Furey models with Division Algebras

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arivero

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Just noticed a new article of C Furey not arxived, but readable in SCOAP

Phys.Lett. B742 (2015)
http://inspirehep.net/record/1342971?ln=es
Charge quantization from a number operator

Well, it is not new but I have not found a thread mentioning it. It seems to continue the quest from her previous papers

http://arxiv.org/abs/1405.4601 Generations: Three Prints, in Colour
http://arxiv.org/abs/1002.1497 Unified Theory of Ideals

Basically it plays again the Fermion cube, now considering how the octonions and clifford algebra can be used to build a charge operator layering the cube. There is also an intriguing announcement about a generalisation to Pati-Salam.
 
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

arivero

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Has anybody took time to look at Furey's papers, specially from the point of view of group representation theory? Her work on generations extract from the octonions a Clifford algebra that seems complex Cl6, is it?
 
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Yes, I have red the three papers today. There is only one main and obsessional idea behind the three papers.

But for me, the most interesting, also the strangest and the most unconventional, idea is exposed in the inspire paper (see résumé).

If the intention of the author wouldn't have been to look for the reason why the standard model has its actually well-accepted mathematical structure, I suppose that that kind of paper would never have been published!

But it has been... and that's really ... (citation): "a new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state...(end of the citation)".

My crazy questions:
- "Do you really think it is a realistic/serious hypo-thesis: neutrinos as fundamental stones of the vacuum?" I have some doubts because if I believe what I have seen in diverse lectures that I have made on the topic, the neutrinos cannot completely explain the (dark) matter we are looking for.
- "Do you think neutrinos can really be obtained by a Lamb-Retherford effect in vacuum?"
 

arivero

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Thinking aloud and naively, first thing I wonder about is what group do we get from the complex clifford algebra of six components. I'd expect SO(6) so SU(4). But the whole Clifford algebra has 64 elements, are we to assume that they partition over SO(6) as 1,6,15,20,15,6,1? Not as sum of spinor irreps? Or perhaps the history is about SO(12) or SO(14)?

The point is, ok, lets assume that we are seeing the article on color+generations is seeing the branching of SU(4) down to SU(3) x U(1).
It could be it is only seeing branchings of one 15 and four 4,
4 = (1)(−3) + (3)(1)
15 = (1)(0) + (3)(4) + (3)(−4) + (8)(0)
plus one singlet. Neglecting the question of the U(1) charge, it looks as three generations indeed.

Question is, what representations is the math seeing here and how to obtain them from a GUT theory?

For reference:
4 × 4 = 1 + 15
4 × 4 = 6 + 10
6 = (3)(−2) + (3)(2)
10 = (1)(−6) + (3)(−2) + (6)(2)

The most elemental attempt could be get 64 out of squaring 4+4 but then we get su(3) sextets from the 10:
(4+4)x(4+4)=6+10+ 1+ 15+ 15+ 1 + 10 + 6
 
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arivero

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:smile: Today an alternative view of Cl(6) in the arxiv:

https://arxiv.org/abs/1702.04336
The Standard Model Algebra by Ovidiu Cristinel Stoica

Same criticism: I miss some links with the theory of irreducible representations of Spin(6)

even without su(4), only with su(3) and u(1), I'd expect the clifford aka exterior algebra to show things as:
(1+3) x (1 + 3) = 1 + 1 + 3 + 1 +3 + 3 x 3 = 1 + 3 + 3 + 8
(1+3) x (1 + 3) = 1 + 3 + 3 + 6 + 3
(1+3 + 1 + 3) x (1+3 + 1 + 3)= 6*1 + 5*3 + 5*3 + 2*8 + 2*6
How do the sextets move to make triplet+antitriplet plus six singlets?

(and yes, it could be a good thing if the sextet were the top quark plus the neutrinos. Alternatively, a mechanism breaking 6 into 3+3 and some triplets into neutrinos in a way compatible with the previous post, which also divides the quarks in five plus one)
 
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arivero

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For reference, the code of the multiple SU(3) representation over the 32 elements that Furey locates in this paper. As I said, the 5+1 structure of the quark sector looks interesting, as it opens a possibility to explain why the top is peculiar.

My trouble here is if some of the triplets could be higher dimensional representations in disguise, and we are not seeing them because we are using a representation of the ladder operators only valid for SU(2) doublets. On other hand, does it matter? Not sure.

