously true, and the ghost of Kuhn will taunt the bones of all the stringers.