Exploring the Symmetry Groups of SU(3)xSU(2)xU(1): Questions and Theories

[ChrisVer]

Do you have the Higgs-mass-predicting paper that deals with gravity?
I don't know, have people (sucessfully) tried to gauge group-ize gravity?
Then I think any GUT introduces new "forces" let's say... especially if the breakings happen subsequently (you remain with other groups + SM)

 

[ohwilleke]

Do you have the Higgs-mass-predicting paper that deals with gravity?
I don't know, have people (sucessfully) tried to gauge group-ize gravity?
Then I think any GUT introduces new "forces" let's say... especially if the breakings happen subsequently (you remain with other groups + SM)

The paper is http://www.sciencedirect.com/science/article/pii/S0370269309014579

 

[Urs Schreiber]

have people (sucessfully) tried to gauge group-ize gravity?

Gravity has a natural formulation in terms of Cartan geometry over the spacetime symmetry group. This is colloquially called the result of "locally gauging" the spacetime symmetry (see here), but even if one may call the result a kind of gauge theory, it is not a Yang-Mills-type gauge theory like that which governs the other forces. In particular the coupling constants of the Yang-Mills forces are on a different footing than those of gravity.

 

[samalkhaiat]

In particular the coupling constants of the Yang-Mills forces are on a different footing than those of gravity.

This has been long over due problem. May be it is about time to formulate a new gravity theory and get rid of Newton constant which appears in the Einstein action $$\mathcal{L}^{g}_{EH} \sim \frac{1}{G_{N}} R \ .$$
This can be done in Yang-Mills type gravity. Getting rid of dimension-full parameters (i.e., scale) has, “recently”, led many people to reconsider Yang-Mills gravity model based on the non-compact group of conformal transformations $SO(2,4)$ $$\mathcal{L}^{YMg}_{SO(2,4)} \sim \frac{1}{\alpha^{2}_{YM}} F^{2} .$$
It is still early to tell, but such model comes with immediate bonuses: 1) since the coupling $\alpha_{YM}$ is dimensionless, renormalization may come with less embarrassment compared to $\mathcal{L}^{g}_{EH}$. 2) observers will no longer need measuring sticks and clocks, all information are obtained by light rays.

We had similar situation in the past: Notice the similarity (in form and problems) between Einstein gravity theory $\mathcal{L}^{g}_{EH}$ and Fermi theory of weak interaction $$\mathcal{L}^{w}_{F} \sim \frac{1}{G_{F}} J_{\mu}J^{\mu} \ ,$$ and recall that Yang-Mills theory of weak interaction $$\mathcal{L}^{YMw}_{SU(2)} \sim \frac{1}{g^{2}_{YM}} F^{2} \ ,$$ proved to be the needed cure for renormalizibility and unitarity.

 

[lpetrich]

Witten observed that another popular GUT group, SU(4)xSU(2)xSU(2) was similar to SO(6)xSO(4) and then to the group of isometries of S5 x S3. ...

I must note that the algebra SU(4) is isomorphic to SO(6) and SU(2)*SU(2) to SO(4).

Some authors distinguish the groups and the algebras by typography: group SO(n) vs. algebra so(n), etc.

I note that the various proposed GUT gauge groups also have numerous subgroups. So if our Universe has an underlying GUT gauge symmetry, the problem is why did it break down to the Standard Model's symmetry group and not some other one. String theory does not seem to have helped. Though one can get at least the spectrum of the Standard Model from string theory, it is far from unique, and I do not know of any progress in resolving that extreme non-uniqueness.

 

[gdixon]

Re:
"Alas, the group SU(3) is not the unit octonions. The unit octonions do not form a group since they aren't associative. SU(3) is related to the octonions more indirectly. The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group G2, which contains SU(3). To get SU(3), we can take the subgroup of G2 that preserves a given unit imaginary octonion... say e1. This is how Dixon relates SU(3) to the octonions. However, why should one unit imaginary octonion be different from the rest? Some sort of "symmetry breaking", presumably? It seems a bit ad hoc."
=======
The complexified octonions (which I label S = C⊗O) is not a division algebra. Its identity can be resolved into a pair of orthogonal idempotents, and it is from this resolution of the identity that much of what's interesting and beautiful about S arises. This resolution requires a direction in imaginary O space be chosen, and the subgroup of G2 leaving this direction invariant is SU(3). With respect to this resolution S splits into 4 SU(3) multiplets: singlet; antisinglet; triplet; antitriplet. This notion was used by Gürsey at Yale in the 1970s. There's nothing ad hoc about it. It's an ineluctable and beautiful part of the maths. There's nothing vague about it, as two books and numerous papers have attempted to demonstrate ad nauseam. Sigh.

 

[arivero]

This resolution requires a direction in imaginary O space be chosen, and the subgroup of G2 leaving this direction invariant is SU(3)

I wonder if this is related to the branched covering of S4 by CP2. (see atiyah an also google https://www.google.es/search?sourceid=chrome-psyapi2&ion=1&espv=2&ie=UTF-8&client=ubuntu&q=S4 branched covering CP2&oq=S4 branched covering CP2&aqs=chrome..69i57.5235j0j7 ). CP2 fibered with S3 should have an isometry group very as the standard model group.

 

[gdixon]

Well, if you have a house, and inside that house there is a very nice and perfectly functional WC, why go outside and dig a hole in the ground to serve the same purpose, but not as well?

I haven't figured out how to include a quote from a previous comment, but this is in reply to arivero's comment above.

