Since leptons and quarks are successfully associated with a group of 3-ribbon braids (Helon model of Bilson-Thompson), one could try to relate this group to the fundamental group of a node that lies on a 3-dimensional torus, and on a 2-dimensional torus it is a trefoil knot. It is remarkable that a 3-torus spanned by a 3-sphere with punctured poles has the symmetry SU(3). Consequently, the nodes can be interpreted as closed vector fields lying on the 3-sphere.
Dear Igor Bayak,
In my paper I just assigned elementary particles to the e8(8) basis via Furey's ladder system. I had a very vague idea that is relating to braids.
(please see p.15 in my paper.)
At the present moment Gresnight's work is the best answer I think.
There are many things to do, it's fantastic if we can understand E8 Jones-Witten theory more thoroughly.
Following Geoffrey Dixon's suggestion, I recommend Shahn Majid's work
arXiv:math/9802116.
I found your paper on algebras of vector fields, it's interesting work.
Can we figure out direct correspondence between your's and Gresnight's work ?
Dear Shuichiro Teramachi, thanks for the comment. I will definitely look at the work you indicated. As for the connection between my work and work and Gresnight's work, I think that my work is still closer to philosophy, and its to physics." However, I still have an unpublished paper on the winding of a sphere that is closer to physics.
Preprint Chaotic dynamics of an electron
There are many ways of describing the gauge groups of the Standard Model, so having found another one isn't news. They're all different ways of expressing the same information. The mathematical fact that the symmetry group of a certain knot on the 3-sphere is SU(3) is totally independent of the physical fact that SU(3) is the symmetry group of the strong interactions. Whether the former is a useful way of performing the calculations of the properties of quarks remains to be seen-and it is up to the people that make the claim to provide the examples-especially given the fact that the way of computing all properties of the quarks is known.
Regarding the technical obstacles of using E8 to describe the known quarks and leptons, cf. Article There is No “Theory of Everything” Inside E8
(it was known already since the 1980s, when people were working on grand unified theories, that, since E8 can't describe chiral fermions, it wasn't relevant for the Standard Model and its possible extensions).
Regarding (some of) the issues involved in using knots to describe particles more generally, cf. Article Knots and Particles
and the papers that cite it.This doesn't mean that knots aren't interesting for describing many situations in physics-just that the properties of quarks and leptons, that are sensitive to knot-like structures are very specific.
Contrary to popular belief, it's not necessary for a concept to be applicable to the Standard Model of elementary particles itself for it to describe interesting physics-so it's not necessary-and, actually, distracting-to focus on quarks and leptons for issues that don't have anything to do with them. It just has to provide a mathematically consistent description of a particular physical situation. It's unfortunate that many people don't realize this.
In E8 theory, the space of fermions is "not" a representation space of E8.
Rather, they are living in E8 itself by introducing a susy-like structure.
In E8, all the interactions are represented by its Lie brackets [ , ]_8.
Hence as everyone knows,
[ Boson , Fermion ]_8 = Fermion ( emission & absorption )
namely fermions are representation of the bosonic part of E8 (U1×SU2×SU3), while
[ Fermion , Fermion ]_8 = Boson ( annihilation & pair production ).
Thus, obviously the space of fermions can not be an E8 representation space.
This is a reason why we can treat chiral fermions easily in E8.
Unexpectedly, to treat both of chiral left & right fermions adjoint representation of e8 should be "real" as it is.
For any group particle states are described by a representation-by definition.
The obstruction with using E8 to describe the known fermions isn't with defining fermions-it's with defining chiral fermions. And to do that complex, not real, representations are required.
Any representation of E8, that contains the correct representation of the Lorentz group and (a direct product of) the Standard Model gauge group, is real, not complex. This is described in detail in Article There is No “Theory of Everything” Inside E8
That's why, in particular, the statement about using the real form of E8(8) to describe anything to do with the Standard Model is wrong.
Once more, all of this was known since the 1980s-but then it was hoped that the additional particles were a prediction; unfortunately, they haven't appeared where they were expected, that's what eliminates the GUT E8.
If one, nonetheless, wishes to work with E8, it's the E8 x E8 heterotic string theory that's more relevant: https://members.ift.uam-csic.es/auranga/lect3.pdf
This is, now, standard knowledge. How to obtain chiral fermions, in this framework, is, now, well known. One further problem, however, is that the theory is strongly coupled, so obtaining a field theoretic description isn't at all obvious, since the low energy action isn't known in full generality. These issues and more are described in the lecture notes.
