Can you extend the Standard Model symmetry group to U(1)×SU(2)×SU(3)×...×SU(N)?

[MTd2]

So, I was thinking, SM symmetry group is U(1)XSU(2)XSU(3), so, why not going on, that is:

U(1)XSU(2)XSU(3)XSU(4)XSU(5)X...XSU(N), N-> infinity

Did anyone ever try that?

 

[Haelfix]

I'm not entirely positive, you should ask on the math forum, but isn't that just homomorphic to SU(N) N--> infinity anyway? Which is just the planar gauge theory.

 

[Coin]

I would like to note that I have previously derived a much sillier GUT theory than this, outlined in the comments to this post.

Haelfix, what would the "planar gauge theory" act like when you try to apply it as a particle gauge theory? (If this is a sensible question)

 

[Haelfix]

Actually it doesn't behave particularly "particulate" at all. Instead its quite stringy, and remarkable simplifications occur at the quantum level leading some people to expect it to be exactly soluble. Nevertheless it was discovered by 'T Hooft in the 70s before string theory was really popular.

Its also one of the founding and motivating examples for AdS/CFT (where you take a large N gauge theory over a Riemann surface in a lower dimension).

 

[MTd2]

In fact, what I mean is really simple, silly indeed. As you go to higher and higher enegies, you would find that the best description of the particle zoo up to a point would be a new direct product with the next j of SU(j). I didn't mean an aproximation, but an almost exact aproximation at each enegry level and that's why I didnt ask in the math forums.

 

[Haelfix]

That won't work, b/c for finite j you will have some leading order groups that admits only self conjugate irreps and are nonchiral. The effective theories thus cannot be broken up in such a way.

 

[MTd2]

I don't see the problem in having some of them non chiral. The standard model will always be a subgroup of the sequence.

 

[Haelfix]

You always need a chiral gauge group. One physical reason it matters is b/c if you didn't you would have fermion masses completely at odds with experiment. Generically all the fermions in the theory would pick up large gauge invariant masses at some high energy scale instead of being massless (until electroweak symmetry breaking).

Thats one of the biggest constraints in GUT model building and why we don't use groups like say E7 for model building (even if it contains the SM as a subgroup)

 

[Bob_for_short]

N cannot go to infinity: there is no particle heavier than the total Universe energy.

Bob.

 

[MTd2]

You always need a chiral gauge group.

These are just embendings of smaller groups into largers ones until infinity. It's not like searching a smaller group inside a bigger one, but making a bigger extending the smallerm, and see what happens. And isn't every SU(j) gauge group chiral invariant?

 

[Haelfix]

I don't really understand your first sentence. For instance E7 * G(standard model) is still unsuitable for phenomenology even if you are deadset on trying to complicate the high energy landscape for no suitable rewards!

Actually, the problem with SU(N) with N finite, is actually a little more complicated than I said above. In order to have a physically viable candidate, you need both a complex representation (and hence a chiral spectrum), which SU(N) does have, except that you also need to satisfy the additional requirement for anomaly cancellation. The only completely antisymmetric m fold representation for SU(N) then only contains the familiar SU(5) representation 1, 5, 5bar, 10bar. Which you can build up to get higher versions.

Unfortunately there again you run into certain choices for some fixed N, with a fixed representation r where you will have to explain why either fermions don't receive large masses or why the theory doesn't possesses possible gauge anomalies..

For N --> infinity, well that's a different story b/c of the aforementioned simplifications. But then that's not a candidate GUT either.

 

[MTd2]

The only completely antisymmetric m fold representation for SU(N) then only contains the familiar SU(5) representation 1, 5, 5bar, 10bar. Which you can build up to get higher versions.

I don't get any of this.

Hmm. Fermions haing a huge mass is a desired effect, because I am adding a new force for every SU(j) added. They shouldn't be seen that easily.

I'd like to know how to unify that kind of inifinite sequence of forces.