Kaluza-Klein Theory: Unifying Weak Force and Gravity

[Rezaderex]

Hello everyone!

Im new on this forum, have been lurking around paying attention to all the interesting stuff you guys talk about, and now iv joined up!

Firstly, I am not sure if I am posting in the right place - there isn't really a distinct place for a question like this. Hopefully I am in the right place.

So basically, here is the question, recently i have been working through kaluza klein theory, more specifically the original one. i.e. S^1 compactified under the usual U(1) gauge, we gain a gauge field in which we quantise around the circle, we truncate our final action and gain a low-energy action whereby coupled EM and GR field equations come out. I do know there are issues with this truncation as, we can't really truncate massive to massless fields without actually losing massless fields.

However, I am trying to figure out what would happen in a sphere. What I am basically starting off with is some generalised arbitrary Yang-Mills theory, SU(2), and trying to unify this with gravity. However, some opinions have arisen. Essentially what i would like to do is gain coupled equations for the weak nuclear field and einstein field equations. So I am sort of trying to equate SO(3) and SU(2) (I know that is said without rigor). I have a set of killing vectors for SO(3), of which i assume make make the lie algebra, also i have the yang mills theory, i.e. a generalised non-abelian theory, i have lie algebra and the field strength for this too.

The problem is that, i have been told that it might not work, as the Ricci flatness equation will not come out. This being due to S^2 having positive curvature, whilst in the original theory we only had a S^1 which in flat... iv been told the theory works better with a torus ie. U(1)^n. The issue i have with this is that i wouldve hope to try an unite weak force and gravity, and under U(1)^n i would just be getting EM in some higher dimension.

any help would be greatly appreciated, hopefully one of you guys can tell me what's up.

cheers

 

[arivero]

You need Witten's "Realistic Kaluza Klein Theory", and any book including a reprint of this article.

The strategy to obtain a group G is to look for "symmetric spaces" G/H with G acting nontrivially. The first example is SU(2), acting in the space SU(2)/U(1). This is the 2 dimensional sphere (note SU(2) has, as manifold, dimension 3, and U(1) has dimension 1). Of course SO(3) is the group of symmetries leaving invariant the sphere S2, so you are right.

You can see other examples I am trying in the thread https://www.physicsforums.com/showthread.php?t=358142

 

[arivero]

As for the theory working better in flat spaces, it is the full theory, with supersymmetry, strings and all that. Point is that the product space of minkowsky times extra dimensions is not granted to be a solution of the equations of full (with extra dimensions) general relativity. In some cases, you can make it to be a solution, but with a cosmological constant. It is up to you to discuss how troublesome the situation is. Nowadays, the only extant argument against a cosmological constant in the 11D Lagrangian is that it breaks supersymmetry.

 

[Rezaderex]

hey, thanks very much for your reply, really useful stuff, i think I am going to have a read of wittens paper

 

[arivero]

hey, thanks very much for your reply, really useful stuff, i think I am going to have a read of wittens paper

Besides the reading, it is good to show your friends. In some articles, it appears as if the only content of the paper were the "non go theorem" for chiral fermions on classical manifolds, so that at the end nobody reads the paper and nobody worries about why the coincidence that D=11 is the dimension of the Kaluza Klein standard model.

I forget: a good place where the paper is reprinted is in "Modern Kaluza-Klein Theories", by Appelquist Chodos Freund. Another one is "The World in 11 Dimensions", by Duff et al.

 

[arivero]

A help request: if you find some interesting article on the topic of symmetry breaking for kaluza klein theories (not "induced by", but the case where the geometry generates both the symmetry and the breaking) please tell me.

 

[arivero]

Other interesting reference is chapter 8 of volume 1 of Polchinski string book. In section 3.3 he shows how the massless strings (and the the supergravity states too) rearrange under Kaluza Klein compactification. The most relevant phenomena here is that a total derivative does not cancel and then the descendants are richer than in point-particle kaluza klein (albeit point particle Kaluza Klein is a field theory too). Next he speaks of enhanced symmetries: there is some common element with the enhancement in Witten's classification of 7-manifolds, namely that the enhancement is associate to singular points of the parameter space of the compactification ("space of the moduli", let's say). Kaluza Klein Witten gauge enhancement occurs when the symmetry group moves from SU(2)xU(1) to U(1)xSU(2), passing across SU(2)xSU(2) or SO(4). String theory enhancement happens when the group moves across $$R=\alpha'$$. The point is that in both sides the theories are the same, "dual" and equal, so this point is "termination" in the parameter space. But beyond that, the mechanism seems to be different.

Two footnotes:
1) superstrings do not enhance because T duality changes boundary conditions and the theories are not really equal in both sides, only "dual".
2) Witten parametrisation of manifolds is very similar to Weinberg angle. But GSW does not seem to enhance at 45 degrees of mixing.