Code:
from sympy import *
#MV is the multivector from the ga package, not standard anymore in sympy.
from mv import MV
e1, e2, e3, e4, e5, e6 = MV.setup('e1 e2 e3 e4 e5 e6', metric='[-1, -1, -1, -1, -1, -1]')
e7=e1*e2*e3*e4*e5*e6
nu=(1+I*e7)/2
nuc=(1-I*e7)/2

L1=I/2*(e1*e5-e3*e4)*nu
L2=I/2*(-e1*e4-e3*e5)*nu
L3=I/2*(-e1*e3+e4*e5)*nu
L4=I/2*( e2*e5+e4*e6)*nu
L5=I/2*(-e2*e4+e5*e6)*nu
L6=I/2*( e1*e6+e2*e3)*nu
L7=I/2*( e1*e2+e3*e6)*nu
L8=I/(2*sqrt(3))*( e1*e3+e4*e5-2*e2*e6)*nu

Traise=(L1+I*L2)/2
Tlower=(L1-I*L2)/2
Vraise=(L4+I*L5)/2
Vlower=(L4-I*L5)/2
Uraise=(L6+I*L7)/2
Ulower=(L6-I*L7)/2
I3=L3/2
Y=L8/sqrt(3)

qR1=(-I*e1*e2-e1*e6+e2*e3+I*e3*e6)*nu
qG1=(-I*e2*e4-e2*e5+e4*e6-I*e5*e6)*nu
qB1=( I*e1*e4+e1*e5+e3*e4-I*e3*e5)*nu

aqR2=( I*e1*e2-e1*e6+e2*e3-I*e3*e6)*nu
aqG2=( I*e2*e4-e2*e5+e4*e6+I*e5*e6)*nu
aqB2=(-I*e1*e4+e1*e5+e3*e4+I*e3*e5)*nu
aqR3=( I*e4+e5+e1*e3*e4-I*e1*e3*e5)*nu
aqG3=(I*e1 + e3 + e1*e2*e6 + e1*e4*e5)*nu
aqB3=(I*e2 + e6 -e1*e2*e3 - I*e1*e3*e6)*nu

aqR4=(I*e1 - e3 + e1*e2*e6 - e1*e4*e5)*nu
aqG4=(-I*e4 + e5 + e1*e3*e4 + I*e1*e3*e5)*nu
aqB4=(I*e1*e2*e4 - e1*e2*e5 - e1*e4*e6 - I*e1*e5*e6)*nu

aqR5=(-I*e2 + e6 + e1*e2*e3 - I*e1*e3*e6)*nu
aqG5=(I*e1*e2*e4 - e1*e2*e5 + e1*e4*e6 + I*e1*e5*e6)*nu
aqB5=(I*e4 - e5 + e1*e3*e4 + I*e1*e3*e5)*nu

aqR6=(I*e1*e2*e4 + e1*e2*e5 + e1*e4*e6 - I*e1*e5*e6)*nu
aqG6=(I*e2 - e6 + e1*e2*e3 - I*e1*e3*e6)*nu
aqB6=(-I*e1 + e3 + e1*e2*e6 - e1*e4*e5)*nu

l1=(1 + I*e1*e3 + I*e2*e6 + I*e4*e5)*nu
l2=(3 - I*e1*e3 - I*e2*e6 - I*e4*e5)*nu
l3=(-I*e1*e2*e4 - e1*e2*e5 + e1*e4*e6 - I*e1*e5*e6)*nu
l4=(-I*e1 - e3 + e1*e2*e6 + e1*e4*e5)*nu
l5=(I*e2 + e6 + e1*e2*e3 + I*e1*e3*e6)*nu
l6=(I*e4 + e5 - e1*e3*e4 + I*e1*e3*e5)*nu

L1*qR1-qR1*L1-qG1
for x in (Traise,Tlower,Vraise,Vlower,Uraise,Ulower): print (x*qR1-qR1*x==0)
 

arivero

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The other intriguing thing in this family of papers, and in all the octonion-related SM, is that they get ti extract an unified SM with less dimensions than GUT groups.

I mean, SO(10) is the symmetry group of the 9-sphere. Pati Salam (ok, not a GUT really) is so(6) times so(4), the simmetry group of S5xS3 product of spheres. And SU(5) is the symmetry group of the projective space CP4. Classical group theory GUT seems to live in objects of dimension 8, 9 or higher. Octonions are barely of dimension 8, and most times they are related to the 7-sphere.