 

[mitchell porter]

gdixon, what do you do for dynamics?

 

[gdixon]

Dynamics, hmm. A possibly poor metaphor: it's like asking Mendeleev, after he presents you with the periodic table, where's the chemistry? A perfectly valid question, and Mendeleev certainly would have been able to answer that. My attitude has always been this:

Adding O to P=C⊗H, yielding T=C⊗H⊗O, results ultimately in an expansion of the Dirac algebra and its associated spinor space, which is where the particle fields reside. As I understand it, one starts building dynamics into Dirac maths by building a Lagrangian density. All the pieces needed to do that are present in T-maths, they're just bigger. So, if one wishes one can construct a Lagrangian density for this more complicated spinor space, and its associated 1,9-spacetime Clifford algebra (the expanded Dirac algebra). I did this in my first book 22 years ago, guaging the result. Very nicely one can read from this all viable particle interactions, for it is not hard to pick out from T^2 the bits that represent individual quarks, leptons, anti-quarks, and anti-leptons.

And then there is the matter of the quarks themselves, and the extra 6 space dimensions, neither of which are seeable in any conventional sense, and both of which are associated with the 6 octonions units that do not occur in the resolution of the identity. As I showed in my last published paper, taking this unseeableness to its logical mathematical conclusion implies that from our 1,3-spacetime we can also not see the anti-matter part of the full 1,9-spacetime, so we appear to live in a universe dominantly matter.

Anyway, I'm retired now. I'll present my last paper at a conference this summer, then I'm dropping the mike.

 

[gdixon]

Keep in mind: what I did was a proof of concept. My hope is that reality is more subtle than this.

 

[kneemo]

"Why SU(3)xSU(2)xU(1)? A truly fundamental theory should explain where this precise set of symmetry groups is coming from.

Greetings

One idea to keep in mind is phase transitions. For example, water in a steam state or liquid state is much more symmetric than an ice state with flaws and cracks. If you rotate the ice with streaks and cracks, it's easy to see you lost rotational symmetry.

Similarly, imagine spacetime itself, as we perceive it now, in an "ice-like" state, with temperatures far cooler than the big bang. The big bang temperature would be akin to "boiling" temperature for spacetime. And this is why one studies quantum gravity, in hopes of finding a deeper theory that can describe the extreme phase transitions expected to occur at black hole event horizons and the big bang itself. Essentially, anywhere one would expect singularities in Einstein's general relativity (akin to an effective spacetime hydrodynamics), this would be the realm of quantum gravity and unified field theory.

Hence, from the perspective of phase transitions, SU(3)xSU(2)xU(1) would be the result of a broken higher symmetry, due to the Universe entering an "ice-like" state after 13.82 billion years. This is reasonable, as my living room is nothing like a black hole event horizon.

The whole point of building huge particle accelerators is to reproduce extreme energies, forcing a phase transition to a more symmetric state for a brief time, and take a snap shot of the resulting symmetry breaking that occurs after a high energy collision.

The appeal of E8, for example, is that it is a unique Lie algebra (mathematically) which gives elegant rules for scattering a robust set of particles (bosonic & fermionic), in a closed manner, where the observed inelegant symmetries of the standard model can be seen as part of a larger symmetric whole.

The quest continues...

 

[gdixon]

https://www.quantamagazine.org/20161215-proton-decay-grand-unification/

 

[gdixon]

PS

"Keep in mind: what I did was a proof of concept. My hope is that reality is more subtle than this."

By this I do not intend to cast doubt on the necessity of C⊗H⊗O as a basis
for any viable TOE. I have no doubt. None. The algebra is necessary
because parallelizable spheres are necessary. And once you have it,
the mathematics gives you everything else.

http://7stones.com/7_new/7_Why.html

 

[arivero]

By the way, what is the status on looking for family structure emerging from octonions and/or trialities? I see Dixon's 2004 paper http://7stones.com/Homepage/123cho.pdf, also some idea in Dray-Manogue https://arxiv.org/pdf/hep-th/9910010.pdf, and then some newcomers, namely Furey and Dubois-Violette. Not sure if Farnsworth-Boyle try generations in some publication; it is mentioned section 5.3.3 of Fansworth's thesis, in https://inspirehep.net/record/1419192/files/Farnsworth_Shane.pdf

I am particularly fascinated by the point that Dubois-Violette produces an extra chiral fermion in each generation, coming from the diagonal of the exceptional jordan algebra, and I wonder if other approaches also derive similar extra "dark" particles.

 

[lpetrich]

Hence, from the perspective of phase transitions, SU(3)xSU(2)xU(1) would be the result of a broken higher symmetry, due to the Universe entering an "ice-like" state after 13.82 billion years.

After that amount of time? It would be early in the Universe's history. Extrapolation of the Standard Model's coupling constants with increasing interaction energy reveals that they meet at around 2*10¹⁶ GeV.

But the overall principle is correct, I think, and a variety of higher-symmetry Grand Unified Theories have been proposed.

 

[arivero]

Others think that U(1) is because of the Complex numbers, SU(2) is due to the quaternions, and SU(3) is due to the octonians. Although, I don't think this is completely worked out yet. And more effort needs to be done to resolve it.

Is this the kind of iteration you were thinking about in the thread about Furey's models?

 

[friend]

Is this the kind of iteration you were thinking about in the thread about Furey's models?

That's classified. Actually, it's a little too involved to state here. I don't know how to summarize without sounding speculative as defined in these forums. If you're interested PM me.