These properties of E8 are, however, utterly irrelevant for the question of how to use knots to represent particles and, equally, irrelevant for discussing the very technical point that Furey was making about grand unification using SU(5). Unfortunately, the problems with using SU(5) to unify the strong with the electroweak interactions aren't limited to proton decay. (Not to mention the fact that it isn't by any means obvious that the constraints imposed on the classical action-i.e. on the vacuum state of the free oscillators-in the way proposed are consistent with quantum corrections, i.e. when quantum effects of interactions are taken into account.) And the solution proposed to this issue doesn't address the other problems.
Please note that in E8 "chiral" fermions are not (real) representation of E8,
just they are belonging to (complex) representation of its "subgroup" U1×SU2×SU3 instead.
It will suffice to derive standard model's property qualitatively.
But the point is that E8 doesn't have complex representations. So any way of decomposing the fermions into representations of the Standard Model gauge group leads to non-chiral fermions, since both chiralities contribute equally-that's what ``real representation'' means for E8. That's why E8 is useless for describing the Standard Model and its extensions-it implies, necessarily, that the fermions don't have only one chirality, whereas the fermions of the Standard Model do. There's no way to isolate one chirality only, when using fermions that belong to E8 representations, since E8 describes operations that mix both chiralities-and such operations don't correspond to anything in the Standard Model.
As mentioned, these are (a) well known facts and (b) irrelevant for knots, octonions; nor do they have anything to do with the constraints imposed on the SU(5) GUT (beyond mentioning the number eight! Not everything that refers to the number eight is related...).
No, E8 fermions cannot be any decomposition of E8 representation.
Since for any fermion F in e8, ad F = [F , ]_8 maps e8-fermions to e8-bosons
(as I wrote 2days ago).
E8 fermions cannot belong any representation of E8 itself.
(Sigh... it seems Stam Nicolis faded out with a parting shot without understanding this simple thing.)
I'd like to add some basics.
About : There is No “Theory of Everything” Inside E8 arXiv:0905.2658
we can easily see their assumption (ToE 3) is simply wrong for G=E8(8),
with only single calculation of e8 Lie bracket like [ anti-up , down ]_8 = W^-.
Before I found a particle representation via Vinberg's magic-square construction of e8, it was not clear.
Unfortunately, in Lisi's original proposal G=E8(-24) 3-generations were assigned to "triality". In this case for some of fermions (F)
[ F_1 , F_2 ]_8 = F_3.
It was just contradictory for me, but even in this case I believe [ anti-F , F ]_8 should be boson.
How they struggled to find that something is wrong, I recommend to read Distler’s blog
https://golem.ph.utexas.edu/~distler/blog/archives/001505.html
Now we can understand this more directly. The problem was in the interpretation of “triality” not in the representation issue.
In the next time we should go back to Igor’s original question.
You might also look at my latest preprint, attached below. It seems that rational tangles can reproduce all known elementary particles and all their Feynman diagrams.
Regards
Christoph
The development of the theme will add a few words. What connects nodes and elementary particles? So, my answer is elementary particles are the topological features of the vector field, in which the current lines of the vector field are closed into nodes. And in order for a vector field to be closed, it must be set in a semi-closed extended Minkowski space. To do this, we take an 8-dimensional space with the square of the metric interval: x2+y2+z2-t2-x*2-y*2-z*2+t*2 and each pseudo-Euclidean plane (x,x*), (y,y*), (z,z*), (t,t*) is rolled into a tube so that one of the isotropic lines of this plane is wound on a circle.Then in this semi-closed extended Minkowski space it is possible to define a vacuum vector field as a flow of a current line, which coincide with the coordinate time line t. At the same time, the perturbations of the vacuum vector field can be arbitrary, and in the case of the closure of the current lines of this vector field, it forms elementary particles. I think that's a good explanation for the connection between the nodes and the particles. Moreover, the group of symmetries of the 3-torus entwined around the pole punctured 3-sphere coincides with the group SU(3).
Igor,
it seems that 3+1 dimensions might be sufficient. In this preprintPreprint A Conjecture on Deducing General Relativity and the Standard...
it is suggested that tangles (i.e., open knots, i.e., knots with ropes going to "far away") might be used to deduce the Dirac equation, U(1), broke, SU(2) and SU(3) - all quite naturally. In addition, the different tangles explain the particle spectrum. In comparison to helons, tangles have the advantage that they yield 3 generations, plus the gauge symmetries, plus quantum theory, plus all the known Feynman diagrams. On the other hand, both helons and tangles show that algebraic structures are important to understand particle physics.- Badges
- Science topic