On other hand the standard model, as we know it, seems to have a limit -say, move the higgs to Planck Scale- where it is just SU(3)xU(1), a simpler symmetry which can live in CP2xS1 or in S5 (because SU(4) can). One could think that the Higgs field parametrized some move from d=8 to d=5. And one could hope that non extreme values of these parameters have some mathematical meaning as symmetries in d=7 or d=6. In fact Witten pointed out that d=7 can be obtained by quotienting the Pati-Salam spheres by an U(1) action. But when we try to do calculations in d=7 we are always using either SO(8) over a seven sphere or SU(5) over a product of spheres. I pondered this in a question to MO time ago http://mathoverflow.net/questions/75875/why-su3-is-not-equal-to-so5 Of this three Dynkin diagrams:
Code:
       o                  o                         o
      /                                      
     /                                        
o----o    SO(8)    o----o     SU(3)xSO(4)    o====o     SO(5)xSO(4)
     \
      \
       o                  o                         o
we want to get the middle one. It seems that octonion tecniques -or some other non commutative alternatives- can pivot between SO(8) and the SM in a way that group theory can not. It is peculiar.
 

arivero

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It seems that Furey has uploaded a lecture on this

 
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Furey has another paper today.

Something I don't understand about these alternative theories based on Clifford algebras, is when and how they become quantum field theory. I see discussion of symmetries, representations, transformations - OK. Where is the dynamics? How do we describe a process, like decay or scattering? Or even just a free particle moving through space? I don't know if I'm missing something, or if they (authors of such theories) are missing something.
 

arivero

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My guess is that the proponents of this kind of models expect to shape some sort of [relativistic] quantum mechanics, and then field theory follows from the usual "first quantisation is a mistery, second quantisation is a functor". But given that they are very near of the study of the geometry of the 7-sphere, it could also be a sort of Kaluza-Klein model.
 
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Is Dr. Furey going from causal sets to the algebra she's working with? If so, I did not see how she did it. If not, why did she bother to mention it?
 
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Is there a single book that teaches the connection between the division algebras and the symmetry groups that Dr Jurey seems to be using?
 
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arivero

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Some books on representation theory of exceptional groups happen to use division algebras as a technique.
Can you name a few books with this feature?
 
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By the way, Wolchover has just published a journalistic note on Furey's work https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/
Thanks for the link. I read that this effort seems to entail some risk to ones career. I suppose this is most likely caused because everyone who works on it is going on a hunch that there is some deep connection, but no one has proved it yet. I became interested in this view for the opposite reason. I've come up with a theory that seems to predict the normed division algebras from first principles and am now starting to see how particles can come from it. It's not been peer reviewed, so it's considered speculative at this point. But I'm willing to share with anyone via Private Message if they wish.
 
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arivero

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The "first principles" direction is not so opposite, and has been explored too, that should be Dixon work, and Adler too perhaps.


As for an example of text pivoting on octonions, lets say
Adams J. F.
Lectures on Exceptional Lie Groups
Univ. Chicago Press (1996)

but you can find a lot by googling around site:arxiv.org, words exceptional and octonions. Also, search this cite for ancient, closed threads naming octonions.
 
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The "first principles" direction is not so opposite, and has been explored too, that should be Dixon work, and Adler too perhaps.
It seems to me that right now it's more of a mere coincidence that the division algebras correlated to particles. We need some principle to require the division algebras first, then the particles can be considered inevitable. I have some ideas on that but it is speculative.
 
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Thanks for the link. Lubos Motl is not very generous here. The reason her works seems like numerology is because it does not seem to be justified on more basic grounds. The division algebras can be derived by an iterative process, Caley-Dickson construction. So if we found some iterative process in the quantum formalism that also gave the division algebras, then there would be no argument with it. I think I found such an iterative process (it needs review), and that's why I'm excited about her work.

I also wonder if her work panned out, would that eliminate other people's favorite research projects? For example, (and please correct me if I'm wrong), wouldn't her work require the existence of all the different particles and forces no matter how high the energy? Wouldn't that mean that they don't achieve unity at the gut scale, where they are all seen as the same thing? And wouldn't that mean that the proton would not decay?
 
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arivero

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I also wonder if her work panned out, would that eliminate other people's favorite research projects?
Nah, division algebras are very frequent in mainstream research, it is just that they are not a good guiding rule and usually found a posteriori. It is similar that Connes's theory; if it "panned out", it would immediately be absorbed into the corpus. This is hinted when you see criticism about most of the results being trivial or well-known. I have liked specially the comment about 12 being half of 24... but then 14 is described by Motl as being 2/3 of 21, and not half of 28. Point being, there is a lot of representation theory to juggle with.


“Octonions are to physics what the Sirens were to Ulysses,” says Pierre Ramond, in the Quanta Magazine article. But there are sirens within sirens, and what is interesting about Furey work is that they are being carefully avoided: Bott periodicity, Evans supersymmetry, Atiyah CP2, Tits–Freudenthal Square, Duff brane scan ... to name some.
 

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