John Baez and John Huerta Paper

[John86]

The Algebra of Grand Unified Theories

John C. Baez, John Huerta
(Submitted on 9 Apr 2009)

Abstract: The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify forces and particles by extending the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves finite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.

http://arxiv.org/abs/0904.1556


I been reading this paper yesterday and although an amateur i found it really interesting. correct me if i am wrong but do Baez en Huerta say that the standard model of particle physics is a subgroup of a deeper more fundemantal symmetry. And is there any relation with the Howard Georgi's idea of unparticle physics.

 

[Vanadium 50]

. correct me if i am wrong but do Baez en Huerta say that the standard model of particle physics is a subgroup of a deeper more fundemantal symmetry.

Well, it wasn't Baez et al. who proposed it. As the paper points out, this was an idea that at least four people had in 1974.

And is there any relation with the Howard Georgi's idea of unparticle physics.

No.

 

[apeiron]

On another tack, the natural presumption is that the observed U(1), SU(2), SU(3) symmetries unify within some larger single supersymmetry. But are there any interesting arguments along the lines that symmetries may not all fold into each other so neatly? That nature works a different way?

 

[f-h]

apeiron, John Baez has argued along these lines, that instead of trying to unfiy it to a simple symmetry and then having to contend with an ugly (often unnatural) breaking process we should look for mathematical structures that contain this group naturally.

 

[apeiron]

It is sort of a platonic argument. All kinds of symmetries exist as possibilities. But for the "cooler" universe-creating symmetries to be expressed, there has to be some cooling mechanism. A dissipative structure.

So then, must the cool universe-creating symmetries (U(1) x SU(2) x SU(3)) be a nested hierarchy of symmetries, all part of the same broken symmetry, or could they just be a set of unrelated slots so to speak.

There are all these higher dimensional symmetries seemingly scattered about, sporadic like the extraordinary lie groups. So either all these symmetries are related in some deep hidden way, or perhaps they are actually randomly scattered - independent instants of order in a chaos of possibility.

I've not seen the two alternative views debated. And maybe that is because it is not even a relevant kind of question.

Anyway, this Baez/Huerta paper argues for a nested hierarchy of symmetries. I cannot judge how well the maths holds up, but it presents a plausible view.

We have so far been unable to fit a higher symmetry to U(1) x SU(2) x SU(3). But we may have two lineages that account for opposing aspects of what we measure. SU(5) unifies the fermions and spin(10) that unifies the bosons. We would then need to step back further to find the symmetry that unifies them - U(ΛC^5) Baez calls it.

Do others find this an attractive idea? To me, it says both spin and rotational symmetries are "slots" and cooling would naturally find a way to express both.

So a cascade of symmetry breakings from a single larger symmetry seems more likely on this argument.

 

[Vanadium 50]

On another tack, the natural presumption is that the observed U(1), SU(2), SU(3) symmetries unify within some larger single supersymmetry.

Supersymmetry is a bad word here. It means something different. "Higher symmetry" is perhaps better.

But are there any interesting arguments along the lines that symmetries may not all fold into each other so neatly? That nature works a different way?

Well, U(1) and SU(2) are already unified, so you'll never get those separated again.

 

[apeiron]

Well, U(1) and SU(2) are already unified, so you'll never get those separated again.

Yes, the lowest level symmetries have a definite nested relationship. But does that nesting extend all the way up to include all symmetries?

You would think it probably would. But what kind of argument could be employed?

I would draw a parallel with period doubling and the onset of chaos - Feigenbaum’s constant and the logistic map. At the "cold" end, when doubling starts, you get bifurcation then a rapid descent into chaos. But then briefly a return to order with three oscillations - a so-called island of stability. Followed by descent into chaos. Then by further breaks with six then 12 oscillations.

Each of these "higher symmetries" appear more vaguely, more fleetingly. Perhaps they even run to infinity. Before total chaos kicks in and we reach the asymptote of the Feigenbaum constant.

An example of a logistic map with islands of stability:

http://en.wikipedia.org/wiki/File:LogisticMap_BifurcationDiagram.png

So as with a deterministic chaos model, there may also be a hidden order behind the disorderly occurence of lie symmetries. One that seems clear at the cool end of the spectrum and hard to see at the hot end.

I don't know the maths well enough to say if there is a similar story here. Which was why I asked what people might have said about it.

 

[john baez]

apeiron wrote:

"Anyway, this Baez/Huerta paper argues for a nested hierarchy of symmetries."

We're not really arguing for anything. We're just explaining some of the most well-known grand unified theories in a way that mathematicians might enjoy. But yes, one of the fun features is how these theories fit together in a nested way.